Weak Convergence Theorems for a System of Mixed Equilibrium Problems and Nonspreading Mappings in a Hilbert Space

Somyot Plubtieng,Kamonrat Sombut

doi:10.1155/2010/246237

Somyot Plubtieng, Kamonrat Sombut

Open Access

https://doi.org/10.1155/2010/246237

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Abstract

We introduce an iterative sequence and prove a weak convergence theorem for finding a solution of a system of mixed equilibrium problems and the set of fixed points of a quasi-nonexpansive mapping in Hilbert spaces. Moreover, we apply our result to obtain a weak convergence theorem for finding a solution of a system of mixed equilibrium problems and the set of fixed points of a nonspreading mapping. The result obtained in this paper improves and extends the recent ones announced by Moudafi (2009),Iemoto and Takahashi (2009), and many others. Using this result, we improve and unify several results in fixed point problems and equilibrium problems.

Highlights

Let C be a nonempty closed convex subset of a real Hilbert space H
A mapping T of C into itself is said to be nonexpansive if T x − T y ≤ x − y for all x, y ∈ C, and a mapping F is said to be firmly nonexpansive if Fx − Fy 2 ≤ x − y, Fx − Fy for all x, y ∈ C
A mapping S : C → C is said to be nonspreading if φ Sx, Sy φ Sy, Sx ≤ φ Sx, y φ Sy, x

Summary

Introduction

Let C be a nonempty closed convex subset of a real Hilbert space H. Let x1 u ∈ C, and {xn} are given by yn PC xn − μBxn , 1.11 xn 1 αnu βnxn γnT PC yn − λAyn , n ∈ N They proved that the iterative sequence {xn} converges strongly to a common element of the set of fixed points of a nonexpansive mapping and a general system of variational inequalities with inverse-strongly monotone mappings under some parameters controlling conditions. We introduce an iterative sequence and prove a weak convergence theorem for finding a solution of a system of mixed equilibrium problems and the set of fixed points of a quasi-nonexpansive mapping in Hilbert spaces. We apply our result to obtain a weak convergence theorem for finding a solution of a system of mixed equilibrium problems and the set of fixed points of a nonspreading mapping

Preliminaries

Main Result