Convergence Analysis for a System of Generalized Equilibrium Problems and a Countable Family of Strict Pseudocontractions

Suthep Suantai,Prasit Cholamjiak

doi:10.1155/2011/941090

Abstract

We introduce a new iterative algorithm for a system of generalized equilibrium problems and a countable family of strict pseudocontractions in Hilbert spaces. We then prove that the sequence generated by the proposed algorithm converges strongly to a common element in the solutions set of a system of generalized equilibrium problems and the common fixed points set of an infinitely countable family of strict pseudocontractions.

Highlights

Let H be a real Hilbert space with the inner product ·, · and inducted norm ·
In 2008, Moudafi 23 introduced an iterative method for approximating a common element of the fixed points set of a nonexpansive mapping S and the solutions set of a generalized equilibrium problem GEP f, A as follows: a sequence {xn} defined by x0 ∈ C and f yn, y
Takahashi 24 introduced another modification iterative method of 1.8 for finding a common element of the fixed points set of a nonexpansive mapping and the solutions set of a generalized equilibrium problem in the framework of a real Hilbert space

Summary

Introduction

Let H be a real Hilbert space with the inner product ·, · and inducted norm ·. In 2008, Moudafi 23 introduced an iterative method for approximating a common element of the fixed points set of a nonexpansive mapping S and the solutions set of a generalized equilibrium problem GEP f, A as follows: a sequence {xn} defined by x0 ∈ C and f yn, y. Takahashi 24 introduced another modification iterative method of 1.8 for finding a common element of the fixed points set of a nonexpansive mapping and the solutions set of a generalized equilibrium problem in the framework of a real Hilbert space. We prove a strong convergence theorem of the iteration process 1.14 for a system of generalized equilibrium problems and a countable family of strict pseudocontractions in a real Hilbert space. Our results extend the main results obtained by Yao et al 25 in several aspects

Preliminaries

G: μnTn

Path Convergence Results

Strong Convergence Results