Hybrid and Relaxed Mann Iterations for General Systems of Variational Inequalities and Nonexpansive Mappings

L C Ceng,A A N Abdou,A E Al-Mazrooei,A Latif

doi:10.1155/2013/102820

Abstract

We introduce hybrid and relaxed Mann iteration methods for a general system of variational inequalities with solutions being also common solutions of a countable family of variational inequalities and common fixed points of a countable family of nonexpansive mappings in real smooth and uniformly convex Banach spaces. Here, the hybrid and relaxed Mann iteration methods are based on Korpelevich’s extragradient method, viscosity approximation method, and Mann iteration method. Under suitable assumptions, we derive some strong convergence theorems for hybrid and relaxed Mann iteration algorithms not only in the setting of uniformly convex and 2-uniformly smooth Banach space but also in a uniformly convex Banach space having a uniformly Gateaux differentiable norm. 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∗
Cai and Bu [11] constructed an iterative algorithm for solving general system of variational inequalities (GSVI) (13) and a common fixed point problem of a countable 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 2uniformly smooth Banach space X

Summary

Introduction

Let X be a real Banach space whose dual space is denoted by X∗. The normalized duality mapping J : X → 2X∗ is defined by. Cai and Bu [11] constructed an iterative algorithm for solving GSVI (13) and a common fixed point problem of a countable 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 2uniformly 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∗. 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, [8, 10, 11, 14, 33, 38]

Preliminaries

Hybrid Mann Iterations and Their Convergence Criteria

Relaxed Mann Iterations and Their Convergence Criteria