Hybrid Viscosity Approaches to General Systems of Variational Inequalities with Hierarchical Fixed Point Problem Constraints in Banach Spaces

Lu-Chuan Ceng,Abdul Latif,Saleh A Al-Mezel

doi:10.1155/2014/945985

Abstract

The purpose of this paper is to introduce and analyze hybrid viscosity methods for a general system of variational inequalities (GSVI) with hierarchical fixed point problem constraint in the setting of real uniformly convex and 2-uniformly smooth Banach spaces. Here, the hybrid viscosity methods are based on Korpelevich’s extragradient method, viscosity approximation method, and hybrid steepest-descent method. We propose and consider hybrid implicit and explicit viscosity iterative algorithms for solving the GSVI with hierarchical fixed point problem constraint not only for a nonexpansive mapping but also for a countable family of nonexpansive mappings inX, respectively. We derive some strong convergence theorems under appropriate conditions. Our results extend, improve, supplement, and develop the recent results announced by many authors.

Highlights

Let X be a real Banach space whose dual space is denoted by X∗
The purpose of this paper is to introduce and analyze hybrid viscosity methods for a general system of variational inequalities (GSVI) with hierarchical fixed point problem constraint in the setting of real uniformly convex and 2-uniformly smooth Banach spaces
We propose and consider hybrid implicit and explicit viscosity iterative algorithms for solving the GSVI with hierarchical fixed point problem constraint for a nonexpansive mapping and for a countable family of nonexpansive mappings in X, respectively

Summary

Introduction

Let X be a real Banach space whose dual space is denoted by X∗. Let U = {x ∈ X : ‖x‖ = 1} denote the unit sphere of X.A Banach space X is said to be uniformly convex if, for each ε ∈ 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) ≠ 0, the set Fix(T) is a sunny nonexpansive retract of C.

Objectives

Results

Conclusion