Hybrid Extragradient Methods for Finding Zeros of Accretive Operators and Solving Variational Inequality and Fixed Point Problems in Banach Spaces

Lu-Chuan Ceng,Ching-Feng Wen

doi:10.1155/2013/894926

Abstract

We introduce and analyze hybrid implicit and explicit extragradient methods for finding a zero of an accretive operator and solving a general system of variational inequalities and a fixed point problem of an infinite family of nonexpansive self-mappings in a uniformly convex Banach spaceXwhich has a uniformly Gateaux differentiable norm. We establish some strong convergence theorems for hybrid implicit and explicit extra-gradient algorithms under suitable assumptions. Furthermore, we derive the strong convergence of hybrid implicit and explicit extragradient algorithms for finding a common element of the set of zeros of an accretive operator and the common fixed point set of an infinite family of nonexpansive self-mappings and a self-mapping whose complement is strictly pseudocontractive and strongly accretive inX. The results presented in this paper improve, extend, supplement, and develop the corresponding results announced in the earlier and very recent literature.

Highlights

Let X be a real Banach space whose dual space is denoted by X∗
We introduce and analyze hybrid implicit and explicit extragradient methods for finding a zero of an accretive operator and solving a general system of variational inequalities and a fixed point problem of an infinite family of nonexpansive self-mappings in a uniformly convex Banach space X which has a uniformly Gateaux differentiable norm
Cai and Bu [10] constructed an iterative algorithm for solving general system of variational inequalities (GSVI) (14) and a common fixed point problem of an infinite family of nonexpansive mappings in a uniformly convex and 2-uniformly smooth Banach space

Summary

Introduction

Let X be a real Banach space whose dual space is denoted by X∗. Let U = {x ∈ X : ‖x‖ = 1} denote the unite sphere of X. Cai and Bu [10] constructed an iterative algorithm for solving GSVI (14) and a common fixed point problem of an infinite family of nonexpansive mappings in a uniformly convex and 2-uniformly smooth Banach space They proved the strong convergence of the proposed algorithm by virtue of the following inequality in a 2-uniformly smooth Banach space X. where κ is the 2-uniformly smooth constant of X and J is the normalized duality mapping from X into X∗. Motivated and inspired by the research going on this area, we introduce and analyze hybrid implicit and explicit extragradient methods for finding a zero of an accretive operator A ⊂ X × X such that D(A) ⊂ C ⊂ ⋂r>0 R(I + rA) and solving GSVI (14) and a fixed point problem of an infinite family of nonexpansive self-mappings on C. The results presented in this paper improve, extend, supplement, and develop the corresponding results announced in the earlier and very recent literature; see, for example, [5, 10, 13, 16]

Preliminaries

Hybrid Implicit Extragradient Algorithm

Hybrid Explicit Extragradient Algorithm