Iterative Algorithms for a System of Generalized Mixed Equilibrium Problems

Abstract

The purpose of this paper is to present an iterative algorithm for a system of generalized mixed equilibrium problems(for short, denoted by SGMEP) in Banach space. We prove strong convergence theorems of the iterative algorithm for finding a common element of the fixed point set of relatively nonexpansive mappings and the solution set of the SGMEP in Banach space under some suitable conditions. Keywords—System of generalized mixed equilibrium problems; Iterative algorithm; Relatively nonexpansive mappings; Strong convergence I.INTRODUCTION The equilibrium problem, which was first introduced by Blum and Oettli[1], provides a unified model of many problems such as optimization problems, variational inequality problems, complementarity problems, fixed point problems and so on. And the equilibrium problem is the special case of the generalized mixed equilibrium problem. The generalized mixed equilibrium problem plays an important role in economic, management and engineering. Iterative methods for nonexpansive mappings have recently been applied with generalized mixed equilibrium problem. In this paper, we introduce an SGMEP and an iterative algorithm for the SGMEP is suggested for finding a common element of the fixed point set of relatively nonexpansive mappings and the solution set of the SGMEP in Banach space under some suitable conditions. The results obtained here extend and improve the corresponding results of [2-4]. In this paper, we consider the SGMEP of finding x C ∈ such that ( ) ( ) ( ) ( ) ( ) ( )        ∈ ∀ ≥ + ∈ ∀ ≥ + ∈ ∀ ≥ + , , 0 , , , , , 0 , , , , , 0 , , ,