IJIAM-Spl-Issue-ICAA 2019

INDIAN JOURNAL OF INDUSTRIAL AND APPLIED MATHEMATICS
Vol. 10, No. 1 (Special Issue), Jan-June 2019
EDITORIAL BOARD
Patron of ISIAM: Prof. K.R. Sreenivasan
EDITOR-IN-CHIEF
Prof. Abul Hasan Siddiqi
Sharda University, President, Indian Society of Industrial and Applied Mathematics, (ISIAM), India,
siddiqi.abulhasan@gmail.com
MANAGING EDITORS
1.
2.
3.
Prof. Pammy Manchanda, Department of Mathematics, Guru Nanak Dev University, Amritsar, India,
pmanch2k1@yahoo.co.in
Prof. Rashmi Bhardwaj, USBAS, Nonlinear Dynamics Research Lab, Guru Gobind Singh Indraprastha University, New
Delhi, India, rashmib22@gmail.com
Prof. A. A. Khan, Rochester Institute of Technology, Rochester, U.S.A., aaksma@rit.edu
EDITORIAL BOARD
International
1. Prof. Zafer Aslan, Istanbul Ayadin University, Turkey, zaferaslan@aydin.edu.tr
2. Prof. Martin Brokate, Technical University Munich, Germany, brokate@ma.tum.de
3. Prof. O. Christensen, Technical University of Denmark, ochr@dtu.dk
4. Prof. Iain S. Duff, Rutherford Appleton Laboratory, U.K., iain.duff@stfc.ac.uk
5. Prof. Pavel Exner, Department of Theoretical Physics, Nuclear Physics Institute, Academy of Sciences, Near
Prague, Czech Republic, exner@ujf.cas.cz
6. Prof. Graeme Fairweather, American Mathematical Society 416 Fourth Street, Ann Arbor, MI, U.S.A., gxf@ams.org
7. Prof. Yuri Farkov, Russian Presidential Academy, Moscow, Russia, farkov@list.ru
8. Prof. H.G. Feichtinger, University of Vienna, Austria, hans.feichtinger@univie.ac.at
9. Prof. Sandor Fridli, Budapest, Hungary, fridli@numanal.inf.elte.hu
10. Prof. A. Gupta, University of British Columbia, Canada, arvind@mitacs.ca
11. Prof. Rene Lozi, University of Nice Sophia Antipolis, France, Rene.LOZI@unice.fr
12. Prof. M. Lawati, Sultan Qaboos University, Oman (Ex. Chairman), mohamed.allawati@gmail.com
13. Prof. Guenter Leugering, Friedrich-Alexander-Universität Erlangen-Nürnberg, Germany, guenter.leugering@fau.de
14. Prof. P. Maass, Universität Bremen, FB 3, Mathematik und Informatik Zentrum für Technomathematik, Germany,
pmaass@math.uni-bremen.de
15. Prof. M. Moonis, Department of Neurology Worcester, MA 01655, U.S.A., majaz.moonis@umassmemorial.org
16. Prof. M. Zuhair Nashed, University of Central Florida, U.S.A., znashed@mail.ucf.edu
17. Prof. Fahima Nekka, Biotechnology Focus, Montreal University, Canada, fahima.nekka@umontreal.ca
18. Prof. M. Yu. Rasulova, Institute of Nuclear Physics, Uzbekistan, m.yu.rasulova@live.com
19. Prof. M. Skopina, Euler Institute of Mathematical Sciences, Saint Petersburg State University, Russia,
skopinama@gmail.com
20. Prof. D. Walnut, George Mason University, U.S.A., dwalnut@gmu.edu
21. Prof. A.I. Zayed, DePaul University, U.S.A., azayed@condor.depaul.edu
Indian
22. Prof. R.C. Singh, Department of Physics, School of Basic Sciences and Research, Sharda University, Greater Noida,
UP, India, rcsingh@sharda.ac.in
23. Prof. G.D.V. Gowda, TIFR Centre for Applicable Mathematics Sharada Nagar, Chikkabommasandra, India,
gowda@math.tifrbng.res.in
24. Prof. U.B. Desai, IIT Hyderabad, India, ubdesai@iith.ac.in
25. Prof. Narendra K. Gupta, Department of Applied Mechanics Indian Institute of Technology, Delhi, India,
narinder_gupta@yahoo.com
26. Prof. S. Kesavan, Institute of Mathematical Sciences, Chennai, India, kesh@imsc.res.in
27. Prof. A.K. Pani, IIT Bombay, India, amiya.pani08@gmail.com
28. Prof. G. Rangarajan, IISc, Bangaluru, India, rangaraj@math.iisc.ernet.in
29. Prof. U.P. Singh, (Ex. VC, Purvanchal University) Rajendra Nagar, East Gorakhpur, India,
upsingh300@gmail.com
Indexed/Abstracted with: NAAS Rating 2017-3.98; EBSCO Discovery; INFOBASE INDEX (IB Factor 2017-3.1);
CNKI Scholar; Summon (ProQuest); Google Scholar; Cite Factor; ISRA-JIF; J-Gate; IIJIF; DRJI; Indian Citation Index
(ICI)
Preface
An International Conference on Analysis and its Applications (ICAA-2017) was held at Department
of Mathematics, Aligarh Muslim University, Aligarh, India during November 20-22, 2017, under the
DRS (SAP-II) programme of the University Grants Commission, Government of India. This conference
focused on wide range of topics related to analysis and its applications, namely, Nonlinear Analysis,
Operator Theory, Fixed Point Theory, Sequence Spaces and Matrix Transformations, Modern Methods
in Summability and Approximation Theory, Set-valued Analysis, Variational Analysis including
Variational Inequalities, Convex Analysis, Smooth and Non-smooth Analysis, Wavelet Analysis, Fourier
Analysis, etc..
This conference was an intense academic affair where approximately 208 participants were actively
engaged which include 30 foreign delegates from 20 countries of the world, 90 out-station participants
beside 88 internal ones. We had excellent speakers from India as well as abroad.
Following such very successful conference, we decided to publish a special issue dedicated to this
event. We have received more than 40 research articles for this conference proceeding. After peer
review process, only 18 research articles have been accepted for the publication in this special issue.
Sincere thanks to the following sponsors of the conference for their continuous support: Aligarh
Muslim University (AMU), University Grants Commission (UGC), Science and Engineering Research
Board-Department of Science and Technology (DST-SERB), National Board of Higher Mathematics
(NBHM), Council of Scientific and Industrial Research (CSIR) and the Indian National Science Academy
(INSA). Finally, we appreciate Indian Society of Industrial and Applied Mathematics for doing excellent
job in bringing out this special issue.
Editors
Rais Ahmad
Mijanur Rahaman
Ordering Information
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contemporary work on applications of mathematics to emerging areas. Editorial board comprising
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mathematics is the mother of all technologies. Peer reviewed research papers, and popular review
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Indian Journal of Industrial and Applied Mathematics
Vol. 10, No. 1 (Special Issue), Jan-June 2019
Contents
1.
On the Zeros of a Polynomial with Restricted Coefficients
Bilal Dar
1-10
2.
Projection Method for Bounded and Unbounded Solution of Hypersingular Integral
Equations of the First Kind
Z.K. Eshkuvatov and Anvar Narzullaev
11-37
3.
Discrete Legendre Projection Methods for the Fredholm Integral Equations of the
Second Kind
Jitendra Kumar Malik and Bijaya Laxmi Panigrahi
38-50
4.
Some Curvature Conditions on Nearly Cosymplectic Manifolds
G¨ulhan Ayar, Pelin Tekin and Nesip Aktan
51-58
5.
Second-order Characterization of Invex Functions and Its Applications in Optimization
Problems
M.T. Nadi and J. Zafarani
59-70
6.
Approximation of Periodic Functions via Statistical β-summability and Its Applications to
Approximation Theorems
B.B. Jena, S.K. Paikray and U.K. Misra
71-86
7.
On Semi-infinite Optimization Problem Under the Generalized Convexity
Pushkar Jaisawal, Vivek Laha and S.K. Mishra
87-108
8.
A Strict Constraint Qualification in Vector Optimization
Muskan Kapoor and C.S. Lalitha
109-120
9.
Fixed Point Theorems for Generalized [a, b, c] p-nonexpansive Mappings in Banach Spaces 121-144
D.R. Sahu and Satyendra Kumar
10.
On Generalized Operator Mixed Vector Quasi-equilibrium Problem
Rais Ahmad, Haider Abbas Rizvi, Mijanur Rahaman and Javid Iqbal
145-151
11.
On Approximation of Signals in the Generalized Zygmund Class Using (E, 1)
(N, pn)-summability Means of Fourier Series
Tejaswini Pradhan, Susanta Kumar Paikray and Umakanta Misra
152-164
12.
n-tupled Coincidence Point Results for Nonlinear Contraction in Partially Ordered
Complete Metric Spaces
Mohammad Imdad, Aftab Alam, Anupam Sharma and Mohammad Arif
165-181
13.
On Starlikeness and Convexity Properties of the Generalized Bessel-Hypergeometric
Function
Nusrat Raza and Eman S.A. AbuJarad
182-201
14.
A New Iterative Class for Finding Common Fixed Points of a Finite Family of Generalized
Total Asymptotically Nonexpansive and Multivalued Mappings in Hyperbolic Spaces
Shamshad Husain and Nisha Singh
202-219
15.
On a Class of Generalised ( p, q) Bernstein Operators
Lakshmi Narayan Mishra, Shikha Pandey and Vishnu Narayan Mishra
220-233
16.
Dhage Iteration Method for Approximating Solutions of IVPs of Nonlinear Second
Order Hybrid Neutral Functional Differential Equations
Bapurao C. Dhage
234-246
17.
A New Version of KKM Theorem with Geodesic Convex Hull with an Application on
Hadamard Manifold
J.C. Yao
247-254
18.
The Relaxed Resolvent Operator For Solving Fuzzy Variational Inclusion Problem
Involving Ordered RME Set-Valued Mapping
Iqbal Ahmad, Abul H. Siddiqi and Rais Ahmad
255-268
Printed & Published by: Diva Enterprises Pvt. Ltd. on behalf of Indian Society of Industrial and Applied
Mathematics, Printed at: Spectrum, 208 A/14A, Savitri Nagar, New Delhi-110017, Published at: Diva Enterprises
Pvt.Ltd. B-9, A-Block, L.S.C., Naraina Vihar, New Delhi-110028, India, Editor in Chief: Prof. A.H. Siddiqi
INDIAN JOURNAL OF INDUSTRIAL AND APPLIED MATHEMATICS
Vol. 10, No. 1 (Special Issue), Jan–June 2019, pp. 1–10
DOI:
On the Zeros of a Polynomial with Restricted
Coefficients
Bilal Dar
Department of Mathematics Govt. Degree College Boys, Pulwama-192301
Email: darbilal67@gmail.com, darbilal@dcpulwama.edu.in
Abstract: This paper focuses on the problem concerning the location and the number of zeros
of polynomials in a specific region when their coefficients are restricted with special conditions.
We obtain extensions of some classical results concerning the number of zeros of a polynomial in a
prescribed regions by subjecting the real and imaginary parts of its coefficients to certain restrictions.
Mathematics Subject Classifications (2010): 30A99, 30E10, 41A10.
Keywords: Polynomial, Zeros, Eneström-Kakeya theorem.
1. INTRODUCTION
If P(z) = nj=0 a j z j is a polynomial of degree n such that an ≥ an−1 ≥ ... ≥ a1 ≥ a0 > 0, then
P(z) has all its zeros in |z| ≤ 1. This famous result is known as Eneström-Kakeya theorem, for reference see (section 8.3 of 11). In the literature, for example see [1–12], there exist various extensions
and generalizations of Eneström-Kakeya theorem. Taking account of the restrictions on the coefficients of a polynomial allows for establishing improved bounds and here, in this paper, we impose
some restrictions on the coefficients of polynomials in order to count the number of zeros in a certain region. The following result concerning the number of zeros of a polynomial in a closed disk
can be found in Titchmarsh’s classic ”The Theory of Functions”, see [13, page 171, 2nd edition].
Theorem A. Let F(z) be analytic in |z| ≤ R. Let |F(z)| ≤ M in |z| ≤ R and suppose F(0) = 0.
Then for 0 < δ < 1, the number of zeros of F(z) in the disk |z| ≤ Rδ does not exceed
M
1
.
log
1
|F(0)|
log δ
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Vol. 10, No. 1 (Special Issue), Jan–June 2019
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Bilal Dar
Regarding the number of zeros in |z| ≤ 12 and by putting a restriction on the coefficients of a polynomial similar to that of the Eneström-Kakeya theorem, Mohammad [9] used a special case of
Theorem A to prove the following result
Theorem B. If P(z) = nj=0 a j z j is a polynomial of degree n such that 0 < a0 ≤ a1 ≤ ... ≤ an ,
then the number of zeros of P(z) in |z| ≤ 12 does not exceed
1+
an
1
.
log
log2
a0
The above result of Mohammad [9] was generalized in different ways for example see [1–3,5,6,11].
Recently Ahmad, Rasool and Liman [1] proved the following result under less restrictive conditions
on the coefficients of a polynomial.
Theorem C. If P(z) = nj=0 a j z j is a polynomial of degree n with complex coefficients such
that for some λ ≥ 1 and 0 ≤ k ≤ n, |an | ≤ |an−1 | ≤ ... ≤ |ak+1 | ≤ λ|ak | ≥ |ak−1 | ≥ ... ≥ |a0 | and
|arg a j − β| ≤ α ≤ π2 , 1 ≤ j ≤ n, for some real α and β. then the number of zeros of P(z) in
|z| ≤ 12 does not exceed
M
1
log
,
log2
|a0 |
where
M =2λ|ak |cosα + 2(λ − 1)|ak |sinα + |an |(sinα − cosα + 1)
+ 2sinα
n−1
|a j | + 2(λ − 1)|ak |.
j=0
In this paper, we further weaken the hypotheses of the above results by considering a larger class
of polynomials and obtain extensions of some classical results concerning the number of zeros of a
polynomial in a prescribed region.
Theorem 1. If P(z) = nj=0 a j z j is a polynomial of degree n such that for some t > 0, λ ≥ 1 and
0 ≤ k ≤ n, 0 < |a0 | ≤ t|a1 | ≤ ... ≤ t k−1 |ak−1 | ≤ λt k |ak | ≥ t k+1 |ak+1 | ≥ ... ≥ t n |an | and |arg a j −
β| ≤ α ≤ π2 , 1 ≤ j ≤ n for some real α and β. Then for 0 < δ < 1, the number of zeros of P(z) in
the disk |z| ≤ δt does not exceed
M
1
,
log
t|a0 |
log 1δ
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Vol. 10, No. 1 (Special Issue), Jan–June 2019
On the Zeros of a Polynomial with Restricted Coefficients
3
where
M =t|a0 |(1 − cosα − sinα) + 2sinα
n−1
|a j |t j+1 + t n+1 |an |(1 + sinα − cosα)
j=0
+ 2t
k+1
|ak | λcosα + (λ − 1)(1 + sinα) .
In particular taking t = 1 in Theorem 1, we get the following:
Corollary 1. If P(z) = nj=0 a j z j is a polynomial of degree n with complex coefficients such
that for some λ ≥ 1 and 0 ≤ k ≤ n, 0 < |a0 | ≤ |a1 | ≤ ... ≤ |ak−1 | ≤ λ|ak | ≥ |ak+1 | ≥ ... ≥ |an |
and |arg a j − β| ≤ α ≤ π2 , 1 ≤ j ≤ n, for some real α and β. Then for 0 < δ < 1, the number
of zeros of P(z) in the disk |z| ≤ δ does not exceed
1
M
log
|a0 |
log 1δ
where
M =|a0 |(1 − cosα − sinα) + 2sinα
n−1
|a j | + |an |(1 + sinα − cosα)
j=0
+ 2|ak | λcosα + (λ − 1)(1 + sinα) .
Taking δ =
exceed
1
2
in the Corollary 1, we get the number of zeros of P(z) in the disk |z| ≤
1
2
does not
1
M
,
log
log2
|a0 |
where the value of M is same as in Corollary 1.
Since 0 ≤ α ≤ π2 , we have 1 − cosα − sinα ≤ 0. So the value of M given in the Corollary 1
is less than or equal to
2sinα
n−1
j=0
|a j | + |an |(1 + sinα − cosα)
+ 2|ak | λcosα + (λ − 1)(1 + sinα)
and hence Corollary 1 implies Theorem C. With k = n and λ = 1 in Corollary 1, the hypothesis becomes
0 < |a0 | ≤ |a1 | ≤ ... ≤ |an | and the value of M becomes |a0 |(1 − cosα − sinα) +
2sinα n−1
j=0 |a j | + |an |(1 + sinα + cosα) and which by the same reasoning as above is less or
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Vol. 10, No. 1 (Special Issue), Jan–June 2019
4
Bilal Dar
equal to
|an |(1 + sinα + cosα) + 2sinα
n−1
|a j |.
j=0
This shows Corollary 1 implies a result of Pukhta [10, Theorem 1].
Remark 1. For λ = 1, Theorem 1 reduces to a result of Gardnar and Shields [5, Theorem 1].
Theorem 2. If P(z) = nj=0 a j z j is a polynomial of degree n with complex coefficients. If
Re(a j ) = α j , I m(a j ) = β j , 0 ≤ j ≤ n and for some t > 0, λ ≥ 1, we have 0 = α0 ≤ tα1 ≤ ... ≤
t k−1 αk−1 ≤ λt k αk ≥ t k+1 αk+1 ≥ ... ≥ t n αn , 0 ≤ k ≤ 1, then for 0 < δ < 1, the number of zeros of
P(z) in the disk |z| ≤ δt does not exceed
M
1
,
log
t|a0 |
log 1δ
where
M =(|αn | − αn )t n+1 + (|α0 | − α0 )t
+ 2λαk t k+1 + 2(λ − 1)|αk |t k+1 + 2
n
|β j |t j+1 .
j=0
Taking t = 1 in Theorem 2, we get the following:
Corollary 2. If P(z) = nj=0 a j z j is a polynomial of degree n with complex coefficients. If
Re(a j ) = α j , I m(a j ) = β j , 0 ≤ j ≤ n and for some λ ≥ 1, we have 0 = α0 ≤ α1 ≤ ... ≤ αk−1 ≤
λαk ≥ αk+1 ≥ ... ≥ αn , 0 ≤ k ≤ 1, then for 0 < δ < 1, the number of zeros of P(z) in the disk
|z| ≤ δ does not exceed
1
M
,
log
|a0 |
log 1δ
where
M =(|αn | − αn ) + (|α0 | − α0 )
+ 2λαk + 2(λ − 1)|αk | + 2
n
|β j |.
j=0
Corollary 2, the hypothesis becomes 0 < α0 ≤ α1 ≤ ... ≤ αn and
With k = n, λ = 1 and α0 > 0 in the value of M becomes 2(αn + nj=0 |β j |), therefore, Corollary 2 reduces to a result of Pukhta
[10]. For β j = 0, 1 ≤ j ≤ n and δ = 12 , Corollary 2 reduces to a result of Dewan and Bidkham [3].
Remark 2. For δ =
Theorem 2.1].
1
2
and t = 1, Theorem 2 reduces to a result of Rasool, Ahmad and Liman [12,
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Vol. 10, No. 1 (Special Issue), Jan–June 2019
On the Zeros of a Polynomial with Restricted Coefficients
5
Theorem 3. If P(z) = nj=0 a j z j is a polynomial of degree n with complex coefficients. If
Re(a j ) = α j , I m(a j ) = β j , 0 ≤ j ≤ n. Suppose that for some t > 0, λ ≥ 1 and 0 ≤ k ≤ n, we have
0 = α0 ≤ tα1 ≤ ... ≤ t k−1 αk−1 ≤ λt k αk ≥ t k+1 αk+1 ≥ ... ≥ t n αn and for some μ ≥ 1, 0 ≤ l ≤ n,
we have β0 ≤ tβ1 ≤ ... ≤ t l−1 βl−1 ≤ μt l βl ≥ t l+1 βl+1 ≥ ... ≥ t n βn . Then for 0 < δ < 1, the number of zeros of P(z) in the disk |z| ≤ δt does not exceed
1
M
,
log
t|a0 |
log 1δ
where
M =(|αn | − αn )t n+1 + (|α0 | − α0 )t + (|βn | − βn )t n+1 + (|β0 | − β0 )t
+ 2λαk t k+1 + 2μβl t l+1 + 2(λ − 1)|αk |t k+1 + 2(μ − 1)|βl |t l+1 .
We have some consequences of Theorem 3. If we take t = λ = μ = 1 in Theorem 3, we get the
following result.
Corollary 3. If P(z) = nj=0 a j z j is a polynomial of degree n with complex coefficients. If
Re(a j ) = α j , I m(a j ) = β j , 0 ≤ j ≤ n. Suppose that for 0 ≤ k ≤ n, we have 0 = α0 ≤ α1 ≤ ... ≤
αk−1 ≤ αk ≥ αk+1 ≥ ... ≥ αn and for 0 ≤ l ≤ n, we have β0 ≤ β1 ≤ ... ≤ βl−1 ≤ βl ≥ βl+1 ≥ ... ≥
βn . Then for some 0 < δ < 1, the number of zeros of P(z) in the disk |z| ≤ δ does not exceed
1
M
,
log
|a0 |
log 1δ
where
M =(|αn | − αn ) + (|α0 | − α0 ) + (|βn | − βn ) + (|β0 | − β0 )
+ 2αk + 2βl .
With k = l = n and α0 > 0 and β0 > 0 in Corollary 3, the hypothesis becomes 0 < α0 ≤ α1 ≤ ... ≤
αn and 0 < β0 ≤ β1 ≤ ... ≤ βn and the value of M becomes 2(αn + βn ) and hence the number of
zeros of P(z) in the disk |z| ≤ δ, 0 < δ < 1, does not exceed
1
2(αn + βn )
.
log
|a0 |
log 1δ
With t = λ = μ = 1 and k = l = 0, Theorem 3 gives the following result.
Corollary 4. If P(z) = nj=0 a j z j is a polynomial of degree n with complex coefficients. If
Re(a j ) = α j , I m(a j ) = β j , 0 ≤ j ≤ n. Suppose 0 = α0 ≥ α1 ≥ ... ≥ αn and β0 ≥ β1 ≥ ... ≥ βn .
Then for 0 < δ < 1, the number of zeros of P(z) in the disk |z| ≤ δ does not exceed
1
M
,
log
|a0 |
log 1δ
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Vol. 10, No. 1 (Special Issue), Jan–June 2019
6
Bilal Dar
where
M = (|αn | − αn ) + (|α0 | − α0 ) + (|βn | − βn ) + (|β0 | − β0 ).
Besides obtaining bounds for the zeros of polynomials with restriction on their coefficients, it is
equally important to locate the zeros of polynomials with arbitrary coefficients. Here, we have been
able to prove the following result.
m
j=0
|a j |
1p
, p > 1, then all the zeros of the polynomial P(z) = z n +
1
m
m−1
am z + am−1 z
+ ... + a1 z + a0 are contained in the disk |z| ≤ k, where k ≥ max 1, |am | n−m
Theorem 4. Let β =
p
is the largest positive root of the equation
n−m
1
1
q
+ = 1,
z
− |am | z q − 1 − β q = 0,
p q
where 0 ≤ m ≤ n − 1.
Remark 3. The special case of the above theorem for m = n − 1 was proved by Joyal, Labelle and
Rahman [7, Theorem 2].
2. PROOFS OF THEOREMS
We need the following lemma for the proofs of theorems.
Lemma 1. For any two complex numbers b0 and b1 such that |b0 | ≥ |b1 | and |arg b j − β| ≤ α ≤
π
, j = 0, 1 for some real β, then
2
|b0 − b1 | ≤ (|b0 | − |b1 |)cosα + (|b0 | + |b1 |)sinα.
The above lemma is due to Govil and Rahman [6].
Proof of Theorem 1. Consider the polynomial
F(z) = (t − z)P(z)
= (t − z)
n
ajz j
j=0
=
n
ta j z −
j
j=0
= ta0 +
a j z j+1
j=0
n
j=1
= ta0 +
n
n
ta j z j −
n
a j−1 z j − an z n+1
j=1
(ta j − a j−1 )z j − an z n+1 .
j=1
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Vol. 10, No. 1 (Special Issue), Jan–June 2019
On the Zeros of a Polynomial with Restricted Coefficients
7
For |z| = t, we have
|F(z)| ≤ t|a0 | +
n
|ta j − a j−1 |t j + |an |t n+1
j=1
= t|a0 | +
k−1
|ta j − a j−1 |t j + |tak − ak−1 |t k
j=1
+ |tak+1 − ak |t k+1 +
n
|a j−1 − ta j |t j + |an |t n+1
j=k+2
= t|a0 | +
k−1
|ta j − a j−1 |t j + t k |tak + λtak − λtak − ak−1 |
j=1
+t
k+1
|tak+1 + λak − λak − ak | +
n
|a j−1 − ta j |t j + |an |t n+1
j=k+2
≤ t|a0 | +
k−1
|ta j − a j−1 |t j + t k |λtak − ak−1 | + t k+1 |λak − tak+1 |
j=1
+ 2(λ − 1)|ak |t
k+1
+
n
|a j−1 − ta j |t j + |an |t n+1
j=k+2
k−1
(t|a j | − |a j−1 |)cosα + (|a j−1 | + t|a j |)sinα t j
≤ t|a0 | +
j=1
+ (λt|ak | − |ak−1 |)cosα + (λt|ak | + |ak−1 |)sinα t k
+ (λ|ak | − t|ak+1 |)cosα + (λ|ak | + t|ak+1 |)sinα t k+1
+ 2(λ − 1)|ak |t k+1 +
n
(|a j−1 | − t|a j |)cosα + (|a j−1 | + t|a j |)sinα t j
j=k+2
+ |an |t
n+1
(by using Lemma 1)
= t|a0 | − (t|a0 | + t n+1 |an |)cosα + 2λt k+1 |ak |cosα
+ 2(λ − 1)t k+1 |ak | − (t|a0 | + 2|ak |t k+1 )sinα
+ 2sinα
n−1
|a j |t j+1 + t n+1 |an |sinα + 2λt k+1 |ak |sinα + |an |t n+1
j=0
= t|a0 |(1 − cosα − sinα) + 2sinα
n−1
|a j |t j+1 + t n+1 |an |(1 + sinα − cosα)
j=0
+ 2t
k+1
|ak | λcosα + (λ − 1)(1 + sinα)
= M.
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Bilal Dar
Now F(z) is analytic in |z| ≤ t and |F(z)| ≤ M for |z| = t and F(0) = 0. So by Theorem A and
the Maximum Modulus Theorem, the number of zeros of F (and hence of P) in |z| ≤ δt does not
exceed
M
1
.
log
t|a0 |
log 1δ
Hence the Theorem 1 follows.
Proof of Theorem 2. Consider the polynomial
F(z) = (t − z)P(z)
= (t − z)
n
ajz j
j=0
n
= ta0 +
(ta j − a j−1 )z j − an z n+1
j=1
= (α0 + iβ0 )t +
n
(α j + iβ j )t − (α j−1 + iβ j−1 ) z j − (αn + iβn )z n+1
j=1
= (α0 + iβ0 )t +
n
n
(α j t − α j−1 )z + i
(β j t + β j−1 )z j − (αn + iβn )z n+1 .
j
j=1
j=1
For |z| = t, we have
|F(z)| ≤ (|α0 | + |β0 |)t +
n
|α j t − α j−1 |t j
j=1
+
n
(|β j |t + |β j−1 |)z j + (|αn | + |βn |)t n+1
j=1
= (|α0 | + |β0 |)t +
k−1
n
(α j t − α j−1 )t j + |tαk − αk−1 |t k +
(α j−1 − tα j )t j
j=1
+ |tαk+1 − αk |t k+1 +
j=k+2
n−1
|β j |t j+1 + |βn |t n+1
j=1
+ |β0 |t +
n−1
|β j |t j+1 + (|αn | + |βn |)t n+1
j=1
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On the Zeros of a Polynomial with Restricted Coefficients
= t|α0 | +
k−1
(tα j − α j−1 )t j + t k |tαk + λtαk − λtαk − αk−1 | +
j=1
n
9
(α j−1 − tα j )t j
j=k+2
+ t k+1 |tαk+1 + λαk − λαk − αk | + 2
n
|β j |t j+1 + |αn |t n+1
j=0
≤ t|α0 | +
k−1
(tα j − α j−1 )t j + t k (λtαk − αk−1 ) + t k+1 (λαk − tαk+1 )
j=1
+ 2(λ − 1)|αk |t k+1 +
n
(α j−1 − tα j )t j + 2
j=k+2
n
|β j |t j+1 + |αn |t n+1
j=0
= (|α0 | − α0 )t + (|αn | − αn )t n+1 + 2λt k+1 αk + 2(λ − 1)t k+1 |αk | + 2
n
|β j |t j+1
j=0
= M.
The result follows as in the proof of theorem 1.
Proof of Theorem 3. Proceeding on the same lines of proof of Theorem 2, the proof of Theorem 3
follows.
Proof of Theorem 4. We have
P(z) = z n + am z m + am−1 z m−1 + ... + a1 z + a0 ,
so that
|P(z)| ≥ |z|n − |am ||z|m − |am−1 z m−1 + ... + a1 z + a0 |.
(2.1)
By using the well-known Holder’s inequality, we have
|am−1 z m−1 + ... + a1 z + a0 |
1p m−1
q1
m−1
p
jq
≤
|a j |
|z|
j=0
|z| − 1
|z|q − 1
β|z|m
<
1 .
(|z|q − 1) q
=β
mq
q1
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j=0
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Bilal Dar
From (2.1), we get
|P(z)| > |z|n − |am ||z|m −
=
|z|m
(|z|q − 1)
1
q
β|z|m
1
(|z|q − 1) q
n−m
1
− |am |)(|z|q − 1) q − β .
(|z|
Hence |P(z)| > 0 if
1
(|z|n−m − |am |)(|z|q − 1) q − β ≥ 0
i.e, if
(|z|n−m − |am |)q (|z|q − 1) − β q ≥ 0.
Define
F(z) = (z n−m − |am |)q (z q − 1) − β q .
(2.2)
1
We observe that F |am | n−m < 0 and F(1) < 0. Also limz→+∞ F(z) =
+∞, we see from (2.2)
1
that the largest positive zero k of F(z) is greater than or equal to max 1, |am | n−m . This implies
|P(z)| > 0 if F(|z|) > 0 and hence |z| > k.
This proves the result.
REFERENCES
[1] I. Ahmad, T. Rasool and Abdul Liman, Zeros of certain polynomials and analytic functions with restricted coefficients,
J. Class. Anal., 4(2014), 149–157.
[2] K. K. Dewan, Extremal properties and coefficient estimates for polynomials with restricted zeros and on location of
zeros of polynomials, Ph.D. Thesis, Indian Institute of Technology, Delhi, 1980.
[3] K. K. Dewan and M. Bidkham, On the Eneström-Kakeya theorem, J. Math. Anal. Appl., 180(1993), 29—36.
[4] R. B. Gardner and N. K. Govil, On the location of the zeros of a polynomial, J. Approx. Theory, 78(1994), 286–292.
[5] R. Gardner and Brett Shields, The number of zeros of a polynomial in a disk, J. Class. Anal., 3(2013), 167–176.
[6] N. K. Govil and Q. I. Rahman, On the Eneström-Kakeya theorem, Tôhoku Math. Jour., 20(1968), 126–136.
[7] A. Joyal, G. Labelle and Q. I. Rahman, On the location of zeros of polynomials, Canad. Math. Bull., 10(1967), 53–63.
[8] M. Marden, Geometry of Polynomials, Math. Surveys, No.3, Amer, Math. Soc., Providence, R.I., 1966.
[9] Q. G. Mohammad, On the zeros of the polynomials, Amer. Math. Monthly, 72(1965), 631–633.
[10] M. S. Pukhta, On the zeros of a polynomial, Appl. Math, 2(2011), 1356–1358.
[11] Q. I. Rahman and G. Schmeisser, Analytic Theory of Polynomials, Oxford University Press, 2002.
[12] T. Rasool, I. Ahmad and Abdul Liman, On zeros of polynomials with restricted coefficients, Kyungpook Math. J.,
55(2015), 807–816.
[13] E. C. Titchmarsh, The Theory of Functions, 2nd Edition, Oxford University Press, London, 1939.
I NDIAN J OURNAL OF I NDUSTRIAL AND A PPLIED M ATHEMATICS
Vol. 10, No. 1 (Special Issue), Jan–June 2019
INDIAN JOURNAL OF INDUSTRIAL AND APPLIED MATHEMATICS
Vol. 10, No. 1 (Special Issue), Jan–June 2019, pp. 11–37
DOI:
Projection Method for Bounded and Unbounded
Solution of Hypersingular Integral Equations of
the First Kind
Z.K. Eshkuvatov1,2∗ and Anvar Narzullaev1
1
Faculty of Science and Technology, Universiti Sains Islam Malaysia (USIM), Nigeri Sembilan,
Malaysia
2
Institute for Mathematical Research, Universiti Putra Malaysia, Selangor, Malaysia
∗
Corresponding author Email: zainidin@usim.edu.my
Abstract: In this note, we consider a hypersingular integral equations (HSIEs) of the first kind
on the interval [−1, 1] with the assumption that kernel of the hypersingular integral is constant on
the diagonal of the domain D = [1, −1] × [−1, 1]. Projection method together with Chebyshev
polynomials of the first and second kinds are used to find bounded and unbounded solutions of
HSIEs respectively. Exact calculations of hypersingular and singular integrals for Chebyshev
polynomials allows us to obtain high accurate approximate solution. Gauss-Chebyshev quadrature
with Gauss-Lobotto nodes are presented as the high accurate computation of regular kernel
integrals. Six examples are provided to verify the validity and accuracy of the proposed method.
Comparisons with other methods are also given. Numerical examples reveals that approximate
solutions are exact if solution of HSIEs is of the polynomial forms with corresponding weights. It
is worth to note that proposed method works well for large value of node points n and errors are
drastically decreases. Comparisons of SPU times are also shown to demonstrate effectiveness of
the method and less complexity computations.
Mathematics Subject Classification: 65R20, 45E05.
Keywords: Integral equations, Hypersingular integral equations, Chebyshev polynomials, Approximation, Convergence.
1. INTRODUCTION
It is known that a closed-form solution of singular and hypersingular integral equations (HSIEs) of
the form
a(t)ϕ(t) +
1
b(t) 1 ϕ(t)
=
dt
+
L(x, t)ϕ(t)dt = f (x), p = {1, 2}, −1 < x < 1,
π −1 (t − x) p
−1
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Z.K. Eshkuvatov and Anvar Narzullaev
is only possible in exceptional cases. Comprehensive presentations and extensive literature survey
associated with all methods of solutions of singular integral equations (SIEs) (the case p = 1 in
Equation (1.1)) of the first and second kinds are found in [1–17] and literatures cited there in.
The methods of solution of HSIEs (the case p = 2 in Equation (1.1)) are much less elaborated.
In [18–31] the quadrature, projection, semi-analytical solution, Galerkin and collocation methods,
homotopy perturbation method, reproducing kernel method, for solving HSIEs of the first and second kinds (1.1) were developed and justified by imposing some conditions on the kernels and righthand sides of the equations.
In several physical problems such as aerodynamics, hydrodynamics, and elasticity, one encounters the integral equations of the form (see [22])
K (x, t)
1 1
+
L
(x,
t)
dt = f (x) , −1 < x < 1.
(1.2)
= ϕ(t)
1
π −1
(t − x)2
Hadamard ([20], 1952) was first to introduce the HSIEs in solving Cauchy’s problem for hyperbolic
differential equations. He defined the HSIEs as the finite part of a divergent integral whose singularities are arranged at the endpoints of the integration interval for one-dimensional integrals or on
the boundary of the domain for multidimensional integrals. These integrals came to be known as
Hadamard’s finite part integrals or simply Hadamard’s integrals.
For HSIEs: Chan et al. ([21], 2003) investigate HSIEs of the form
Iα (Tn , m, r ) =
Iα (Un , m, r ) =
1
−1
1
−1
m−1/2
Tn (s) 1 − s 2
ds, |r | < 1, m ∈ N,
(s − r )α
m−1/2
Un (s) 1 − s 2
ds, |r | < 1,
(s − r )α
where Tn (x) and Un (x) are the Chebyshev polynomials of the first and second kinds, respectively, and the exact solution are derived for the cases α = {1, 2, 3, 4} and m = {0, 1, 2, 3}. Mandal
and Bera [22] considered Equation (1.2) with K (x, t) = 1 and solved it by polynomial expansion
method and proved the exactness of the method for linear density function. Mandal and Bhattacharya [23] proposed approximate numerical solutions of some classes of integral equations by
using Bernstein polynomials as basis. The method was explained with illustrative examples. As
well as the convergence of the method is established for each class of integral equations. Martin [24] obtain the analytic solution to the simplest one-dimensional hypersingular integral equation
i.e. the case of K (x, t) = 1 and L(x, t) = 0 in Equation (1.2). Abdulkawi et al. [26], considered
the finite part integral equation (1.2) with K (x, t) = 1 and showed the exactness of the proposed
method for the linear density function and illustrated it with examples. Nik and Eshkuvatov [27]
have used the complex variable function method to formulate the multiple curved crack problems
into hypersingular integral equations of the first kind in more general case and these hypersingular
integral equations are solved numerically for the unknown function, which are later used to find the
stress intensity factor (SIF).
For the SIEs: Capobianco et al. [3, 4] studied collocation and quadrature methods for singular integro-differential equations of Prandtl’s type in weighted Sobolev spaces by using Langrange polynomials as basis. Elliot [7,8] proposed classical collocation method for the approximate
I NDIAN J OURNAL OF I NDUSTRIAL AND A PPLIED M ATHEMATICS
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Projection Method and Error Bound for HSIEs
13
solution of complete singular integral equations with Cauchy kernel taken over the arc (-1, 1) and
proved that under reasonable conditions, the approximate solutions converge to the solution of the
original equation as well as Petrov-Galerkin methods is applied to solve SIEs of second kind and
proved convergence of the proposed method, respectively. Ioakimidis [11, 12] proposed a classical
collocation and Galerkin methods for the numerical solution of SIEs of the first kind (1.2) involving
a finite part integrals with double pole singularity in the case of K (x, t) = 1. As an application of
the method to the problem of a straight crack under an exponential normal loading distribution is
also made and showed the rapid convergence of the obtained numerical results for the stress intensity factors at the crack tips. Eshkuvatov et al. [13, 14] consider SIEs (1.1) in the case p = 1, of the
first and second kind and proposed approximate methods based on Chebyshev polynomials for all
type of solutions of the first kind and bounded solution of the second kind respectively. Exactness of
the method is also shown. Boykov et al. [28, 29] proposed method asymptotically optimal and optimal in order algorithms for numerical evaluation of one-dimensional hypersingular integrals with
fixed and variable singularities and described a spline-collocation method and its justification for the
solution of one-dimensional hypersingular integral equations, polyhypersingular integral equations,
and multi-dimensional hypersingular integral equations, respectively.
Golberg [19] consider Equation (1.2) with the kernel K (x, t) = 1 and proposed projection
method with the truncated series of Chebyshev polynomials of the second kind together with
Galerkin and collacation methods. The author established the uniform convergence and obtained
convergence rates for their algorithms for solving a class of Hadamard singular integral equations.
By extending the results of Golberg [19] for the Equation (1.2), where the kernel K (x, t) is a constant on the diagonal of the domain D = [−1, 1] × [−1, 1], we outline collocations and Galerkin
methods together with Gauss-Chebyshev quadrature. For the unique solution of the bounded and
unbounded cases the following conditions
ϕ(−1) = ϕ(1) = 0,
1 1
ϕ(x)d x = C.
π −1
(1.3)
are imposed respectively.
The structure of this note is arranged as follows: In Section 2.1, the detail derivations of the projection method is presented followed by Gauss-Chebyshev quadrature in Section 2.2 with adaptation
weighted kernel integrals. Section 3 deals with six examples to verify the validity and accuracy of
the proposed method and outlines computation of complexity and economization of CPU time by
comparing with other methods. Finally, the conclusion and acknowledgment are presented in Section 4.
2. DESCRIPTION AND DISCRETIZATION OF THE METHOD
2.1. Description of the method
Since kernel in Equation (1.2) is constant on the diagonal we can write it as follows
K (x, t) = c0 + (t − x)Q(x) + (t − x)2 Q 1 (x, t), c0 = 0.
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Z.K. Eshkuvatov and Anvar Narzullaev
where Q(x) is smooth function and Q 1 (x, t) is square integrable kernel. Taking into account Equation (2.1) we are able to write Equation (1.2) in the form
c0 1 ϕ(t)
1 1
Q(x) 1 ϕ(t)
=
−
dt
+
dt
+
L(x, t)ϕ(t)dt = f (x), −1 < x < 1, (2.2)
π −1 (t − x)2
π
π −1
−1 t − x
where L(x, t) = Q 1 (x, t) + L 1 (x, t), and the first integral in Equation (2.2) is being understood as
the Hadamard finite-part.
Main aim is to find bounded and unbounded solution of Equation (2.2) satisfying condition
(1.3). Hence, we search solution in the form
ϕ(x) = ωi (x)u(x), i = {1, 2},
(2.3)
√
1 − x 2 , ω2 (x) =
(2.4)
where
ω1 (x) =
√ 1 ,
1−x 2
Substituting (2.3) into (2.2) yields
c0 1 ωi (t)
Q(x) 1 ωi (t)
u(t) dt
u(t) dt +
π −1 (t − x)2
π
−1 t − x
1 1
+
ωi (x)L(x, t) u(t) dt = f (x),
π −1
−1 < x < 1,
(2.5)
Introducing notations
c0 1 ωi (x)
=
Hi u =
u(t)dt,
π −1 (t − x)2
Ci u =
Q(x) 1 ωi (x)
−
u(t)dt,
π
−1 t − x
Li u =
1
π
(2.6)
1
−1
L(x, t)ωi (x)u(t)dt,
leads to the operator equation
Hi u + Ci u + L i u = f, i = {1, 2}, f ∈ L 2ρ , u ∈ L 1ρ ,
(2.7)
where the spaces L 2ρ and L 1ρ are defined as weighted Hilbert space and subspace of Hilbert respectively.
It is known that the hypersingular operator Hg can be considered as differential Cauchy operator
i.e.,
1
1
ω(t)
d
d
Cg u =
−
u(t)dt .
(2.8)
Hg u =
dx
d x π −1 t − x
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Projection Method and Error Bound for HSIEs
15
Therefore, Equation (2.7) may be viewed as an integro-differential Prandtl’s type equation. On the
other hand we know that (Mason [32])
√
1 1 1 − t 2 Un (t)
C g Un (x) =
−
dt = −Tn+1 (x), n = 1, 2, ...,
π −1
(t − x)
1 1
−
C g Tn (x) =
π −1
(2.9)
Tn (t)
1 − t 2 (t − x)
dt = Un−1 (x), n = 1, 2, ...,
For n = 0 we have
C g U0 (x) = −T1 (x), C g T0 (x) = 0,
where Tn (x) and Un (x) are the Chebyshev polynomials of the first and second kinds respectively,
defined by
Tn (x) = cos(nθ ), Un (x) =
sin(n + 1)θ
, x = cos θ, 0 ≤ θ ≤ π.
sin θ
By differentiating Equation (2.9) (see Chan [21]), we obtain
√
1 1 1 − t 2 Un (t)
Hg Un (x) =
dt = −(n + 1)Un (x), n = 1, 2, ...,
−
π −1 (t − x)2
1 1
Hg Tn (x) =
−
π −1
Tn (t)
1
dt =
2
2
1
−
x2
1 − t (t − x)
n+1
n−1
Un−2 (x) −
Un (x) , n = 1, 2, ...,
2
2
(2.10)
for n = 0
Hg U0 (x) = −U0 (x), Hg T0 (x) = 0,
Moreover,
Tn+1 (x) =
1
[Un+1 (x) − Un−1 (x)] , n = 0, 1, 2, ...,
2
xUn (x) =
1
[Un+1 (x) + Un−1 (x)] , n = 0, 1.2, ...
2
Un (x) =
(2.11)
1
[Tn (x) − Tn+2 (x)], n = 0, 1, 2, ...,
2(1 − x 2 )
where U−1 (x) = 0.
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Z.K. Eshkuvatov and Anvar Narzullaev
It can easily be shown from (2.6), (2.9), (2.10) and (2.11) that
H1 φn,1 (x) = −c0 (n + 1) φn,1 (x), n = 1, 2, . . .
n+1
c0
n−1
φn−2,1 (x) −
φn,1 (x) , n = 1, 2, . . .
H2 φn,2 (x) =
1 − x2
2
2
Q(x)
φn+1,1 (x) − φn−1,1 (x) , n = 1, 2, . . . ,
C1 φn,1 (x) = −
2
Q(x)
C2 φn,2 (x) = −
φn−1,1 (x), n = 1, 2, . . .
2
(2.12)
where φ−1,1 (x) = 0 and orthonormal polynomials are defined as
φi,1 (x) =
φi,2 (x) =
⎧
⎪
⎪
⎪
⎨
2
Ui (t), i = 0, 1, ..., n
π
⎪
⎪
⎪
⎩
2
Ti (x), i = 1, 2, ..., n
π
1
T0 (x), i = 0,
π
(2.13)
Equations (2.12) -(2.13) are crucial to the rest of our analysis.
To find an approximate solution of Equation (2.5), u (t) is approximated by
u(t) ∼
= u n,i (t) =
n
b(n)
j,i φ j,i (t), i = {1, 2}.
(2.14)
j=0
Therefore approximate solution of Equation (2.2) can be written as
ϕ(x) ≈ ϕn,i (x) = ωi (x)
n
b(n)
j,i φ j,i (x), i = {1, 2},
(2.15)
j=0
where unknown coefficients b(n)
j,i need to be defined.
To do these end let us consider two cases:
r Case 1, (i = 1). This is the case where the solution of Equation (2.7) is bounded at the end of
the interval [−1, 1]. Substituting (2.14) into (2.5) and using Equation (2.12) - (2.13) yields
n Q(x) (n)
(n)
(n)
−c0 ( j + 1)b(n)
b
U
U
(x)
+
−
b
(x)
+
b
ψ
(x)
j
j,1 j
j+1,1
j−1,1
j,1 j,1
2
j=0
+
Q(x) (n)
b Un+1 (x) =
2 n,1
π
f (x),
2
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(2.16)
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Projection Method and Error Bound for HSIEs
17
n
n
where U−1 (x) = 0, b−1,1
= bn+1,1
= 0 and
1
ψ j,1 (x) =
π
1
L(x, t) 1 − t 2 U j (t)dt.
(2.17)
1
n
such as roots
(i) For the collocation method we choose the suitable collocation points {xi }i=1
2
of Un+1 (x) or Tn+1 (x) or (1 − x )Un−1 (x). Then Equation (2.16) leads to a system of linear
equation
n Q(xk ) (n)
(n)
(n)
(n)
b j+1,1 − b j−1,1 U j (xk ) + b j,1 ψ j,1 (xk )
−c0 ( j + 1)b j,1 U j (xk ) +
2
j=0
+
Q(x) (n)
π
bn,1 Un+1 (xk ) = f (xk ), k = 0, 1, ..., n,
2
2
(2.18)
(n)
(n)
provided that b−1,1
= bn+1,1
= 0.
Solving the system of Equation (2.18) for the unknown coefficients b(n)
j,1 , j = 0,. . . , n and
(n)
substituting the values of b j,1 into Equation (2.15) yields the numerical solution of Equation
(2.2).
(ii) For the Galerkin’s method by taking into account (2.12) and (2.14), we consider the
following residual
√
√
c0 1 1 − t 2 u n,1 (t)
Q(x) 1 1 − t 2
=
−
u n,1 (t)dt
dt +
Rn,1 (x) =
π −1
(t − x)2
π
−1 t − x
+
1
π
1
L(x, t) 1 − t 2 u n,1 (t)dt − f (x)
−1
n 2
Q(x) (n)
(n)
(n)
(n)
−c0 ( j + 1)b j,1 U j (x) +
(b j+1,1 − b j−1 )U j (x) + b j,1 ψ j,1 (x)
π j=0
2
=
+
2 Q(x) (n)
b Un+1 (x) − f (x),
π 2 n,1
(2.19)
in which satisfies the conditions
1
−1
1 − τ 2 Rn,1 (τ )φk,1 (τ )dτ = 0, k = 0, ..., n,
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Z.K. Eshkuvatov and Anvar Narzullaev
It is known (Mason [32]) that
1
2
π
1 − x 2 φm,1 (t)φn,1 (t)dt =
−1
1
−1
1
1 − x 2 Um (t)Un (t)dt =
−1
⎧
1 1
⎪
⎪
⎨
φm,2 (t)φn,2 (t)
π −1
dt =
√
2
⎪
2 1
1−x
⎪
⎩
π −1
1,
0,
T0 (t)T0 (t)
dt,
√
1,
1 − x2
=
Tm (t)Tn (t)
0,
dt.
√
1 − x2
m = n,
m = n
(2.21)
m = n,
m = n.
(2.22)
Due to orthogonality conditions (2.21), residual in Equation (2.20) reduces to the system of
algebraic equation for finding the unknown coefficients {bnj,1 }nj=0 .
1 (n)
1
(n)
(n)
Q(x) bk+1,1
+
− bk−1,1
(b
− b(n)
j−1,1 ) Qφ j,1 , φk,1
2
2 j=0 j+1,1
n
(n)
− c0 (k + 1)bk,1
+
+
n
b(n)
j,1 ψ j,1 , φk,1
ρ
j=0
1 (n)
+ bn,1
Qφn+1,1 , φk,1
2
(n)
(n)
with b−1,1
= bn+1,1
= 0 and f, g
ρ
ρ
Qφ j,1 , φk,1
ρ
=
= f, φk,1 ρ , k = 0, ...n,
(2.23)
denotes the inner product with weight function i.e.,
f, φk,1
ρ
ρ
=
1
−1
1
1 − x 2 f (x)φk,1 (x)d x.
−1
1 − x 2 (Q(t) − Q(x))φ j,1 (x)φk,1 (x)d x.
r Case 2 (i = 2). In this case, solution of Equation (2.7) will be unbounded at the end of the
interval [−1, 1]. Substitute (2.14) into (2.5) and use Equation (2.12) to get
n
j=1
b(n)
j,2
c0
·
1 − x2
j +1
j −1
U j−2 (x) −
U j (x) + Q(x)U j−1 (x) + ψ j,2 (x)
2
2
(n)
ψ0,2 (x) =
+ b0,2
π
f (x),
2
(2.24)
T j (t)
L(x, t) √
dt.
1 − t2
(2.25)
where U−1 (x) = 0 and
ψ j,2 (x) =
1
π
1
1
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Projection Method and Error Bound for HSIEs
19
To have unique solution of Equation (2.24) we impose condition (1.3) to yield
C=
1
π
1
−1
ϕ(t)dt =
n
1 (n) 2 1 T j (t)dt
b j,2
√
π j=0
π −1 1 − t 2
2 (n) 2
b j,2
+
π
π
j=1
n
(n)
= b0,2
1
−1
T j (t)T0 (t)
(n)
dt = b0,2
√
2
1−t
2
.
π
From this it follows that
(n)
=C
b0,2
π
.
2
(2.26)
Substituting (2.26) into (2.24) yields
n
b(n)
j,2
j=1
c0
·
1 − x2
j +1
j −1
U j−2 (x) −
U j (x) + Q(x)U j−1 (x) + ψ j,2 (x) = f 1 (x),
2
2
(2.27)
π
f (x) + ψ0,2 (x) · C
2
(i) To solve Equation (2.27) for unknown parameters b(n)
j,2 using collocation method we
n
such as roots of Un (x) or Tn (x) or (1 − x 2 )Un−2 (x).
choose the suitable node points {xi }i=1
Then Equation (2.27) leads to a system of linear equation
where f 1 (x) =
n
j=1
b(n)
j,2
c0
·
1 − xk2
j +1
j −1
U j−2 (xk ) −
U j (xk ) + Q(x)U j−1,1 (xk ) + ψ j,2 (xk )
2
2
= f 1 (xk ), k = 1, 2, ..., n.
(2.28)
Solving the system of Equation (2.28) for the unknown coefficients b(n)
j,2 , j = 1, . . . , n and
(n)
substituting the values of b j,2 , j = 0, 1, ..., n into Equation (2.15) yields the numerical
solution of Equation (2.2).
(ii) In the case of unbounded solution for the Galerkin’s method we consider the following
residual
u n,2 (t)
u n,2 (t)
c0 1
Q(x) 1
= √
− √
dt
dt +
Rn,2 (x) =
2
2
π −1 1 − t (t − x)
π
1 − t 2 (t − x)
−1
+
1
π
1
u n,2 (t)
L(x, t) √
dt − f (x).
1 − t2
−1
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(2.29)
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Z.K. Eshkuvatov and Anvar Narzullaev
From (2.12), (2.14), (2.26) and (2.29) it follows that
n
2 (n) c0
j +1
j −1
U
U
b j,2
·
(x)
−
(x)
j−2
j
π j=1
1 − x2
2
2
Rn,2 (x) =
2 (n)
b ψ0,2 (x) · C − f (x)
+Q(x)U j−1,1 (x) + ψ j,2 (x) +
π 0,2
⎧
⎡
⎤
n−1 3
j
+
3
2 ⎨ c0
n
−
1
j
−
1
(n)
(n)
(n)
(n)
bn,2 Un (x) +
b j+2,2 −
b j,2 U j (x)⎦
· ⎣ b2,2 U0 (x) −
=
π ⎩ 1 − x2
2
2
2
2
j=1
⎫
n−1
⎬
(n) +
b j,2 Q(x)U j−1 (x) + ψ j,2 (x) + C · ψ0,2 (x) − f (x).
(2.30)
⎭
j=1
Since rational function seats on the denominator, residual in (2.30) satisfies the conditions
1
(1 − τ 2 )3/2 Rn,2 (τ )φk,1 (τ )dτ = 0, k = 1, ..., n.
(2.31)
−1
Equation (2.31) reduces to the system of algebraic equation for finding the unknown coefficients {bnj,2 }nj=1 , due to (2.12) and (2.21),
c0
n
k + 3 (n)
k − 1 (n)
bk+2,2 −
bk,2 +
b(n)
j,2 Qφ j−1,1 , φk,1
2
2
j=1
+ C ψ0,2 , φk,1
ρ
= f, φk,1 ,
ρ
ρ
+ ψ j,2 , φk,1
k = 1, ...n − 1,
ρ
(2.32)
where b−1,2 = bn+1,2 = 0. For k = n we have
n − 1 (n) (n) bn,2 +
b j,2 Qφ j−1,1 , φn,1
2
j=1
n
− c0
ρ
+ ψ j,2 , φn,1
ρ
+ C ψ0,2 , φn,1
ρ
= f, φn,1 ,
ρ
(2.33)
where f, g
ρ
denotes the inner product with weight function i.e.,
1
f, φk,1 ρ =
(1 − x 2 )3/2 f (x)φk,1 (x)d x,
−1
1
(1 − x 2 )3/2 Q(x)φ j,1 (x)φk,1 (x)d x,
Qφ j , φk,1 ρ =
−1
1
(1 − x 2 )3/2 ψ j,2 (x)φk,1 (x)d x.
ψ j,2 , φk,1 ρ =
−1
Solving the system of Equation (2.33) for the unknown coefficients b(n)
j,2 , j = 1, . . . , n and
into
Equation
(2.15)
yields
the
numerical
solution of Equation
substituting the values of b(n)
j,2
(2.2).
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Projection Method and Error Bound for HSIEs
21
2.2. Discretization of the method
In the Section 2.1, we have described two methods which deals with many form of weighted kernel
integrals. In this section, we develop Gauss-Chebyshev quadrature formula with Gauss-Lobotto
nodes for weighted kernel integrals. It is known that many weighted kernel integrals have not exact
solution. So that we need suitable quadrature for numerical computation of weighted kernel integrals
In Kythe [35], states that the Gauss quadrature formula of the form
b
w(x) f (x)d x =
a
n
Ai f (xi ),
(2.34)
i=0
is exact for all f ∈ P2n+1 if the weights Ai and the nodes xi can be found for different orthogonal
polynomials approximation of f (x) on the interval [a, b].
In particular, if [a, b] = [−1, 1] and ω1 (x), ω2 (x) are defined by (2.4), the orthogonal polynomials are the Chebyshev polynomials Tn (t), Un (t) of the first and second kind respectively then
resulting formulas of Equation (2.34) are known as Gauss-Chebyshev rule.
Let us define the nodes ξi,1 and ξi,2 as the zeros of Tn+1 (x) and Un+1 (x) respectively,
ξi,1 = cos
(2i − 1)π
2(n + 1)
ξi,2 = cos
, i = 1, 2, ..., n + 1
(2.35)
iπ
, i = 1, 2, ..., n + 1.
n+2
(2.36)
Lemma 1. Open Gauss-Chebyshev rule (in some literature it is called Meller quadrature) is given
as
1
−1
where Ak,1 =
1
f (t)dt =
Ak,1 f (tk ) + Rn.1 ( f ).
√
1 − t2
k=1
n+1
(2.37)
π
and tk is defined by (2.35). Similarly
n+1
1
−1
1 − t 2 f (t)dt =
n+1
Ak,2 f (tk ) + Rn.2 ( f ),
(2.38)
k=1
π (1 − tk2 )
and tk is defined by (2.36). The word “open” is used for not including
n+2
endpoints. We usually omit “open” since all Gaussian rules with positive weight function are of the
open type.
where Ak,2 =
Proper Derivation. Let us show proper derivation of Gauss-Chebyshev quadrature formulas (QF)
(2.37)-(2.38) which is different than Kythe [35] and Israilov [36].
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Z.K. Eshkuvatov and Anvar Narzullaev
It is known that (see Israilov [36, pp. 338]) coefficients of the interpolation quadrature formula with
weight function
b
ρ(t) f (t)dt =
a
n
Ak f (xk ),
(2.39)
Pn (t)
dt,
(t − tk )
(2.40)
k=1
are defined as
1
Ak =
Pn (tk )
b
a
ρ(t)
where tk are the zeros of interpolation polynomial Pn (t).
In view of (2.39)-(2.40) and (2.9), the coefficients of the QF (2.37) is
Ak,1 =
1
Tn+1 (tk )
1
−1
1
1
Tn+1 (t)
dt =
πUn (tk ),
√
2
Tn+1 (tk )
1 − t (t − tk )
(2.41)
It is easy to check that
Tn+1 (x) = (n + 1)Un (x),
therefore
Ak,1 =
π
.
n+1
To derive QF (2.38) we take tk , k = 1, 2, ..., n + 1 as the zeros of Chebyshev polynomials of the
second kind Un+1 (t) = 0 which is defined by (2.36). In a similar way as (2.41), by applying (2.9),
we obtain
1 √
1
1 − t 2 Un (t)
1
dt =
(−π Tn+1 (tk )).
(2.42)
Ak,2 =
Un (tk ) −1
(t − tk )
Un (tk )
Since tk , k = 1, 2, ..., n + 1 are the zeros of Un+1 (x) and
1
(Un+1 (x) − Un−1 (x)),
2
d
n+2
1
n
Un (x) =
U
U
(x)
−
(x)
,
n−1
n+1
dx
1 − x2
2
2
Tn+1 (x) =
(2.43)
(2.44)
π (1 − tk2 )
.
n+2
The proof of main theorem based on the following theorems given in Kythe [35].
we have Ak,2 =
Theorem 1 (Johnson and Riess 1977). Gaussian QF has precision 2n + 1 only if the points xi , i =
0, 1, ..., n are the zeros of φn+1 (x), where φn+1 (x) are orthogonal polynomials.
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Vol. 10, No. 1 (Special Issue), Jan–June 2019
Projection Method and Error Bound for HSIEs
Theorem 2. If f ∈ C 2n+2 [a, b], then the error of Gaussian QF is given by
f 2n+2 (ξ ) b
2
Rn ( f ) = Iab ( f ) − In ( f ) =
ρ(x)Wn+1
(x)d x, ξ ∈ [a, b]
(2n + 2)! a
23
(2.45)
where Wn+1 (x) is the polynomials of degree n + 1 with n + 1 distinct zeros anf ρ(x) is a weight
function. Main theorem is formulated as follows
Theorem 3. If f ∈ C 2n+2 [−1, 1], then the error of Gauss-Chebyshev QFs (2.37) and (2.38) are
given by
π
Rn,1 ( f ) =
f 2n+2 (ξ1 )
, ξ1 ∈ [−1, 1],
(2n + 2)!
22n+1
π
Rn,2 ( f ) =
(2.46)
f 2n+2 (ξ2 )
, ξ2 ∈ [−1, 1].
(2n + 2)!
22n+3
Proof. In Israilov [36] shown the relationship between polynomial Wn+1 in (2.45) and Chbyshev
polynomials of the first and second kind as fallows
Tn+1 (x)
Wn+1 (x) =
,
2n
(2.47)
Un+1 (x)
Wn+1 (x) =
.
2n+1
On the other hand it is known that
Tn+1 , Tn+1 =
1
−1
Un+1 , Un+1 =
T 2 (t)
π
√n+1 dt = ,
2
2
1−t
(2.48)
1
−1
2
1 − t 2 Un+1
(t)dt =
π
.
2
From (2.47) - (2.48) it follows that
Rn,1 ( f ) =
f 2n+2 (ξ1 )
(2n + 2)!
f 2n+2 (ξ2 )
Rn,2 ( f ) =
(2n + 2)!
1
−1
√
1 2
π f 2n+2 (ξ1 )
,
Tn+1 (t)dt = 2n+1
2n
2
(2n + 2)!
1 − t2 2
1
(2.49)
1
−1
1 − t2
1
22n+2
2
Un+1
(t)dt =
π
22n+3
f 2n+2 (ξ2 )
.
(2n + 2)!
Now we extend Gauss-Chebyshev QF (2.37) - (2.38) for the weight kernel integrals (2.6) which is
given in Section 2.1. In many problems of HSIEs regular kernel L(x, t) will be given as convolution
type
m
L(x, t) =
ci (x)di (t).
(2.50)
i=1
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Z.K. Eshkuvatov and Anvar Narzullaev
In the case of convolution type kernel (2.50), for the regular kernel in (2.6), the Gauss-Chebyshev
QF in Lemma 1 has the form
1
L 1 (u) =
π
−1
−1
L(x, t) 1 − t 2 u(t)dt
m
n
2 1 1
ci (x)b(n)
j,1
π i=1 j=0
π −1
=
m n n+1
1 − t 2 di (t)U j (t)dt
(2.51)
ci (x)b(n)
j,1 Ak,1 f 1i, j (tk ).
i=1 j=0 k=1
Similarly
L 2 (u) =
1
π
−1
L(x, t) √
−1
1
1 − t2
u(t)dt
m
n
2 1 1
1
ci (x)b(n)
di (t)T j (t)dt
√
j,2
π i=1 j=0
π −1 1 − t 2
=
m n n+1
(2.52)
ci (x)b(n)
j,2 Ak,2 f 2i, j (tk ).
i=1 j=0 k=1
where
Ak,1 =
1 − tk2
, f 1i, j (tk ) = di (tk )U j (tk ),
n+2
Ak,2 =
1
, f 2i, j (tk ) = di (tk )T j (tk ).
n+1
For non convolution regular kernel L(x, t) case, we have the following Gauss-Chebyshev QF
L 1 (u) =
1
π
=
−1
−1
L(x, t) 1 − t 2 u(t)dt
n
2 (n) 1 1
b j,1
π j=0
π −1
n n+1
1 − t 2 L(x, t)U j (t)dt
(2.53)
b(n)
j,1 Ak,1 f 1(x, tk ).
j=0 k=1
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Projection Method and Error Bound for HSIEs
25
In the same way we obtain
1
L 2 (u) =
π
=
−1
−1
1
L(x, t) √
u(t)dt
1 − t2
n
2 (n) 1 1
1
b j,2
L(x, t)T j (t)dt
√
π j=0
π −1 1 − t 2
n n+1
(2.54)
b(n)
j,1 Ak,2 f 2(x, tk ).
j=0 k=1
where
Ak,1 =
1 − tk2
, f 1 (x, tk ) = L(x, tk )U j (tk ),
n+2
Ak,2 =
1
, f 2 (x, tk ) = L(x, tk )T j (tk ).
n+1
3. NUMERICAL RESULTS
We very often need to use polynomial values of the Chebyshev polynomials for the numerical
computation.
3.1. Case 1, i = 1. Bounded solution
Example 1. Consider the following HSIEs
1
π
1
−1
ϕ(t)
1
dt +
(t − x)2
π
1
−1
16x 3 t 3 ϕ(t)dt = 16x − 31x 3 ,
(3.1)
with the conditions
ϕ(±1) = 0.
It is not hard to verify that the exact solution of Equation (3.1) is
ϕ(x) =
1 − x 2 (8x 3 − 4x).
(3.2)
Solution: We search bounded solution of HSIEs in the form
ϕ(x) ∼
= ϕn (x) =
2
π
1 − x2
I NDIAN J OURNAL OF I NDUSTRIAL AND A PPLIED M ATHEMATICS
n
b(n)
j,1 U j (x).
(3.3)
j=0
Vol. 10, No. 1 (Special Issue), Jan–June 2019
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Z.K. Eshkuvatov and Anvar Narzullaev
Table 1. Chebyshev polynomials of the first and second kind
n
Tn (x)
Un (x)
0
1
1
1
x
2x
2
2x 2 − 1
4x 2 − 1
3
4x 3
8x 4
4
− 3x
− 8x 2
8x 3 − 4x
+1
16x 4
16x 5 − 20x 3 + 5x
5
64x 7
7
32x 5 − 32x 3 + 6x
− 48x 4
+ 18x 2
−1
64x 6 − 80x 4 + 24x 2 − 1
− 112x 5
+ 56x 3
− 7x
128x 7 − 192x 5 + 80x 3 − 8x
32x 6
6
− 12x 2 + 1
Since c0 = 1, Q(x) = 0 and L(x, t) = 16x 3 t 3 , from the Equation (2.16) and (2.17) with n = 3 it
follows that
3
π
(16x − 31x 3 ),
2
b(n)
j,1 −( j + 1)U j (x) + ψ j,1 (x) =
j=0
where ψ j,1 (x) is defined as
ψ j,1 (x) =
1
π
1
−1
(3.4)
(16x 3 t 3 ) 1 − t 2 U j (t)dt.
Since
t3 =
1
[U3 (t) + 2U1 (t)] ,
8
(3.5)
we obtain
ψ0,1 (x) = 0, ψ1,1 (x) = 2x 3 , ψ2,1 (x) = 0, ψ3,1 (x) = x 3 , ψ j,1 (x) = 0, j > 3.
(3.6)
Substituting (3.6) into (3.4) and taking into account Table 1, yields
(n)
(n)
(n)
(n)
− b0,1
+ 2(x 3 − 2x)b1,1
− 3(4x 2 − 1)b2,1
+ (16x − 31x 3 )b3,1
=
π
(16x − 31x 3 ).
2
(3.7)
Equating the same power of x yields
(n)
(n)
(n)
(n)
b0,1
= b1,1
= b2,1
= 0, b3,1
=
π
.
2
Substituting all values of b(n)
j,1 , j = {0, 1, 2, 3} into (3.3), gives
ϕ(x) =
1 − x 2 U3 (x) =
1 − x 2 (8x 3 − 4x) ,
which is identical to the exact solution (3.2).
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Projection Method and Error Bound for HSIEs
27
Let us apply Galerkin method for Example 1. Since c0 = 1, Q(x) = 0, and φ j,1 (x) are defined by
(2.13), residual in Equation (2.19) is
Rn,1 (x) =
n
2
(−( j + 1))b(n)
j,1 U( x) +
π j=0
2 3 (n)
(n)
2x b1,1 + x 3 b3,1
− 16x − 31x 3 .
π
In view of (3.5) and 2x = U1 (x) with n = 3, it can be written as
2
21
(n)
(n)
(U3 (x) + 2U1 (x))
2b1,1
(−( j + 1))b(n)
+ b3,1
j,1 U( x) +
π j=0
π8
31
− 8U1 (x) − (U3 (x) + 2U1 (x)) .
8
3
Rn,1 (x) =
(3.8)
Multiplying (3.8) by φk,1 (τ ), and integrating over [−1, 1] yields
1
−1
1 − τ 2 R3,1 (τ )φk,1 (τ )τ = 0, k = {0, 1, 2, 3}.
(3.9)
From (3.9) it follows that
(3)
(3)
b0,1
= b2,1
= 0,
(3.10)
and
⎧
1
1
⎪
(n)
(n)
(n)
⎪
⎪
−2b1,1
2b1,1
−
+
+ b3,1
⎪
⎨
4
4
π
= 0,
2
(3.11)
⎪
⎪
31
1
⎪
(n)
(n)
(n)
⎪
⎩ −4b3,1
2b1,1
+
+
+ b3,1
8
8
π
= 0.
2
Solving Equation (3.11), gives
π
.
2
(n)
(n)
= 0, b3,1
=
b1,1
(n)
Exact solution can be obtained by substituting the values of bi,1
,
(3.3).
(3.12)
j = {0, 1, 2, 3} into Equation
Example 2. Let us investigate the following HSIEs
1
π
1
−1
(1 + 2(t − x))
1
ϕ(t)dt +
(t − x)2
π
1
−1
I NDIAN J OURNAL OF I NDUSTRIAL AND A PPLIED M ATHEMATICS
1 2x 3
e t ϕ(t)dt = f (x),
2
(3.13)
Vol. 10, No. 1 (Special Issue), Jan–June 2019
28
Z.K. Eshkuvatov and Anvar Narzullaev
where
f (x) = −16x 4 − 40x 3 + 4x 2 + 22x + 1 +
π 2x
e .
32
The exact solution of Equation (3.13) is
ϕ(x) =
1 − x 2 (8x 3 + 4x 2 − 4x − 1).
(3.14)
Solution: Comparing (3.13) with (2.2) we get
c0 = 1, Q(x) = 2, L(x, t) =
1 2x 3
e t .
2
From (2.16) and (2.17) we obtain
n (n)
(n)
(n)
(n)
−( j + 1)b(n)
j,1 + b j+1,1 − b j−1,1 U j (x) + b j,1 ψ j,1 (x) + bn,1 Un+1 (x) =
j=0
π
f (x), (3.15)
2
n
n
where U−1 (x) = 0, b−1,1
= bn+1,1
= 0 and
1
ψ j,1 (x) =
π
1
1
1 2x 3
e t
2
1 − t 2 U j (t)dt.
(3.16)
From (3.16) and orthogonality conditions (2.21) it follows that
ψ0,1 (x) = 0, ψ1,1 (x) =
1 2x
1 2x
e , ψ2,1 (x) = 0, ψ3,1 (x) =
e , ψ j,1 (x) = 0, j > 3.
16
32
(3.17)
Substituting (3.17) into (3.15) and choosing collocation points xi as the root of Tn+1 (x) which is
given in Equation (2.35), we have the following system of algebraic equations
n (n)
(n)
−( j + 1)b(n)
+
b
−
b
j,1
j+1,1
j−1,1 U j (x i ) + bn,1 Un+1 (x i )
j=0
+
1
(n)
(n)
2b1,1
e2xi =
+ b3,1
32
π
f (xi ), i = 0, ..., n,
2
(3.18)
Solving Equation (3.18) at the collocation points (2.35) for the different value of n, we obtain
the numerical solution of Equation (3.13). The errors of numerical solution of Equation (3.13) are
summarized in Table 2.
Table 2 reveals that method proposed here is very accurate and stable. The large value of n =
{50, 500} shows that proposed method is exact for HSIEs (3.13). Actually, it can be shown that
proposed method is exact for n = 3 only.
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Vol. 10, No. 1 (Special Issue), Jan–June 2019
Projection Method and Error Bound for HSIEs
29
Table 2. Numerical solution of Example 2
Exact value of HSIEs (3.13)
Errors Rn , n = 5
Errors Rn , n = 50
Errors Rn , n = 500
−0.01412481476
5.0480 × 10−016
4.7878 × 10−016
4.8121 × 10−016
−0.00011175527
1.0973 × 10−015
4.9398 × 10−015
2.4761 × 10−015
-0.725
0.65698032427
1.5543 × 10−015
3.5527 × 10−015
2.5313 × 10−015
-0.436
0.75715136458
1.2212 × 10−015
1.0123 × 10−015
1.0214 × 10−015
−1.4433 × 10−015
−1.8874 × 10−015
x
-0.9999
-0.901
-0.015
0.015
0.436
× 10−016
−0.93902134227
2.2204e
−1.05895384758
4.4409 × 10−016
1.0014 × 10−016
−1.5543 × 10−015
−1.18843460478
−1.1102 × 10−015
−2.6645 × 10−015
−3.5527 × 10−015
0.725
0.86171092460
1.0014 × 10−016
−2.8866e × 10−015
−3.6637 × 10−015
0.901
1.94987172916
1.7764 × 10−015
−2.4425 × 10−015
−3.5527 × 10−015
0.09895288143
9.7145 × 10−16
1.0024 × 10−016
−8.3267 × 10−016
0.9999
Example 3. Solve HSIEs of the form
1
π
1
−1
K (x, t)
+ L(x, t) ϕ(t)dt = f (x),
(t − x)2
(3.19)
where
K (x, t) = 2 + t x(t − x), L 1 (x, t) =
1
1
+
,
t +2 x +2
and
√
√
√
√
20 3
10x 2
10 √
10(2 − 3)
f (x) = −
(2 − 3 + x) + 10(2 − 3)x + (2 3 − 3) +
.
−
(2 + x)2
x +2
3
x +2
The exact solution of Equation (3.19) is
ϕ(x) =
1 − x2
10
.
x +2
(3.20)
Remark. In this example, main kernel K (x, t) is given as convolution form but on the diagonal
K (x, x) = const. On the other hand regular kernel L(x, t) is not convolution type. Here we present
experimentally that the proposed method can work well even the regular kernel L(x, t) in Equation
(3.19) is not in the convolution type and solution is not of the polynomial form.
