Approximation of Common Solutions to System of Mixed Equilibrium Problems, Variational Inequality Problem, and Strict Pseudo-Contractive Mappings

Poom Kumam,Chaichana Jaiboon

doi:10.1155/2011/347204

Abstract

We introduce an iterative algorithm for finding a common element of the set of fixed points of strict pseudocontractions mapping, the set of common solutions of a system of two mixed equilibrium problems and the set of common solutions of the variational inequalities with inverse strongly monotone mappings. Strong convergence theorems are established in the framework of Hilbert spaces. Finally, we apply our results for solving convex feasibility problems in Hilbert spaces. Our results improve and extend the corresponding results announced by many others recently.

Highlights

Throughout this paper, we denote by N and R the sets of positive integers and real numbers, respectively
Yao 30, we introduce a new approximation iterative scheme for finding a common element of the set of fixed points of strict pseudo-contractions, the set of common solutions of the system of a mixed equilibrium problem, and the set of common solutions of the variational inequalities with inverse strongly monotone mappings in Hilbert spaces
We apply our results for solving convex feasibility problems in Hilbert spaces

Summary

Introduction

Throughout this paper, we denote by N and R the sets of positive integers and real numbers, respectively. Wangkeeree 34 introduced a general iterative method for finding a common element of the set of solutions of the mixed equilibrium problems, the set of fixed point of a k-strict pseudo-contraction mapping, and the set of solutions of the variational inequality for an inverse strongly monotone mapping in Hilbert spaces They obtained a strong convergence theorem except the condition H for the sequences generated by these processes. Yao 30 , we introduce a new approximation iterative scheme for finding a common element of the set of fixed points of strict pseudo-contractions, the set of common solutions of the system of a mixed equilibrium problem, and the set of common solutions of the variational inequalities with inverse strongly monotone mappings in Hilbert spaces. The results in this paper extend and improve some well-known results in 17, 30, 32, 34, 35

Preliminaries

Suppose a mapping

Main Results

22 Notice that