System of variational inequalities and an accretive operator in Banach spaces

Lu-Chuan Ceng,Ching-Feng Wen

doi:10.1186/1687-1812-2013-249

Abstract

In this paper, we introduce composite Mann iteration methods for a general system of variational inequalities with solutions being also common fixed points of a countable family of nonexpansive mappings and zeros of an accretive operator in real smooth Banach spaces. Here, the composite Mann iteration methods are based on Korpelevich’s extragradient method, viscosity approximation method and the Mann iteration method. We first consider and analyze a composite Mann iterative algorithm in the setting of uniformly convex and 2-uniformly smooth Banach space, and then another composite Mann iterative algorithm in a uniformly convex Banach space having a uniformly Gâteaux differentiable norm. Under suitable assumptions, we derive some strong convergence theorems. 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∗
A Banach space X is said to be smooth if the limit x + ty – x lim t→
Let T : C → C be a nonexpansive mapping with Fix(T) = ∅

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. ) and a common fixed point problem of a countable family of nonexpansive mappings in a uniformly convex and -uniformly smooth Banach space. If C is a nonempty closed convex subset of a strictly convex and uniformly smooth Banach space X, and if T : C → C is a nonexpansive mapping with the fixed point set Fix(T) = ∅, the set Fix(T) is a sunny nonexpansive retract of C. Let C be a nonempty closed convex subset of a Banach space X, and let T : C → C be a nonexpansive mapping with Fix(T) = ∅. ]) Let C be a nonempty closed convex subset of a real -uniformly smooth Banach space X.

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