Solution. Suitable changes in Equation (3.19) leads to
2
π
1
−1
ϕ(t)dt
x2
+
(t − x)2
π
1
−1
1
ϕ(t)dt
+
t−x
π
1
−1
x+
1
1
+
ϕ(t)dt = f (x),
x +2 t +2
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(3.21)
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Z.K. Eshkuvatov and Anvar Narzullaev
Table 3. Numerical solution of Example 3
x
Exact, (3.20) Errors Rn , n = 6 Errors [25], n, m = 6, Errors Rn , n = 26, Errors [25], n = m = 26,
0.1414037
3.5663 × 10−4
1.0852 × 10−4
5.2180 × 10−015
1.4040 × 10−10
3.9473984
1.2235 × 10−4
3.5502 × 10−4
−1.2879 × 10−015
1.8203 × 10−9
-0.725
5.4019519
2.4011 × 10−3
1.88998 × 10−4
1.7764 × 10−015
0.6194 × 10−9
-0.436
5.7541347
2.2201 × 10−3
3.0648 × 10−4
−1.5987 × 10−014
0.2622 × 10−9
5.0372166
−1.6201 × 10−3
0.8678 × 10−4
−1.4211 × 10−014
0.1739 × 10−9
4.9622208
−1.601 × 10−3
1.9992 × 10−4
−1.1546 × 10−014
1.8552 × 10−9
0.436
3.6943623
1.3001 × 10−3
3.3917 × 10−4
−9.7700 × 10−015
0.1970 × 10−9
0.725
2.5275188
−0.9487 × 10−3
0.6333 × 10−4
−2.6645 × 10−015
0.2347 × 10−9
0.901
1.4954122
2.0388 × 10−4
1.2746 × 10−4
−4.8850 × 10−015
0.1340 × 10−9
0.0471408
1.0831 × 10−4
0.3333 × 10−4
1.4225 × 10−015
3.5400 × 10−10
-0.9999
-0.901
-0.015
0.015
0.9999
So that c0 = 2, Q(x) = x 2 and L(x, t) = x +
1
1
+
. From Equations (2.15) - (2.17) and
x +2 t +2
(3.21) we obtain
1
1
x2
x2
(n)
(n)
x+
+ g0,1 b0,1 + −2 +
b1,1
+ Un+1 (x)bn,1
2
x +2
2
2
n x 2 (n)
(n)
(n)
b
−2( j + 1)b(n)
U
=
+
+
−
b
(x)
+
b
g
j
j,1
j,1
j+1,1
j−1,1
j,1
2
j=1
where b−1,1 = bn+1,1 = 0 and g j,1
1
=
π
π
f (x),
2
(3.22)
1 √
−1
1 − t2
U j (t)dt.
t +2
We choose the collocation points xi as in Equation (2.35). Solving Equation (3.22) for the unknown
coefficients b(n)
j,1 for different values of n and substituting it into (2.15), we obtain the numerical
solution of Equation (3.19). Errors of numerical solution of Equation (3.19) and comparisons with
the method presented in Eshkuvatov [25] are given in Table 3.
where n is a number of collocation points and m denotes the number of selection functions.
Table 3 shows that when x comes close to the end points of the interval (−1, 1) or in the middle
of the interval, errors decreases drastically and when n = 26 the errors reached to almost zero. It
reveals that proposed method is suitable for HSIEs when solution is bounded. On the other hand
proposed method is dominated over the method proposed in Eshkuvatov at al. [25]. In [25] n stands
for number of nodes and m is for number of selection function.
Example 4 (Mandal and Bera [22]). Let us consider the following HSIEs
1
π
1
−1
ϕ(t)
1
dt +
(t − x)2
π
1
−1
(t + x) ϕ(t)dt = 1 + 2x,
I NDIAN J OURNAL OF I NDUSTRIAL AND A PPLIED M ATHEMATICS
(3.23)
Vol. 10, No. 1 (Special Issue), Jan–June 2019
Projection Method and Error Bound for HSIEs
31
The exact solution of Equation (3.23) is
4
1 − x 2 (9 + 10x).
31
ϕ(x) = −
(3.24)
Solution: Comparing (3.23) with (2.2) we get
c0 = 1, Q(x) = 0 L(x, t) = x + t.
Equations (2.16) and (2.17) yields
n
#
"
b(n)
j,1 −( j + 1)U j (x) + ψ j,1 (x) =
j=0
π
(1 + 2x),
2
(3.25)
where
1
ψ j,1 (x) =
π
1
(t + x) 1 − t 2 U j (t)dt.
1
Due to orthogonality condition (2.21), we obtain
ψ0,1 (x) =
x
1
, ψ1,1 (x) =
ψ j,1 (x) = 0, j ≥ 2.
2
4
(3.26)
Substituting (3.26) into (3.25) and choosing collocation points xi as given in Equation (??), the
system of algebraic equations (3.25) has the form
n
xi
1 (n)
b0,1 + b1,1
(−( j + 1))b(n)
=
j,1 U j (x i ) +
2
4
j=0
π
(1 + 2xi ), i = 0, ..., n,
2
(3.27)
Solving Equation (3.27) at the collocation points (2.35) for the different value of n, we obtain the
numerical solution of Equation (3.23). The comparison errors of Equation (3.23) are summarized in
Table 4.
Table 4, reveals that proposed method (2.18) and Mandal’s method [22] are very accurate to this
example for small value of n but Table 5 shows that CPU time of proposed method is much more
less than Mandal’s method [22]. On the other hand for large value of n computational complexity
of Mandal’s method is much more higher than the proposed method. On the other hand proposed
method can be used for any value of n. In Table 4, we are able to compute for “n = {3, 7}” only. It
can be shown that the method proposed here is exact for Example 4 with only n = 2.
I NDIAN J OURNAL OF I NDUSTRIAL AND A PPLIED M ATHEMATICS
Vol. 10, No. 1 (Special Issue), Jan–June 2019
32
Z.K. Eshkuvatov and Anvar Narzullaev
Table 4. Comparison results of Example 4
x
Exact solution (3.24)
Error of proposed method (2.15)
Errors of Mandal and Bera [22]
0.0079350
1.73 × 10−018
5.55 × 10−017
-0.688
−0.19851699
1.11 × 10−016
−1.94 × 10−016
-0.118
−1.00198275
2.22 × 10−016
0.00 × 10+00
−1.30437141
2.22 × 10−016
0.00 × 10+00
−1.48700461
2.22 × 10−016
4.44 × 10−016
−0.15481294
2.27 × 10−017
8.33 × 10−017
0.0079350
6.94 × 10−018
3.09 × 10−016
-0.688
−0.19851699
5.55 × 10−017
3.61 × 10−016
-0.118
−1.00198275
2.22 × 10−016
0.00 × 10+00
−1.30437141
2.22 × 10−016
0.00 × 10+00
−1.48700461
0.00 × 10+00
−8.88 × 10−016
−0.15481294
2.27 × 10−017
−6.11 × 10−016
n=3
-0.998
0.118
0.688
0.998
n=7
-0.998
0.118
0.688
0.998
Table 5. CPU time (in seconds). Comparisons for Example 4
Number of points n
CPU Prop. method (3.23)
Mandal and Bera [22]
3
0.3121
0.6022
7
0.9409
1.5328
10
1.6526
20
5.204
30
11.3804
50
32.4334
3.2. Case 2 i = 2. Unbounded solution
Example 5. Consider the following HSIEs with corresponding condition
1
π
1
π
1
−1
ϕ(t)
1
dt +
(t − x)2
π
1
−1
16x 3 t 3 ϕ(t)dt = 1,
(3.28)
1
3
ϕ(t)dt = .
2
−1
It is not hard to verify that the exact solution of Equation (3.28) is
ϕ(x) = √
1
1−
x2
(1 + x 2 ).
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Vol. 10, No. 1 (Special Issue), Jan–June 2019
Projection Method and Error Bound for HSIEs
33
Solution: Approximate solution is searched as
n
1
ϕ(x) = ϕn (x) = √
b(n)
j,2 φ j,2 (x).
1 − x 2 j=0
(3.29)
Since c0 = 1, Q(x) = 0 and L(x, t) = 16x 3 t 3 , from the Equation (2.27) it follows that
n
b(n)
j,2
j=1
1
·
1 − xk2
j +1
j −1
(n)
ψ0,2 (x) =
U j−2 (xk ) −
U j (xk ) + ψ j,2 (x) + b0,2
2
2
π
, (3.30)
2
where U−1 (x) = 0 and
1
ψ j,2 (x) =
π
1
−1
T j (t)
(16x 3 t 3 ) √
dt, j = 0, 1, ...
1 − t2
It is easy to check that
t3 =
1
(T3 (t) + 3T1 (x).
4
(3.31)
Applying (3.31) and using orthogonality condition (2.22), we obtain
ψ0,2 (x) = 0, ψ1,2 (x) = 6x 3 , ψ2,2 (x) = 0, ψ3,2 (x) = 2x 3 , ψ j,2 (x) = 0, j ≥ 4.
With these values of ψ j,2 (x), Equation (3.30) with n = 3 has the form
3
j=1
b(n)
j,2
1
·
1 − xk2
j +1
j −1
U j−2 (xk ) −
U j (xk )
2
2
+ 6b1,2 x 3 + 2b3,2 x 3 =
π
,
2
(3.32)
and taking into account Table 1, it follows that
6b1,2 x 3 + 2b2,2 + b3,2 (8x + 2x 3 ) =
π
.
2
(3.33)
By equating the same powers of x in Equation (3.33) and take into account (2.26), we arrive at
b0,2 =
π3
, b1,2 = 0, b2,2 =
22
π1
, b3,2 = 0.
22
(3.34)
Exact solution is achieved when we substitute Equation (3.34) into (3.29) i.e.
1
ϕn (x) = √
1 − x2
$
3
2
1
2
+
π
2
2
T2 (x)
π
%
I NDIAN J OURNAL OF I NDUSTRIAL AND A PPLIED M ATHEMATICS
1
π
=√
(1 + x 2 ).
2
1 − x2
Vol. 10, No. 1 (Special Issue), Jan–June 2019
34
Z.K. Eshkuvatov and Anvar Narzullaev
Example 6. Let HSIEs with corresponding condition be given by
1
π
1
π
1
−1
1
−1
(1 + 2(t − x))
1
ϕ(t)dt +
2
(t − x)
π
1
−1
1 2x 3
e t ϕ(t)dt = 4 + 8x,
2
(3.35)
ϕ(t)dt = 1.
The exact solution of Equation (3.35) is
1
(4x 2 − 1).
ϕ(x) = √
1 − x2
(3.36)
Solution: Comparing Equation (3.35) with (2.2) we obtain c0 = 1, Q(x) = 2 and L(x, t) =
Let approximate solution be searched as (3.29), then Equation (2.28) becomes
n
b(n)
j,2
j=1
1
·
1 − x2
e2x t 3
.
2
j +1
j −1
U j−2 (x) −
U j (x) + 2U j−1,1 (x) + ψ j,2 (x)
2
2
(3.37)
+b0,2 ψ0,2 (x) =
π
(4 + 8x),
2
where
ψ j,2 (x) =
1
π
−1
1
e2x t 3 T j (t)
dt, j = 0, 1, ...
√
2
1 − t2
Taking into account (3.31) and orthogonality conditions (2.22), we get
ψ0,2 (x) = 0, ψ1,2 (x) =
3 2x
1 2x
e , ψ2,2 (x) = 0, ψ3,2 (x) =
e , ψ j,2 (x) = 0, j ≥ 4.
16
16
With these values of ψ j,2 (x), Equation (3.37) takes the form
n
j=1
b(n)
j,2
1
·
1 − x2
j +1
j −1
U j−2 (x) −
U j (x) + 2U j−1 (x)
2
2
3e2x
e2x
+ b3,2
=
+b1,2
16
16
π
(4 + 8x),
2
(3.38)
For n = 3 and applying Table 1, we arrive at
(3)
(3)
(3)
(3)
2b1,2
+ b2,2
(2 + 4x) + b3,2
(8x + 2(4x 2 − 1)) + b1,2
I NDIAN J OURNAL OF I NDUSTRIAL AND A PPLIED M ATHEMATICS
2x
3e2x
(3) e
+ b3,2
=
16
16
π
(4 + 8x). (3.39)
2
Vol. 10, No. 1 (Special Issue), Jan–June 2019
Projection Method and Error Bound for HSIEs
35
Table 6. Numerical solution of Example 6
Exact of HSIEs (3.35)
Errors Rn , n = 3
Errors Rn , n = 12
Errors Rn , n = 30
212.0807707
1.13687 × 10−13
2.2991 × 10−11
4.99057 × 10−11
1.600728589
8.88178 × 10−16
1.49089 × 10−14
6.02045 × 10−15
-0.436
−0.266255779
1.1657 × 10−15
1.42511 × 10−13
6.66389 × 10−13
-0.015
−0.999212418
3.33067 × 10−16
1.17653 × 10−13
7.672987 × 10−13
−0.999212418
5.55112 × 10−16
1.14435 × 10−13
8.02681 × 10−13
−0.266255779
7.77156 × 10−16
1.145391 × 10−13
2.04732 × 10−13
0.725
1.600728589
4.44089 × 10−16
1.08045 × 10−13
1.64476 × 10−13
0.9999
212.0807707
1.13687 × 10−13
5.60651 × 10−12
2.77061 × 10−11
x
-0.999
-0.725
0.015
0.436
Equating like powers of x from both sides of Equation (3.39), we get
π
.
2
(3)
(3)
(3)
b1,2
= b3,2
= 0, b2,2
=2
(3.40)
(3)
To find b0,2
we impose second condition of (3.35) to yield
(3)
=
b0,2
π
.
2
(3.41)
Substitute Equations (3.40) - (3.41) into (3.29)
ϕ(x) = √
1
1−
x2
(T0 (x) + 2T2 (x)) = √
1
1 − x2
(4x 2 − 1).
(3.42)
which is identical with the exact solution (3.36).
Let us solve the problem 6 numerically. To do end this we solve Equation (3.38) at the collocation points (2.36) and substitute it into (3.29). Results are summarized in Table 6.
4. CONCLUSIONS
In this note, we have developed projection method for solving HSIEs of the first kind, where the
main kernel K (x, t) is const on the diagonal of domain D = [−1, 1] × [−1, 1]. Collocation and
Galerkin method are used to obtain a system of algebraic equations for the unknown coefficients.
Stable computational scheme and high accurate Gauss-Chebyshev QF leads to high accurate and
stable method. Developed method is suitable for both bounded and unbounded solutions of HSIEs.
Examples 1 − 4 verify that the developed method is very accurate and stable for almost any kernel
K (x, t) of HSIEs of the first kind. Especially, Example 3 shows that method converges very fast
even if exact solution is in the rational form. Examples 5-6, show that developed method is exact
whenever solution of HSIEs is the product of wight function and polynomials. Table 6 shows that
approximate solution is stable for large value of n.
I NDIAN J OURNAL OF I NDUSTRIAL AND A PPLIED M ATHEMATICS
Vol. 10, No. 1 (Special Issue), Jan–June 2019
36
Z.K. Eshkuvatov and Anvar Narzullaev
ACKNOWLEDGEMENT
This work was supported by Universiti Sains Islam Malaysia (USIM) under Research Management
Center (RMC), Project code: PPP/USG-0216/FST/30/15316. Authors are grateful for sponsor and
financial support of the RMC of USIM.
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Vol. 10, No. 1 (Special Issue), Jan–June 2019
INDIAN JOURNAL OF INDUSTRIAL AND APPLIED MATHEMATICS
Vol. 10, No. 1 (Special Issue), Jan–June 2019, pp. 38–50
DOI:
Discrete Legendre Projection Methods for the
Fredholm Integral Equations of the Second Kind
Jitendra Kumar Malik∗ and Bijaya Laxmi Panigrahi1
Department of Mathematics, Sambalpur University, Odisha-768019, India
(∗ Corresponding author) Email: ∗ jitu.malik100@gmail.com, 1 blpanigrahi@suniv.ac.in
Abstract: In this paper, we consider the discrete Galerkin and discrete collocation methods to solve
the Fredholm integral equations of the second kind using Legendre polynomial bases. Using sufficiently accurate numerical quadrature rule, we obtain the error bounds for discrete Legendre solutions and iterated discrete Legendre solution for L 2 norm in both Legendre Galerkin and Legendre
collocation method. We also obtain the superconvergence results for iterated Legendre Galerkin
solution over Legendre Galerkin solution in L 2 -norm. Numerical example is presented to illustrate
the theoretical results.
Keywords: Discrete projection methods, Fredholm integral equations, Legendre polynomial bases.
2010 Mathematical Subject Classifications: 45B05, 65R20, 47G10.
1. INTRODUCTION
Consider the following integral operator K defined on X = L 2 ([−1, 1]) or C([−1, 1]) by
1
Ku(s) =
k(s, t)u(t) dt, s ∈ [−1, 1].
(1.1)
−1
We are interested to solve the Fredholm integral equations of the second kind
(I − K)u = f.
(1.2)
It is required to obtain approximate solutions because the above problem can not be solved exactly.
There are so many different methods have been developed to find the approximate solutions of
equations (1.2), see for example [6, 9, 10]. Spectral methods have developed rapidly in the past
two decades and this method applied successfully in many fields. The Galerkin, collocation and
their discretized versions are the commonly used spectral projection methods for finding numerical
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Discrete Legendre Projection Methods for the Fredholm Integral Equations of the Second Kind
39
solutions of various integral equations, see for example [3,4,7,12] and references therein. Legendre
spectral approximation method for eigenvalue problem of a compact integral operator is developed
in [13]. In this paper, we use discrete Legendre spectral projection methods to solve the Fredholm
integral equations of the second kind and evaluate the error bounds for approximate solutions and
the iterated solutions with the exact solutions. We show that the iterated solution converges faster
than approximate solution in discrete Legendre Galerkin method.
We organize this paper as follows. In Section 2, we discuss the discrete Legendre Galerkin and
discrete Legendre collocation methods for Fredholm integral equations. In Section 3, we discuss
the convergence rates for the projection (Galerkin and collocation) methods for Fredholm integral
equations. In Section 4, we present numerical examples.
2. DISCRETE LEGENDRE PROJECTION METHODS
Let L 2 be the space of complex valued square integrable functions on [−1, 1] with the inner product
f, g =
1
−1
f (t) g(t)dt, f, g ∈ L 2 ,
1
and norm f L 2 = f, f 2 .
Let X = C[−1, 1] ⊂ L 2 [−1, 1] be the space of complex valued continuous functions on [−1, 1].
Consider the Fredholm integral equations of the second kind
u(s) −
1
−1
k(s, t)u(t)dt = f (s), s ∈ [−1, 1],
(2.1)
where the kernel k(s, t) ∈ C([−1, 1] × [−1, 1]) and f are known functions and u is the unknown
function in X to be determined. Let
1
Ku(s) =
k(s, t)u(t) dt, s ∈ [−1, 1].
(2.2)
−1
Then K is a compact linear integral operator on C[−1, 1] and L 2 [−1, 1]. Then the equation (2.1)
can be written in an operator form as
(I − K)u = f,
(2.3)
where K is a compact linear integral operator, f is a given function, u is an unknown to be determined and I be the identity operator defined on X.
To describe the discrete Legendre projection methods for the solution of Fredholm integral
equations of the second kind (2.3), we let Xn = span{φ0 , φ1 , · · · , φn } be the sequence of Legendre
polynomial subspaces of X of degree ≤ n, where {φ0 , φ1 , · · · , φn } forms an orthonormal basis for
Xn . The φi ’s are given by
2i + 1
L i (s), i = 0, 1, . . . , n,
φi (s) =
2
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Jitendra Kumar Malik and Bijaya Laxmi Panigrahi
where L i ’s are the Legendre polynomials of degree ≤ i. The Legendre polynomials can be generated
by the following recurrence relation
L 0 (s) = 1, L 1 (s) = s, s ∈ [−1, 1],
and for i = 1, 2, · · · , n − 1,
(i + 1)L i+1 (s) = (2i + 1)s L i (s) − i L i−1 (s), s ∈ [−1, 1].
Since φi and φ j ’s are polynomials, note that
φi , φ j =
1
−1
φi (t)φ j (t)dt =
1
−1
φi (t)φ j (t)dt = δi, j ,
(2.4)
for i, j = 0, 1, . . . , n.
Now we will discuss on discrete methods. We choose a numerical integration scheme
1
f (t)dt
−1
M(n)
w p f (t p ),
(2.5)
p=1
where M(n) is a constant depend upon n, and
(i) w p are the weights such that
w p > 0, p = 1, 2, · · · , M(n).
(2.6)
(ii) the above quadrature rule has degree of precision d, which is atleast 2n, that is
1
−1
f (t)dt =
M(n)
w p f (t p )
(2.7)
p=1
for all polynomial of degree ≤ 2n ≤ d.
For the notational convenience, from now on we set M(n) = M. Using the above quadrature
rule (2.5), we define the discrete inner product
f, g M =
M
w p f (t p )g(t p ), f, g ∈ C[−1, 1].
(2.8)
p=1
For the approximation of K, using the rule (2.5), the Nyström operator Kn is defined by
(Kn u)(s) =
M
w p k(s, t p )u(t p ).
(2.9)
p=1
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41
Now, we set the following notations. Let C r [−1, 1] denote the space of r -times continuously
differentiable complex valued function on [−1, 1]. For u ∈ C r [−1, 1], let
ur,∞ = max{u (i) ∞ : 1 ≤ i ≤ r },
where u (i) denote the i-th derivative of u. Assume k(., .) ∈ C d [−1, 1] × [−1, 1], where d is the
degree of precision of the numerical quadrature rule and d ≥ 2n > n ≥ r > 1. For fixed s ∈
[−1, 1], we denote ks (t) = k(s, t). Since w j > 0 and
2=
1
−1
ds =
M
wi ,
(2.10)
i=1
it follows that for j = 0, 1, . . . , d,
M
∂j
(Kn u)( j) ∞ = sup |(Kn u)( j) (s)| = sup w p j ks (t p )u(t p ) ≤ 2u∞ k j,∞ ,
∂s
s∈[−1,1]
s∈[−1,1] p=1
where k j,∞ =
∂ i+l
i l ks (t). Then
∂s
∂t
s,t∈[−1,1],0≤i,l≤ j
sup
Kn ud,∞ = max{(Kn u)( j) ∞ : 0 ≤ j ≤ d} ≤ ckd,∞ u∞ ,
j
(2.11)
where c is a constant independent of n.
Also, for j = 0, 1, . . . , d, we have
(Ku)( j) ∞ = sup |(Ku)( j) (s)| = sup s∈[−1,1]
s∈[−1,1]
1
−1
∂j
k
(t)u(t)dt
≤ 2u∞ k j,∞ .
s
∂s j
Thus,
Kud,∞ ≤ cu∞ kd,∞ ,
(2.12)
where c is a constant independent of n.
We quote the following theorem, which gives the error bound of Nyström operator (2.9) with
the operator K (2.2).
Theorem 2.1. ( [7]) Let k(., .) ∈ C d [−1, 1] × [−1, 1], then for any u ∈ C d [−1, 1], we have
(Kn − K)u∞ ≤ cn −d kd,∞ ud,∞ ,
(2.13)
where c is a constant independent of n.
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Jitendra Kumar Malik and Bijaya Laxmi Panigrahi
Lemma 2.2. Let Kn be the Nyström operator defined by (2.9). Assume that k(., .) ∈ C d ([−1, 1] ×
[−1, 1]) and d ≥ 2n > n ≥ r > 1, then
(K − Kn )K∞ = O(n −d ).
Proof. By replacing u with Ku in equation (2.13) and using the estimate (2.12), we obtain
(K − Kn )Ku∞ ≤ cn −d kd,∞ Kud,∞ ≤ cn −d k2d,∞ u∞ ,
(2.14)
where c is a constant independent of n. This completes the proof.
Lemma 2.3. ([2]) Let X be a Banach space and S ⊂ X is a relatively compact set. Assume that T
and Tn are bounded linear operators from X into X satisfying Tn ≤ c, for all n ∈ N, and for
each x ∈ S,
Tn − T → 0 as n → ∞,
where c is a constant independent of n. Then Tn − T → 0 uniformly for all x ∈ S.
Discrete Legendre orthogonal projection operator: To discuss on the discrete Legendre Galerkin
methods, we need to introduce the discrete orthogonal projection operator. Discrete orthogonal projection namely hyper interpolation operator QnG : X → Xn (Sloan [14]) is defined by
QnG u =
n
u, φ j M φ j , u ∈ X,
(2.15)
j=0
for j = 0, 1, . . . , n and QnG satisfy
QnG u, φ M = u, φ M , for all φ ∈ Xn .
Now we quote some properties of QnG from ( [7, 14]).
Lemma 2.4. Let QnG : X → Xn , be the hyperinterpolation operator defined as above. Then the
following results hold
(i) For any u ∈ X,
√
QnG u L 2 ≤ 2u∞ ,
(2.16)
and
√
QnG u − u L 2 ≤ 2 2 inf u − u n ∞ → 0, as n → ∞.
u n ∈Xn
(ii) In particular, for u ∈ C ([−1, 1]),
r
QnG u − u L 2 ≤ cn −r u (r ) ∞ ,
(2.17)
where c is a constant independent of n and n ≥ r .
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Discrete Legendre Projection Methods for the Fredholm Integral Equations of the Second Kind
43
Note that for any u ∈ C r [−1, 1], using Jackson’s theorem ( [?]) and the estimate (2.10), we get
1
1
2
2
u − QnG u, u − QnG u M
= min u − χ , u − χ M
χ∈Xn
=
=
min
M
χ∈Xn
M
12
i=1
wi
1
2
i=1
≤
wi (u − χ )2 (ti )
inf u − χ ∞
χ∈Xn
√
2cn −r ur,∞ ,
(2.18)
where c is a constant independent of n and n ≥ r .
Using orthogonal projection QnG , the discrete Legendre Galerkin method for the Fredholm
integral equations of the second kind is
u nG − QnG Kn u nG = QnG f,
(2.19)
where u nG ∈ Xn is the approximation of u. We define the iterated solution as ũ nG = Kn u nG + f .
Discrete Legendre interpolatory projection operator: Let {τ0 , τ1 , . . . , τn } be the zeros of the
Legendre polynomial of degree n + 1 and define the interpolatory projection QCn : X → Xn by
QCn u ∈ Xn , QCn u(τi ) = u(τi ), i = 0, 1, . . . , n, u ∈ X.
(2.20)
Lemma 2.5. ( [4, 8]) Let QCn : X → Xn be the interpolatory projection operator defined by (2.20).
Then the following conditions hold:
(i) QCn u L 2 ≤ cu∞ , u ∈ C[−1, 1], where c is a constant independent of n.
(ii) There exists a constant c > 0 such that for any u ∈ X,
u − QCn u L 2 ≤ c inf u − φ∞ → 0, as n → ∞.
φ∈Xn
(iii) For any u ∈ C r ([−1, 1]), there exists a constant c independent of n such that
u − QCn u L 2 ≤ cn −r u (r ) ∞ .
(2.21)
Using interpolatory projection QCn , the discrete Legendre collocation method for the Fredholm
integral equations of the second kind is
u Cn − QCn Kn u Cn = QCn f,
(2.22)
where u Cn ∈ Xn is the approximation of u. We define the iterated solution as ũ Cn = Kn u Cn + f .
Remark 2.6. If M = n + 1 and the quadrature points used in the discrete inner product (2.8) and
the collocation nodes in (2.20) are the same, the discrete orthogonal projection operator QnG reduces
to the interpolatory projection operator, i.e., in such case QnG and QCn are the same.
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Jitendra Kumar Malik and Bijaya Laxmi Panigrahi
3. CONVERGENCE RATES
In this section, we will discuss on the convergence rates of the approximate and iterated approximate solutions with the exact solution for the Fredholm integral equations of the second kind using
discrete Legendre Galerkin and discrete Legendre collocation methods.
At first, we will evaluate the convergence rates by using discrete Legendre Galerkin methods.
3.1. Discrete Legendre Galerkin Method:
The discrete Legendre Galerkin method for the Fredholm integral equations of the second kind is
u nG − QnG Kn u nG = QnG f.
First, we will prove that the operator QnG Kn is ν-convergent to K in L 2 -norm, then we will analyze
the existence and error bounds of the approximate and iterated approximate solutions by using
discrete Legendre Galerkin methods.
Theorem 3.1. QnG Kn is ν-convergent to K in L 2 -norm.
Proof. Consider
QnG Kn L 2 ≤ p1 Kn ∞ ,
where p1 is a constant independent of n.
From Theorem-2.1, we see that {Kn } converges to K pointwise. Hence, {Kn } is pointwise
bounded and since X is Banach space, by Uniform Boundedness principle we have {Kn } is uniformly bounded, i.e., Kn ∞ ≤ p2 , where p2 is a constant independent of n. This shows that
QnG Kn L 2 is uniformly bounded.
By using the estimate (2.17), we obtain
(QnG − I)Ku L 2 ≤ cn −r (Ku)(r ) ∞ ≤ cn −r kr,∞ u∞ ,
(3.1)
where c is a constant independent of n.
Next consider
|(QnG Kn − K)u(t)| = |(QnG Kn − QnG K + QnG K − K)u(t)|
≤ |QnG (Kn − K)u(t)| + |(QnG − I)Ku(t)|.
Now using the estimates (2.16), (3.1) and (2.13), we see
(QnG Kn − K)u L 2
≤ QnG (Kn − K)u L 2 + (QnG − I)Ku L 2
≤ c(Kn − K)u∞ + (QnG − I)Ku L 2
≤ cn −r kr,∞ ur,∞ + cn −r kr,∞ u∞ → 0,
(3.2)
as n → ∞ and c is a constant independent of n. This gives that QnG Kn is pointwise converges to K.
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45
Let B = {u ∈ X : u ≤ 1} be a closed unit ball in C([−1, 1]). Since K is compact operator,
the set S = {Ku : u ∈ B} is a relatively compact set in C([−1, 1]). Then by Lemma-2.3, we have
(QnG Kn − K)K L 2
= sup{(QnG Kn − K)Ku L 2 : u ∈ B}
= sup{(QnG Kn − K)u L 2 : u ∈ S} → 0 as n → ∞.
Since QnG is uniformly bounded in L 2 norm over Kn u, u ∈ B and Kn is compact, S = {QnG Kn u :
u ∈ B} is a relatively compact set. Thus
(QnG Kn − K)QnG Kn L 2
= sup{(QnG Kn − K)QnG Kn u L 2 : u ∈ B}
= sup{(QnG Kn − K)u L 2 : u ∈ S} → 0 as n → ∞.
Combining all these results leads to the first result that QnG Kn is ν-convergent to K in L 2 -norm.
This completes the proof.
Since QnG Kn is ν-convergent to K, for all small n, the spectrum of QnG Kn inside consists of
G
G
G
, λn,2
, · · · , λn,m
counted accordingly to their algebraic multiplicities inside
m eigenvalues, say λn,1
. Let
λnG =
G
G
G
λn,1
+ λn,2
+ · · · + λn,m
m
denote their arithmetic mean and we approximate λ by λnG . Let P S and PnS be the spectral projections
of K and QnG Kn , respectively, associated with their corresponding spectral inside .
Lemma 3.2. Assume (I − K)−1 exists on X. If the approximate operator QnG Kn is ν-convergent to
K in L 2 -norm, then there is a positive integer N such that for all n ≥ N , the inverse (I − QnG Kn )−1
exists as linear operator defined on X and there exist positive constants c1 independent of n such
that for all n ≥ N , (I − QnG Kn )−1 L 2 ≤ c1 .
Proof. The proof is very obvious from [1] with Theorem-3.1.
Theorem 3.3. Let k(., .) ∈ C d [−1, 1], d ≥ 2n > n ≥ r > 1 and u ∈ X be the exact solution of
equation (2.3). Let u nG be the discrete Legendre Galerkin approximation of u defined by (2.19).
Then the following holds
u − u nG L 2 = O(n −r ).
Proof. From the equation (2.3) and (2.19), we obtain
u nG − u
=
(I − QnG Kn )−1 Qn f − (I − K)−1 f
=
(I − QnG Kn )−1 [QnG − QnG K − I + QnG Kn ]u.
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(3.3)
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Then by taking the norm on both sides and using the last Theorem with the estimates (2.17), (2.16)
and (2.13) we obtain
u nG − u L 2
≤
≤
c1 (QnG − I)u L 2 + QnG (Kn − K)u L 2 )
√
cn −r u (r ) ∞ + c1 2(Kn − K)u∞
≤
cn −r u (r ) ∞ + cn −d kd,∞ ud,∞ ,
where c and c1 are the constants independent of n.
Thus,
u nG − u L 2 = O(n − min{r,d} ) = O(n −r ).
This completes the proof.
Theorem 3.4. Let k(., .) ∈ C d [−1, 1], d ≥ 2n > n ≥ r > 1 and ũ nG = Ku nG + f be the iterated discrete Legendre Galerkin approximation of u. Then the following holds
u − ũ nG L 2 = O(n −2r ).
Proof. We have ũ nG = Kn u nG + f . It follows that QnG ũ nG = u nG . Now by using the estimates (2.3)
and (2.13), we get
u − ũ nG L 2
=
=
Ku + f − ( f + Kn QnG ũ nG ) L 2
Ku − Kn QnG ũ nG L 2
≤
(K − Kn )u L 2 + Kn u − Kn QnG ũ nG L 2
≤
cn −d kd,∞ ud,∞ + Kn (u − u nG ) L 2 ,
(3.4)
where c is a constant independent of n.
By using the estimate (3.8) and the Lemma-3.2, we obtain
Kn (u − u nG ) L 2
=
Kn (I − QnG Kn )−1 (QnG − QnG K − I + QnG Kn )u L 2
≤
(I − QnG Kn )−1 L 2 Kn (QnG − QnG K − I + QnG Kn )u L 2
≤
c1 Kn (QnG − I)u L 2 + c1 Kn (QnG Kn − QnG K)u L 2 ,
(3.5)
where c1 is a constant independent of n.
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Now the first term in the right hand side of the above equation, we use the orthogonality of QnG
|Kn (QnG − I)u(s)| =
M
w p k(s, t p )(QnG − I)u(t p )
p=1
= (QnG − I)u, ks M = (QnG − I)u, (QnG − I)ks M M
= w p (QnG − I)u(t p )(QnG − I)ks (t p )
p=1
Set ks = ls . Now by using Cauchy Schwarz inequality, we obtain
|Kn (QnG − I)u(s)| ≤
M
w p |(QnG − I)ls (t p )|2
M
1/2 p=1
≤
≤
w p |(QnG − I)u(t p )|2
1/2
p=1
1/2
(QnG − I)ls , (QnG − I)ls M (QnG
cn −2r ls(r ) ∞ u (r ) ∞ ,
1/2
− I)u, (QnG − I)u M
where c is a constant independent of n.
Thus,
Kn (QnG − I)u L 2 ≤ cn −2r kr,∞ u (r ) ∞ .
(3.6)
Now the second term of the right hand side of (3.5),
|Kn (QnG Kn − QnG K)u(s)| =
M
w p k(s, t p )QnG (Kn − K)u(t p )
p=1
≤
QnG (Kn − K)u L 2
M
w 2p |k(s, t p )|2
1/2
p=1
≤
c(Kn − K)u∞ ks ∞ ≤ cn −d ks ∞ kd,∞ ud,∞ ,
where c is a constant independent of n.
Thus,
Kn (QnG Kn − QnG K)u L 2 ≤ cn −d kr,∞ kd,∞ ud,∞ .
(3.7)
Now by using equation (3.6), (3.7) and (3.5) in (3.4), we obtain
u − ũ nG L 2 = O(n −min{d,2r } ) = O(n −2r ).
This completes the proof.
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3.2. Discrete Legendre collocation methods:
The discrete Legendre collocation methods for the Fredholm integral equations of the second kind
is
u Cn − QCn Kn u Cn = QCn f.
First, we will prove that the operator QCn Kn is ν-convergent to K in L 2 -norm, then we will analyze
the existence and error bounds of the approximate and iterated approximate solutions by using
discrete Legendre collocation methods.
Theorem 3.5. QCn Kn is ν-convergent to K in L 2 -norm.
Proof. The proof is similar to the proof of Theorem-3.1. So, we omit it.
Since QCn Kn is ν-convergent to K, for all small n, in L 2 -norm, the spectrum of QCn Kn inside consists of m eigenvalues say λCn,1 , λCn,2 , . . . , λCn,m counted accordingly to their algebraic multiplicities inside ( Chatelin [5], Osborn [11]). Let
λ̂Cn =
λCn,1 + λCn,2 + · · · + λCn,m
m
,
denote their arithmetic mean and we approximate λ by λ̂Cn . Let PnC be the spectral projections of
QCn Kn , associated with their corresponding spectral inside .
Lemma 3.6. Assume (I − K)−1 exists on X. If the approximate operator QCn Kn is ν-convergent to
K, then there is a positive integer N such that for all n ≥ N , the inverse (I − QCn Kn )−1 exists as
linear operator defined on X and there exist positive constants c2 independent of n such that for all
n ≥ N , (I − QCn Kn )−1 L 2 ≤ c2 .
Proof. The proof is very obvious from [1] with Theorem-3.5.
Theorem 3.7. Let k(., .) ∈ C d [−1, 1] and u be exact solution of equation (2.3). Let u Cn be the discrete Legendre collocation approximation of u defined by (2.19). Then the following holds
u − u Cn L 2 = O(n −r ).
Proof. From the equation (2.3) and (2.19), we obtain
u Cn − u
= (I − QCn Kn )−1 Qn f − (I − K)−1 f
= (I − QCn Kn )−1 [QCn − QCn K − I + QCn Kn ]u.
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(3.8)
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49
Table 1.
n
u − unG L 2
u − ũ nG L 2
u − unC L 2
2
3
4
3.201944e-03
1.230043e-05
4.512378e-06
4.716913e-05
2.145267e-07
1.237801e-09
3.918328e-03
8.281007e-05
2.178939e-07
5
6
9.120781e-07
1.456890e-08
2.267891e-10
8.776551e-12
3.456712e-08
2.345689e-09
Then by taking the norm on both sides and using the last Theorem with the estimates (2.17), (2.16)
and (2.13) we obtain
u Cn − u L 2
≤ c1 (QCn − I)u L 2 + QCn (Kn − K)u L 2 )
√
≤ cn −r u (r ) ∞ + c1 2(Kn − K)u∞
≤ cn −r u (r ) ∞ + cn −d kd,∞ ud,∞
where c1 and c are the constants independent of n.
Thus,
u Cn − u L 2 = O(n − min{r,d} ) = O(n −r ).
This completes the proof.
Remark 3.8. By using a sufficiently accurate numerical quadrature rule, we obtained that the discrete Legendre Galerkin and iterated discrete Legendre Galerkin solution converges with the orders
O(n −r ) and O(n −2r ), respectively in L 2 norm, where r being the smoothness of the solution and
d ≥ 2n be the degree of the precision of the numerical quadrature. Also, we obtained that the discrete Legendre collocation solution converges with the order O(n −r ) in L 2 norm.
4. NUMERICAL RESULT
In this section, we present a numerical example. Choose the approximating subspace Xn to be the
Legendre polynomial subspaces of degree less than equal to n.
In Table 1, we present the errors for discrete Legendre Galerkin, iterated discrete Legendre
Galerkin and discrete Legendre collocation solutions with exact solution for Fredholm integral
equations of the second kind (2.3) in L 2 -norm. We denote u nG , ũ nG and u Cn be the discrete Legendre
Galerkin, iterated discrete Legendre Galerkin and discrete Legendre collocation solution, respectively.
For different values of n, we compute u nG , ũ nG , u Cn . The computed errors in L 2 -norm are presented in the following Table.
Example 4.1. We consider the following integral equation
1
k(s, t)u(t)dt = f (s), s ∈ [−1, 1],
u(s) −
−1
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Jitendra Kumar Malik and Bijaya Laxmi Panigrahi
with the kernel k(s, t) = st and the function f (s) = s, where the exact solution is given by u(s) =
3s.
ACKNOWLEDGEMENT
The second author is supported by a research grant from Science and Engineering Research Board,
DST, India-SR/FTP/MS-010/2012.
REFERENCES
[1] M. Ahues, A. Largillier and B. V. Limaye, Spectral computations for bounded operators, Chapman and Hall/CRC,
New York, 2001.
[2] K. E. Atkinson, The Numerical solution of Integral Equations of the Second Kind. Cambridge University
Press,,Cambridge, 1997.
[3] Guo Ben-yu, Spectral methods and their applications, World Scientific, 1998.
[4] C. Canuto and M. Y. Hussaini and A. Quarteroni and T.A. Zang, Spectral methods, Springer-Verlag Berlin Heidelberg,
2006.
[5] F. Chatelin, Spectral approximation of linear operators, Academic Press, New York, 1983.
[6] Z. Chen, G. Long and G. Nelakanti, The discrete multi-projection method for Fredholm integral equations of the
second kind, J. Integral Equations and Appl., 19 (2007), 143–162.
[7] P. Das, G. Nelakanti and G. Long. Discrete Legendre spectral projection methods for Fredholm-Hammerstein integral
equations. J. Comp. Appl. Math., 278 (2015) 293–305.
[8] M. A. Golberg, C.S. Chen, Discrete projection methods for integral equations, Computational Mechanics Publications,
Southampton (1997).
[9] R.P. Kulkarni, G. Nelakanti, Spectral approximation using iterated discrete Galerkin method, Numer. Funct. Anal.
Optim., 23 (2002) 91–104.
[10] G. Long, G. Nelakanti, B.L. Panigrahi and M.M. Sahani, Discrete multi-projection methods for eigen-problems of
compact integral operators, Appl. Math. Comput., 217 (2010), 3974–3984.
[11] J. E. Osborn, Spectral approximation for compact operators, Math. Comp., 29 (1975), 712–725.
[12] B. L. Panigrahi and G. Nelakanti, Legendre multi-projection methods for solving eigenvalue problems for a compact
integral operator, J. Comput. Appl. Math., 239 (2013), 135–151.
[13] B. L. Panigrahi and G. Nelakanti, Superconvergence of Legendre projection methods for the eigenvalue problem of a
compact integral operator, J. Comput. Appl. Math., 235 (2011), 2380–2391.
[14] I. H. Sloan, Polynomial interpolation and hyperinterpolation over general regions, Journal of Approximation Theory,
83 (2) (1995), 238–254.
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Vol. 10, No. 1 (Special Issue), Jan–June 2019
INDIAN JOURNAL OF INDUSTRIAL AND APPLIED MATHEMATICS
Vol. 10, No. 1 (Special Issue), Jan–June 2019, pp. 51–58
DOI:
Some Curvature Conditions on Nearly
Cosymplectic Manifolds
Gülhan Ayar1∗ , Pelin Tekin2 and Nesip Aktan3
1
Karamanoğlu Mehmetbey University,Kamil Ö zdağ Science Faculty, Department of Mathematics
Trakya University, Faculty of Science, Department of Mathematics, Edirne/TURKEY
3
Necmettin Erbakan University, Faculty of Sciences, Department of Mathematics-Computer Sciences,
Konya/TURKEY
(∗ Corresponding author) Email: ∗ gulhanayar@gmail.com, 2 pelintekin@trakya.edu.tr,
3
nesipaktan@gmail.com
2
Abstract: The aim of this study is to show that η−Einstein nearly cosymplectic manifolds are constant curvature manifolds. Also in this work, we present a complete connected nearly cosymplectic
manifold is M−projectively flat if and only if it is either isometric to the sphere S n or the real
projective space.
2000 Mathematics Subject Classiffication. 53C15, 53D15, 53C25.
Keywords: Contact Manifolds, Cosymplectic Manifolds, Nearly Cosymplectic manifolds
1. INTRODUCTION
Cosymplectic manifold is an odd dimensional counterpart of a Kähler manifold which is defined by
Lipperman 1959 [12] and Blair 1967 [5]. An almost contact metric structure (ϕ, ξ, η, g) is said to
be cosymplectic if it is normal and both and η are closed [2].
In 1971, Blair [1] defined an almost contact manifolds with Killing structure tensors as a nearly
cosymplectic manifold. The other study of Blair with Showers in 1972, was on the topological
aspect of this manifolds.
Nearly Kähler manifolds were defined by Gray [9] as almost Hermitian manifolds (M, J, g)
such that the covariant derivative of the almost complex structure with respect to the Levi-Civita
connection is skew-symmetric, that is
(∇ X J )X = 0,
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Gülhan Ayar, Pelin Tekin and Nesip Aktan
for every vector field X on M.
Notice that in the defining condition of a nearly Kähler manifold, only the symmetric part of
∇ J vanishes, in contrast to the Kähler case where ∇ J = 0. Nearly cosymplectic manifolds were
defined in the same spirit starting from cosymplectic or sometimes called coKähler manifolds. A
smooth manifold M endowed with an almost contact metric structure (ϕ, ξ, η, g) is said to be nearly
cosymplectic if
(∇ X ϕ)X = 0,
(1.1)
for every vector field X on M [7].
Recently, nearly Sasakian and nearly cosymplectic manifolds was studied by CappellettiMontano, B., Dileo, G. in [6]. In that paper, it is proved for five-dimensional nearly cosymplectic manifolds that any such manifold is Einstein with positive scalar curvature. It is also worth
remarking that (1-parameter families of) examples of both nearly Sasakian and nearly cosymplectic
structures are provided by every 5-dimensional manifold endowed with a Sasaki-Einstein SU (2)structure.
In the light of literature studies, in this paper, after given some basic concepts we show that an
η−Einstein nearly cosymplectic manifolds are constant curvature manifolds and a nearly cosymplectic manifold M n is M−projectively flat if and only if it is either locally isometric to the elliptic
space or has constant scalar curvature. Also we show that a complete connected nearly cosymplectic manifold is M−projectively flat if and only if it is either isometric to the sphere S n or the real
projective space.
2. PRELIMINARIES
Let (M, ϕ, ξ, η, g) be an n = (2m + 1)- dimensional almost contact Riemannian manifold, where
ϕ is a (1, 1)-tensor field, ξ is the structure vector field, η is a 1-form and g is the Riemannian metric.
It is well known that the (ϕ, ξ, η, g)-structure satisfies the conditions [4]
ϕξ = 0, η(ϕ X ) = 0,
ϕ 2 X = −X + η(X )ξ,
η(ξ ) = 1,
(2.1)
η(X ) = g(X, ξ ),
(2.2)
g(ϕ X, ϕY ) = g(X, Y ) − η(X )η(Y ),
(2.3)
for any vector fields X and Y on M.
From the above definition, ϕ is skew-symmetric with respect to g, so that the bilinear form :=
g(., ϕ.) defines a 2-form on M, called fundamental 2-form. An almost contact metric manifold such
that dη = 2 is called a contact metric manifold. In this case η is a contact form, i.e., η ∧ (dη)n = 0
everywhere on M [7].
A nearly cosymplectic manifold is an almost contact metric manifold (M, ϕ, ξ, η, g) such that
(∇ X ϕ)Y + (∇Y ϕ)X = 0,
(2.4)
for all vector fields X, Y on M [7]. Clearly, this condition is equivalent to (∇ X ϕ)X = 0. It is known
that in a nearly cosymplectic manifold the Reeb vector field ξ is Killing and satisfies ∇ξ ξ = 0 and
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Some Curvature Conditions on Nearly Cosymplectic Manifolds
53
∇ξ η = 0. The tensor field h of type (1, 1) defined by
∇X ξ = h X
(2.5)
is skew symmetric and anti-commutes with ϕ. It satisfies hξ = 0, η ◦ h = 0 and
∇ξ ϕ = ϕh =
1
Lξ ϕ.
3
Also the following formulas hold [7, 8]:
g((∇ X ϕ)Y, h Z ) = η(Y )g(h 2 X, ϕ Z ) − η(X )g(h 2 Y, ϕ Z ),
(2.6)
(∇ X h)Y = g(h 2 X, Y )ξ − η(Y )h 2 X,
(2.7)
tr (h 2 ) = constant,
(2.8)
R(Y, Z )ξ = η(Y )h 2 Z − η(Z )h 2 Y,
(2.9)
S(ξ, Z ) = −η(Z )tr (h 2 ),
(2.10)
S(ϕY, Z ) = S(Y, ϕ Z ), ϕ Q = Qϕ,
(2.11)
S(ϕY, ϕ Z ) = S(Y, Z ) + η(Y )η(Z )tr (h 2 ),
(2.12)
g((∇ X ϕ)ϕY, Z ) = g(∇ X ϕ)Y, ϕ Z ) + η(Z )g(h X, Y ) + η(Y )g(h X, Z ),
(2.13)
g(∇ϕ X ϕ)Y, Z ) = g(∇ X ϕ)Y, ϕ Z ) + η(Z )g(h X, Y ) + η(X )g(h Z , Y ),
(2.14)
g(∇ϕ X ϕ)ϕY, Z ) = −g(∇ X ϕ)Y, Z ) + η(X )g(h Z , ϕY ) + η(Y )g(h X, ϕ Z ).
(2.15)
3. η−EINSTEIN NEARLY COSYMPLECTIC MANIFOLDS
Definition 1. Let M be a nearly cosymplectic manifold, for every X, Y ∈ χ (M). If M satisfies the
condition
S(X, Y ) = ag(X, Y ) + bη(X )η(Y )
(3.1)
then M is an η−Einstein manifold where a, b : M −→ R is a function.
Proposition 1. Let M be an n dimensional nearly cosymplectic manifold. If M is an η−Einstein
manifold then a + b = −tr (h 2 ).
Proof. From (3.1) and (2.4), S(ξ, ξ ) = a + b = −tr (h 2 ) = constant can be obtained easily.
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Gülhan Ayar, Pelin Tekin and Nesip Aktan
Proposition 2. Let M be a nearly cosymplectic manifold of dimension n. If M is η−Einstein manifold, then a and b are constant.
Proof. M is a n dimensional η−Einstein nearly cosymplectic manifold and let {ei }, i = 1, 2, . . . , n
be orthonormal basis on every point of the tangent space. If we write X = Y = ei in (3.1) for
1 ≤ i ≤ n, we get
n
S(ei , ei ) = a
i=1
n
g(ei , ei ) + b
i=1
n
η(ei )η(ei )
i=1
and scalar curvature r as follows,
r
a+b
= (2m + 1)a + b
= −tr (h 2 )
r
= 2ma − tr (h 2 ).
(3.2)
If we take the derivative of r,
∇Y r
=
2
=
2
=
2
=
2
g((∇ei Q)Y, ei )
g(∇ei QY − Q∇ei Y, ei )
g(∇ei (aY + bη(Y )ξ ) − a∇ei Y − bη(∇ei Y )ξ, ei )
g(ei (a)Y + a∇ei Y + ei (b)η(Y )ξ + bη(∇ei Y )ξ + bg(Y, ∇ei ξ )ξ
=
+bη(Y )∇ei ξ − a∇ei Y − bη(∇ei Y )ξ, ei )
2 g(ei (a)Y + ei (b)η(Y )ξ + bg(Y, hei )ξ + bη(Y )hei , ei )
2 ei (a)g(Y, ei ) + ei (b)η(Y )η(ei ) + bg(hei , Y )η(ei ) + bη(Y )g(hei , ei )
2 g(∇a, ei )g(Y, ei ) + ξ (b)η(Y ) − bg(hY, ξ ) + bη(Y )tr (h 2 )
=
2Y (a) + 2ξ (b)η(Y ).
=
=
Finally writing this equation as 2mY (a) = 2Y (a) + 2ξ (b)η(Y ), let us take Y = ξ then,
mξ (a) =
(m − 1)ξ (a) =
=⇒
ξ (a) + ξ (b)
ξ (b)
ξ (a) = 0, ξ (b) = 0.
If we consider a + b = −tr (h 2 ),
a=
r + tr (h 2 )
,
n−1
b=
−tr (h 2 )n + r
n−1
(3.3)
so we have a and b are constant.
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Corollary 1. Let M be a n dimensional η−Einstein nearly cosymplectic manifold. The condition
η−Einstein for nearly cosymplectic manifolds are characterized by the equality
r + tr (h 2 )
−n tr (h 2 ) + r
S(X, Y ) =
g(X, Y ) +
η(X )η(Y ).
n−1
n−1
(3.4)
Proof. The proof is clear from (3.1) and (3.3)
Theorem 1. Let M be a n dimensional η−Einstein nearly cosymplectic manifold. The Ricci tensor
of M is η−parallel if and only if M is a constant curvature manifold.
Proof. By using (3.4) and (2.3), we obtain
(∇U S)(Y, Z ) =
dr (U )
−n tr (h 2 ) + r
[g(ϕY, ϕ Z )] +
[g(Y, hU )η(Z ) + η(Y )g(z, hU )]
n−1
n−1
and then
(∇U S)(ϕY, ϕ Z ) =
dr (U )
[g(ϕY, ϕ Z )] .
n−1
Hence we have desired result.
4. THE M-PROJECTIVE CURVATURE TENSOR
In 1971, Pokhariyal and Mishra [13] defined a tensor field W * on a m-dimensional Riemannian
manifold as
W * (X, Y )Z = R(X, Y )Z −
1
[S(Y, Z )X − S(X, Z )Y + g(Y, Z )Q X − g(X, Z )QY ]
2(m − 1)
(4.1)
so that
W * (X, Y, Z , U )de f = g(W * (X, Y )Z , U ) = W * (Z , U, X, Y )
(4.2)
Wi*jkl wi j w kl = Wi jkl wi j w kl
(4.3)
and
where Wi*jkl and Wi jkl are components of W * and W respectively and wkl is a skew-symmetric
tensor [14, 18]. Such a tensor field W * is known as M−projective curvature tensor. The properties
of M−projective curvature tensor in Sasakian and Kähler manifolds are defined and studied in
[14,15]. He has also shown that it bridges the gap between conformal curvature tensor, conharmonic
curvature tensor and concircular curvature tensor on one side and H −projective curvature tensor on
the other.
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Gülhan Ayar, Pelin Tekin and Nesip Aktan
The Weyl projective curvature tensor W , concircular curvature tensor C and conformal curvature tensor V are given by [16]
1
{S(Y, Z )X − S(X, Z )Y },
m−1
(4.4)
r
{g(Y, Z )X − g(X, Z )Y }
m(m − 1)
(4.5)
W (X, Y )Z = R(X, Y )Z −
on the M-projective curvature tensor
C(X, Y )Z = R(X, Y )Z −
and
1
V (X, Y )Z = R(X, Y )Z − (m−2)
[S(Y, Z )X − S(X, Z )Y + g(Y, Z )Q X − g(X, Z )QY ]
r
+ (m−1)(m−2) {g(Y, Z )X − g(X, Z )Y }.
(4.6)
Following theorems and corollary are given in [20].
Theorem 2. The M−projective and Weyl projective curvature tensors of the Riemannian manifold
M are linearly dependent if and only if M is an Einstein manifold.
Theorem 3. The necessary and sufficient condition for a Riemannian manifold to be an Einstein
manifold is that the M−projective curvature tensor W * and concircular curvature tensor C are linearly dependent.
Theorem 4. A Riemannian manifold becomes an Einstein manifold if and only if conformal and
M−projective curvature tensors of the manifold are linearly dependent.
Corollary 2. In a Riemannian manifold M, the followings are equivalent
i)
M is an Einstein manifold
ii) M-projective and Weyl projective curvature tensors are linearly dependent
iii) M-projective and concircular curvature tensors are linearly dependent
iv) M-projective curvature and conformal curvature tensors are linearly dependent.
5. M−PROJECTIVELY FLAT NEARLY COSYMPLECTIC MANIFOLDS
In view of W * = 0, (4.1) becomes
R(X, Y )Z =
1
[S(Y, Z )X − S(X, Z )Y + g(Y, Z )Q X − g(X, Z )QY ].
2 (n − 1)
(5.1)
Replacing Z by ξ in (5.1) and then using some curvature relations of nearly cosymplectic manifolds,
we obtain
1
2(n−1)
η(X )h 2 Y − η(Y )h 2 X ) =
2
−η(Y )X tr (h ) − η(X )Y + tr (h 2 ) + η(Y )Q X − η(X )QY .
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(5.2)
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57
Again putting Y = ξ in above relation and using some curvature relations of nearly cosymplectic
manifolds, we have
Q X = tr (h 2 )X − 2 (n − 1) (h 2 )X ⇔ S(X, Y ) = tr (h 2 )g(X, Y ) − 2 (n − 1) g(h 2 X, Y )
(5.3)
and
r = (n − 2)tr (h 2 ).
(5.4)
We know that the eigenvalues of the symmetric operator h 2 are constant and since h is skewsymmetric, the nonvanishing eigenvalues of h 2 are negative [7]. So, if we write h 2 (X ) = −λ2 X ,
then we obtain tr (h 2 ) = −nλ2 .
Putting this equation in the above relations; we obtain
Q X = (n − 2) (λ2 )X
(5.5)
r = n(n − 2)λ
(5.6)
and
In consequence of (5.5), (5.1) becomes
R(X, Y )Z =
(n − 2) (λ2 )
[g(Y, Z )X − g(X, Z )Y ]
(n − 1)
(5.7)
2
)
We can easily see that (n−2)(λ
> 0.
(n−1)
A space form is said to be elliptic if and only if the sectional curvature tensor is positive [24].
Thus, we can state:
Theorem 5. A n−dimensional nearly cosymplectic manifold M is M−projectively flat if and only
if it is either locally isometric to the elliptic space E n (C) or M has constant scalar curvature (n −
2
2)tr (h 2 ) where C = (n−2)λ
.
(n−1)
Corollary 3. A n−dimensional complete connected nearly cosymplectic manifold
M is
(n−1)λ2
n
M−projectively flat if and only if it is either isometric to the sphere S of radius
or the
n−1
real projective space S n / {±I } .
ACKNOWLEDGEMENT
This work is supported by Necmettin Erbakan University Scientific Research Projects Coordination
Unit.
REFERENCES
[1] Blair, D. E., Almost Contact Manifolds with Killing Structure, Tensors I. Pac. J. Math., 39(1971), 285–292.
[2] Blair, D. E., The theory of quasi-Sasakian structures, J. Diff. Geom., 1(1967), 331–345.
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Gülhan Ayar, Pelin Tekin and Nesip Aktan
[3] Blair, D. E., Showers, D.K., Almost Contact Manifolds with Killing Structures Tensors II. J. Differ. Geom., 9(1974),
577–582.
[4] Blair, D. E., Contact manifolds in Riemannian geometry, Lecture Notes in Mathematics, 509(1976), Springer-Verlag,
Berlin.
[5] Blair, D. E., Goldberg S. I.: Topology of almost contact manifolds, J. Differential Geometry, 1 (1967), 347–354.
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(2017). https://doi.org/10.1007/s10231-017-0671-2
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Coll. G´eom. Diff. Globale, pp. 37–59.
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INDIAN JOURNAL OF INDUSTRIAL AND APPLIED MATHEMATICS
Vol. 10, No. 1 (Special Issue), Jan–June 2019, pp. 59–70
DOI:
Second-order Characterization of Invex Functions
and Its Applications in Optimization Problems
M.T. Nadi1 and J. Zafarani2∗
1
University of Isfahan, Isfahan, 81745-163, Iran
University of Isfahan and Sheikhbahaee University, Isfahan, Iran
∗
Corresponding author Email: jzaf@zafarani.ir
2
Abstract: We give a second-order characterization for invex functions by using the Clarke subdifferentials, analogous to the positive semidefinite property for convex functions. Furthermore, some
applications of these characterizations in optimization problems are obtained.
Keywords: Invexity; Second-order characterization; Second-order optimality conditions; Clarke
subdifferential.
2010 Mathematics Subject Classification. 47J05; 90C33.
1. INTRODUCTION AND PRELIMINARIES
Convexity, monotonicity and their generalizations are very useful and applicable in optimization,
economy, engineering and many other sciences. The notion of invexity is a very important generalization of convexity which was introduced by Hanson as a condition that guarantees Kuhn-Tuker
conditions to be sufficient for optimality in non-linear programming [12]. This concept, named afterwards by Carven [8], which is an abbreviation of invariant convex. Some extensions of invexity and
their applications presented also by others. For example, Kaul and Kaur [21] introduced quasiinvex
and pseudoinvex functions. The concept of preinvex functions and its applications have studied by
Jeyakumar and Weir [16]. See, also [24, 30] and the references therein for other generalizations of
invexity and their properties. Also, the notion of invexity have been utilized frequently as a useful
tool in optimization; see, e.g., [27–29].
Invariant monotonicity, some of its extension and applications have been studied by many
authors; see, e.g., [14,15,19,20,22]. The relationships between various kinds of generalized invexity
and generalized monotonicity of corresponding subdifferentials are very impressive; see, e.g., [13].
It seems that the second-order characterization (characterization of a kind of generalized convex function by its corresponding second-order subdifferential) is more useful. It is well known in
the classic mathematical analysis that the second-order differential of a function f : Rn −→ R at its
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M.T. Nadi and J. Zafarani
optimal point is positive semi-definite and conversely, positive definiteness of second-order differential (Hessian matrix) at a critical point (where ∇ f vanishes), ensures the optimality of the reference
point. So, the second order differential of a function can be used as a strong tool in optimization
and characterization of a convex function. The concept of Hessian, for non-differentiable case, generalized by Mordukhovich [25], by using the coderivative of set-valued mappings. Afterwards, this
generalization used for characterization of maximal monotone mappings by Poliquin and Rockafellar [31]. Also, the second order characterization of a convex function and some of its generalizations,
have been studied by many authors. Crouziex and Ferland [9], characterized the quasiconvex and
pseudoconvex functions, by their second order differential. Luc and Schaible, investigated this characterization for class of C 1,1 functions [23]. For, more study in this field, see [3–5]. As a useful
study in second order characterization, we can mention [4], which gives a characterization for a
continuously differentiable convex function on the Hilbert space, by its second order Fréchet and
Limiting subdifferentials. Also recently, we obtained some characterizations of the convexity in
terms of second order subdifferentials [26].
The second order generalized directional derivative, also have been studied by many authors,
for obtaining some necessary and sufficient conditions of optimality; see, e.g., [2, 7, 10, 11, 17, 18].
Here, our work is based on generalized second order Clarke subdifferential, and generalized
Hessian which introduced by Luc and Schaible [23]. Indeed, we obtain an invariant case which is a
generalization of Hessian.
Throughout this paper, we suppose X is an arbitrary Hilbert space.
Definition 1.1. Let f : X → R be a locally Lipschitz function at x ∈ X and v any vector in X . The
Clarke generalized directional derivative of f at x in direction v, denoted by f ◦ (x, v), is defined by
f ◦ (x, v) = lim sup
y→x,t↓0
f (y + tv) − f (y)
.
t
Definition 1.2. Let f : X → R be a locally Lipschitz function at x ∈ X and v any vector in X . The
Clarke generalized subdifferential of f at x, denoted by ∂ f (x), and defined by
∂ f (x) = {ξ ∈ X : f ◦ (x, v) ≥ ξ, v , for any v ∈ X }.
For more details and properties of the Clarke subdifferential, we refer the reader to [6].
Let K be a nonempty subset of X , η : X × X → X be a vector valued function and F : X ⇒ X
be a set-valued mapping.
Definition 1.3. A set K is said to be invex with respect to η : X × X → X , when for any x, y ∈ K
and 0 ≤ λ ≤ 1,
y + λη(x, y) ∈ K .
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Definition 1.4. A vector valued function η : X × X → X is said to be skew, if
η(x, y) + η(y, x) = 0, for any x, y ∈ X.
The following assumptions are frequently used in the literature:
ASSUMPTION A: Let K be an invex set with respect to η, and f : K → R. Then
f (y + η(x, y)) ≤ f (x) for any x, y ∈ K .
ASSUMPTION C: Let η : X × X → X . Then, for any x, y ∈ X and for any λ ∈ [0, 1],
η(y, y + λη(x, y)) = −λη(x, y),
η(x, y + λη(x, y)) = (1 − λ)η(x, y).
Remark 1.1. [33] It is easy to see that Assumption C implies
η(y + λη(x, y), y) = λη(x, y).
Definition 1.5. Let K be an invex subset of X with respect to η : X × X → X . The function f :
K → R is said to be preinvex with respect to η, if for any x, y ∈ K and λ ∈ [0, 1],
f (y + λη(x, y)) ≤ λ f (x) + (1 − λ) f (y).
Definition 1.6. A differentiable function f : X → R is said to be invex with respect to η, if for any
x, y ∈ K , one has
∇ f (y), η(x, y) ≤ f (x) − f (y).
(1.1)
Definition 1.7. A locally Lipschitz function f : K ⊆ X → R is said to be invex with respect to η,
if for any x, y ∈ K and any ξ ∈ ∂ f (y), one has
ξ, η(x, y) ≤ f (x) − f (y).
(1.2)
Remark 1.2. A function f is said to be invex in a neighborhood of x, if either (5) or (6) holds for
all y such that y − x is sufficiently small.
Remark 1.3. Note that, in the above definitions by letting η(x, y) = x − y, we reduce to the convex
case. Indeed, both preinvex and invex functions reduce to convex function, and invex set, to convex
set.
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Definition 1.8. [13] A set valued mapping F : X ⇒ X ∗ is said to be invariant monotone on K with
respect to η, if for any x, y ∈ K and any u ∈ F(x), v ∈ F(y), one has
v, η(x, y) + u, η(y, x) ≤ 0.
In Section 2, we give some second order necessary and sufficient conditions for nondifferentiable invex functions, and in Section 3, we present some applications of our results in
optimization.
2. MAIN RESULTS
We begin with a necessary condition for a twice differentiable function which guarantees the invexity and afterwards we obtain a similar result for C 1,1 functions (i.e. functions with locally Lipschitz
differential). In the following, we denote the differential of η(., y) (differential of η, relative to the
first argument) by ηx (., y).
Proposition 2.1. Let f : Rn → R be an invex function with respect to η : Rn × Rn → Rn , which
is skew, f be twice differentiable at x ∈ Rn and η(., x) be differentiable at x. Then ηx (x, x)u, D 2
f (x)u ≥ 0, for any u ∈ Rn .
Proof. The differential mapping F = f is η − monotone, since f is η − invex. Suppose on the
contrary that, ηx (x, x)u, D f (x)u < 0 for some u ∈ Rn . By definitions of D F(x)u and ηx (x, x)u,
we have
η(x + tu, x) − η(x, x)
F(x + tu) − F(x)
and D F(x)u = lim
.
t→0
t→0
t
t
ηx (x, x)u = lim
Thus, for sufficiently small t > 0, we conclude that
η(x + tu, x) − η(x, x) F(x + tu) − F(x)
,
< 0,
t
t
which implies that
η(x + tu, x) − η(x, x), F(x + tu) − F(x) < 0.
By letting y = x + tu, we have η(y, x), F(y) − F(x) < 0, which contradicts η − monotonicit y
of F. Therefore, when D F(x) exists, for any u ∈ Rn we conclude that ηx (x, x)u, D f (x)u ≥ 0.
Now, we present an analogous second-order necessary condition for C 1,1 functions. We denote
the Clarke subdifferential of the mapping F : Rn → Rn at x, by ∂ F(x).
Theorem 2.1. Suppose that f : Rn → R is C 1,1 , invex function with respect to an skew η : Rn ×
Rn → Rn , where η is differentiable in the first argument at x and continuous. Then ηx (x, x)u,
x ∗ u ≥ 0, for any u ∈ Rn and x ∗ ∈ ∂ f (x).
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Proof. Since f = F : Rn → Rn is a locally Lipschitz mapping, by Rademacher Theorem [6], we
have:
∂ F(x) = conv{lim D F(xi ) : xi → x and F is differentiable at xi }.
Thus, for any x ∗ ∈ ∂ F(x), we can find x1∗ , x2∗ ∈ Rn and two sequences xi1 → x and xi2 → x such
that D F(xi1 ) → x1∗ , D F(xi2 ) → x2∗ and x ∗ is a convex combination of x1∗ and x2∗ , which means
x ∗ = λx1∗ + (1 − λ)x2∗ , for some λ ∈ [0, 1].
First, we show that ηx (x, x)u, D F(xi1 )u ≥ 0 for sufficiently large i. Suppose, on the contrary
that, there exists a subsequence (xi1k ) of (xi1 ) such that ηx (x, x)u, D F(xi1k )u < 0. Therefore, by
definition of differential, we have
F(xi1k + tu) − F(xi1k )
η(x + tu, x) − η(x, x)
and D F(xi1k )u = lim
.
t→0
t→0
t
t
ηx (x, x)u = lim
Thus, for each xi1k we can find sufficiently small tk > 0 such that
η(x + tk u, x) − η(x, x), F(xi1k + tk u) − F(xi1k ) < 0.
By continuity of η at both arguments, for sufficiently large k, we conclude that
η(xi1k + tk u, xi1k ) − η(xi1k , xi1k ), F(xi1k + tk u) − F(xi1k ) < 0,
which contradicts η-monotonicity of F, since we know that f is invex with respect to η.
We can use the foregoing argument for x2∗ and (xi2 ), and we conclude for sufficiently large i, that
ηx (x, x)u, D f (xi2 )u ≥ 0.
Therefore, we have
ηx (x, x)u, x1∗ u ≥ 0 and ηx (x, x)u, x2∗ u ≥ 0,
which implies that ηx (x, x)u, x ∗ u = ηx (x, x)u, (λx1∗ + (1 − λ)x2∗ )u ≥ 0.
The following example, shows that some times to characterize the invexity of a function by
second order condition is easier than checking by the first order condition.
Example 2.1. Consider the following C 1,1 function f : R → R,
−x 2 + x, x ≤ 0
f (x) =
x > 0.
x 2 + x,
Let η(x, y) = x 3 − y 3 . An easy calculation implies that
⎧
x <0
⎨ −2,
[−2, 2], x = 0
∂ f (x) =
⎩
2,
x > 0,
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which means that ηx (x, x)u, x ∗ u = 3x 2 u 2 x ∗ < 0, by letting x = −1 and any arbitrary u ∈ R. The
foregoing function is not also invex with respect to η(x, y) = x y 2 − yx 2 , by using a similar calculation for x = 1. It is more difficult to check the invexity of this function with respect to each of the
above η, by using the first order characterization conditions.
Note that we can find some η, such that f is invex with respect to η, since f has not any stationary
f (y)
. When η is not
point ( where f vanishes). Indeed, f is invex with respect to η(x, y) = f (x)−
f (y)
differentiable relative to the first argument, but f is twice differentiable, we can also, present an
analogous of the above Theorem, by a similar argument, as bellow. We denote the Clarke subdifferential of η, relative to the first argument by ∂x η.
Proposition 2.2. Let f : Rn → R be an invex function with respect to η : Rn × Rn → Rn , which
is skew, f be twice differentiable at x ∈ Rn and η(., x) be Lipschitz in the first argument. Then
ξ u, D 2 f (x)u ≥ 0, for any u ∈ Rn and ξ ∈ ∂x η(x, x). The following example illustrate that we
can use the above result for denying the invexity of a twice differentiable function, when η is Lipschitz with respect to the first argument.
Example 2.2. Consider the twice differentiable function f : R → R as
f (x) =
x 3 + x 2,
−x 3 + x 2 ,
x ≤0
x > 0,
and η : R × R → R as bellow:
⎧
−x + y,
⎪
⎪
⎨
x − y,
η(x, y) = |x| − |y| =
x + y,
⎪
⎪
⎩
−x − y,
x
x
x
x
≤ 0, y
≥ 0, y
> 0, y
< 0, y
≤0
≥0
<0
> 0.
An easy calculation implies that ∂x η(0, 0) = [−1, 1]. Since we can find some ξ ∈ ∂x η(0, 0) such that
ξ u, D 2 f (0)u = 2ξ u 2 < 0 for any arbitrary u ∈ R, therefore the second order necessary condition
in Proposition 2.2 dose not hold. Hence, f is not invex with respect to η. Although we can find
some η such that f is invex with respect to it.
Remark 2.1. The above results are the natural extensions of convex case. In fact by replacing
η(x, y) with x − y, we have the classical form of Hessian.
In the following, we give some second-order sufficient conditions for invex functions. We begin
with a sufficient condition for twice differentiable case in the next theorem. First, we need the
following lemma, which is an extension of monotone case. We follow a similar technique as in
Lemma 3.4 of [5], to establish our lemma.
Definition 2.1. We say that a set valued mapping F : Rn ⇒ Rn is semi-locally invariant monotone
at x̄ ∈ dom F with respect to η, if there exists a neighborhood U of x̄ such that for any x, y ∈ U
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and any u ∈ F(x), v ∈ F(y), one has
v, η(x, y) + u, η(y, x) ≤ 0.
Also, we say that F is semi-locally monotone on an invex set K ⊆ Rn , if it is semi-locally invariant
monotone at any point of K .
We need the following useful concept to prove the following lemma.
Definition 2.2. [1] Let K ⊆ Rn be a nonempty invex set with respect to η, and x and u, two arbitrary
points of K . A set Puv is said to be a closed η − path joining the points u and v = u + η(x, u)
(contained in K ) if
Puv := {y = u + λη(x, u) : λ ∈ [0, 1]}.
Analogously, an open η − path joining the points u and v = u + η(x, u) (contained in K) is the
following set
◦
:= {y = u + λη(x, u) : λ ∈ (0, 1)}.
Puv
Lemma 2.1. Let F : Rn ⇒ Rn be a semi-locally η − monotone mapping on Rn and η be skew
satisfying Assumption C and η(., y) be onto for any y ∈ Rn . Then F is η − monotone on Rn .
Proof. Suppose that u 1 and u 2 be arbitrary, v1 ∈ F(u 1 ) and v2 ∈ F(u 2 ). We will show that v2 −
v1 , η(u 2 , u 1 ) ≥ 0. Since η(., u 1 ) is onto, there exists u ∈ X such that u 2 = u 1 + η(u, u 1 ). First, we
show that v2 − v1 , η(u, u 1 ) ≥ 0.
Consider η − path, Pu 1 u 2 := {u 1 + λη(u, u 1 ) : 0 ≤ λ ≤ 1}. By semi-local η-monotonicity of
F, for any x ∈ Pu 1 u 2 , there exists γx > 0 such that
y2 − y1 , η(x2 , x1 ) ≥ 0 for any x1 , x2 ∈ Bγx (x) and y1 ∈ F(x1 ), y2 ∈ F(x2 ).
