Strong convergence theorems for mixed equilibrium problems and Bregman relatively nonexpansive mappings in reflexive Banach spaces

Volume 14, Issue 2, pp 63--79 http://dx.doi.org/10.22436/jnsa.014.02.02

Publication Date: July 03, 2020 Submission Date: October 22, 2019 Revision Date: February 11, 2020 Accteptance Date: February 19, 2020

Download PDF

Download XML

• Export citations
  • CITATIONS & REFERENCES
  • EndNote (.ENW)
  • Mendeley, JabRef (.BIB)
  • Papers, Zotero, Reference Manager, RefWorks (.RIS)
  • ARTICLE CITATION
  • EndNote (.ENW)
  • Mendeley, JabRef (.BIB)
  • Papers, Zotero, Reference Manager, RefWorks (.RIS)
  • REFERENCES
  • EndNote (.ENW)
  • Mendeley, JabRef (.BIB)
  • Papers, Zotero, Reference Manager, RefWorks (.RIS)

• 1150 Downloads
• 2387 Views

Authors

Kittisak Jantakarn - Department of Mathematics, Faculty of Science, Naresuan University, Phitsanulok 65000, Thailand. Anchalee Kaewcharoen - Department of Mathematics, Faculty of Science, Naresuan University, Phitsanulok 65000, Thailand.

Abstract

In this paper, we propose a new iterative method for solving the mixed equilibrium problems and the fixed point problems for a countable family of Bregman relatively nonexpansive mappings in reflexive Banach spaces. We prove that the sequence generated by the proposed iterative algorithm converges strongly to a common solution of the mentioned problems. Further, a numerical example of the iterative algorithm supporting our main result is presented.

Share and Cite

Keywords

• Mixed equilibrium problems
• Bregman relatively nonexpansive mappings
• reflexive Banach spaces

MSC

•  47H10
•  54H25

References

• [1] R. P. Agarwal, J.-W. Chen, Y. J. Cho, Strong convergence theorems for equilibrium problems and weak Bregman relatively nonexpansive mappings in Banach spaces, J. Inequal. Appl., 2013 (2013), 16 pages
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [2] Y. I. Alber, Metric and generalized projection operators in Banach spaces: Properties and applications, In: Theory and Applications of Nonlinear Operators of Accretive and Monotone Type, 1996 (1996), 15--50
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [3] H. H. Bauschke, J. M. Borwein, P. L. Combettes, Bregman monotone optimization algorithms, SIAM J. Control Optim., 42 (2003), 596--636
  • View Article
  • MathSciNet
  • Google Scholar


• [4] H. H. Bauschke, P. L. Combettes, J. M. Borwein, Essential Smoothness, Essential Strict Convexity, and Legendre functions in Banach Spaces, Commun. Contemp. Math., 3 (2001), 615--647
  • View Article
  • Google Scholar
  • MATH


• [5] E. Blum, W. Oettli, From optimization and variational inequalities to equilibrium problems, Math. Student, 63 (1994), 123--145
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [6] L. M. Bregman, The relaxation method of finding the common point of convex sets and its application to the solution of problems in convex programming, USSR Comput. Math. Phys., 7 (1967), 200--217
  • View Article
  • Google Scholar


• [7] R. E. Bruck, S. Reich, Nonexpansive projections and resolvents of accretive operators in Banch spaces, Houston J. Math., 3 (1977), 459--470
  • Google Scholar


• [8] D. Butnariu, A. N. Iusem, Totally Convex Functions for Fixed Points Computation and Infinite Dimensional Optimization, Kluwer Academic Publ., Dordrecht (2000)
  • View Article
  • Google Scholar
  • MATH


• [9] D. Butnariu, A. N. Iusem, C. Zalineacu, On uniform convexity, total convexity and convergence of the proximal point and outer Bregman projection algorithms in Banach spaces, J. Convex Anal., 10 (2003), 35--61
  • View Article
  • MathSciNet
  • Google Scholar


• [10] D. Butnariu, S. Reich, A. J. Zaslavski, Asymptotic behavior of relatively nonexpansive operators in Banach spaces, J. Appl. Anal., 7 (2001), 151--174
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [11] D. Butnariu, E. Resmerita, Bregman distances, totally convex functions, and a method for solving operator equations in Banach spaces, Abstr. Appl. Anal., 2006 (2006), 39 pages
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [12] 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
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [13] J. W. Chen, Z. P. Wan, L. Y. Yuan, Y. Zheng, Approximation of fixed points of weak Bregman relatively nonexpansive mappings in Banach spaces, Int. J. Math. Math. Sci., 2011 (2011), 23 pages
  • View Article
  • Google Scholar
  • MATH


• [14] P. Cholamjiak, Y. J. Cho, S. Suantai, Composite iterative schemes for maximal monotone operators in reflexive Banach spaces, Fixed Point Theory Appl., 2011 (2011), 10 pages
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [15] W. Cholamjiak, P. Cholamjiak, S. Suantai, Convergence of iterative schemes for solving fixed point of multi-valued nonself mappings and equilibrium problems, J. Nonlinear Sci. Appl., 8 (2015), 1245--1256
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [16] P. Cholamjiak, S. Suantai, Iterative methods for solving equilibrium problems, variational inequalities and fixed points of nonexpansive semigroups, J. Global Optim., 57 (2013), 1277--1297
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [17] V. Darvish, A new algorithm for mixed equilibrium problem and Bregman strongly nonexpansive mapping in Banach spaces, arXiv, 2015 (2015), 20 pages
  • View Article
  • Google Scholar


• [18] K. R. Kazmi, R. Ali, S. Yousuf, Generalized equilibrium and fixed point problems for Bregman relatively nonexpansive mappings in Banach spaces, J. Fixed Point Theory Appl., 20 (2018), 21 pages
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [19] R. P. Phelps, Convex Functions, Monotone Operators, and Differentiability, Springer-Verlag, Berlin (1993)
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [20] S. Reich, A weak convergence theorem for the alternating method with Bregman distances, In: Theory and Applications of Nonlinear Operators, 1996 (1996), 313--318
  • View Article
  • Google Scholar


• [21] 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
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [22] S. Reich, S. Sabach, Two strong convergence theorems for a proximal method in reflexive Banach spaces, Numer. Funct. Anal. Optim., 31 (2010), 22--44
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [23] S. Reich, S. Sabach, A projection method for solving nonlinear problems in reflexive Banach spaces, J. Fixed Point Theory Appl., 9 (2011), 101--116
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [24] 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, 2011 (2011), 301--316
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [25] S. Sabach, Products of finitely many resolvents of maximal monotone mappings in reflexive Banach spaces, SIAM J. Optim., 21 (2011), 1289--1308
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [26] N. Shahzad, H. Zegeye, Convergence theorem for common fixed points of a finite family of multi-valued Bregman relatively nonexpansive mappings, Fixed Point Theory Appl., 2014 (2014), 14 pages
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [27] S. Suantai, Y. J. Cho, P. Cholamjiak, Halpern's iteration for Bregman strongly nonexpansive mappings in reflexive Banach spaces, Comput. Math. Appl., 64 (2012), 489--499
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [28] G. C. Ugwunnadi, B. Ali, I. Idris, M. S. Minjibir, Strong convergence theorem for quasi-Bregman strictly pseudocontractive mappings and equilibrium problems in Banach spaces, Fixed Point Theory Appl., 231 (2014), 1--16
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [29] C. Zalinescu, Convex Analysis in General Vector Spaces, World Scientific Publishing Co., River Edge (2002)
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH