Viscosity Approximation Methods with Errors and Strong Convergence Theorems for a Common Point of Pseudocontractive and Monotone Mappings: Solutions of Variational Inequality Problems

Yan Tang

doi:10.1155/2014/871026

Abstract

We introduce two proximal iterative algorithms with errors which converge strongly to the common solution of certain variational inequality problems for a finite family of pseudocontractive mappings and a finite family of monotone mappings. The strong convergence theorems are obtained under some mild conditions. Our theorems extend and unify some of the results that have been proposed for this class of nonlinear mappings.

Highlights

In many problems, for example, convex optimization, linear programming, monotone inclusions, elliptic differential equations, and variational inequalities, it is quite often to seek a proximal point of a given nonlinear problem
In this paper, motivated and inspired by the above results, we introduce two iterations with perturbations which converge strongly to a common element of the set of fixed points of a finite family of pseudocontractive mappings more general than nonexpansive mappings and the solution set of a finite family of monotone mappings more general than αinverse strongly monotone mappings or maximal monotone mappings
First we prove that {xn} is, PC are nonexpansive; we have that

Summary

Introduction

For example, convex optimization, linear programming, monotone inclusions, elliptic differential equations, and variational inequalities, it is quite often to seek a proximal point of a given nonlinear problem. Tang [22] introduced the following sequence and obtained the strong convergence theorems: m yn = λnxn + (1 − λn) ∑μiFirn xn, i=1 (11)

Results

Conclusion