Now,
n the compactness of Pu 1 u 2 implies that, there exist x1 , x2 , ..., xn ∈ Pu 1 u 2 such that Pu 1 u 2 ⊆
i=1 (int Bγxi (x i )). We select ū j ∈ Pu 1 u 2 as ū j = u 1 + λ j η(u, u 1 ) such that
0 = λ0 < λ1 < ... < λm = 1, j ∈ {0, 1, ..., m − 1},
and
[ū j+1 , ū j ] ⊆ int Bγxi (xi j ), with some i j ∈ {1, ..., n}.
j
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For arbitrary v̄ j ∈ F(ū j ) where j ∈ {1, ..., m − 1} and for v̄0 = v1 and v̄m = v2 , by semi-local η −
monotonicit y of F and Assumption C, we have:
v̄ j+1 − v̄ j , η(ū j+1 , ū j ) = v̄ j+1 − v̄ j , η(u 1 + λ j+1 η(u, u 1 ), u 1 + λ j η(u, u 1 ))
= v̄ j+1 − v̄ j , η(u 1 + λ j+1 η(u, u 1 ), u 1 + λ j+1 η(u, u 1 ) + (λ j − λ j+1 )η(u, u 1 ))
= v̄ j+1 − v̄ j , η(u 1 + λ j+1 η(u, u 1 ), u 1 + λ j+1 η(u, u 1 )
λ j − λ j+1
η(u, u 1 + λ j+1 η(u, u 1 ))
1 − λ j+1
λ j+1 − λ j
= v̄ j+1 − v̄ j ,
η(u, u 1 + λ j+1 η(u, u 1 ))
1 − λ j+1
+
= (λ j+1 − λ j ) v̄ j+1 − v̄ j , η(u, u 1 ) ≥ 0.
Therefore, we conclude that
v̄ j+1 − v̄ j , η(u, u 1 ) ≥ 0 for any j ∈ {0, 1, ..., m − 1}.
Thus, we have
v2 − v1 , η(u, u 1 ) = v̄m − v̄m−1 , η(u, u 1 ) + v̄m−1 − v̄m−2 , η(u, u 1 )
+ ... + v̄1 − v̄1 , η(u, u 1 ) ≥ 0.
Now, by using Assumption C again,
v2 − v1 , η(u 2 , u 1 ) = v2 − v1 , η(u 1 + λη(u, u 1 ), u 1 )
= v2 − v1 , η(u, u 1 ) ≥ 0.
But, this means that F is invariant monotone with respect to η.
Theorem 2.2. Let F : Rn → Rn be a differentiable mapping, η : Rn × Rn → Rn be skew satisfying Assumption C and η(., y) be onto for any y ∈ Rn . If ηx (x, x)u, D F(x)u ≥ 0, for any
x, u ∈ Rn , then F is invariant monotone with respect to η.
Proof. Suppose on the contrary that F is not η − monotone. Therefore, by Lemma 2.1, F is not
semi-locally invariant monotone. So, we can find x ∈ X, a sequence tk → 0 and yk = x + tk u,
which converges to x, such that
F(yk ) − F(x), η(yk , x) < 0, for any k.
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By definition of directional derivative,
F(x + tu) − F(x)
t→0
t
F(x + tk u) − F(x)
η(x + tk u, x), η(x + tk u, x)
×
= lim
k→∞
tk
η(x + tk u, x), η(x + tk u, x)
F(x + tk u) − F(x), η(x + tk u, x)
η(x + tk u, x) − η(x, x)
× lim
.
= lim
k→∞
k→∞
η(x + tk u, x), η(x + tk u, x)
tk
D F(x)u = lim
Therefore, we have D F(x)u = Mηx (x, x)u, for some M ≤ 0. But, this contradicts the assumption
ηx (x, x)u, D F(x)u ≥ 0, which means that F is η − monotone.
Now, we give a sufficient second-order condition for invexity of a twice differentiable function
f : Rn → R.
Corollary 2.1. Let f : Rn → R be a twice differentiable function, f and η satisfy Assumptions A
and C, η(., y) be onto for any y ∈ Rn and skew. If ηx (x, x)u, ∇ 2 f (x)u ≥ 0, for any x, u ∈ Rn ,
then f is invex with respect to η.
Proof. By Theorem 2.2, ∇ f (x) is invariant monotone with respect to η. Now, it suffices to use
Theorem 3.1 of [13] which implies that f is invex.
We can also, give another second-order sufficient condition for invexity of a function f : X →
R, by using the generalization of Taylor expansion for invex case. Although, the invex case of Taylor
expansion in theorem 17 of [1] is given for Rn , but we can use it for a Hilber space X as bellow.
Proposition 2.3. Suppose that f : X → R is a twice differentiable function which satisfies
Assumption A, and η is skew and continuous. Moreover, suppose that D 2 f (x)(η(y, x), η(y, x)) ≥
0, for any x, y ∈ X. Then f is invex in a neighborhood of x with respect to η.
Proof. By Taylor-Young expansion of order 2, for an arbitrary and fixed x ∈ X and any h ∈ X, we
have:
f (x + h) = f (x) + D f (x)(h) +
1 2
D f (x)(h, h) + h 2 (h),
2
with limh→0 (h) = 0.
By letting h = η(y, x) and using the assumption D 2 f (x)(η(y, x), η(y, x)) ≥ 0, we conclude that
f (x + η(y, x)) − f (x) ≥ D f (x)(η(y, x)),
for y ∈ X which η(y, x) is sufficiently closed to zero. Since η is skew and continuous in the first
argument, η(y, x) is sufficiently closed to zero, when y is sufficiently closed to x. Now, by using
Assumption A, we have f (x + η(y, x)) ≤ f (y) and therefore f (y) − f (x) ≥ D f (x)(η(y, x)),
which means that f is invex in a neighborhood of x.
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M.T. Nadi and J. Zafarani
In particular, for X = Rn we have the following result:
Corollary 2.2. Let f : Rn → R be a twice differentiable function which satisfying Assumption A,
and η be skew and continuous. Moreover, suppose that
∇ 2 f (x)(η(y, x), η(y, x)) ≥ 0, for any y ∈ Rn . Then f is invex in a neighborhood of x with respect
to η.
3. APPLICATIONS IN OPTIMIZATION
The notion of invexity introduced by Hanson [12], and afterwards appeared frequently for solving
the optimization problems in literature. Due to the definition of invexity, it is easy to see that when
K is an invex set and f : K → R is a differentiable invex function, then ∇ f (x̄) = 0 implies that x̄
is a local minimum point of f . Specially, in constrained problems, invexity plays an important role
for attaining sufficient conditions of optimality. So, the above second-order sufficient conditions
can be used in constrained and unconstrained optimization problems. Now, we give the following
second-order conditions in optimization.
Proposition 3.1. Let f : Rn → R be a twice differentiable function which satisfies Assumption A
with respect to some η, and ∇ f (x̄) = 0. Moreover, suppose that one of the following holds:
(i) ηx (x, x)u, ∇ 2 f (x)u ≥ 0, for any x, u ∈ Rn , where η is skew satisfies Assumption C and η(., y)
is onto for any y ∈ Rn .
(ii) η(y, x̄), ∇ 2 f (x̄)η(y, x̄) ≥ 0, for any y ∈ Rn .
Then x is a local minimizer of f .
Proof. By Corollaries 2.1 and 2.2, conditions (i) and (ii) imply the invexity of f in a neighborhood
of x̄. So, ∇ f (x̄) = 0 implies that x̄ is a local minimizer of f.
Consider the following constrained optimization problem:
min f 0 (x) subject to f i (x) ≤ 0 (i = 1, ..., m),
(3.1)
which f 0 , f 1 , ..., f m are twice differentiable functions defined on Rn .
Let f (x) = ( f 1 (x), ..., f m (x)). We know that the existence of a vector λ = (λ1 , ..., λm ) ∈ Rm which
satisfies the following conditions, which named Kuhn-Tucker conditions are necessary for a point x̄
to be a solution of this problem:
m
∇ f 0 (x̄) + i=1
λi ∇ f i (x̄) = 0
λ, f (x̄) = 0
(3.2)
(3.3)
λ1 , ..., λm ≥ 0.
(3.4)
Hanson [12] showed that the Kuhn-Tucker conditions are also sufficient for x̄ to be a solution of
(3), when each f i is invex with respect to the same η. Indeed, only the invexity in a neighborhood
of x̄ for each f i guarantees that the foregoing conditions are sufficient [8].
Now, we give some second-order sufficient conditions for constrained optimization problem,
by using our results.
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Proposition 3.2. Suppose the constrained optimization problem (3). If the Kuhn-Tucker conditions
hold in x̄ and each f i satisfies Assumption A, and one of the following second-order conditions
holds (with respect to the same η):
(i) ηx (x, x)u, ∇ 2 f i (x)u ≥ 0, for any x, u ∈ Rn , where η is skew satisfies Assumption C and η(., y)
is onto for any y ∈ Rn .
(ii) η(y, x̄), ∇ 2 f i (x)η(y, x̄) ≥ 0 for any y ∈ Rn ,
then x̄ is a solution for constrained optimization problem (3).
Proof. By using each of conditions (i) or (ii), Corollaries 2.1 and 2.2 imply that each fi is invex in
a neighborhood of x̄ with respect to η. Now, by Theorem 1 of [8], Kuhn-Tucker conditions imply
minimality of x̄, for constrained optimization problem.
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Vol. 10, No. 1 (Special Issue), Jan–June 2019
INDIAN JOURNAL OF INDUSTRIAL AND APPLIED MATHEMATICS
Vol. 10, No. 1 (Special Issue), Jan–June 2019, pp. 71–86
DOI:
Approximation of Periodic Functions via Statistical
B-summability and Its Applications to
Approximation Theorems
B.B. Jena1 , S.K. Paikray2∗ and U.K. Misra3
1,2
Department of Mathematics, Veer Surendra Sai University of Technology, Burla-768018, Odisha,
India
3
Department of Mathematics, National Institute of Science and Technology, Pallur Hills,
Golanthara-761008, Odisha, India
(∗ Corresponding author) Email: ∗ skpaikray math@vssut.ac.in, 1 bidumath.05@gmail.com,
3
umakanta misra@yahoo.com
Abstract: The A-statistical summability is stronger than A-statistical convergence which was introduced by Edely and Mursaleen [7] (see [Edely and Mursaleen, On statistical A-summability, Math.
Comput. Model. 49, 672-680). In this paper, by using the concept of statistical B-summability we
establish a result on Korovkin-type approximation theorem for periodic functions defined on a
Banach space C ∗ (R), which is also stronger than its statistical A-summability version. Furthermore,
we demonstrate a theorem for the rate of the B-statistical convergence for same set of functions with
the help of the modulus of continuity.
(2010) Mathematics Subject Classification. Primary 40A05, 41A36; Secondary 40G15.
Keywords: Statistical B-summability; statistical A-summability; B-statistical convergence; rate of
convergence; Korovkin type approximation theorem.
1. INTRODUCTION, DEFINITIONS AND MOTIVATION
The notion of statistical convergence was introduced and studied by Fast [8] and Steinhaus [25].
Recently, statistical convergence has been a dynamic research area due basically to the fact that it
is more general than classical convergence and such theory is discussed in the study in the areas of
(for instance) Fourier Analysis, Number Theory, Functional Analysis and Approximation Theory.
For more details, see the recent works [1, 2, 5, 6], [10–14] and [17–23].
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B.B. Jena, S.K. Paikray and U.K. Misra
Let N be the set of natural numbers and let K ⊆ N. Also let
Kn = {k : k n
k ∈ K}
and
and suppose that |Kn | be the cardinality of Kn . Then the natural density of K is defined by
d(K) = lim
n→∞
|Kn |
1
= lim |{k : k n
n→∞ n
n
and
k ∈ K}|,
provided that the limit exists.
A given sequence (xn ) is said to be statistically convergent to a number if, for each > 0,
K = {k : k ∈ N
and
|xk − | }
has zero natural density (see [8], [25]). This means that, for each > 0, we have
d(K ) = lim
n→∞
|K |
1
= lim |{k : k n
n→∞ n
n
and
|xk − | }| = 0.
In this case, we write
stat lim xn = .
n→∞
We present below an example to illustrate that every convergent sequence is statistically convergent
but the converse is not true.
Example 1. Let us consider a sequence x = (xn ) by
⎧ 1
(n = m 2 , m ∈ N)
⎨ 2
xn =
⎩ n
(otherwise).
n+1
Here, the sequence (xn ) is statistically convergent to 1 even if it is not classically convergent.
Let X and Y be two sequence spaces and let A = (an,k ) be a non-negative regular matrix. If for
each xk ∈ X the series,
An x =
∞
an,k xk
k=1
converges for each n ∈ N and the sequence (An x) belongs to Y , then the matrix A maps X into Y .
Here, the symbol (X, Y ) denotes the set of all matrices which map X into Y .
A matrix A is said to be regular, if
lim An x = L
n→∞
whenever
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lim xk = L .
k→∞
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Approximation of Periodic Functions via Statistical B-summability
73
The well-known Silverman-Toeplitz theorem (see, details [4]) asserts that A = (an,k ) is regular if
and only if the conditions,
(i) sup
n→∞
∞
|an,k | < ∞,
k=1
(ii) lim an,k = 0 for each k
n→∞
and
(iii) lim
n→∞
∞
an,k = 1 hold true.
k=1
Freedman and Sember [9] extended the definition of statistical convergence with the help
of the non-negative regular matrix A = (an,k ), which he called it A-statistical convergence. Let
for any non-negative regular matrix A, we say that a sequence (xn ) is A-statistical convergent to a
number if, for each > 0, we have,
d A (K ) = 0,
where
K = {k : k ∈ N
This means that, for each > 0, we have
lim
n→∞
|xk − | }.
and
an,k = 0.
k:|xk −|
In this case, we write
stat A lim xn = .
Now we recall statistical A-summability for a nonnegative regular matrix A.
Definition 1. Let A = (an,k ) be a non-negative regular matrix and we say that the sequence
(xn ) is statistical A-summable to a number if, for each > 0,
K = {k : k ∈ N
and
|Ak (x) − | }
has zero natural density (see [7, 17]).
This means that, for each > 0, we have
|K |
1
= lim |{k : k n
n→∞ n
n→∞ n
d(K ) = lim
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and
|Ak (x) − | }| = 0.
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B.B. Jena, S.K. Paikray and U.K. Misra
In this case, we write
stat lim An (x) = or Astat lim xn = .
n→∞
n→∞
In the year 1998, the concept of A-statistical convergence was extended by Kolk [15] to B-statistical
convergence with respect to FB -convergence (or B-summable) due to Steiglitz [24].
Suppose that B = (Bi ) is a sequence of infinite matrices with Bi = (bn,k (i)). Then the given
sequence (xn ) ∈ ∞ is said to be B-summable to the value B- lim (xn ), if
n→∞
lim (Bi x)n = lim
n→∞
n→∞
∞
bn,k (i)(x)k = B- lim (xn )
uniformly in i (n, i = 0, 1, 2, ...).
n→∞
k=0
The method (Bi ) is regular if and only if the following conditions hold true (see, for details, [3]
and [24]):
(i) B = sup
∞
|bn,k (i)| < ∞;
n,i→∞ k=0
(ii) lim bn,k (i) = 0
n→∞
(iii) lim
n→∞
∞
for each k ∈ N;
bn,k (i) = 1.
k=0
Let K = {ki } ⊂ N (ki < ki+1 ) for all i. Then the B-density of K is defined by
dB (K) = lim
∞
n→∞
bn,k (i).
k=0
Let R+ denotes the set of all regular methods B with bn,k (i) 0 for all n, k and i. Also let B ∈ R+ .
We say that a sequence (xn ) is B-statistically convergent to a number if, for each > 0, we have
dB (K ) = 0,
where
K = {k : k ∈ N
This means that, for each > 0, we have,
lim
and
n→∞
|xk − | }.
bn,k (i) = 0.
k:|xk −|
In this case, we write
statB lim xn = .
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Definition 2. Let B = (Bn,k (i)) be a non-negative regular matrix and we say that the sequence (xn )
is statistically B-summable to a number if, for each > 0,
K = {k : k ∈ N
and
|B(x) − | }
has zero natural density (see [13], [15]). This means that, for each > 0, we have
d(K ) = lim
n→∞
|K |
1
= lim |{k : k n
n→∞
n
n
and
|B(x) − | }| = 0.
In this case, we write
stat lim B(x) = or Bstat lim xn = .
n→∞
n→∞
Remark 1. Upon replacing the matrices Bi by the matrix A for all i, the B-statistical convergence
reduces to the A-statistical convergence. Furthermore, by choosing B = (C, 1) (the Cesàro matrix
of order one), the B-statistical convergence coincides with the statistical convergence (see [8]
and [25]). Finally if B = I (identity matrix), then the B-statistical convergence coincides with the
classical convergence.
Example 2. Let us consider the infinite matrices B = (Bi ) with Bi = (bn,k (i)) given by (see [13])
Bi =
⎧
⎪
⎨
(i k i + n)
1
n+1
⎪
⎩
0
(otherwise)
⎧
⎪
⎨ 1
(n = odd)
and let a sequence x = (xn ) is given by
xn =
⎪
⎩
(1.1)
−1
(n = even).
Then, we observe that Bi = (bn,k (i)) is a non-negative regular matrix and for the given sequence
(xn ),
stat lim Bn (x) = 0.
n→∞
Here the sequence (xn ) is neither convergent nor statistically convergent. However, (xn ) is statistically B-summable to 0 but it is not B-statistical convergent to 0.
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2. A KOROVKIN TYPE THEOREM VIA STATISTICAL
B-SUMMABILITY
In recent years, quite a few researchers worked toward extending or generalizing the Korovkin type
theorems in many ways based on several aspects, including (for example) function spaces, abstract
Banach lattices, Banach algebras, Banach spaces, and so on. This theory is extremely valuable
in Real Analysis, Functional Analysis, Harmonic Analysis, Measure Theory, Probability Theory,
Summability Theory and Partial Differential Equations, and many other fields. In this section, by
using the concept of statistical B-summability, we prove a Korovkin type approximation theorem
(see for details [16]) for periodic functions. Also by using Fejér operators, we show that our proposed method is stronger than that of classical and statistical versions of Korovkin’s theorem.
We denote C ∗ (R), the space of all real valued 2π periodic continuous functions defined on R.
We recall that if a function f has period 2π for all x ∈ R,
f (x + 2π k) = f (x)
holds for (k = 0, ±1, ±2, ...).
This space is equipped with supremum norm. That is
f C ∗ (R) = sup | f (x)|,
f ∈ C ∗ (R).
x∈R
Let L be a linear operator from C ∗ (R) into C ∗ (R). Then, as usual, we say that L is a positive
linear operator provided that
f 0 implies L( f ) 0.
Also, we denote the value of L( f ) at a point x ∈ R by L( f (u); x) or, briefly, L( f ; x).
Throughout this paper, we use the following test functions
f 0 (x) = 1,
f 1 (x) = cos x and f 2 (x) = sin x.
Theorem 1. Let B = (bn,k (i)) be a nonnegative regular matrix and let (L n ) (n ∈ N) be a sequence
of positive linear operators from C ∗ (R) into itself and let f ∈ C ∗ (R). Then
∞
bn,k (i)L n ( f ; x) − f (x)
= 0, f ∈ C ∗ (R)
(2.1)
stat − lim n C ∗ (R)
n=0
if and only if
∞
bn,k (i)L n ( f j ; x) − f j (x)
stat − lim n n=0
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= 0 ( j = 0, 1, 2.).
(2.2)
C ∗ (R)
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77
Proof. Since each of the functions given by
f 0 (x) = 1,
f 1 (x) = cos x
f 2 (x) = sin x
and
are in C ∗ (R), the implication
(2.1) =⇒ (2.2)
is fairly obvious. In order to complete the proof of the Theorem 1, we first assume that (2.2) holds
true. Let f ∈ C ∗ (R), ∀ x ∈ C ∗ (R) and let I be a closed subinterval of length 2π of R. Fix x ∈ I .
By the continuity of f at x, for given > 0 there exists δ > 0 such that
| f (t) − f (x)| < |t − x| < δ
whenever
(2.3)
for all t, x ∈ R.
Since f (x, y) is bounded on R, then there exists a constant f C ∗ (R) > 0 such that
(∀ x ∈ R),
| f (x)| f C ∗ (R)
which implies that
| f (t) − f (x)| 2 f C ∗ (R)
(t, x ∈ R).
(2.4)
Clearly, f is a continuous function on R, and for a given > 0, there exists δ = δ() > 0; Moreover
from equation (2.3) and (2.4), we get
| f (t) − f (x)| < +
2 f C ∗ (R)
ϕ(t),
sin2 2δ
(2.5)
where
ϕ(t) = sin2
t−x
2
.
Since the function f ∈ C ∗ (R), the inequality (2.5) holds for t, x ∈ R.
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B.B. Jena, S.K. Paikray and U.K. Misra
Now, since the operator L n ( f ; x) is monotone and linear, by applying this operator to the
inequality in (2.5), we have
∞
bn,k (i)L n ( f (t); x) − f (x) =
n=0
∞
bn,k (i)L n ( f (t) − f (x); x) + f (x)
n=0
∞
∞
2 f C ∗ (R)
+
ϕ(t); x
sin2 2δ
bn,k (i)L n
n=0
+
∞
∞
bn,k (i)L k ( f 0 ; x) − f 0
bn,k (i)L k (1; x) − 1
n=0
+ | f (x)
∞
bn,k (i)L n (1; x) − 1
n=0
bn,k (i)L n ( f 0 ; x) − f 0 (x) + f C ∗ (R)
n=0
+
n=0
bn,k (i)L n (| f (t) − f (x)|; x) + | f (x)|
n=0
∞
∞
bn,k (i)L n ( f 0 ; x) − f 0 (x)
n=0
∞
2 f C ∗ (R) bn,k (i)L n (ϕ(t); x).
sin2 2δ n=0
(2.6)
After some simple calculations, we get
ϕ(t) =
1
(1 − cos t cos x − sin t sin x).
2
Next, we have
∞
1
bn,k (i)L n (ϕ(t); x) 2
n=0
∞
bn,k (i)L n ( f 0 ; x) − f 0 (x) + | cos x|
n=0
∞
bn,k (i)L n ( f 1 ; x) − f 1 (x)
n=0
∞
1
| sin x|
bn,k (i)L n ( f 2 ; x) − f 2 (x) . (2.7)
+
2
n=0
Then using (2.7), we have
∞
2 f C ∗ (R)
bn,k (i)L n (ϕ(t); x) − f (x) +
ϕ(t); x
sin2 2δ
n=0
+| cos x|
∞
∞
bn,k (i)L n ( f 0 ; x) − f 0 (x)
n=0
bn,k (i)L n ( f 1 ; x) − f 1 (x)
n=0
+| sin x|
∞
bn,k (i)L n ( f 2 ; x) − f 2 (x) .
(2.8)
n=0
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79
Further, taking supx,y∈R , in both side of (2.8), we get
∞
bn,k (i)L n (ϕ(t); x) − f (x)
+M
∞
C ∗ (R)
n=0
bn,k (i)L n ( f 0 ; x) − f 0 (x)
C ∗ (R)
n=0
∞
+| cos x|
bn,k (i)L n ( f 1 ; x) − f 1 (x)
C ∗ (R)
n=0
+| sin x|
∞
bn,k (i)L n ( f 2 ; x) − f 2 (x)
(2.9)
C ∗ (R)
n=0
where,
2 f C ∗ (R)
M = +
ϕ(t); x .
sin2 2δ
Now, for a given r > 0, we choose > 0, such that 0 < < r . Then, upon setting
∞
bn,k (i)L n ( f (t); x) − f (x) r
An = n : n ∈ N and
n=0
and
A j,n =
n:n∈N
and
∞
r −
bn,k (i)L n ( f j (t); x) − f j (x) 3M
n=0
( j = 0, 1, 2)
we easily find from (2.9) that
An 3
A j,n .
j=0
Thus, we have
A j,n C ∗ (R)
An C ∗ (R)
.
n
n
j=0
3
(2.10)
Finally, by using the above assumption about the implication in (2.2) and Definition 2, the right-hand
side of (2.10) is seen to tend to zero (n → ∞). Consequently, we get
∞
stat − lim bn,k (i)L n ( f j (t); x) − f j (x)
= 0.
n→∞ n=0
C ∗ (R)
Hence, the implication (2.1) holds true. The proof of Theorem 1 is thus completed.
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Remark 4. If B = (I ) (identity matrix) in our Theorem 1, then we obtain classical version of
Korovkin-type approximation theorem (see [16]). Also, if B = (C, 1), Cesàro matrix of order 1, in
our Theorem 1, then we obtain statistical version of Korovkin type approximation theorem (see [8].
Furthermore, if B = (A), in our Theorem 1, then we obtain statistical A-summability version of
Korovkin-type approximation theorem (see [17]).
We now present below an illustrative example for Theorem 1 based on following Fejér
operators.
Example 3. Let I = [−π, π]. For a function f ∈ C ∗ (I ), the operators
1
Fn ( f ; x) =
kπ
π
−π
sin2 k2 (t − x)
f (t)
dt.
2 sin2 [ t−x
]
2
(2.11)
Also, observe that
Fn ( f 0 ; x) = 1, Fn ( f 1 ; x) =
k−1
k−1
cos x, Fn ( f 2 ; x) =
sin x.
k
k
Now we consider consider the following positive linear operators
L n : C ∗ (I ) → C ∗ (I )
such that
L n ( f ; x) = (1 + xn )Fn ( f ; x),
(2.12)
where (xn ) is a sequence defined as in Example 2. It is clear that the sequence (L n ) satisfies the
conditions (2.2) of our Theorem 1, thus we obtain:
∞
stat − lim bn,k (i)L n ( f j ; x) − f j (x)
n→∞ = 0.
C ∗ [−π,π]
n=0
Therefore by Theorem 1, we have
∞
stat − lim bn,k (i)L n ( f ; x) − f (x)
n→∞ n=0
= 0.
C ∗ [−π,π]
However, since (xn ) is not statistically weighted A-summable, so the result of Karakuş et al. ( [7],
p. 162 Theorem 2.1) does not hold true for our operators defined by (2.12). Moreover, since (xn ) is
statistical B-summable, therefore we conclude that our Theorem 1 works for the same operators.
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3. RATE OF THE B-STATISTICAL CONVERGENCE
In this section, we introduce the rate of the B-statistical convergence for a sequence of positive
linear operators defined on C ∗ (R) into itself by using the modulus of continuity. We first present
Definition 3 below.
Definition 3. Let B ∈ R+ , and also let (u n ) is positive and non-decreasing sequence. We say that
the sequence (xn ) is B-statistical convergent to a number with rate o(u n ) if, for each > 0,
1 bm,k (i) = 0,
lim
n→∞ u n
k∈K
where
K = {k : k n and |xk − | }.
In this case, we write
xn − = statB − o(u n ).
We now state and prove Lemma 1 as follows.
Lemma 1. Let (u n ) and (vn ) are two positive non-decreasing sequences. Assume that B ∈ R+ , and
let x = (xm ) and y = (ym ) be two sequences such that
xk − 1 = statB − o(u n )
and
yk − 2 = statB − o(vn ).
Then each of the following assertions hold true:
(i) (xk − L 1 ) ± (yk − 2 ) = statB − o(wn );
(ii) (xk − L 1 )(yk − 2 ) = statB − o(u n vn );
(iii) γ (x k − 1 ) = statB − o(u n ) (for any scalar γ );
√
(iv) |xk − 1 | = statB − o(u n ),
where wn = max{u n , vn }.
Proof. In order to prove the assertion (i) of Lemma 1, we choose > 0 and define the following
sets
Tn = k : k N and | (xk + yk ) − (1 + 2 )| ,
T0;n =
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and
T1,n =
k : k N and |yk − 2 | .
2
Clearly, we have
Tn ⊆ T0,n ∪ T1,n
and this implies, for n ∈ N, that
lim
bm,k (i) lim
bm,k (i) + lim
bm,k (i).
n→∞
n→∞
k∈Tn
n→∞
k∈T0,n
(3.1)
k∈,T 1,n
Moreover, since
wn = max{u n , vn },
(3.2)
by (3.1), we get
lim
n→∞
1 1 1 bm,k (i) lim
bm,k (i) + lim
bm,k (i).
n→∞
n→∞
wn k∈T
u n k∈T
vn k∈,T
n
0,n
(3.3)
1,n
Also, by applying the Theorem 1, we obtain
1 bm,k (i) = 0,
n→∞ wn
k∈T
lim
(3.4)
n
which proves the assertion (i) of Lemma 1.
The other assertion (ii) to (iv) of Lemma 1 are similar to (i), so it is not difficult to prove these
assertions along similar lines. This, evidently completes the proof Lemma 1.
Recalling that the modulus of continuity of a function f (x) ∈ C ∗ [π, π ] is defined by
| f (t) − f (x)| : (t − x)2 δ
(δ > 0),
(3.5)
ω( f ; δ) =
sup
(t,x)∈[−π,π]
which implies
| f (t) − f (x)| ω f ; (t − x)2 .
(3.6)
Now we propose a theorem to get the rates of B-statistical convergence by using the modulus of
continuity in (3.5).
Theorem 2. Let B ∈ R+ , and let L n : C ∗ [−π, π ] → C ∗ [−π, π ] be sequences of positive
linear operators. Also let (u n ) and (vn ) be the positive non-decreasing sequences. Suppose that the
following conditions are satisfied:
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(i) L n (1; x) − 1C ∗ [−π,π] = statB − o(u n );
(ii) ω( f, λn ) = statB − o(vn ) on [−π, π],
where
λn =
L n (ϕ 2 (t, x)C ∗ [−π,π]
ϕ(t) = sin2
and
(t − x)
.
2
Then, for all f ∈ C ∗ [−π, π], the following assertion holds true:
L n ( f ; x) − f (x)C ∗ [−π,π] = statB − o(wn ),
(3.7)
where (wn ) is given by (3.2).
Proof. Let f ∈ C ∗ [−π, π] and by using (3.6), we have
|L n ( f ; x) − f (x)| L n (| f (t) − f (x)|; x) + | f (x)||L n (1; x) − 1|,
(t − x)2
+ 1; x ω( f, δ) + N |L n (1; x) − 1|,
Ln
δ
L n (1; x) +
1
L n (ϕ(t); x) ω( f, δ) + N |L n (1; x) − 1|,
δ2
where N = f C B (D) .
Taking the supremum over (x, y) ∈ D on both sides, we have
L n ( f ; x) − f (x)C B (D) ω( f, δ)
Now, putting δ = λn =
1
L
(ϕ(t);
x)
+
L
(1;
x)
−
1
+
1
n
C
(D)
n
C
(D)
B
B
δ2
+N L n (1; x) − 1C B (D) .
L n (ϕ 2 ; x), we get
L n ( f ; x) − f (x)C B (D) ω( f, λn ) L n (1; x) − 1C B (D) + 2 + N L n (1; x) − 1C B (D)
ω( f, λn )L n (1; x) − 1C B (D) + 2ω( f, λn ) + N L n (1; x) − 1C B (D) .
So, we have
L m ( f ; x) − f (x)C B (D) μ ω( f, λn )L n (1; x) − 1C B (D) + ω( f, λn ) + L n (1; x) − 1C B (D)
where
μ = max{2, N }.
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For a given > 0, we consider the following sets:
Hn = n : n ∈ N and L n ( f ; x) − f (x)C B (D) ;
H0,n = n : n ∈ N and ω( f, λn )Tn ( f ; x) − f (x)C B (D) ;
3μ
H1,n = n : n ∈ N and ω( f, λn ) 3μ
and
H2,n
(3.8)
(3.9)
(3.10)
.
= n : n ∈ N and L n (1; x) − 1C B (D) 3μ
(3.11)
Finally, in view of the conditions (i) and (ii) of Theorem 2 in conjunction with Lemma 1, this last
inequalities (3.8)-(3.11) leads us to the assertion (3.7) of Theorem 2.
This completes the proof of Theorem 2.
4. CONCLUDING REMARKS AND OBSERVATIONS
In this concluding section of our investigation, we present several further remarks and observations
concerning the various results which we have proved here.
Remark 5. Let (xn )n∈N be the sequence given in Example 2. Then, since
C ∗ (R),
Bstat lim xn → 0 on
n→∞
we have
∞
stat lim bn,k (i)L n ( f j ; x) − f j (x)
n = 0,
f ∈ C ∗ (R).
(4.1)
= 0,
f ∈ C ∗ (R)
(4.2)
C ∗ (R)
n=0
Therefore, by applying Theorem 1, we write
∞
stat lim bn,k (i)L n ( f ; x) − f (x)
n n=0
C ∗ (R)
where
f 0 = 1, f 1 = cos x and f 2 = sin x.
However, since (xn ) is not ordinarily convergent and so also it does not converge uniformly in the
ordinary sense. Thus, the classical Korovkin Theorem does not work here for the operators defined
by (2.12). Hence, this application clearly indicates that our Theorem 1 is a non-trivial generalization
of the classical Korovkin-type theorem (see [16]).
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Remark 6. Let (xn )n∈N be the sequence as given in Example 2. Then, since
Bstat lim xn → 0 on C ∗ (R),
n→∞
so (4.1) holds. Now by applying (4.1) and our Theorem 1, condition (4.2) holds. However, since
(xn ) does not statistical A-summable, so we can say that the result of Edely and Mursaleen ( [7],
p. 162, Theorem 2.1) does not hold true for our operator defined in (2.12). Thus, our Theorem 1
is also a non-trivial extension of Edely and Mursaleen ( [7], p. 162, Theorem 2.1) and [16]. Based
upon the above results, it is concluded here that our proposed method has successfully worked for
the operators defined in (2.12) and therefore it is stronger than the classical and statistical version
of the Korovkin type approximation theorem (see [16] and [25] ) established earlier.
Remark 7. Let us suppose we replace the conditions (i) and (ii) in our Theorem 2 by the
following condition:
|L n ( f j ; x) − f j (x)|C ∗ (R) = statB − o(u n j )
( j = 0, 1, 2).
(4.3)
Now, we can write
L n (ϕ 2 ; x) = N
2
L n ( f j (t); x) − f j (x)C ∗ (R) ,
(4.4)
j=0
where
N = 1 + f 1 C ∗ (R) + f 2 C ∗ (R) , ( j = 0, 1, 2).
It now follows from (4.3), (4.4) and Lemma 1 that
λn = L n (ϕ 2 ) = statB − o(dn ) on C ∗ (R),
(4.5)
where
o(dn ) = max{u n 0 , u n 1 , u n 2 }.
Hence, we get
ω( f, δ) = statB − o(dn ) on C ∗ (R).
By using (4.5) in Theorem 2, we immediately see for all f ∈ CB (D) that
L n ( f ; x) − f (x) = statB − o(dn ) on C ∗ (R).
(4.6)
Therefore, if we use the condition (4.3) in Theorem 2 instead of conditions (i) and (ii), then we
obtain the rates of the statistical B-summability of the sequence (L n ) of positive linear operators in
Theorem 1.
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[3] H. T. Bell, Order summability and almost convergence, Proc. Am. Math. Soc. 38 (1973), 548–553.
[4] J. Boos, Classical and Modern Methods in Summability, Clarendon (Oxford University) Press, Oxford, London and
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[14] V. Karakaya and T. A. Chishti, Weighted statistical convergence, Iranian. J. Sci. Technol. Trans., A 33 (A3)(2009),
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[15] E. Kolk, Matrix summability of statistically convergent sequences, Analysis, 13 (1993), 77–83.
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[25] H. Steinhaus, Sur la convergence ordinaire et la convergence asymptotique, Colloq. Math., 2 (1951), 73–74.
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Vol. 10, No. 1 (Special Issue), Jan–June 2019
INDIAN JOURNAL OF INDUSTRIAL AND APPLIED MATHEMATICS
Vol. 10, No. 1 (Special Issue), Jan–June 2019, pp. 87–108
DOI:
On Semi-infinite Optimization Problem Under the
Generalized Convexity
Pushkar Jaisawal∗ , Vivek Laha and S.K. Mishra
Department of Mathematics, Institute of Science, Banaras Hindu University, Varanasi-221005, India
∗
Corresponding author Email: pushkarjaisawal2@gmail.com
Abstract: In this paper, we study nonsmooth semi-infinite optimization problems involving infinite
number of inequality constraints. We derive sufficient optimality conditions for a point to be an
optimal solution of the problem under generalized convexity assumptions. We also formulate Wolfe
and Mond-Weir type dual models and establish several duality results.
Keywords: Sufficient optimality condition, Nonsmooth optimization, Semi-infinite programming,
Wolfe type dual, Mond-Weir type dual, Generalized convexity
1. INTRODUCTION
An optimization problem in which infinite (countable or uncountable) number of inequality constraints present with finite number of variables is known as Semi-Infinite Optimization problem.
We examine the semi-infinite optimization programming (SIP) problem which is nonsmooth and
presented in Kanzi and Nobakhtian [15]
(SIP)
min f (x), subject to gi (x) ≤ 0, ∀i ∈ I,
x∈Rn
where f : Rn → R ∪ {+∞} , gi : Rn → R ∪ {+∞}, ∀i ∈ I , all are locally Lipschitz functions
and I is the index set which is infinite (countable or uncountable).
Using Clarke’s subdifferential [4] Kanzi and Nobakhtian [15] collected the necessary and sufficient optimality conditions and Mishra et al. [21] established the Wolfe [27] and Mond-Weir [23]
dual problems and related results like weak duality, strong duality and strict converse duality
results under invexity assumptions for nonsmooth (SIP). Semi-infinite programming problems are
used in engineering, optimal control theory, robust optimization, social work and different fields
of mathematics etc. Because of this, many scholars are interested in working in this field, see
e.g., [8–10, 12, 20, 24]. We have also seen the work of past decades, which have been done by
Dinh et al. [6], Canovas et al. [2, 3], Shapiro [24, 25] and also see this [16–19] for the application of
(SIP).
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Sufficient optimality results play an important role in optimization problems. Wolfe duality [27]
and Mond-Weir duality [23] also play a very important role for optimization. Wolfe [27] in 1961
gave the Wolfe type dual problem and Mond and Weir [23] in 1981 gave the Mond-Weir type dual
problem, see e.g., Jeyakumar [13], Daldoul and Baccari [5] and Wang et al. [26]. In the above (SIP)
problem Kanzi and Nobakhtian [15] and Mishra et al. [21] gave the results for invex functions.
The requirement of invex function is limited to the same kernal function for all the constraints and
objective functions. For that reason, in this paper, we establish the sufficient optimality conditions
and duality results for (SIP) under V −invex function. The concept of V −invex function was first
established in 1992 by Jeyakumar and Mond [14].
The outline of this paper is as follows: in Section 2, we give some well-known definitions
and KKT condition given by Kanzi and Nobakhtian [15]. In Section 3, we give some definitions
which will be used in the sequel, we derive sufficient optimality conditions for (SIP) involving
V −invex function and generalized V −invex function both using Clarke’s subdifferential. In Section
4, we derive a new type of Wolfe dual model and Mod-Weir dual model for r −invex and V −invex
functions, respectively, and establish weak duality, strong duality and strict converse duality results
for both the dual models under V −invexity and generalized V −invexity assumptions with the use
of Clarke’s subdifferential. And lastly in Section 5, we conclude the results present in this paper.
2. PRELIMINARIES
Let S be a nonempty and open subset of Rn , we denote by cone(S), S̄ and conv(S), the convex cone
generated by S containing the origin of Rn , closure of S and convex hull of S. The strictly negative
and negative polar cones of S are denoted by S S N and S N , respectively, and defined by:
S S N = {d ∈ Rn : x, d < 0, ∀x ∈ S}
S N = {d ∈ Rn : x, d ≤ 0, ∀x ∈ S}
Definition 1. (Contingent cone) Let x̄ ∈ S̄. The contingent cone of S at x̄ is denoted by K (S, x̄)
and defined by
x t − x̄
t
t
t
K (S, x̄) := lim
:
λ
→
0
,
x
∈
S,
x
→
x̄
.
+
t→∞
λt
Definition 2. (Lipschitz function) Let f be a function defined by f : Rn → R, if there exist a k,
where k is a positive constant such that
| f (x) − f (y)| ≤ k||x − y||
then f is said to be locally Lipschitz at z ∈ S where x, y ∈ N , N is a neighbourhood of z and S be
an open subset of Rn . Also, if it satisfies for every z ∈ S then f is said to be Lipschitz on S.
Clarke [4] introduced the concept of Clarke’s tangent cone and Clarke’s subdifferential.
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Definition 3 [4]. (Clarke’s subdifferential) A vector x ∗ ∈ Rn is a Clarke’s subdifferential vector of
f at x̄ ∈ S if it satisfies
f 0 (x̄; d) ≥ x ∗ , d , ∀d ∈ Rn
where f 0 (x̄; d) is a generalized Clarke’s directional derivative of a locally Lipschitz function f at
x̄ ∈ S in the direction d and it is defined by
f 0 (x̄, d) = lim sup
y→x̄, t↓0
f (y + td) − f (y)
.
t
The collection of all such vectors is called the Clarke’s subdifferential at x̄ ∈ S. It is denoted by
∂ C f (x̄) and defined by
∂ C f (x̄) = {x ∗ ∈ Rn : f 0 (x̄; d) ≥ x ∗ , d , ∀d ∈ Rn }.
Definition 4 [4]. (Clarke’s tangent cone) Collection of all the vectors v ∈ Rn is called the Clarke’s
tangent cone of S at x̄ ∈ S which satisfies
d S0 (x̄; v) = 0.
It is denoted by T (S, x̄) and mathematically express by
T (S, x̄) = v ∈ Rn : d S0 (x̄; v) = 0 ,
where d S (x) is the distance function related to S and defined by
d S (x) = inf{||x − y|| : y ∈ S}.
In 1981 Hanson [11] gave the concept of differential invex function. The generalization of invex
function was introduced by Jeyakumar and Mond [14] which is known as V −invex function. The
generalization of V −invex function for the nonsmooth case was given by Egudo and Hanson [7]
using the Clarke’s generalized subdifferential [4].
Definition 5 [7]. (Clarke V −invex function) The function f defined by f := ( f1 , ..., f m ) : S ⊆
Rn → Rm is locally Lipschitz near x̄ ∈ S. The function f is said to be Clarke V −invex at x̄ ∈ S
with respect to η : Rn × Rn → Rn and αi : Rn × Rn → R+ \ {0}, for all i ∈ M = {1, ..., m} over
S, iff for all i ∈ M, x ∈ S and x̄i∗ ∈ ∂ C f i (x̄), we have
f i (x) − f i (x̄) ≥ αi (x, x̄) x̄i∗ , η(x, x̄) .
The function f is said to be Clarke V −invex on S with respect to η and αi , for i ∈ M, iff f is
Clarke V −invex at x̄ ∈ S over S with respect to η and αi , for all i ∈ M and for all x̄ ∈ S.
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Definition 6 [7]. (Clarke V −strictly-invex function) The function f defined by f := ( f 1 , ..., f m ) :
S ⊆ Rn → Rm is locally Lipschitz near x̄ ∈ S. The function f is said to be Clarke V −strictlyinvex at x̄ ∈ S with respect to η : Rn × Rn → Rn and αi : Rn × Rn → R+ \ {0}, for all i ∈ M =
{1, ..., m} over S, iff for all i ∈ M, x ∈ S, x = x̄ and for all x̄i∗ ∈ ∂ C f i (x̄), we have
f i (x) − f i (x̄) > αi (x, x̄) x̄i∗ , η(x, x̄) .
The function f is said to be Clarke V −strictly-invex on S with respect to η and αi , for all i ∈ M,
iff f is Clarke V −strictly-invex at x̄ ∈ S over S with respect to η and αi , for all i ∈ M and for all
x̄ ∈ S.
In 1996 Mishra and Mukherjee [22] introduced the following concept of V −pseudo-invex and
V −quasi-invex functions.
Definition 7 [22]. (Clarke V −pseudo-invex function) The function f defined by f := ( f1 , ..., f m ) :
S ⊆ Rn → Rm is locally Lipschitz near x̄ ∈ S. The function f is said to be Clarke V −pseudoinvex at x̄ ∈ S if there exist functions η : Rn × Rn → Rn and αi : Rn × Rn → R+ \ {0}, for all
i ∈ M = {1, ..., m} such that for x, x̄ ∈ S
αi (x, x̄) f i (x) <
αi (x, x̄) f i (x̄),
i∈M
⇒ for any
i∈M
x̄i∗
∈ ∂ f i (x̄),
C
x̄i∗ , η(x, x̄) < 0.
i∈M
The function f is said to be Clarke V −pseudo-invex on S with respect to η and αi , for all i ∈ M,
iff f is Clarke V −pseudo-invex at x̄ ∈ S over S with respect to η and αi , for all i ∈ M and for all
x̄ ∈ S.
Definition 8 [22]. (Clarke V −quasi-invex function) The function f defined by f := ( f1 , ..., f m ) :
S ⊆ Rn → Rm is locally Lipschitz near x̄ ∈ S. The function f is said to be Clarke V −quasi-invex
at x̄ ∈ S if there exist functions η : Rn × Rn → Rn and αi : Rn × Rn → R+ \ {0}, for all i ∈ M =
{1, ..., m} such that for x, x̄ ∈ S
αi (x, x̄) f i (x) ≤
αi (x, x̄) f i (x̄),
i∈M
⇒ for any
i∈M
x̄i∗
∈ ∂ f i (x̄),
C
x̄i∗ , η(x, x̄) ≤ 0.
i∈M
The function f is said to be Clarke V −quasi-invex on S with respect to η and αi , for all i ∈ M, iff
f is Clarke V −quasi-invex at x̄ ∈ S over S with respect to η and αi , for all i ∈ M and for all x̄ ∈ S.
Definition 9. (Clarke V −strictly-pseudo-invex function) The function f defined by f :=
( f 1 , ..., f m ) : S ⊆ Rn → Rm is locally Lipschitz near x̄ ∈ S. The function f is said to be Clarke
V −strictly-pseudo-invex at x̄ ∈ S if there exist functions η : Rn × Rn → Rn and αi : Rn × Rn →
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R+ \ {0}, for all i ∈ M = {1, ..., m} such that for x, x̄ ∈ S
αi (x, x̄) f i (x) ≤
αi (x, x̄) f i (x̄),
i∈M
⇒ for any
i∈M
x̄i∗
∈ ∂ f i (x̄),
C
x̄i∗ , η(x, x̄) < 0.
i∈M
The function f is said to be Clarke V −strictly-pseudo-invex on S with respect to η and αi , for all
i ∈ M, iff f is Clarke V −strictly-pseudo-invex at x̄ ∈ S over S with respect to η and αi , for all
i ∈ M and for all x̄ ∈ S.
Consider the following constraint qualification from [15].
Definition 10. (ACQ) The Abadie constraint qualification (ACQ) holds at x̄ ∈ S if
B 0 (x̄) ⊆ K (S, x̄),
where B(x̄) := i∈I (x̄) ∂ C gi (x̄) and I (x̄) is the active constraints.
The KKT condition established by Kanzi and Nobakhtian [15] in terms of Clarke’s subdifferentials is as follows:
Theorem 1. (KKT Condition) Suppose that x̄ is an optimal solution of (SIP), A(x̄) := cone(B(x̄))
is closed, (ACQ) holds and I (x̄) = φ. Then there exist λi ≥ 0, i ∈ I (x̄), with λi = 0, for finitely
many indexes such that,
λi ∂ C gi (x̄).
(2.1)
0 ∈ ∂ C f (x̄) +
i∈I (x̄)
3. SUFFICIENT CONDITIONS
We consider the following (SIP) problem
(SIP)
min f (x), subject to gi (x) ≤ 0, ∀i ∈ I,
x∈Rn
where f, gi : Rn → R ∪ {+∞}, for all i ∈ I , all are locally Lipschitz functions and I is the index
set which is infinite (countable or uncountable).
Let S = {x ∈ Rn : gi (x) ≤ 0} and I (x̄) = {i ∈ I : gi (x̄) = 0}, where x̄ ∈ S. We consider the
following possibilities for the functions ( f, g I ).
(a) The function ( f, g I ) is said to be the V −invex at x̄ ∈ S, iff there exist β, αi : Rn × Rn → R+ \
{0}, for all i ∈ I and η : Rn × Rn → Rn such that
f (x) − f (x̄) ≥ β(x, x̄) x̄ ∗ , η(x, x̄) , ∀x̄ ∗ ∈ ∂ C f (x̄)
(3.1)
gi (x) − gi (x̄) ≥ αi (x, x̄) x̄i∗ , η(x, x̄) , ∀x̄i∗ ∈ ∂ C gi (x̄), ∀i ∈ I.
(3.2)
and
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(b) The function ( f, g I ) is said to be the V −strictly-invex at x̄ ∈ S, iff there exist β, αi : Rn × Rn →
R+ \ {0}, for all i ∈ I and η : Rn × Rn → Rn such that
f (x) − f (x̄) > β(x, x̄) x̄ ∗ , η(x, x̄) , ∀x̄ ∗ ∈ ∂ C f (x̄)
(3.3)
gi (x) − gi (x̄) ≥ αi (x, x̄) x̄i∗ , η(x, x̄) , ∀x̄i∗ ∈ ∂ C gi (x̄), ∀i ∈ I.
(3.4)
and
(c) The function ( f, g I ) is said to be the V −pseudo-quasi-invex at x̄ ∈ S, iff there exist β, αi :
Rn × Rn → R+ \ {0}, for all i ∈ I and η : Rn × Rn → Rn such that
β(x, x̄) f (x) < β(x, x̄) f (x̄) ⇒ for any x̄ ∗ ∈ ∂ C f (x̄),
and
αi (x, x̄)gi (x) ≤
i∈I
αi (x, x̄)gi (x̄) ⇒ for any x̄i∗ ∈ ∂ C gi (x̄),
i∈I
x̄ ∗ , η(x, x̄) < 0
x̄i∗ , η(x, x̄) ≤ 0.
(3.5)
(3.6)
i∈I
(d) The function ( f, g I ) is said to be the V −strictly-pseudo-quasi-invex at x̄ ∈ S, iff there exist
β, αi : Rn × Rn → R+ \ {0}, for all i ∈ I and η : Rn × Rn → Rn such that
β(x, x̄) f (x) ≤ β(x, x̄) f (x̄) ⇒ for any x̄ ∗ ∈ ∂ C f (x̄),
and
i∈I
αi (x, x̄)gi (x) ≤
αi (x, x̄)gi (x̄) ⇒ for any x̄i∗ ∈ ∂ C gi (x̄),
i∈I
x̄ ∗ , η(x, x̄) < 0
x̄i∗ , η(x, x̄) ≤ 0.
(3.7)
(3.8)
i∈I
In this section, we analyse sufficient optimality conditions of the above taken programming problem.
Theorem 1. Let x̄ ∈ S and I (x̄) = φ. Suppose that there exist λi ≥ 0, i ∈ I (x̄), with λi = 0, for
finitely many indexes, and
0 ∈ ∂ C f (x̄) +
λi ∂ C gi (x̄)
(3.9)
i∈I (x̄)
holds, where I (x̄) = {i ∈ I : gi (x̄) = 0}. Moreover, suppose that ( f, g I (x̄) ) are V −invex at x̄. Then
x̄ is a global solution of (SIP).
Proof. The given condition is
0 ∈ ∂ C f (x̄) +
λi ∂ C gi (x̄),
i∈I (x̄)
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then there exist x̄ ∗ ∈ ∂ C f (x̄) and x̄i∗ ∈ ∂ C gi (x̄), such that
0 = x̄ ∗ +
λi x̄i∗ .
93
(3.10)
i∈I (x̄)
Let x ∈ S, then for each i ∈ I (x̄), we have
gi (x) ≤ 0 = gi (x̄), ∀i ∈ I (x̄) and ∀x ∈ S,
then,
gi (x) − gi (x̄) ≤ 0, ∀i ∈ I (x̄) and ∀x ∈ S.
(3.11)
By the V −invexity of ( f, g I (x̄) ) at x̄, we obtain,
f (x) − f (x̄) ≥ β(x, x̄) x̄ ∗ , η(x, x̄) , ∀x̄ ∗ ∈ ∂ C f (x̄) and ∀x ∈ S,
gi (x) − gi (x̄) ≥ αi (x, x̄) x̄i∗ , η(x, x̄) , ∀i ∈ I (x̄), ∀x ∈ S and ∀x̄i∗ ∈ ∂ C gi (x̄).
(3.12)
(3.13)
By (3.11), we obtain
0 ≥ gi (x) − gi (x̄) ≥ αi (x, x̄) x̄i∗ , η(x, x̄) , ∀i ∈ I (x̄), ∀x ∈ S and ∀x̄i∗ ∈ ∂ C gi (x̄),
(3.14)
that is,
0 ≥ αi (x, x̄) x̄i∗ , η(x, x̄) , ∀x ∈ S and ∀x̄i∗ ∈ ∂ C gi (x̄), ∀i ∈ I (x̄),
then, there exist λi ≥ 0, for all i ∈ I (x̄), we get,
λi x̄i∗ , η(x, x̄) ≤ 0, ∀x̄i∗ ∈ ∂ C gi (x̄), ∀x ∈ S.
(3.15)
(3.16)
i∈I (x̄)
Using (3.10), we get,
x̄ ∗ , η(x, x̄) ≥ 0, ∀x ∈ S and ∃ x̄ ∗ ∈ ∂ C f (x̄).
(3.17)
β(x, x̄) x̄ ∗ , η(x, x̄) ≥ 0, ∀x ∈ S and ∃ x̄ ∗ ∈ ∂ C f (x̄).
(3.18)
Then,
By (3.12), we get,
f (x) − f (x̄) ≥ 0, ∀x ∈ S.
Hence, we obtain the result.
This case is validated by the following example.
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Example 1. Examine the specified semi-infinite programming problem
(P) min f (x)
subject to gi (x) ≤ 0, ∀i ∈ N ,
and g j (x) ≤ 0, ∀ j ∈ N ,
x ∈ R2 ,
x1
1
1
, x2 = 0, gi (x) = x2 − x1 − and g j (x) = −x2 − . It is easy to verify that
x2
i
j
f (x), gi (x) and g j (x), for all i, j ∈ N all are V −invex functions with respect to β(x, y) =
y2
, αi (x, y) = α j (x, y) = 1, and η(x, y) = x − y, respectively, at for all x1 = x2 points in the feax2
sible set, where x = (x1 , x2 ), y = (y1 , y2 ) ∈ R2 .
Using the above theorem, we can easily to see that all the feasible points x = (x1 , x2 ) which are
like x1 = x2 are optimal solution of the problem.
where f (x) =
Theorem 2. Let x̄ ∈ S and I (x̄) = φ. Suppose that there exist λi ≥ 0, i ∈ I (x̄), with λi = 0, for
finitely many indexes, such that (3.9) holds. Moreover, Suppose that ( f, g I (x̄) ) is V −pseudo-quasiinvex at x̄. Then, x̄ is a global solution of (SIP).
Proof. Since (3.9) holds, then, there exist x̄ ∗ ∈ ∂ C f (x̄) and x̄i∗ ∈ ∂ C gi (x̄), for all i ∈ I (x̄), such
that,
λi x̄i∗ .
(3.19)
0 = x̄ ∗ +
i∈I (x̄)
Let x ∈ S. Then, for each i ∈ I (x̄), we have,
gi (x) ≤ 0 = gi (x̄), ∀x ∈ S.
(3.20)
Then, there exist αi (x, x̄) ∈ R+ \ {0}, such that,
αi (x, x̄)gi (x) ≤ αi (x, x̄)gi (x̄), ∀i ∈ I (x̄), ∀x ∈ S,
which implies that,
i∈I (x̄)
αi (x, x̄)gi (x) ≤
αi (x, x̄)gi (x̄), ∀x ∈ S.
(3.21)
i∈I (x̄)
Since ( f, g I (x̄) ) is V −pseudo-quasi-invex at x̄, then, there exist β(x, x̄), αi (x, x̄) ∈ R+ \ {0}, for all
i ∈ I (x̄) and for all x ∈ S, such that,
β(x, x̄) f (x) < β(x, x̄) f (x̄) ⇒ for any x̄ ∗ ∈ ∂ C f (x̄),
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(3.22)
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and
αi (x, x̄)gi (x) ≤
i∈I (x̄)
αi (x, x̄)gi (x̄) ⇒ for any x̄i∗ ∈ ∂ C gi (x̄),
i∈I (x̄)
x̄i∗ , η(x, x̄) ≤ 0.
i∈I (x̄)
(3.23)
Using (3.21) in (3.23), we get,
x̄i∗ , η(x, x̄) ≤ 0, ∀x̄i∗ ∈ ∂ C gi (x̄), ∀x ∈ S,
(3.24)
i∈I (x̄)
then, there exist λi ≥ 0, for all i ∈ I (x̄), we get,
λi x̄i∗ , η(x, x̄) ≤ 0, ∀x̄i∗ ∈ ∂ C gi (x̄), ∀x ∈ S.
(3.25)
i∈I (x̄)
By (3.19), we get,
x̄ ∗ , η(x, x̄) ≥ 0, for ∃ x̄ ∗ ∈ ∂ C f (x̄), ∀x ∈ S.
(3.26)
Assume f (x) < f (x̄), for all x ∈ S, and there exist β(x, x̄) ∈ R+ \ {0}, then,
β(x, x̄) f (x) < β(x, x̄) f (x̄), ∀x ∈ S.
(3.27)
x̄ ∗ , η(x, x̄) < 0, ∀x̄ ∗ ∈ ∂ C f (x̄), ∀x ∈ S.
(3.28)
By (3.22), we get
Which is contradiction of (3.26). Hence the result.
4. DUALITY
In this section, we can develop Wolfe and Mond-Weir type dual models and establish the weak,
strong and strict converse duality results for Wolfe type dual model and Mond-Weir type dual
model under the r −invexity and V −invexity assumptions, respectively. Also, we can derive these
results for Mond-Weir dual model in terms of generalized V −invexity assumptions.
Antczak [1] introduced a class of (scalar) Lipschitz r −invex functions, which is
Definition 1. [1] (r −invex functions) Let f : Rn → R be a Lipschitz function on a nonempty set
S ⊆ Rn , and let r be an arbitrary real number. If, for all x ∈ S, the inequality
1.
2.
≥ r1 er f (x̄) [1 + r x̄ ∗ , η(x, x̄) ] (> and x = x̄) for r = 0,
f (x) − f (x̄) ≥ x̄ ∗ , η(x, x̄) (> and x = x̄) for r = 0,
1 r f (x)
e
r
holds, then, f is said to be (strict) r −invex with respect to η at x̄ on S.
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4.1. Wolfe-type Dual
In this subsection, first of all, we develop a dual model for the nonsmooth semi-infinite programming
problem taken up using the Clarke’s subdifferentials, which is called the Wolfe-type dual problem
(WSID). After that, we get some fundamental results, which is a weak duality, strong duality and
strictly converse duality theorems.
We start with Wolfe-type dual problem which is given by
(WSID)
λi gi (u)
Maximize f (u) +
i∈I
subject to 0 ∈ ∂ C ( f (u) +
λi gi (u)),
i∈I
where λi ≥ 0, for i ∈ I and λi = 0, for finitely many i ∈ I (u).
The feasible solution of the (WSID) problem is denoted by W. Further, we denote by SW the
following set SW = {u ∈ Rn : (u, λ) ∈ W }.
Theorem 1. (Weak Duality) Let x be feasible solution of the (SIP) and (x̄, λ) be feasible for
(WSID). If the Lagrangian i.e., ( f (.) + i∈I λi gi (.)) is r −invex with respect to η at x̄ over S ∪ SW .
Then, we have, f (x) ≥ f (x̄) + i∈I λi gi (x̄).
Proof. Let x be feasible for (SIP) and (x̄, λ) be feasible for (WSID) and λi ≥ 0, for all i ∈ I.
Then, we have,
gi (x) ≤ 0, ∀i ∈ I,
which implies that,
λi gi (x) ≤ 0, ∀i ∈ I,
(4.1)
and
0 ∈ ∂ C ( f (x̄) +
λi gi (x̄)),
i∈I
then, there exist x̄ ∗ ∈ ∂ C f (x̄) and x̄i∗ ∈ ∂ C gi (x̄), for all i ∈ I, such that
0 = x̄ ∗ +
λi x̄i∗ .
(4.2)
i∈I
We start with the contradiction. Suppose that
f (x) < f (x̄) +
λi gi (x̄),
(4.3)
i∈I
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by (4.1) and (4.3), we get
f (x) +
λi gi (x) < f (x̄) +
i∈I
λi gi (x̄).
(4.4)
i∈I
By assumptions, the Lagrangian is r −invex with respect to η at x̄ over S ∪ SW , then,
1 r( f +
e
r
i∈I
λi gi )(x)
1
≥ er ( f +
r
i∈I
λi gi )(x̄)
∗
× 1 + r x̄ +
λi x̄i∗ , η(x, x̄)
, ∀x ∈ S ∪ SW .
(4.5)
i∈I
Then, by (4.4) and (4.5), we obtain
x̄ ∗ +
λi x̄i∗ , η(x, x̄) < 0,
(4.6)
i∈I
holds for each x̄ ∗ ∈ ∂ C f (x̄) and x̄i∗ ∈ ∂ C gi (x̄), for all i ∈ I. Which is the contradiction. Hence our
supposition is wrong. Hence,
f (x) ≥ f (x̄) +
λi gi (x̄).
i∈I
Hence the result.
The following theorem gives strong duality results between (SIP) and (WSID) under the
assumptions that the dual objective function is Clarke regular.
Theorem 2. (Strong Duality) Let x̄ be an optimal solution for (SIP) at which a constraint qualification is satisfied. Then, there exist λ̄ = (λ̄i ), i ∈ I, such that λ̄i gi (x̄) = 0, for all i ∈ I and (x̄, λ̄) is
feasible for (WSID) . If weak duality holds between (SIP) and (WSID), then, (x̄, λ̄) is optimal for
(WSID) and the respective objective values are equal.
Proof. Let x̄ is an optimal solution for (SIP) and a suitable constraints qualification is satisfied,
hence there exist λ̄ = (λ̄i ), i ∈ I such that the KKT conditions for (SIP) are satisfied. That is,
0 ∈ ∂ C ( f (x̄) +
λ̄i gi (x̄)),
(4.7)
i∈I
therefore (x̄, λ̃) is a feasible solution for (WSID).
By assumptions,
λ̄i gi (x̄) = 0.
(4.8)
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We proceed by contradiction. Suppose that (x̄, λ̄) is not an optimal solutions for (WSID). Then,
there exist at least one element of the feasible set in (WSID), which is ( ỹ, λ̃) such that
f (x̄) +
λ̄i gi (x̄) < f ( ỹ) +
i∈I
λ̃i gi ( ỹ),
(4.9)
i∈I
using (4.8), we get the inequality
f (x̄) < f ( ỹ) +
λ̃i gi ( ỹ),
(4.10)
i∈I
which contradicts the weak duality theorem for (WSID). Hence our supposition is wrong, i.e., (x̄, λ̄)
is an optimal solution for (WSID) and the respective objective values are equal. Hence the result.
Theorem 3. (Strict Converse Duality) Let x̃ be an optimal solutions for (SIP) and (x̄, λ̄) be an
optimal solution for (WSID). If a constraint qualification is satisfied for (SIP) and the Lagrange
function is strictly r −invex at x̄ on S ∪ SW , then, x̄ = x̃.
Proof. We proceed by contradiction. Assume x̃ = x̄. By strong duality theorem for (WSID) there
exist λ∗ = (λi∗ ), i ∈ I and λi∗ ≥ 0 such that (x̃, λ∗ ) is optimal for (WSID).
Hence,
λi∗ gi (x̃) = f (x̄) +
(4.11)
λ̄i gi (x̄).
f (x̃) = f (x̃) +
i∈I
i∈I
Again, by
0 ∈ ∂ C ( f (x̄) +
λ̄i gi (x̄)),
i∈I
then, there exist x̄ ∗ ∈ ∂ C f (x̄) and x̄i∗ ∈ ∂ C gi (x̄), such that
0 = x̄ ∗ +
λ̄i x̄i∗ .
(4.12)
i∈I
Now the Lagrangian function is strictly r −invex at x̄ on S ∪ SW , then, we get
1 r( f +
e
r
i∈I
λ̄i gi )(x)
1 r( f +
e
r
≥
i∈I
λ̄i gi )(x̄)
× 1 + r x̄ ∗ +
λ̄i x̄i∗ , η(x, x̄) , ∀x ∈ S ∪ SW . (4.13)
i∈I
then, by (4.12), we get
(f +
i∈I
λ̄i gi )(x) > ( f +
λ̄i gi )(x̄), ∀x ∈ S ∪ SW ,
(4.14)
i∈I
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then, for x̃ ∈ S, we get
(f +
λ̄i gi )(x̃) > ( f +
i∈I
λ̄i gi )(x̄),
(4.15)
i∈I
which implies that,
f (x̃) > ( f +
λ̄i gi )(x̄),
(4.16)
i∈I
which contradicts the equation (4.11). Hence our supposition is wrong. Hence x̃ = x̄. This completes the proof.
4.2. Mond-Weir type dual
In this subsection, first of all, we develop a dual model for the nonsmooth semi-infinite programming
problem taken up using the Clarke’s subdifferentials, which is called the Mond-Weir-type dual problem (MWSID). After that, we get some fundamental results, which are weak duality, strong duality
and strictly converse duality theorems.
We start with Mond-Weir-type dual problem which is given by
(MWSID)
Maximize f (u)
λi ∂ C gi (u),
subject to 0 ∈ ∂ C f (u) +
i∈I
and
λi gi (u) ≥ 0.
i∈I
Where λi ≥ 0, for i ∈ I and λi = 0, for finitely many i ∈ I (u).
Theorem 4. (Weak Duality) Let x be feasible for (SIP) and (u, λ I ) where λ I = (λi ), i ∈ I, be
feasible for (MWSID). Let ( f, g I ) be V −invex at u, where u ∈ Rn with respect to β : Rn × Rn →
R+ \ {0} and αi : Rn × Rn → R+ \ {0}, for every i ∈ I.
Then, we have, f (x) ≥ f (u).
Proof. Let x be feasible for (SIP) and (u, λ I ) be feasible for (MWSID). Then, we have
gi (x) ≤ 0, ∀i ∈ I,
λi ∂ C gi (u)
0 ∈ ∂ C f (u) +
(4.17)
(4.18)
i∈I
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and
λi gi (u) ≥ 0.
(4.19)
i∈I
Hence, there exist u ∗ ∈ ∂ C f (u) and u i∗ ∈ ∂ C gi (u), i ∈ I such that
λi u i∗ .
0 = u∗ +
(4.20)
i∈I
Since ( f, g I ) are V −invex at u with respect to η and (β, α I ) where β : Rn × Rn → R+ \ {0}, and
αi : Rn × Rn → R+ \ {0}, with λi ≥ 0, i ∈ I, we have
f (x) − f (u) ≥ β(x, u) u ∗ , η(x, u) , ∀u ∗ ∈ ∂ C f (u) and ∀x ∈ S,
then,
f (x) − f (u)
≥ u ∗ , η(x, u) , ∀u ∗ ∈ ∂ C f (u) and ∀x ∈ S.
β(x, u)
(4.21)
And
gi (x) − gi (u) ≥ αi (x, u) u i∗ , η(x, u) , ∀u i∗ ∈ ∂ C gi (u), ∀i ∈ I and ∀x ∈ S,
then,
gi (x) − gi (u)
≥ u i∗ , η(x, u) , ∀u i∗ ∈ ∂ C gi (u), ∀i ∈ I and ∀x ∈ S,
αi (x, u)
which implies that,
i∈I
λi
λi
gi (x) −
gi (u) ≥
λi u i∗ , η(x, u) , ∀u i∗ ∈ ∂ C gi (u) and ∀x ∈ S,
αi (x, u)
α
(x,
u)
i
i∈I
i∈I
we have,
λi
λi
gi (x) −
gi (u) ≥
λi u i∗ , η(x, u) , ∀u i∗ ∈ ∂ C gi (u) and ∀x ∈ S,
α
(x,
u)
α
(x,
u)
i
i
i∈I
i∈I
i∈I
(4.22)
from (4.17), we get
0≥
i∈I
λi
gi (u) +
λi u i∗ , η(x, u) , ∀u i∗ ∈ ∂ C gi (u) and ∀x ∈ S.
αi (x, u)
i∈I
(4.23)
Adding (4.21) and (4.23), we get
f (x) − f (u) λi
≥
gi (u) + u ∗ +
λi u i∗ , η(x, u) , ∀u i∗ ∈ ∂ C gi (u) and ∀x ∈ S,
β(x, u)
α
(x,
u)
i
i∈I
i∈I
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then, by (4.20), we get
f (x) − f (u) λi
≥
gi (u), ∀x ∈ S.
β(x, u)
αi (x, u)
i∈I
With the help of (4.19), we get,
f (x) − f (u)
≥ 0, ∀x ∈ S.
β(x, u)
Hence, we get the required results, which is
f (x) ≥ f (u).
This completes the proof.
Theorem 5. (Strong Duality) Let x̄ be an optimal solution for (SIP) at which a constraint qualification is satisfied. Then, there exist λ̄ = (λ̄i ), i ∈ I, such that λ̄i gi (x̄) = 0, for all i ∈ I and (x̄, λ̄) is
feasible for (MWSID) . If weak duality holds between (SIP) and (MWSID), then, (x̄, λ̄) is optimal
for (MWSID) and the respective objective values are equal.
Proof. From the KKT conditions, there exist λ̄ = (λ̄i ), i ∈ I, such that
0 ∈ ∂ C f (x̄) +
λ̄i ∂ C gi (x̄),
i∈I
and by assumptions,
λ̄i gi (x̄) = 0, i ∈ I.
Therefore, (x̄, λ̄) is feasible for (MWSID).
On the other hand by weak duality theorem, we have
f (x̄) ≥ f (u),
for any feasible solution (u, λ) for (MWSID).
Thus, (x̄, λ̄) is optimal for (MWSID) and the respective objective values are equal. Hence the
results.
Theorem 6. (Strict Converse Duality) Let ( f, g I ) are V −invex at x̄. let x̃ be an optimal solution
for (SIP) and (x̄, λ̄) be an optimal solution for (MWSID). If a constraint qualification is satisfied for
(SIP), and ( f, g I ) is V −strictly-invex for (SIP) at x̄, then, x̃ = x̄.
Proof. We proceed by contradiction. Assume x̃ = x̄. By strong duality theorem for (MWSID) there
exist λ̄ = (λ̄i ), i ∈ I and λ̄i ≥ 0 such that (x̄, λ̄) is optimal for (MWSID).
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Hence,
f (x̃) = f (x̄).
(4.24)
Again, by (4.2) there exist x̄ ∗ ∈ ∂ C f (x̄) and x̄i∗ ∈ ∂ C gi (x̄) such that
0 = x̄ ∗ +
λ̄i x̄i∗ .
(4.25)
i∈I
Now by V −strict-invexity of ( f, g I ) at x̄, with λi ≥ 0, such that
f (x̃) − f (x̄) > β(x̃, x̄) x̄ ∗ , η(x̃, x̄) , ∀x̄ ∗ ∈ ∂ C f (x̄) and x̃ ∈ S,
then,
f (x̄)
f (x̃)
−
> x̄ ∗ , η(x̃, x̄) , ∀x̄ ∗ ∈ ∂ C f (x̄) and x̃ ∈ S,
β(x̃, x̄) β(x̃, x̄)
(4.26)
and
gi (x̃) − gi (x̄) ≥ αi (x̃, x̄) x̄i∗ , η(x̃, x̄) , ∀x̄i∗ ∈ ∂ C gi (x̄), ∀i ∈ I and x̃ ∈ S,
then,
gi (x̄)
gi (x̃)
−
≥ x̄i∗ , η(x̃, x̄) , ∀x̄i∗ ∈ ∂ C gi (x̄), ∀i ∈ I and x̃ ∈ S,
αi (x̃, x̄) αi (x̃, x̄)
since λ̄i ≥ 0, then, we get,
λ̄i gi (x̃)
λ̄i gi (x̄)
−
≥ λ̄i x̄i∗ , η(x̃, x̄) , ∀x̄i∗ ∈ ∂ C gi (x̄), ∀i ∈ I and x̃ ∈ S,
αi (x̃, x̄) αi (x̃, x̄)
then,
λ̄i gi (x̃) λ̄i gi (x̄)
−
≥
λ̄i x̄i∗ , η(x̃, x̄) , ∀x̄i∗ ∈ ∂ C gi (x̄) and x̃ ∈ S.
α
(
x̃,
x̄)
α
(
x̃,
x̄)
i
i
i∈I
i∈I
i∈I
(4.27)
As, gi (x̃) ≤ 0 and λ̄i ≥ 0, we have
0≥
λ̄i gi (x̄)
+
λ̄i x̄i∗ , η(x̃, x̄) , ∀x̄i∗ ∈ ∂ C gi (x̄) and x̃ ∈ S.
α
(
x̃,
x̄)
i
i∈I
i∈I
(4.28)
By Adding (4.26) and (4.28), we get,
λ̄i gi (x̄)
f (x̃)
f (x̄)
−
>
+ x̄ ∗ +
λ̄i x̄i∗ , η(x̃, x̄) , ∀x̄i∗ ∈ ∂ C gi (x̄) and x̃ ∈ S.
β(x̃, x̄) β(x̃, x̄)
α
(
x̃,
x̄)
i
i∈I
i∈I
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Then, by using (4.25), we get
λ̄i gi (x̄)
f (x̃)
f (x̄)
−
>
, x̃ ∈ S.
β(x̃, x̄) β(x̃, x̄)
αi (x̃, x̄)
i∈I
By the feasibility of (MWSID), we get
f (x̃)
f (x̄)
−
> 0, x̃ ∈ S,
β(x̃, x̄) β(x̃, x̄)
(4.29)
then,
f (x̃) > f (x̄).
