Iterative Methods for Family of Strictly Pseudocontractive Mappings and System of Generalized Mixed Equilibrium Problems and Variational Inequality Problems

Yekini Shehu

doi:10.1155/2011/852789

Abstract

AbstractWe introduce a new iterative scheme by hybrid method for finding a common element of the set of common fixed points of infinite family of "Equation missing"-strictly pseudocontractive mappings and the set of common solutions to a system of generalized mixed equilibrium problems and the set of solutions to a variational inequality problem in a real Hilbert space. We then prove strong convergence of the scheme to a common element of the three above described sets. We give an application of our results. Our results extend important recent results from the current literature.

Highlights

Let K be a nonempty closed and convex subset of a real Hilbert space H
A mapping A : K → H is called inverse-strongly monotone see, e.g., 1, 2 if there exists a positive real number λ such that Ax − Ay, x − y ≥ λ Ax − Ay 2, for all x, y ∈ K
Motivated by the results of Takahashi et al 33, Kumam 28 studied the problem of approximating a common element of set of solutions to an equilibrium problem, set of solutions to variational inequality problem and the set of fixed points of a nonexpansive mapping in a real Hilbert space

Summary

Introduction

Let K be a nonempty closed and convex subset of a real Hilbert space H. Ceng and Yao 25 introduced a new iterative scheme of approximating a common element of the set of solutions to mixed equilibrium problem and set of common fixed points of finite family of nonexpansive mappings in a real Hilbert space H In their results, they imposed the following condition on a nonempty closed and convex subset K of H:. Motivated by the results of Takahashi et al 33 , Kumam 28 studied the problem of approximating a common element of set of solutions to an equilibrium problem, set of solutions to variational inequality problem and the set of fixed points of a nonexpansive mapping in a real Hilbert space

Preliminaries

Main Results