Korean J. Math.  Vol 26, No 4 (2018)  pp.777-798
DOI: https://doi.org/10.11568/kjm.2018.26.4.777

A new algorithm for solving mixed equilibrium problem and finding common fixed points of Bregman strongly nonexpansive mappings

Nader Biranvand, Vahid Darvish

Abstract


In this paper, we study a new iterative method for solving mixed equilibrium problem and a common fixed point of a finite family of Bregman strongly nonexpansive mappings in the framework of reflexive real Banach spaces. Moreover, we prove a strong convergence theorem for finding common fixed points which also are solutions of a mixed equilibrium problem.

Keywords


Banach space, Bregman projection, Bregman distance, Bregman strongly nonexpansive mapping, fixed point, mixed equilibrium problem.

Subject classification

47H05, 47J25, 58C30

Sponsor(s)



Full Text:

PDF

References


Y.I. Alber, Metric and generalized projection operators in Banach spaces: properties and applications, in: A.G. Kartsatos (Ed.), Theory and Applications of Nonlinear Operator of Accretive and Monotone Type, Marcel Dekker, New York, (1996) 15–50. (Google Scholar)

M. Aslam Noor, Generalized mixed quasi-equilibrium problems with trifunction, Appl. Math. Lett. 18 (2005) 695–700. (Google Scholar)

M. Aslam Noor, W. Oettli, On general nonlinear complementarity problems and quasi equilibria, Matematiche (Catania) 49 (1994) 313–331. (Google Scholar)

E. Blum, W. Oettli, From optimization and variational inequalities to equilibrium problems, Math. Student 63 (1994) 123–145. (Google Scholar)

H. H. Bauschke, J. M. Borwein, P. L. Combettes, Essential smoothness, essential strict convexity, and Legendre functions in Banach spaces, Commun. Contemp. Math. 3 (2001) 615–647. (Google Scholar)

J. F. Bonnans, A. Shapiro, Perturbation Analysis of Optimization Problem, Springer, NewYork (NY), 2000. (Google Scholar)

R.E. Bruck, S. Reich, Nonexpansive projections and resolvents of accretive operators in Banach spaces, Houston J. Math. 3 (1977) 459–470. (Google Scholar)

D. Butnariu, E. Resmerita, Bregman distances, totally convex functions and a method for solving operator equations in Banach spaces, Abstr. Appl. Anal. Art. ID 84919 (2006) 1–39. (Google Scholar)

D. Butnariu, A. N. Iusem, Totally Convex Functions for Fixed Points Computation and Infinite Dimensional Optimization, Applied Optimization, 40 Kluwer Academic, Dordrecht 2000. (Google Scholar)

L.C. Ceng, J.C. Yao, A hybrid iterative scheme for mixed equilibrium problems and fixed point problems, J. Comput. Appl. Math. 214 (2008) 186–201. (Google Scholar)

Y. Censor, A. Lent, An iterative row-action method for interval convex programming, J. Optim. Theory Appl. 34 (1981) 321–353. (Google Scholar)

O. Chadli, N.C. Wong, J.C. Yao, Equilibrium problems with applications to eigenvalue problems, J. Optim. Theory Appl. 117 (2003) 245–266. (Google Scholar)

O. Chadli, S. Schaible, J.C. Yao, Regularized equilibrium problems with an application to noncoercive hemivariational inequalities, J. Optim. Theory Appl. 121 (2004) 571–596. (Google Scholar)

J. B. Hiriart-Urruty, C. Lemar ́echal, Grundlehren der mathematischen Wis-senschaften, in: Convex Analysis and Minimization Algorithms II, 306, Springer-Verlag, (1993). (Google Scholar)

G. Kassay, S. Reich, S. Sabach, Iterative methods for solving systems of variational inequalities in reflexive Banach spaces, SIAM J. Optim. 21 (2011) 1319– 1344. (Google Scholar)