Which is contradiction. Hence x̃ = x̄. This completes the proof.
4.3. Mond-Weir duality using generalized V − invexity
In this subsection, we prove weak duality, strong duality and strict converse duality results under
the generalized V −invex functions.
Theorem 7. (Weak Duality) Let x be feasible for (SIP) and (u, λ) be feasible for (MWSID). If
( f, λ I g I ) is V −pseudo-quasi-invex at u. Then, f (x) ≥ f (u).
Proof. Since x be feasible for (SIP), then, gi (x) ≤ 0, for all i ∈ I, λi ≥ 0, for i ∈ I and λi = 0, for
finitely many i ∈ I (u). Then,
λi gi (x) ≤ 0,
then, there exist αi (x, u) ∈ R+ \ {0} such that,
αi (x, u)λi gi (x) ≤ 0, ∀i ∈ I, ∀x ∈ S,
then,
αi (x, u)λi gi (x) ≤ 0, ∀x ∈ S.
(4.30)
i∈I
Since, (u, λ) be feasible for (MWSID) then,
λi gi (u) ≥ 0, ∀i ∈ I,
then,
αi (x, u)λi gi (u) ≥ 0, ∀i ∈ I, ∀x ∈ S,
which implies that,
αi (x, u)λi gi (u) ≥ 0, ∀x ∈ S.
(4.31)
i∈I
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From (4.30) and (4.31), we get,
αi (x, u)λi gi (x) ≤
αi (x, u)λi gi (u), ∀x ∈ S,
i∈I
(4.32)
i∈I
Since ( f, λ I g I ) is V −pseudo-quasi-invex at u, then, there exist β(x, u), αi (x, u) ∈ R+ \ {0}, for all
i ∈ I and for all x ∈ S, such that,
β(x, u) f (x) < β(x, u) f (u) ⇒ for any u ∗ ∈ ∂ C f (u),
and
αi (x, u)λi gi (x) ≤
i∈I
u ∗ , η(x, u) < 0
αi (x, u)λi gi (u) ⇒ for any u i∗ ∈ ∂ C gi (u),
i∈I
(4.33)
λi u i∗ , η(x, u) ≤ 0.
i∈I
(4.34)
From (4.32), we get,
λi u i∗ , η(x, u) ≤ 0, ∀u i∗ ∈ ∂ C gi (u), ∀x ∈ S.
(4.35)
i∈I
Let,
0 ∈ ∂ C f (u) +
λi ∂ C gi (u),
i∈I
then, there exist u ∗ ∈ ∂ C f (u) and u i∗ ∈ ∂ C gi (u), i ∈ I, such that,
0 = u∗ +
λi u i∗ ,
i∈I
then,
0 = u ∗ , η(x, u) +
λi u i∗ , η(x, u) , ∀x ∈ S.
i∈I
Then, by (4.35), we get,
u ∗ , η(x, u) ≥ 0, for some u ∗ ∈ ∂ C f (u), ∀x ∈ S.
Assume that f (x) < f (u), since β(x, u) ∈ R+ \ {0}, we get,
β(x, u) f (x) < β(x, u) f (u), ∀x ∈ S.
Then, by (4.33), we get,
u ∗ , η(x, u) < 0, for any u ∗ ∈ ∂ C f (u), ∀x ∈ S.
Which is contradiction. Hence the results.
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Theorem 8. (Strong Duality) Let x̄ be an optimal solution for (SIP) at which a constraint qualification is satisfied. Then, there exist λ̄ = (λ̄i ), i ∈ I, such that λ̄i gi (x̄) = 0, for all i ∈ I and (x̄, λ̄) is
feasible for (MWSID). If weak duality holds between (SIP) and (MWSID), then, (x̄, λ̄) is optimal
for (MWSID) and the respective objective values are equal.
Proof. From the KKT conditions, there exist λ̄ = (λ̄i ), i ∈ I, such that
0 ∈ ∂ C f (x̄) +
λ̄i ∂ C gi (x̄),
i∈I
and by assumptions,
λ̄i gi (x̄) = 0, i ∈ I.
Therefore, (x̄, λ̄) is feasible for (MWSID).
On the other hand by weak duality theorem, we have
f (x̄) ≥ f (u),
for any feasible solution (u, λ) for (MWSID).
Thus, (x̄, λ̄) is optimal for (MWSID) and the respective objective values are equal. Hence the
results.
Theorem 9. (Strict Converse Duality) Let (SIP) have an optimal solution x̃ at which a constraint
qualification is satisfied. Assume that ( f, λ¯I g I ) are V −pseudo-quasi-invex function. If (x̄, λ̄) is an
optimal solution for (MWSID) and ( f, λ¯I g I ) is V −strictly-pseudo-quasi-invex at x̄, then, x̄ = x̃.
Proof. We prove this by the contradiction. Assume that x̃ = x̄. Then, by strong duality theorem
there exist λ∗ = (λi∗ ), i ∈ I such that (x̃, λ∗ ) is optimal for (MWSID). Hence
f (x̃) = f (x̄).
(4.36)
Since x̃ is an optimal solution for (SIP) then, gi (x̃) ≤ 0, for all i ∈ I, λ̄i ≥ 0, for i ∈ I and λ̄i = 0,
for finitely many i ∈ I (x̄). Then,
λ̄i gi (x̃) ≤ 0,
and there exist αi (x, x̄) ∈ R+ \ {0}, such that,
αi (x, x̄)λ̄i gi (x̃) ≤ 0, ∀i ∈ I, ∀x ∈ S,
then,
αi (x, x̄)λ̄i gi (x̃) ≤ 0, ∀x ∈ S.
(4.37)
i∈I
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Since (x̄, λ̄) is an optimal solution for (MWSID) then,
λ̄i gi (x̄) ≥ 0, ∀i ∈ I,
then,
αi (x, x̄)λ̄i gi (x̄) ≥ 0, ∀i ∈ I, ∀x ∈ S,
which implies that,
αi (x, x̄)λ̄i gi (x̄) ≥ 0, ∀x ∈ S.
(4.38)
i∈I
From (4.37) and (4.38), we get,
αi (x, x̄)λ̄i gi (x̃) ≤
αi (x, x̄)λ̄i gi (x̄), ∀x ∈ S,
i∈I
i∈I
since x̃ ∈ S, then,
αi (x̃, x̄)λ̄i gi (x̃) ≤
i∈I
αi (x̃, x̄)λ̄i gi (x̄),
(4.39)
i∈I
Since ( f, λ¯I g I ) is V −strictly-pseudo-quasi-invex at x̄, then, there exist β(x, x̄), αi (x, x̄) ∈ R+ \
{0}, for all i ∈ I and for all x ∈ S, such that,
β(x, x̄) f (x) ≤ β(x, x̄) f (x̄) ⇒ for any x̄ ∗ ∈ ∂ C f (x̄),
and
αi (x, x̄)λ̄i gi (x) ≤
i∈I
x̄ ∗ , η(x, x̄) < 0
αi (x, x̄)λ̄i gi (x̄) ⇒ for any x̄i∗ ∈ ∂ C gi (x̄),
i∈I
(4.40)
λ̄i x̄i∗ , η(x, x̄) ≤ 0.
i∈I
(4.41)
From (4.39), we get,
λ̄i x̄i∗ , η(x̃, x̄) ≤ 0, ∀x̄i∗ ∈ ∂ C gi (x̄), x̃ ∈ S.
(4.42)
i∈I
Let,
0 ∈ ∂ C f (x̄) +
λi ∂ C gi (x̄),
i∈I
then, there exist x̄ ∗ ∈ ∂ C f (x̄) and x̄i∗ ∈ ∂ C gi (x̄), i ∈ I, such that,
0 = x̄ ∗ +
λi x̄i∗ ,
i∈I
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then,
0 = x̄ ∗ , η(x, x̄) +
λ̄i x̄i∗ , η(x, x̄) , ∀x ∈ S,
i∈I
which implies that,
0 = x̄ ∗ , η(x̃, x̄) +
λ̄i x̄i∗ , η(x̃, x̄) , x̃ ∈ S.
i∈I
Then, by (4.42), we get,
x̄ ∗ , η(x̃, x̄) ≥ 0, for some x̄ ∗ ∈ ∂ C f (x̄), x̃ ∈ S.
(4.43)
Assume f (x̃) ≤ f (x̄), then,
β(x̃, x̄) f (x̃) ≤ β(x̃, x̄) f (x̄), for x̃ ∈ S,
then, by (4.40), we get
x̄ ∗ , η(x̃, x̄) < 0, ∀x̄ ∗ ∈ ∂ C f (x̄), for x̃ ∈ S,
which is contradiction of (4.43), then, this is not possible.
Hence,
f (x̃) > f (x̄),
which is a contradiction of (4.36), i.e., contradiction of x̃ = x̄. Hence the results.
5. CONCLUSIONS
In this paper, we have taken a semi-infinite programming problem involving V −invex and generalized V −invex functions. We have derived sufficient optimality conditions under V −invexity and
generalized V −invexity both using the help of Clarke’s subdifferential. We have formulated Wolfe
type duality model and Mond-Weir type duality model and gave weak duality, strong duality and
strict converse duality results for r −invex and V −invex functions with the use of Clarke’s subdifferential.
ACKNOWLEDGEMENTS
The authors are thankful to the anonymous referees of this paper for their useful suggestions which
helped to modify the papers in its present form. The research of the first author is supported by
the Council of Scientific and Industrial Research, New Delhi, Ministry of Human Resources Development, Government of India Grant [09/013(0672)/2017 − EMR − I]. The research of the second
author is supported by UGC-BSR start up grant by University grant Commission, New Delhi, India
(Letter No. F. 30 − 370/2017(BSR)) (Project No. M−14 − 40).
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INDIAN JOURNAL OF INDUSTRIAL AND APPLIED MATHEMATICS
Vol. 10, No. 1 (Special Issue), Jan–June 2019, pp. 109–120
DOI:
A Strict Constraint Qualification in
Vector Optimization
Muskan Kapoor1∗ and C.S. Lalitha2
1
Department of Mathematics, University of Delhi, Delhi-110007, India
Department of Mathematics, University of Delhi South Campus, New Delhi-110021, India
(∗ Corresponding author) Email: ∗ muskankapoor22@yahoo.com, 2 cslalitha@maths.du.ac.in
2
Abstract: The main aim of this paper is to present the notion of cone-continuity property introduced by Andreani et al. (Andreani, R., Martı́nez, J.M., Ramos, A. and Silva, P.J.S., A conecontinuity constraint qualification and algorithmic consequences, SIAM J. Optim., 26(2016), 96–
110) in the context of vector optimization. We extend this property and establish it to be a strict
regularity condition while dealing with efficient solutions. This property is the weakest condition
which ensures that weak approximate Karush-Kuhn-Tucker conditions imply weak Karush-KuhnTucker conditions. Further, cone-continuity is established to be a strict constraint qualification while
dealing with weak and proper efficient solutions.
Keywords: Vector optimization, Karush-Kuhn-Tucker conditions, approximate Karush-KuhnTucker conditions, strict constraint qualification, cone-continuity property
1. INTRODUCTION
The problem of identifying an optimal solution of a nonlinear programming problem led to the
development of various types of optimality conditions. These conditions are also important in
designing numerical methods for finding the solutions. One such optimality conditions namely Fritz
John optimality conditions are satisfied at an optimal solution, but there are cases for which the corresponding multiplier of the objective function is zero. It is important to ensure the positivity of the
multiplier corresponding to the objective function as this leads to the identification of the solutions
of the problem. This is achieved by imposing certain assumptions on the constraints called constraint qualification (CQ). The optimality conditions with positive multipliers are referred to as the
Karush-Kuhn-Tucker (KKT) conditions.
The main approach to derive optimality in vector optimization has been through scalarization
of the vector optimization problem. This is achieved by translating the vector optimization problem
into a scalar optimization problem and by using the optimality conditions of the scalar problem to
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Muskan Kapoor and C.S. Lalitha
deduce KKT type conditions for the vector problem. Hence, in such cases it becomes a necessity to
impose conditions on the constraints and on all the objective functions except possibly on one, while
deriving KKT conditions from Fritz John conditions. In this situation one uses the term regularity
condition rather than constraint qualification (see [5, 9, 13]). When at least one of the multipliers,
corresponding to the objective functions, is positive we say that weak KKT conditions hold for
the problem. On the other hand strong KKT conditions are said to hold when all the multipliers
corresponding to the objective functions are positive.
Practical methods for solving constrained optimization problems are usually iterative. At each
iteration, one must decide whether it is sensible to terminate the execution of the algorithm or not.
Since testing true optimality is very difficult, the obvious idea is to terminate when a necessary
optimality condition is approximately satisfied. In literature there are many papers dealing with
sequential optimality conditions in vector optimization (see [2–4,7,14]). Such conditions are termed
as Approximate Karush-Kuhn-Tucker (AKKT) conditions in [2–4] which can be tested by most of
numerical optimization solvers. Sequential optimality conditions provide adequate theoretical tools
to justify stopping criteria. A local minimizer can be approximated by a sequence of AKKT points.
Strict constraint qualification (SCQ) is a property of feasible points that guarantees AKKT points to
satisfy KKT conditions. Andreani et al. [2,3] introduced relaxed constant positive linear dependence
and constant positive generator strict constraint qualifications. Andreani et al. [4] introduced a conecontinuity property (CCP) and showed it to be weakest possible SCQ that guarantees AKKT implies
KKT for constrained optimization problems.
The main aim of the paper is to bring the cone-continuity property into the scenario of vector
optimization both as a strict regularity and strict constraint qualification. This is done by using some
well-known scalarization schemes available in literature for characterizing efficient, weak efficient
and properly efficient solutions.
The paper is organized as follows. In Section 2, we provide the cone-continuity notion for
scalar problems and scalarization schemes for a vector optimization problem with both inequality
and equality constraints. In Section 3, we introduce weak AKKT and a cone-continuity property
for the vector problem and establish that cone-continuity is a strict regularity condition for efficient
solutions. Also, we establish that this property is the weakest condition which ensures that weak
AKKT implies weak KKT conditions. In Section 4, we establish that the cone-continuity considered
in [4] is a strict constraint qualification while considering weak and proper efficient solutions.
2. PRELIMINARIES
We first consider the following scalar minimization problem
(CP)
minimize f (x)
subject to gi (x) ≤ 0, i = 1, 2, . . . , m,
h i (x) = 0, i = 1, 2, . . . , l,
where f : Rn → R, g : Rn → Rm , h : Rn → Rl are continuously differentiable on Rn . Let
S = {x ∈ Rn : gi (x) ≤ 0, i = 1, 2, . . . , m, h i (x) = 0, i = 1, 2, . . . , l}
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be the feasible set of (CP). For a point x ∈ S we denote the set of active indices of g by I (x), that
is, I (x) = {i ∈ {1, 2, . . . , m} : gi (x) = 0}.
We next recall sequential optimality AKKT conditions [1, 4, 6] for problem (CP).
Definition 2.1 [4, Definition 1.2]. A point x̄ ∈ S is said to satisfy AKKT conditions if there exist
k
l
k
sequences {x k } ⊆ Rn , {α k } ⊆ Rm
+ , {μ } ⊆ R such that x → x̄,
lim
k→∞
∇ f (x k ) +
m
i=1
k
αik ∇gi (x k ) +
l
μik ∇h i (x k ) = 0,
(2.1)
i=1
lim min{αik , −gi (x )} = 0, ∀ i = 1, 2, . . . , m.
k→∞
The AKKT conditions have been rewritten in an equivalent form [1, 4] as follows.
Lemma 2.1 [1]. The AKKT conditions hold at x̄ ∈ S if and only if there exist sequences {x k } ⊆ Rn ,
k
k
l
/ I (x̄) such that x k → x̄ and (2.1) holds.
{α k } ⊆ Rm
+ , {μ } ⊆ R with αi = 0 for i ∈
It is known that every local minimizer of (CP) satisfies AKKT conditions. One may refer to
Theorem 2.1 in [1] for more details.
We next recall the notion of the cone-continuity property (CCP) introduced by Andreani et
al. [4] to establish that it is the weakest possible strict constraint qualification.
Definition 2.2 [4]. A point x̄ ∈ S satisfies cone-continuity property (CCP) if the set-valued mapping
K : Rn → Rn defined by
K (x) :=
⎧
⎨
⎩
i∈I (x̄)
αi ∇gi (x) +
l
i=1
⎫
⎬
μi ∇h i (x) : αi ∈ R+ , i ∈ I (x̄), μi ∈ R, i = 1, 2, . . . , l ,
⎭
is outer semicontinuous at x̄, that is, lim sup K (x) ⊆ K (x̄) where
x→x̄
lim sup K (x) := { ȳ ∈ Rn : ∃ (x k , y k ) → (x̄, ȳ) such that y k ∈ K (x k )}.
x→x̄
The following theorem establishes the fact that CCP is the weakest SCQ which guarantees that
AKKT implies KKT for the constrained scalar problem (CP).
Theorem 2.1 [4, Definition 3.2]. The CCP condition at x̄ ∈ S is the weakest property under which
AKKT conditions at x̄ implies KKT conditions at x̄, for every continuously differentiable objective
function f that attains a minimum at x̄.
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We study the role of cone-continuity property in vector optimization. We consider the following
vector optimization problem
(VP) minimize f (x) = ( f 1 (x), f 2 (x), . . . , f p (x))
subject to x ∈ S
where f : Rn → R p , g : Rn → Rm , h : Rn → Rl , are continuously differentiable functions.
We now give the following well-known solution concepts in vector optimization.
Definition 2.3. A point x̄ ∈ S is said to be
1.
2.
an efficient solution of (VP) if there does not exist any other x ∈ S such that fi (x) ≤ f i (x̄),
for all i = 1, 2, . . . , p, fr (x) < fr (x̄), for some r ;
a weak efficient solution of (VP) if there does not exist any x ∈ S such that fi (x) < f i (x̄),
for all i = 1, 2, . . . , p.
The optimality conditions for a vector problem are usually established through the scalar problem associated with it. The efficient solution of (VP) having p objectives can be characterized in
terms of the optimal solutions of p scalar problems considered by Chankong and Haimes [10].
Lemma 2.2 [10]. A point x̄ ∈ S is an efficient solution of (VP) if and only if x̄ is an optimal solution
of (Pr (x̄)) for each r = 1, 2, . . . , p, where (Pr (x̄)) is defined as
(Pr (x̄)) minimize fr (x)
subject to f i (x) ≤ f i (x̄), for all i = 1, 2, . . . , p, i = r,
gi (x) ≤ 0, i = 1, 2, . . . , m,
h i (x) = 0, i = 1, 2, . . . , l.
The weak efficient solutions have been characterized by the optimal solutions of the associated
weighted scalar problem. For this we first recall the notion of cone convexlike function from Illés
and Kassay [12] by taking cone as the nonnegative orthant in the finite dimensional setting.
Definition 2.4 [12]. A function f : Rn → R p is said to be convexlike on Rn if for there exists
t ∈ [0, 1] such that for every x1 , x2 ∈ Rn , there exists x3 ∈ Rn such that
p
(1 − t) f (x1 ) + t f (x2 ) − f (x3 ) ∈ R+ .
The following lemma is a consequence of the Alternative Theorem 3.1 in Illés and Kassay [12]
for cone convexlike by taking the cone as nonegative orthant.
Lemma 2.3. Let f be convexlike on Rn . A point x̄ ∈ S is a weak efficient solution of (VP) if and
p
only if there exists λ ∈ R+ \ {0} such that x̄ minimizes the following scalar problem
(Pλ )
minimize
p
λi f i (x)
i=1
subject to x ∈ S.
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A notion of proper efficiency has been introduced by Geoffrion [11] for a vector maximization
problem which is now formulated for the minimization problem (VP).
Definition 2.5 [11]. A point x̄ ∈ S is a properly efficient solution of (VP) if it is efficient and if
there exists a scalar M > 0 such that for each i and x ∈ S satisfying fi (x) < f i (x̄) we have
f i (x̄) − f i (x)
≤M
f j (x) − f j (x̄)
for some j satisfying f j (x) > f j (x̄).
Now we formally state the Geoffrion criteria for scalarization in the form of the following
lemma.
Lemma 2.4 [11]. Let S be a convex set and fi , for i = 1, 2, . . . , p be convex on S. Then x̄ ∈ S
is properly efficient solution of (VP) if and only if x̄ is optimal for (Pλ ) for some λ with strictly
positive components.
The multipliers corresponding to all the components of the objective functions are not always
positive. Two types of KKT conditions, namely weak and strong KKT conditions, are usually considered in literature while dealing with a vector optimization problem.
Definition 2.6 [8]. A point x̄ ∈ S is said to satisfy
1.
p
l
weak KKT conditions if there exist λ ∈ R+ \ {0}, α ∈ Rm
+ , μ ∈ R such that
p
i=1
λi ∇ f i (x̄) +
m
αi ∇gi (x̄) +
i=1
l
μi ∇h i (x̄) = 0
αi gi (x̄) = 0, ∀ i = 1, 2, . . . , m;
2.
(2.2)
i=1
(2.3)
p
l
strong KKT conditions if there exist λ ∈ intR+ , α ∈ Rm
+ , μ ∈ R such that (2.2) and (2.3)
hold.
3. CONE-CONTINUITY PROPERTY AS STRICT REGULARITY
CONDITION
In this section we establish that cone-continuity property is the weakest property under which weak
AKKT conditions imply weak KKT conditions. Moreover, we prove that cone-continuity property
is a strict regularity condition, that is, they guarantee that weak AKKT implies KKT optimality
conditions for (VP).
In view of Lemma 2.1 we define a notion of weak AKKT conditions for (VP).
Definition 3.1. Let x̄ ∈ S. The sequential weak AKKTr conditions with respect to r ∈ {1, 2, . . . , p}
p−1
k
k
l
hold at x̄, if there exist sequences {x k } ⊆ Rn , λk ⊆ R+ , {α k } ⊆ Rm
+ , {μ } ⊆ R with αi = 0 for
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i∈
/ I (x̄) such that x k → x̄,
⎡
⎤
⎢
lim ⎣∇ fr (x k ) +
λik ∇ f i (x k ) +
p
k→∞
m
αik ∇gi (x k ) +
i=1
i=1
i =r
l
⎥
μik ∇h i (x k )⎦ = 0.
i=1
Since the cone-continuity property defined below involves all objective function except the r th
objective for some r ∈ {1, 2, . . . , p} the cone-continuity property CCPr will be shown to be a strict
regularity condition.
Definition 3.2. A point x̄ ∈ S satisfies cone-continuity property (CCPr ) with respect to r ∈
{1, 2, . . . , p}, if the set-valued mapping K r : Rn → Rn defined by
⎧
⎫
⎪
⎪
p
l
⎨
⎬
λi ∇ f i (x) +
αi ∇gi (x) +
μi ∇h i (x) : λi ∈ R+ , αi ∈ R+ , μi ∈ R ,
K r (x) :=
⎪
⎪
⎩ i=1
⎭
i∈I (x̄)
i=1
i =r
is outer semicontinuous at x̄, that is, lim sup K r (x) ⊆ K r (x̄).
x→x̄
We now give an example to illustrate the cone-continuity property.
Example 3.1. Consider the vector optimization where the functions are defined on R2
minimize f (x1 , x2 ) = ( f 1 (x1 , x2 ), f 2 (x1 , x2 ))
2
= (x13 , (x1 )2 e(x2 ) )
subject to g1 (x1 , x2 ) = x1 e x2 ≤ 0.
The constraints are active at the feasible point x̄ = (0, 0) and
K 1 (x) = {λ2 ∇ f 2 (x) + α1 ∇g1 (x) : λ2 ≥ 0, α1 ≥ 0}
2
2
= {(2λ2 x1 e(x2 ) + α1 e x2 , 2λ2 (x1 )2 x2 e(x2 ) + α1 x1 e x2 ) : λ2 ≥ 0, α1 ≥ 0},
K 2 (x) = {λ1 ∇ f 1 (x) + α1 ∇g1 (x) : λ1 ≥ 0, α1 ≥ 0}
= {(3λ1 (x1 )2 + α1 e x2 , α1 x1 e x2 ) : λ1 ≥ 0, α1 ≥ 0}.
Hence, K 1 (x̄) = K 2 (x̄) = R+ × {0}.
We now show that x̄ satisfies CCP2 . Let z̄ = (z̄ 1 , z̄ 2 ) = lim sup K 2 (x), hence there exists a
x→x̄
sequence (x1k , x2k , z 1k , z 2k ) → (0, 0, z̄ 1 , z̄ 2 ) such that
k
k
(z 1k , z 2k ) = (3λk1 (x1k )2 + α1k e x2 , α1k x1k e x2 )
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for some λk1 ≥ 0, α1k ≥ 0. We next show that z̄ ∈ K 2 (x̄). Suppose on the contrary that z̄ ∈
/ K 2 (x̄).
Then, from (3.1) it is clear that z̄ 2 = 0 and hence for sufficiently large k we have
k
|z 2k | = α1k |x1k |e x2 >
|z̄ 2 |
> 0.
2
Thus, from (3.1) it follows that
k
k
z 1k = 3λk1 (x1k )2 + α1k e x2 ≥ α1k e x2 >
|z̄ 2 |
.
2|x1k |
Taking limits we obtain z 1k → ∞, which is a contradiction. Thus, x̄ satisfies CCP2 .
However, in this example we show that x̄ fails to satisfy CCP1 . Define x1k = − 1k , x2k = k1 , λk2 =
k
k, α1 = k1 . Then for every k we have
k 2
k
k 2
k
(y1k , y2k ) = (2λk2 x1k e(x2 ) + α1k e x2 , 2λk2 (x1k )2 x2k e(x2 ) + α1k x1k e x2 )
1 1 2 1 2
1 1
1 2
= −2e( k ) + e k , 2 e( k ) − 2 e k ∈ K 1 (x k )
k
k
k
but its limit (−2, 0) ∈
/ K 1 (x̄).
Theorem 3.1. If CCPr condition holds at x̄ ∈ S for some r ∈ {1, 2, . . . , p} then sequential AKKTr
at x̄ implies weak KKT at x̄ with λr > 0.
Proof. Let us first show that if x̄ satisfies CCPr , that is, lim sup K r (x) ⊆ K r (x̄), then the sequential
x→x̄
AKKTr condition at x̄ implies the weak KKT condition at x̄ independently of the objective function.
Let fr be the objective function such that sequential AKKTr condition holds at x̄, then there are
p−1
k
l
k
sequences {x k } ⊆ Rn , {λk } ⊆ R+ , {α k } ⊆ Rm
+ , {μ } ⊆ R such that x → x̄,
⎡
⎤
⎢
lim ⎣∇ fr (x k ) +
λik ∇ f i (x k ) +
p
k→∞
m
αik ∇gi (x k ) +
i=1
i=1
i =r
l
⎥
μik ∇h i (x k )⎦ = 0.
i=1
Then
yrk =
p
λik ∇ f i (x k ) +
m
αik ∇gi (x k ) +
l
i=1
i=1
i =r
μik ∇h i (x k ) ∈ K r (x k ).
i=1
Since x̄ satisfies CCPr we have
lim yrk = lim (−∇ fr (x k )) = −∇ fr (x̄) ∈ K r (x̄)
k→∞
k→∞
that is, x̄ satisfies weak KKT condition at x̄ with λr > 0.
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Using the above theorem we show that cone-continuity property CCPr is a strict regularity
condition.
Theorem 3.2. If x̄ is an efficient solution of (VP), CCPr condition holds at x̄ for some r ∈
{1, 2, . . . , p} then x̄ satisfies weak KKT condition with λr > 0.
Proof. By Lemma 2.2 it is clear that x̄ is an optimal solution of (Pr (x̄)) which implies that x̄ satisfies
AKKTr and hence the conclusion follows from Theorem 3.1.
Theorem 3.3. If x̄ is an efficient solution of (VP), CCPr condition holds at x̄ for each r =
1, 2, . . . , p then x̄ satisfies strong KKT condition.
Proof. By Theorem 3.2 it follows that x̄ satisfies weak KKT condition for each r = 1, 2, . . . , p,
p
r
l
r
hence there exist λr ∈ R+ \ {0}, αr ∈ Rm
+ , μ ∈ R such that λr > 0,
p
i=1
λri ∇ f i (x̄) +
m
αir ∇gi (x̄) +
i=1
l
μri ∇h i (x̄) = 0,
i=1
αir gi (x̄) = 0, ∀ i = 1, 2, . . . , m.
Summing the above inequalities over r and setting λi =
p
r =1
λri , αi =
that x̄ satisfies strong KKT conditions with multipliers λ, α and μ.
p
r =1
αir , μi =
p
r =1
μri , it follows
Remark 3.1. Since a properly efficient solution is an efficient solution the above two theorems also
hold for properly efficient solutions.
The next theorem establishes that CCPr is the weakest condition which guarantees that weak
AKKT implies weak KKT conditions.
Theorem 3.4. If weak AKKTr implies weak KKT condition with λr > 0 for every continuously
differentiable function fr that attains a minimum at x̄ ∈ S then CCPr condition holds at x̄.
Proof. Let ȳr ∈ lim sup K r (x), hence there exists a sequence (x k , yrk ) → (x̄, ȳr ) such that yrk ∈
x→x̄
K r (x k ). Define fr (x) = − ȳr , x for all x ∈ Rn . Since ∇ fr (x k ) + yrk = − ȳr + yrk → 0 it follows
that sequential AKKTr holds at x̄ and hence by hypothesis we have that weak KKT condition holds
at x̄ with λr > 0 which further implies that −∇ fr (x̄) ∈ K r (x̄), that is, ȳr ∈ K r (x̄).
4. CONE-CONTINUITY PROPERTY AS STRICT CONSTRAINT
QUALIFICATION
While dealing with weak efficient solutions it is known that the scalarization scheme of Chankong
and Haimes [10] fails to hold. In order to show that cone-continuity is a strict constraint qualification
p
or a strict regularity condition we need to consider scalarized problem (Pλ ) for some λ ∈ R+ \ {0}.
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In such situations one need to consider CCP rather than CCPr . Also, there are situations where CCP
holds but CCPr fails to hold as illustrated in the following example.
Example 4.1. Consider the vector optimization where the functions are defined on R2
minimize f (x1 , x2 ) = ( f 1 (x1 , x2 ), f 2 (x1 , x2 ))
= (−x1 , −(x1 x2 )2 )
subject to g1 (x1 , x2 ) = x13 ≤ 0
g2 (x1 , x2 ) = x1 e x2 ≤ 0.
Both the constraints are active at the feasible point x̄ = (0, 0) and
K (x) = {α1 ∇g1 (x) + α2 ∇g2 (x) : α1 ≥ 0, α2 ≥ 0}
= {(3α1 (x1 )2 + α2 e x2 , α2 x1 e x2 ) : α1 ≥ 0, α2 ≥ 0},
K 1 (x) = {λ2 ∇ f 2 (x) + α1 ∇g1 (x) + α2 ∇g2 (x) : λ2 ≥ 0, α1 ≥ 0, α2 ≥ 0}
= {(−2λ2 x1 (x2 )2 + 3α1 (x1 )2 + α2 e x2 , −2λ2 (x1 )2 x2 + α2 x1 e x2 ) : λ2 ≥ 0,
α1 ≥ 0, α2 ≥ 0},
K 2 (x) = {λ1 ∇ f 1 (x) + α1 ∇g1 (x) + α2 ∇g2 (x) : λ1 ≥ 0, α1 ≥ 0, α2 ≥ 0}
= {(−λ1 + 3α1 (x1 )2 + α2 e x2 , α2 x1 e x2 ) : λ1 ≥ 0, α1 ≥ 0, α2 ≥ 0}.
Hence, K (x̄) = K 1 (x̄) = R+ × {0} and K 2 (x̄) = R × {0}.
We now show that x̄ satisfies CCP. Let ȳ = ( ȳ1 , ȳ2 ) ∈ lim sup K (x), hence there exists a
x→x̄
sequence (x1k , x2k , y1k , y2k ) → (0, 0, ȳ1 , ȳ2 ) such that
k
k
(y1k , y2k ) = (3α1k (x1k )2 + α2k e x2 , α2k x1k e x2 )
(4.1)
for some α1k ≥ 0, α2k ≥ 0. We next show that ȳ ∈ K (x̄). Suppose on the contrary that ȳ ∈
/ K (x̄).
Then, from (4.1) it is clear that ȳ2 = 0 and hence for sufficiently large k we have
k
|y2k | = α2k |x1k |e x2 >
| ȳ2 |
> 0.
2
Thus, from (4.1) it follows that
k
k
y1k = 3α1k (x1k )2 + α2k e x2 ≥ α2k e x2 >
| ȳ2 |
.
2|x1k |
Taking limits we obtain y1k → ∞, which is a contradiction. Thus, x̄ satisfies CCP.
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We next show that x̄ does not satisfy CCP1 . Define x1k = x2k = k1 , λk2 =
α2k = 0. Then for every k we have
1
,
2(x1k )2 x2k
k
α1k =
x2k
,
3(x1k )3
k
(y1k , y2k ) = (−2λk2 x1k (x2k )2 + 3α1k (x1k )2 + α2k e x2 , −2λk2 (x1k )2 x2k + α2k x1k e x2 )
= (0, −1) ∈ K 1 (x k )
but its limit (0, −1) ∈
/ K 1 (x̄).
Similarly, we next show that x̄ does not satisfy CCP2 . Define x1k = x2k = k1 , α1k = k, α2k =
x2k
1
k
x1k e x2
,
λk1 = 3α1k (x1k )2 + α2k e . Then for every k we have
k
k
(y1k , y2k ) = (−λk1 + 3α1k (x1k )2 + α2k e x2 , −α2k x1k e x2 )
= (0, 1) ∈ K 2 (x k )
but its limit (0, 1) ∈
/ K 2 (x̄).
The following theorem establishes that CCP is a strict constraint qualification for a weak efficient solution.
Theorem 4.1. If f is convexlike on Rn , x̄ is a weak efficient solution of (VP), CCP condition holds
at x̄ then x̄ satisfies weak KKT condition.
Proof. Since f is convexlike on Rn and x̄ is a weak efficient solution of (VP), then by Lemma 2.3
p
there exists λ ∈ R+ \ {0} such that x̄ is a minimizer of (Pλ ). Since x̄ is a minimizer of (Pλ ) there are
k
k
k
l
/ I (x̄) such that x k → x̄,
sequences {x } ⊆ Rn , {α k } ⊆ Rm
+ , {μ } ⊆ R with αi = 0 for i ∈
lim
k→∞
p
λi ∇ f i (x ) +
k
i=1
m
αik ∇gi (x k )
+
i=1
l
μik ∇h i (x k )
= 0.
i=1
Then
yk =
m
αik ∇gi (x k ) +
i=1
l
μik ∇h i (x k ) ∈ K (x k ).
i=1
Since x̄ satisfies CCP we have
lim y = lim
k
k→∞
k→∞
−
p
λi ∇ f i (x ) = −
k
i=1
p
λi ∇ f i (x̄) ∈ K (x̄)
i=1
that is, x̄ satisfies weak KKT condition.
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Remark 4.1. Since every properly efficient solution is a weak efficient solution it follows from
the above theorem that even for properly efficient solutions CCP is a strict constraint qualification
which leads to weak KKT conditions.
The following theorem establishes that strict constraint qualification CCP in fact leads to strong
KKT conditions for properly efficient solutions.
Theorem 4.2. If S is a convex set, fi is convex on S for i = 1, 2, . . . , p, x̄ is a properly efficient
solution of (VP) and CCP condition holds at x̄ then x̄ satisfies strong KKT condition.
Proof. Since S is a convex set and fi , for i = 1, 2, . . . , p, is convex on S and x̄ is a properly efficient
p
solution of (P), then by Lemma 2.4 there exists λ ∈ int R+ such that x̄ is a minimizer of (Pλ ). Since
k
k
n
k
l
x̄ is a minimizer of (Pλ ) there are sequences {x } ⊆ R , {α k } ⊆ Rm
+ , {μ } ⊆ R with αi = 0 for
k
i∈
/ I (x̄) such that x → x̄,
p
m
l
λi ∇ f i (x k ) +
αik ∇gi (x k ) +
μik ∇h i (x k ) = 0.
lim
k→∞
i=1
i=1
i=1
Then
yk =
m
αik ∇gi (x k ) +
i=1
Since x̄ satisfies CCP we have
lim y = lim
k
k→∞
k→∞
l
μik ∇h i (x k ) ∈ K (x k ).
i=1
−
p
λi ∇ f i (x ) = −
k
i=1
p
λi ∇ f i (x̄) ∈ K (x̄)
i=1
that is, x̄ satisfies strong KKT condition.
REFERENCES
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Optimization, 60, 627–641 (2011).
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qualification and applications. Math. Program., Ser. A 135, 255–273 (2012).
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[8] Burachik, R.S., Rizvi, M.M.: On weak and strong Kuhn-Tucker conditions for smooth multiobjective optimization. J.
Optim. Theory Appl., 155, 477–491 (2012).
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[9] Chandra, S., Dutta, J., Lalitha, C.S.: Regularity conditions and optimality in vector optimization. Numer. Funct. Anal.
Optim., 25, 479–501 (2004).
[10] Chankong, V., Haimes, Y.Y.: Vector Decision Making: Theory and Methodology, Amsterdam: North-Holland (1983).
[11] Geoffrion, A.M.: Proper efficiency and the theory of vector maximization. J. Math. Anal. Appl., 22, 618–630 (1968).
[12] Illés, T., Kassay, G.: Theorems of the alternative and optimality conditions for convexlike and general convexlike
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Optim. Theory Appl., 142, 385–398 (2009).
[14] Sun, X.-K., Long, X.-J., Chai, Y.: Sequential Optimality Conditions for Fractional Optimization with Applications to
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INDIAN JOURNAL OF INDUSTRIAL AND APPLIED MATHEMATICS
Vol. 10, No. 1 (Special Issue), Jan–June 2019, pp. 121–144
DOI:
Fixed Point Theorems for Generalized
[a, b, c] p -nonexpansive Mappings in
Banach Spaces
D.R. Sahu∗ and Satyendra Kumar1
Department of Mathematics, Institute of Science, Banaras Hindu University, Varanasi-221005, India
(∗ Corresponding author) Email: ∗ drsahudr@gmail.com, 1 saty.maths1986@gmail.com
Abstract: The purpose of this paper is to introduce a new class of generalized nonexpansive mappings and establish a fixed point existence result in a uniformly convex Banach space. The convergence analysis of Mann iteration and S-iteration process for this class of mappings has been
studied in Banach spaces. By some examples, it is shown that S-iteration process is faster than that
of Mann and Ishikawa iteration processes. Our result extends and improves various existing results
in literature in the context of S-iteration process.
Keywords: Generalized nonexpansive mapping, S-iteration process, Mann iteration process.
Mathematics Subject Classification (2010) 47J25.47H09.47H10
1. INTRODUCTION
Let C be a nonempty subset of a Banach space X and T : C → X a mapping. Let Fi x(T ) be the
set of all fixed points of mapping T . The mapping T is said to be:
(1) nonexpansive if
T x − T y ≤ x − y for all x, y ∈ C,
(2) quasi-nonexpansive if Fi x(T ) = ∅ and
T x − z ≤ x − z for all x ∈ C and z ∈ Fi x(T ),
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(3) firmly nonexpansive [1] if
T x − T y ≤ r (x − y) + (1 − r )(T x − T y) for all r > 0 and x, y ∈ C.
(4) Zamfirescu operator [16] if there exist real numbers, a, b, and c satisfying 0 < a < 1, 0 <
b, c < 12 such that for each pair x, y ∈ X , at least one of the following is true:
(Z 1 ) T x − T y ≤ ax − y,
(Z 2 ) T x − T y ≤ b(x − T x + y − T y),
(Z 3 ) T x − T y ≤ c(x − T y + y − T x).
Clearly, every firmly nonexpansive mapping is nonexpansive mapping. More details on firmly
nonexpansive mappings can be found in [2–5]. The generalization of firmly nonexpansive mappings
was studied by various authors (see [6–9]). Recently, nearly firmly nonexpansive sequence is studied in Sahu, Ansari and Yao [10]. In 2010, Takahashi [11] introduced the class of hybrid mappings
which is an important generalization of firmly nonexpansive mappings in Hilbert spaces. The class
of λ−hybrid mappings was introduced by Aoyama et al. [12] in 2010. Aoyama et al. [12] proved
some fixed point theorems and ergodic theorems for λ-hybrid mappings. The class of λ−hybrid
mappings contains the classes of nonexpansive mappings, nonspreading mappings and hybrid mappings in Hilbert spaces.
In 2011, Aoyama and Kohsaka [13] introduced the concept of α−nonexpansive mappings in
Banach spaces. They generalized some results of Aoyama et al. [12] in Banach spaces.
Let C be a nonempty convex subset of a Banach space X and T : C → C be a mapping. For
x1 ∈ C, the Mann iteration process [14] {xn } is defined as follows:
xn+1 = (1 − αn )xn + αn T xn for all n ∈ N,
where {αn } is a sequence in [0, 1].
The Ishikawa iteration process [15] is defined as follows:
xn+1 = (1 − αn )xn + αn T yn ;
yn = (1 − βn )xn + βn T xn for all n ∈ N,
(1.1)
(1.2)
where {αn } and {βn } are a sequences in [0, 1].
In 2007, Agarwal, O’Regan and Sahu [18] introduced the S-iteration process which is defined
as follows:
xn+1 = (1 − αn )T xn + αn T yn ;
(1.3)
yn = (1 − βn )xn + βn T xn for all n ∈ N,
where {αn } and {βn } are sequences in (0, 1).
In 2011, Sahu [19], introduced the Normal S-iteration as follows:
xn+1 = T [(1 − αn )xn + αn T xn ] for all n ∈ N,
(1.4)
where {αn } is a sequence in (0, 1).
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It is remarkable that S-iteration process is independent of Mann and Ishikawa iteration processes, also S-iteration process is faster than Mann and Ishikawa iteration processes for contraction
mappings (see, [18, 19]) and it work well for nonexpansive mappings. The S-iteration process is
faster than the Mann and Ishikawa iteration processes, and normal S-iteration process is faster than
the Picard, Mann, Ishikawa iteration processes for Zamfirescu operators (see, [35, 36]). The normal
S-iteration process is applicable for finding solutions of constrained minimization problems and
split feasibility problems (see, [19]).
In 2013, Naraghirad, Wang and Yao [22] studied some existence, weak and strong convergence
theorems for a fixed point of an α-nonexpansive mapping by Ishikawa iteration process in uniformly convex Banach spaces and CAT(0) spaces. In 2016, Song et. al. [23] studied the existence
of monotone α-nonexpansive mappings in uniformly convex Banach spaces by Mann iteration process, also the convergence analysis of Mann iteration process for monotone α-nonexpansive have
been discussed in ordered Banach spaces.
Motivated and inspired by work of Agarwal, O’Regan and Sahu [18], Aoyama and Kohsaka
[13], Naraghirad, Wang and Yao [22], Song et. al. [23], in this paper we introduce a new class
of generalized [a, b, c] p -nonexpansive mappings in Banach spaces and study the existence theorem and convergence analysis of Mann iteration, S−iteration and Normal S-iteration processes for
generalized [a, b, c]-nonexpansive mappings. As the S-iteration process and normal S-iteration process is faster than that of Mann and Ishikawa iteration processes for various mappings, in this paper
we will also compare these iteration processes for new class of generalized [a, b, c]-nonexpansive
mappings with some examples.
2. PRELIMINARIES
A Banach space X is said to be strictly convex [17] if
x, y ∈ S X with x = y ⇒ (1 − λ)x + λy < 1 for all λ ∈ (0, 1),
where S X = {x ∈ X : x = 1}. The modulus of convexity of X [17] is defined by
x + y
: x, y ≤ 1, x − y ≥ for all ∈ [0, 2].
δ X () = inf 1 −
2
The space X is said to be uniformly convex if δ X (0) = 0 and δ X () > 0 for all 0 < ≤ 2.
The space X is said to be p-uniformly convex, if there is a constant c p > 0 such that δ X () ≥
c p p . Every Hilbert space is 2-uniformly convex and for p > 1 the space L p is max{ p, 2}-uniformly
convex. Let C be a nonempty subset of a Banach space X . A function f : C → R is said to be
coercive, if f (z n ) → ∞ whenever {z n } is a sequence in C such that z n → ∞.
Let ∞ be the Banach space of all bounded real sequences with the supremum norm. A linear functional μ on ∞ is called a mean if μ(e) = μ = 1, where e = (1, 1, 1, ...). For x =
(x1 , x2 , x3 , ...), the value μ(x) is also denoted by μn xn . A Banach limit on ∞ is an invariant mean
i.e., μn (xn+1 ) = μn (xn ). If μ is a Banach limit on ∞ , then for x = (x1 , x2 , x3 , ...) ∈ ∞ ,
lim inf xn ≤ μn xn ≤ lim sup xn .
n→∞
n→∞
Clearly, if limn→∞ xn = x, the Banach limit of {xn } is also x, i.e., μn xn = x (see, [17]).
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Definition 2.1. Let C be a nonempty closed and convex subset of a Banach space X. Let {xn } be a
sequence in C. A mapping T : C → X is said to be demiclosed, if {xn } weakly converges to x and
{T xn } converges to y imply T x = y.
Definition 2.2. (Opial condition) [21] A Banach space X is said to satisfy the Opial property, if for
every weakly convergent sequence xn x in X, we have
lim sup xn − x < lim sup xn − y
n→∞
n→∞
for all y ∈ X with x = y.
In above definition, lim sup can be replaced by lim inf .
Remark 2.1. Every Hilbert spaces, finite dimensional Banach spaces and the Banach spaces p (1 ≤
p < ∞) satisfy the Opial property, but the uniformly convex Banach spaces L p [0, 2π ]( p = 2) do
not satisfy Opial property.
Lemma 2.1. [24] Let C be a nonempty closed and convex subset of a reflexive Banach space X and
let f : C → R be a convex continuous and coercive function. Then there exists x ∗ ∈ C such that
f (x ∗ ) = inf x∈C f (x).
Lemma 2.2. [26] The Banach space X is uniformly convex iff .2 is uniformly convex on bounded
convex sets i.e., for each r > 0 and ∈ (0, 1], there exist δ > 0 such that
λx + (1 − λ)y2 ≤ λx2 + (1 − λ)y2 − λ(1 − λ)δ,
for all λ ∈ (0, 1) and for all x, y ∈ Br [0] := {u ∈ X : u ≤ r } with x − y ≥ .
Lemma 2.3. [17] Let p > 1 be a given real number and X be a Banach space. Then X is puniformly convex iff there exist a constant c p > 0 such that
λx + (1 − λ)y p ≤ λx p + (1 − λ)y p − (λ p (1 − λ) + λ(1 − λ) p )c p x − y p ,
for all λ ∈ (0, 1) and for all x, y ∈ X.
Lemma 2.4. [25] Let r > 0 be a fixed real number. If X is a uniformly convex Banach space, then
there exists a continuous strictly increasing convex function g : [0, ∞) → [0, ∞) with g(0) = 0
such that
λx + (1 − λ)y2 ≤ λx2 + (1 − λ)y2 − λ(1 − λ)g(x − y)
for all x, y ∈ Br [0] and λ ∈ [0, 1].
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Lemma 2.5. [17] Let X be a uniformly convex Banach space with modulus of convexity δ X . If
r > 0 and x, y ∈ X with x ≤ r, y ≤ r, then
x − y
λx + (1 − λ)y ≤ r 1 − 2 min{λ, 1 − λ}δ X
for all λ ∈ (0, 1).
r
Lemma 2.6. [20] If C is a closed convex subset of a strictly convex normed linear space, and
T : C → C is quasi-nonexpansive, then Fi x(T ) = { p : p ∈ C and T p = p} is a nonempty closed
convex set on which T is continuous.
Let C be a nonempty convex subset of a normed linear space X and T : C → C a mapping
with Fi x(T ) = ∅. Let {xn } be a sequence in C. Then the sequence {xn } is said to satisfy
(D1 ) limit existence property (or LE property) if limn→∞ xn − z exists for all z ∈ Fi x(T ),
(D2 ) approximate fixed point property (or AF point property) if limn→∞ xn − T xn = 0.
(D3 ) LEAF point property if {xn } has both LE property and AF point property (see [19]).
The properties (D1 ) and (D2 ) play important role in the approximation of fixed points of nonexpansive and asymptotically nonexpansive mappings by means of the Mann and S-iteration processes
(see [27–34]) in Banach spaces under suitable geometric structures.
3. MAIN RESULTS
3.1. Generalized [a, b, c] p -nonexpansive mappings
Definition 3.1. [12] Let C be a nonempty subset of a real Hilbert space H and λ a real number. A
mapping T : C → H is said to be λ-hybrid if
T x − T y2 ≤ x − y2 + 2(1 − λ) x − T x, y − T y for all x, y ∈ C.
Definition 3.2. [13] Let C be a nonempty subset of a Banach space X and let α be a real number
such that α < 1. A mapping T : C → X is said to be α-nonexpansive if
T x − T y2 ≤ αT x − y2 + αx − T y2 + (1 − 2α)x − y2 for all x, y ∈ C.
It can be noted that 0−nonexpansive mapping is nonexpansive and every firmly nonexpansive
mapping is α−nonexpansive for all real number α such that 0 ≤ α ≤ 12 . Every α−nonexpansive
mapping T with Fi x(T ) = ∅ is quasi-nonexpansive.
Let C be a nonempty subset of a Hilbert space H and let T : C → H be a mapping. Let λ
. Then T is λ−hybrid iff T is α−nonexpansive
be a real number such that λ < 2 and put α = 1−λ
2−λ
(see [13]).
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We now introduced the generalized [a, b, c] p -nonexpansive mapping as follows:
Definition 3.3. Let p > 1 and let C be a nonempty subset of a Banach space X . A mapping T :
C → X is said to be generalized [a, b, c] p -nonexpansive mapping if there exists real numbers a, b
and c with a < 1, b < 1 and a + b + c = 1 such that
T x − T y p ≤ aT x − y p + bx − T y p + cx − y p for all x, y ∈ C.
Remark 3.1.
(i) If p = 2, then we denote generalized [a, b, c] p -nonexpansive mapping by generalized
[a, b, c]-nonexpansive mapping.
(ii) It follows from definition that c > −1.
(iii) If a = b = α and p = 2, then generalized [a, b, c] p -nonexpansive mapping is αnonexpansive mapping.
(iv) Generalized [0, 0, 1] p -nonexpansive mapping is nonexpansive mapping.
Example 3.1. Let X = R with usual norm and C = [0, 1]. Let T : [0, 1] → [0, 1] be defined by
x − x 2 , if x ∈ [0, 1);
Tx = 1
(3.1)
, if x = 1.
2
Then we have the following:
(a) T is quasi-nonexpansive with Fi x(T ) = {0}.
(b) T is generalized [0.1, 0.6, 0.3]-nonexpansive.
(c) T is not 0.1-nonexpansive.
Proof.
(a) It is obvious.
(b) Consider the following cases:
Case I. x, y ∈ [0, 1). In this case
T x − T y2 = (x − x 2 ) − (y − y 2 )2 = x − y2 1 − (x + y)2 ≤ x − y2 .
Case II. x = y = 1. In this case
T x − T y = 0.
Case III. x ∈ [0, 1) and y = 1. Define F(x) = T x − 12 2 , f (x) = x − 12 2 , g(x) = T x − 12
and h(x) = x − 12 for all x ∈ [0, 1). From Figure 1, it is clear that
F(x) ≤ 0.1 f (x) + 0.6g(x) + 0.3h(x) for all x ∈ [0, 1).
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Figure 1. Graphical representation of F(x), 0.1 f (x) + 0.6g(x) + 0.3h(x) and 0.1 f (x) + 0.1g(x) + 0.8h(x)
It means that
1
1
T x − 2 ≤ 0.1x − 2 + 0.6T x − 12 + 0.3x − 12 for all x ∈ [0, 1).
2
2
From cases I, II and III, we conclude that the mapping T is generalized [0.1, 0.6, 0.3]-nonexpansive.
(c) From Figure 1, it is also clear that
F(x) 0.1 f (x) + 0.1g(x) + (1 − 0.2)h(x) for all x ∈ [0, 1).
It means that the mapping T is not 0.1-nonexpansive.
Remark 3.2. From Remark 3.1 and Example 3.1, we observe that the class of generalized [a, b, c]nonexpansive mappings is essentially wider than class of α-nonexpansive mapping.
Proposition 3.1. Let X be a Banach space with norm ., define norm on X × X × X by
|(x1 , y1 , z 1 ) − (x2 , y2 , z 2 )| = (x1 − x2 2 + y1 − y2 2 + z 1 − z 2 2 )1/2 ,
for all (x1 , y1 , z 1 ), (x2 , y2 , z 2 ) ∈ X × X × X. Let Ti : X → X be generalized [a, b, c]nonexpansive mappings (i = 1, 2, 3). Define a mapping T : X × X × X → X × X × X by
T (x1 , x2 , x3 ) = (T1 x1 , T2 x2 , T3 x3 ) for all x1 , x2 , x3 ∈ X.
Then T is generalized [a, b, c]-nonexpansive.
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Proof. Let x = (x1 , y1 , z 1 ), y = (x2 , y2 , z 2 ) ∈ X × X × X. Then
T x − T y2 = T (x1 , y1 , z 1 ) − T (x2 , y2 , z 2 )2
= (T1 x1 , T2 y1 , T3 z 1 ) − (T1 x2 , T2 y2 , T3 z 2 )2
= T1 x1 − T1 x2 2 + T2 y1 − T2 y2 2 + T3 z 1 − T3 z 2 2
≤ aT1 x1 − x2 2 + bx1 − T1 x2 2 + cx1 − x2 2
+aT2 y1 − y2 2 + by1 − T2 y2 2 + cy1 − y2 2
+aT3 z 1 − z 2 2 + bz 1 − T3 z 2 2 + cz 1 − z 2 2
= a(T1 x1 − x2 , T2 y1 − y2 , T3 z 1 − z 2 )2 + b(x1 − T1 x2 , y1 − T2 y2 , z 1 − T3 z 2 )2
+c(x1 − x2 , y1 − y2 , z 1 − z 2 )2
= a(T1 x1 , T2 y1 , T3 z 1 ) − (x2 , y2 , z 2 )2 + b(x1 , y1 , z 1 ) − (T1 x2 , T2 y2 , T3 z 2 )2
+c(x1 , y1 , z 1 ) − (x2 , y2 , z 2 )2
= aT x − y2 + bx − T y2 + cx − y2 .
This shows that T is generalized [a, b, c]-nonexpansive mapping.
Lemma 3.1. Let p > 1 and let C be a nonempty subset of a Banach space X and T : C → X a
generalized [a, b, c] p -nonexpansive mapping with Fi x(T ) = ∅. Then T is quasi-nonexpansive.
Proof. For x ∈ C and z ∈ F(T ), we have
T x − z p
≤
aT x − z p + bx − T z p + cx − z p
= aT x − z p + (b + c)x − z p .
This implies that
(1 − a)T x − z p ≤ (b + c)x − z p .
Thus, T is quasi-nonexpansive mapping.
Proposition 3.2. Let p > 1 and let C be a nonempty closed convex subset of a p-uniformly convex
Banach space X . Let T : C → C be a generalized [a, b, c] p -nonexpansive mapping. If Fi x(T ) = ∅,
then Fi x(T ) is closed convex subset of C.
Proof. Following Lemma 3.1 we have that if Fi x(T ) = ∅, then T is quasi-nonexpansive. By
Lemma 2.6, Fi x(T ) is closed convex subset of C.
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Proposition 3.3. Let C be a nonempty subset of a Banach space X and T : C → X a generalized
[a, b, c]-nonexpansive mapping. Then for all x, y ∈ C the following assertions holds:
(i) If 0 ≤ a, b < 1, then
x − T y2 ≤
1+a
2
x − T x2 +
(ax − y + T x − T y)x − T x + x − y2 .
1−b
1−b
(ii) If a, b < 0, then
x −T y2 ≤
1−a
2
x −T x2 +
((−a)y −T x+T x −T y)x −T x+x − y2 .
1−b
1−b
Proof. (i) Suppose that 0 ≤ a, b < 1. Let x, y ∈ C. Then
x − T y2
= x − T x + T x − T y2
≤
(x − T x + T x − T y)2
= x − T x2 + T x − T y2 + 2x − T xT x − T y
≤
x −T x2 +aT x − y2 + bx − T y2 + cx − y2 + 2x − T xT x − T y
≤
x − T x2 + a(T x − x + x − y)2 + bx − T y2 + cx − y2
+2x − T xT x − T y
= x − T x2 + aT x − x2 + ax − y2 + 2aT x − xx − y
+bx − T y2 + cx − y2 + 2x − T xT x − T y
= (1 + a)x − T x2 + (a + c)x − y2 + bx − T y2
+2aT x − xx − y + 2x − T xT x − T y.
This implies that
(1−b)x −T y2 ≤ (1+a)x −T x2 + 2(ax − y + T x − T y)x − T x + (a + c)x − y2 .
Therefore, we get
x − T y2 ≤
1+a
2
x − T x2 +
(ax − y + T x − T y)x − T x + x − y2 .
1−b
1−b
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(ii) Suppose that a, b < 0. Let x, y ∈ C. Then
x − T y2
= x − T x + T x − T y2
≤ (x − T x + T x − T y)2
= x − T x2 + T x − T y2 + 2x − T xT x − T y
≤
x − T x2 + aT x − y2 + bx − T y2 + cx − y2
+2x − T xT x − T y
= x − T x2 + aT x − y2 + bx − T y2 + (1 − b)x − y2
+(−a)x − y2 + 2x − T xT x − T y
≤
x − T x2 + aT x − y2 + bx − T y2 + (1 − b)x − y2
+(−a)(x − T x + T x − y)2 + 2x − T xT x − T y
= x − T x2 + aT x − y2 + bx − T y2 + (1 − b)x − y2
+(−a)(x − T x2 + T x − y2 + 2x − T xT x − y)2
+2x − T xT x − T y
= (1 − a)x − T x2 + bx − T y2
+2((−a)T x − y + T x − T y)x − T x + (1 − b)x − y2 .
This implies that
(1−b)x −T y2 ≤ (1−a)x −T x2 +2((−a)T x − y+T x −T y)x −T x+(1 − b)x − y2 .
Therefore, we get
x − T y2 ≤
1−a
2
x − T x2 +
((−a)T x − y + T x − T y)x − T x + x − y2 .
1−b
1−b
Lemma 3.2. (Demicloseness Principle) Let C be a nonempty closed convex subset of a Banach
space X with the Opial property. Let T : C → C be a generalized [a, b, c]-nonexpansive mapping.
Then (I − T ) is demiclosed at zero.
Proof. Let {xn } be a sequence in C such that it converges weakly to x ∗ and limn→∞ xn − T xn =
0. Then, sequences {xn } and {T xn } are bounded. Let M1 := sup{xn , T xn , x ∗ , T x ∗ : n ∈
N} < ∞. For 0 ≤ a, b < 1, we have
xn − T x ∗ 2
≤
≤
1+a
2
xn − T xn 2 +
(axn − x ∗ + T xn − T x ∗ )xn − T xn 1−b
1−b
+xn − x ∗ 2
1+a
4M1 (1 + a)
xn − T xn 2 +
xn − T xn + xn − x ∗ 2 .
1−b
1−b
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Next, for a, b < 0, we have
xn − T x ∗ 2
≤
≤
1−a
2
xn − T xn 2 +
((−a)T xn − x ∗ + T xn − T x ∗ )xn − T xn 1−b
1−b
+xn − x ∗ 2
1−a
4M1 (1 − a)
xn − T xn 2 +
xn − T xn + xn − x ∗ 2 .
1−b
1−b
Therefore, in each cases, we have
lim sup xn − T x ∗ ≤ lim sup xn − x ∗ .
n→∞
n→∞
From Opial’s property, we get T x ∗ = x ∗ .
3.2. Existence
In this section, we established the existence result of fixed points of generalized [a, b, c]nonexpansive mappings. We begin with the following proposition:
Proposition 3.4. Let p > 1 and let C be a nonempty subset of a Banach space X . Let T : C → C
be a generalized [a, b, c] p -nonexpansive mapping. Suppose that {T n x} is bounded for some x ∈ C.
Then μn T n x − T y p ≤ μn T n x − y p for all Banach limit μ and for all y ∈ C.
Proof. Let μ be a Banach limit and y ∈ C. Since T is generalized [a, b, c] p -nonexpansive mapping,
we have
T n+1 x − T y p ≤ aT n+1 x − y p + bT n x − T y p + cT n x − y p .
Since μ is Banach limit, we get that
μn T n x − T y p ≤ aμn T n x − y p + bμn T n x − T y p + cμn T n x − y p ,
which implies that
(1 − b)μn T n x − T y p ≤ (a + c)μn T n x − y p .
Theorem 3.1. Let p > 1 and let C be a nonempty closed convex subset of a p-uniformly convex
Banach space X. Let T : C → C be a generalized [a, b, c] p -nonexpansive mapping. Suppose there
exists x ∈ C such that {T n x} is bounded, then Fi x(T ) = ∅.
Proof. Let f : C → R be a function defined by f (y) = μn T n x − y p for all y ∈ C. We claim
that f is a convex, continuous and coercive function. Convexity of function f follows from Lemma
2.3.
For continuity of f , let {ym } be a sequence in C such that limm→∞ ym = y. Then by mean value
theorem, we have
|T n x − ym p − T n x − y p |
=
p−1
|T n x − ym − T n x − y| | pcm,n
|,
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for all m, n ∈ N, where
min{T n x − ym , T n x − y} ≤ cm,n ≤ max{T n x − ym , T n x − y}.
From (3.2), we get
|T n x − ym p − T n x − y p | ≤ |T n x − ym − T n x − y| | p(T n x − ym + T n x − y) p−1 |
≤ ym − y sup{ p(T n x − ym + T n x − y) p−1 : m, n ∈ N}.
This shows that the function g : C → ∞ defined by
g(z) = {T 1 x − z p , T 2 x − z p , T 3 x − z p , ...} for all z ∈ C
is continuous. Hence the mapping f = μog is continuous.
Now to show that f is coercive, let {z m } is a sequence in C such that z m → ∞. Then we
have
T n x − z m p ≥ (|z m − T n x|) p .
This implies that f (z m ) → ∞. Hence f is coercive.
Since f is convex, continuous and coercive, from Lemma 2.1 there exists x * in C such that
*
f (x ) = inf x∈C f (x). By the uniform convexity of X , one can show that such a point x * is unique.
From Proposition 3.4, we have f (T x * ) ≤ f (x * ). Hence x * ∈ Fi x(T ).
Corollary 3.1. Let p > 1 and let C be a nonempty closed convex subset of a p-uniformly convex
Banach space X. Let T : C → C be a α-nonexpansive mapping for some α < 1. Suppose there
exists x ∈ C such that {T n x} is bounded, then Fi x(T ) = ∅.
In view of Lemma 2.2, Proposition 3.4 and Theorem 3.1, we obtain the following result:
Theorem 3.2. Let C be a nonempty closed convex subset of a uniformly convex Banach space X .
Let T : C → C be a generalized [a, b, c]-nonexpansive mapping. Suppose there exist x ∈ C such
that {T n x} is bounded, then Fi x(T ) = ∅.
Corollary 3.2. Let C be a nonempty closed convex subset of a uniformly convex Banach space X .
Let T : C → C be an α-nonexpansive mapping for some α < 1. Suppose there exist x ∈ C such
that {T n x} is bounded, then Fi x(T ) = ∅.
3.3. Convergence
We begin with following proposition:
Proposition 3.5. Let C be a nonempty closed convex subset of a Banach space X . Let T : C → C
be a generalized [a, b, c]-nonexpansive mapping such that Fi x(T ) = ∅. Let {αn } be sequences in
(0, 1), and let {xn } be a sequence in C defined by Mann iteration process (1.1). Then for all z ∈
Fi x(T ), we have the following:
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(a) xn+1 − z ≤ xn − z for all n ∈ N.
(b) The sequence {xn } holds limit existence property (LE property) for T , i.e., limn→∞ xn −
z exists.
(c) limn→∞ d(xn , Fi x(T )) exists, where d(xn , Fi x(T )) = inf z∈Fi x(T ) xn − z.
Proof. (a) Let z ∈ Fi x(T ). From (1.1) and using Lemma 3.1, we get
xn+1 − z
=
(1 − αn )xn + αn T xn − z ≤ (1 − αn )xn − z + αn T xn − z
≤
(1 − αn )xn − z + αn xn − z = xn − z.
(3.3)
Therefore, we obtain xn+1 − z ≤ xn − z for all n ∈ N. (b) From (3.3), it is clear that the
sequence {xn − z} is a monotonic decreasing sequence of nonnegative numbers. Therefore, it
is convergent, i.e., limn→∞ xn − z exist. (c) This part follows from part (b).
Theorem 3.3. Let C be a nonempty closed convex subset of a uniformly convex Banach space X.
Let T : C → C be a generalized [a, b, c]-nonexpansive mapping such that Fi x(T ) = ∅. Let {αn }
be sequence in (0, 1), and let {xn } be a sequence in C defined by the Mann iteration process (1.1).
Then we have the following:
(a) If {xn } is bounded and lim inf n→∞ xn − T xn = 0, then Fi x(T ) = ∅.
(b) Assume that Fi x(T ) = ∅. Then {xn } is bounded and following holds:
(i) If lim supn→∞ αn (1 − αn ) > 0, then lim inf n→∞ xn − T xn = 0.
(ii) If lim inf n→∞ αn (1 − αn ) > 0, then the sequence {xn } holds approximate fixed point
property (AF point property) for T , i.e., limn→∞ xn − T xn = 0.
Proof. (a) Let {xn } is bounded and lim inf n→∞ xn − T xn = 0. Then there exists a subsequence
{xn k } of {xn } such that limk→∞ xn k − T xn k = 0. Suppose that A(C, {xn k }) = {z}, and let M2 =
supk∈N {xn k , T xn k , z, T z} < ∞. Then for 0 ≤ a, b < 1, we have
1+a
2
xn k −T xn k 2 +
(axn k − z + T xn k −T z)xn k −T xn k +xn k − z2
1−b
1−b
1+a
4M2 (1 + a)
xn k − T xn k 2 +
xn k − T xn k + xn k − z2 .
≤
1−b
1−b
xn k − T z2 ≤
Also for a, b < 0, we have
xn k − T z2
≤
≤
1−a
2
xn k − T xn k 2 +
((−a)z − T xn k + T xn k − T z)xn k − T xn k 1−b
1−b
+xn k − z2
1−a
2M2 (1 − a)
xn k − T xn k 2 +
xn k − T xn k + xn k − z2 .
1−b
1−b
Therefore in each case, we get
lim sup xn k − T z2 ≤ lim sup xn k − z2 .
k→∞
k→∞
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This implies that
r (T z, xn k ) =
lim sup xn k − T z ≤ lim sup xn k − z = r (z, xn k ).
k→∞
k→∞
This shows that T z ∈ A(C, {xn k }). Since X is uniformly convex Banach space, therefore we get that
T z = z i.e., Fi x(T ) = ∅.
(b) (i) Assume that Fi x(T ) = ∅ and z ∈ Fi x(T ). From Proposition 3.5, we have that
limn→∞ xn − z exists and hence {xn } is bounded. Let
lim xn − z = r > 0.
n→∞
(3.4)
In view of Lemma 2.4, there exists a continuous strictly increasing convex function g : [0, ∞) →
[0, ∞) with g(0) = 0 such that
xn+1 − z2
=
=
(1 − αn )xn + αn T xn − z2
(1 − αn )(xn − z) + αn (T xn − z)2
≤
≤
(1 − αn )xn − z2 + αn T xn − z2 − αn (1 − αn )g(xn − T xn )
(1 − αn )xn − z2 + αn xn − z2 − αn (1 − αn )g(xn − T xn )
=
xn − z2 − αn (1 − αn )g(xn − T xn ).
This implies that
αn (1 − αn )g(xn − T xn ) ≤ xn − z2 − xn+1 − z2 .
(3.5)
Thus, if lim supn→∞ αn (1 − αn ) > 0, then we have lim inf n→∞ g(xn − T xn ) = 0. Since g(0) = 0,
we get that if lim supn→∞ αn (1 − αn ) > 0, then lim inf n→∞ xn − T xn = 0.
(ii) From (3.5), in same manner, we obtain that if lim inf n→∞ αn (1 − αn ) > 0, then
limn→∞ xn − T xn = 0.
Corollary 3.3. Let C be a nonempty closed convex subset of a uniformly convex Banach space X.
Let T : C → C be an α-nonexpansive mapping for some α < 1. Let {αn } be sequence in (0, 1), and
let {xn } be a sequence in C defined by the Mann iteration process (1.1). Then we have the following:
(a) If {xn } is bounded and lim inf n→∞ xn − T xn = 0, then Fi x(T ) = ∅.
(b) Assume that Fi x(T ) = ∅. Then {xn } is bounded and following holds:
(i) If lim supn→∞ αn (1 − αn ) > 0, then lim inf n→∞ xn − T xn = 0.
(ii) If lim inf n→∞ αn (1 − αn ) > 0, then limn→∞ xn − T xn = 0.
Theorem 3.4. Let C be a nonempty closed convex subset of a uniformly convex Banach space
X with the Opial property. Let T : C → C be a generalized [a, b, c]-nonexpansive mapping with
Fi x(T ) = ∅. Let {αn } be sequence in (0, 1) such that lim inf n→∞ αn (1 − αn ) > 0, and let {xn } be a
sequence in C defined by the Mann iteration process (1.1). Then {xn } converges weakly to a fixed
point of T .
Proof. From Theorem 3.3, it follows that the sequence {xn } is bounded and limn→∞ xn − T xn =
0. Since X uniformly convex Banach space, therefore X is reflexive. Then there exists a subsequence
{xn k } of {xn } such that xn k z ∈ C as k → ∞. It follows Lemma 3.2 that z ∈ Fi x(T ). We claim
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that xn z ∈ C as n → ∞. If possible, suppose that there exists another subsequence {xn j } of
{xn } such that xn j z ∈ C as j → ∞ and z = z . From Lemma 3.2 we again conclude that z ∈
Fi x(T ). By Proposition 3.5 we have that limn→∞ xn − z and limn→∞ xn − z exists. Using
Opial property, we have
lim xn − z =
n→∞
=
<
lim xn k − z < lim xn k − z k→∞
k→∞
lim xn − z = lim xn j − z n→∞
j→∞
lim xn j − z = lim xn − z,
j→∞
n→∞
which is a contradiction. Thus we have z = z . This proves the theorem.
Corollary 3.4. Let C be a nonempty closed convex subset of a uniformly convex Banach space
X with the Opial property. Let T : C → C be an α-nonexpansive mapping for some α < 1 with
Fi x(T ) = ∅. Let {αn } be sequence in (0, 1) such that lim inf n→∞ αn (1 − αn ) > 0, and let {xn } be a
sequence in C defined by the Mann iteration process (1.1). Then {xn } converges weakly to a fixed
point of T .
Theorem 3.5. Let C be a nonempty compact convex subset of a uniformly convex Banach space X .
Let T : C → C be a generalized [a, b, c]-nonexpansive mapping. Let {αn } be a sequence in (0, 1)
such that lim supn→∞ αn (1 − αn ) > 0, and let {xn } be a sequence in C defined by the Mann iteration
process (1.1). Then {xn } converges strongly to a fixed point of T .
Proof. By compactness of C it follows that C is bounded and from Theorem 3.2 we get that
Fi x(T ) = ∅. Since lim supn→∞ αn (1 − αn ) > 0, therefore from Proposition 3.3, we have that {xn }
is bounded and lim inf n→∞ xn − T xn = 0. The compactness of C implies that there exists a subsequence {xn k } of {xn } such that xn k → z as k → ∞ for some z ∈ C and limk→∞ xn k − T xn k = 0.
Let M3 = sup{xn k , T xn k , z, T z : k ∈ N} < ∞. For 0 ≤ a, b < 1, from Proposition
3.3, we have
1+a
2
xn k −T xn k 2 +
(axn k −z+T xn k − T z)xn k − T xn k + xn k − z2
1−b
1−b
1+a
4M3 (1 + a)
xn k − T xn k 2 +
+ xn k − z2 .
≤
1−b
1−b
xn k −T z2 ≤
Also, for a, b < 0, from Proposition 3.3, we have
xn k − T z2
≤
≤
1−a
2
xn k − T xn k 2 +
((−a)z − T xn k + T xn k − T z)xn k − T xn k 1−b
1−b
+xn k − z2
1−a
4M3 (1 − a)
xn k − T xn k 2 +
xn k − T xn k + xn k − z2 .
1−b
1−b
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Thus in each cases, we have
lim sup xn k − T z2 ≤ lim sup xn k − z2 .
k→∞
k→∞
This implies that limk→∞ xn k − T z = 0 and we have T z = z. By Proposition 3.5, limn→∞ xn −
z exists. Thus, we have limn→∞ xn − z = 0.
Corollary 3.5. Let C be a nonempty compact convex subset of a uniformly convex Banach space
X . Let T : C → C be an α-nonexpansive mapping for some α < 1. Let {αn } be a sequence in (0, 1)
such that lim supn→∞ αn (1 − αn ) > 0, and let {xn } be a sequence in C defined by the Mann iteration
process (1.1). Then {xn } converges strongly to a fixed point of T .
Proposition 3.6. Let C be a nonempty closed convex subset of a Banach space X . Let T : C → C
be a generalized [a, b, c]-nonexpansive mapping such that Fi x(T ) = ∅. Let {αn } and {βn } be two
sequences in (0, 1), and let {xn } be a sequence in C defined by S−iteration process (1.3). Then for
all z ∈ Fi x(T ), we have the following:
(a) max{xn+1 − z, yn − z} ≤ xn − z for all n ∈ N.
(b) The sequence {xn } holds limit existence property (LE property) for T, i.e., limn→∞ xn −
z exists.
(c) limn→∞ d(xn , Fi x(T )) exists, where d(xn , Fi x(T )) = inf z∈Fi x(T ) xn − z.
