Strong Convergence Iterative Algorithms for Equilibrium Problems and Fixed Point Problems in Banach Spaces

Shenghua Wang,Shin Min Kang

doi:10.1155/2013/619762

Abstract

We first introduce the concept of Bregman asymptotically quasinonexpansive mappings and prove that the fixed point set of this kind of mappings is closed and convex. Then we construct an iterative scheme to find a common element of the set of solutions of an equilibrium problem and the set of common fixed points of a countable family of Bregman asymptotically quasinonexpansive mappings in reflexive Banach spaces and prove strong convergence theorems. Our results extend the recent ones of some others.

Highlights

Let E be a real reflexive Banach space with norm ‖ ⋅ ‖ and E∗ the dual space of E equipped with the inducted norm ‖ ⋅ ‖∗
Let K be a nonempty, closed and convex subset of int dom f and T : K → K a closed Bergman asymptotically quasi-nonexpansive mapping with the sequence {kn} ⊂ [1, +∞) such that kn → 1 as n → ∞
(1) Resfg is single-valued; (2) Resfg is a Bregman firmly nonexpansive mapping; (3) the set of fixed points of Resfg is the solution set of the equilibrium problem, that is, F(Resfg) = EP(g); (4) EP(g) is a closed and convex subset of C; (5) for all x ∈ E and u ∈ F(Resfg), one has

Summary

Introduction

Let E be a real reflexive Banach space with norm ‖ ⋅ ‖ and E∗ the dual space of E equipped with the inducted norm ‖ ⋅ ‖∗. The mapping T : K → K is said to be Bregman asymptotically quasi-nonexpansive if there exists a sequence {kn} ⊂ [1, ∞) satisfying limn → ∞kn = 1 such that, for every n ≥ 1, Df (V, Tnx) ≤ knDf (V, x) , ∀V ∈ F (T) , x ∈ K.

Results

Conclusion