Strong Convergence Theorems for an Infinite Family of Equilibrium Problems and Fixed Point Problems for an Infinite Family of Asymptotically Strict Pseudocontractions

Shenghua Wang,Shinmin Kang,Youngchel Kwun

doi:10.1155/2011/859032

Abstract

We prove a strong convergence theorem for an infinite family of asymptotically strict pseudo-contractions and an infinite family of equilibrium problems in a Hilbert space. Our proof is simple and different from those of others, and the main results extend and improve those of many others.

Highlights

Let C be a closed convex subset of a Hilbert space H
Let Φ be a bifunction from C × C to Ê, where Ê is the set of real numbers
Takahashi 5 first introduced an iterative scheme by the viscosity approximation method for finding a common element of the set of solutions of the equilibrium problem and the set of fixed points of a nonexpansive mapping in a Hilbert space H and proved a strong convergence theorem which is connected with Combettes and Hirstoaga’s result 6 and Wittmann’s result 7

Summary

S is called nonexpansive if

It is clear that every asymptotically nonexpansive mapping is an asymptotically 0strict pseudo-contraction and every κ-strict pseudo-contraction is an asymptotically κ-strict pseudo-contraction with γn 0 for all n ≥ 1. Takahashi 5 first introduced an iterative scheme by the viscosity approximation method for finding a common element of the set of solutions of the equilibrium problem and the set of fixed points of a nonexpansive mapping in a Hilbert space H and proved a strong convergence theorem which is connected with Combettes and Hirstoaga’s result 6 and Wittmann’s result 7. In 8 , Tada and Takahashi proposed a hybrid algorithm to find a common element of the set of fixed points of a nonexpansive mapping and the set of solutions of an equilibrium problem and proved the following strong convergence theorem. In this paper, motivated by 3, 8 , we propose a new algorithm for finding a common element of the set of fixed points of an infinite family of asymptotically strict pseudocontractions and the set of solutions of an infinite family of equilibrium problems and prove a strong convergence theorem. Our proof is simple and different from those of others, and the main results extend and improve those Kim and Xu 3 , Tada and Takahashi 8 , and many others

Preliminaries

Main Results