Proof. (a) Let z ∈ Fi x(T ). From (1.3) and using Lemma 3.1, we get
yn − z = (1 − βn )xn + βn T xn − z
≤ (1 − βn )xn − z + βn T xn − z
≤ (1 − βn )xn − z + βn xn − z
= xn − z.
(3.6)
From (1.3), (3.6) and Lemma 3.1, we get
xn+1 − z = (1 − αn )T xn + αn T yn − z
≤
≤
(1 − αn )T xn − z + αn T yn − z
(1 − αn )xn − z + αn yn − z
≤ (1 − αn )xn − z + αn xn − z
= xn − z.
(3.7)
Therefore, we obtain max{xn+1 − z, yn − z} ≤ xn − z for all n ∈ N. (b) From (3.7), it is
clear that the sequence {xn − z} is a monotonic decreasing sequence of nonnegative numbers.
Therefore, it is convergent, i.e., limn→∞ xn − z exists. (c) This part follows from part (b).
Theorem 3.6. Let C be a nonempty closed convex subset of a uniformly convex Banach space X.
Let T : C → C be a generalized [a, b, c]-nonexpansive mapping such that Fi x(T ) = ∅. Let {αn }
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and {βn } be sequences in (0, 1) such that limn→∞ βn (1 − βn ) > 0, and let {xn } be a sequence in C
defined by the S−iteration process (1.3). Then we have the following:
(a) If {xn } is bounded and lim inf n→∞ xn − T xn = 0, then Fi x(T ) = ∅.
(b) Assume that Fi x(T ) = ∅. Then {xn } is bounded and following holds:
(i) If lim supn→∞ αn (1 − αn ) > 0, then lim inf n→∞ xn − T xn = 0.
(ii) If lim inf n→∞ αn (1 − αn ) > 0, then the sequence {xn } satisfy approximate fixed point
property (AF point property), i.e., limn→∞ xn − T xn = 0.
Proof. (a) This part follows from Theorem 3.3.
(b) (i) Assume that Fi x(T ) = ∅ and z ∈ Fi x(T ). From Proposition 3.6, we have that
limn→∞ xn − z exists and hence {xn } is bounded. Let
lim xn − z = r > 0.
(3.8)
n→∞
In view of Lemma 2.4, there exists a continuous strictly increasing convex function g : [0, ∞) →
[0, ∞) with g(0) = 0 such that
xn+1 − z2
= (1 − αn )T xn + αn T yn − z2
= (1 − αn )(T xn − z) + αn (T yn − z)2
≤ (1 − αn )T xn − z2 + αn T yn − z2 − αn (1 − αn )g(T xn − T yn )
≤
≤
(1 − αn )xn − z2 + αn yn − z2 − αn (1 − αn )g(T xn − T yn )
(1 − αn )xn − z2 + αn xn − z2 − αn (1 − αn )g(T xn − T yn )
= xn − z2 − αn (1 − αn )g(T xn − T yn ).
This implies that
αn (1 − αn )g(T xn − T yn ) ≤ xn − z2 − xn+1 − z2 .
(3.9)
If lim supn→∞ αn (1 − αn ) > 0. Then, from (3.9), we have lim inf n→∞ g(T xn − T yn ) = 0. g(0) =
0 implies that
lim inf T xn − T yn = 0.
n→∞
(3.10)
Now xn+1 − T xn = (1 − αn )T xn + αn T yn − T xn = αn T xn − T yn . Using (3.10), we
get
lim inf xn+1 − T xn = 0.
n→∞
(3.11)
Again xn+1 − T yn ≤ xn+1 − T xn + T xn − T yn . Therefore, from (3.10) and (3.11), we get
lim inf xn+1 − T yn = 0.
n→∞
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Next xn+1 − z ≤ xn+1 − T yn + T yn − z ≤ xn+1 − T yn + yn − z. Using (3.8) and
(3.12), we have
r ≤ lim inf yn − z.
(3.13)
n→∞
Notice that yn − z ≤ xn − z for all n ∈ N. Therefore, from (3.8), we obtain that
lim sup yn − z ≤ r.
(3.14)
lim yn − z = r.
(3.15)
n→∞
From (3.13) and (3.14), we have
n→∞
Since yn − z ≤ xn − z and T xn − z ≤ xn − z. Therefore, from (3.15) and using Lemma
2.5, we get
r
=
=
≤
lim yn − z = lim (1 − βn )xn + βn T xn − z
n→∞
n→∞
lim (1 − βn )(xn − z) + βn (T xn − z)
n→∞
xn − T xn .
lim xn − z 1 − 2βn (1 − βn )δ X
n→∞
xn − z
This implies that
2 lim xn − zβn (1 − βn )δ X
n→∞
xn − T xn xn − z
≤ 0.
Notice that limn→∞ xn − z = r > 0 and limn→∞ βn (1 − βn ) > 0. Therefore, we get
xn − T xn = 0.
lim δ X
n→∞
xn − z
Hence, by property of δ X we conclude that if lim supn→∞ αn (1 − αn ) > 0, then lim inf n→∞ xn −
T xn = 0.
(ii) In the same manner as above, from (3.9), we can obtain that if lim inf n→∞ αn (1 − αn ) >
0, then limn→∞ xn − T xn = 0.
Corollary 3.6. Let C be a nonempty closed convex subset of a uniformly convex Banach space X.
Let T : C → C be a α-nonexpansive mapping for some α < 1. Let {αn } and {βn } be sequences in
(0, 1) such that limn→∞ βn (1 − βn ) > 0, and let {xn } be a sequence in C defined by the S−iteration
process (1.3). Then we have the following:
(a) If {xn } is bounded and lim inf n→∞ xn − T xn = 0, then Fi x(T ) = ∅.
(b) Assume that Fi x(T ) = ∅. Then {xn } is bounded and following holds:
(i) If lim supn→∞ αn (1 − αn ) > 0, then lim inf n→∞ xn − T xn = 0.
(ii) If lim inf n→∞ αn (1 − αn ) > 0, then limn→∞ xn − T xn = 0.
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Theorem 3.7. Let C be a nonempty closed convex subset of a uniformly convex Banach space
X with the Opial property. Let T : C → C be a generalized [a, b, c]-nonexpansive mapping with
Fi x(T ) = ∅. Let {αn } and {βn } be sequences in (0, 1) such that lim inf n→∞ αn (1 − αn ) > 0 and
limn→∞ βn (1 − βn ) > 0, and let {xn } be a sequence in C defined by the S-iteration process (1.3).
Then {xn } converges weakly to a fixed point of T .
Corollary 3.7. Let C be a nonempty closed convex subset of a uniformly convex Banach space
X with the Opial property. Let T : C → C be an α-nonexpansive mapping for some α < 1 with
Fi x(T ) = ∅. Let {αn } and {βn } be sequences in (0, 1) such that lim inf n→∞ αn (1 − αn ) > 0 and
limn→∞ βn (1 − βn ) > 0, and let {xn } be a sequence in C defined by the S-iteration process (1.3).
Then {xn } converges weakly to a fixed point of T .
Theorem 3.8. Let C be a nonempty compact convex subset of a uniformly convex Banach space X .
Let T : C → C be a generalized [a, b, c]-nonexpansive mapping. Let {αn } and {βn } be sequences in
(0, 1) such that lim supn→∞ αn (1 − αn ) > 0 and limn→∞ βn (1 − βn ) > 0, and let {xn } be a sequence
in C defined by the S-iteration process (1.3). Then {xn } converges strongly to a fixed point of T .
Corollary 3.8. Let C be a nonempty compact convex subset of a uniformly convex Banach space X .
Let T : C → C be an α-nonexpansive mapping for some α < 1. Let {αn } and {βn } be sequences in
(0, 1) such that lim supn→∞ αn (1 − αn ) > 0 and limn→∞ βn (1 − βn ) > 0, and let {xn } be a sequence
in C defined by the S-iteration process (1.3). Then {xn } converges strongly to a fixed point of T .
Proposition 3.7. Let C be a nonempty closed convex subset of a Banach space X . Let T : C → C
be a generalized [a, b, c]-nonexpansive mapping such that Fi x(T ) = ∅. Let {αn } be a sequence
in (0, 1), and let {xn } be a sequence defined by Normal S−iteration process (1.4). Then for all
z ∈ Fi x(T ), we have the following:
(a) xn+1 − z ≤ xn − z for all n ∈ N.
(b) The sequence {xn } satisfy limit existence property (LE property), i.e., limn→∞ xn − z
exists.
(c) limn→∞ d(xn , Fi x(T )) exists, where d(xn , Fi x(T )) = inf z∈Fi x(T ) xn − z.
Proof. (a) Let z ∈ Fi x(T ). From (1.4) and using Lemma 3.1, we get
xn+1 − z = T [(1 − αn )xn + αn T xn ] − z
≤
≤
[(1 − αn )xn + αn T xn − z
(1 − αn )xn − z + αn T xn − z
≤
=
(1 − αn )xn − z + αn xn − z
xn − z.
(3.16)
Therefore, we obtain xn+1 − z ≤ xn − z for all n ∈ N. (b) From (3.7), it is clear that the
sequence {xn − z} is a monotonic decreasing sequence of nonnegative numbers. Therefore, it is
convergent i.e., limn→∞ xn − z exist. (c) This part follows from part (b).
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Theorem 3.9. Let C be a nonempty closed convex subset of a uniformly convex Banach space X.
Let T : C → C be a generalized [a, b, c]-nonexpansive mapping such that Fi x(T ) = ∅. Let {αn }
be a sequence in (0, 1), and let {xn } be a sequence in C defined by the Normal S−iteration process
(1.4). Then we have the following:
(a) If {xn } is bounded and lim inf n→∞ xn − T xn = 0, then Fi x(T ) = ∅.
(b) Assume that Fi x(T ) = ∅. Then {xn } is bounded and following holds:
(i) If lim supn→∞ αn (1 − αn ) > 0, then lim inf n→∞ xn − T xn = 0.
(ii) If lim inf n→∞ αn (1 − αn ) > 0, then the sequence {xn } satisfy approximate fixed point
property (AF point property), i.e., limn→∞ xn − T xn = 0.
Proof. (a) This part follows from Theorem 3.3.
(b) (i) Assume that Fi x(T ) = ∅ and z ∈ Fi x(T ). From Proposition 3.7, we have that
limn→∞ xn − z exists and hence {xn } is bounded. Let
lim xn − z = r > 0.
n→∞
(3.17)
In view of Lemma 2.4, there exists a continuous strictly increasing convex function g : [0, ∞) →
[0, ∞) with g(0) = 0 such that
xn+1 − z2
=
T [(1 − αn )xn + αn T xn ] − z2
≤
≤
(1 − αn )xn + αn T xn − z2
(1 − αn )(xn − z) + αn (T xn − z)2
≤
≤
(1 − αn )xn − z2 + αn T xn − z2 − αn (1 − αn )g(xn − T xn )
(1 − αn )xn − z2 + αn xn − z2 − αn (1 − αn )g(xn − T xn )
=
xn − z2 − αn (1 − αn )g(xn − T xn ).
This implies that
αn (1 − αn )g(xn − T xn ) ≤ xn − z2 − xn+1 − z2 .
(3.18)
From (3.18), it is easy to show that
if lim sup αn (1 − αn ) > 0, then lim inf xn − T xn = 0.
n→∞
n→∞
(ii) From (3.18), we can show that
if lim inf αn (1 − αn ) > 0, then lim xn − T xn = 0.
n→∞
n→∞
Corollary 3.9. Let C be a nonempty closed convex subset of a uniformly convex Banach space X.
Let T : C → C be a α-nonexpansive mapping for some α < 1. Let {αn } be a sequence in (0, 1),
and let {xn } be a sequence in C defined by the Normal S−iteration process (1.4). Then we have the
following:
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141
(a) If {xn } is bounded and lim inf n→∞ xn − T xn = 0, then Fi x(T ) = ∅.
(b) Assume that Fi x(T ) = ∅. Then {xn } is bounded and following holds:
(i) If lim supn→∞ αn (1 − αn ) > 0, then lim inf n→∞ xn − T xn = 0.
(ii) If lim inf n→∞ αn (1 − αn ) > 0, then limn→∞ xn − T xn = 0.
Theorem 3.10. Let C be a nonempty closed convex subset of a uniformly convex Banach space
X with the Opial property. Let T : C → C be a generalized [a, b, c]-nonexpansive mapping with
Fi x(T ) = ∅. Let {αn } and be a sequence in (0, 1) such that lim inf n→∞ αn (1 − αn ) > 0, and let {xn }
be a sequence in C defined by the Normal S-iteration process (1.4). Then {xn } converges weakly to
a fixed point of T .
Corollary 3.10. Let C be a nonempty closed convex subset of a uniformly convex Banach space
X with the Opial property. Let T : C → C be an α-nonexpansive mapping for some α < 1 with
Fi x(T ) = ∅. Let {αn } be a sequence in (0, 1) such that lim inf n→∞ αn (1 − αn ) > 0, and let {xn } be
a sequence in C defined by the Normal S-iteration process (1.4). Then {xn } converges weakly to a
fixed point of T .
Theorem 3.11. Let C be a nonempty compact convex subset of a uniformly convex Banach space
X . Let T : C → C be a generalized [a, b, c]-nonexpansive mapping. Let {αn } be a sequence in
(0, 1) such that lim supn→∞ αn (1 − αn ) > 0, and let {xn } be a sequence in C defined by the Normal
S-iteration process (1.4). Then {xn } converges strongly to a fixed point of T .
Corollary 3.11. Let C be a nonempty compact convex subset of a uniformly convex Banach space
X . Let T : C → C be an α-nonexpansive mapping for some α < 1. Let {αn } be a sequence in (0, 1)
such that lim supn→∞ αn (1 − αn ) > 0, and let {xn } be a sequence in C defined by the Normal Siteration process (1.4). Then {xn } converges strongly to a fixed point of T .
4. NUMERICAL EXAMPLE
Example 4.1. Let X = R with usual norm and C = [0, 1]. Let T : [0, 1] → [0, 1] be a mapping
defined by (3.1). Set α = 0.1, a = 0.1, b = 0.6 and c = 0.3. It is shown in Example 3.1 that, T is
not α-nonexpansive but it is generalized [a, b, c]-nonexpansive mapping.
1
1
) and βn = ( 18 + 4n
) for all n ∈ N. For initial points 0.1, 0.4, 0.7 and 1.0, the
Set αn = ( 61 + 4n
convergence of Mann, Ishikawa, S-iteration and Normal S-iterations processes upto 60000 iterations
are shown in Figure 2. Set stopping criteria xn − x ∗ ≤ 10−4 . The effect of parameters αn and
βn on convergence of different iteration processes are given in Table 1 for the following set of
parameters:
(i) αn
(ii) αn
(iii) αn
(iv) αn
1
1
= ( 16 + 4n
) and βn = ( 81 + 4n
).
1
1
1
= (1 − 2n ) and βn = ( 2 + 3n ).
= (1 − 2n1 2 ) and βn = 12 .
n
= n+1
and βn = n−1
.
n+1
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Figure 2. Convergence of iteration processes with different initial points
Table 1. Number of iterations required to obtain fixed point with different initial points and parameters
1
1
αn = ( 61 + 4n
) and βn = ( 18 + 4n
) for all n ∈ N
0.1
0.4
0.7
59917
59960
59966
53263
53303
53309
9780
9786
9786
31
36
36
1
1
αn = (1 − 2n
) and βn = ( 21 − 3n
) for all n ∈ N
Initial point
0.1
0.4
0.7
Mann iteration
9990
9996
9996
Ishikawa iteration
6664
6669
6669
S-iteration
6661
6665
6665
Normal S-iteration
6
6
7
αn = (1 − 2n12 ) and βn = 12 for all n ∈ N
Initial point
0.1
0.4
0.7
Mann iteration
9985
9992
9992
Ishikawa iteration
6659
6663
6663
S-iteration
6658
6662
6662
Normal S-iteration
5
5
5
n
αn = n+1
and βn = n−1
for
all
n
∈
N
n+1
Initial point
0.1
0.4
0.7
Mann iteration
9994
10000
10000
Ishikawa iteration
5009
5012
5013
S-iteration
5005
5008
5007
Normal S-iteration
7
7
7
Initial point
Mann iteration
Ishikawa iteration
S-iteration
Normal S-iteration
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1.0
59969
53311
9787
36
1.0
9997
6669
6666
7
1.0
9992
6664
6663
5
1.0
10001
5013
5008
7
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143
Using stopping criteria xn − x ∗ ≤ 10−4 and above set of parameters, we have studied convergence of different iteration processes for initial points 0.1, 0.4, 0.7 and 1.0. We observe that
the S-iteration process is faster than Mann and Ishikawa iteration processes, and the Normal Siteration process is faster than Mann, Ishikawa and S-iteration processes for generalized [a, b, c]nonexpansive mappings. Also it is stable with respect to the parameters αn and βn .
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Vol. 10, No. 1 (Special Issue), Jan–June 2019
INDIAN JOURNAL OF INDUSTRIAL AND APPLIED MATHEMATICS
Vol. 10, No. 1 (Special Issue), Jan–June 2019, pp. 145–151
DOI:
On Generalized Operator Mixed Vector
Quasi-equilibrium Problem
Rais Ahmad1 , Haider Abbas Rizvi2 , Mijanur Rahaman1 and Javid Iqbal3
1
Department of Mathematics, Aligarh Muslim University, Aligarh-202002, India
Department of Applied Mathematics, ZHCET, Aligarh Muslim University, Aligarh-202002, India
3
Department of Mathematical Sciences, Baba Ghulam Shah Badshah University, Rajouri, India
Email: 1 raisain 123@rediffmail.com; 2 haider.alig.abbas@gmail.com; 1 mrahman96@yahoo.com;
3
javid2iqbal@yahoo.co.in
2
Abstract: In this paper, we study a new operator equilibrium problem which is a conjunction of two
problems i.e., an operator vector quasi-equilibrium problem and an operator vector quasi-variational
inequality problem. We call our problem as generalized operator mixed vector quasi-equilibrium
problem. We prove some existence results for our problem in topological vector spaces by using
1-person game theorem and core of a set of operators. We do not use KKM-theory and generalized
p-quasi convexity as used by other authors in related works. Some special cases of generalized
operator mixed quasi-equilibrium problem are also discussed.
Keywords: Existence, Game Theorem, Core, Operator, Equilibrium.
AMS subject classification: 2010 49J40, 47H19, 47H10
1. INTRODUCTION
The important generalizations of a variational inequality problem is a vector variational inequality
problem as well as a vector equilibrium problem. It is well known in the theory of applied sciences
that equilibrium problems and their generalizations provides us powerful tools to study a wide range
of problems arising in nonlinear analysis, optimization, economics, finance and game theory, etc..
The equilibrium problems includes many mathematical programming problems,complementarity
problems, variational inequality problems, fixed point problems, minimax inequality problem, Nash
equilibrium problems in non-cooperative games, etc., see [3, 5, 8, 15, 16].
Extending the concept of scalar and vector variational inequalities, Domokos and Kolumban [7] introduced the concept of operator variational inequalities and shown that scalar and
vector variational inequalities are the special cases of operator variational inequalities, also see
[1, 2, 4, 9, 11, 12, 17–20] and references therein. On the other hand operator equilibrium problem for
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146
Rais Ahmad, Haider Abbas Rizvi, Mijanur Rahaman and Javid Iqbal
single-valued operators was studied by Kazmi and Raouf [10] and for multi-valued operators by
Kum and Kim [19].
The above mentioned research works motivated us to introduce and study a generalized operator mixed vector quasi-equilibrium problem which involves an operator quasi vector variational
inequality problem as well as an operator quasi vector equilibrium problem. The concept of core of
a set [3] is extended for core of a set of operators and a related proposition is proved. We prove two
existence results for generalized operator mixed vector quasi equilibrium problem by using 1-person
game theorem and core of a set of operators. Our results are more general than others available in
the literature.
2. PRELIMINARIES
Let X and Y be the Hausdorff topological vector spaces, L(X, Y ) be the space of all continuous linear operator from X into Y equipped with the topology of pointwise convergence and K ⊂ L(X, Y )
a nonempty convex set. A nonempty subset C of X is called a convex cone if λC ⊆ C, for all λ > 0
and C + C = C. We denote by 2 K the family of all subsets of K , int X K the interior of K in X ,
cl X K the closure of K in X , and coK the convex hull of K.
Suppose that T : K → X , F : K × K → Y are the single-valued operators such that F( f, f ) =
0, for all f ∈ K . Let A : K → 2 K and C : K → 2Y be multi-valued operators such that C( f ) is a
convex cone with intC( f ) = φ and 0 ∈
/ C( f ), for all f ∈ K . We consider the following problem
of finding f ∈ K such that f ∈ cl K A( f ) and
f − g, T ( f ) + F( f, g) ∈
/ −C( f ), for all g ∈ A( f ).
(2.1)
We call problem 2.1 as generalized operator mixed vector quasi-equilibrium problem.
Below are some special cases of problem 2.1.
T = 0 and A( f ) = K , then from problem 2.1, we get an operator equilibrium problem
mentioned in Kazmi and Rauf [10].
(2) If F = 0 and A( f ) = K , then problem 2.1 reduces to the operator variational inequality
problem considered by Domokos and Kolumbán [7].
(3) If F = 0, A( f ) = K and T is a multi-valued operator, then problem 2.1 coincides with
the operator vector variational inequality studied by Kum and Kim [13].
(4) If T = 0 and F is a multi-valued operator, then problem 2.1 reduces to the generalized
operator quasi-equilibrium problem considered by Kum and Kim [14].
(1)
We need the following lemma, which is a variant form of Theorem 2 of Ding, Kim and Tan [6].
Lemma 2.1. Let K be a nonempty compact convex subset of a Hausdorff topological vector space
X . Suppose that A, cl X A, P : K → 2 K are multi-valued operator such that for each x ∈ K , A(x) is
nonempty convex set, for each y ∈ K , A−1 (y) is open set in K , cl X A is upper semicontinuous, for
each x ∈ K , x ∈
/ CoP(x) and for each y ∈ K , P −1 (y) in open in K . Then there exists x ∗ ∈ K such
∗
that x ∈ cl K A(x ∗ ) and A(x ∗ ) ∩ P(x ∗ ) = φ.
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The following definition is an extension of definition of Core of a set defined by Blum and
Oettli [3] for operators.
Definition 2.1. Let K and D be the convex subsets of L(X, Y ) and D ⊂ K . The core of D relative
to K , denoted by cor e K f D, is the set defined by f ∈ cor e K f D if and only if f ∈ D and D ∩ {λ f +
(1 − λ)g} = φ, λ ∈ (0, 1), for all g ∈ K \ D.
Definition 2.2. An operator T : → X is said to be C( f )-convex if for any f, g ∈ K and λ ∈ [0, 1],
T (λg + (1 − λ f )) ∈ λT (g) + (1 − λ)T ( f ) − C( f ).
Definition 2.3. Let B be a subset of K . A multi-valued operator W : K → 2Y is said to have a
closed graph with respect to B if for every net { f α }α∈τ ⊂ K and {yα }α∈τ ⊂ Y such that yα ∈ C( f α ),
f α converges to f ∈ B and yα converges to y ∈ Y, then y ∈ C( f ).
3. EXISTENCE RESULTS
In this section, we prove some existence results for generalized operator mixed vector quasiequilibrium problem 2.1.
Theorem 3.1. Let K be a nonempty compact convex subset of L(X, Y ), where L(X, Y ) is equipped
with the topology of pointwise convergence. Let T : K → X , F : K × K → Y be the single-valued
operators and C : K → 2Y be the multi-valued operator such that for each f ∈ K , C( f ) is closed
convex cone in Y with intC( f ) = φ and 0 ∈
/ C( f ).
Assume that
For each f ∈ K , F( f, f ) = 0,
the operator T is continuous,
the operator F is continuous in the first argument and affine in the second argument,
the operator W : K → 2Y defined by W ( f ) = Y \ −C( f ) has a closed graph in K × Y ,
the multi-valued operator A : K → 2Y such that A( f ) is nonempty convex subset of K and
A−1 (g) is open in K for each f, g ∈ K ,
6. the mapping cl K A( f ) : K → 2 K is upper semicontinuous.
1.
2.
3.
4.
5.
Then the generalized operator mixed vector quasi-equilibrium problem 2.1 admits a solution.
Proof. As L(X, Y ) is equipped with the topology of pointwise convergence, it is a locally convex
Hausdorff topological vector space. We define a multi-valued operator P : K → 2 K by
P( f ) = {g ∈ K : f − g, T ( f ) + F( f, g) ∈ −C( f )}, for each f ∈ K .
Initially, we show that f ∈
/ CoP( f ). Suppose to the contrary that there exists f ∈ K such
n that f ∈
CoP( f ). Then there exists a finite subset {g1 , g2 , ....gn } of K and λi ≥ 0, such that i=1
λi = 1
n
λi gi .
and f = i=1
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That is
f − gi , T ( f ) + F( f, gi ) ∈ −C( f ).
Hence
n
λi f − gi , T ( f ) +
i=1
n
F( f, gi ) ∈ −C( f ).
i=1
It is clear from the hypothesis that
f −
n
i=1
λi gi , T ( f ) + F( f,
n
λi gi ) ∈ −C( f ),
i=1
i.e., f − f, T ( f ) + F( f, f ) ∈ −C( f ).
Since F( f, f ) = 0, we have,
0 ∈ −C( f ),
which contradicts that 0 ∈
/ C( f ) and hence f ∈
/ CoP( f ).
Now, we show that P −1 (g) is open in K , which is equivalent to show that [P −1 (g)]c = K \
/
P −1 (g) is closed. In fact, let { f λ } be a net in [P −1 (g)]c convergent to f ∈ K . Then, we have g ∈
P( f λ ) and hence
f λ − g, T ( f λ ) + F( f λ , g) ∈
/ −C( f λ ),
thus
f λ − g, T ( f λ ) + F( f λ , g) ∈ W ( f λ ).
As W has a closed graph in K × Y , T is continuous and F is continuous in the first argument, we
have
f − g, T ( f ) + F( f, g) ∈ W ( f ),
i.e., f − g, T ( f ) + F( f, g) ∈
/ −C( f ).
Hence f ∈ [ p −1 (g)]c and [ p −1 (g)]c is closed, i.e., P −1 is open in K . Thus all the condition of
Lemma 2.1 are satisfied. Hence there exists f ∈ K such that f ∈ cl K A( f ) and A( f ) ∩ P( f ) = φ,
i.e.,
f − g, T ( f ) + F( f, g) ∈
/ −C( f ), for all g ∈ A( f ).
Thus the generalized operator mixed vector quasi-equilibrium problem 2.1 admits a solution.
The following corollaries can be derived from Theorem 3.1, easily.
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Corollary 3.1. If T = 0 and F( f, g) = ϕ( f ) − ϕ(g), where ϕ : K → Y is a mapping, then there
exists f ∈ K such that the following operator minimization problem has a solution.
/ −C( f ), for all g ∈ A( f ).
f ∈ cl K A( f ) and ϕ( f ) − ϕ(g) ∈
Corollary 3.2. If F = 0, then there exists f ∈ K such that the following operator variational
inequality problem has a solution f ∈ cl K A( f ) and
f − g, T ( f ) ∈
/ −C( f ), for all g ∈ A( f ).
The following Lemma is an extension of Lemma 4 of Blum and Oettli [3] for operators.
Lemma 3.1. Let K and D be the convex subsets of L(X, Y ) and D ⊂ K . Assume that ϕ : K →
L(X, Y ) is C( f )-convex, f 0 ∈ cor e K f D, ϕ( f 0 ) ∈ −C( f ) and ϕ(g) ∈ C( f ), for all g ∈ D. Then
ϕ(g) ∈ C( f ), for all g ∈ K .
Proof. On the contrary, suppose that ϕ(h) ∈
/ C( f ), for some h ∈ K \ D. Suppose for some λ ∈
(0, 1), f̄ = λ f 0 + (1 − λ)h. By using the C( f )-convexity of ϕ, we have
ϕ(λ f 0 + (1 − λ)h)
∈
λϕ( f 0 ) + (1 − λ)ϕ(h) − C( f )
i.e., ϕ( f̄ )
∈
λϕ( f 0 ) + (1 − λ)ϕ(h) − C( f )
∈
−C( f ) + [Y \ C( f )] − C( f )
= [Y \ C( f )],
That is ϕ( f̄ ) ∈
/ C( f ).
Since f 0 ∈ cor ek f D and D ∩ {λ f 0 + (1 − λh)} = φ, so we have g ∈ D ∩ {λ f 0 + (1 − λh)}
such that ϕ(g) ∈
/ C( f ), which is a contradiction to hypothesis. Hence
ϕ(g) ∈ C( f ), ∀g ∈ K .
Without using compactness assumption and employing above Lemma 3.1, we prove the following result for generalized operator mixed vector quasi-equilibrium problem 2.1.
Theorem 3.2. Let K be a nonempty convex subset of L(X, Y ). Let T : K → X and F : K × K →
Y be the single-valued operators and C : K → 2Y and A : K → 2 K be the multi-valued operators
such that intC( f ) = φ and 0 ∈ C( f ), for each f ∈ K , satisfying all the conditions of Theorem 3.1.
In addition, the following condition is satisfied : there exists a non-empty compact convex subset D
of K such that for f ∈ D \ cor e K f D, h ∈ cor e K f D ∩ A( f ), such that
f ∈ cl D A( f ), f − h, T ( f ) + F( f, h) ∈ −C( f ).
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Then, there exists f ∈ K such that f ∈ cl K A( f ) and
f − g, T ( f ) + F( f, g) ∈
/ −C( f ), for all g ∈ A( f ).
Proof. By Theorem 3.1, it follows that there exists f ∈ D such that f ∈ cl D A( f ) and
f − g, T ( f ) + F( f, g) ∈
/ −C( f ), ∀ g ∈ A( f ).
Set ϕ(g) = f − g, T ( f ) + F( f, g). Then ϕ(g) is C( f )-convex and ϕ(g) ∈ C( f ), for all g ∈ D.
If f ∈ cor e K f D, then choose f 0 = f. If f ∈ D \ Cor e K f D, then choose f 0 = h, where h is same
as in the hypothesis of the theorem. In both cases, f 0 ∈ Cor e K f D and φ( f 0 ) ∈ −C( f ). Applying
Lemma 3.1, it follows that φ(g) ∈ C( f ), for all g ∈ K , which implies that there exist f ∈ K such
that
f − g, T ( f ) + F( f, g) ∈ C( f ), for all g ∈ K .
As cl K A : K → 2 K is upper semicontinuous with compact values, there exists f ∈ K such that
f ∈ cl K A( f ) and f − g, T ( f ) + F( f, g) ∈
/ −C( f ), for all g ∈ A( f ).
REFERENCES
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[9] N. Hadjisavas, S. Schiable, From scalar to vector equilibrium problems in the quasi monotone case, J. optim. Theory
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[10] K.R. Kazmi, A. Raouf, A class of operator equilibrium problems, J. Math. Anal. Appl., 308(2005), 554–564.
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[12] I.V. Konnov and J.C. Yao, Existence of solutions for generalized vector equilibrium problems, J. Math. Anal. Appl.,
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[13] S. Kum and W.K. Kim, An extension of vector variational inequalities with operator solutions, Nonlinear and Convex
Analysis (Edited by T. Tanaka) 2004.4, Kokyuroku, Kyoto University, Japan.
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565.
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[16] A. Moudafi, Mixed equilibrium problems: Sensitivity analysis and algorithmic aspect, Computers and Mathematics
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INDIAN JOURNAL OF INDUSTRIAL AND APPLIED MATHEMATICS
Vol. 10, No. 1 (Special Issue), Jan–June 2019, pp. 152–164
DOI:
On Approximation of Signals in the Generalized
Zygmund Class Using (E, 1)(N, pn )-summability
Means of Fourier Series
Tejaswini Pradhan1 , Susanta Kumar Paikray2∗ and Umakanta Misra3
1,2
Department of Mathematics, Veer Surendra Sai University of Technology, Burla, Odisha-768018,
India
3
Department of Mathematics, National Institute of Science and Technology, Palur Hills, Golanthara,
Odisha-761008, India
(∗ Corresponding author) Email: ∗ skpaikray math@vssut.ac.in, 1 tejaswini.bini@gmail.com,
3
umakanta misra@yahoo.com
Abstract: In this paper, we investigate the degree of approximation of a function in the generalized
Zygmund class Z r(ω) (r ≥ 1) by (E, 1)(N , pn )-summability means of trigonometric Fourier series.
Keywords: Degree of approximation; generalized Zygmund class; trigonometric Fourier series;
(E, 1)-summability means; (N , pn )-summability means; (E, 1)(N , pn )-summability means
AMS Subject classification (2010): 41A24; 41A25; 42B05; 42B08
1. INTRODUCTION AND PRELIMINARIES
Signal analysis is concerned with the reliable estimation, detection and classification of signals
(functions) which are subject to random fluctuations. Signal analysis has its roots in probability theory, mathematical statistics and, more recently, approximation theory and communications theory.
These approximation analysis of signals (functions) have great importance in the field of science and
engineering. And it has given a new dimensions due to their wide applications in signal analysis,
system design in modern telecommunications, radar and image processing system. The error estimation of functions in various function spaces such as Lipschitz, Hölder, Zygmund, Besov spaces
using different summability techniques of trigonometric fourier series has been received a growing
interest of several researchers in last decades. Functions in L r (r ≥ 1)-spaces assumed to be most
practicable in signal analysis. Particularly, L 1 , L 2 and L ∞ spaces are used by engineers for designing digital filters. Very recently, the generalized different Lipschitz classes have been established
in various summability means. For more details, see the current works [5, 6, 7, 8] and [9]. Subsequently, the generalized Zygmund class Z r(ω) (r ≥ 1) is a generalization of Z (α) , Z (α),r , Z (ω) -classes.
The generalized Zygmund class Z r(ω) (r ≥ 1) is investigated by Leindler [4], Moricz [11], Moricz
and Nemeth [12]. Very recently in 2017 Singh et al. [14] proved approximation of functions in the
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Approximation of Signals in the Generalized Zygmund Class
153
generalized Zygmund class using Hausdörff means. Lal and Shireen [2] proved best approximation
of functions of generalized Zygmund class by matrix-Euler summability mean of Fourier series.
To get best approximation and advance study in this direction, In the proposed paper, we investigate the degree of approximation of a function in the generalized Zygmund class Z r(ω) (r ≥ 1) by
(E, 1)(N , pn ) means of trigonometric Fourier series.
Let f be a 2π -periodic and integrable functions belonging to
2π
L r [0, 2π ] = f : [0, 2π ] → R;
| f (x)|r d x < ∞ .
0
The Fourier series of f (x) is given by
∞
u n (x) =
n=0
∞
1
(an cos nx + bn sin nx)
a0 +
2
n=1
(1.1)
with its partial sum
Sn ( f ; x) =
1
π
π
−π
f (x + t)Dn (t)dt,
where
Dn (t) =
sin(n + 12 )t
.
2 sin 2t
The L r norm of a function f (x) is defined by
⎧
2π
⎪
1
r
⎪
⎪
⎨ 2π 0 | f (x)| d x
f r =
⎪
⎪
⎪
⎩ess sup | f (x)|
1
r
(1 ≤ r < ∞)
(r = ∞).
0<x≤2π
Zygmund modulus of continuity of f (x) is defined by
ω( f, h) =
sup
| f (x + t) + f (x − t) − 2 f (x)| (see [16]).
0≤h, x∈R
Let C2π denote the Banach space of all 2π -periodic continuous functions defined on [0, 2π ]
under the supremum norm. For 0 < α ≤ 1, the function space
Z (α) = { f ∈ C2π : | f (x + t) + f (x − t) − 2 f (x)| = O(|t|α )}
is a Banach space under the norm · (α) is defined by
| f (x + t) + f (x − t) − 2 f (x)|
.
|t|α
x,t =0
f (α) = sup | f (x)| + sup
0≤x≤2π
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Tejaswini Pradhan, Susanta Kumar Paikray and Umakanta Misra
For f ∈ L r [0, 2π ] (r ≥ 1) the integral Zygmund modulus of continuity is defined by
ωr ( f, h) = sup
0<t≤h
1
2π
2π
| f (x + t) + f (x − t) − 2 f (x)| d x
r1
r
.
0
For f ∈ C2π and r = ∞,
ω∞ ( f, h) = sup max | f (x + t) + f (x − t) − 2 f (x)|.
0<t≤h
x
It is known that
ωr ( f, h) → 0 as r → 0.
Now define
Z (α),r =
f ∈ L r [0, 2π ] :
2π
| f (x + t) + f (x − t) − 2 f (x)| d x
r
r1
α
= O(|t| ) .
0
The space Z (α),r , r ≥ 1, 0 < α ≤ 1 is a Banach space under the norm · (α),r .
f (α),r = f r + sup
t =0
f (· + t) + f (· − t) − 2 f (·)r
.
|t|α
The class of function Z (ω) is defined as
Z (ω) = { f ∈ C2π : | f (x + t) + f (x − t) − 2 f (x)| = O(ω(t))}
where ω is a Zygmund modulus of continuity, that is, ω is positive, non-decreasing continuous
function with the sub linearity property that is,
(i) ω(0) = 0
(ii) ω(t1 + t2 ) ≤ ω(t1 ) + ω(t2 )
Let ω : [0, 2π ] → R be an arbitrary function with ω(t) > 0 for 0 ≤ t < 2π and let
limt→0+ ω(t) = ω(0) = 0, define
f (· + t) + f (· − t) − 2 f (·)r
(ω)
Z r = f ∈ L r : 1 ≤ r ≤ ∞, sup
<∞
ω(t)
t =0
where
f r(ω) = f r + sup
t =0
f (· + t) + f (· − t) − 2 f (·)r
, r ≥ 1.
ω(t)
Clearly, · r(ω) is a norm on Z r(ω) . As we know L r (r ≥ 1) is complete, the space Z r(ω) is also
complete. Hence we can say Z r(ω) is a Banach space under the norm · r(ω) .
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Approximation of Signals in the Generalized Zygmund Class
Let ω(t) and v(t) denotes the Zygmund moduli of continuity such that
decreasing, then
ω(2π )
f r(ω) ≤ ∞.
f r(v) ≤ max 1,
v(2π )
Thus, we have
ω(t)
v(t)
155
be positive, non-
(1.2)
Z r(ω) ⊆ Z r(v) ⊆ L r (r ≥ 1).
Remark 1.
(i)
If we take r → ∞ then the class Z r(ω) reduces to the Z (ω) class.
(ii) If we take ω(t) = t α in Z r(ω) class, then it reduces to Z (α),r class.
(iii) If we take ω(t) = t α , the Z (ω) class reduces to Z (α) class.
Let u n be a given infinite series with the sequence of partial sum {sn }. Let { pk } for k = 0, 1, 2, ....
be a sequence of nonnegative integers such that p0 > 0 and
n
Pn =
pk → ∞ as n → ∞.
(1.3)
k=0
Let the sequence to sequence transformation
n
1 τnN =
pk sk , n = 0, 1, 2, ...,
Pn k=0
(1.4)
u n is said to be
defines (N , pn ) mean of {sn } generated by the sequence { pk }. The series
summable to s by (N , pn ) method if, limn→∞ τnN → s as n → ∞ and this (N , pn ) method is regular
[1].
The sequence to sequence transformation,
E n1
n 1 k
sk ,
= n
2 k=0 v
(1.5)
defines the (E, 1) transform of the sequence {sn }. The series u n is summable to s with respect to
(E, 1) summability, If E n1 → s as n → ∞. Also (E, 1) method is regular [1].
Now we define a new composite transform using the product (E, 1)(N , pn ) transform. As the
(N , pn ) and (E, 1) summability methods are regular, the product (E, 1)(N , pn ) method is also regular.
Let
n k
1 n
1 EN
p v sv ,
(1.6)
τn = n
2 k=0 k
Pk v=0
defines the (E, 1)(N , pn ) transform of the sequence {sn }. We say that series
by product (E, 1)(N , pn ) transform, if τnE N → s as n → ∞.
I NDIAN J OURNAL OF I NDUSTRIAL AND A PPLIED M ATHEMATICS
u n is summable to s
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Tejaswini Pradhan, Susanta Kumar Paikray and Umakanta Misra
We use the following notations through out this paper;
φ(x, t) = f (x + t) + f (x − t) − 2 f (x)
k
n 1
sin(v
+
)t
n
1
1
2
K nE N (t) =
pv
.
π 2n+1 k=0 k
Pk v=0
sin( 2t )
2. KNOWN THEOREMS
Lal and Shireen [2] proved the following theorems using Matrix-Euler summability means of
Fourier series.
Theorem 1. Let the lower triangular matrix A = (an,k ) satisfying the following conditions:
n
an,k ≥ 0 (n = 0, 1, 2, ...; k = 0, 1, 2, ...);
n−1
|an,k | = O
k=0
1
n+1
an,k = 1,
(2.1)
k=0
and (n + 1)an,n = O(1).
(2.2)
if f : [0, 2π ] → R be a 2π periodic, Lebesgue integrable and belonging to generalized Zygmund
class Z r(ω) (r ≥ 1); ω and v be Zygmund modulus of continuity and ω(t)
be positive, non-decreasing
v(t)
then the best approximation of f by triangular matrix-Euler means
tnE =
n
an,k
k=0
k 1 k
sv
2k v=0 v
of its Fourier series (1.1) is given by
E n ( f ) = inf
tnE
tnE
−
f r(v)
=O
1
n+1
π
1
n+1
ω(t)
.
t 2 v(t)
(2.3)
Theorem 2. Let A = (an,k ) be lower triangular matrix satisfying conditions (2.7) and (2.8) and in
ω(t)
is non-increasing. For f ∈ Z r(ω) (r ≥ 1), its best approximation by
addition to Theorem 1, tv(t)
triangular matrix-Euler means tnE satisfies
1
ω( n+1
)
En = O
log(n + 1)π .
(2.4)
1
v( n+1
)
3. MAIN RESULTS
The main objective of this paper is to prove the following theorems.
Theorem 1. Let f : [0, 2π ] → R be a 2π periodic, Lebesgue integrable and belonging to generalized Zygmund class Z r(ω) (r ≥ 1). Then the degree of approximation of signal (function) f, using
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product (E, 1)(N , pn ) mean of Fourier series (1.1), is given by
E n ( f ) = inf τnE N − f rv = O
τnE N
π
1
n+1
ω(t)
,
t v(t)
where ω(t) and v(t) denotes the Zygmund moduli of continuity such that
ing.
157
(3.1)
ω(t)
v(t)
is positive and increas-
Theorem 2. Let f : [0, 2π ] → R be a 2π periodic, Lebesgue integrable and belonging to generalized Zygmund class Z r(ω) (r ≥ 1). Then the degree of approximation of signal (function) f, using
(E, 1)(N , pn ) means of Fourier series (1.1), is given by
1
ω( n+1
)
EN
v
E n ( f ) = inf τn − f r = O
(π (n + 1) − 1) ,
(3.2)
1
(n + 1)2 v( n+1
)
τnE N
where w(t) and v(t) denotes the Zygmund moduli of continuity such that
decreasing.
w(t)
tv(t)
is positive and
To prove the above theorems, we need the following Lemmas.
Lemma 1. |K nE N (t)| = O(n) for 0 ≤ t ≤
Proof. For 0 ≤ t ≤
1
n+1
1
.
n+1
and sin nt ≤ n sin t.
|K nE N (t)|
1
≤
π 2n+1
n n
k
1
≤
π 2n+1
n n
k
k=0
k
1 sin(v + 12 )t
pv
Pk v=0
sin( 2t )
k=0
k
1 (2v + 1) sin( 2t )t
pv
Pk v=0
sin( 2t )
n k
1 n
2k + 1 =
pv n+1
k
π2
P
k
k=0
v=0
n 2n + 1 n =
k π 2n+1 k=0
2n + 1
=
π 2n+1
∵ 2n =
n n
k=0
= O(n).
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k
(3.3)
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Lemma 2. |K nE N (t)| = O( 1t ) for
Proof. For
1
n+1
1
n+1
≤ t ≤ π.
≤ t ≤ π , sin nt ≤ 1 and sin 2t ≥ πt .
n n
1
1
|K nE N (t)| ≤
π 2n+1 k=0 k
Pk
n 1 n
1
≤
π 2n+1 k=0 k
Pk
n 1
π n
≤
· π 2n+1 t k=0 k
n 1 n ≤
t2n+1 k=0 k 1
= O
.
t
k
sin(v + 12 )t
pv
sin( 2t )
v=0
k
1
pv
t
sin(
)
2
v=0
k
1 pv Pk
v=0
(3.4)
Lemma 3 (see [2. )] Let f ∈ Z r(ω) then for 0 < t ≤ π,
(i)
φ(·, t)r = O(ω(t))
O(ω(t))
O(ω(y))
(iii) If ω(t) and v(t) defined as in Theorem 1, then
(ii)
φ(· + y, t) + φ(· − y, t) − 2φ(·, t)r =
ω(t)
φ(· + y, t) + φ(· − y, t) − 2φ(·, t)r = O v(y)
v(t)
where
φ(x, t) = f (x + t) + f (x − t) − 2 f (x).
4. PROOF OF THEOREM 3
Let sk ( f ; x) denotes the k th partial sum of the series (1.1) and following [15], we have
π
1
sin(k + 1/2)t
dt.
φ(x; t)
sk ( f ; x) − f (x) =
2π 0
sin(t/2)
Therefore using (1.4), the (N , pn ) transform of sk ( f ; x) is given by
π
n
n
1 1
1 sin(k + 1/2)t
dt.
pk (sk ( f ; x) − f (x)) =
φ(x; t)
pk
Pn k=0
2π 0
Pn k=0
sin(t/2)
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(4.1)
(4.2)
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159
Now denoting the (E, 1)(N , pn ) transform of sk ( f ; x) by τnE N , then we write
v
n 1 π φ(x; t) 1 − f (x) =
pv sin(v + 1/2)tdt.
π 2n+1 k=0 0 sin(t/2) Pk k=0
τnE N
(4.3)
Let
Ln (x) = τnE N − f (x) =
π
0
φ(x; t)K nE N (t)dt.
(4.4)
Then
Ln (x + y) + Ln (x − y) − 2Ln (x) =
π
0
[φ(x + y; t) + φ(x − y; t) − 2φ(x; t)] K nE N (t)dt. (4.5)
Using generalized Minkowski’s inequality we have
Ln (· + y) + Ln (· − y) − 2Ln (·)r
=
=
1
2π
1
2π
≤
0
π
1/r
2π
|Ln (x + y) + Ln (x − y) − 2Ln (x)|r d x
0
π
r 1/r
EN
[φ(x + y; t) + φ(x − y; t) − 2φ(x; t)]K n (t)dt d x
2π
0
1
2π
0
2π
0
π
=
0
r 1/r
dt.
[φ(x + y; t) + φ(x − y; t) − 2φ(x; t)]K nE N (t)d x (|K nE N (t)|)1/r
+φ(x − y; t) −
π
=
0
0
+
π
1
n+1
0
2π
[φ(x + y; t)
r 1/r
EN
2φ(x; t)]K n (t) d x
dt
φ(· + y; t) + φ(· − y; t) − 2φ(·; t)r |K nE N (t)|dt
1
n+1
=
1
2π
φ(· + y; t) + φ(· − y; t) − 2φ(·; t)r |K nE N (t)|dt
φ(· + y; t) + φ(· − y; t) − 2φ(·; t)r |K nE N (t)|dt
= I1 + I2 (say).
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Now, using Lemma 1, Lemma 3 and monotonicity of (ω(t)/v(t)) with respect to t, we have
I1
1
n+1
=
≤
φ(· + y; t) + φ(· − y; t) − 2φ(·; t)r |K nE N (t)|dt
ω(t)
O(n)dt
O v(y)
v(t)
0
1
n+1 ω(t)
dt .
O n v(y)
v(t)
0
=
0
1
n+1
By using 2nd mean value theorem of integral, we have
1
1
ω( n+1
) n+1
I1 ≤ O nv(y)
dt
1
v( n+1
) 0
1
)
ω( n+1
n
= O
v(y)
1
n+1
v( n+1
)
1
)
ω(
n
∵
=
O(1)
.
= O v(y) n+1
1
n+1
v( n+1
)
(4.7)
Next, using Lemma 2 and Lemma 3, we get
π
φ(· + y; t) + φ(· − y; t) − 2φ(·; t)r |K nE N (t)|dt
I2 =
1
≤
=
ω(t) 1
dt
v(y)
1
v(t) t
n+1
π
ω(t)
O v(y)
dt .
1
tv(t)
n+1
n+1
π
(4.8)
From (4.20), (4.21) and (4.22), we have
Ln (· + y) + Ln (· − y) − 2Ln (·)r = O v(y)
1
ω( n+1
)
+ O v(y)
1
v( n+1
)
π
1
n+1
ω(t)
dt .
tv(t)
(4.9)
Therefore, we have
Ln (· + y) + Ln (· − y) − 2Ln (·)r
=O
sup
v(y)
y =0
1
ω( n+1
)
1
v( n+1
)
+O
π
1
n+1
ω(t)
dt .
tv(t)
(4.10)
Clearly,
φ(x; t) = | f (x + t) + f (x − t) − 2 f (x)|.
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161
Now applying Minkowski’s inequality, we have
φ(·, t)r
=
=
f (x + t) + f (x − t) − 2 f (x)r
O(ω(t)).
(4.11)
Now using Lemma 1, Lemma 2 and (4.25), we obtain
Ln (·)r
≤
1
n+1
+
1
n+1
0
=
=
=
=
π
φ(·, t)r |K nE N (t)|dt
ω(t)
dt
ω(t)dt + O
1
t
0
n+1
1
π
n+1
1
w(t)
O nω(
)
dt
ω(t)dt + O
1
n+1 0
t
n+1
π
n
1
ω(t)
O
w(
) +O
dt
1
n+1
n+1
t
n+1
π
1
ω(t)
O ω(
) +O
dt .
1
n+1
t
n+1
O n
1
n+1
π
(4.12)
From (4.24) and (4.26), we have
Ln (·)rv
Ln (· + y) + Ln (· − y) − 2Ln (·)r
v(y)
y =0
1
π
π
ω( n+1
)
ω(t)
ω(t)
1
) +O
dt + O
dt .
O ω(
+O
1
1
1
n+1
t
tv(t)
v( n+1
)
n+1
n+1
= Ln (·)r + sup
=
=
4
Ji .
(4.13)
i=1
Now we write J1 in terms of J3 and further J2 , J3 in terms of J4 .
In view of monotonicity of v(t) for 0 < t ≤ π , we have
ω(t)
ω(t)
.v(t) ≤ v(π )
.v(t) = O
ω(t) =
v(t)
v(t)
ω(t)
.
v(t)
Therefore, we can write
J1 = O(J3 ).
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Tejaswini Pradhan, Susanta Kumar Paikray and Umakanta Misra
Again by using monotonicity of v(t), v
π
ω(t)
J2 =
dt
1
t
n+1
1
n+1
≤
=
Using the fact that
ω(t)
v(t)
π
ω(t)
v(t)dt
tv(t)
π
ω(t)
dt
v(π )
1
tv(t)
n+1
=
O(J4 ).
(4.15)
is positive and increasing, we have
π
1
1
ω( n+1
ω( n+1
) π dt
)
ω(t)
dt =
≥
J4 =
.
1
1
1
1
tv(t)
t
v( n+1 ) n+1
v( n+1 )
n+1
(4.16)
Therefore, we can write
J3 = O(J4 ).
Now combining (4.27) and (4.31), we have
Ln (·)rv
= O(J4 ) = O
En ( f ) =
ω(t)
.
tv(t)
π
1
n+1
Hence,
inf Ln (·)rv
n
(4.17)
=O
π
1
n+1
ω(t)
.
tv(t)
(4.18)
(4.19)
This completes the proof of Theorem 3.
PROOF OF THEOREM 4
Following the proof of Theorem 3, we have
E n ( f ) = inf Ln (·)rv = O
n
In Theorem 4 we assumed
π
1
n+1
ω(t)
dt .
tv(t)
(4.20)
ω(t)
tv(t)
is positive and decreasing in t. Thus, we have
π 1
ω( n+1
)
v
dt
E n ( f ) = inf Ln (·)r = O
1
n
1
(n + 1)v( n+1
) n+1
1
ω( n+1
)
= O
[t]π 1
1
(n + 1)v( n+1
) n+1
1
ω( n+1
)
= O
(π (n + 1) − 1) .
1
(n + 1)2 v( n+1
)
(4.21)
This completes the proof of Theorem 4.
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163
5. APPLICATIONS
The following corollaries can be obtained from Theorem 3.
Corollary 1. If we replace (E, 1)(N , pn ) mean by (E, 1)(C, 1) mean in Theorem 3, then the degree
of approximation of a function f ∈ Z r(ω) by (E, 1)(C, 1) mean
τnEC
n k
1 1 n
= n
sv (see [13])
2 k=0 k k + 1 v=0
of Fourier series (1.1) is given by is given by
En ( f ) = O
π
1
n+1
ω(t)
dt .
tv(t)
(5.1)
Corollary 2. If we replace (E, 1)(N , pn ) mean by (E, q)(N , pn , qn ) mean in Theorem 3, then the
degree of approximation of a function f ∈ Z r(ω) by (E, q)(N , pn , qn ) mean
n k
n n−k 1 1
EN
τn =
q
pk−v qv sv (see [10])
(1 + q)n k=0 k
Rk v=0
of Fourier series (1.1) is given by
En ( f ) = O
π
1
n+1
ω(t)
dt .
tv(t)
(5.2)
Corollary 3. If we replace (E, 1)(N , pn ) mean by Euler-Hausdörff mean in Theorem 3, then the
degree of approximation of a function f ∈ Z r(ω) by Euler-Hausdörff mean
τnE H =
n k
n n−k 1
q
h k,v sv (see [3])
(1 + q)n k=0 k
v=0
of Fourier series (1.1) is given by
En ( f ) = O
1
n+1
π
1
n+1
ω(t)
dt .
t 2 v(t)
(5.3)
REFERENCES
[1] Hardy G. H., Divergent Series, Oxford University Press, First Edition, (1949).
[2] Lal S. and Shireen, Best approximation of functions of generalized Zygmund class by Matrix-Euler summability mean
of Fourier series.Bull. Math. Anal. Appl., 5(4) (2013), 1–13.
[3] Lal S. and Mishra A., Euler-Hausdörff matrix summability operator and trigonometric approximation of the conjugate
of a function belonging to the generalized Lipschitz class, J. Inequal. Appl., 2013, 59 (2013).
[4] Leindler L., Strong approximation and generalized Zygmund class.Acta Sci. Math., 43 (1981), 301–309.
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[5] Mishra V. N., Khatri K. and Mishra L. N., Approximation of Functions belonging to Li p(ξ (t), r ) class by
(N , pn )(E, q)-summability of Conjugate Series of Fourier series, J. Inequal. Appl., 2012 (2012), Article ID: 296.
[6] Mishra L. N., Mishra V. N., Khatri K. and Deepmala, On the Trigonometric approximation of signals belonging
to generalized weighted Lipschitz W (L r , ξ (t))(r ≥ 1)− class by matrix (C 1 .N p ) Operator of conjugate series of its
Fourier series, Appl. Math. Comput., 237 (2014), 252–263.
[7] Mishra V. N., Khatri K. and Mishra L. N., Product (N , pn )(C, 1)-summability of a sequence of Fourier coefficients,
Math. Sci., 2012 2012, DOI: 10.1186/2251-7456-6-38.
[8] Mishra V. N., Khatri K., Mishra L. N. and Deepmala, Trigonometric approximation of periodic signals belonging to
generalized weighted Lipschitz W (L r , ξ (t)), (r ≥ 1)− class by Nörlund-Euler (N , pn )(E, q) operator of conjugate
series of its Fourier series, Journal of J. Class. Anal., 5 (2014), 91–105.
[9] Mishra V. N. and Mishra L. N., Trigonometric Approximation of Signals (Functions) in L p ( p ≥ 1)-norm, International Journal of Contemporary Mathematical Sciences, 7 2012, 909–918.
[10] Misra M., Palo P., Padhy B. P., Samanta P., Misra U. K., Approximation of Fourier series of a function of Lipchitz
class by product means, J. Adv. Math., 9 (4) (2014), 2475–2483.
[11] Moricz F., Enlarged Lipschitz and Zygmund classes of functions and Fourier transforms. East J. Approx., 16 (3)
(2010), 259–271.
[12] Moricz F. and Nemeth J., Generalized Zygmund classes of functions and strong approximation by Fourier series, Acta
Sci. Math., (Szeged), 73 (2007), 637–647.
[13] Nigam H. K. and Sharma K., On (E,1)(C,1) summability of Fourier series and its conjugate series, Int. J. Pure Appl.
Math., 82 (3) (2013), 365–375.
[14] Singh M. V., Mittal M. L. and Rhoades B. E., Approximation of functions in the generalized Zygmund class using
Hausdorff means, J. Inequal. Appl., 2017, 101 (2017), DOI 10.1186/s13600-017-1361-8.
[15] Titechmalch E. C., The Theory of Functions, Oxford University Press, (1939).
[16] Zygmund A., Trigonometric series, 2nd rev. ed., I, Cambridge Univ. Press, Cambridge, 51 (1968).
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INDIAN JOURNAL OF INDUSTRIAL AND APPLIED MATHEMATICS
Vol. 10, No. 1 (Special Issue), Jan–June 2019, pp. 165–181
DOI:
n-tupled Coincidence Point Results for Nonlinear
Contraction in Partially Ordered Complete Metric
Spaces
Mohammad Imdad1 , Aftab Alam2 , Anupam Sharma3∗ and Mohammad Arif4
1,2,4
Department of Mathematics, Aligarh Muslim University, Aligarh-202 002, India
Department of Mathematics and Statistics, Indian Institute of Technology, Kanpur-208016, India
(∗ Corresponding author) Email: ∗ anupam@iitk.ac.in, 1 mhimdad@yahoo.co.in, 2 aafu.amu@gmail.com,
4
mohdarif154c@gmail.com
3
Abstract: In this paper, we establish n-tupled coincidence point theorems for a pair of mappings
satisfying a nonlinear contractivity condition in partially ordered complete metric spaces. Our
results extend and generalize several existing results in the literature: [Bhaskar and Lakshmikantham: Nonlinear Anal. 65(7) (2006), 1379-1393], [Lakshmikantham, and Ćirić: Nonlinear Anal.
70 (2009), 4341-4349], [Berinde and Borcut: Nonlinear Anal. 74(15) (2011), 4889-4897], [Borcut
and Berinde: App. Math. Comp. 218(10) (2012), 5929-5936], [Borcut: App. Math. Comp. 218(14)
(2012), 7339-7346], [Borcut: HJMC], [Gordji and Ramezani: preprint].
2010 Mathematics Subject Classiffication. 47H10, 54H25.
Keywords: Partially ordered metric space; nonlinear -contraction; MCB property; n-tupled coincidence point.
1. INTRODUCTION
The classical Banach fixed point theorem [2] and its applications are well known. In the recent past,
many authors extended this theorem by considering relatively more general contractive mappings
on various types of metric spaces. In 2004, Ran and Reurings [17] extended Banach fixed point
theorem to partially ordered complete metric spaces. Thereafter, Nieto and López [15] modified the
Ran and Reuring’s fixed point theorem. Nieto and López’s fixed point theorem further generalized
by many authors, for example ( [1, 16]).
The idea of coupled fixed point was initiated by Guo and Lakshmikantham [9] in 1987 which is
also followed by Bhaskar and Lakshmikantham [4] wherein authors introduced the notion of mixed
monotone property for a weakly linear contraction mapping F : X 2 → X, (where X is a partially
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M. Imdad, A. Alam, A. Sharma and M. Arif
ordered metric space) and utilized the same to prove some theorems on existence and uniqueness
of coupled fixed points, which are viewed as the coupled formulation of certain results of Nieto and
López [15]. In 2009, Lakshmikantham and Ćirić [14] generalized these results for nonlinear contraction mappings by introducing the notions of coupled coincidence point and mixed g-monotone
property. In recent years, the notion of coupled fixed point is extended to higher dimensions by many
authors. In an attempt to extend the definition from X 2 to X 3 , Berinde and Borcut [3] introduced
the concept of tripled fixed point and utilize the same to prove some tripled fixed point theorems.
Karapinar and Luong [13] introduced the notion of quadrupled fixed point and utilize the same to
prove some quadrupled fixed point theorems for nonlinear contractions satisfying mixed monotone
property.
In fact, in 2010, Samet and Vetro [18] introduced the notion of fixed point of n-order (or n-tupled
fixed point), n ∈ N and n ≥ 3, as natural extension of coupled as well as quadrupled fixed points
and established some results for n-tupled fixed point in complete metric spaces, using a new concept
of F-invariant set. Here it can be pointed out that the notion of tripled fixed point due to Berinde
and Borcut [3] is different from the one defined by Samet and Vetro [18] for n = 3 in the case of
partially ordered metric spaces in order to keep the mixed monotone property properly working.
Afterwards, Imdad et al. [10] generalized the notion of n-tupled fixed point for a pair of mappings
F and g by defining n-tupled coincidence point and introduced the general concept of mixed gmonotone property on X n , for even n. For more details see [11, 12, 19]. Also they employed the
same to prove an even-tupled coincidence theorem for nonlinear contraction mappings satisfying
mixed g-monotone property. Basically this result is true for only even n but not for odd ones.
Recently, Gordji and Ramezani [8] introduced the concept of a new n-tupled fixed point, which is an
extension of coupled as well as tripled fixed points and established some n-tupled fixed point results
for linear contractions in complete metric spaces. However, the notion of n-fixed point introduced by
Gordji and Ramazani [8] is different from n-tupled fixed point in the sense of Samet and Vetro [18],
but it holds for a general natural number n (odd or even) in order to keep mixed monotone property
properly working. Gordji and Ramezani [8] say this type n-tupled fixed point as n-fixed point.
2. PRELIMINARIES
Throughout the paper, n stands for a general natural number, In := {1, 2, ..., n}, i, j ∈ In . Let X
denotes a non-empty set and X n denotes the product set X × X × ... × X (n times) and {On , E n }
denotes a partition of In , where
n+1
,
On = 2 p − 1 : p ∈ 1, 2, ...,
2
that is, the set of all odd natural numbers in In .
n
,
E n = 2 p : p ∈ 1, 2, ...,
2
that is, the set of all even natural numbers in In . It is clear that On ∪ E n = In and On ∩ E n =
∅. Also, m ∈ N ∪ {0}, {x m } and {U m } denote sequences in X and X n respectively, where U m =
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(x1m , x2m , . . . , xnm ) so that {x1m }, {x2m }, . . . , {xnm } are sequences in X. Let F : X n → X and g : X →
X be two mappings.
Definition 2.1 [16]. A triplet (X, d, ) is called a partially ordered metric space if X is a nonempty
set, d is a metric on X and is a partial order relation on X .
Moreover, if d is a complete metric on X , we say that (X, d, ) is a partially ordered complete
metric space.
Definition 2.2 [1]. Let (X, d, ) be a partially ordered metric space. We say that (X, d, ) has
MCB (Monotone-Convergence-Boundedness) property if it satisfies the following conditions:
(a)
every non-decreasing convergent sequence {x m } in X is bounded above by its limit (as an
upper bound), i.e.,
xm
(b)
d
x m+1 and x m −
→ x ⇒ xm
x, ∀ m ∈ N ∪ {0}.
every non-increasing convergent sequence {x m } in X is bounded below by its limit (as a
lower bound), i.e.,
xm
d
x m+1 and x m −
→ x ⇒ xm
x, ∀ m ∈ N ∪ {0}.
Definition 2.3 [1]. Let (X, d, ) be a partially ordered metric space. We say that (X, d, ) has
g-MCB property if it satisfies the following conditions:
(a)
g-image of every non-decreasing convergent sequence {x m } in X is bounded above by
g-image of its limit (as an upper bound), i.e.,
xm
(b)
d
x m+1 and x m −
→ x ⇒ g(x m )
g(x), ∀ m ∈ N ∪ {0}.
g-image of every non-increasing convergent sequence {x m } in X is bounded below by
g-image of its limit (as a lower bound), i.e.,
xm
d
x m+1 and x m −
→ x ⇒ g(x m )
g(x), ∀ m ∈ N ∪ {0}.
If g is the identity mapping on X, then Definition 2.3 reduces to Definition 2.2.
Definition 2.4 [10]. Let (X, ) be a partially ordered set. Then the mapping F : X n → X is said to
have the mixed monotone property if F is monotone nondecreasing in its odd position arguments
and monotone nonincreasing in its even position arguments, that is, for any x1 , x2 , ..., xi−1 , xi , xi+1 ,
..., xn ∈ X and for all i ∈ In ,
xi , xi ∈ X, xi
xi ⇒ F(x1 , x2 , ..., xi−1 , xi , xi+1 , ..., xn )
F(x1 , x2 , ..., xi−1 , xi , xi+1 , ..., xn ) ∀ i ∈ On ,
xi , xi ∈ X, xi
xi ⇒ F(x1 , x2 , ..., xi−1 , xi , xi+1 , ..., xn )
F(x1 , x2 , ..., xi−1 , xi , xi+1 , ..., xn ) ∀ i ∈ E n .
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Definition 2.5 [10]. Let (X, ) be a partially ordered set. Then the mapping F is said to have the
mixed g-monotone property if F is monotone g-nondecreasing in its odd position arguments and
monotone g-nonincreasing in its even position arguments, that is, for any x1 , x2 , ..., xi−1 , xi , xi+1 ,
..., xn ∈ X and for all i ∈ In ,
xi , xi ∈ X, gxi
gxi ⇒ F(x1 , x2 , ..., xi−1 , xi , xi+1 , ..., xn )
F(x1 , x2 , ..., xi−1 , xi , xi+1 , ..., xn ) ∀ i ∈ On ,
xi , xi ∈ X, gxi
gxi ⇒ F(x1 , x2 , ..., xi−1 , xi , xi+1 , ..., xn )
F(x1 , x2 , ..., xi−1 , xi , xi+1 , ..., xn ) ∀ i ∈ E n .
Definition 2.6 [8]. Let X be a non-empty set. Then an element (x1 , x2 , ..., xn ) ∈ X n is called ntupled fixed point of the mapping F if
xi = F(xi , xi−1 , ..., x2 , x1 , x2 , ..., xn−i+1 ) ∀ i ∈ In .
Definition 2.7. Let X be a non-empty set. Then an element (x1 , x2 , ..., xn ) ∈ X n is called n-tupled
coincidence point of the mappings F and g if
gxi = F(xi , xi−1 , ..., x2 , x1 , x2 , ..., xn−i+1 ) ∀ i ∈ In .
Remark 2.1. Notice that if g = I, the identity mapping on X, Definition 2.5 reduces to Definition
2.4 while Definition 2.7 reduces to Definition 2.6.
Definition 2.8. Let X be a non-empty set. Then an element (x1 , x2 , ..., xn ) ∈ X n is called common
n-tupled fixed point of the mappings F and g if
xi = gxi = F(xi , xi−1 , ..., x2 , x1 , x2 , ..., xn−i+1 ) ∀ i ∈ In .
Definition 2.9 [10]. Let X be a non-empty set. The mappings F and g are said to be commuting if
g(F(x1 , x2 , ..., xn )) = F(gx1 , gx2 , ..., gxn ) ∀ x1 , x2 , ..., xn ∈ X.
Definition 2.10 [5]. A function ψ : [0, ∞) → [0, ∞) is said to be a D-function if it satisfies the
following conditions:
1.
2.
3.
ψ is non-decreasing,
ψ(t) < t, for all t > 0,
lim+ ψ(r ) < t, for all t > 0.
r →t
The class of all functions ψ : [0, ∞) → [0, ∞) is denoted by .
3. EXISTENCE OF N-TUPLED COINCIDENCE POINT
Theorem 3.1. Let (X, d, ) be a partially ordered complete metric space. Let F : X n → X and
g : X → X be two mappings. Suppose that the following conditions are satisfied:
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(i)
(ii)
(iii)
(iv)
(v)
(vi)
169
F(X n ) ⊆ g(X ),
F has mixed g-monotone property,
(F, g) is a commuting pair,
g is continuous,
either F is continuous or (X, d, ) has g-MCB property,
there exists a D-function ψ ∈ such that
d(F(U ), F(V )) ≤ ψ(max d(gxi , gxi )) ∀ U = (x1 , x2 , ..., xn ), V = (y1 , y2 , ..., yn ) ∈ X n ,
i∈In
(3.1)
such that gxi gyi ∀ i ∈ On and gxi
(vii) there exist x10 , x20 , ..., xn0 ∈ X such that
gxi0
gxi0
gyi ∀ i ∈ E n ,
0
0
F(xi0 , xi−1
, ..., x20 , x10 , x20 , ..., xn−i+1
) ∀ i ∈ On
0
0
0
0
0
0
F(xi , xi−1 , ..., x2 , x1 , x2 , ..., xn−i+1 ) ∀ i ∈ E n .
(3.2)
Then F and g have an n-tupled coincidence poiint.
Proof. In view of hypothesis (i), we construct the sequences {x1m }, {x2m }, ..., {xnm } in X as follows:
m
m
g(xim+1 ) = F(xim , xi−1
, ..., x2m , x1m , x2m , ..., xn−i+1
) ∀ i ∈ In .
(3.3)
By induction, we prove that for all m ∈ N ∪ {0},
gxim
gxim+1 ∀ i ∈ On and gxim
gxim+1 ∀ i ∈ E n .
(3.4)
On using (3.2) and (3.3), we obtain
gxi0
0
0
F(xi0 , xi−1
, ..., x20 , x10 , x20 , ..., xn−i+1
) = gxi1 ∀ i ∈ On ,
gxi0
0
0
F(xi0 , xi−1
, ..., x20 , x10 , x20 , ..., xn−i+1
) = gxi1 ∀ i ∈ E n .
So (3.4) holds for m = 0. Suppose (3.4) holds for some m = k > 0. On using mixed g-monotone
property of F, for i ∈ On , we have
k
k
gxik+1 = F(xik , xi−1
, ..., x2k , x1k , x2k , ..., xn−i+1
)
k
k
, ..., x2k , x1k , x2k , ..., xn−i+1
)
F(xik+1 , xi−1
k+1
k
, ..., x2k , x1k , x2k , ..., xn−i+1
)
F(xik+1 , xi−1
..
.
k+1
k+1
, ..., x2k+1 , x1k+1 , x2k+1 , ..., xn−i+1
) = gxik+2 ,
F(xik+1 , xi−1
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and for i ∈ E n , we have
k
k
, ..., x2k , x1k , x2k , ..., xn−i+1
)
gxik+1 = F(xik , xi−1
k
k
F(xik+1 , xi−1
, ..., x2k , x1k , x2k , ..., xn−i+1
)
k+1
k
, ..., x2k , x1k , x2k , ..., xn−i+1
)
F(xik+1 , xi−1
..
.
k+1
k+1
, ..., x2k+1 , x1k+1 , x2k+1 , ..., xn−i+1
) = gxik+2 .
F(xik+1 , xi−1
Thus (3.4) holds for m = k + 1 and hence by induction (3.4) holds for all m ∈ N ∪ {0}. For m ∈
N ∪ {0}, let us denote
δm = max d(gxim , gxim+1 ).
i∈In
(3.5)
Let us consider the case δm 0 = 0 for some m 0 ∈ N ∪ {0}. Then, gxim 0 = gxim 0 +1 ∀ i ∈ In , which on
using (3.3) implies that
m0
m0
, ..., x2m 0 , x1m 0 , x2m 0 , ..., xn−i+1
)
gxim 0 = F(xim 0 , xi−1
so that (x1m 0 , x2m 0 , ..., xnm 0 ) is an n-tupled coincidence point of F and g and in this case we are done.
On the other hand, assume that δm > 0, ∀ m ∈ N ∪ {0}, then we show that for each i ∈ In and for
all m ∈ N ∪ {0},
d(gxim+1 , gxim+2 ) ≤ ψ(δm ).
(3.6)
To prove this, first we consider i ∈ On . Denote
JOn ,i = {i, i − 2, . . . , 3, 1, 3, . . . , n − i (if n is even) or, n − i + 1 (if n is odd)}
JO n ,i = {i − 1, i − 3, . . . , 4, 2, 4, . . . , n − i (if n is odd) or, n − i + 1 (if n is even)}.
Clearly, JOn ,i ⊆ On and JO n ,i ⊆ E n . Thus by using (3.4), we get
gx mj
gx m+1
∀ j ∈ JOn ,i and gx mj
j
gx m+1
∀ j ∈ JO n ,i .
j
Hence by using (3.1), (3.2) and monotone property of ψ,
m
m
, ..., x2m , x1m , x2m , ..., xn−i+1
),
d(gxim+1 , gxim+2 ) = d(F(xim , xi−1
m+1
m+1
, ..., x2m+1 , x1m+1 , x2m+1 , ..., xn−i+1
))
F(xim+1 , xi−1
)),
≤ ψ(max d(gx mj , gx m+1
j
j∈J
where J = JOn ,i ∪ JO n ,i
≤ ψ(max d(gxim , gxim+1 )) = ψ(δm ).
i∈In
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Now, we consider i ∈ E n . Denote
JEn ,i = {i, i − 2, . . . , 4, 2, 4, . . . , n − i (if n is even) or, n − i + 1 (if n is odd)}
JE n ,i = {i − 1, i − 3, . . . , 3, 1, 3, . . . , n − i (if n is odd) or, n − i + 1 (if n is even)}.
Clearly, JEn ,i ⊆ E n and JE n ,i ⊆ On . Thus by using (3.4), we get
gx m+1
j
gx mj ∀ j ∈ JE n ,i .
gx mj ∀ j ∈ JEn ,i and gx m+1
j
Hence by using symmetric property of d, (3.1), (3.3) and monotone property of , we get
d(gxim+1 , gxim+2 ) = d(gxim+2 , gxim+1 )
m+1
m+1
= d(F(xim+1 , xi−1
, ..., x2m+1 , x1m+1 , x2m+1 , ..., xn−i+1
),
m
m
F(xim , xi−1
, ..., x2m , x1m , x2m , ..., xn−i+1
))
where J = JEn ,i ∪ JE n ,i
, gx mj )),
≤ ψ(max d(gx m+1
j
j∈J
≤ ψ(max d(gxim+1 , gxim )) = ψ(δm ).
i∈In
Hence (3.6) holds for each i and for all m. Now, we will show that {δm } is a monotone decreasing
sequence. On taking maximum over i ∈ In on both sides of inequality (3.6), we get
max d(gxim+1 , gxim+2 )) = ψ(δm ).
i∈In
⇒ δm+1 ≤ ψ(δm ).
