On solutions of a system of variational inequalities and fixed point problems in Banach spaces

Lu-Chuan Ceng,Jen-Chih Yao,Abdul Latif

doi:10.1186/1687-1812-2013-176

Abstract

Abstract In this paper, considering the problem of solving a system of variational inequalities and a common fixed point problem of an infinite family of nonexpansive mappings in Banach spaces, we propose a two-step relaxed extragradient method which is based on Korpelevich’s extragradient method and viscosity approximation method. Strong convergence results are established. MSC:49J30, 47H09, 47J20.

Highlights

In the last three decades, the theory of variational inequalities has been used as a tool to study the Nash equilibrium problem for a finite or infinite number of players; see, for example, [ – ] and the references therein
There are two ways to study the Nash equilibrium problem by using variational inequality technique: ( ) system of variational inequalities; ( ) variational inequalities defined over the product of sets
We introduce two-step relaxed extragradient method for solving system of variational inequalities (SVI) ( . ) and the common fixed point problem of an infinite family {Sn} of nonexpansive mappings of C into itself

Summary

Introduction

In the last three decades, the theory of variational inequalities has been used as a tool to study the Nash equilibrium problem for a finite or infinite number of players; see, for example, [ – ] and the references therein. If C is a nonempty closed convex subset of a strictly convex and uniformly smooth Banach space X and if T : C → C is a nonexpansive mapping with the fixed point set Fix(T) = ∅, the set Fix(T) is a sunny nonexpansive retract of C. (see [ ]) Let C be a nonempty closed convex subset of a real smooth Banach space X. Let C be a nonempty closed convex subset of a Banach space X and let T : C → C be a nonexpansive mapping with Fix(T) = ∅. Let C be a nonempty closed convex subset of X, let T : C → C be a nonexpansive mapping with Fix(T) = ∅, and f ∈ ΞC.

Then a mapping

We note that