Common Solutions of Generalized Mixed Equilibrium Problems, Variational Inclusions, and Common Fixed Points for Nonexpansive Semigroups and Strictly Pseudocontractive Mappings

Poom Kumam,Usa Hamphries,Phayap Katchang

doi:10.1155/2011/953903

Abstract

We introduce a new iterative scheme by shrinking projection method for finding a common element of the set of solutions of generalized mixed equilibrium problems, the set of common solutions of variational inclusion problems with set‐valued maximal monotone mappings and inverse‐strongly monotone mappings, the set of solutions of fixed points for nonexpansive semigroups, and the set of common fixed points for an infinite family of strictly pseudocontractive mappings in a real Hilbert space. We prove that the sequence converges strongly to a common element of the above four sets under some mind conditions. Furthermore, by using the above result, an iterative algorithm for solution of an optimization problem was obtained. Our results improve and extend the corresponding results of Martinez‐Yanes and Xu (2006), Shehu (2011), Zhang et al. (2008), and many authors.

Highlights

Throughout this paper, we assume that H is a real Hilbert space with inner product and norm are denoted by ·, · and ·, respectively
We introduce a new iterative scheme by shrinking projection method for finding a common element of the set of solutions of generalized mixed equilibrium problems, the set of common solutions of variational inclusion problems with set-valued maximal monotone mappings and inverse-strongly monotone mappings, the set of solutions of fixed points for nonexpansive semigroups, and the set of common fixed points for an infinite family of strictly pseudocontractive mappings in a real Hilbert space
In this paper, motivated by the above results, we present a new general iterative scheme for finding a common element of the set of solutions for a system of generalized mixed equilibrium problems, the set of common solutions of variational inclusion problems with set-valued maximal monotone mappings and inverse-strongly monotone mappings, the set of solutions of fixed points for nonexpansive semigroup mappings, and the set of common fixed points for an infinite family of strictly pseudocontractive mappings in a real Hilbert space

Summary

Introduction

Throughout this paper, we assume that H is a real Hilbert space with inner product and norm are denoted by ·, · and · , respectively. Shehu 19 introduced a new iterative scheme by hybrid method for finding a common element of the set of common fixed points of infinite family of k-strictly pseudocontractive mappings, the set of common solutions to a system of generalized mixed equilibrium problems, and the set of solutions to a variational inequality problem in Hilbert spaces. In this paper, motivated by the above results, we present a new general iterative scheme for finding a common element of the set of solutions for a system of generalized mixed equilibrium problems, the set of common solutions of variational inclusion problems with set-valued maximal monotone mappings and inverse-strongly monotone mappings, the set of solutions of fixed points for nonexpansive semigroup mappings, and the set of common fixed points for an infinite family of strictly pseudocontractive mappings in a real Hilbert space. The results presented in this paper extend and improve the results of Martinez-Yanes and Xu 21 , Shehu 19 , Zhang et al 4 , and many authors

Preliminaries

Wn is nonexpansive and F Wn n i

Main Result

Applications