(3.7)
Since ψ(t) < t for all t > 0, therefore δm+1 < δm ∀ m. Hence {δm } is a monotone decreasing
sequence of nonnegative real numbers. Since it is bounded below by 0, there exists some δ ≥ 0
such that lim δm = δ + . Now, we will show that δ = 0. Suppose on contrary that δ > 0. Taking
m→∞
limit as m → ∞ of both sides of (3.7) and keeping in mind that lim+ ψ(r ) < t ∀ t > 0, we have
r →t
δ = lim δm+1 ≤ lim ψ(δm ) = lim + ψ(δm ) < δ,
m→∞
m→∞
δm →δ
which is a contradiction so that δ = 0, yielding thereby
lim δm = lim max d(gxim , gxim+1 ) = 0.
m→∞
⇒
m→∞ i∈In
(3.8)
lim d(gxim , gxim+1 ) = 0 ∀ i ∈ In .
m→∞
Next, we show that {gxim }, i ∈ In are Cauchy sequences. If possible, suppose that at least one
of {gxim }, i ∈ In is not a Cauchy sequence. Then there exists an > 0 and sequences of positive
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integers {σ (k)} and {μ(k)} such that for all positive integers k with μ(k) > σ (k) ≥ k,
tk = max d(gxiσ (k) , gxiμ(k) ) ≥ .
(3.9)
i∈In
We may choose μ(k), corresponding to σ (k), such that it is the smallest integer satisfying (3.9) and
μ(k) > σ (k) ≥ k. Hence
μ(k)−1
max d(gxiσ (k) , gxi
i∈In
) < .
(3.10)
On using the triangular inequality and (3.10), we have
μ(k)
d(gxiσ (k) , gxi
μ(k)−1
) ≤ d(gxiσ (k) , gxi
μ(k)−1
< + d(gxi
μ(k)−1
) + d(gxi
μ(k)
, gxi
μ(k)
, gxi
)
).
Taking the maximum over i ∈ In on both the sides of above inequality, we have
max d(gxiσ (k) , gxiμ(k) ) < + max d(gxiμ(k)−1 , gxiμ(k) ).
i∈In
i∈In
Using (3.5) and (3.9), we get
≤ tk < + δμ(k)−1 .
Letting k → ∞ in the above inequality and using (3.8), we get
lim tk = .
(3.11)
k→∞
On the other hand, we have
μ(k)
d(gxiσ (k) , gxi
μ(k)+1
) ≤ d(gxiσ (k) , gxiσ (k)+1 ) + d(gxiσ (k)+1 , gxi
μ(k)
⇒ max d(gxiσ (k) , gxi
i∈In
μ(k)+1
) + d(gxi
μ(k)
, gxi
),
) ≤ max d(gxiσ (k) , gxiσ (k)+1 ),
i∈In
μ(k)+1
+ max d(gxiσ (k)+1 , gxi
i∈In
μ(k)+1
) + max d(gxi
i∈In
tk ≤ δσ (k) + δμ(k) + max d(gxiσ (k)+1 , gxiμ(k)+1 ).
i∈In
μ(k)
, gxi
).
(3.12)
Now, we have to show that for each i ∈ In ,
d(gxiσ (k)+1 , gxiμ(k)+1 ) ≤ ψ(tk ).
(3.13)
As σ (k) < μ(k), on using (3.4), we get
gxiσ (k)
μ(k)
gxi
∀ i ∈ On and gxiσ (k)
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μ(k)
gxi
∀ i ∈ En .
(3.14)
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The proof of (3.13) is similar as the proof of (3.6). First we consider i ∈ On , then in view of (3.14),
we have
μ(k)
gx σj (k)
gx j
μ(k)
∀ j ∈ JOn ,i and gx σj (k)
gx j
∀ j ∈ JO n ,i .
On using (3.1), (3.3) and monotone property of ψ, we have
σ (k)
σ (k)
, ..., x2σ (k) , x1σ (k) , x2σ (k) , ..., xn−i+1
),
d(gxiσ (k)+1 , gxiμ(k)+1 ) = d(F(xiσ (k) , xi−1
μ(k)
F(xi
μ(k)
μ(k)
, xi−1 , ..., x2
μ(k)
≤ ψ(max d(gx σj (k) , gx j
j∈J
μ(k)
≤ ψ(max d(gxiσ (k) , gxi
i∈In
μ(k)
, x1
μ(k)
, x2
μ(k)
, ..., xn−i+1 ))
where J = JOn ,i ∪ JO n ,i
))
)) = ψ(tk ).
Next, we consider i ∈ E n . Then again in view of (3.14), we have
μ(k)
μ(k)
gx σj (k) ∀ j ∈ JEn ,i and gx j
gx j
gx σj (k) ∀ j ∈ JEn ,i .
Again on using (3.1), (3.3) and monotone property of ψ, we have
μ(k)+1
d(gxiσ (k)+1 , gxi
μ(k)+1
) = d(gxi
μ(k)
= d(F(xi
, gxiσ (k)+1 )
μ(k)
μ(k)
, xi−1 , ..., x2
μ(k)
, x1
μ(k)
, x2
μ(k)
, ..., xn−i+1 ),
σ (k)
σ (k)
F(xiσ (k) , xi−1
, ..., x2σ (k) , x1σ (k) , x2σ (k) , ..., xn−i+1
))
μ(k)
≤ ψ(max d(gx j
j∈J
, gx σj (k) ))
μ(k)
≤ ψ(max d(gxiσ (k) , gxi
i∈In
where J = JEn ,i ∪ JE n ,i
)) = ψ(tk ).
Thus (3.13) holds for each i and for all m. Now, on using (3.13) in (3.12), we get
tk ≤ δσ (k) + δμ(k) + ψ(tk ).
Letting k → ∞ in above relation and using (3.8) & (3.11), we have
≤ lim ψ(tk ) = lim+ ψ(tk ) < ,
k→∞
tk →
which is a contradiction. Therefore {gxim }, i ∈ In are Cauchy sequences in X. Since X is complete,
there exist xi ∈ X, i ∈ In such that
lim gxim = xi ∀ i ∈ In .
(3.15)
lim g(gxim ) = gxi ∀ i ∈ In .
(3.16)
m→∞
By continuity of g and (3.15), we get
m→∞
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From (3.3) and commutativity of F with g, we get
m
m
, ..., x2m , x1m , x2m , ..., xn−i+1
))
g(gxim+1 ) = g(F(xim , xi−1
m
m
m
m
m
m
= F(gxi , gxi−1 , ..., gx2 , gx1 , gx2 , ..., gxn−i+1
), ∀ i ∈ In .
(3.17)
Now, we will show that F and g have n-tupled coincidence point. To accomplish this we use
condition (v) of our hypotheses. Firstly, we assume that F is continuous. Then on using (3.15)-(3.17)
and continuity of F, we obtain
gxi = lim g(gxim+1 )
m→∞
m
= lim F(gxim , gxim−1 , ..., gx2m , gx1m , gx2m , ..., gxn−i+1
)
m→∞
m
= F( lim gxim , lim gxim−1 , ..., lim gx2m , lim gx1m , lim gx2m , ..., lim gxn−i+1
)
m→∞
m→∞
m→∞
m→∞
m→∞
m→∞
= F(xi , xi−1 , ..., x2 , x1 , x2 , ..., xn−i+1 ) ∀ i ∈ In ,
that is, (x1 , x2 , ..., xn ) is an n-tupled coincidence point of F and g. Otherwise (X, d, ) has g-MCB
property. Since {gxim } is monotone non-decreasing for each i ∈ On and monotone non-increasing
for each i ∈ E n and gxim → xi as m → ∞, we have
g(gxim )
g(xi ) ∀ i ∈ On and g(gxim )
g(xi ) ∀ i ∈ E n .
Now, similar as earlier, two cases arise whenever i ∈ On and i ∈ E n . For both cases, using (3.1),
(3.17), triangular inequality and monotonicity of ψ, we get
d(gxi , F(xi , xi−1 , ..., x2 , x1 , x2 , ..., xn−i+1 ))
≤ d(gxi , gxim+1 ) + d(g(gxim+1 ), F(xi , xi−1 , ..., x2 , x1 , x2 , ..., xn−i+1 ))
m
m
= d(gxi , gxim+1 ) + d(F(gxim , gxi−1
, ..., gx2m , gx1m , gx2m , ..., gxn−i+1
), F(xi , xi−1 , ..., x2 , x1 , x2 , ..., xn−i+1 ))
≤ d(gxi , gxim+1 ) + ψ(max d((gxim , gxi )).
i∈In
Letting m → ∞ in above relation and using (3.16), we have
gxi = F(xi , xi−1 , ..., x2 , x1 , x2 , ..., xn−i+1 ) ∀ i ∈ In .
Thus (x1 , x2 , .., xn ) is an n-tupled coincidence point of F and g. This completes the proof of the
theorem.
Theorem 3.1 immediately yields the following Corollaries:
Corollary 3.1. Theorem 3.1 remains true if we replace the contraction condition (vi) by the following condition besides retaining the rest of the hypotheses:
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(vi ) there exists c ∈ [0, 1) such that
d(F(U ), F(V )) ≤ c max d(gxi , gyi ) ∀ U = (x1 , x2 , ..., xn ), V = (y1 , y2 , ..., yn ) ∈ X n
i∈In
such that gxi
gyi for all i ∈ On and gxi
gyi for all i ∈ E n .
Proof. Taking ψ(t) = ct with c ∈ [0, 1) in Theorem 3.1.
Corollary 3.2. Theorem 3.1 remains true if we replace the contraction condition (vi) by the following condition besides retaining the rest of the hypotheses:
n
αi < 1 such that
(vi ) there exist α1 , α2 , ..., αn ∈ [0, 1) where
i=1
d(F(U ), F(V )) ≤
n
αi d(gxi , gyi ) ∀ U = (x1 , x2 , ..., xn ), V = (y1 , y2 , ..., yn ) ∈ X n
i=1
such that gxi
gyi for all i ∈ On and gxi
Proof. Taking c =
n
gyi for all i ∈ E n .
αi < 1 in Corollary 3.1. Then, we have
i=1
d(F(U ), F(V )) ≤
≤
n
i=1
n
αi d(gxi , gyi )
αi max d(gx j , gy j )
i=1
j∈In
= c max d(gxi , gyi ).
i∈In
Thus (vii ) implies (vii ) and hence we obtain Corollary 3.2 from Corollary 3.1.
Corollary 3.3. Theorem 3.1 remains true if we replace the contraction condition (vi) by the following condition besides retaining the rest of the hypotheses:
(vi ) there exists α ∈ [0, 1) such that
d(F(U ), F(V )) ≤
such that gxi
n
α
d(gxi , gyi ) ∀ U = (x1 , x2 , ..., xn ), V = (y1 , y2 , ..., yn ) ∈ X n
n i=1
gyi for all i ∈ On and gxi
Proof. Taking αi =
α
n
gyi for all i ∈ E n .
for all i ∈ In , where α ∈ [0, 1) in Corollary 3.2.
On setting g = I, the identity mapping on X in Theorem 3.1, Corollaries 3.1, 3.2 and 3.3, we obtain
their corresponding n-tupled fixed point results which are the following (Corollaries 3.4, 3.5, 3.6
and 3.7 respectively).
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Corollary 3.4. Let (X, d, ) be a partially ordered complete metric space. Let F : X n → X be a
mapping. Suppose that the following conditions are satisfied:
(viii) F has mixed monotone property,
(ix) either F is continuous or (X, d, ) has MCB property,
(x) there exists a D-function ψ ∈ such that
d(F(U ), F(V )) ≤ ψ(max d(xi , yi )) ∀ U = (x1 , x2 , ..., xn ), V = (y1 , y2 , ..., yn ) ∈ X n ,
i∈In
such that xi yi ∀ i ∈ On and xi yi ∀ i ∈ E n ,
(xi) there exists x10 , x20 , ..., xn0 ∈ X such that
xi0
0
0
F(xi0 , xi−1
, ..., x20 , x10 , x20 , ..., xn−i+1
) ∀ i ∈ On
xi0
0
0
F(xi0 , xi−1
, ..., x20 , x10 , x20 , ..., xn−i+1
) ∀ i ∈ En .
Then F has an n-tupled fixed point.
Corollary 3.5. Corollary 3.4 remains true if we replace the contraction condition (x) by the following contraction condition besides retaining the rest of the hypotheses:
(x ) there exists c ∈ [0, 1) such that
d(F(U ), F(V )) ≤ c max d(xi , yi ) ∀ U = (x1 , x2 , ..., xn ), V = (y1 , y2 , ..., yn ) ∈ X n
i∈In
such that xi
yi for all i ∈ On and xi
yi for all i ∈ E n .
Corollary 3.6. Corollary 3.4 remains true if we replace the contraction condition (x) by the following contraction condition besides retaining the rest of the hypotheses:
n
(x ) there exist α1 , α2 , ..., αn ∈ [0, 1) with
αi < 1 such that
i=1
d(F(U ), F(V )) ≤
n
αi d(xi , yi ) ∀ U = (x1 , x2 , ..., xn ), V = (y1 , y2 , ..., yn ) ∈ X n
i=1
such that xi
yi for all i ∈ On and xi
yi for all i ∈ E n .
Corollary 3.7. Corollary 3.4 remains true if we replace the contraction condition (x) by the following contraction condition besides retaining the rest of the hypotheses:
(x ) there exists α ∈ [0, 1) such that
d(F(U ), F(V )) ≤
such that xi
n
α
d(xi , yi ) ∀ U = (x1 , x2 , ..., xn ), V = (y1 , y2 , ..., yn ) ∈ X n
n i=1
yi for all i ∈ On and xi
yi for all i ∈ E n .
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Notice that Corollary 3.6 is the main result of [8].
Now, we point that previous results of this section yield several known results proved on coupled as
well as tripled fixed and coincidence points as special cases. Some of them are listed below:
(1) By setting n = 3 in Theorem 3.1, we get Theorem 5 contained in Borcut [5].
(2) By setting n = 3 in Corollary 3.1, we get Corollary 1 contained in Borcut [5].
(3) By setting n = 3 in Corollary 3.2, we get Theorem 4 contained in Borcut and Berinde [6].
(4) By setting n = 3 in Corollary 3.6, we get Theorems 7 and 8 contained in Berinde and
Borcut [3].
(5) By setting n = 3 in Corollary 3.7, we get main results of Borcut [7].
(6) By setting n = 2 in Corollary 3.3, we get Corollary 2.1 contained in Lakshmikantham and
Ćirić [14].
(7) By setting n = 2 in Corollary 3.7, we get Theorem 2.4 contained in Bhaskar and Lakshmikantham [4].
4. UNIQUENESS OF N-TUPLED COINCIDENCE POINT
Let (X, ) be a partially ordered set. Equip the product set X n with the following partial ordering
n
n : for all U = (x 1 , x 2 , ..., x n ), V = (y1 , y2 , ..., yn ) ∈ X
U
n
V ⇔ xi
yi ∀ i ∈ On and xi
yi ∀ i ∈ E n .
We say that two elements U and V of X n are comparable if either U
n
V or U
n
V.
Now, we state and prove the corresponding result regarding the uniqueness of n-tupled coincidence
points.
Theorem 4.1. In addition to the hypotheses of Theorem 3.1, suppose for all n-tupled coincidence
points U = (x1 , x2 , ..., xn ), V = (y1 , y2 , ..., yn ) of F and g, there exists W = (z 1 , z 2 , ..., z n ) such
that (gz 1 , gz 2 , ..., gz n ) is comparable to (gx1 , gx2 , ..., gxn ) and (gy1 , gy2 , ..., gyn ), then F and g
have a unique n-tupled coincidence point. Moreover, this unique n-tupled coincidence point is a
unique common n-tupled fixed point of mappings F and g.
Proof. In view of Theorem 3.1, the set of n-tupled coincidence points is nonempty. If U =
(x1 , x2 , ..., xn ) and V = (y1 , y2 , ..., yn ) ∈ X n are two n-tupled coincidence points of F and g, then
gxi = F(xi , xi−1 , ..., x2 , x1 , x2 , ..., xn−i+1 ) ∀ i ∈ In ,
(3.18)
gyi = F(yi , yi−1 , ..., y2 , y1 , y2 , ..., yn−i+1 ) ∀ i ∈ In .
(3.19)
gxi = gyi ∀ i ∈ In .
(3.20)
We have to show that
By assumption, there exists W = (z 1 , z 2 , ..., z n ) ∈ X n such that (gz 1 , gz 2 , ..., gz n ) is comparable to
(gx1 , gx2 , ..., gxn ) and (gy1 , gy2 , ..., gyn ). Suppose that
(gx1 , gx2 , ..., gxn )
n
(gz 1 , gz 2 , ..., gz n ) and (gy1 , gy2 , ..., gyn )
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(gz 1 , gz 2 , ..., gz n ).
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The other cases are similar. Now, put z i0 = z i ∀ i ∈ In . Since F(X n ) ⊆ g(X ), similar as on the lines
of the proof of the Theorem 3.1, we can inductively define sequences {zim }, i ∈ In such that
m
m
, ..., z 2m , z 1m , z 2m , ..., z n−i+1
) ∀ i ∈ In .
g(z im+1 ) = F(z im , z i−1
(3.21)
As F has mixed g-monotone property, by a similar region as in the proof of Theorem 3.1, we have
gz im+1 ∀ i ∈ On and gz im
gz im
Since (gx1 , gx2 , ..., gxn )
gxi
n
gz im
gz im+1 ∀ i ∈ E n .
(gz 1 , gz 2 , ..., gz n ), we have
gz im+1 ∀ i ∈ On and gxi
gz im
gz im+1 ∀ i ∈ E n .
(3.22)
Now, we claim that for each i ∈ In and for all m ≥ 1 that
d(gxi , gz im+1 ) ≤ ψ(γm ),
(3.23)
where γm = max d(gxi , gz im+1 ).
i∈In
Now, similar on the lines of the proof of Theorem 3.1, two cases arise, whenever i ∈ E n as well as
i ∈ On . For both cases, on using (3.1), (3.18), (3.21) and (3.22) and monotonicity of ψ, we get
d(gxi , gz im+1 ) = d((xi , xi−1 , ..., x2 , x1 , x2 , ..., xn−i+1 ),
m
m
F(z im , z i−1
, ..., z 2m , z 1m , z 2m , ..., z n−i+1
))
m
≤ ψ(max{d(gxi , gz im ), d(gxi−1 , gz i−1
), ..., d(gx2 , gz 2m ),
m
d(gx1 , gz 1m ), d(gx2 , gz 2m ), ..., d(gxn−i+1 , gz n−i+1
)})
≤ ψ(max d(gxi , gz im )) = ψ(γm ).
i∈I
It follows that for all m ≥ 1,
γm+1 = max d(gxi , gz im+1 )) ≤ ψ(γm ) ≤ ψ 2 (γm+1 ) . . . ≤ ψ m (γ1 ) ≤ ψ m+1 (γ0 ).
i∈In
(3.24)
Now, on using properties of ψ, we can obtain
lim ψ m+1 (t) = 0 ∀ t > 0.
m→∞
(3.25)
Taking the limit as m → ∞ in (3.24) and using (3.25), we get
lim γm = 0.
m→∞
Consequently, we have
lim d(gxi , gz im ) = 0 ∀ i ∈ In .
m→∞
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Similarly, we can show that
lim d(gyi , gz im ) = 0 ∀ i ∈ In .
m→∞
(3.27)
By triangular inequality, (3.26) and (3.27), we get for each i ∈ In ,
d(gxi , gyi ) ≤ d(gxi , gz im ) + d(gz im , gyi ) → 0 as m → ∞,
which implies that gxi = gyi ∀ i ∈ In . Thus (3.20) is proved.
On using (3.18) and commutativity of F and g, we get
g(gxi ) = g(F(xi , xi−1 , ..., x2 , x1 , x2 , ..., xn−i+1 ))
= F(gxi , gxi−1 , ..., gx2 , gx1 , gx2 , ..., gxn−i+1 ).
(3.28)
Write gxi = x i for each i ∈ In , then from (3.28), we get
g(x i ) = F(x i , x i−1 , ..., x 2 , x 1 , x 2 , ..., x n−i+1 ).
Thus (x 1 , x 2 , ..., x n ) is also n-tupled coincidence point of F and g. Now, on using (3.20), we get
x i = g(x i ) = F(x i , x i−1 , ..., x 2 , x 1 , x 2 , ..., x n−i+1 ).
Hence is (x 1 , x 2 , ..., x n ) is a common n-tupled fixed point of F and g. To prove uniqueness, assume
that (u 1 , u 2 , ..., u n ) is another common n-tupled fixed point. Then by (3.20), for each i ∈ In , we
have
u i = g(u i ) = g(x i ) = x i .
This completes the proof.
For any U = (x1 , x2 , ..., xn ) ∈ X n and for each i ∈ In , denote by U(i) the fixed partially ordered
element of X n relative to U as
U(i) = (xi , xi−1 , ..., x2 , x1 , x2 , ..., xn−i+1 ).
Clearly, U(1) = U.
Theorem 4.2. In addition to the hypotheses of Theorem 3.1, suppose for all n-tupled coincidence
points U = (x1 , x2 , ..., xn ), V = (y1 , y2 , ..., yn ) of F and g, there exists W = (z 1 , z 2 , ..., z n ) such
that (F(W(1) ), F(W(2) ), ..., F(W(n) )) is comparable to (gx1 , gx2 , ..., gxn ) and (gy1 , gy2 , ..., gyn ),
then F and g have a unique common n-tupled fixed point.
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Proof. In this case, we observe that,
gxi
F(W(i) ) = F(z i , z 1−1 , . . . , z 2 , z 1 , z 2 , . . . , z n−i+1 )
0
0
= F(z i0 , z 1−1
, . . . , z 20 , z 10 , z 20 , . . . , z n−i+1
)
= g(z i1 ) ∀ i ∈ On ,
gxi
F(W(i) ) = F(z i , z 1−1 , . . . , z 2 , z 1 , z 2 , . . . , z n−i+1 )
0
0
= F(z i0 , z 1−1
, . . . , z 20 , z 10 , z 20 , . . . , z n−i+1
)
= g(z i1 ) ∀ i ∈ E n .
Hence by induction process, we find that
gxi
gz im
gz im+1 ∀ i ∈ On and gxi
gz im
gz im+1 ∀ i ∈ E n .
Thus, the proof is done on the similar lines of Theorem 4.1.
REFERENCES
[1] Alam, A. and Imdad, M.: Some Coincidence Theorems for Nonlinear φ−cotraction in partially ordered Metric Spaces
with Applications, Preprint.
[2] Banach, S.: Sur les operations dans les ensembles abstraits et leur application aux quations intgrales, Fund. Math., 3
(1922), 133–181.
[3] Berinde, V. and Borcut, M.: Tripled fixed point theorems for contractive type mappings in partially ordered metric
spaces, Nonlinear Anal., 74 (15) (2011), 4889–4897.
[4] Bhaskar, T. G. and Lakshmikantham, V.: Fixed point theorems in partially ordered metric spaces and applications,
Nonlinear Anal., 65 (7) (2006), 1379–1393.
[5] Borcut, M.: Tripled coincidence theorems for contractive type mappings in partially ordered metric spaces, App. Math.
Comp., 218 (14) (2012), 7339–7346.
[6] Borcut, M. and Berinde, V.: Tripled coincidence theorems for contractive type mappings in partially ordered metric
spaces, App. Math. and Comp., 218 (10) (2012), 5929–5936.
[7] Borcut, M.: Tripled fixed point theorems in partially ordered metric spaces, Hacettepe Journal of Mathematics and
Statistics, (submitted).
[8] Gordji, M. E. and Ramezani, M.: N -fixed point theorems in partially ordered metric spaces, Preprint.
[9] Guo, D. and Lakshmikantham, V.: Coupled fixed points of nonlinear operators with applications, Nonlinear Analysis,
11 (5) (1987), 623–632.
[10] Imdad, M., Soliman, A. H., Choudhury, B. S. and Das, P.: On n-tupled coincidence point results in metric spaces,
Journal of Operators, 2013 (2013), Article ID 532867, 8 pages.
[11] Imdad, M., Sharma, A. and Rao, K. P. R.: n-tupled coincidence and common fixed point results for weakly contractive
mappings in complete metric spaces, Bull. Math. Anal. Appl., Volume 5, Issue 4, (2013), 19–39.
[12] Imdad, M., Sharma, A. and Rao, K. P. R.: Generalized n-tupled fixed point theorems for nonlinear contraction mapping, Afrika Matematika, 26 (3), (2015), 443–455.
[13] Karapinar, E. and Luong, N. V.: Quadruple fixed point theorems for nonlinear contractions, Computers & Mathematics
with Applications, 64 (6) 2012, 1839–1848.
[14] Lakshmikantham, V. and Ćirić, L.: Coupled fixed point theorems for nonlinear contractions in partially ordered metric
spaces, Nonlinear Anal., 70 (2009), 4341–4349.
[15] Nieto, J. J. and López, R. R.: Contractive mapping theorems in partially ordered sets and applications to ordinary
differential equations, Order, 22 (3) (2005), 223–239.
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[16] O’Regan, D. and Petruşel, A.: Fixed point theorems for generalized contractions in ordered metric spaces, J. Math.
Anal. Appl., 341 (2) (2008), 1241–1252.
[17] Ran, A. C. M. Ran and Reurings, M. C. B.: A fixed point theorem in partially ordered sets and some applications to
matrix equations, Proc. Amer. Math. Soc., 132 (2004), 1435–1443.
[18] Samet, B. and Vetro, C.: Coupled fixed point, F-invariant set and fixed point of N -order, Ann. Funct. Anal., 1 (2)
(2010), 46–56.
[19] Sharma, A., Imdad, M. and Alam, A.: Shorter proofs of some recent even tupled coincidence theorems for weak
contractions in ordered metric spaces, Mathematical Sciences, 8 (4), (2015), 131–138.
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INDIAN JOURNAL OF INDUSTRIAL AND APPLIED MATHEMATICS
Vol. 10, No. 1 (Special Issue), Jan–June 2019, pp. 182–201
DOI:
On Starlikeness and Convexity Properties of the
Generalized Bessel-Hypergeometric Function
Nusrat Raza1∗ and Eman S.A. AbuJarad2
1
Mathematics Section Women’s College Aligarh Muslim University, India
Department of Mathematics Aligarh Muslim University, India
(∗ Corresponding author) Email: ∗ nraza.maths@gmail.com, 2 emanjarad2@gmail.com
2
Abstract: In this paper, we introduce the Generalized Bessel-Hypergeometric function by making
convolution of the Generalized Bessel function and the Hypergeometric function. Also we obtain
the necessary and sufficient conditions for the Generalized Bessel-Hypergeometric function to be β
- uniformly starlike and β - uniformly convex functions of order α. Further, we introduce a function
in terms of integral involving the Generalized Bessel-Hypergeomtric function and obtain its order
of starlikness and convexity.
Keywords: Generalized Bessel function, Hypergeometric function, univalent function, Starlike
function, convex function, integral operator, β-uniformly starlike function, β-uniformly convex
function.
1. INTRODUCTION AND PRELIMINARIES
An important property of analytic functions, is that these functions can be locally represented as
power serieses. Certain geometric properties, for example, starlikness and convexity of some analytic functions like Generalized Bessel functions and Hypergeometric functions, are studied by Baricz [3], Carlson and Schaffer [4], respectively.
Let A be the class of analytic functions of the form
f (z) = z +
∞
an z n ,
(1.1)
n=2
where f is analytic in the open unit disc U = {z ∈ C : |z| < 1}.
This research article is funded by University Grant commission, New Delhi, India, under UGC-BSR Research Start-Up-Grant No.
F.30-129(A)/2015(BSR).
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A function f is said to be univalent [6] in a domain D ⊂ C, if it never takes on the same values
twice, that is, f (z 1 ) = f (z 2 ) for z 1 , z 2 ∈ D implies that z 1 = z 2 , in other words, f is one to one on
D.
Let B be the class of functions which are normalized by the condition f (0) = f (0) − 1 = 0,
where f is univalent in an open unit disc U . We note that, B is a subclass of A.
Let T be the class of functions of the form [17]:
f (z) = z −
∞
an z n (an ≥ 0, z ∈ U ).
(1.2)
n=2
It is clear that T is a subclass of A.
Let g ∈ A be another function, given by the following power series:
g(z) = z +
∞
bn z n ,
(1.3)
n=2
then the Hadamard product (or Convolution) of the analytic functions f (z), given by equation (1.1)
and g(z), given by equation (1.3), denoted by ( f *g) (z), is defined as [9]:
( f *g) (z) = z +
∞
an bn z n .
(1.4)
n=2
A set ε is said to be starlike with respect to ω0 ∈ ε, if the line segment joining ω0 to every other
point ω ∈ ε lies entirely in ε . If the function f (z) maps a domain D ⊂ C onto a domain that is
starlike with respect to ω0 , then f (z) is said to be starlike with respect to ω0 .
Also, the set ε is said to be convex, if the line segment joining any two points of ε lies entirely
in ε . Equivalently t z 1 + (1 − t)z 2 ∈ ε for every z 1 , z 2 ∈ ε and 0 ≤ t ≤ 1.
Schild [15] introduced the classes S(α) and K (α), which are given by the following definitions:
Definition 1.1. A function f (z) in A is said to be starlike of order α (0 ≤ α < 1), if and only if
z f (z)
>α
(z ∈ U ).
Re
f (z)
We denote the class of such functions by S(α), we note that S(α) ⊆ S(0), where S = S(0) is the
class of starlike function with respect to the origin in the open unit disc U .
Definition 1.2. A function f (z) in A is said to be convex of order α (0 ≤ α < 1), if and only if
z f (z)
> α (z ∈ U ).
Re 1 + f (z)
We denote the class of such functions by K (α), we note that K (α) ⊆ K (0), where K = K (0) is the
class of convex function with respect to the origin in the open unit disc U .
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According to Alexander [1], if f is analytic in U with f (0) = f (0) − 1 = 0, then f ∈ K (0) if
and only if z f ∈ S(0), and thus f ∈ K (α) if and only if z f ∈ S(α).
The class β − U C V was introduced and its geometric definition, connections with the conic
domains were considered by Kanas and Wisniowska [8]. The class β − U C V was defined pure
geometrically as a subclass of univalent functions, that the image of every circular arc γ contained
in the open unit disc U with a center ζ , |ζ | ≤ β (0 ≤ β < 1) is convex arc. The notion of β uniformly convex function is a natural extension of the classical convexity.
We note that, if β = 0, then the center ζ is the origin and the class β − U C V reduces to the class
of convex univalent functions K . Moreover for β = 1 corresponds to the class of uniformly convex
functions U C V introduced by Goodman [7] and studied extensively by Rønning [14]. Alexander [1]
proved that the relation between class β − U C V and class β − S P which is equivalent to the relation
between the classes of convex K (α) and starlike S(α).
The classes S P (α, β) and U C V (α, β) are defined in the following definitions:
Definition 1.3. A function f ∈ A is said to be in the class β-uniformly starlike functions of order
α, which is denoted by S P (α, β), for −1 < α ≤ 1 and β ≥ 0 if it satisfies the following condition:
z f (z)
z f (z)
(z ∈ U ).
(1.5)
−α >β
− 1
Re
f (z)
f (z)
Definition 1.4. A function f ∈ A is said to be in the class β-uniformly convex functions of order α,
which is denoted by U C V (α, β), for −1 < α ≤ 1 and β ≥ 0 if it satisfies the following condition:
z f (z) z f (z)
(z ∈ U ).
(1.6)
Re 1 + − α > β f (z)
f (z) It follows from (1.5) and (1.6) that f ∈ U C V (α, β) if and only if z f ∈ S P (α, β).
Remark 1.1. It is clear that S P (α, β) and U C V (α, β) are subclasses of A and U C V (α, 0) = K (α)
and S P (α, 0) = S(α).
Now, we define the following subclasses S P (λ, α, β) and U C V (λ, α, β) of A introdued by
Murugusundaramoorthy and Magesh [11], respectively:
Definition 1.5. A function f ∈ A is said to be in the class S P (λ, α, β), for 0 ≤ α < 1, 0 ≤ λ < 1
and β ≥ 0 if it satisfies the following condition:
z f (z)
z f (z)
(z ∈ U ).
(1.7)
−α >β
− 1
Re
(1 − λ) f (z) + λz f (z)
(1 − λ) f (z) + λz f (z)
Definition 1.6. A function f ∈ A is said to be in the class U C V (λ, α, β), for 0 ≤ α < 1, 0 ≤ λ < 1
and β ≥ 0 if it satisfies the following condition:
f (z) + z f (z)
f (z) + z f (z)
Re
(z ∈ U ).
(1.8)
−
α
>
β
−
1
f (z) + λz f (z)
f (z) + λz f (z)
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Murugusundaramoorthy and Magesh [11] introduced T S P (λ, α, β) and U C T (λ, α, β) as:
T S P (λ, α, β) = S P (λ, α, β) ∩ T
and
U C T (λ, α, β) = U C V (λ, α, β) ∩ T.
Certain characterization properties for the classes S P (λ, α, β), T S P (λ, α, β), U C V (λ, α, β) and
U C T (λ, α, β) are as follows [11]:
Lemma 1.1. Let 0 ≤ α < 1, 0 ≤ λ < 1 and β ≥ 0, then a function f (z) of the form (1.1) is in
S P (λ, α, β) if [11]
∞
[n(1 + β) − (α + β)(1 + nλ − λ)] |an | ≤ 1 − α.
(1.9)
n=2
Lemma 1.2. Let 0 ≤ α < 1, 0 ≤ λ < 1 and β ≥ 0, then a function f (z) of the form (1.2) is in
T S P (λ, α, β) if and only if [11]
∞
[n(1 + β) − (α + β)(1 + nλ − λ)] |an | ≤ 1 − α.
(1.10)
n=2
Lemma 1.3. Let 0 ≤ α < 1, 0 ≤ λ < 1 and β ≥ 0, then a function f (z) of the form (1.1) is in
U C V (λ, α, β) if [11]
∞
n [n(1 + β) − (α + β)(1 + nλ − λ)] |an | ≤ 1 − α.
(1.11)
n=2
Lemma 1.4. Let 0 ≤ α < 1, 0 ≤ λ < 1 and β ≥ 0, then a function f (z) of the form (1.2) is in
U C T (λ, α, β) if and only if [11]
∞
n [n(1 + β) − (α + β)(1 + nλ − λ)] |an | ≤ 1 − α.
(1.12)
n=2
The Bessel functions are associated with a wide range of problems in important areas of mathematical physics and Engineering. These functions appear in the solutions of heat transfer and other
problems in cylindrical and spherical coordinates. Rainville [13] discussed the properties of the
Bessel function.
The Generalized Bessel functions wν,b,d (z) are defined as [2]:
wν,b,d (z) =
∞
n=0
z
(−d)n
b+1
2
n!
ν+n+
2
2n+ν
,
(1.13)
where ν, b, d, z ∈ C.
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The Generalized Bessel function wν,b,d (z) is the solution of the following differential equation
[2]:
z 2 w (z) + bzw (z) + dz 2 − ν 2 + (1 − b)ν w(z) = 0.
(1.14)
Orhan, Deniz and Srivastava [5] defined the function ϕν,b,d (z) : U → C as:
ϕν,b,d (z) = 2ν
ν+
ν
√
b + 1 1−
z 2 wν,b,d ( z),
2
(1.15)
by using the Generalized Bessel function wν,b,d (z), given by equation(1.14).
The power series representation for the function ϕν,b,d (z) is as follows [5]:
ϕν,b,d (z) =
∞
(−d/4)n
n=0
z n+1 ,
(c)n n!
(1.16)
b+1
> 0, ν, b, d ∈ R and z ∈ U = {z ∈ C : |z| < 1}.
where c = ν +
2
Now, we discuss the following special case for the function ϕν,b,d (z):
For −b = d = −1, the Generalized Bessel function wν,b,d (z) reduces to the modified Bessel
function. Therefore taking −b = d = −1 in equation (1.13), we get the following series definition
for the modified Bessel function Hν (z) [13]:
Hν (z) =
∞
n=0
z
1
n! (ν + n + 1) 2
2n+ν
(z ∈ U, ν ∈ R).
(1.17)
Also, in view of equations (1.15) and (1.16), we get
ϕν,1,−1 (z) = 2
ν
(ν + 1) z
1−
ν
∞
√
2 Hν ( z) =
n=0
4n
1
z n+1 .
(ν + 1)n n!
(1.18)
The Hypergeometric functions appear as the solutions of certain problems in classical and quantum physics, engineering and applied mathematics .Rainville [13] discussed the Hypergeometric
function F(a, p; q; z) and its properties. The Hypergeometric functions F(a, p; q; z) are defined
by [13]:
F(a, p; q; z) =
∞
(a)n ( p)n
n=0
(q)n n!
zn ,
(1.19)
where a, p, q ∈ R and q > 0.
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2. MAIN RESULTS
In this section, we introduce the Generalized Bessel-Hypergeometric function by making convolution of the Generalized Bessel function and the Hypergeometric function. Also we discuss the
starlikness and convexity of this function.
a,d, p
We define the function Ic,q (z) : C → C by making convolution of the function ϕν,b,d (z), given
by equation (1.16) and F(a, p; q; z), given by equation (1.19) as:
a,d, p
Ic,q
(z) = ϕv,b,d (z)*z F(a, p; q; z)
=
∞
(−d/4)n (a)n ( p)n
(q)n (c)n ((n)!)2
n=0
=z+
∞
(−d/4)n−1 (a)n−1 ( p)n−1
n=2
where c = ν +
z n+1
(q)n−1 (c)n−1 ((n − 1)!)2
(2.1)
zn ,
b+1
> 0, a, b, d, p, q, ν ∈ R and q > 0.
2
Remark 2.1. Since the series representing F(a, p, q, 1) (a, p, q ∈ R) converges [13], if q > a +
a,d, p
p. Therefore the series on the right hand side of equation (2.1), which represent Ic,q (1), for z = 1,
also converges, if q > a + p.
Now, we obtain the following result involving the Generalized Bessel-Hypergeometric function
a,d, p
Ic,q (z):
Theorem 2.1. If a, c, d, p, q ∈ R, q > 0, c > 0 and q > a + p, then
a,d, p
a,d, p
z Ic,q
(z) − Ic,q
(z) =
∞
(−d/4)n (a)n ( p)n n+1
z
(z ∈ C).
(q)n (c)n (n)!(n − 1)!
n=1
(2.2)
Also,
a,d, p
(z)
z 2 Ic,q
a,d, p
a,d, p
− z Ic,q
(z) + Ic,q
(z) =
∞
(−d/4)n (a)n ( p)n n+1
z
(z ∈ C).
(q)n (c)n ((n − 1)!)2
n=1
(2.3)
Proof. Differentiating equation (2.1) with respect to z and then multiplying the resultant equation
with z, we get
a,d, p
(z) =
z Ic,q
∞
(n + 1)(−d/4)n (a)n ( p)n
n=0
(q)n (c)n ((n)!)2
z n+1
∞
∞
(−d/4)n (a)n ( p)n n+1 (−d/4)n (a)n ( p)n n+1
z
=
+
z ,
(q)n (c)n (n − 1)!(n)!
(q)n (c)n ((n)!)2
n=1
n=0
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Nusrat Raza and Eman S.A. AbuJarad
which are again using equation (2.1), gives
∞
(−d/4)n (a)n ( p)n n+1
a,d, p
+ Ic,q
(z).
z
(q)
(c)
(n
−
1)!(n)!
n
n
n=1
a,d, p
(z) =
z Ic,q
Hence, we obtained assertion (2.2).
Again, differentiating equation (2.2) with respect to z and then multipling with z, we get
a,d, p
(z)
z 2 Ic,q
a,d, p
a,d, p
+ z Ic,q
(z) − z Ic,q
(z) =
∞
(n + 1)(−d/4)n (a)n ( p)n
(q)n (c)n (n − 1)!(n)!
n=1
z n+1 ,
or, equivalently
z
2
a.d, p
Ic,q
(z)
∞
∞
(−d/4)n (a)n ( p)n n+1 n(−d/4)n (a)n ( p)n n+1
z ,
=
z
+
(q)n (c)n ((n − 1)!)2
(q)n (c)n (n − 1)!(n)!
n=1
n=1
which on again using equation (2.2), gives
a,d, p
(z)
z 2 Ic,q
=
∞
(−d/4)n (a)n ( p)n n+1
a,d, p
a, p
z
+ z Ic,q
(z) − Ic,q
(z).
2
(q)
(c)
((n
−
1)!)
n
n
n=1
Hence, we obtained assertion (2.3).
a,d, p
Next, we define the function θc,q (z) : C → C as :
a,d, p
a,d, p
θc,q
(z) = 2z − Ic,q
(z),
which on using equation (2.1), gives
a,d, p
(z) = z −
θc,q
∞
(−d/4)n−1 (a)n−1 ( p)n−1
n=2
(q)n−1 (c)n−1 ((n − 1)!)2
zn .
(2.4)
a,d, p
Now, we obtain the following necessary and sufficient condition for the function θc,q (z) to be in
the classes T S P (λ, α, β) and U C T (λ, α, β):
a,d, p
Theorem 2.2. If a, b, p, q, ν ∈ R, d < 0, q > 0, c > 0 and q > a + p, then θc,q (z) ∈
T S P (λ, α, β) if and only if
a,d, p
a,d, p
(1) + [(α + β)(λ − 1)] Ic,q
(1) ≤ 2(1 − α),
[(1 + β) − λ(α + β)] Ic,q
(2.5)
a,d, p
where 0 ≤ α < 1, 0 ≤ λ < 1, β ≥ 0 and Ic,q (z) is defined by equation (2.1).
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189
a,d, p
Proof. Since d < 0, therefore in view of Lemma 1.2, θc,q (z) ∈ T S P (λ, α, β) if and only if
∞
[n(1 + β) − (α + β)(1 + nλ − λ)]
n=2
(−d/4)n−1 (a)n−1 ( p)n−1
≤ (1 − α) .
(q)n−1 (c)n−1 ((n − 1)!)2
(2.6)
Now,
∞
[n(1 + β) − (α + β)(1 + nλ − λ)]
n=2
=
∞
(−d/4)n−1 (a)n−1 ( p)n−1
(q)n−1 (c)n−1 ((n − 1)!)2
[(n − 1) [1 + β − λ(α + β)] + (1 − α)]
n=2
= [1 + β − λ(α + β)]
∞
(n − 1)(−d/4)n−1 (a)n−1 ( p)n−1
n=2
= [1 + β − λ(α + β)]
(−d/4)n−1 (a)n−1 ( p)n−1
(q)n−1 (c)n−1 ((n − 1)!)2
(q)n−1 (c)n−1 ((n − 1)!)2
+ (1 − α)
∞
(−d/4)n−1 (a)n−1 ( p)n−1
n=2
(q)n−1 (c)n−1 ((n − 1)!)2
∞
∞
(−d/4)n (a)n ( p)n
(−d/4)n (a)n ( p)n
+ (1 − α)
.
(q)n (c)n (n)!(n − 1)!
(q)n (c)n ((n)!)2
n=1
n=1
Since q > a + p, therefore in view of Remark 2.1 and using equations (2.1) and (2.2), for z = 1, in
the above equation, we get
∞
[n(1 + β) − (α + β)(1 + nλ − λ)]
n=2
(−d/4)n−1 (a)n−1 ( p)n−1
(q)n−1 (c)n−1 ((n − 1)!)2
= [1 + β − λ(α + β)]
a,d, p
a,d, p
a,d, p
Ic,q
(1) − Ic,q
(1) + (1 − α) Ic,q
(1) − 1 ,
or, equivalently
∞
n=2
[n(1 + β) − (α + β)(1 + nλ − λ)]
(−d/4)n−1 (a)n−1 ( p)n−1
(q)n−1 (c)n−1 ((n − 1)!)2
a,d, p
a,d, p
= [1 + β − λ(α + β)] Ic,q
(1) + [(α + β)(λ − 1)] Ic,q
(1) − (1 − α).
(2.7)
Using equation (2.7) is condition (2.6), we get the necessary and sufficient condition (2.5) for the
a,d, p
function θc,q (z) to be in T S P (λ, α, β).
Further, taking −b = d = −1, c = ν +
(1.18), we get the following result:
b+1
= ν + 1 in assertion (2.5) and using equation
2
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a,−1, p
Corollary 2.1. If c = ν + 1 > 0, a, p, q, ν ∈ R, q > a + p and q > 0 , then θν+1,q (z) = 2z −
a,−1, p
Iν+1,q (z) ∈ T S P (λ, α, β) if and only if
a,−1, p
[(1 + β) − λ(α + β)] Iν+1,q (1)
a,−1, p
+ [(α + β)(λ − 1)] Iν+1,q (1) ≤ 2(1 − α),
(2.8)
where 0 ≤ α < 1, 0 ≤ λ < 1, β ≥ 0 and
a,−1, p
Iν+1,q (z) = ϕν,1,−1 (z)*z F(a, p; q; z) =
∞
n=0
(a)n ( p)n
z n+1 .
4n (q)n (ν + 1)n ((n)!)2
(2.9)
a,d, p
Next, we obtain the following necessary and sufficient conditions for θc,q (z) to be β- uniformly convex of order α:
a,d, p
Theorem 2.3. If a, b, p, q, c ∈ R, d < 0, q > 0, c > 0 and q > a + p, then θc,q (z) ∈
U C T (λ, α, β) if and only if
a,d, p
(1)
[(1 + β) − λ(α + β)] Ic,q
a,d, p
+ (1 − α) Ic,q
(1) ≤ 2(1 − α),
(2.10)
where 0 ≤ α < 1, 0 ≤ λ < 1 and β ≥ 0.
a,d, p
Proof. Since d < 0, therefore, in view of Lemma 1.4, θc,q (z) ∈ U C T (λ, α, β) if and only if
∞
n [n(1 + β) − (α + β)(1 + nλ − λ)]
n=2
(−d/4)n−1 (a)n−1 ( p)n−1
≤ 1 − α.
(q)n−1 (c)n−1 ((n − 1)!)2
(2.11)
Now, we consider the left hand side of relation (2.11). On shifting index, we get
∞
n [n(1 + β) − (α + β)(1 + nλ − λ)]
n=2
=
(−d/4)n−1 (a)n−1 ( p)n−1
(q)n−1 (c)n−1 ((n − 1)!)2
∞
(−d/4)n (a)n ( p)n
(n + 1) [(n + 1)(1 + β) − (α + β)(1 + (n + 1)λ − λ)]
(q)n (c)n ((n)!)2
n=1
= [1 + β − λ(α + β)]
∞
∞
(−d/4)n (a)n ( p)n
(n + 1)(−d/4)n (a)n ( p)n
(n + 1)2
−
(α
+
β)(1
−
λ)
.
2
(q)n (c)n ((n)!)
(q)n (c)n ((n)!)2
n=1
n=1
On simplifying the above equation, we get
∞
n=2
n [n(1 + β) − (α + β)(1 + nλ − λ)]
(−d/4)n−1 (a)n−1 ( p)n−1
(q)n−1 (c)n−1 ((n − 1)!)2
∞
∞
∞
(−d/4)n (a)n ( p)n
(−d/4)n (a)n ( p)n
(−d/4)n (a)n ( p)n
+
= [1 + β − λ(α + β)]
+
2
2
(q)n (c)n (n)!(n − 1)! n=1 (q)n (c)n ((n)!)2
(q)n (c)n ((n − 1)!)
n=1
n=1
∞
∞
(−d/4)n (a)n ( p)n
(−d/4)n (a)n ( p)n
− [(α + β)(1 − λ)]
+
.
(q)n (c)n (n − 1)!(n)! n=1 (q)n (c)n ((n)!)2
n=1
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Since q > a + p, therefore in view of Remark 2.1 and using equations (2.1) and (2.3), for z = 1, in
the above equation, we get
∞
n [n(1 + β) − (α + β)(1 + nλ − λ)]
n=2
= [1 + β − λ(α + β)]
a,d, p
Ic,q
(1)
(−d/4)n−1 (a)n−1 ( p)n−1
(q)n−1 (c)n−1 ((n − 1)!)2
a,d, p
a,d, p
+ Ic,q
(1) − 1 − [(α + β)(1 − λ)] Ic,q
(1) − 1 ,
or, equivalently
∞
(−d/4)n−1 (a)n−1 ( p)n−1
(q)n−1 (c)n−1 ((n − 1)!)2
n [n(1 + β) − (α + β)(1 + nλ − λ)]
n=2
a,d, p
= [(1 + β) − λ(α + β)] Ic,q
(1)
(2.12)
a,d, p
+ (1 − α) Ic,q
(1) − (1 − α).
Using equation (2.12) is condition (2.11), we get the necessary and sufficient condition (2.10) for
a,d, p
the function θc,q (z) to be in U C T (λ, α, β)
Finally, taking −b = d = −1, c = ν +
(1.18), we get the following result:
b+1
= ν + 1 in assertion (2.10) and using equation
2
a,−1, p
Corollary 2.2. If c = ν + 1 > 0, a, p, q, ν ∈ R, q > a + p and q > 0, then θν+1,q (z) = 2z −
a,−1, p
Iν+1,q (z) ∈ U C T (λ, α, β) if and only if
a,−1, p
[(1 + β) − λ(α + β)] Iν+1,q (1)
a,−1, p
+ (1 − α) Iν+1,q (1)
≤ 2(1 − α),
(2.13)
a,−1, p
where 0 ≤ α < 1, 0 ≤ λ < 1, β ≥ 0 and Iν+1,q (z), given by equation (2.9).
3. CONVEXITY OF MORE GENERALIZED FUNCTIONS
In this section, we obtained the conditions for uniformly convexity of the convolution of the Generalized Bessel-Hypergeometric function with some analytic functions in class A.
a,d, p
We defined the function L c,q (z) : C → C by making convolution of the Generalized Bessela,d, p
Hypergeometric function Ic,q (z), given by equation (2.1) with the function f (z), given by equation
(1.1) as:
p
a,d, p
L a,d,
c,q (z) = Ic,q (z)*z f (z),
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which on using equations (1.1) and (2.1), gives
p
L a,d,
c,q (z) =
∞
(−d/4)n (a)n ( p)n
n=0
=z+
(q)n (c)n ((n)!)2
an z n+1
∞
(−d/4)n−1 (a)n−1 ( p)n−1
n=2
(q)n−1 (c)n−1 ((n − 1)!)2
(3.1)
an z n ,
b+1
where c = ν +
> 0, a, b, d, p, q, ν ∈ R and q > 0.
2
Pal [12], defined the class R σ (m, e) (σ ∈ C {0} , −1 ≤ e < m ≤ 1) as:
Definition 3.1. A function f (z) is said to be in the class R σ (m, e) (σ ∈ C {0} , −1 ≤ e < m ≤ 1),
if it satisfies the following inequality:
f (z) − 1
(z ∈ U ).
(3.2)
(m − e)σ − e [ f (z) − 1] < 1
a,d, p
To obtain the condition for the function L c,q (z) to be in the class U C T (λ, α, β), we require the
following Lemma [13]:
Lemma 3.1. The function f (z), given by equation (1.1) belong to the class R σ (m, e) (σ ∈ C
{0} , −1 ≤ e < m ≤ 1) if it satisfies the following inequality:
|an | (m − e)
|σ |
n
(n ∈ N {1}) .
(3.3)
Now, we obtain the following result:
Theorem 3.1. If f ∈ R σ (m, e) (σ ∈ C {0} , −1 ≤ e < m ≤ 1) and the following condition is
satisfied
a,d, p
a,d, p
(m − e) |σ | [(1 + β) − λ(α + β)] (Ic,q
(1)) + [(α + β)(λ − 1)] Ic,q
(1) ≤ 2(1 − α),
(3.4)
a,d, p
where f (z) and Ic,q (z) (a, b, p, q, ∈ R, d < 0, q > 0, c > 0, q > a + p) are given by equations
a,d, p
(1.1) and (2.1), respectively. Then L c,q (z) ∈ U C T (λ, α, β) (0 ≤ α < 1, 0 ≤ λ < 1, β ≥ 0).
a,d, p
Proof. Since d < 0, therefore in view of Lemma (1.4), L c,q (z) ∈ U C T (λ, α, β) if the following
inequality holds:
∞
n=2
n [n(1 + β) − (α + β)(1 + nλ − λ)]
(−d/4)n−1 (a)n−1 ( p)n−1
|an | ≤ 1 − α.
(q)n−1 (c)n−1 ((n − 1)!)2
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(3.5)
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Since f ∈ R σ (m, e), then using inequality (3.3) in the left hand side of inequality (3.5), we have
∞
n [n(1 + β) − (α + β)(1 + nλ − λ)]
n=2
≤
∞
(−d/4)n−1 (a)n−1 ( p)n−1
|an |
(q)n−1 (c)n−1 ((n − 1)!)2
n [n(1 + β) − (α + β)(1 + nλ − λ)]
n=2
(−d/4)n−1 (a)n−1 ( p)n−1
(q)n−1 (c)n−1 ((n − 1)!)2
(m − e)
|σ |
.
n
(3.6)
Now, we consider the right hand side of inequality (3.6) and simplify it to get
∞
(−d/4)n−1 (a)n−1 ( p)n−1
n [n(1 + β) − (α + β)(1 + nλ − λ)]
(q)n−1 (c)n−1 ((n − 1)!)2
n=2
= (m − e) |σ |
= (m − e) |σ |
∞
n=2
∞
[n(1 + β) − (α + β)(1 + nλ − λ)]
|σ |
(m − e)
n
(−d/4)n−1 (a)n−1 ( p)n−1
(q)n−1 (c)n−1 ((n − 1)!)2
[(n − 1) (1 + β − λ(α + β)) + (1 − α)]
n=2
(−d/4)n−1 (a)n−1 ( p)n−1
.
(q)n−1 (c)n−1 ((n − 1)!)2
Further, simplifying the above equation, gives
∞
n=2
n [n(1 + β) − (α + β)(1 + nλ − λ)]
(−d/4)n−1 (a)n−1 ( p)n−1
(q)n−1 (c)n−1 ((n − 1)!)2
(m − e)
|σ |
n
∞
∞
(−d/4)n (a)n ( p)n
(−d/4)n (a)n ( p)n
+ (1 − α)
= (m − e) |σ | (1 + β − λ(α + β))
,
(q)n (c)n (n − 1)!n!
(q)n (c)n ((n)!)2
n=1
n=1
Since q > a + p, therefore in view of Remark 2.1 and using the equations (2.1) and (2.2), for z = 1,
in the above equation, gives
∞
n [n(1 + β) − (α + β)(1 + nλ − λ)]
n=2
(−d/4)n−1 (a)n−1 ( p)n−1
|an |
(q)n−1 (c)n−1 ((n − 1)!)2
(3.7)
a,d, p
a,d, p
a,d, p
≤ (m − e) |σ | (1 + β − λ(α + β)) (Ic,q
(1)) − Ic,q
(1) + (1 − α) Ic,q
(1) − 1 .
If
a,d, p
a,d, p
a,d, p
(m − e) |σ | (1 + β − λ(α + β)) (Ic,q
(1)) − Ic,q
(1) + (1 − α) Ic,q
(1) − 1
≤ (1 − α),
(3.8)
then, in view of inequality (3.7), we get inequality (3.5). Inequality (3.8) gives assertion (3.4).
Now, taking −b = d = −1, c = ν +
(1.18), we get the following result:
b+1
= ν + 1 in assertion (3.4) and using equation
2
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Corollary 3.1. If f ∈ R σ (m, e) (σ ∈ C {0} , −1 ≤ e < m ≤ 1) and the following condition
holds:
a,−1, p
a,−1, p
(m − e) |σ | [(1 + β) − λ(α + β)] (Iν+1,q (1)) + [(α + β)(λ − 1)] Iν+1,q (1) ≤ 2(1 − α), (3.9)
a,−1, p
where 0 ≤ α < 1, 0 ≤ λ < 1, β ≥ 0 and Iν+1,q (z) (c = ν + 1 > 0, a, p, c, q, ν ∈ R, d < 0, q >
a,−1, p
0, q > a + p), given by equation (2.9), then L ν+1,q (z) ∈ U C T (λ, α, β) (0 ≤ α < 1, 0 ≤ λ <
1, β ≥ 0).
a,d, p
Next, we define the function Pc,q (z) : C → C as:
a,d, p
Pc,q
(z)
=
z
2−
0
a,d, p
Ic,q (t)
dt,
t
(3.10)
a,d, p
where a, b, p, q, ∈ R, d < 0, q > 0, c > 0, q > a + p and Ic,q (z) is given by equation (2.1).
Using equation (2.1), for z = t, in equation (3.10) and then integrating, we get
a,d, p
Pc,q
(z) = z −
∞
(−d/4)n−1 (a)n−1 ( p)n−1 z n
n=2
(q)n−1 (c)n−1 (n − 1)!(n)!
.
(3.11)
a,d, p
Now, we obtain the following necessary and sufficient condition for the function Pc,q (z) to be βuniformly convex of order α :
a,d, p
Theorem 3.2. If a, b, d, p, q ∈ R, d < 0, q > 0 and c > 0, then Pc,q (z) ∈ U C T (λ, α, β) if and
only if
a,d, p
a,d, p
(1) + [(α + β)(λ − 1)] Ic,q
(1) ≤ 2(1 − α).
[(1 + β) − λ(α + β)] Ic,q
(3.12)
a,c, p
where 0 ≤ α < 1, 0 ≤ λ < 1, β ≥ 0 and Ic,q (z) is defined by equation (2.1).
a,d, p
Proof. Since d < 0, therefore in view of Lemma 1.4, Pc,q (z) ∈ U C T (λ, α, β), if the following
inequality holds:
∞
n [n(1 + β) − (α + β)(1 + nλ − λ)]
n=2
(−d/4)n−1 (a)n−1 ( p)n−1
≤ 1 − α,
(q)n−1 (c)n−1 (n − 1)!n!
or, equivalently
∞
n=2
[n(1 + β) − (α + β)(1 + nλ − λ)]
(−d/4)n−1 (a)n−1 ( p)n−1
≤ 1 − α.
(q)n−1 (c)n−1 ((n − 1)!)2
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Simplifying the left hand side of relation (3.13), we get
∞
n [n(1 + β) − (α + β)(1 + nλ − λ)]
n=2
= [1 + β − λ(α + β)]
(−d/4)n−1 (a)n−1 ( p)n−1
(q)n−1 (c)n−1 (n − 1)!n!
∞
∞
(−d/4)n (a)n ( p)n
(−d/4)n (a)n ( p)n
.
+ (1 − α)
(q)n (c)n (n)!(n − 1)!
(q)n (c)n ((n)!)2
n=1
n=1
(3.14)
Since q > a + p, therefore in view of Remark 2.1 and using equations (2.1) and (2.2), for z = 1, in
inequality (3.14), we get
∞
n [n(1 + β) − (α + β)(1 + nλ − λ)]
n=2
= [(1 + β) − λ(α + β)]
(−d/4)n−1 (a)n−1 ( p)n−1
(q)n−1 (c)n−1 (n − 1)!n!
a, p
Ic,q
(1)
+ [(α + β)(λ − 1)]
(3.15)
a, p
Ic,q
(1)
− (1 − α).
Using equation (3.15) in inequality (3.13), we get the necessary and sufficient condition (3.12) for
a,d, p
Pc,q (z) to be in U C T (λ, α, β).
4. AN INTEGRAL OPERATOR
In this section, we discuss the starlikness and convexity properties of the function obtained by using
an integral operator introduced by Seenivasagan and Breaz [16]. The integral operator for analytic
functions f i ∈ A, (i = 1, 2, 3, ...) are defined as [16]:
1
⎧
⎫
1
ρ
⎪
⎪
n ⎨ z
f i (t) αi ⎬
F(z) := Fα1 , α2 , ...αn , ρ (z) = ρ
t ρ−1
dt
,
⎪
⎪
t
⎩ 0
⎭
i=1
where α1 , α2 , ...αn , ρ ∈ C.
We define the function G α1 , α2 , ...αn , ρ (z) in terms of the integral involving the generalized
a,d, p
Bessel-Hypergeometric function Ic,q (z):
1
⎧
⎫
1
ρ
⎪
⎪
a,d, p ⎪
⎪
n
⎨ z
Ic,q (t) αi ⎬
ρ−1
G(z) := G α1 , α2 , ...αn , ρ (z) = ρ
,
t
dt
⎪
⎪
t
⎪
⎪
i=1
⎩ 0
⎭
where α1 , α2 , ...αn , ρ ∈ R, c = ν +
(4.1)
b+1
> 0, a, b, d, p, q, ν ∈ R and q > 0.
2
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a,d, p
Now, according to Definition 1.2, the function Ic,q (z) is starlike function of order α (0 ≤
α < 1) if the following inequality holds:
⎧ ⎫
a,d, p
⎪
⎨ z Ic,q
⎬
(z) ⎪
Re
> α (z ∈ U, 0 ≤ α < 1).
(4.2)
a,d, p
⎪
⎩ Ic,q (z) ⎪
⎭
Since, for the function f : C → C, we have Re( f (z)) < | f (z)|, therefore from relation (4.2), we
get
a,d, p
z Ic,q
(z) (z ∈ U, 0 ≤ α < 1).
(4.3)
>α
a,d, p
Ic,q
(z) We need to mention the following definitions, defined by Miller [10] to establish the starlikeness
property of the function G(z), given by equation (4.1):
Definition 4.1. Let u = u 1 + iu 2 and v = v1 + iv2 and let be the set of functions ψ(u, v) satisfying the following conditions:
(i)
(ii)
ψ(u, v)is continuous in a domain D ⊂ C2 ,
(1, 0) ∈ D and Reψ(1, 0) > 0,
(iii)
Reψ(iu 2 , v1 ) ≤ 0, whenever (iu 2 , v1 ) ∈ D and v1 ≤
−1
(1 + u 22 ).
2
Definition 4.2. Let h(z) = 1 + c1 z + c2 z 2 + ... be regular in a unit disc and let ψ ∈ with
corresponding domain D. We denote by P(ψ) those functions h(z) that satisfy:
(i) (h(z), zh (z)) ∈ D,
(ii) Reψ(h(z), zh (z)) > 0,
when z ∈ . Also, we need to mention the following result obtained by Miller [10]:
Lemma 4.1. If ψ ∈ , with corresponding domain D, and if (h(z), zh (z)) ∈ D, then
Re(ψh(z), zh (z)) > 0 =⇒ Re(h(z)) > 0.
(4.4)
Next, we establish the following result:
Theorem 4.1. Let
zG (z)
= (1 − δ) p(z) + δ,
G(z)
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197
where the function G(z) is defined by equation (4.1), 0 < δ < 1, and p(z) = 1 + p1 z + p2 z 2 +
... ( p1 , p2 , ... ∈ R). If p(z) ∈ P(ψ), where ψ(∈ ) is given by
ψ(u, v) =
n
−(1 − δ)v
1
+ (1 − α)
+ ρ(1 − δ)(u − 1)
(1 − δ)u − δ
α
i=1 i
(u, v) ∈ D ⊂ C2 , u =
δ
,
1−δ
(4.6)
and
1
zp (z) ≥ .
2
(4.7)
Then G(z) is starlike function of order δ satisfying
− 2(1 − α)
n
i=1
2
n 1
2(1 − α) i=1
− 2ρ + 1 + 8ρ
αi
1
− 2ρ + 1 +
αi
1
,
2
(4.8)
where α1 , α2 , ...αn , ρ positive real numbers and α is the order of starlikness of the function
b+1
a,d, p
> 0, a, b, d, p, q, ν ∈ R, q > 0, q > a + p), defined by equation (2.1).
Ic,q (z), (c = ν +
2
0<δ≤
4ρ
, δ=
Proof. From equation (4.1), we get
ρ
(G(z))
=
ρ
z
t ρ−1
0
n
a,d, p
Ic,q (t)
1
αi
dt.
t
i=1
(4.9)
Differentiating both sides of equation (4.9), we obtain
zG (z)
=
G(z)
z
!n
ρ
i=1
1
a,d, p
Ic,q (z) αi
z
(G(z))ρ
.
(4.10)
Now, from equations (4.5) and (4.10), we get
zρ
(1 − δ) p(z) + δ =
!n
i=1
a,d, p
Ic,q (z)
z
(G(z))ρ
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αi
.
(4.11)
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Taking Logarithm of equation (4.11), then differentiating with respect to z and multiplying the
resultant equation with z, yields
⎛ ⎞
a,d, p
n
n
(z)
z
I
c,q
(1 − δ)zp (z)
1
zG (z)
⎜
⎟ 1
=ρ+⎝
.
(4.12)
−
−ρ
⎠
a,d,
p
(1 − δ) p(z) + δ
α
α
G(z)
Ic,q (z)
i=1 i
i=1 i
Using equation (4.5) in the right hand side of equation (4.12), we get
⎛ ⎞
a,d, p
n
n
z Ic,q
(z) 1
1
(1 − δ)zp (z)
⎜
⎟
+
=
+ ρ(1 − δ) [ p(z) − 1] .
⎝
⎠
a,d, p
α
(1
−
δ)
p(z)
+
δ
α
i
Ic,q (z)
i=1
i=1 i
(4.13)
Since each αi > 0, i = 1, 2, 3, ..., n , therefore taking modulus of both the sides of equation (4.13),
we have
n
a,d, p
n
z Ic,q
(1 − δ)zp (z)
(z) 1
1
+
=
+
ρ(1
−
δ)
p(z)
−
1]
[
.
a,d, p
Ic,q
(1 − δ) p(z) + δ
α
(z) i=1 αi
i=1 i
From equation (4.3), we get
n
n
(1 − δ)zp (z)
1
1
+
+ ρ(1 − δ) [ p(z) − 1] > α
.
(1 − δ) p(z) + δ
α
α
i
i=1
i=1 i
Using the inequality |z 1 | − |z 2 | ≤ |z 1 + z 2 | ≤ |z 1 | + |z 2 | (z 1 , z 2 ∈ C), we obtain
n
(1 − δ) zp (z)
1
+ (1 − α)
+ ρ(1 − δ) (| p(z)| − 1) > 0.
(1 − δ) | p(z)| − δ
α
i=1 i
(4.14)
(4.15)
(4.15).
Taking u = | p(z)| and v = − zp (z) in equation (4.6), we get the left hand side of relation
Now, we proceed to verify that ψ(u, v) , for u = | p(z)| + 0i and v = − zp (z) + 0i, satisfies the conditions of Definition 4.1. Since, it is clear that ψ(u, v) is continuous in its domain D.
Therefore the condition (i) is satisfied.
Further, taking u = 1 and v = 0 in equation (4.6), we get
ψ(1, 0) = (1 − α)
n
1
.
α
i=1 i
(4.16)
Since 0 ≤ α < 1, α1 , α2 , ...αn positive real numbers, therefore in view of equation (4.16), we have
Re (ψ(1, 0)) > 0. Thus condition (ii) of Definition 4.1 is satisfied.
Next, we verify the last condition of Definition 4.1. Here
(4.17)
v1 = Re(v) = − zp (z) .
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Using inequality (4.7) in equation (4.17), we get
v1 ≤
−1
,
2
(4.18)
or, equivalently
v1 ≤
−1
1 + u 22
2
(u 2 = I m(u) = 0).
Now,
ψ(iu 2 , v1 ) = ψ(0, v1 ) =
n
1
(1 − δ)v1
− ρ(1 − δ) + (1 − α)
.
δ
α
i=1 i
(4.19)
Since the right hand side of equation (4.19) is real, therefore we get
Re (ψ(0, v1 )) =
n
1
(1 − δ)v1
.
− ρ(1 − δ) + (1 − α)
δ
α
i=1 i
(4.20)
Using inequality (4.18) in equation (4.20), we get
Re (ψ(0, v1 )) ≤
n
1
−(1 − δ)
− ρ(1 − δ) + (1 − α)
,
2δ
α
i=1 i
or, equivalently
Re (ψ(0, v1 )) ≤
−(1 − δ) − 2δρ(1 − δ) + 2δ (1 − α)
n
2δ
i=1
1
αi
.
(4.21)
Since 0 < δ < 1, therefore, for the condition (iii) of Definition 4.1 to be satisfied, δ must satisfy the
following inequality:
−(1 − δ) − 2δρ(1 − δ) + 2δ (1 − α)
n
1
≤ 0,
α
i=1 i
or, equivalently
n
1
2ρδ + δ 2(1 − α)
− 2ρ + 1 − 1 ≤ 0.
α
i=1 i
2
(4.22)
On solving inequality (4.22) for δ, we get the assertion (4.8) of Theorem 4.1.
Further, since assertion (4.8) holds, therefore in view of relations (4.21) and (4.22), the condition (iii) of Definition 4.1 is satisfied and hence ψ ∈ .
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Since, the function p(z) ∈ P(ψ), therefore, it satisfies all the conditions of Definition 4.2 .
Thus, in view of Lemma 4.1, we have Re( p(z)) > 0. Therefore from equation (4.5), we have
zG (z)
> δ,
(4.23)
Re
G(z)
which proves that, G(z) is starlike function of order δ satisfying the relation (4.8).
Now, we obtain the order of convexity of the function G(z), defined by (4.1) in the following result:
b+1
a,d, p
> 0, a, b, d, p, q, ν ∈ R, q > 0, q >
Theorem 4.2. If the function Ic,q (z), (c = ν +
2
a + p), defined by equation (2.1) is starlike function of order α, (0 ≤ α < 1) and the function
G(z), defined by equation (4.1), is starlike function of order δ, given by inequality (4.8), then the
function G(z) is convex of order η, given by
η = (α − 1)
n
1
+ ρ − δ(ρ − 1),
α
i=1 i
(4.24)
where α1 , α2 , ...αn , ρ are positive numbers.
Proof. Taking Logarithm of equation (4.9), then differentiating with respect to z, and multiplying
the resultant equation with z, yields
⎛ ⎞
a,d, p
n
n
(z)
z
I
(z)
c,q
1
zG
zG (z)
⎜
⎟ 1
+
ρ
−
1 + (z) = ⎝
−
(ρ
−
1)
.
(4.25)
⎠
a,d,
p
G
α
G(z)
α
Ic,q (z)
i=1 i
i=1 i
Taking the real parts of both sides of equation (4.25), we get
⎛ ⎞
a,d, p
(z) n
n
(z) (z)
z
I
c,q
zG
1
zG
⎜
⎟ 1
Re 1 + (z) = Re ⎝
+
ρ
−
−
(ρ
−
1)Re
.
⎠
a,d,
p
G
α
G(z)
α
Ic,q (z)
i=1 i
i=1 i
Using relations (4.2) and (4.23), we get
n
1
zG (z)
+ ρ − δ(ρ − 1),
Re 1 + (z) > (α − 1)
G
α
i=1 i
this shows that, G(z) is convex function of order η, given by equation (4.24).
REFERENCES
[1] J. W. Alexander, Functions which map the interior of the unit circle upon simple regions, The Annals of Mathematics,
17(1) (1915) 12–22.
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[2] Á. Baricz, Generalized Bessel functions of the first kind, Springer, (2010).
[3] Á. Baricz, Geometric properties of generalized Bessel functions, Publ Math-Debrecen, 73(1-2) (2008) 155–178.
[4] B. Carlson, D. B. Shaffer, Starlike and prestarlike hypergeometric functions, SIAM J. Numer. Anal., 15(4) (1984)
737–745.
[5] E. Deniz, H. Orhan, H. Srivastava, Some sufficient conditions for univalence of certain families of integral operators
involving generalized Bessel functions, Taiwan. J. Math., 15(2) (2011) 883–917.
[6] P. L. Duren, Univalent functions, Vol. 259, Springer Science & Business Media, 2001.
[7] A. Goodman, On uniformly convex functions, Ann. Polon. Math., 56(1) (1991) 87–92.
[8] S. Kanas, A. Wisniowska, Conic regions and k-uniform convexity, J. Comput. Appl. Math., 105(1) (1999) 327–336.
[9] J.-L. Liu, Certain convolution properties of multivalent analytic functions associated with a linear operator, Anal. Appl,
292 (2004) 470–483.
[10] S. Miller, Differential inequalities and Caratheodory functions, Bull. Am. Math. Soc., 81(1) (1975) 79–81.
[11] G. Murugusundaramoorthy, On certain subclasses of analytic functions associated with hypergeometric functions,
Appl. Math. Lett., 24(4) (2011) 494–500.
[12] K. D. K. Pal, On a class of univalent functions related to complex order, Indian J. Pure Appl. Math., 26(9) (1995)
889–896.
[13] E. D. Rainville, Special functions, Chelsea (1971).
[14] F. Rønning, Uniformly convex functions and a corresponding class of starlike functions, Proc. Am. Math. Soc., 118(1)
(1993) 189–196.
[15] A. Schild, On starlike functions of order α, Am. J. Math., 87(1) (1965) 65–70.
[16] N. Seenivasagan, D. Breaz, Certain sufficient conditions for univalence, Gen. Math, 15(4) (2007) 7–15.
[17] H. Silverman, Univalent functions with negative coefficients, Proc. Am. Math. Soc., 51(1) (1975) 109–116.
I NDIAN J OURNAL OF I NDUSTRIAL AND A PPLIED M ATHEMATICS
Vol. 10, No. 1 (Special Issue), Jan–June 2019
INDIAN JOURNAL OF INDUSTRIAL AND APPLIED MATHEMATICS
Vol. 10, No. 1 (Special Issue), Jan–June 2019, pp. 202–219
DOI:
A New Iterative Class for Finding Common Fixed
Points of a Finite Family of Generalized Total
Asymptotically Nonexpansive and Multivalued
Mappings in Hyperbolic Spaces
Shamshad Husain1∗ and Nisha Singh2
Department of Applied Mathematics, Z H College of Engineering and Technology Aligarh Muslim
University, Aligarh-202002, India
(∗ Corresponding author) Email: ∗ s_husain68@yahoo.com, 2 nishasingh096@gmail.com
Abstract: The purpose of the paper is to introduce a general iteration process for the finite families
of total asymptotically nonexpansive mappings and generalized nonexpansive multivalued mappings in hyperbolic space which include Hadamard manifold and CAT(0) space as special cases.
