Strong Convergence of a New Iterative Method for Infinite Family of Generalized Equilibrium and Fixed-Point Problems of Nonexpansive Mappings in Hilbert Spaces

Baohua Guo,Shenghua Wang

doi:10.1155/2011/392741

Abstract

We introduce an iterative algorithm for finding a common element of the set of solutions of an infinite family of equilibrium problems and the set of fixed points of a finite family of nonexpansive mappings in a Hilbert space. We prove some strong convergence theorems for the proposed iterative scheme to a fixed point of the family of nonexpansive mappings, which is the unique solution of a variational inequality. As an application, we use the result of this paper to solve a multiobjective optimization problem. Our result extends and improves the ones of Colao et al. (2008) and some others.

Highlights

Let H be a real Hilbert space and T be a mapping of H into itself
The set of fixed points of T is denoted by F T
The proof method of the main result of this paper is different with ones of others in the literatures and our result extends and improves the ones of Colao et al 5 and some others

Summary

Introduction

Let H be a real Hilbert space and T be a mapping of H into itself. T is said to be nonexpansive if. For finding a common element of the set of a finite family of nonexpansive mappings and the set of solutions of an equilibrium problem, by combining the schemes 1.11 and 1.17 , Colao et al 5 proposed the following explicit scheme: 1.20 xn 1 nγ f xn βxn 1 − β I − nA Wnun, ∀n ≥ 1, and proved under some certain hypotheses that both sequences {xn} and {un} converge strongly to a point x∗ ∈ F which is an equilibrium point for G and is the unique solution of the variational inequality:. We consider a new iterative scheme for obtaining a common element in the solution set of an infinite family of generalized equilibrium problems and in the common fixed-point set of a finite family of nonexpansive mappings in a Hilbert space. The proof method of the main result of this paper is different with ones of others in the literatures and our result extends and improves the ones of Colao et al 5 and some others

Preliminaries

Main Results

Applications for Multiobjective Optimization Problem