Strong convergence theorems for solutions of fixed point and variational inequality problems

Lanxiang Yu,Jianmin Song

doi:10.1186/1029-242x-2014-215

Abstract

The purpose of this paper is to investigate viscosity approximation methods for finding a common element in the set of fixed points of a strict pseudocontraction and in the set of solutions of a generalized variational inequality in the framework of Banach spaces.

Highlights

Let C be a nonempty, closed and convex subset of a real Hilbert space H, and let PC be the metric projection of H onto C
It is clear that variational inequality problem ( . ) is equivalent to a fixed point problem. u is a solution of the above inequality iff it is a fixed point of the mapping PC(I – rA), where I is the identity and r is some positive real number
Variational inequality problems have emerged as an effective and powerful tool for studying a wide class of problems which arise in economics, finance, image reconstruction, ecology, transportation, and network

Summary

Introduction

Let C be a nonempty, closed and convex subset of a real Hilbert space H, and let PC be the metric projection of H onto C. Many authors have investigated the problems of finding a common element in the set of solution of variational inequalities for an inverse-strongly monotone mapping and in the set of fixed points of nonexpansive mappings or strict pseudocontractions; see [ – ] and the references therein.

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Results

Conclusion