Some important properties of the new defined iterative process are analyzed. Finally, - convergence and strong convergence for the iterative process in hyperbolic space are established. The
results presented in this paper extend and improve some existing results.
Keywords: common fixed point; asymptotically nonexpansive mapping; strong convergence; convergence; hyperbolic space.
AMS Subject Classification: 47H10. 47H09. 49M05. 54E70.
1. INTRODUCTION
In nonlinear functional analysis, one of the most productive tools is the fixed point theory, which
has numerous applications in many quantitative disciplines such as biology, chemistry, computer
science and in many branches of engineering.
The study of metric spaces without linear structure has played a vital role in various branches
of pure and applied sciences. In the past two decades, the metric fixed point theory has been investigated extensively by numerous mathematicians. Takahashi [17] introduced the concept of convexity
in a metric space (X,d) as follows:
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A convex structure in a metric space (X, d) is a mapping W : X × X × [0, 1] → X such that,
for all x, y, u ∈ X and all λ ∈ [0, 1],
d(u, W (x, y, λ)) ≤ λd(u, x) + (1 − λ)d(u, y).
A metric space together with a convex structure is called a convex metric space. A nonempty subset
K of X is said to be convex if W (x, y, λ) ∈ K for all (x, y, λ) ∈ K × K × [0, 1].
Recently, Kohlenbach [8] introduced the concept of convex metric space by defining hyperbolic
space.
A hyperbolic space [8], (X, d, W ) is a metric space (X, d) together with a convexity mapping
W : X × X × [0, 1] → X such that:
i.
ii.
iii.
iv.
d(u, W (x, y, α)) ≤ αd(u, x) + (1 − α)d(u, y)
d(W (x, y, α), W (x, y, β)) = |α − β|d(x, y)
W (x, y, α) = W (y, x, (1 − α))
d(W (x, z, α), W (y, w, α)) ≤ αd(x, y) + (1 − α)d(z, w)
for all x, y, z, w ∈ X and α, β ∈ [0, 1]. If (X, d, W ) satisfies only (i), then it coincides with the
convex metric space which is introduced by Takahashi [17].
The concept of hyperbolic spaces defined above is more restrictive than the hyperbolic type
defined by Goebel and Kirk [4]. But it is more general than the hyperbolic space defined by Reich
and Shafrir [14].
All normed spaces and their subsets are the examples of hyperbolic spaces as well as convex
metric spaces. It is remarked that every CAT(0) and Banach spaces are special cases of hyperbolic
space.
In 2008, Zhao et al. [21] introduced the generalized implicit iterative process for the common
fixed points of the finite family {T j : j ∈ J }.
In 2011, Khan [6] generalized the results of Zhao et al [21] for two finite families of nonexpansive mappings.
In 2012, Khan et al [7] studied implicit iteration for two finite families of nonexpansive mappings in a hyperbolic space.
Later on some authors discussed the convergence of the iterative process in hyperbolic spaces
[1, 3, 5, 15, 19].
Remark: A hyperbolic space (X,d,W) is said to be uniformly convex if for any r > 0 and ∈ (0, 2]
and for all u, x, y ∈ X , there exists δ ∈ (0, 1] such that,
1
, u ≤ (1 − δ)r,
d W x, y,
2
provided max{d(x, u), d(y, u)} ≤ r and d(x, y) ≥ r [9, 16].
The class of uniformly convex hyperbolic spaces includes both uniformly convex normed
spaces and CAT(0) spaces as special cases.
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Markin [11] and Nadler [12] introduced the study of fixed points for multivalued contractions
and nonexpansive mappings using the Hausdorff metric. Shimizu and Takahashi [16] proved the
existence of fixed points for multivalued nonexpansive mappings in convex metric spaces, that is,
every multivalued mapping T : X → C(X ) has a fixed point in a bounded, complete and uniformly
convex metric space (X,d), where C(X) is the family of all compact subsets of X.
Inspired and motivated by the above work, we define a new algorithm as follows:
Let K be a nonempty closed and convex subset of a hyperbolic space X . For j ∈ J , let T j :
j
j
j
K → K be a ({μn }, {ξn }, ρ j )-total asymptotically nonexpansive mapping with limn→∞ μn = 0
j
and limn→∞ ξn = 0 and a strictly increasing continuous function ρ j : [0, +∞) → [0, +∞) satj
j
isfying ρ j (0) = 0 and let S j : K → K be a ({μ̂n }, {ξ̂n }, ρ̂ j )-total asymptotically nonexpansive
j
j
mapping with limn→∞ μ̂n = 0 and limn→∞ ξ̂n = 0 and a strictly increasing continuous function
j
j
ρ̂ : [0, +∞) → [0, +∞) satisfying ρ̂ (0) = 0 and Q j : K → K be an asymptotically nonexpan
j
sive mapping such that ∞
k < +∞ and R j : K → K be an asymptotically nonexpansive map∞ j n=1 n
ping such that n=1 k̂n < ∞. Let T̂ : K → P(K ) be a multivalued mapping such that F(T̂ ) = ∅
and PT̂ is a nonexpansive mapping. We consider the following general iterative sequence {xn }:
⎧
⎪
δr n
⎪
,
β
⎨ yn = W Trn xn , W xn , Q rn u n , 1−β
rn
rn
⎪
θr n
⎪
,
α
⎩xn+1 = W Srn yn , W u n , Rrn vn , 1−α
rn ,
rn
(1.1)
where u n ∈ PT̂ (xn ) and vn ∈ PT̂ (yn ) and suppose that {αr n }, {βr n }, {δr n } and {θr n } are the real
sequences in [a, b] for some a, b ∈ (0, 1) where F = nj=1 [F(T j ) F(S j ) F(R j )
F(Q j )] F(T̂ ) is the set of all common fixed points.
Further, we study some results concerning -convergence as well as strong convergence for
the above defined iteration process. The results presented in the paper improve and extend the corresponding results given by [6, 7, 13, 21]
2. PRELIMINARIES
From now, N denotes the set of natural numbers and J = {1, 2, . . . , N }, the set of first N natural
numbers.
Definition 2.1. Let T : K → 2 K be a multivalued mapping. An element x ∈ K is said to be a fixed
point of T , if x = T x. Denote by F(T ), the set of fixed points of T .
Definition 2.2. A subset K ⊂ X = ∅ is said to be proximal, if for each x ∈ X , there exists an
element y ∈ K such that,
d(x, y) = dist(x, K ) := inf{d(x, z) : z ∈ K }.
It is well known that each weakly compact convex subset of a Banach space is proximal as well as
each closed convex subset of a uniformly convex Banach space is also proximal.
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Definition 2.3. Let C B(K ) be the collection of all nonempty and closed bounded subsets and P(K )
be the collection of all nonempty proximal bounded and closed subsets of K , respectively. Let
H (., .) be the Hausdorff distance on C B(K ) is defined by:
H (A, B) := max{sup dist(x, B), sup dist(y, A)}, ∀A, B ∈ C B(X ).
x∈A
y∈B
Definition 2.4. A mapping T : K → K is said to be:
(i)
(ii)
(iii)
semi-compact if every bounded sequence {xn } ⊂ K , satisfying d(xn , T xn ) → 0 as n →
∞, has a convergent subsequence;
nonexpansive if d(T x, T y) ≤ d(x, y) for any x, y ∈ K ;
asymptotically nonexpansive if there exists a sequence {kn } ⊂ [0, +∞) and
limn→∞ kn = 0 such that
d(T n x, T n y) ≤ (1 + kn )d(x, y), ∀x, y ∈ K , n ≥ 1;
(iv)
({μn }, {ξn }, ρ)-total asymptotically nonexpansive, if there exist nonnegative sequences
{μn }, {ξn } with μn → 0, ξn → 0 and a strictly increasing continuous function ρ :
[0, +∞) → [0, +∞) with ρ(0) = 0 such that
d(T n x, T n y) ≤ d(x, y) + μn ρ(d(x, y)) + ξn , ∀x, y ∈ K , n ≥ 1;
(v)
uniformly L-Lipschitzian if there exists a constant L > 0 such that
d(T n x, T n y) ≤ Ld(x, y), ∀x, y ∈ K , n ≥ 1.
Definition 2.5. Let {xn } be a bounded sequence in a hyperbolic space X . For x ∈ X , we define a
continuous functional r (., {xn }) : X → [0, +∞) by:
r (x, {xn }) = lim sup d(x, xn ).
n→∞
(i)
The asymptotic radius r̂ ({xn }) of {xn } is given by
r̂ ({xn }) = inf{r (x, {xn }) : x ∈ X }.
(ii)
The asymptotic center of a bounded sequence {xn } with respect to K ⊂ X is defined as
follows:
A K ({xn }) = {x ∈ X : r (x, {xn }) ≤ r (y, {xn }), ∀y ∈ K },
which is the set of minimizers for (., {xn }).
It is simply denoted by A({xn }) when the asymptotic center is taken with respect to X and a sequence
{xn } in X is said to be -convergent to x ∈ X if x is the unique asymptotic center of {u n } for every
subsequence {u n } of {xn }. In this case, we write -limn→∞ xn = x and call x the -limit of {xn }.
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Lemma 2.1 [10]. Let (X, d, W ) be a complete uniformly convex hyperbolic space with monotone
modulus of uniform convexity. Then every bounded sequence {xn } in X has a unique asymptotic
center with respect to any nonempty closed convex subset K of X.
Lemma 2.2 [20]. Let (X, d, W ) be a uniformly convex hyperbolic space with monotone modulus of
uniform convexity η. Let x ∈ X and {αn } be a sequence in [a, b] for some a, b ∈ (0, 1). If {xn } and
{yn } are sequences in X such that for some c ≥ 0, limn→∞ sup d(xn , x) ≤ c, limn→∞ sup d(yn , x) ≤
c, limn→∞ d(W (xn , yn , αn ), x) = c. Then limn→∞ d(xn , yn ) = 0.
Lemma 2.3 [20]. Let K be a nonempty closed convex subset of a uniformly convex hyperbolic
space and {xn } a bounded sequence in K such that A({xn }) = {y} and r ({xn }) = ρ. If {ym } is another
sequence in K such that limm→∞ r (ym , {xn }) = ρ, then limm→∞ ym = y.
Lemma 2.4 [18]. Let {αn }, {βn } and {γn } be nonnegative real sequences satisfying
αn+1 ≤ (1 + γn )αn + βn , ∀n ≥ 1.
∞
If ∞
n=1 γn < ∞ and
n=1 βn < ∞, then the limn→∞ αn exist. If there exists a subsequence {αn i } ⊂
{αn } such that αni → 0, then limn→∞ αn = 0.
3. MAIN RESULTS
Theorem 3.1. Let K be a nonempty closed convex subset of a hyperbolic space X . For j ∈ J ,
let T j , S j : K → K be a total asymptotically nonexpansive mapping and Q j , R j : K → K be an
asymptotically nonexpansive mapping and let T̂ : K → P(K ) be a multivalued mapping. Assume
that F = ∅ and for each j ∈ J , the following holds:
(i)
∞
n=1
∞
j
μn < +∞;
j
n=1 kn
(ii)
< +∞;
∞
n=1
∞
j
μ̂n < +∞;
j
n=1 k̂n
∞
j
n=1 ξn
∞
j
n=1 ξ̂n
< +∞;
< +∞;
< +∞;
there exists a constant M ∗ > 0 such that
ρ j (r ) ≤ M ∗r,
ρ̂ j (r ) ≤ M ∗r, ∀r > 0.
Let {xn } be the sequence defined in (1.1), then limn→∞ d(xn , p) exists for all p ∈ F.
j
j
j
j
j
j
Proof. Let p ∈ F. Set μn = max j∈J {μn , μ̂n };
ξ̂n }; kn = max j∈J {k
n , k̂n } and
ξn = max j∈J {ξn ,
∞
∞
μ
<
+∞,
ξ
<
+∞
and
ρ = max j∈J {ρ j , ρ̂ j }. By (i), we know that ∞
n
n
n=1
n=1
n=1 kn <
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New Iterative Class for Finding Common Fixed Points of a Finite Family
+∞. Then from (1.1), we have
d(yn , p)
≤
δr n
,p
1 − βr n
βr n d(Trn xn , p) + δr n d(xn , p) + (1 − βr n − δr n )d(Q rn u n , p)
≤
βr n [d(xn , p) + μrn ρ r (d(xn , p)) + ξnr ] + δr n d(xn , p)
≤
βr n d(Trn xn ,
207
p) + (1 − βr n )d W xn , Q rn u n ,
+(1 − βr n − δr n )knr d(u n , p)
≤
βr n [d(xn , p) + μn ρ(d(xn , p)) + ξn ] + δr n d(xn , p)
≤
βr n [d(xn , p) + μn M ∗ d(xn , p) + ξn ] + δr n d(xn , p)
+(1 − βr n − δr n )kn dist(u n , PT̂ ( p))
+(1 − βr n − δr n )kn H (PT̂ (xn ), PT̂ ( p))
≤
βr n [(1 + μn M ∗ )d(xn , p) + ξn ] + δr n d(xn , p)
+(1 − βr n − δr n )kn d(xn , p)
≤
Next
d(xn+1 , p)
(1 + μn M ∗ )kn d(xn , p) + ξn .
(3.1)
≤
θr n
,p
1 − αr n
αr n d(Srn yn , p) + θr n d(u n , p) + (1 − αr n − θr n )d(Rrn vn , p)
≤
αr n [d(yn , p) + μ̂rn ρ̂ r (d(yn , p)) + ξ̂nr ] + θr n d(u n , p)
≤
αr n d(Srn yn , p) + (1 − αr n )d W u n , Rrn vn ,
+(1 − αr n − θr n )k̂nr d(vn , p)
≤
αr n [d(yn , p) + μn ρ(d(yn , p)) + ξn ] + θr n dist(u n , PT̂ ( p))
+(1 − αr n − θr n )kn dist(vn , PT̂ ( p))
≤
αr n [d(yn , p) + μn M ∗ d(yn , p) + ξn ] + θr n H (PT̂ (xn ), PT̂ ( p))
+(1 − αr n − θr n )kn H (PT̂ (yn ), PT̂ ( p))
≤
αr n [(1 + μn M ∗ )d(yn , p) + ξn ] + θr n d(xn , p)
≤
αr n [(1 + μn M ∗ )d(yn , p) + ξn ] + θr n d(xn , p)
+(1 − αr n − θr n )kn d(yn , p)
+(1 − αr n − θr n )kn [(1 + μn M ∗ )kn d(xn , p) + ξn ]
≤
αr n [(1 + μn M ∗ )2 kn d(xn , p) + ξn (1 + μn M ∗ + ξn ] + θr n d(xn , p)
+(1 − αr n − θr n )kn [(1 + μn M ∗ )kn d(xn , p) + ξn ]
≤
≤
(1 + μn M ∗ )2 kn2 d(xn , p) + (1 + μn M ∗ )ξn
2
2
μn M ∗ +
d(xn , p)kn2 1 +
(μn M ∗ )2 + (1 + μn M ∗ )ξn
1
2
≤
(1 + an2 μn )d(xn , p) + (1 + μn M ∗ )ξn
≤
(1 + M1 μn )d(xn , p) + M2 ξn ,
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(3.2)
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where an2 = 21 M ∗ + 22 μn (M ∗ )2 and from (i), there exists positive constants M1 and M2 such
that an2 ≤ M1 , (1 + μn M ∗ ) ≤ M2 for each n ≥ 1. Now from Lemma 2.4, we have limn→∞ d(xn , p)
exists for each p ∈ F.
In 1993, Bruck et al [2] introduced a notion of asymptotically nonexpansive mapping in
the intermediate sense. A mapping T : K → K is said to be asymptotically nonexpansive mapping in the intermediate sense, provided that T is uniformly continuous and lim supn→∞ supx,y∈K
{d(T n x, T n y) − d(x, y)} ≤ 0.
The following corollaries can be obtained immediately from Theorem 3.1.
Corollary 3.1. Let K be a nonempty closed convex subset of a hyperbolic space X . For j ∈ J ,
j
let T j : K → K be a {ξn }-asymptotically nonexpansive mapping in the intermediate sense and S j :
j
j
K → K be a {ξ̂n }-asymptotically nonexpansive mapping in the intermediate sense. If ∞
n=1 ξn <
∞ j
+∞,
n=1 ξ̂n < +∞ and Q j , R j : K → K be an asymptotically nonexpansive mapping and F =
∅, then for the sequence {xn } defined in (1.1), limn→∞ d(xn , p) exists for all p ∈ F.
Corollary 3.2. Let K be a nonempty closed convex subset of a hyperbolic
space X . For j ∈ J let
j
j
T j : K → K be a {ln }-asymptotically nonexpansive mapping with ∞
l
n=1 n < +∞ and S j : K →
∞ j
j
K be a {l̂n }-asymptotically nonexpansive mapping with n=1 l̂n < +∞ and Q j , R j : K → K be
an asymptotically nonexpansive mapping and F = ∅, then for the sequence {xn } defined in (1.1),
limn→∞ d(xn , p) exists for all p ∈ F.
Theorem 3.2. Let K be a nonempty closed convex subset of a uniformly convex hyperbolic space
X with monotone modulus of uniform convexity η. For j ∈ J , let T j : K → K be a uniformly
L j -Lipschitzian and total asymptotically nonexpansive mapping and let S j : K → K be a uniformly L̂ j -Lipschitzian and total asymptotically nonexpansive mapping and Q j , R j : K → K be an
asymptotically nonexpansive mapping and let T̂ : K → P(K ) be a multivalued mapping. Assume
that F = ∅ and the conditions (i) and (ii) in Theorem 3.1 hold. Then for j ∈ J , the sequence {xn }
generated by (1.1), we have
lim d(xn , T j xn ) = lim d(xn , S j xn ) = lim d(xn , R j xn ) = lim d(xn , Q j xn ) = 0.
n→∞
n→∞
n→∞
n→∞
Proof. From Theorem 3.1, it follows that limn→∞ d(xn , p) exists for each p ∈ F. Assume that
lim d(xn , p) = c > 0.
n→∞
(3.3)
Since μn → 0 and ξn → 0 as n → ∞, so taking lim sup on both sides of (3.1), we have
lim sup d(yn , p) ≤ c.
(3.4)
lim inf d(yn , p) ≥ c.
(3.5)
n→∞
Also taking lim inf in (3.2), we get
n→∞
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From (3.4) and (3.5), we get
lim d(yn , p) = c.
(3.6)
n→∞
From (3.6), we get
lim d W Trn xn , W xn , Q rn u n ,
n→∞
δr n
, βr n , p = c.
1 − βr n
(3.7)
Now,
d W xn , Q rn u n ,
δr n
,p
1 − βr n
≤
≤
≤
≤
≤
≤
δr n
d(xn , p) +
1 − βr n
δr n
d(xn , p) +
1 − βr n
δr n
d(xn , p) +
1 − βr n
δr n
d(xn , p) +
1 − βr n
δr n
d(xn , p) +
1 − βr n
d(xn , p),
1−
1−
1−
1−
1−
δr n
1 − βr n
δr n
1 − βr n
δr n
1 − βr n
δr n
1 − βr n
δr n
1 − βr n
d(Q rn u n , p)
knr d(u n , p)
kn dist(u n , PT̂ ( p))
kn H (PT̂ (xn ), PT̂ ( p))
kn d(xn , p)
which implies that
lim sup d W xn , Q rn u n ,
n→∞
δr n
, p ≤ c.
1 − βr n
(3.8)
Further,
lim sup d(Trn xn , p) ≤ c.
(3.9)
n→∞
From (3.7)-(3.9) and Lemma 2.2, we have
lim d Trn xn , W xn , Q rn u n ,
n→∞
δr n
1 − βr n
= 0.
(3.10)
Next,
lim d(xn+1 , p) = lim d W Srn yn , W u n , Rrn vn
n→∞
n→∞
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θr n
, αr n , p = c.
1 − αr n
(3.11)
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Next from (3.1), we have
d W u n , Rrn vn ,
θr n
θr n
θr n
,p ≤
d(u n , p) + 1 −
d(Rrn vn , p)
1 − αr n
1 − αr n
1 − αr n
θr n
θr n
d(u n , p) + 1 −
k̂n d(vn , p)
≤
1 − αr n
1 − αr n
θr n
θr n
dist(u n , PT̂ ( p)) + 1 −
kn dist(vn , PT̂ ( p))
≤
1 − αr n
1 − αr n
θr n
θr n
H (PT̂ (xn ),PT̂ ( p))+ 1−
kn H (PT̂ (yn ),PT̂ ( p))
≤
1−αr n
1−αr n
θr n
θr n
≤
d(xn , p) + 1 −
kn d(yn , p)
1 − αr n
1 − αr n
θr n
θr n
d(xn , p) + 1 −
kn
≤
1 − αr n
1 − αr n
[(1 + μn M ∗ )kn d(xn , p) + ξn ]
≤ (1 + μn M ∗ )kn2 d(xn , p) + ξn ,
which implies that
lim sup d W u n , Rrn vn ,
n→∞
θr n
, p ≤ c.
1 − αr n
(3.12)
Also,
lim sup d(Srn yn , p) ≤ c.
(3.13)
n→∞
From (3.11)- (3.13) and Lemma 2.2, we have
lim d
n→∞
Srn yn , W
u n , Rrn vn ,
θr n
1 − αr n
= 0.
(3.14)
Further,
d(xn+1 , Srn yn )
θr n
n
, αr n , Sr yn
1 − αr n
θr n
n
n
n
n
, Sr yn ,
αr n d(Sr yn , Sr yn ) + (1 − αr n )d W u n , Rr vn ,
1 − αr n
n
= d W Sr yn , W u n , Rrn vn ,
≤
and using (3.14), we get
lim d(xn+1 , Srn yn ) = 0.
n→∞
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(3.15)
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Next, we have
d(xn+1 , p)
≤
≤
≤
θr n
,p
1 − αr n
θr n
n
n
,p
αr n d(xn+1 , p) + αr n d(xn+1 , Sr yn ) + (1 − αr n )d W u n , Rr vn ,
1 − αr n
αr n
θr n
,p .
(3.16)
d(xn+1 , Srn yn ) + d W u n , Rrn vn ,
1 − αr n
1 − αr n
αr n d(Srn yn ,
p) + (1 − αr n )d W u n , Rrn vn ,
Taking lim inf on both sides of (3.16) and using (3.15), we have
lim inf d W u n , Rrn vn ,
n→∞
θr n
, p ≥ c.
1 − αr n
(3.17)
From (3.12) and (3.17), we get
d W u n , Rrn vn ,
θr n
, p = c.
1 − αr n
(3.18)
Using Lemma 2.2 and (3.18), we get
lim d(u n , Rrn vn ) = 0.
n→∞
(3.19)
Further,
d(yn , Trn xn )
=
≤
δr n
n
, βr n , Tr xn
1 − βr n
δr n
n
n
n
n
, Tr xn .
βr n d(Tr xn , Tr xn ) + (1 − βr n )d W xn , Q r u n ,
1 − βr n
n
d W Tr xn , W xn , Q rn u n ,
Using (3.10), we get
lim d(yn , Trn xn ) = 0.
n→∞
(3.20)
Next, we have
d(yn , p)
≤
≤
≤
δr n
n
,p
p) + (1 − βr n )d W xn , Q r u n ,
1 − βr n
δr n
n
n
,p
βr n d(yn , p) + βr n d(Tr xn , yn ) + (1 − βr n )d W xn , Q r u n ,
1 − βr n
βr n
δr n
,p .
(3.21)
d(Trn xn , yn ) + d W xn , Q rn u n ,
1 − βr n
1 − βr n
βr n d(Trn xn ,
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Taking lim inf on both sides of (3.21) and using (3.20), we get
lim inf d W xn , Q rn u n ,
n→∞
δr n
, p ≥ c.
1 − βr n
(3.22)
From (3.8) and (3.22), we get
lim d W xn , Q rn u n ,
n→∞
δr n
, p = c.
1 − βr n
(3.23)
Using Lemma 2.2 and (3.23), we get
lim d(xn , Q rn u n ) = 0.
n→∞
(3.24)
Further,
d(xn , Trn xn )
≤ d(xn , yn ) + d(yn , Trn xn )
≤
≤
≤
≤
δr n
n
, xn + d(yn , Trn xn )
+ (1 − βr n )d W xn , Q r u n ,
1 − βr n
1
δr n
n
, xn +
d(yn , Trn xn )
d W xn , Q r u n ,
1 − βr n
1 − βr n
δr n
1
δr n
n
d(xn , xn ) + 1 −
d(Q r u n , xn ) +
d(yn , Trn xn )
1 − βr n
1 − βr n
1 − βr n
1
δr n
n
d(Q r u n , xn ) +
d(yn , Trn xn ).
1−
1 − βr n
1 − βr n
βr n d(xn , Trn xn )
From (3.24) and (3.20), we have
lim d(xn , Trn xn ) = 0.
n→∞
(3.25)
Moreover,
d(xn , yn ) ≤
≤
δr n
, xn
1 − βr n
βr n d(xn , Trn xn ) + (1 − βr n − δr n )d(xn , Q rn u n ).
βr n d(xn , Trn xn ) + (1 − βr n )d W xn , Q rn u n ,
From (3.24) and (3.25), we have
lim d(xn , yn ) = 0.
n→∞
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(3.26)
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Let L = max j∈J {L j , L̂ j }, where L j and L̂ j are Lipschitz constants for T j and S j for j ∈ J , respectively. Since each T j is uniformly L-Lipschitzian for j ∈ J , we have
d(xn , T jn xn )
≤ d(xn , yn ) + d(yn , T jn xn )
≤ d(xn , yn ) + d(yn , T jn yn ) + d(T jn yn , T jn xn )
≤ d(xn , yn ) + d(yn , T jn yn ) + Ld(yn , xn )
≤ (1 + L)d(xn , yn ) + d(yn , T jn yn ).
(3.27)
Further
d(yn , T jn yn )
≤
d(yn , T jn xn ) + d(T jn xn , T jn yn )
≤
d(yn , T jn xn ) + Ld(xn , yn ).
From (3.20) and (3.28) we get
lim d(yn , T jn yn ) = 0.
n→∞
(3.28)
Using (3.26) and (3.28) in (3.27), we get
lim d(xn , T jn xn ) = 0 ∀ j ∈ J.
n→∞
(3.29)
Further,
d(xn , T j xn )
≤
d(xn , T jn xn ) + d(T jn xn , T jn yn ) + d(T jn yn , T j xn )
≤
d(xn , T jn xn ) + Ld(xn , yn ) + Ld(T jn−1 yn , xn )
≤
d(xn , T jn xn ) + 2Ld(xn , yn ) + Ld(T jn−1 yn , yn ).
From (3.26), (3.28) and (3.29), we get
lim d(xn , T j xn ) = 0, ∀ j ∈ J.
n→∞
Similarly, we show that
lim d(xn , S j xn ) = 0, ∀ j ∈ J.
n→∞
Next, we show that
lim d(xn , Q j xn ) = 0.
n→∞
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Now, we have
d(xn , Q rn xn )
≤ d(xn , Q rn u n ) + d(Q rn u n , Q rn xn )
≤ d(xn , Q rn u n ) + knr d(u n , xn )
≤ d(xn , Q rn u n ) + kn dist(u n , PT̂ (xn ))
≤ d(xn , Q rn u n ) + kn H (PT̂ (xn ), PT̂ (xn ))
≤ d(xn , Q rn u n ) + kn d(xn , xn ).
So from (3.24), we have
lim d(xn , Q rn xn ) = 0.
n→∞
(3.30)
Further,
d(xn , Q j xn ) ≤
≤
d(xn , Q nj xn ) + d(Q nj xn , Q j xn )
d(xn , Q nj xn ) + kn d(Q n−1
j x n , x n ).
From (3.35), we get
lim d(xn , Q j xn ) = 0.
n→∞
Similarly, we show that
lim d(xn , R j xn ) = 0.
n→∞
Further, we have
d(u n , p) = dist(u n , PT̂ ( p)) ≤ H (PT̂ (xn ), PT̂ ( p)) ≤ d(xn , p).
(3.31)
Taking the lim sup of both sides of (3.31), we have
lim sup d(u n , p) ≤ c.
(3.32)
lim sup d(vn , p) ≤ c.
(3.33)
n→∞
Similarly, we have
n→∞
Since limn→∞ d(xn+1 , p) = c, so from (3.32) and (3.33) and using Lemma 2.2, we have
lim d(u n , vn ) = 0.
(3.34)
lim d(u n , xn ) = 0.
(3.35)
n→∞
By using (3.4) and Lemma 2.2, we have
n→∞
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From
d(xn , T̂ xn )
≤ d(xn , PT̂ (xn ))
≤ d(xn , u n ).
(3.36)
So, we have
lim d(xn , T̂ xn ) = 0.
(3.37)
n→∞
Corollary 3.3. Let K be a nonempty closed convex subset of a uniformly convex hyperbolic space
X with monotone modulus of uniform convexity η. For j ∈ J , let T j : K → K be a uniformly
j
L j -Lipschitzian and {ξn }-asymptotically nonexpansive mapping in the intermediate sense and let
j
S j : K → K be a uniformly L̂ j -Lipschitzian and {ξ̂n }-asymptotically nonexpansive mapping in the
∞ j
∞ j
intermediate sense. If n=1 ξn < +∞ and n=1 ξ̂n < +∞ and Q j , R j : K → K be an asymptotically nonexpansive mapping and F = ∅. Then for j ∈ J , the sequence {xn } generated by (1.1), we
have
lim d(xn , T j xn ) = lim d(xn , S j xn ) = lim d(xn , R j xn ) = lim d(xn , Q j xn ) = 0.
n→∞
n→∞
n→∞
n→∞
Corollary 3.4. Let K be a nonempty closed convex subset of a uniformly convex hyperbolic space
X with monotone modulus of uniform convexity η. For j ∈ J , let T j : K → K be a uniformly L j j
j
Lipschitzian and {ln }-asymptotically nonexpansive mapping with ∞
Sj : K →
n=1 l n < +∞ and let
j
j
K be a uniformly L̂ j -Lipschitzian and {l̂n }-asymptotically nonexpansive mapping with ∞
n=1 l̂ n <
+∞ and Q j , R j : K → K be an asymptotically nonexpansive mapping and F = ∅. Then for j ∈ J ,
the sequence {xn } generated by (1.1), we have
lim d(xn , T j xn ) = lim d(xn , S j xn ) = lim d(xn , R j xn ) = lim d(xn , Q j xn ) = 0.
n→∞
n→∞
n→∞
n→∞
4. STRONG CONVERGENCE
Now we establish -convergence and strong convergence of the iteration process (1.1).
Theorem 4.1. Let K be a nonempty closed convex subset of a uniformly convex hyperbolic space
X with monotone modulus of uniform convexity η. For j ∈ J , let T j : K → K be a uniformly
j
j
L j -Lipschitzian and ({μn }, {ξn }, ρ j )-total asymptotically nonexpansive mapping and S j : K → K
j
j
be a uniformly L̂ j -Lipschitzian and ({μ̂n }, {ξ̂n }, ρ̂ j )-total asymptotically nonexpansive mapping.
Let Q j , R j : K → K be an asymptotically nonexpansive mapping and let T̂ : K → P(K ) be a
multivalued mapping. Assume that F = ∅ and the conditions (i) and (ii) in Theorem 3.1 hold. Then
for j ∈ J , the sequence {xn } generated by (1.1), -converges to a common fixed point of F.
Proof. Since the sequence {xn } is bounded, therefore from Lemma 2.1, {xn } has a unique asymptotic
center in K , i.e., A({xn }) = {x}. Let {vn } be any subsequence of {xn } such that A({vn }) = {v}. Then
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from Theorem 3.2 ∀ j ∈ J , we have
lim d(vn , T j vn ) = lim d(vn , S j vn ) = lim d(vn , R j vn ) = lim d(vn , Q j vn ) = 0.
n→∞
n→∞
n→∞
n→∞
(4.1)
Now, we claim that v is the common fixed point of {T j }rj=1 ; {S j }rj=1 ; {Q j }rj=1 and {R j }rj=1 . For each
j ∈ J , define a sequence {z m } in K by z m = T jm v. Then,
d(z m , vn ) ≤
≤
d(T jm v, T jm vn ) + d(T jm vn , T jm−1 vn ) + . . . + d(T j vn , vn )
[d(v, vn ) + μmj ρ j (d(v, vn )) + ξmj ] +
m−1
d(T ji+1 vn , T ji vn ).
j=0
Since each T j is uniformly L j -Lipschitzian for j ∈ J , so we have
d(z m , vn ) ≤ [(1 + μm M ∗ )d(v, vn ) + ξm ] + m Ld(T j vn , vn ),
(4.2)
where L = max j∈J {L j , L̂ j }.
Taking lim sup on both sides of (4.2) and using (4.1), we have
r (z m , {vn }) = lim sup d(z m , vn ) ≤ lim sup d(v, vn ) = r (v, {vn }),
n→∞
n→∞
which implies that |r (z m , {vn }) − r (v, {vn })| → 0 as m → ∞. From Lemma 2.3, it follows that
limm→∞ T jm v = v, by the uniform continuity of T j , we have
T j (v) = T ( lim T jm v) = lim T jm+1 v = v.
m→∞
m→∞
From the arbitrariness of j ∈ J , we have v is the common fixed point of {T j }rj=1 . Similarly, we
show that v is the common fixed point of {S j }rj=1 ; {Q j }rj=1 and {R j }rj=1 . Hence v ∈ F.
Next, we claim that v is the unique asymptotic center for each subsequence {vn } of {xn }.
On the contrary, let v = x. From Theorem 3.1, limn→∞ d(xn , v) exists and by the uniqueness
of asymptotic centers, we have
lim sup d(vn , v)
n→∞
<
<
lim sup d(vn , x) < lim sup d(xn , x)
n→∞
n→∞
lim sup d(xn , v) = lim sup d(vn , v),
n→∞
n→∞
which is a contradiction. Therefore, v = x. Since {vn } is an arbitrary subsequence of {xn }, A({vn }) =
{x} for all subsequence {vn } of {xn }, so {xn }-converges to a common fixed point x of
{T j }rj=1 ; {S j }rj=1 ; {Q j }rj=1 and {R j }rj=1 .
Corollary 4.1. Let K be a nonempty closed convex subset of a uniformly convex hyperbolic
space X with monotone modulus of uniform convexity η. For j ∈ J , let T j : K → K be a unij
formly L j -Lipschitzian and {ξn }-asymptotically nonexpansive mapping in the intermediate sense
j
and S j : K → K be a uniformly L̂ j -Lipschitzian and {ξ̂n }-asymptotically nonexpansive mapping
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∞ j
j
in the intermediate sense. If ∞
n=1 ξn < +∞ and
n=1 ξ̂n < +∞ and if Q j , R j : K → K be an
asymptotically nonexpansive mapping and F = ∅. Then for j ∈ J , the sequence {xn } generated by
(1.1), -converges to a common fixed point of F.
Corollary 4.2. Let K be a nonempty closed convex subset of a uniformly convex hyperbolic space
X with monotone modulus of uniform convexity η. For j ∈ J , let T j : K → K be a uniformly L j j
j
Lipschitzian and {ln }-asymptotically nonexpansive mapping with ∞
:K →K
n=1 l n < +∞ and S j
j
j
be a uniformly L̂ j -Lipschitzian and {l̂n }-asymptotically nonexpansive mapping with ∞
n=1 l̂ n <
+∞. If Q j , R j : K → K be an asymptotically nonexpansive mapping and F = ∅. Then for j ∈ J ,
the sequence {xn } generated by (1.1), -converges to a common fixed point of F.
In order to prove strong convergence of the iteration (1.1) in the hyperbolic space we define the
following condition:
() There exists a nondecreasing self mapping on [0, +∞) with f (0) = 0 and f (t) > 0 for all
t ∈ (0, +∞) such that d(x, T x) ≥ f (d(x, F(T ))) for all x ∈ K , where T : K → K is a nonlinear
mapping with F(T ) = ∅ and d(x, F(T )) = inf{d(x, y) : y ∈ F(T )}.
Lemma 4.1. Let K , X, {T j }rj=1 , {S j }rj=1 , {Q j }rj=1 and {R j }rj=1 be as in Theorem 3.1. Then {xn }
converges strongly to some p ∈ F if and only if
lim inf d(xn , F) = 0.
n→∞
Proof. If {xn } converges strongly to p ∈ F, then limn→∞ d(xn , p) = 0. Since 0 ≤ d(xn , F) ≤
d(xn , p), we have limn→∞ inf d(xn , F) = 0.
Conversely, suppose that limn→∞ inf d(xn , F) = 0. It follows from Theorem 3.1 that
limn→∞ d(xn , F) exists. Now limn→∞ inf d(xn , F) = 0 implies that limn→∞ d(xn , F) = 0.
Next, we show that {xn } is a Cauchy sequence. From (3.2), we get
d(xn+1 , p) ≤ (1 + M1 μn )d(xn , p) + M2 ξn .
Taking infimum on p ∈ F on both sides in the above inequality, we have
d(xn+1 , F) ≤ (1 + M1 μn )d(xn , F) + M2 ξn .
∞
∞
M1 ∞
n=1 μn = M. Since lim
Since
n→∞ d(x n , F) = 0, for any
n=1 μn < ∞,
n=1 ξn < ∞, set e
given > 0, there exists a positive integer n 0 such that
d(xn o , F) <
∞
and
ξn <
.
4(M + 1)
2M M2
n=n
o
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Above inequality implies that there exists po ∈ F such that d(xn o , po ) <
n ≥ n o and m ≥ 1, we have
d(xn o +m , xn o )
≤
≤
Hence, for any
d(xn o +m , po ) + d(xn o , po )
M1
[e
n o +m−1
μk
k=n o
+ξn o +m−3 e
≤
.
2(M+1)
+ 1]d(xn o , po ) + M2 [ξn o +m−1 + ξn o +m−2 e M1 μno +m−1
M1
n o +m−1
μk
+ . . . + ξn e
∞
(M + 1)d(xn o , po ) + M M2
ξn
k=n o +m−2
M1
n o +m−1
k=n o +1
μk
]
n=n o
+ M M2
< (M + 1)
= ,
2(M + 1)
2M M2
which implies that {xn } is a Cauchy sequence in X . Since K is a closed subset of a complete
hyperbolic space X , so it is complete. Now we assume that limn→∞ xn = q and q ∈ K . Since
limn→∞ d(xn , F) = 0, we get q ∈ F.
Theorem 4.2. Suppose that K , X, {T j }rj=1 , {S j }rj=1 , {Q j }rj=1 and {R j }rj=1 be the same as in Theorem and satisfies (). Then the sequence {xn } defined in (1.1) converges strongly to some p ∈ F.
Corollary 4.3. Suppose that K , X, {T j }rj=1 , {S j }rj=1 , {Q j }rj=1 and {R j }rj=1 be the same as in Corollary 3.1 and satisfies (). Then the sequence {xn } defined in (1.1) converges strongly to some p ∈ F.
Corollary 4.4. Suppose that K , X, {T j }rj=1 , {S j }rj=1 , {Q j }rj=1 and {R j }rj=1 be the same as in Corollary 3.2 and satisfies (). Then the sequence {xn } defined in (1.1) converges strongly to some p ∈ F.
Remarks: In order to prove -convergence and strong convergence of the iteration (1.1) in hyperbolic spaces, we gave and analyzed some important properties related to the general iterative processes (1.1) and proposed some results presented in this paper, which show that our results extend
and improve the corresponding results of iterative approximation for asymptotically nonexpansive
mapping, multivalued nonexpansive mapping of all normed linear spaces, Hadamard manifolds and
CAT(0) spaces as special cases.
It is well known that iterative processes are the main tool for approximation of fixed points of
generalizations of nonexpansive mappings. Furthermore, the analysis of general iterative processes,
in a more general setting, is a problem of interest in theoretical numerical analysis.
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Analysis and Optimization, 513, (2010), 193–209.
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INDIAN JOURNAL OF INDUSTRIAL AND APPLIED MATHEMATICS
Vol. 10, No. 1 (Special Issue), Jan–June 2019, pp. 220–233
DOI:
On a Class of Generalised ( p, q) Bernstein
Operators
Lakshmi Narayan Mishra1∗ , Shikha Pandey2 and Vishnu Narayan Mishra3
1
Department of Mathematics, School of Advanced Sciences, Vellore Institute of Technology (VIT)
University, Vellore-632 014, Tamil Nadu, India
2
Department of Applied Mathematics & Humanities, Sardar Vallabhbhai National Institute of
Technology, Ichchhanath Mahadev Dumas Road, Surat-395 007 (Gujarat), India
3
Department of Mathematics, Indira Gandhi National Tribal University, Lalpur, Amarkantak-484 887,
Madhya Pradesh, India
(∗ Corresponding author) Email: ∗ lakshminarayanmishra04@gmail.com, 2 sp1486@gmail.com,
3
vishnunarayanmishra@gmail.com
Abstract: In the present paper, we introduce a generalised class of ( p, q) Bernstein operators. The
study proves that this generalised operator gives better approximation results comparison to the
( p, q) Bernstein operators defined in [11]. We estimate the rate of approximation using modulus of
continuity and prove Voronovskaya type theorem as well. In the end we have proved the statistical
and weighted approximation results for this operator.
Keywords: ( p, q)-integers; ( p, q)-Bernstein operators; modulus of continuity; linear positive operators.
2010 AMS Subject Classification: Primary 41A25, 41A36.
1. INTRODUCTION AND PRELIMINARIES
For a function f (x) defined on the closed interval [0, 1], the expression
n k
n
k
n−k
Bn ( f ; x) =
f
x (1 − x)
k
n
k=0
(1.1)
is called the Bernstein polynomial of order n of the function f (x) defined by S.N. Bernstein [1] in
1912. Since then various mathematicians have studied different generalizations of Bernstein polynomials (See [13]). Indefinite work has been done in the field of approximation of linear positive
operators (See [7, 8, 12]).
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221
The study of (p,q) calculus has its origin in the theory of hopf algebra. In hopf algebra we
study a structure named as quantum groups which are basicly symmetry group of noncommutative spaces. This is one reason they have been investigated in physics and mathematical physics
(noncommutative spaces arise as quantization of commutative ones). The two parameter quantizations (or (p,q)-analogues) of the general linear group GL(n) is the quantum group with the algebraic
operations that are resulted into (p,q) calculus.
Let’s go through some basic definitions of ( p, q) calculus. For any n ∈ N
(1)
(p, q)-Integer:
[0] p,q := 0 and [n] p,q =
(2)
pn − q n
if n ≥ 1,
p−q
where 0 < q < p ≤ 1.
(p, q)-Factorial:
[0] p,q ! := 1 and [n]! p,q = [1] p,q [2] p,q · · · [n] p,q if n ≥ 1.
(3)
(p, q)-Binomial coefficient :
[n] p,q !
n
for all n, k ∈ N with n ≥ k.
=
k p,q
[k] p,q ! [n − k] p,q !
(4)
(p, q) binomial expansion:
(ax +
by)np,q
:=
n
k=0
n
k
a n−k bk x n−k y k .
p,q
(x + y)np,q := (x + y)( px + qy)( p 2 x + q 2 y) . . . ( p n−1 x + q n−1 y).
(5)
(p, q)-Derivative: Let f : R → R, then the ( p, q)-derivative of function f is defined by
∗
(D p,q f )(x) = f p,q
(x) =
(6)
(7)
f ( px) − f (q x)
∗
, x = 0, f p,q
(0) := f (0)
( p − q)x
provided that f is differentiable at 0. It happens clearly that (x n )∗p,q = [n] p,q x n−1 .
The following product rules hold:
( f · h)∗p,q (x) =
∗
f p,q
(x) · h(q x) + h ∗p,q (x) · f ( px),
( f · h)∗p,q (x) =
∗
h ∗p,q (x) · f (q x) + f p,q
(x) · h( px).
The definite integrals of the function f are given by
0
a
k ∞
pk
p
f (x)d p,q x = (q − p)a
f
a ,
k+1
k+1
q
q
k=0
I NDIAN J OURNAL OF I NDUSTRIAL AND A PPLIED M ATHEMATICS
p
< 1,
q
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On a Class of Generalised ( p, q) Bernstein Operators
and
a
f (x)d p,q x = ( p − q)a
0
k ∞
qk
q
f
a ,
k+1
p
p k+1
k=0
p
> 1.
q
Further ( p, q) analysis can be found in [2].
Recently, Mursaleen et al. [11] studied Bernstein polynomials based on ( p, q) integers defined
as
Bn, p,q ( f ; x) =
1
p
n(n−1)
2
n−k
n n−k−1
k(k−1)
p [k] p,q
n
k
s
s
2
p
x
( p − q x) f
,
k p,q
[n] p,q
k=0
s=0
(1.2)
where f ∈ C[0, 1], x ∈ [0, 1].
In this paper we study a new generalization of ( p, q) Bernstein polynomials. Let a, b ∈ N.
[n+a] p,q
,
For f ∈ C 0, [n+b] p,q
n n−k−1 k(k−1)
[n + b] p,q n 1 n
k
s [n + a] p,q
s
2
p
p
x
−q x
n(n−1)
[n + a] p,q
[n + b] p,q
p 2 k=0 k p,q
s=0
n−k
p [k] p,q [n + a] p,q
(1.3)
f
[n] p,q [n + b] p,q
a,b
Bn,
p,q (
f ; x)
=
For a = b the above operator converts into (1.2), and for p = q = 1, a = b then it converts into
[n+a] p,q
(1.1). With the help of MAPLE, we can plot the [n+b] p,q
with respect to different value of n. So for
a > b and a < b, we have
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Lakshmi Narayan Mishra, Shikha Pandey and Vishnu Narayan Mishra
223
Thus for a < b we can increase the upper bound of the approximated function. So, considering
a < b throughout the paper in order to increase the interval of approximation.
2. MAIN RESULTS
[n+a]
p,q
, a < b, n ∈ N the following
Lemma 1. Assume ei = t i , i = 1, 2, 3, .... For all x ∈ 0, [n+b] p,q
equalities holds
1.
2.
a,b
Bn,
p,q (1; x) = 1,
a,b
Bn,
p,q (e; x) = x,
3.
a,b
2
2
Bn,
p,q (e ; x) = x +
Proof.
pn−1 x([n+a] p,q −[n+b] p,q x)
.
[n] p,q [n+b] p,q
n n−k−1 k(k−1)
[n + b] p,q n 1 n
k
s [n + a] p,q
s
2
p
x
−
q
x
p
n(n−1)
[n + a] p,q
[n + b] p,q
p 2 k=0 k p,q
s=0
n n
[n + a] p,q
[n + b] p,q
=
−x+x
[n + a] p,q
[n + b] p,q
=1
a,b
Bn,
p,q (1; x) =
a,b
Bn,
p,q (e; x) =
=
=
[n + b] p,q
[n + a] p,q
[n + b] p,q
[n + a] p,q
=x
=
=
=
[n + b] p,q
[n + a] p,q
[n + b] p,q
[n + a] p,q
x
[n] p,q
k
k=1
k−1
k
p
[n] p,q p
n(n−5)
2
[n] p,q p
n−2 p
n
k=0
k
k−1
s=0
k(k−1)
2
n−k−2 x k+1
ps
s=0
p
k(k−1)
2
p n−k [k] p,q [n + a] p,q
[n] p,q [n + b] p,q
[n + a] p,q
− qs x
[n + b] p,q
n−k−1 xk
ps
s=0
p,q
p
k(k−5)
2
n−1
k=0
k
[n + a] p,q
− qs x
[n + b] p,q
n−k−1 [k] p,q x k
ps
s=0
p,q
n−1
p n−1
[n + a] p,q
− qs x
[n + b] p,q
ps
p,q
k=1
(n−1)(n−2)
2
xk
[n + a] p,q
− qs x
[n + b] p,q
n−1
n−1
n 2 −5n+4
2
k(k−3)
2
p
n
1
ps
s=0
n
n(n−1)
2
n−k−1 xk
n−k−1 p,q
k=0
1
k(k−1)
2
p,q
p
n−1
n
p
n−1
1
[n + b] p,q
[n + a] p,q
k=0
n(n−1)
2
[n + b] p,q
+x−x
[n + a] p,q
[n + b] p,q
[n + a] p,q
n−2
n
n−1
(n−1)(n−2)
2
n−1 n
n
1
p
1
p
n(n−3)
2
n−1
n−2
n
1
p
a,b
2
Bn,
p,q (e ; x) =
n−1
[n + b] p,q
[n + a] p,q
=x
[n + b] p,q
[n + a] p,q
k(k−3)
2
n−k−2 x k+1 ( p k + q[k] p,q )
n−1
n−1
k=0
k
ps
s=0
p,q
p
p,q
k(k−1)
2
n−k−2 xk
ps
s=0
I NDIAN J OURNAL OF I NDUSTRIAL AND A PPLIED M ATHEMATICS
p n−k [k] p,q [n + a] p,q
[n] p,q [n + b] p,q
[n + a] p,q
− qs x
[n + b] p,q
p
[n + a] p,q
− qs x
[n + b] p,q
[n + a] p,q
− qs x
[n + b] p,q
2
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On a Class of Generalised ( p, q) Bernstein Operators
+
n−1 n−k−2 k(k−3)
q[n − 1] p,q n − 2
k
s [n + a] p,q
s
2
p
p
x
−
q
x
n 2 −5n+4
k − 1 p,q
[n + b] p,q
p 2
k=1
=
x
[n] p,q
[n + b] p,q
[n + a] p,q
s=0
n−2 p n−1
[n + b] p,q
+x−x
[n + a] p,q
n−1
n−2 n−k−3 k(k−1)
q[n − 1] p,q n − 2
k+1
s [n + a] p,q
s
2
p
+ (n−2)(n−3)
p
x
−q x
k
[n + b] p,q
p 2
p,q
k=0
=
p n−1 x
[n] p,q
= x2 +
[n + a] p,q
[n + b] p,q
s=0
+ qx2
[n − 1] p,q
[n] p,q
p n−1 x([n + a] p,q − [n + b] p,q x)
[n] p,q [n + b] p,q
[n+a]
p,q
a,b
Theorem 1. limn→∞ Bn,
p,q ( f ; x) = f (x), ∀ f ∈ C 0, [n+b] p,q .
Proof. From Lemma 1 we have
lim
n→∞
a,b
i
i
max
⎤ |Bn, p,q (e ; x) − x | = 0, i = 0, 1
[n + a] p,q
⎦
x∈⎣0,
[n + b] p,q
⎡
and for i = 2,
a,b
2
2
max
max
⎤ |Bn, p,q (e ; x) − x | = lim
⎤
⎡
n→∞
n→∞
[n + a] p,q
[n + a] p,q
⎦
⎦
x∈⎣0,
x∈⎣0,
[n + b] p,q
[n + b] p,q
lim
⎡
p n−1 ( p + q − 1)
= lim
n→∞
[2]2p,q [n] p,q
p n−1 x([n + a] p,q − [n + b] p,q x)
[n] p,q [n + b] p,q
[n + a] p,q
[n + b] p,q
2
=0
According to Bohman-Korovkin theorem [9], we obtain the desired result.
[n+a]
p,q
, then we have the following inequality of rate of convergence
Theorem 2. If f ∈ C 0, [n+b] p,q
[n
+
a]
1
1
p,q
a,b
|Bn,
( p + q − 1) ω f ; .
p,q ( f ; x) − f (x)| ≤ 1 +
2 [n + b] p,q
[n] p,q
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(2.1)
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225
Proof. Using the definition of modulus of continuity, for any sequence of positive numbers δn we
have,
|t − x|
ω( f ; δn )
| f (t) − f (x)| ≤ 1 +
δn
Operating both sides with the linear positive operator (1.3), we get
1 a,b
a,b
|Bn, p,q ( f ; x) − f (x)| ≤ 1 + Bn, p,q ((t − x); x) ω( f ; δn )
δn
Using Cauchy-Schwartz inequality, Lemma 1 we get
1
a,b
a,b
2
|Bn, p,q ( f ; x) − f (x)| ≤ 1 +
Bn, p,q ((t − x) ; x) ω( f ; δn )
δn
1 p n−1 x([n + a] p,q − [n + b] p,q x)
= 1+
ω( f ; δn )
δn
[n] p,q [n + b] p,q
1 p n−1 ( p + q − 1) [n + a] p,q 2
≤ 1+
ω( f ; δn )
δn
[2]2p,q [n] p,q
[n + b] p,q
[n + a] p,q 1
1
≤ 1+
( p + q − 1) ω( f ; δn )
δn [2] p,q [n] p,q [n + b] p,q
Put δn = √ 1
[n] p,q
, then we get the result.
Theorem 3. (Voronovskaya Type Theorem)
[n+a] p,q
[n+a] p,q
∀ f ∈ C 2 0, [n+b] p,q
such that f, D p,q ( f ), D2p,q ( f ) ∈ C 0, [n+b] p,q
, we have
a,b
lim [n] p,q (Bn,
p,q ( f ; x) − f (x)) =
n→∞
x(1 − x) 2
D p,q ( f ).
[2] p,q
(2.2)
[n+a]
p,q
, Define
Proof. Let f ∈ C 2 0, [n+b] p,q
f (y)− f (x)−(y−x)D
ψ(y, x) =
p,q (
f )− [2]1p,q (y−x)2p,q D 2p,q ( f )
(y−x)( py−q x)
if y = x,
if y = x.
0
[n+a]
p,q
. Taylor’s theorem states that
One can check that ψ(x, x) = 0 and ψ(·, x) ∈ C 0, [n+b] p,q
f (y) = f (x) + (y − x)D p,q ( f ) +
1
(y − x)2p,q D 2p,q ( f ) + (y − x)2p,q ψ(y, x), where lim ψ(y, x) = 0.
y→x
[2] p,q
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On a Class of Generalised ( p, q) Bernstein Operators
Thus,
a,b
[n](Bn,
p,q ( f ; x) − f (x))
=
+
[n] a,b
B
((y − x)2p,q ; x)D 2p,q ( f )
[2] n, p,q
a,b
2
[n]Bn,
(2.3)
p,q ((y − x) p,q ψ(y, x); x).
a,b
[n]Bn,
p,q ((y − x); x)D p,q ( f ) +
Using Cauchy-Schwartz inequality we can get the last term as
1
1
a,b
2
2 a,b
4
a,b
2
2
2
[n]Bn,
p,q ((y − x) p,q ψ(y, x); x) ≤ ([n] Bn, p,q ((y − x) p,q ; x)) (Bn, p,q (ψ (y, x); x)) .
[n+a]
p,q
Let η(y, x) := ψ 2 (y, x), which implies that η(x, x) = 0 and η(·, x) ∈ C 0, [n+b] p,q
.
From Theorem 1 we get
a,b
2
a,b
lim Bn,
p,q (ψ (y, x); x) = lim Bn, p,q (η(y, x); x) = η(x, x) = 0.
n→∞
n→∞
Using these results in (2.3) and taking limit for large values of n, we get
[n] a,b
B
((y − x)2p,q ; x)D 2p,q ( f )
[2] n, p,q
[n] p n−1 x([n + a] p,q − [n + b] p,q x) 2
D p,q ( f )
= lim
n→∞ [2]
[n] p,q [n + b] p,q
x(1 − x) 2
D p,q ( f )
=
[2] p,q
a,b
lim [n](Bn,
p,q ( f ; x) − f (x)) = lim
n→∞
n→∞
which is the required result.
3. STATISTICAL APPROXIMATION
The statistical version of Korovkin theorem for sequence of positive linear operators has been given
by Gadjiev and Orhan [5, 6].
Let K be a subset of the set N of natural numbers. Then, the asymptotic density δ(K ) of K
is defined as δ(K ) = limn n1 |{k ≤ n : k ∈ K }| and | · | represents the cardinality of the enclosed set.
A sequence x = (xk ) said to be statistically convergent to the number L if for each > 0, the set
K () = {k ≤ n : |xk − L| > } has asymptotic density zero, i.e.,
1
lim |{k ≤ n : k ∈ K }| = 0
n n
In this case, we write st − lim x = L. Let us recall the following theorem:
Theorem 1 [5]. Let An be the sequence of linear positive operators from C[0, 1] to C[0, 1] satisfies the conditions st − limn An (t ν ; x) − x ν C[0,1] = 0 for ν = 0, 1, 2. then for any function
f ∈ C[0, 1],
st − lim An ( f ) − f
n
C[0,1]
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= 0.
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227
3.1. Korovkin Type statistical approximation properties
Theorem 2. In order to get statistical approximation properties, consider two sequences p = pn
and q = qn s.t 0 < qn < pn ≤ 1, and st − limn qn = 1, st − limn pn = 1, and st − limn→∞ pnn =
[n+a]
p,q
A, st − limn→∞ qnn = B, with 0 < A, B ≤ 1 then for any function f ∈ C 0, [n+b] p,q
a,b
st − lim Bn,
pn ,qn ( f, ·) − f = 0.
n
Proof. Clearly for ν = 0, 1
a,b
a,b
Bn,
p,q (1; x) = 1, Bn, p,q (t; x) = x
which implies
a,b
a,b
st − lim Bn,
pn ,qn (1; x) − 1 = 0 and st − lim Bn, pn ,qn (t; x) − x = 0
n
n
For ν = 2
[n + a] p,q
[n + b] p,q
[n − 1] p,q
− x2
[n] p,q
p n−1 [n + a] p,q
q[n − 1] p,q
=
x+
− 1 x2
[n] p,q [n + b] p,q
[n] p,q
a,b
2
2
Bn,
≤
pn ,qn (t ; x) − x
q[n−1] p,q
[n] p,q
− 1 and βn =
+ qx2
q[n − 1] p,q
p n−1 [n + a] p,q
x
−1 +
[n] p,q
[n] p,q [n + b] p,q
≤
Choose αn =
p n−1 x
[n] p,q
pn−1 [n+a] p,q
.
[n] p,q [n+b] p,q
Clearly
st − lim αn = st − lim βn = 0.
n
n
Given > 0 we can define:
a,b
2
2
≥ 0,
U := Bn,
pn ,qn (t ; x) − x
U1 := {n : αn ≥
} and U2 := {n : βn ≥ }.
2
2
One can easily check U ⊆ U1 ∪ U2 , Hence we can get
a,b
2
2
δ{K ≤ n : Bn,
≥ }
pn ,qn (t ; x) − x
δ{K ≤ n : αn ≥
} + δ{K ≤ n : βn ≥ }
2
2
Thus by Theorem 1 we can say that the following holds true.
a,b
st − lim Bn,
pn ,qn ( f, ·) − f = 0.
n
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On a Class of Generalised ( p, q) Bernstein Operators
3.2. Rate of Statistical Convergence
In this section, we will obtain the rate of statistical convergence using modulus of continuity and
Lipschitz type maximal functions.
Theorem 3. Let the sequence p := pn and q := qn satisfy for 0 < qn < pn ≤ 1, so we have
a,b
|Bn,
pn ,qn ( f (t), x) − f (x)| ≤ 2ω( f ; δn (x)),
where
1
a,b
2
2
δn (x) = [Bn,
pn ,qn ((t − x) ; x)] .
(3.1)
Proof. Using the definition of modulus of continuity, we have
|t − x|
| f (t) − f (x)| ≤ 1 +
ω( f ; δ)
δ
Therefore
1 a,b
a,b
B
|Bn,
(
f
(t);
x)
−
f
(x)|
≤
1
+
(|t
−
x|;
x)
ω( f ; δ)
pn ,qn
δ n, pn ,qn
Using Cauchy-Schwartz inequality
1
1
1
a,b
2
a,b
2
2
[Bn,
f (t); x) − f (x)| ≤ ω( f ; δ(x)) 1 +
pn ,qn ((t − x) ; x)] [Bn, pn ,qn (1; x)]
δ(x)
1
1
a,b
2
2
[B
≤ ω( f ; δ(x)) 1 +
((t − x) ; x)]
δ(x) n, pn ,qn
a,b
|Bn,
pn ,qn (
Choosing δ(x) = δn (x) as in the (3.1), we get the result.
As by Theorem 2 st − limn δn = 0 and by the definition we have st − limn ω( f ; δ) = 0. This
a,b
gives pointwise rate of convergence of the operators Bn,
pn ,qn ( f ; x) to f (x).
Theorem 4. Let f ∈ Li p M (θ ), M > 0, 0 < θ < 1 is a continuous bounded function on [0, 1] × E
then we have
a,b
θ
|Bn,
pn ,qn ( f (t); x) − f (x)| ≤ M(δn (x)) .
Proof. Since f is a Lipschitz function, we have
a,b
a,b
|Bn,
pn ,qn ( f (t); x) − f (x)| ≤ Bn, pn ,qn (| f (t) − f (x)|; x)
a,b
θ
≤ M Bn,
pn ,qn (|t − x| ; x).
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Lakshmi Narayan Mishra, Shikha Pandey and Vishnu Narayan Mishra
229
Applying Hölder inequality we get
a,b
θ2
a,b
2
|Bn,
.
pn ,qn ( f (t); x) − f (x)| ≤ M Bn, pn ,qn ((t − x) ; x)
Taking δn (x) as in (3.1), we get the result.
4. WEIGHTED APPROXIMATION
In this section, with the help of Korovkin type theorem proved by Gadjiev in [3, 4], we give
a,b
approximation properties of the operators Bn,
p,q of the weighted spaces of continuous functions
+
on R0 = [0, ∞). For this purpose, we consider the following weighted spaces of functions which
are defined on the R+
0 . Let ρ(x) be the weight function and M f be a positive constant, we define the
weighted space of functions as
1.
+
Bρ (R+
0 ) be the space of functions f defined on R0 satisfying | f (x)| ≤ M f ρ(x).
+
2. Cρ (R+
0 ) be the subspace of all continuous functions in Bρ (R0 ).
+
3. Cρ∗ (R+
0 ) is the subspace of functions f ∈ C ρ (R0 ) for which limx→∞
f (x)
ρ(x)
is finite.
Note that the space Bρ (R+
0 ) is a normed linear space with the norm
f
ρ
= sup
x∈R+
0
| f (x)|
.
ρ(x)
In order to calculate the rate of convergence consider the weighted modulus of continuity defined as
( f ; δ) =
sup
x≥0, 0<h≤δ
| f (x + h) − f (x)|
ρ(x + h)2
f or f ∈ Cρ∗ (R+
0 ).
Lemma 1. The weighted modulus of continuity has following properties as defined in [10],
1.
( f ; δ) is a monotonic increasing function of δ.
2. limδ→0+ ( f ; δ) = 0.
3. For any λ ≥ 0, ( f ; λδ) ≤ (1 + λ) ( f ; δ).
We consider sequences ( pn ) and (qn ) for 0 < qn < pn ≤ 1 satisfying
lim pn = lim qn = 1, lim pnn = a and lim qnn = c where a, c ∈ (0, 1),
n→∞
n→∞
n→∞
n→∞
Moreover st − lim pn = st − lim qn = 1, st − lim pnn = d and st − lim qnn = e
n
n
where 0 < d, e ≤ 1.
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n→∞
n→∞
(4.1)
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On a Class of Generalised ( p, q) Bernstein Operators
Lemma 2. Let ( pn ),(qn ) be two sequences with the property (4.1) and ρ(x) = 1 + x 2 be a weight
function. If f ∈ Cρ (R+
0 ), then
a,b
Bn,
pn ,qn (ρ; x)
ρ
≤ 1 + M.
Proof. Using Lemma 1, we have
a,b
Bn,
pn ,qn (ρ; x)
=
a,b
a,b
2
Bn,
pn ,qn (1; x) + Bn, pn ,qn (t ; x)
[n − 1] pn ,qn
pnn−1 x [n + a] pn ,qn
+ qn x 2
.
= 1+
[n] pn ,qn [n + b] pn ,qn
[n] pn ,qn
Now multiplying the both-sides of the above equality by
we deduce that
and taking the supremum over x ≥ 0,
1
1+x 2
a,b
Bn,
pn ,qn (ρ; x) ρ
1
pnn−1 x [n + a] pn ,qn
2 [n − 1] pn ,qn
1
+
+
q
= sup
x
n
2
[n] pn ,qn [n + b] pn ,qn
[n] pn ,qn
x≥0 1 + x
n−1
[n + a] pn ,qn
[n − 1] pn ,qn
p
+ qn
≤ 1+ n
[n] pn ,qn [n + b] pn ,qn
[n] pn ,qn
Since limn→∞
1
[n] pn ,qn
= 0 , there exists a positive M such that
a,b
Bn,
pn ,qn (ρ; x)
ρ
≤ 1 + M.
Thus, the proof is completed.
+
+
a,b
It clearly indicates that the operator Bn,
pn ,qn act from C ρ (R0 ) to Bρ (R0 ).
a,b
Theorem 1. Let Bn,
pn ,qn be the sequence of positive linear operators defined by (1.3) and ρ(x) =
2
1 + x , then for each f ∈ Cρk (R+
0)
lim
n→∞
a,b
Bn,
pn ,qn ( f ; x) − f (x)
ρ
= 0.
Proof. It is sufficient to prove that the conditions of the weighted Korovkin type theorem [3, 4] are
satisfied. From Lemma 1, one obtains
n→∞
lim
a,b
Bn,
pn ,qn (1; x) − 1
ρ
= 0,
lim
a,b
Bn,
pn ,qn (t; x) − x
ρ
= 0,
n→∞
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Lakshmi Narayan Mishra, Shikha Pandey and Vishnu Narayan Mishra
231
and and in the last
a,b
2
2
Bn,
pn ,qn (t ; x) − x ρ
[n + a] pn ,qn
[n − 1] pn ,qn x 2
pnn−1
x
x2
lim sup
+
q
−
n
n→∞
[n] pn ,qn [n + b] pn ,qn 1 + x 2
[n] pn ,qn 1 + x 2
1 + x2
x∈R+
0
n−1 [n + a] pn ,qn
[n − 1] pn ,qn
pn
≤ lim
+ qn
−1 .
n→∞ [n] p ,q
[n
+
b]
[n] pn ,qn
pn ,qn
n n
lim
n→∞
=
which implies
lim
n→∞
a,b
2
2
Bn,
pn ,qn (t ; x) − x
ρ
= 0.
ρ
= 0,
Thus, as per weighted Korovkin theorem, we have
lim
n→∞
a,b
i
i
Bn,
pn ,qn (t ; x) − x
which is the desired result.
2
Theorem 2. If f ∈ Cρ∗ (R+
0 ), and ρ(x) = 1 + x then as n → ∞ we have
a,b
|Bn,
pn ,qn ( f ; x) − f (x)|
≤
(1 + x 2 )(a2 + a4 x 2 + x + 1)
( f ; δ)
≤ a(1 + x 2+λ )
( f ; δ),
√ ∗
where λ ≥ 1, δn = μ (n) and a is a positive constant independent of f and n.
Proof. By the definition of weighted modulus of continuity and Lemma 1, we get
|t − x|
( f ; δ)
| f (t) − f (x)| ≤ 1 + (x + |t − x|)2 1 +
δ
|t − x|
( f ; δ).
≤ 1 + (2x + t)2 1 +
δ
Operating with (1.3) both sides of inequality and using Cauchy-Schwarz inequality we get
a,b
|Bn,
p,q ( f ; x) − f (x)|
a,b
2
a,b
2 |t − x|
≤
Bn, p,q ( f ; x) 1 + (2x + t) ; x + Bn, p,q ( f ; x) 1 + (2x + t)
;x
δ
1
a,b
a,b
2
2 2;x
≤
Bn,
(
f
;
x)
1
+
(2x
+
t)
;
x
+
B
(
f
;
x)
1
+
(2x
+
t)
n,
p,q
p,q
δn
a,b
2
( f ; δ).
× Bn,
p,q ( f ; x) (t − x) ; x
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232
On a Class of Generalised ( p, q) Bernstein Operators
For large values of n, from Lemma 1, we can obtain
a,b
2
Bn,
p,q ( f ; x)(1 + t ; x)
1 + x2
≤ 1 + a1 ,
(4.2)
where a1 is a positive constant. From (4.2) we can get
a,b
2
2
Bn,
p,q ( f ; x)(1 + (2x + t) ; x) ≤ a2 (1 + x ), a2 > 0.
In the same way we can get
a,b
4
Bn,
p,q ( f ; x)(1 + t ; x)
1 + x4
≤ 1 + a3 , a3 > 0
and hence for large values of n,
a,b
2
2 2
Bn,
p,q ( f ; x)((1 + (2x + t) ) ; x) ≤ a4 (1 + x ), a4 > 0.
Hence, we have
1
a,b
2
∗ (n)
|Bn,
(
f
;
x)
−
f
(x)|
≤
(1
+
x
)
a
+
a
(1
+
x)
μ
( f ; δ).
2
4
pn ,qn
δn
√
If we choose δn = μ∗ (n), then
a,b
2
∗
|Bn,
p,q ( f ; x) − f (x)| ≤ (1 + x )(a2 + a4 (1 + x)) ( f ; μ (n))
≤ a(1 + x 2+λ ) ( f ; μ∗ (n)),
where a := a2 + a4 .
5. GRAPHICAL RESULTS
In this section, with the help of MAPLE, we show comparisons and some illustrative
graphics for
1
j
the convergence of operators (1.2) and (1.3) to the Weierstrass function f (x) = 100
j
j=0 5 cos(29 π x)
under different choices for the parameters. We have found it to be convenient to investigate our series
only for finite sums n = 20. More powerful equipments with higher speed can easily compute the
more complicated infinite series in a similar manner.
Here, in our computations, we take n = 20 and p = 0.99, q = 0.98, a = 20, b = 1 which
gives the interval of approximation as [0, 1.82]. Figure 1. shows that the approximated new modified
polynomial follows the nature of the original function and Figure 2. shows the comparison of the
errors that occur for classical Bernstein, ( p, q)-Bernstein and Modified ( p, q)- Bernstein after the
interval [0, 1].
Conflict of Interest: The authors declare that they have no competing interests regarding the publication of this manuscript.
I NDIAN J OURNAL OF I NDUSTRIAL AND A PPLIED M ATHEMATICS
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Lakshmi Narayan Mishra, Shikha Pandey and Vishnu Narayan Mishra
Figure 1. Plot
233
Figure 2. Comparison Plot
REFERENCES
[1] S.N. Bernstein, Démostration du théoréme de Weierstrass fondée sur le calcul de probabilités, Comm. Soc. Math.
Kharkow, (2), 13 (1912/1913), pp. 1–2.
[2] I. M. Burban, A. U. Klimyk, ( p, q)-differentiation, ( p, q)-integration, and ( p, q)-hypergeometric functions related to
quantum groups, Integral Transforms and Special Functions, 1994, Vol. 2, No. 1, pp. 15–36.
[3] A.D. Gadjiev, The convergence problem for a sequence of positive linear operators on bounded sets and theorems
analogous to that of P.P. Korovkin, Dokl. Akad. Nauk SSSR 218 (5) (1974). Transl. in Soviet Math. Dokl. 15 (5)
(1974) 1433–1436.
[4] A.D. Gadjiev, On P. P. Korovkin type theorems, Mat. Zametki 20 (1976) 781786. Transl. in Math. Notes (5-6) (1978)
995–998.
[5] A. D. Gadjiv, C. Orhan, Some approximzation theorems via statistical convergence, Rocky Mount. J. Math., 32 (2002)
129–138.
[6] O. Duman, C. Orhan (2004), Statistical approximation by positive linear operators. Studia Math, 161(2), 187–197.
[7] G. Içöz, R.N. Mohapatra, Approximation properties by q-Durrmeyer-Stancu operators. Anal. Theory Appl., 29 (2013),
no. 4, 373–383.
[8] G. Içöz, R.N. Mohapatra, Weighted approximation properties of Stancu type modification of q- Szász-Durrmeyer
operators, Commun. Fac. Sci. Univ. Ank. Sér. A1 Math. Stat., Volume 65, Number 1, (2016), pp. 87–103.
[9] P.P. Korovkin, On Convergence of Linear Positive Operators in the Space of Continuous Functions, Dokl. Akad. Nauk,
90(1953) 961–964.
[10] AJ, López-Moreno, Weighted simultaneous approximation with Baskakov type operators, Acta Math. Hung., 104
(1-2), 143–151 (2004).
[11] M. Mursaleen, K.J. Ansari, A. Khan, On ( p, q)-analogue of Bernstein operators, Appl. Math. Comput., 266 (2015),
pp. 874–882 [Erratum: Appl. Math. Comput., 278 (2016) 70–71].
[12] R.N. Mohapatra, Z. Walczak, Remarks on a class of Szász-Mirakyan type operators, East J. Approx., 15 (2) (2009)
197–206.
[13] M.A. Siddique, R.R. Agarwal and N. Gupta, On a Class of New Bernstein Operators, Advanced Studies in Contemporary Mathematics, 2014.
I NDIAN J OURNAL OF I NDUSTRIAL AND A PPLIED M ATHEMATICS
Vol. 10, No. 1 (Special Issue), Jan–June 2019
INDIAN JOURNAL OF INDUSTRIAL AND APPLIED MATHEMATICS
Vol. 10, No. 1 (Special Issue), Jan–June 2019, pp. 234–246
DOI:
Dhage Iteration Method for Approximating
Solutions of IVPs of Nonlinear Second Order
Hybrid Neutral Functional Differential Equations
Bapurao C. Dhage
Kasubai, Gurukul Colony, Ahmepur-413515, Dist. Latur, Maharashtra, India
E-mail: bcdhage@gmail.com
Abstract: In this paper we prove the existence and approximation result for a nonlinear initial value
problem of second order hybrid functional differential equations of neutral type via construction of
an algorithm. The main results rely on the Dhage iteration method embodied in a recent hybrid
fixed point principle of Dhage [7]. An example is also furnished to illustrate the hypotheses and the
abstract result of this paper.
2010 MSC: 34A12, 34A45, 47H07, 47H10.
Keywords: Second order neutral functional differential equation; Hybrid fixed point principle;
Dhage iteration method; Existence and Approximation theorem.
