Composite Iterative Algorithms for Variational Inequality and Fixed Point Problems in Real Smooth and Uniformly Convex Banach Spaces

Ching-Feng Wen,Lu-Chuan Ceng

doi:10.1155/2013/761864

Abstract

We introduce composite implicit and explicit iterative algorithms for solving a general system of variational inequalities and a common fixed point problem of an infinite family of nonexpansive mappings in a real smooth and uniformly convex Banach space. These composite iterative algorithms are based on Korpelevich's extragradient method and viscosity approximation method. We first consider and analyze a composite implicit iterative algorithm in the setting of uniformly convex and 2-uniformly smooth Banach space and then another composite explicit iterative algorithm in a uniformly convex Banach space with 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 literatures.

Highlights

Let X be a real Banach space whose dual space is denoted by X∗
We introduce composite implicit and explicit iterative algorithms for solving a general system of variational inequalities and a common fixed point problem of an infinite family of nonexpansive mappings in a real smooth and uniformly convex Banach space
In this paper we introduce composite implicit and explicit iterative algorithms for solving general system of variational inequalities (GSVI) (9) and the common fixed point problem of an infinite family {Sn} of nonexpansive mappings of C into itself

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. A Banach space X is said to be q-uniformly smooth if there exists a constant c > 0 such that ρ(τ) ≤ cτq for all τ > 0. Let C be a nonempty, closed, and convex subset of a real smooth Banach space X. In this paper we introduce composite implicit and explicit iterative algorithms for solving GSVI (9) and the common fixed point problem of an infinite family {Sn} of nonexpansive mappings of C into itself. We first propose a composite implicit iterative algorithm in the setting of uniformly convex and 2-uniformly smooth Banach space X: yn = αnf (yn) + (1 − αn) SnG (xn) , (14). We propose another composite explicit iterative algorithm in a uniformly convex Banach space X with a uniformly Gateaux differentiable norm: yn = αnG (xn) + (1 − αn) SnG (xn) , (16). The results presented in this paper improve, extend, supplement, and develop the corresponding results announced in the earlier and very recent literatures

Preliminaries

Implicit Iterative Schemes

Explicit Iterative Schemes