Solution of variational inequality problems on fixed point sets of nonexpansive mappings using iterative methods

TL;DR

This paper develops new iterative methods for solving variational inequality problems on fixed point sets of nonexpansive mappings in Banach spaces, extending existing results and providing strong convergence guarantees.

Contribution

Introduction of novel implicit and explicit iterative schemes with strong convergence for variational inequalities in Banach spaces, unifying and generalizing prior results.

Findings

New iterative schemes converge strongly to unique solutions.

Methods extend to common fixed points of pseudocontractive mappings.

Results improve and unify previous theorems by Xu and Yamada.

Abstract

In this paper, we introduce new implicit and explicit iterative schemes which converge strongly to a unique solution of variational inequality problems for strongly accretive operators over a common fixed point set of finite family of nonexpansive mappings in $q$-uniformly smooth real Banach spaces. As an application, we introduce an iteration process which converges strongly to a solution of the variational inequality which is a common fixed point of finite family of strictly pseudocontractive mappings. Our theorems extend, generalize, improve and unify the corresponding results of Xu \cite{27} and Yamada \cite{Yamada} and that of a host of other authors. Our corollaries and our method of proof are of independent interest.