A general composite iterative method for generalized mixed equilibrium problems, variational inequality problems and optimization problems

Jin Sup Jung

doi:10.1186/1029-242x-2011-51

Abstract

In this article, we introduce a new general composite iterative scheme for finding a common element of the set of solutions of a generalized mixed equilibrium problem, the set of fixed points of an infinite family of nonexpansive mappings and the set of solutions of a variational inequality problem for an inverse-strongly monotone mapping in Hilbert spaces. It is shown that the sequence generated by the proposed iterative scheme converges strongly to a common element of the above three sets under suitable control conditions, which solves a certain optimization problem. The results of this article substantially improve, develop, and complement the previous well-known results in this area.2010 Mathematics Subject Classifications: 49J30; 49J40; 47H09; 47H10; 47J20; 47J25; 47J05; 49M05.

Highlights

Let H be a real Hilbert space with inner product 〈·, ·〉 and induced norm || · ||
Let B : C ® H be a nonlinear mapping and : C ® R be a function, and Θ be a bifunction of C × C into R, where R is the set of real numbers
In this article, inspired and motivated by above mentioned results, we introduce a new iterative method for finding a common element of the set of solutions of a generalized mixed equilibrium problem (1.1), the set of fixed points of a countable family of nonexpansive mappings, and the set of solutions of the variational inequality problem

Summary

Introduction

Let H be a real Hilbert space with inner product 〈·, ·〉 and induced norm || · ||. Let C be a nonempty closed convex subset of H and S : C ® C be a self-mapping on C. The following result for the existence of solutions of the variational inequality problem for inversestrongly monotone mappings was given in Takahashi and Toyoda [25].

Results

Conclusion