F. Kohsaka, W. Takahashi, Proximal point algorithms with Bregman functions in Banach spaces, J. Nonlinear Convex Anal. 6 (2005) 505–523. (Google Scholar)

W. Kumam, U. Witthayaratb, P. Kumam, S. Suantai, K. Wattanawitoon, Convergence theorem for equilibrium problem and Bregman strongly nonexpansive mappings in Banach spaces, Optimization 65 (2016) 265–280. (Google Scholar)

I.V. Konnov, S. Schaible, J.C. Yao, Combined relaxation method for mixed equilibrium problems, J. Optim. Theory Appl. 126 (2005) 309–322. (Google Scholar)

V. Martin-Marquez, S. Reich, S. Sabach, Iterative methods for approximating fixed points of Bregman nonexpansive operators, Discrete Contin. Dyn. Syst. Ser. S. 6 (2013) 1043–1063. (Google Scholar)

J. J. Moreau, Sur la fonction polaire dune fonction semi-continue suprieurement [On the polar function of a semi-continuous function superiorly], C. R. Acad. Sci. Paris. 258 (1964) 1128–1130. (Google Scholar)

J. W. Peng, J. C. Yao, Strong convergence theorems of iterative scheme based on the extragradient method for mixed equilibrium problems and fixed point prob- lems, Math. Comp. Model. 49 (2009) 1816–1828. (Google Scholar)

R. P. Phelps, Convex Functions, Monotone Operators, and Differentiability, second ed., in: Lecture Notes in Mathematics, vol. 1364, Springer Verlag, Berlin, 1993. (Google Scholar)

S. Reich, A weak convergence theorem for the alternating method with Bregman distances, in: Theory and Applications of Nonlinear Operators of Accretive and Monotone Type, Marcel Dekker, New York, (1996) 313–318. (Google Scholar)

S. Reich, S. Sabach, A strong convergence theorem for a proximal-type algorithm in reflexive Banach spaces, J. Nonlinear Convex Anal. 10 (2009) 471–485. (Google Scholar)

S. Reich, S. Sabach, Existence and approximation of fixed points of Bregman firmly nonexpansive mappings in reflexive Banach spaces. In: Fixed-Point Algorithms for Inverse Problems in Science and Engineering, Optimization and Its Applications, 49 (2011) 301–316. (Google Scholar)

S. Reich, S. Sabach, Two strong convergence theorems for a proximal method in reflexive Banach spaces, Numer. Funct. Anal. Optim. 31 (2010) 22–44. (Google Scholar)

R. T. Rockafellar, Level sets and continuity of conjugate convex functions, Trans. Amer. Math. Soc. 123 (1966) 46–63. (Google Scholar)

S. Suantai, Y. J. Cho, P. Cholamjiak, Halperns iteration for Bregman strongly nonexpansive mappings in reflexive Banach spaces, Comput. Math. Appl. 64 (2012) 489–499. (Google Scholar)

H. K. Xu, An iterative approach to quadratic optimization, J. Optim. Theory Appl. 116 (2003) 659–678. (Google Scholar)

Y. Yao, M. Aslam Noor, S. Zainab, Y. C. Liou, Mixed equilibrium problems and optimization problems J. Math. Anal. Appl. 354 (2009) 319–329 . (Google Scholar)

C. Z ́alinescu, Convex analysis in general vector spaces, World Scientific, River Edge, (2002). (Google Scholar)

H. Zegeye, Convergence theorems for Bregman strongly nonexpansive mappings in reflexive Banach spaces, Filomat. 7 (2014) 1525–1536. (Google Scholar)


Refbacks

  • There are currently no refbacks.


ISSN: 1976-8605 (Print), 2288-1433 (Online)

Copyright(c) 2013 By The Kangwon-Kyungki Mathematical Society, Department of Mathematics, Kangwon National University Chuncheon 21341, Korea Fax: +82-33-259-5662 E-mail: kkms@kangwon.ac.kr