scholarly journals Strong Convergence Theorem for Bregman Strongly Nonexpansive Mappings and Equilibrium Problems in Reflexive Banach Spaces

2013 ◽  
Vol 2013 ◽  
pp. 1-7 ◽  
Author(s):  
Jinhua Zhu ◽  
Shih-sen Chang ◽  
Min Liu
Keyword(s):  
Banach Space ◽  
Strong Convergence ◽  
Hybrid Method ◽  
Convergence Theorem ◽  
Equilibrium Problems ◽  
Reflexive Banach Space ◽  
Nonexpansive Mappings ◽  
Common Element ◽  
Strong Convergence Theorem ◽  
Reflexive Banach Spaces

By using a new hybrid method, a strong convergence theorem for finding a common element of the set of solutions of an equilibrium problem and the set of fixed points of Bregman strongly nonexpansive mappings in a reflexive Banach space is proved.

scholarly journals Convergence Theorem Based on a New Hybrid Projection Method for Finding a Common Solution of Generalized Equilibrium and Variational Inequality Problems in Banach Spaces

2010 ◽  
Vol 2010 ◽  
pp. 1-25 ◽  
Author(s):  
Siwaporn Saewan ◽  
Poom Kumam ◽  
Kriengsak Wattanawitoon
Keyword(s):  
Banach Space ◽  
Variational Inequality ◽  
Strong Convergence ◽  
Projection Method ◽  
Convergence Theorem ◽  
Nonexpansive Mappings ◽  
Common Element ◽  
Strong Convergence Theorem ◽  
Hybrid Projection Method ◽  
Generalized Equilibrium

The purpose of this paper is to introduce a new hybrid projection method for finding a common element of the set of common fixed points of two relatively quasi-nonexpansive mappings, the set of the variational inequality for anα-inverse-strongly monotone, and the set of solutions of the generalized equilibrium problem in the framework of a real Banach space. We obtain a strong convergence theorem for the sequences generated by this process in a 2-uniformly convex and uniformly smooth Banach space. Base on this result, we also get some new and interesting results. The results in this paper generalize, extend, and unify some well-known strong convergence results in the literature.

scholarly journals A Strong Convergence Theorem for Relatively Nonexpansive Mappings and Equilibrium Problems in Banach Spaces

2012 ◽  
Vol 2012 ◽  
pp. 1-12
Author(s):  
Mei Yuan ◽  
Xi Li ◽  
Xue-song Li ◽  
John J. Liu
Keyword(s):  
Strong Convergence ◽  
Banach Spaces ◽  
Projection Method ◽  
Convergence Theorem ◽  
Projection Operator ◽  
Equilibrium Problems ◽  
Nonexpansive Mappings ◽  
Strong Convergence Theorem ◽  
Relatively Nonexpansive Mappings ◽  
Relatively Nonexpansive

Relatively nonexpansive mappings and equilibrium problems are considered based on a shrinking projection method. Using properties of the generalizedf-projection operator, a strong convergence theorem for relatively nonexpansive mappings and equilibrium problems is proved in Banach spaces under some suitable conditions.

scholarly journals Approximations for Equilibrium Problems and Nonexpansive Semigroups

2014 ◽  
Vol 2014 ◽  
pp. 1-7
Author(s):  
Huan-chun Wu ◽  
Cao-zong Cheng
Keyword(s):  
Iterative Method ◽  
Strong Convergence ◽  
Hilbert Spaces ◽  
Convergence Theorem ◽  
Equilibrium Problems ◽  
Common Fixed Points ◽  
Common Element ◽  
Strong Convergence Theorem ◽  
Nonexpansive Semigroup ◽  
New Iterative Method

We introduce a new iterative method for finding a common element of the set of solutions of an equilibrium problem and the set of all common fixed points of a nonexpansive semigroup and prove the strong convergence theorem in Hilbert spaces. Our result extends the recent result of Zegeye and Shahzad (2013). In the last part of the paper, by the way, we point out that there is a slight flaw in the proof of the main result in Shehu's paper (2012) and perfect the proof.

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