Three-step Mann iterations for a general system of variational inequalities and an infinite family of nonexpansive mappings in Banach spaces

Ching-Feng Wen,Lu-Chuan Ceng

doi:10.1186/1029-242x-2013-539

Abstract

In this paper, let X be a uniformly convex Banach space which either is uniformly smooth or has a weakly continuous duality map. We introduce and consider three-step Mann iterations for finding a common solution of a general system of variational inequalities (GSVI) and a fixed point problem (FPP) of an infinite family of nonexpansive mappings in X. Here three-step Mann iterations are based on Korpelevich’s extragradient method, the viscosity approximation method and the Mann iteration method. We prove the strong convergence of this method to a common solution of the GSVI and the FPP, which solves a variational inequality on their common solution set. We also give a weak convergence theorem for three-step Mann iterations involving the GSVI and the FPP in a Hilbert space. The results presented in this paper improve, extend, supplement and develop the corresponding results announced in the earlier and very recent literature. MSC:49J30, 47H09, 47J20.

Highlights

Let X be a real Banach space whose dual space is denoted by X∗
Motivated and inspired by the research going on in this area, we introduce and analyze three-step Mann iterations for finding a common solution of general system of variational inequalities (GSVI) ( . ) and a fixed point problem (FPP) of an infinite family of nonexpansive self-mappings on C
We prove the strong convergence of this method to a common solution of GSVI ( . ) and the FPP, which solves a variational inequality on their common solution set

Summary

Introduction

Let X be a real Banach space whose dual space is denoted by X∗. The normalized duality mapping J : X → X∗ is defined by. Let C be a nonempty closed convex subset of a real uniformly smooth Banach space X, let T : C → C be a nonexpansive mapping such that Fix(T) = ∅ and f ∈ ΞC with a contractive coefficient ρ ∈ ( , ), where ΞC is the collection of all contractive self-mappings on C. Let C be a nonempty closed convex subset of a real Banach space X, and f ∈ ΞC with a contractive coefficient ρ ∈ ( , ). In , Ceng et al [ ] introduced and analyzed the following hybrid viscosity approximation method for finding a common fixed point of an infinite family of nonexpansive mappings in a strictly convex and reflexive Banach space, which either is uniformly smooth or has a weakly continuous duality map Jφ with gauge φ.

Given sequences

Define a mapping

Since and