1. STATEMENT OF THE PROBLEM
Given the real numbers r > 0 and T > 0, consider the closed and bounded intervals I0 = [−r, 0]
and I = [0, T ] in R and let J = [−r, T ]. By C = C(I0 , R) we denote the space of continuous realvalued functions defined on I0 . We equip the space C with he norm · C defined by
xC = sup |x(θ )|.
(1.1)
−r ≤θ≤0
Clearly, C is a Banach space with this supremum norm and it is called the history space of the
functional differential equation in question.
For any continuous function x : J → R and for any t ∈ I , we denote by xt the element of the
space C defined by
xt (θ ) = x(t + θ ), −r ≤ θ ≤ 0.
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The differential equations involving the history of the dynamic systems are called functional differential equations and the differential equations involving the derivative of history function are called
neutral functional differential equations. It has been recognized long back the importance of such
problems in the theory of differential equations. Since then, several classes of nonlinear functional
differential equations of neutral type have been discussed in the literature for different qualitative
properties of the solutions (see Ntouyas [21] and the references therein). Recently, the study of a
special class of functional differential equations involving maxima is initiated by Dhage [7], Dhage
and Otrocol [16] and Dhage and Dhage [12] via a new Dhage iteration method and established the
existence and approximation results along with algorithm of the solutions. Therefore, it is desirable
to extend this new method to other classes of functional differential equations involving delay in
the arguments. Very recently, the present author in [8] applied this new iteration method to IVPs of
nonlinear first order neutral functional differential equations involving a delay. The present paper is
also an attempt in this direction and extends the Dhage iteration method to ordinary second order
functional differential equations of neutral type.
In this paper, we consider the nonlinear second order hybrid functional differential equations
(in short HFDE) of neutral type
⎫
d x (t) − f (t, x t ) = g(t, xt ), t ∈ I,⎬
dt
(1.3)
⎭
x0 = φ, x (0) = η,
where φ ∈ C and f, g : I × C → R are continuous functions.
Definition 1.1. A function x ∈ C(J, R) is said to be a solution of the HFDE (1.3) if
(i)
x0 ∈ C,
(ii) xt ∈ C for each t ∈ I , and
(iii) the function t → [x (t) − f (t, x t )] is continuously differentiable on I and satisfies the
equations in (1.3),
where C(J, R) is the space of continuous real-valued functions defined on J .
The neutral HFDE (1.3) is well-known and is a linear perturbation of second type of functional
differential equations. See Dhage [3, 4] and the references therein) and can be handled with the
hybrid operator theoretic technique involving the sum of two operators in a Banach space (see
Dhage [4] and the references therein). It has been discussed in Ntouyas et al. [21] with usual known
method of Lerray-Schauder fixed point principle and established the existence theorem. The special
cases of it are well-known and extensively discussed in the literature for different aspects of the
solutions (sSee Hale [19], Dhage [8, 9] and the references therein). There is a vast literature on
nonlinear functional differential equations of neutral type for different aspects of the solutions via
different approaches and methods. The method of upper and lower solution or monotone method
is interesting and well-known, however it requires the existence of both the lower as well as upper
solutions as well as certain inequality involving monotonicity of the nonlinearity. In this paper
we prove the existence and approximation theorem for the hybrid functional differential equations
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B. C. Dhage
neutral type (1.3) via a new Dhage iteration method which does not require the existence of both
upper and lower solution as well the related monotonic inequality and also obtain the algorithm for
the solutions under some natural conditions.
The rest of the paper is organized as follows. Section 2 deals with the preliminary definitions
and auxiliary results that will be used in subsequent sections of the paper. The main result and an
illustrative example are given in Section 3.
2. AUXILIARY RESULTS
Throughout this paper, unless otherwise mentioned, let (E, , · ) denote a partially ordered
normed linear space. Two elements x and y in E are said to be comparable if either the relation x y or y x holds. A non-empty subset C of E is called a chain or totally ordered if all
the elements of C are comparable. It is known that E is regular if {xn } is a nondecreasing (resp.
nonincreasing) sequence in E and xn → x ∗ as n → ∞, then xn x ∗ (resp. xn x ∗ ) for all n ∈ N.
The conditions guaranteeing the regularity of E may be found in Guo and Lakshmikantham [18]
and the references therein. Similarly a few details of a partially ordered normed linear space is given
in Dhage [4] while orderings defined by different order cones are given in Deimling [1], Guo and
Lakshmikantham [18], Heikkilaá and Lakshmikantham [20] and the references therein.
We need the following definitions (see Dhage [3,4] and the references therein) in what follows.
A mapping T : E → E is called isotone or nondecreasing if it preserves the order relation ,
that is, if x y implies T x T y for all x, y ∈ E. Similarly, T is called nonincreasing if x y
implies T x T y for all x, y ∈ E. Finally, T is called monotonic or simply monotone if it is either
nondecreasing or nonincreasing on E. A mapping T : E → E is called partially continuous at a
point a ∈ E if for > 0 there exists a δ > 0 such that T x − T a < whenever x is comparable
to a and x − a < δ. T called partially continuous on E if it is partially continuous at every
point of it. It is clear that if T is partially continuous on E, then it is continuous on every chain C
contained in E and vice-versa. A non-empty subset S of the partially ordered Banach space E is
called partially bounded if every chain C in S is bounded. An operator T on a partially normed
linear space E into itself is called partially bounded if T (E) is a partially bounded subset of E. T
is called uniformly partially bounded if all chains C in T (E) are bounded by a unique constant.
A non-empty subset S of the partially ordered Banach space E is called partially compact if every
chain C in S is a compact subset of E. A mapping T : E → E is called partially compact if
T (E) is a partially relatively compact subset of E. T is called uniformly partially compact if T
is a uniformly partially bounded and partially compact operator on E. T is called partially totally
bounded if for any bounded subset S of E, T (S) is a partially relatively compact subset of E.
If T is partially continuous and partially totally bounded, then it is called partially completely
continuous on E.
Remark 2.1. Suppose that T is a nondecreasing operator on E into itself. Then T is a partially
bounded or partially compact if T (C) is a bounded or relatively compact subset of E for each
chain C in E.
Definition 2.1 Dhage [4]. The order relation and the metric d on a non-empty set E are said
to be D-compatible if {xn } is a monotone sequence, that is, monotone nondecreasing or monotone
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nonincreasing sequence in E and if a subsequence {xn k } of {xn } converges to x ∗ implies that the
original sequence {xn } converges to x ∗ . Similarly, given a partially ordered normed linear space
(E, , · ), the order relation and the norm · are said to be D-compatible if and the metric
d defined through the norm · are D-compatible. A subset S of E is called Janhavi if the order
relation and the metric d or the norm · are D-compatible in it. In particular, if S = E, then E
is called a Janhavi metric or Janhavi Banach space.
Definition 2.2 Dhage [4]. An upper semi-continuous and monotone nondecreasing function ψ :
R+ → R+ is called a D-function provided ψ(0) = 0. An operator T : E → E is called partial
nonlinear D-contraction if there exists a D-function ψ such that
T x − T y ≤ ψ x − y
(2.1)
for all comparable elements x, y ∈ E, where 0 < ψ(r ) < r for r > 0. In particular, if ψ(r ) = k r ,
k > 0, T is called a partial Lipschitz operator with a Lipschitz constant k and moreover, if 0 < k <
1, T is called a partial linear contraction on E with a contraction constant k.
The Dhage iteration method embodied in the following applicable hybrid fixed point principle
of Dhage [7] in a partially ordered normed linear space is used as a key tool for our work contained in
this paper. The details of other hybrid fixed point theorems involving the Dhage iteration principle
and method are given in Dhage [3–5, 10, 11], Dhage and Dhage [13], Dhage et.al [14, 15], Dhage
and Otrocol [16] and the references therein.
Theorem 2.1. Let E, , · be a regular partially ordered complete normed linear and let every
compact chain C of E be Janhavi. Let A, B : E → E be two nondecreasing operators such that
(a) A is a partially bounded and partial nonlinear D-contraction,
(b) B is partially continuous and partially compact, and
(c) there exists an element α0 ∈ E such that α0 Aα0 + Bα0 or α0
Aα0 + Bα0 .
Then the operator equation Ax + Bx = x has a solution x ∗ and the sequence {xn } of successive
iterations defined by x0 = α0 , xn+1 = Axn + Bxn , n = 0, 1, . . . ; converges monotonically to x ∗ .
Remark 2.2. The condition that every compact chain of E is Janhavi holds if every partially compact subset of E possesses the compatibility property with respect to the order relation and the
norm · in it. This simple fact is used to prove the main existence results of this paper.
Remark 2.3. The regularity of E in above Theorem 2.1 may be replaced with a stronger continuity
condition of the operator A and B on E which is a result proved in Dhage [3].
3. MAIN RESULTS
In this section, we prove an existence and approximation result for the neutral HFDE (1.3) on a
closed and bounded interval J = [a, b] under mixed partial Lipschitz and partial compactness type
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B. C. Dhage
conditions on the nonlinearities involved in it. We place the neutral HFDE (1.3) in the function
space C(J, R) of continuous real-valued functions defined on J . We define a norm · and the
order relation in C(J, R) by
x = sup |x(t)|
(3.1)
t∈J
and
x y ⇐⇒ x(t) ≤ y(t) for all t ∈ J.
(3.2)
Clearly, C(J, R) is a Banach space with respect to above supremum norm and also partially ordered
with respect to the above partially order relation . It is known that the partially ordered Banach
space C(J, R) is regular and lattice so that every pair of elements of E has a lower and an upper
bound in it (see Dhage [3,4] and the references therein). The following useful lemma concerning the
Janhavi subsets of C(J, R) follows immediately from the Arzelá-Ascoli theorem for compactness
(see Dhage [8] and Dhage and Dhage [12]).
Lemma 3.1 Dhage [8] and Dhage and Dhage [12]. Let C(J, R), , · be a partially ordered
Banach space with the norm · and the order relation defined by (3.1) and (3.2) respectively.
Then every partially compact subset of C(J, R) is Janhavi.
We introduce an order relation C in C induced by the order relation defined in C(J, R).
Thus, for any x, y ∈ C, x C y implies x(θ ) y(θ ) for all θ ∈ I0 . Moreover, if x, y ∈ C(J, R) and
x y, then xt C yt for all t ∈ I .
We need the following definition in what follows.
Definition 3.1. A function u ∈ C(J, R) is said to be a solution of the HFDE (1.3) if
(i) u t ∈ C for each t ∈ I , and
(ii) the function t → [u (t) − f (t, u t )] is continuously differentiable on I and satisfies
⎫
d u (t) − f (t, u t ) ≤ g(t, u t ), t ∈ I,⎬
dt
⎭
u 0 C φ, , u (0) ≤ η.
(∗)
Similarly, a continuously differentiable function v ∈ C(J, R) is called an upper solution of the neutral HFDE (1.3) if the above inequality is satisfied with reverse sign.
We consider the following set of assumptions in what follows:
(H1 )
(H2 )
There exists a constant M f > 0 such that | f (t, x)| ≤ M f for all t ∈ I and x ∈ C;
There exists D-function ϕ : R+ → R+ such that
0 ≤ f (t, x) − f (t, y) ≤ ϕ(x − yC )
for all t ∈ I and x, y ∈ C, x
C
y. Moreover, T ϕ(r ) < r , r > 0.
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(H3 )
(H4 )
(H5 )
239
The function g is bounded on I × C with bound Mg .
The function g(t, x) is nondecreasing in x for each t ∈ I .
The neutral HFDE (1.3) has a lower solution u ∈ C(J, R).
Lemma 3.2. A function x ∈ C(J, R) is a solution of the neutral HFDE (1.3) if and only if it is a
solution of the nonlinear functional integral equation
⎧
⎪
φ(0) + η − f (0, φ) t
⎪
⎪
⎪
t
t
⎨
f (s, xs ) ds +
(t − s)g(s, x s ) ds, if t ∈ I,
(3.3)
x(t) = +
0
0
⎪
⎪
⎪
⎪
⎩φ(t), if t ∈ I .
0
Theorem 3.1. Suppose that hypotheses (H1 )-(H2 ) and (H4 ) hold. Then the neutral HFDE (1.3) has
a solution x * defined on J and the sequence {xn } of successive approximations defined by
x0 = u,
⎧
⎪
φ(0)
+
η
−
f
(0,
φ)
t
⎪
⎪
⎪
t
t
⎨
n
+
f
(s,
x
)
ds
+
(t − s)g(s, x sn ) ds, if t ∈ I,
s
xn+1 (t) =
0
0
⎪
⎪
⎪
⎪
⎩φ(t), if t ∈ I ,
0
(3.4)
where xsn (θ ) = xn (s + θ ), θ ∈ I0 , converges monotonically to x * .
Proof. Set E = C(J, R). Then, in view of Lemma 3.1, every compact chain C in E possesses the
compatibility property with respect to the norm · and the order relation so that every compact
chain C is Janhavi in E.
Define two operators A and B on E by
⎧
t
⎨η − f (0, φ)t +
f (s, xs ) ds if t ∈ I,
Ax(t) =
(3.5)
0
⎩
0, if t ∈ I0 ,
and
Bx(t) =
⎧
⎨φ(0) +
⎩
t
(t − s)g(s, x s ) ds, if t ∈ I,
0
(3.6)
φ(t), if t ∈ I0 .
From the continuity of the functions f , g and the integral, it follows that A and B define the operators A, B : E → E. Applying Lemma 3.2, the neutral HFDE (1.3) is equivalent to the operator
equation
Ax(t) + Bx(t) = x(t), t ∈ J.
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B. C. Dhage
Now, we show that the operators A and B satisfy all the conditions of Theorem 2.1 in a series of
following steps.
Step I: A and B are nondecreasing on E.
let x, y ∈ E be such that x y. Then xt
C
yt for all t ∈ I and by hypothesis (H2 ), we get
⎧
⎨η − f (0, φ)t +
Ax(t) =
⎩
0, if t ∈ I0 ,
⎧
⎨η − f (0, φ)t +
≥
⎩
0, if t ∈ I0 ,
t
f (s, xs ) ds, if t ∈ I,
0
t
f (s, ys ) ds, if t ∈ I,
0
= Ay(t),
for all t ∈ J . This shows that the operator A is also nondecreasing on E.
Similarly, by hypothesis (H4 ), we get
Bx(t) =
⎧
⎨φ(0) +
⎩
t
(t − s)g(s, x s ) ds, if t ∈ I,
0
φ(t), if t ∈ I0 ,
⎧
t
⎨φ(0) +
(t − s)g(s, ys ) ds, if t ∈ I,
≥
0
⎩
φ(t), if t ∈ I0 ,
= By(t),
for all t ∈ J . This shows that the operator B is also nondecreasing on E.
Step II: A is a nonlinear partial D-contraction on E.
Let x, y ∈ E be any two elements such that x y. Then, by hypothesis (H2 ),
t
|Ax(t) − Ay(t)| ≤
| f (s, xs ) − f (s, ys )| ds
0
≤
T ϕ(xs − ys C )
≤
T ϕ(x − y)
(3.8)
for all t ∈ J . Taking the supremum over t, we obtain
Ax − Ay ≤ ψ(x − y)
for all x, y ∈ E, x y, where ψ(r ) = T ϕ(r ) < r for r > 0. As a result A is a nonlinear partial
D-contraction on E in view of Remark 2.2.
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Step III: B is partially continuous on E.
Let {xn }n∈N be a sequence in a chain C such that xn → x as n → ∞. Then xsn → xs as n → ∞.
Since the g is continuous and hypothesis (H3 ) holds, by dominated convergence theorem, we obtain
⎧
t
⎨φ(0) +
(t − s) lim g s, xsn ds, if t ∈ I,
n→∞
lim Bxn (t) =
0
n→∞
⎩
φ(t), if t ∈ I0 ,
⎧
t
⎨φ(0) +
(t − s)g(s, x s ) ds, if t ∈ I,
=
0
⎩
φ(t), if t ∈ I0 ,
= Bx(t)
for all t ∈ J . This shows that Bxn converges to Bx pointwise on J .
Now we show that {Bxn }n∈N is an equicontinuous sequence of functions in E. Now there are
three cases:
Case I: Let t1 , t2 ∈ J with t1 > t2 ≥ 0. Then we have
t2
|Bxn (t2 ) − Bxn (t1 )| ≤ (t2 − s)g s, xsn ds −
≤
0
t2
0
(t2 − s)g s, xsn ds −
+
≤ t1
0
t2
t1
t1
0
t1
0
(t2 − s)g s, xsn ds −
T g s, xsn ds +
T
0
(t1 − s)g s, xsn ds (t2 − s)g s, xsn ds t1
0
(t1 − s)g s, xsn ds |t1 − t2 |g s, xsn ds
≤ 2Mg T |t2 − t1 |
→0
as
t2 → t1 ,
uniformly for all n ∈ N.
Case II: Let t1 , t2 ∈ J with t1 < t2 ≤ 0. Then we have
|Bxn (t2 ) − Bxn (t1 )| = |φ(t2 ) − φ(t1 )| → 0
as
t2 → t1 ,
uniformly for all n ∈ N.
Case III: Let t1 , t2 ∈ J with t1 < 0 < t2 . Then we have
|Bxn (t2 ) − Bxn (t1 )| ≤ |Bxn (t2 ) − Bxn (0)| + |Bxn (0) − Bxn (t1 )| → 0 as t2 → t1 .
uniformly for all n ∈ N.
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Thus in all three cases, we obtain
|Bxn (t2 ) − Bxn (t1 )| → 0
as
t2 → t1 ,
uniformly for all n ∈ N. This shows that the convergence Bxn → Bx is uniform and that B is a
partially continuous operator on E into itself.
Step IV: B is partially compact operator on E.
Let C be an arbitrary chain in E. We show that B(C) is uniformly bounded and equicontinuous
set in E. First we show that B(C) is uniformly bounded. Let y ∈ B(C) be any element. Then there
is an element x ∈ C such that y = T x. By hypothesis (H2 ),
|y(t)| = |Bx(t)|
⎧
t
⎨|φ(0)| +
|t − s| |g(s, xs )| ds, if t ∈ I,
≤
0
⎩
|φ(t)|, if t ∈ I0 ,
≤ φ + M f T 2
= r,
for all t ∈ J . Taking the supremum over t we obtain y ≤ Bx ≤ r for all y ∈ B(C). Hence
B(C) is a uniformly bounded subset of E. Next we show that B(C) is an equicontinuous set in E.
Let t1 , t2 ∈ J , with t1 < t2 . Then proceeding with the arguments that given in Step II it can be shown
that
y(t2 ) − y(t1 ) = |Bx(t2 ) − Bx(t1 )| → 0 as t1 → t2
uniformly for all y ∈ B(C). This shows that B(C) is an equicontinuous subset of E. Now, B(C) is a
uniformly bounded and equicontinuous subset of functions in E and hence it is compact in view of
Arzelá-Ascoli theorem. Consequently B : E → E is a partially compact operator on E into itself.
Step IV: u satisfies the operator inequality u Au + Bu.
By hypothesis (H4 ), the neutral HFDE (1.3) has a lower solution u defined on J . Then we have
⎫
d u (t) − f (t, u t ) ≤ g(t, u t ), t ∈ I,⎬
dt
⎭
u 0 C φ, u (0) ≤ η.
Integrating the above inequality from 0 to t, we get
⎧
φ(0) + η − f (0, φ) t
⎪
⎪
⎪
t
t
⎨
f (s, u s ) ds +
(t − s)g(s, u s ) ds, if t ∈ I,
u(t) ≤ +
⎪
0
0
⎪
⎪
⎩
φ(t), if t ∈ I0 ,
= Au(t) + Bu(t)
for all t ∈ J . As a result we have that u Au + Bu.
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Thus, A and B satisfy all the conditions of Theorem 2.1 and so the operator equation Ax +
Bx = x has a solution. Consequently the integral equation (3.3), and a fortiori the hybrid functional
differential equation (1.3) has a solution x * defined on J . Furthermore, the sequence {xn }∞
n=0 of
successive approximations defined by (3.5) converges monotonically to x * . This completes the
proof.
Remark 3.1. The conclusion of Theorem 3.1 also remains true if we replace the hypothesis (H4 )
and (H7 ) with the following ones:
(H4 )The neutral HFDE (1.3) has an upper solution v ∈ C(J, R).
The proof of Theorem 3.1 under this new hypothesis is similar and can be obtained by closely
observing the same arguments with appropriate modifications.
Example 3.1. Given the closed and bounded intervals I0 = − π2 , 0 and I = [0, 1], consider the
second order neutral hybrid functional differential equation,
⎫
d x (t) − f 1 (t, xt ) = g1 (t, xt ), t ∈ I,⎬
dt
⎭
x0 = φ, x (0) = 1,
(3.9)
where φ ∈ C and f 1 , g1 : I × C → R are continuous functions given by
π φ(t) = sin t, t ∈ − , 0 ,
2
⎧
x
C
⎨
+ 1,
if x ≥C 0, x = 0,
1 + xC
f 1 (t, x) =
⎩
1,
if x C 0,
and
g1 (t, x) =
tanh(xC ) + 1,
1,
if x ≥C 0, x = 0,
if x C 0,
for all t ∈ I .
Clearly, f 1 is continuous and bounded on I × C with bound M f1 = 2. We show that f 1 satisfies
the hypothesis (H2 ). Let x, y ∈ C be such that x C y C 0. Then xC ≥ yC > 0 and therefore,
we have
0 ≤ f 1 (t, x) − f 1 (t, x) =
xC
yC
−
≤ ϕ(x − yC )
1 + xC
1 + yC
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B. C. Dhage
for all t ∈ I , where ϕ(r ) =
obtain
r
< r , r > 0. Again, if x, y ∈ C be such that x C y C 0, then we
1+r
0 ≤ f 1 (t, x) − f 1 (t, x) ≤ ϕ(x − yC )
for all t ∈ I . This shows that the function f 1 (t, x) satisfies the hypothesis (H2 ).
Next, g1 is bounded on I × C with M f1 = 2. Again, let x, y ∈ C be such that x
Then xC ≥ yC > 0 and therefore, we have
C
y
C
0.
g1 (t, x) = tanh(xC ) + 1 ≥ tanh(yC ) + 1 = g1 (t, y)
for all t ∈ I . Again, if x, y ∈ C be such that x C y C 0, then we obtain
g1 (t, x) = 1 = g1 (t, y)
for all t ∈ I0 . This shows that the function g1 (t, x) is nondecreasing in x for each t ∈ I . Finally,
⎧
t
⎪
⎨− , if t ∈ [0, 1],
2
u(t) =
⎪
⎩
sin t, if t ∈ − π2 , 0 ,
is a lower solution of the neutral HFDE (3.9) defined on J . Thus, f 1 satisfies the hypotheses (H1 ),
(H2 ) and (H4 ). Hence we apply Theorem 3.1 and conclude that the neutral HFDE (3.9) has a solution
x ∗ on J and the sequence {xn } of successive approximation defined by
⎧
t
⎪
⎨− , if t ∈ [0, 1],
2
x0 (t) =
⎪
⎩
sin t, if t ∈ − π2 , 0
⎧
t
⎪
⎪
[1
−
f
(0,
φ)]t
+
f 1 (s, xsn ) ds
⎪
⎪
⎪
0
⎨
t
xn+1 (t) = + (t − s)g1 (s, xsn ) ds, if t ∈ [0, 1],
⎪
⎪
0
⎪
⎪
⎪
⎩
sin t, if t ∈ − π2 , 0 ,
for n = 0, 1, . . . , converges monotonically to x * .
Remark 3.2. The conclusion given in Example 3.1 also remains true if we replace the lower solution u with the upper solution v of the neutral HFDE (3.9) defined by
⎧ t
⎪
⎨2t
+ 1 , if t ∈ [0, 1],
2
v(t) =
⎪
⎩
sin t, if t ∈ − π2 , 0 .
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Remark 3.3. We note that if the neutral HFDE (1.3) has a lower solution u as well as an upper
solution v such that u v, then under the given conditions of Theorem 3.1 it has corresponding
solutions x∗ and x ∗ and these solutions satisfy x∗ x ∗ . Hence they are the minimal and maximal
solutions of the neutral HFDE (1.3) respectively in the vector segment [u, v] of the Banach space
E = C(J, R), where the vector segment [u, v] is a set in C(J, R) defined by [u, v] = {x ∈ C(J, R) |
u x v}. This is because the order relation defined by (3.2) is equivalent to the order relation
defined by the order cone K = {x ∈ C(J, R) | x θ } which is a closed set in the Banach space
C(J, R). The existence of extremal solutions for the neutral HFDE (1.3) may be obtained in the
vector segment [u, v] via generalized iteration method given Heikkilá and Lakshmikatham [20], but
in that case we do not get the algorithm for the extremal solutions. Therefore our Dhage iteration
method has some advantages over the iteration method presented in Heikkilá and Lakshmikatham
[20].
ACKNOWLEDGMENT
The author is thankful to the referee for pointing out some misprints/typos in the previous version
of this paper.
REFERENCES
[1] K. Deimling, Nonlinear Functional Analysis, Springer Verlag, 1985.
[2] B.C. Dhage, Quadratic perturbations of periodic boundary value problems of second order ordinary differential equations, Differ. Equ. Appl., 2 (2010), 465–486.
[3] B. C. Dhage, Hybrid fixed point theory in partially ordered normed linear spaces and applications to fractional
integral equations, Differ. Equ. Appl. 5 (2013), 155–184.
[4] B.C. Dhage, Partially condensing mappings in partially ordered normed linear spaces and applications to functional
integral equations, Tamkang J. Math., 45 (4) (2014), 397–427.
[5] B.C. Dhage, A new monotone iteration principle in the theory of nonlinear first order integro-differential equations,
Nonlinear Studies, 22 (3) (2015), 397–417.
[6] B.C. Dhage, Approximating solutions of nonlinear periodic boundary value problems with maxima, Cogent Mathematics, (2016), 3: 1206699.
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(2016), 23–27.
[8] B.C. Dhage, Dhage iteration method for approximating solutions of a IVP of nonlinear first order hybrid functional
differential equations of neutral type, Nonlinear Analysis Forum, 22(1)(2017), 59–70.
[9] B.C. Dhage, Dhage iteration method for approximating solutions of a IVP of nonlinear first order functional differential equations, Malaya J. Mat., 5(4) (1917), 680–689.
[10] B.C. Dhage, Dhage iteration method in the theory of ordinary nonlinear PBVPs of first order functional differential
equations, Commun. Optim. Theory, 2017 (2017), Article ID 32, pp. 22.
[11] B.C. Dhage, The Dhage iteration method for initial value problems of nonlinear first order hybrid functional integrodifferential equations, In “Recent Advances in Fixed Point Theory and Applications” edited by U.C. Gairola and R.
Pant, Nova Publisher, (2017), pp. 177–186.
[12] B.C. Dhage, S.B. Dhage, Approximating solutions of nonlinear first order ordinary differential equations, GJMS
Special Issue for Recent Advances in Mathematical Sciences and Applications-13, GJMS Vol. 2 (2) (2013), 25–35.
[13] S.B. Dhage, B.C. Dhage, Dhage iteration method for Approximating positive solutions of PBVPs of nonlinear hybrid
differential equations with maxima, Intern. Jour. Anal. Appl., 10(2) (2016), 101–111.
[14] S.B. Dhage, B.C. Dhage, J.B. Graef, Dhage iteration method for initial value problem for nonlinear first order integrodifferential equations, J. Fixed Point Theory Appl., 18 (2016), 309–325.
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[15] S.B. Dhage, B.C. Dhage, J.B. Graef, Dhage iteration method for approximating the positive solutions of IVPs for
nonlinear first order quadratic neutral functional differential equations with delay and maxima, Intern. J. Appl. Math.,
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[16] B.C. Dhage, D. Otrocol, Dhage iteration method for approximating solutions of nonlinear differential equations with
maxima, Fixed Point Theory, 19 (2) (2018), 545–556.
[17] L. Erbe, W. Zhicheng, L. Longtu, Boundary value problems for second order mixed type functional differential equations, “Boundary value Problems for Functional Differential Equations”, World scientific 1996, 143–151.
[18] D. Guo, V. Lakshmikantham, Nonlinear Problems in Abstract Cones, Academic Press, New York, London 1988.
[19] J.K. Hale, Theory of Functional Differential equations, Springer-Verlag, New York-Berlin, 1977.
[20] S. Heikkilä, V. Lakshmikantham, Monotone Iterative Techniques for Discontinuous Nonlinear Differential Equations,
Marcel Dekker inc., New York 1994.
[21] S. Ntouyas, Y. Sficas, P. Tsamatos, Existence results for initial value problems for neutral functional differential
equations, J. Diff. Equations, 114 (1994), 527–537.
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INDIAN JOURNAL OF INDUSTRIAL AND APPLIED MATHEMATICS
Vol. 10, No. 1 (Special Issue), Jan–June 2019, pp. 247–254
DOI:
A New Version of KKM Theorem with Geodesic
Convex Hull with an Application on Hadamard
Manifold
J.C. Yao
Research Center for Interneural Computing, China Medical University Hospital, China Medical
University, Taichung, Taiwan
E-mail: yaojc@mail.cmu.edu.tw
Abstract: In this paper, we define a new type of the KKM mapping involving geodesic convex hull
on Hadamard manifold, and we prove a related KKM theorem which we call generalized KKM theorem. A classical equilibrium problem on Hadamard manifold is considered. We prove an existence
result for classical equilibrium problem by applying generalized KKM theorem. An example is also
given in support of our problem.
Keywords: Hadamard manifold, Geodesic convex hull, Generalized KKM theorem, Classical equilibrium problem.
2010 MSC: 49J40; 58E35; 90C33.
1. INTRODUCTION
The famous KKM theorem, Browder fixed point theorem and Ky Fan’s minimax inequality are of
fundamental importance in modern nonlinear analysis, and are mutually equivalent in the sense that
each one can be deduced from another with or without aid of some minor results. In 1929, Knaster,
Kureatowski and Mazurkiewiez [13] formulated the KKM principle (theorem) in finite dimensional
spaces involving certain type of multi-valued mapping which is called the KKM mapping. In 1961,
Fan [7] generalized the classical KKM theorem to infinite dimensional Hausdorff topological vector spaces and proved geometric lemma for multi-valued mappings, which is called Fan’s geometric
lemma. After that, Browder [2] extended Fan’s geometric lemma to Fan-Browder’s fixed point theorem. In [11], Horvath replacing convex hull by contractible subset, gave a purely topological version of the KKM theorem. Very recently, Yang et al. [18] introduced and generalized KKM theorem
without using convex hull.
The theory of variational inequalities was introduced by Hartman and Stampacchia [10],
has important applications, including but not limited to problems in complementarity problems,
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J.C. Yao
boundary value problems, optimization problems etc.. It has been studied extensively in finite and
infinite dimensional linear spaces, see e.g., [6, 8, 9, 12]. Equally important is the area of equilibrium problem initiated by Blum and Oettli [1], which has emerged as an interesting and fascinating
branch of applicable mathematics. They have shown that variational inequality problem can be
viewed as a special realization of the abstract equilibrium problem. This theory has become a rich
source of inspiration and motivation for the study of a wide class of problems arising in economics,
finance, optimization, operation research in a general and unified way. In [1], it was shown that for
f (x, y) = g(x, y) + h(x, y) with f, g, h : K × K → R, where K is subset of a topological vector
space if g = 0, the result becomes a variant of Ky Fans theorem, whereas for h = 0 it becomes a
variant of the Browder-Minty theorem for variational inequalities.
In recent past, several applied problems such as constrained minimization problems, optimization problems, boundary value problems have been extended from Euclidean space to a Riemannian
manifold setting in order to extend the study of convex theory, variational inequality, equilibrium
and fixed point theory. Recently, much attention has been given to study the variational inequalities,
equilibrium and related optimization problems on the Riemannian manifold and Hadamard manifold. Németh [15], Colao et al. [4], Zhou et al. [20] have considered the variational inequalities,
equilibrium problems and optimization problems on Hadamard manifolds, respectively and studied
the existence of solutions for their problems under some suitable conditions.
Due to importance of the problems discussed above, in this paper, we define a new type of
KKM mapping involving geodesic convex hull on Hadamard manifold, and we prove a generalized
KKM theorem. Finally, classical equilibrium problem is solved by applying the generalized KKM
theorem on Hadamard manifold. An example is constructed in support of our problem.
2. PRELIMINARIES
The elementary and primary knowledge about differentiable manifolds can be found in many introductory books on Riemannian geometry, topology and equilibrium problems, see e.g., [5, 16, 17].
Let M be a simply connected m-dimensional manifold and x ∈ M. The tangent space of M
Tx M, which is inherently a
at x is denoted by Tx M and the tangent bundle of M by T M =
x∈M
manifold. A vector field V on M is a mapping of M into T M which relates with each point x ∈ M
to a vector V (x) ∈ Tx M. We always assume that M can be equipped with a Riemannian metric to
become a Riemannian manifold. We denote by ·, ·x the scalar product on Tx M with the associated
norm · x , in the sequel the subscript x will be excluded. For x, y ∈ M, let γ : [a, b] −→ M be
a piecewise smooth curve joining x to y (i.e., γ (a) = x and γ (b) = y). Then the arc-length of γ is
defined by
b
L(γ ) =
γ (t)dt.
a
Definition 2.1. The geodesic distance d(x, y) is the length of minimal geodesic segment between
any two points x, y on a manifold.
Let ∇ be the Levi-Civita connection associated with (M, ·, ·). Let γ be a smooth curve in M.
A vector field V is said to be parallel along γ if ∇γ V = 0. If γ itself is parallel along γ , we say
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that γ is a geodesic, and in this case γ is a constant. If γ = 1, then γ is said to be normalized.
A geodesic joining x to y in M is said to be minimal if its length equals d(x, y).
Definition 2.2. A Hadamard manifold M is a simply connected complete Riemannian manifold of
non-positive sectional curvature.
Definition 2.3. The exponential mapping expx : Tx M −→ M is defined by expx (v) = γv (1, x),
where γv (·, x) = γ (·) is the geodesic starting at x with velocity v. It is clear that expx tv = γv (t, x),
for each real number t (i.e., γ (0) = x, γ (0) = v).
Definition 2.4 [3]. A set X ⊂ M is said to be geodesic convex if for any x, y ∈ X , the minimal
geodesic joining x to y is contained in X , i.e., γx,y (t) = expx (t exp−1
x y) ∈ X , for every t ∈ [0, 1].
Remark 2.1. The exponential mapping and its inverse are continuous on Hadamard manifold, and
exponential mapping is diffeomorphism. It is also remarked that for any x, y ∈ M, the minimal
geodesic joining x and y, expx t exp−1
x y, for t ∈ [0, 1] is unique.
Lemma 2.1. Let x0 ∈ M and {xn } ⊂ M such that xn → x0 . Then the following assertions hold.
(i)
For any y ∈ M,
−1
−1
−1
exp−1
xn y → expx0 y and exp y x n → exp y x 0 ;
(ii)
(iii)
If {vn } is a sequence such that vn ∈ Txn M and vn → v0 , then v0 ∈ Tx0 M;
Given the sequences {u n } and {vn } with u n , vn ∈ Txn M, if u n → u 0 and vn → v0 with
u 0 , v0 ∈ Tx0 M, then u n , vn → u 0 , v0 .
Definition 2.5 [17]. A real-valued mapping g : X −→ R is said to be geodesic convex if and only
if, for any geodesic γ , the composition mapping g ◦ γ : R −→ R is convex, i.e.,
(g ◦ γ )(t x + (1 − t))y ≤ t(g ◦ γ )(x) + (1 − t)(g ◦ γ )(y),
for any x, y ∈ R and t ∈ [0, 1].
Remark 2.2. By Definition 2.4 and Remark 2.1, we can see that a real-valued mapping g : X −→
R is said to be geodesic convex if and only if, for any x, y ∈ X and t ∈ [0, 1], we have
g expx t exp−1
x y ≤ tg(x) + (1 − t)g(y).
Definition 2.6 [19]. Let z be any given point in M. The geodesic convex hull for a set S ⊂ M,
denote by GCoS, is defined as follows:
n
∀x1 , · · · , xn ∈ S; t1 , · · · , tn ∈ [0, 1] .
GCoS = expz
ti exp−1
z xi
i=1
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Lemma 2.2 [15]. If X ⊂ M is nonempty, compact and geodesic convex, then every continuous
mapping h : X −→ X has a fixed point.
3. GENERALIZED KKM THEOREM
We begin this section by introducing a new version of KKM mapping using geodesic convex hull.
Definition 3.1. Let X be a nonempty subset of a Hadamard manifold M. A multi-valued mapping
A : X −→ 2 X is called a generalized KKM mapping if, for any finite subset {x1 , · · · , xn } ⊂ M,
n
GCo{x1 , · · · , xn } ⊆
A(xi ).
i=1
Remark 3.1. If M = Rn and A : X −→ Rn , then geodesic convex hull collapses to linear convex
hull. Then, generalized KKM mapping reduces to classical KKM mapping.
Now, we prove the following generalized KKM theorem.
Theorem 3.1. Let X be a nonempty, compact and geodesic convex subset of a Hadamard manifold
M. Suppose that A : X −→ 2 X be a closed valued generalized KKM mapping. Then
A(x) = ∅.
x∈X
A(x) = ∅. Then
Proof. On contrary, suppose that
x∈X
X=X\
A(x) =
x∈X
[X \ A(x)] .
x∈X
Since X is a nonempty and compact set and A(x) is closed, for all x ∈ X , then there exists a finite
set {x1 , · · · , xn } of X such that
n
X=
[X \ A(xi )] .
i=1
Let {ti : i = 1, · · · , n} be the partition of unity subordinate to the open covering {X \ A(xi ) : i =
1, · · · , n} of X which implies that {ti : i = 1, · · · , n} is the set of continuous mappings with the
following condition:
0 ≤ ti (x) ≤ 1, ∀x ∈ X, i = 1, · · · , n;
and if x ∈ A(x j ), for some j, then t j (x) = 0.
Now, consider a mapping ϕ : X −→ X by
−1
ϕ(x) = expz t1 (x) exp−1
z x 1 + · · · + tn (x) expz x n , ∀x ∈ X.
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Then, ϕ is a continuous mapping. By Lemma 2.2, there exists x̄ ∈ X such that ϕ(x̄) = x̄.
Since A is a generalized KKM mapping, the following inclusion holds
n
GCo{x1 , · · · , xn } ⊆
A(xi ).
i=1
Then, there exists j ∈ {i ∈ {1, · · · , n} : ti (x̄) > 0} such that
−1
x̄ = ϕ(x̄) = expz t1 (x) exp−1
z x 1 + · · · + tn (x) expz x n
= GCo{x1 , · · · , xn }
∈
A(x j ).
It implies that t j (x̄) = 0, which contradicts the fact that t j (x̄) > 0. Therefore,
A(x) = ∅.
x∈X
4. APPLICATION
As an application of generalized KKM theorem, we prove an existence result for classical equilibrium problem, where the function involved is a sum of two functions, and the assumptions are
required separately on each of these functions.
Let X be a nonempty, geodesic convex and compact subset of a Hadamard manifold M. Given
two real-valued mappings g, h : X × X −→ R. We consider the following classical equilibrium
problem of finding x ∈ X such that
g(x, y) + h(x, y) ≥ 0, ∀y ∈ X.
(4.1)
which was introduced by Blum and Oettli [1].
Next, we prove an existence result for classical equilibrium problem (4.1) by applying generalized KKM theorem on Hadamard manifold.
Theorem 4.1. Let X ⊂ M be a nonempty, compact and geodesic convex subset of a Hadamard
manifold M. Suppose that the continuous mappings g, h : X × X −→ R are geodesic convex mappings in the second arguments, respectively, such that g(x, x) = 0 and h(x, x) = 0, for all x ∈ X .
Then, the classical equilibrium problem (4.1) admits a solution.
Proof. For any y ∈ X , define the mapping F : X −→ 2 X by
F(y) = {x ∈ X : g(x, y) + h(x, y) ≥ 0} .
We claim that F(y) is closed in X . Let {xλ } be any net in X with xλ ∈ F(y) and xλ → x0 . Then,
g(xλ , y) + h(xλ , y) ≥ 0.
Since g and h are continuous mappings, we have
g(xλ , y) + h(xλ , y) −→ g(x0 , y) + h(x0 , y).
Therefore, x0 ∈ F(y), and hence F is closed in X .
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In order to apply generalized KKM theorem 3.1, we have to prove that for any choice of
{y1 , · · · , yn } ∈ X ,
n
GCo{y1 , · · · , yn } ⊆
F(yi ).
i=1
On contrary suppose that F(y) is not a generalized KKM mapping. Then, there exists x̄ ∈
GCo{y1 , · · · , yn } such that for all ti ≥ 0, i = 1, · · · , n, we have
n
n
−1
ti expz yi ∈
F(yi ).
x̄ = expz
/
i=1
i=1
Therefore, we have
g(x̄, yi ) + h(x̄, yi ) < 0,
which implies that for any i = 1, · · · , n, we have
yi ∈ {w ∈ X : g(x̄, w) + h(x̄, w) < 0} .
Now, we show that the set {w ∈ X : g(x̄, w) + h(x̄, w) < 0} is geodesic convex. Let x̄ ∈ X be fixed,
and let w1 , w2 ∈ {w ∈ X : g(x̄, w) + h(x̄, w) < 0}, for any λ ∈ [0, 1]. Since g and h are geodesic
convex mappings in the second arguments, respectively, we have
−1
g x̄, expw1 t exp−1
w1 w2 + h x̄, expw1 t expw1 w2
≤ tg (x̄, w1 ) + (1 − t)g (x̄, w2 ) + th (x̄, w1 ) + (1 − t)h (x̄, w2 )
= t {g(x̄, w1 ) + h(x̄, w1 )} + (1 − t) {g(x̄, w2 ) + h(x̄, w2 )}
< 0,
which shows that {w ∈ X : g(x̄, w) + h(x̄, w) < 0} is a geodesic convex set. Therefore, by definition of the geodesic convex hull, we have
x̄ ∈ GCo{y1 , · · · , yn } ⊆ {w ∈ X : g(x̄, w) + h(x̄, w) < 0} ,
and hence
g(x̄, x̄) + h(x̄, x̄) < 0,
which implies that 0 < 0, a contradiction, using the hypotheses g(x, x) = h(x, x) = 0. Thus, F(y)
is a generalized KKM mapping. By applying generalized KKM theorem 3.1, there exists a point
x0 ∈ X such that
x0 ∈
F(y) = ∅,
y∈X
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i.e.,
g(x0 , y) + h(x0 , y) ≥ 0, ∀y ∈ X.
Hence, classical equilibrium problem (4.1) admits a solution. This completes the proof.
We now provide an example to illustrate the existence of solutions of classical equilibrium
problem (4.1).
Example 4.1. Consider the 2-dimensional unit sphere
M = S 2 = (x1 , x2 , x3 ) ∈ R3 : x12 + x22 + x32 = 1 ,
endowed with the standard metric. Let the tangent spaces be defined by
Tx M = p ∈ R3 : p, q = 0 , ∀q ∈ M.
Let
X = {(x1 , x2 , x3 ) ∈ M : x1 , x2 > 0, x3 = 0} .
Then, it is easy to check that X is a nonempty, compact and geodesic convex subset of M. We now
define g, h : X × X −→ R as follows:
g(x, y) =
1
1
(x + y)2 and h(x, y) = (y − x)2 ,
3
2
for all x = (x1 , x2 , x3 ), y = (y1 , y2 , y3 ) ∈ X . Then, g and h are continuous, and geodesic convex
mappings in the second arguments. Hence
by
Theorem 4.1, the classical equilibrium problem (4.1)
admits at least one solution as x0 = 23 , 1, 0 .
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[11] C.D. Horvath, Some results on multi-valued mappings and inequalities without convexity, In: B.L. Lin and S. Simons
(eds.) Nonlinear and Convex Analysis, Lecture Notes in Pure and Applied Math., pp. 99–106, Dekker, New York
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INDIAN JOURNAL OF INDUSTRIAL AND APPLIED MATHEMATICS
Vol. 10, No. 1 (Special Issue), Jan–June 2019, pp. 255–268
DOI:
The Relaxed Resolvent Operator For Solving Fuzzy
Variational Inclusion Problem Involving Ordered
RME Set-Valued Mapping
Iqbal Ahmad1 , Abul H. Siddiqi2 and Rais Ahmad3∗
1
College of Engineering, Qassim University, Buraidah-51452, Al-Qassim, Saudi Arabia
School of Basic Sciences and Research, Sharda University, Greater Noida-201306, India
3
Department of Mathematics, Aligarh Muslim University, Aligarh-202002, India
(∗ Corresponding author) E-mail: ∗ raisain_123@rediffmail.com; 1 iqbal@qec.edu.sa;
2
siddiqi.abulhasan@gmail.com
2
Abstract: In this paper, we consider a relaxed resolvent operator and prove some of its properties.
Using the properties of relaxed resolvent operator, we obtain the solution of a fuzzy variational
inclusion problem with ordered RME set-valued mapping in ordered Hilbert spaces by suggesting
an iterative algorithm. Some special cases are discussed. Some examples are also constructed.
Keywords: Ordered RME Mapping; Ordered Hilbert Space; Inclusion; Algorithm; Fuzzy Mappings.
2010 AMS Subject Classification: 49J40; 47H05; 47S40.
1. INTRODUCTION
In mathematics, a variational inequality is an inequality involving a functional, which has to be
solved for all possible values of a given variable, belonging usually to a convex set. The mathematical theory of variational inequalities has initially developed to deal with equilibrium problems,
precisely the Signorini problem. The formal definition of a variational inequality is as follows:
Let X be a Hilbert space, K ⊂ X a closed convex set and F : K → X be a bounded linear
functional on X. The problem of finding x ∈ K such that
F(x), y − x ≥ 0, ∀ y ∈ K ,
where ., . is the usual inner product on X.
An important generalization of a variational inequality is called a variational inclusion is mainly
due to Hassouni and Moudafi [10]. Using resolvent operator methods, many researchers have
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Iqbal Ahmad, Abul H. Siddiqi and Rais Ahmad
developed iterative algorithms to solve variational inclusions and other equivalent problems. Ansari
[1] and Chang and Zhu [4] at the same time introduced and studied variational inequalities with
fuzzy mappings. A vast literature is available about variational inequalities (inclusions) with fuzzy
mappings, see e.g., [3,5–8,11–14,22,28] and references therein. We also refer to [2,8,20,23,25,26]
for a reasonable study of variational inequalities and its equivalent problems.
It is known that the fixed point theory with applications for nonlinear mappings have been
intensively studied in ordered Banach spaces. A lot of work is done by Li [15–19] to approximate
the solution of a general nonlinear ordered variational inequalities and ordered equations in ordered
Banach spaces. On the other hand the fuzzy set theory due to Zadeh [27] specifically designed to
mathematically represent uncertainty and vagueness and to provide formalized tools for dealing
with the imprecision intrinsic to many problems.
In this paper, we consider a relaxed resolvent operator and prove that it is single-valued, compression as well as Lipschitz continuous. Then these new results are used to solve a fuzzy variational
inclusion problem involving ordered RME set-valued mapping after defining an iterative algorithm.
We claim that all the results of this paper either preliminary or main are the extension of results of
Li [15].
2. PRELIMINARIES
Throughout the paper, we assume that X is a real ordered Hilbert space equipped with a norm ||.||
and an inner product ., .. Let C be a normal cone, “ ≤ " is a partial ordering defined by C and K
be the normal constant of C. For the set of arbitrary elements x, y ∈ X, i.e., {x, y}, we denote the
least upper bound by lub{x, y} and greatest lower bound by glb{x, y}, also we suppose that they
always exist. The operator ⊕ is called an XOR operator if x ⊕ y =lub{x − y, y − x}.
Let F(X ) be a collection of all fuzzy sets over X. A mapping F : X → F(X ) is said to be fuzzy
mapping on X . For each x ∈ X, F(x) (in the sequel, it will be denoted by Fx ) is a fuzzy set on X
and Fx (y) is the membership function of y in Fx .
A fuzzy mapping F : X → F(X ) is said to be closed if for each x ∈ E, the function y → Fx (y)
is upper semi-continuous, that is, for any given net {yα } ⊂ X, satisfying yα → y0 ∈ X, we have
lim sup Fx (yα ) ≤ Fx (y0 ).
α
For B ∈ F(X ) and λ ∈ [0, 1], the set (B)λ = {x ∈ X : B(x) ≥ λ} is called a λ-cut set of B. Let
F : X → F(X ) be a closed fuzzy mapping satisfying the following condition:
Condition (∗) : If there exists a function a : X → [0, 1] such that for each x ∈ X, the set (Fx )a(x) =
{y ∈ X : Fx (y) ≥ a(x)} is a nonempty bounded subset of X.
If F is a closed fuzzy mapping satisfying the condition (∗), then for each x ∈ X, (Fx )a(x) ∈
C B(X ). In fact, let {yα } ⊂ (Fx )a(x) be a net and yα → y0 ∈ X, then (Fx )a(x) ≥ a(x), for each α.
Since F is a closed, we have
Fx (y0 ) ≥ lim sup Fx (yα ) ≥ a(x),
α
which implies that y0 ∈ (Fx )a(x) and so (Fx )a(x) ∈ C B(X ).
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Definition 2.1 [24]. Let X be an ordered Hilbert space and C be a cone with a partial ordered
relation ≤ . For any x, y ∈ X if either x ≤ y or y ≤ x hold, then x and y are said to be comparable
to each other (denoted by x ∝ y).
Definition 2.2 [9]. The cone C is said to be normal if and only if there exists a constant K > 0 such
that for 0 ≤ x ≤ y, we have ||x|| ≤ K ||y||.
Proposition 2.1. If x ∝ y, then lub{x, y} and glb{x, y} exist, x − y ∝ y − x, and 0 ≤ (x − y) ∨
(y − x).
Proposition 2.2 [9]. For any natural number n, if x ∝ yn and yn → y ∗ (n → ∞), then x ∝ y ∗ .
Proposition 2.3 [9]. Let X be an ordered Hilbert space and α, β be two real numbers. Let ⊕ be an
XOR operator and for x, y, u, v ∈ X, the operation x y is defined by x y = (x − y) ∧ (y − x).
Then the following relations hold:
(i)
x y = y x, x x = 0, x y = y x = −(x ⊕ y);
(ii) x 0 ≤ 0, if x ∝ 0;
(iii) 0 ≤ x ⊕ y, if x ∝ y;
(iv) (x + y) (u + v) ≥ (x u) + (y v);
(v)
(x + y) (u + v) ≥ (x v) + (y u);
(vi) αx ⊕ βx = |α − β|x, if x ∝ 0.
Proposition 2.4 [9]. Let C be a normal cone in X with normal constant K , then for each x, y ∈ X,
the following conditions hold:
(i)
(ii)
(iii)
(iv)
||0 ⊕ 0|| = ||0|| = 0;
||x ∨ y|| ≤ ||x|| ∨ ||y|| ≤ ||x|| + ||y||;
||x ⊕ y|| ≤ ||x − y|| ≤ K ||x ⊕ y||;
if x ∝ y, then ||x ⊕ y|| = ||x − y||.
Proposition 2.5. Let C be a normal cone in X, if for x, y ∈ X, they can be compared to each other,
then the following condition holds:
(x + y) ∨ ((−x) + (−y)) ≤ ((x ∨ (−x)) + (y ∨ (−y))).
Definition 2.3 [9]. Let M : X → 2 X be a set-valued mapping such that M(x) is a nonempty closed
subset of X.
Then
(i)
(ii)
M is said to be a comparison mapping, if for any vx ∈ M(x), x ∝ vx , and if x ∝ y, then
for any vx ∈ M(x) and any v y ∝ M(y), vx ∝ v y , ∀ x, y ∈ X ;
a comparison mapping M is said to be ordered rectangular, if for each x, y ∈ X, u x ∈
M(x) and u y ∈ M(y) such that
u x u y , −(x ⊕ y) = 0;
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(iii)
a comparison mapping M is said to be λ-ordered monotone, if there exists a constant
λ > 0 such that
λ(vx − v y ) ≥ x − y, ∀ x, y ∈ X, vx ∈ M(x) and v y ∈ M(y);
(iv)
a comparison mapping M is said to be β-ordered extended, if there exists a constant
β > 0 such that
β(x ⊕ y) ≤ vx ⊕ v y , ∀ x, y ∈ X, vx ∈ M(x), v y ∈ M(y);
(v)
a comparison mapping M is said to be ordered RME with respect to JM,λ , if M is ordered
rectangular and λ-ordered monotone with respect to JM,λ and β-ordered extended, and
(I + λM)(X ) = X, for λ, β > 0, where I stands for the identity mapping on X.
Definition 2.4 [9]. Let A, B : X → X be two single-valued mappings.
Then
(i)
(ii)
A is said to be a comparison, if for each x, y ∈ X, x ∝ y then A(x) ∝ A(y), x ∝ A(x)
and y ∝ A(y).
A and B are said to be comparable to each other, if for each x ∈ X, A(x) ∝ B(x).
Obviously, if A is a comparison mapping, then A ∝ I.
Definition 2.5. Let X be a real ordered Hilbert space, C be a normal cone with a normal constant K in X. A mapping A : X × X → X is said to be β-ordered compression mapping, if A is a
comparison mapping and
A((x, .)) ⊕ A((y, .)) ≤ β(x ⊕ y), f or 0 < β < 1.
Definition 2.6. Let X be a real ordered Hilbert space, C be a normal cone with a normal constant
K in X. A mapping N : X × X → X is said to be (μ, ν)-ordered Lipschitz continuous, if x ∝ y,
u ∝ v, then N (x, u) ∝ N (y, v) and there exist constants μ, ν > 0 such that
N (x, u) ⊕ N (y, v) ≤ μ(x ⊕ y) + ν(u ⊕ v).
Definition 2.7. A set-valued mapping A : X → C B(X ) is said to be D-Lipschitz continuous, if for
each x, y ∈ X , x ∝ y, there exists a constant δ A such that
D(A(x), A(y)) ≤ δ A x ⊕ y, ∀x, y ∈ X.
Definition 2.8. Let M : X → 2 X be a set-valued mapping such that M(x) is a nonempty closed
subset of X and let R : X → X be a single-valued mapping. Then a comparison mapping M is said
(I −R)
, if M is ordered rectangular and λ-ordered monotone
to be ordered RME with respect to JM,λ
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(I −R)
with respect to JM,λ
and β-ordered extended, and [(I − R) + λM](X ) = X, for λ, β > 0, where
I stands for the identity mapping on X.
we construct the following examples.
Example 2.1. Let X = [0, 1] with usual inner product and suppose
(i)
(ii)
the set-valued mapping M : X → 2 X is defined by
[0, 1], i f x =
0,
M(x) =
(0, 1], i f x = 0.
Then, it is easy check that M is ordered rectangular mapping.
the set-valued mapping M : X → 2 X is defined by
{ 1 }, i f x = 0,
M(x) = x
{0}, i f x = 0.
Then, it is easy to check that M is
9
-ordered monotone and 12 -ordered extended mapping.
10
Example 2.2. Let X = R and N : X × X → X is defined by
x + y
+ a, ∀ x, y ∈ X, a > 0.
N (x, y) = sin
5
Then, N is ( 15 , 15 )-ordered Lipschitz continuous mapping.
Using the Proposition 2.5., we have
y + v
x + u
+ a ⊕ sin
+a
N (x, u) ⊕ N (y, v) =
sin
5
5
y + v
y + v
x + u
x + u
− sin
∨ sin
− sin
= sin
5
5
5
5
x + u y + v y + v x + u
−
∨
−
≤
5
5
5
5
x − y u − v x − y u − v +
∨ −
+ −
=
5
5
5
5
x − y x − y u − v u − v ∨ −
+
∨ −
≤
5
5
5
5
1
1
≤
(x ⊕ y) + (u ⊕ v).
5
5
Therefor,
N (x, u) ⊕ N (y, v) ≤
1
1
(x ⊕ y) + (u ⊕ v).
5
5
i.e., N is ( 15 , 15 )-ordered Lipschitz continuous mapping.
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3. FORMULATION OF THE PROBLEM AND SOME BASIC RESULTS
Let A, B : X → F(X ) be the closed fuzzy mappings satisfying the following condition (∗). Then
there exists mappings a, b : X → [0, 1] such that for each x ∈ X, (Ax)a(x) ∈ C B(X ) and (Bx)b(x) ∈
C B(X ). Thus, we define the set-valued mappings by
Ã(x) = (Ax)a(x) and B̃(x) = (Bx)b(x) , ∀ x ∈ X,
where à and B̃ are called the set-valued mappings induced by the fuzzy mappings A and B, respectively. Suppose that M : X → 2 X be an ordered RME set-valued mapping and N : X × X :→ X
be a single-valued mapping. We consider the following problem:
Find x ∈ X, u ∈ (Ax)a(x) , v ∈ (Bx)b(x) such that
0 ∈ N (u, v) + M(x).
(3.1)
Problem (3.1) is called fuzzy variational inclusion problem involving ordered RME set-valued mapping.
Below we have the following special case of problem (3.1) and one can obtain many other cases
for suitable choice of operators involved in the formulation of problem (3.1).
If A, B : X → C B(X ) are the classical set-valued mappings, we can define the fuzzy mappings
A, B by
x → χ A(x) , x → χ B(x) ,
where χ A(x) and χ B(x) are the characteristic functions of A and B, respectively. Taking a(x) =
b(x) = 1, for all x ∈ X, the problem (3.1) converted into the problem.
Find x ∈ X, u ∈ A(x) and v ∈ B(x) such that
0 ∈ N (u, v) + M(x).
(3.2)
Problem (3.2) was studied by Verma [26] and many other researchers in different settings.
Definition 3.1. Let C be a normal cone with normal constant K and M : X → 2 X be a set-valued
ordered rectangular mapping. Let I : X → X be the identity mapping and R : X → X be a single(I −R)
: X → X associated with I, R and M is
valued mapping. The relaxed resolvent operator Jλ,M
defined by
(I −R)
Jλ,M
(x) = [(I − R) + λM]−1 (x), f or all x ∈ X and λ > 0.
(3.3)
If R = 0, then the relaxed resolvent operator defined by (3.3) coincides with the resolvent
operator of Li [15].
Now, we show that the relaxed resolvent operator defined by (3.3) is single-valued, a comparison mapping as well as Lipschitz continuous.
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Proposition 3.1. Let R : X → X be a comparison and γ -ordered compression mapping, and let
(I −R)
M : X → 2 X be the set-valued ordered rectangular mapping. Then the operator Jλ,M
= [(I −
−1
X
R) + λM] : X → 2 is a single-valued, for all λ > 0.
Proof. For any given z ∈ X and a constant λ > 0, let x, y ∈ [(I − R) + λM]−1 (z). Then
1
(z − (I − R)(x)) ∈ M(x),
λ
and
1
(z − (I − R)(y)) ∈ M(y).
λ
Using (iv) and (vi) of Proposition 2.3., we have
=
≥
=
=
=
=
≥
1
1
(z − (I − R)(x)) (z − (I − R)(y))
λ
λ
1
[(z − (I − R)(x)) (z − (I − R)(y))]
λ
1
[z z + (−(I − R)(x)) (−(I − R)(y))]
λ
1
[−(I − R)(x) −(I − R)(y)]
λ
1
− [−(−(I − R)(x)) −(−(I − R)(y))]
λ
1
− [(I − R)(x) ⊕ (I − R)(y)]
λ
1
− [x − R(x) ⊕ y − R(y)]
λ
1
− [x ⊕ y + R(x) ⊕ R(y)].
λ
Thus, we have
1
1
1
(z − (I − R)(x)) (z − (I − R)(y)) ≥ − [x ⊕ y + R(x) ⊕ R(y)].
λ
λ
λ
(3.4)
Since R is γ -ordered compression mapping and using (3.4), we have
1
1
(z − (I − R)(x)) (z − (I − R)(y))
λ
λ
1
− [x ⊕ y + γ (x ⊕ y)]
λ
1
= − (1 + γ )(x ⊕ y).
λ
≥
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(3.5)
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Iqbal Ahmad, Abul H. Siddiqi and Rais Ahmad
Since M is an ordered rectangular mapping and using (3.5), we have
0 =
≥
=
1
1
(z − (I − R)(x)) (z − (I − R)(y)), −(x ⊕ y)
λ
λ
1
− (1 + γ )(x ⊕ y), −(x ⊕ y)
λ
1
(1 + γ )||x ⊕ y||2 .
λ
i.e.,
1
(1 + γ )||x ⊕ y||2 ≤ 0.
λ
(I −R)
= [(I − R) + λM]−1
and consequently x ⊕ y = 0, i.e., we have x = y. Thus, the operator Jλ,M
is a single-valued.
Proposition 3.2. Let M : X → 2 X be a set-valued ordered rectangular, comparison and λ-ordered
(I −R)
. Let R : X → X be a strongly comparison mapping and
monotone mapping with respect to Jλ,M
(I −R)
(I − R) be an strongly comparison mapping with respect to Jλ,M
. Then, the resolvent operator
(I −R)
Jλ,M
: X → X is a comparison mapping.
(I −R)
(I −R)
Proof. Since M : X → 2 X is a comparison mapping with respect to Jλ,M
so that x ∝ Jλ,M
(x).
For any x, y ∈ X, let x ∝ y, and let
vx =
1
(I −R)
(I −R)
(x − (I − R)(Jλ,M
(x))) ∈ M(Jλ,M
(x))
λ
(3.6)
and
1
(I −R)
(I −R)
(y − (I − R)(Jλ,M
(y))) ∈ M(Jλ,M
(y)).
λ
Taking difference of (3.6) and (3.7), we have
1
1
(I −R)
(I −R)
(x − (I − R)(Jλ,M
(y − (I − R)(Jλ,M
(x))) −
(y)))
vx − v y =
λ
λ
1
(I −R)
(I −R)
x − y + (I − R)(Jλ,M (y)) − (I − R)(Jλ,M
(x)) .
=
λ
vy =
(3.7)
As M is a λ-ordered monotone, we have
0
≤
λ(vx − v y ) − (x − y)
(I −R)
(I −R)
(y)) − (I − R)(Jλ,M
(x)) − (x − y)
= (x − y) + (I − R)(Jλ,M
(I −R)
(I −R)
= (I − R)(Jλ,M
(y)) − (I − R)(Jλ,M
(x)),
which implies that
(I −R)
(I −R)
0 ≤ (I − R)(Jλ,M
(y)) − (I − R)(Jλ,M
(x)).
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263
(I −R)
(I −R)
Since (I − R) is strongly comparison mapping with respect to Jλ,M
. Therefore, Jλ,M
(x) ∝
(I −R)
Jλ,M (y). The proof is completed.
(I −R)
Proposition 3.3. Let M : X → 2 X be a ordered RME set-valued mapping with respect to Jλ,M
,
X
for βλ > γ + 1. Let R : X → 2 be a comparison and γ -ordered compression mapping with
(I −R)
. Then the following condition holds:
respect to Jλ,M
(I −R)
(I −R)
(x) ⊕ Jλ,M
(y) ≤
Jλ,M
1
(x ⊕ y).
βλ − γ − 1
(I −R)
(I −R)
Proof. For x, y ∈ X, set u x = Jλ,M
(x), u y = Jλ,M
(y), and let
vx =
1
(I −R)
(I −R)
x − (I − R)(Jλ,M
(x)) ∈ M(Jλ,M
(x))
λ
vy =
1
(I −R)
(I −R)
y − (I − R)(Jλ,M
(y)) ∈ M(Jλ,M
(y)).
λ
and
(I −R)
. That is, M is ordered rectanguAs M be an ordered RME set-valued mapping with respect to Jλ,M
(I −R)
lar, λ-ordered monotone with respect to Jλ,M
and β-ordered extended mapping. From Proposition
(I −R)
3.2., we know that Jλ,M
is a comparison mapping, and as x ∝ y so that vx ∈ M(u x ), v y ∈ M(u y ),
we have u x ∝ u y , vx ∝ v y . By (i x) of the Proposition 2.3, we have
vx ⊕ v y
=
≤
≤
≤
=
1
[(x − (I − R)(u x )) ⊕ (y − (I − R)(u y ))]
λ
1
[x ⊕ y + (I − R)(u x ) ⊕ (I − R)(u y )]
λ
1
[x ⊕ y + u x ⊕ u y + R(u x ) ⊕ R(u y )]
λ
1
[x ⊕ y + u x ⊕ u y + γ (u x ⊕ u y )]
λ
1
[x ⊕ y + (1 + γ )(u x ⊕ u y )].
λ
Since M is β-ordered extended mapping, we have
β(u x ⊕ u y ) ≤ vx ⊕ v y =
It follows that β −
(1+γ )
λ
1
[x ⊕ y + (1 + γ )(u x ⊕ u y )].
λ
(u x ⊕ u y ) ≤ λ1 (x ⊕ y) and consequently, we have
(I −R)
(I −R)
(x) ⊕ Jλ,M
(y) ≤
Jλ,M
1
(x ⊕ y).
βλ − γ − 1
The proof is completed.
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Iqbal Ahmad, Abul H. Siddiqi and Rais Ahmad
4. ITERATIVE ALGORITHM AND EXISTENCE RESULTS
In this section, we define an iterative algorithm to obtain the solution of fuzzy variational inclusion
problem involving ordered ordered RME set-valued mapping (3.1).
Iterative Algorithm 4.1. Let R : X → X, N : X × X → X be the single-valued mappings and
I : X → X is an identity mapping. Let A, B : X → F(X ) be the closed fuzzy mappings satisfying
condition (∗) and Ã, B̃ be the set-valued mappings induced by the fuzzy mappings. Suppose that
M : X → 2 X is the ordered RME set-valued mapping. We define the following scheme:
For any given initial x0 ∈ X, u 0 ∈ (Ax0 )a(x0 ) and v0 ∈ (Bx0 )b(x0 ) , let
(I −R)
[(I − R)(x0 ) − λN (u 0 , v0 )].
x1 = Jλ,M
Since u 0 ∈ (Ax0 )a(x0 ) ∈ C B(X ), v0 ∈ (Bx0 )b(x0 ) ∈ C B(X ), by Nadler’s theorem [21], there exist
u 1 ∈ (Ax1 )a(x1 ) , v1 ∈ (Bx1 )b(x1 ) and suppose that x0 ∝ x1 , u 0 ∝ u 1 , v0 ∝ v1 such that
||u 1 ⊕ u 0 || = ||u 1 − u 0 || ≤ D(A(x1 )a(x1 ) , A(x0 )a(x0 ) )
and
||v1 ⊕ v0 || = ||v1 − v0 || ≤ D(B(x1 )b(x1 ) , B(x0 )b(x0 ) ).
Continuing the above process inductively with the supposition that xn+1 ∝ xn , u n+1 ∝ u n and
vn+1 ∝ vn , for all n ∈ N, we have the following algorithm:
(I −R)
[(I − R)(xn ) − λN (u n , vn )],
xn+1 = Jλ,M
(4.1)
u n+1 ∈ A(xn+1 )a(xn+1 ) , ||u n+1 ⊕ u n || = ||u n+1 − u n || ≤ D(A(xn+1 )a(xn+1 ) , A(xn )a(xn ) ),
(4.2)
vn+1 ∈ B(xn+1 )b(xn+1 ) , ||vn+1 ⊕ vn || = ||vn+1 − vn || ≤ D(B(xn+1 )b(xn+1 ) , B(xn )b(xn ) ),
(4.3)
where λ > 0 is a constant.
Now, we convert our problem (3.1) into a fixed point problem.
Lemma 4.1. Let x ∈ X, u ∈ (Ax)a(x) and v ∈ (Bx)b(x) is a solution of fuzzy variational inclusion
problem involving ordered RME set-valued mapping (3.1) if and only if (x, u, v) satisfies the equation:
(I −R)
[(I − R)(x) − λN (u, v)],
x = Jλ,M
where
(I −R)
Jλ,M
= [(I − R) + λM]−1 ,
and λ > 0 is a constant.
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265
(I −R)
Proof. The proof directly follows from the definition of the relaxed resolvent operator Jλ,M
.
Now, we prove the following existence and convergence result for fuzzy variational inclusion
problem involving ordered RME set-valued mapping (3.1).
Theorem 4.1. Let X be a real ordered Hilbert space and C be a normal cone with normal constant
K . Let R : X → X be a comparison, γ -ordered compression mapping and N : X × X → X be an
(μ, ν)-ordered Lipschitz continuous mapping. Let A, B : X → F(X ) be the closed fuzzy mappings
satisfying the condition (∗) and the let Ã, B̃ : X → C B(X ) be the set-valued mappings induced by
the fuzzy mappings A and B such that à and B̃ are D-Lipschitz continuous mappings with constants
δ A and δ B , respectively. Suppose that M : X → 2 X be an ordered RME set-valued mapping and if
the following condition is satisfied
K λ(μδ A + νδ B ) + (1 + K )(1 + γ ) < βλ,
(4.4)
then the iterative sequences {xn }, {u n } and {vn } generated by Algorithm 4.1. converges strongly to
x, u and v, respectively and (x, u, v) is a solution of fuzzy variational inclusion problem involving
ordered RME set-valued mapping (3.1), where x ∈ X, u ∈ (Ax)a(x) and v ∈ (Bx)b(x) .
Proof. Since R is γ -ordered compression mapping and N is (μ, ν)-ordered Lipschitz continuous
mapping. By Algorithm 4.1., Proposition 2.2 and Proposition 3.3, we have
0
≤
xn+1 ⊕ xn
=
(I −R)
(I −R)
Jλ,M
[(I − R)(xn ) − λN (u n , vn )] ⊕ Jλ,M
[(I − R)(xn−1 ) − λN (u n−1 , vn−1 )]
≤
1
[(I − R)(x n ) − λN (u n , vn ) ⊕ (I − R)(xn−1 ) − λN (u n−1 , vn−1 )]
βλ − γ − 1
≤
1
[(I − R)(x n ) ⊕ (I − R)(xn−1 ) + λ(N (u n , vn ) ⊕ N (u n−1 , vn−1 ))]
βλ − γ − 1
≤
1
[(xn ⊕ xn−1 ) + R(xn ) ⊕ R(xn−1 ) + λ(N (u n , vn ) ⊕ N (u n−1 , vn−1 ))]
βλ − γ − 1
≤
1
[(xn ⊕ xn−1 ) + γ (xn ⊕ xn−1 ) + λ(N (u n , vn ) ⊕ N (u n−1 , vn−1 ))]
βλ − γ − 1
≤
≤
1
[(xn ⊕ xn−1 ) + γ (xn ⊕ xn−1 ) + λ μ(u n ⊕ u n−1 )
βλ − γ − 1
+ν(vn ⊕ vn−1 ) ]
1
[(xn ⊕ xn−1 ) + γ (xn ⊕ xn−1 ) + λμ(u n ⊕ u n−1 ) + λν(vn ⊕ vn−1 )].
βλ − γ − 1
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Iqbal Ahmad, Abul H. Siddiqi and Rais Ahmad
Using Definition 2.2., Proposition 2.3. and D-Lipschitz continuous of à and B̃, we have
||xn+1 − xn || ≤
≤
≤
≤
1
[(xn ⊕ xn−1 ) + γ (xn ⊕ xn−1 ) + λμ(u n ⊕ u n−1 )
βλ − γ − 1
+λν(vn ⊕ vn−1 )]||
K
[(1 + γ )||xn − xn−1 || + λμ||u n − u n−1 ||
βλ − γ − 1
+λν||vn − vn−1 ||]
K
[(1 + γ )||xn − xn−1 || + λμδ A ||xn − xn−1 ||
βλ − γ − 1
+λνδ B ||xn − xn−1 ||]
K
[(1 + γ ) + λ(μδ A + νδ B )]||xn − xn−1 ||.
βλ − γ − 1
K ||
(4.5)
i.e.,
||xn+1 − xn || ≤
||xn − xn−1 ||,
where
=
K [(1 + γ ) + λ(μδ A + νδ B )]
.
βλ − γ − 1
By condition (4.4), we have 0 < < 1, thus {xn } is a cauchy sequence in X and as X is complete,
there exists x ∈ X such that xn → x as n → ∞. From (4.2) and (4.3) of Algorithm 4.1. and DLipschitz continuity of à and B̃, we have
||u n+1 − u n || ≤ D(A(xn+1 )a(xn+1 ) , A(xn )a(xn ) ) ≤ δ A ||xn+1 − xn ||
(4.6)
||vn+1 − vn || ≤ D(B(xn+1 )b(xn+1 ) , B(xn )b(xn ) ) ≤ δ B ||xn+1 − xn ||.
(4.7)
It is clear from (4.6) and (4.7) that {u n } and {vn } are also cauchy sequences in X, so there exist u
and v in X such that u n → u and vn → v as n → ∞. By using the continuity of the operators N ,
I −R
and iterative Algorithm 4.1., we have
Ã, B̃, Jλ,M
(I −R)
[(I − R)x − λN (u, v)].
x = Jλ,M
By Lemma 4.1, we conclude that (x, u, v) is a solution of problem (3.1). It remains to show that
u ∈ (Ax)a(x) and v ∈ (Bx)b(x) . In fact
d(u, (Ax)a(x) )
≤ u − u n + d(u n , (Ax)a(x) )
≤ u − u n + D((Axn )a(xn ) , (Ax)a(x) )
≤
u − u n + δ A xn − x → 0, as n → ∞.
Hence u ∈ (Ax)a(x) . Similarly, we can show that v ∈ (Bx)b(x) . This completes the proof.
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267
Remark 4.1. If N , R, Ã, B̃ are zero mappings, then from Theorem 4.1. one can obtain Theorem
2.6. of Li [18] for solving the problem 0 ∈ M(x).
5. CONCLUSION
The aim of this paper is to introduced a relaxed resolvent operator and we demonstrate some of
its properties. The relaxed resolvent operator is used to define an iterative algorithm for solving a
fuzzy variational inclusion problem involving ordered RME set-valued mapping in ordered Hilbert
spaces. Some preliminary results are proved to obtain the main result. We remark that our results
may be extended for other related problems involving fuzzy mappings and also one may consider
some higher ordered dimensional structure.
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