Viscosity iterative algorithm for variational inequality problems and fixed point problems in a real q-uniformly smooth Banach space

Gang Cai

doi:10.1186/s13663-015-0316-3

Abstract

The purpose of this paper is to study a viscosity iterative algorithm for finding a common element of the set of solutions of a general variational inequality problem for two inverse strongly accretive operators and the set of fixed points of a δ-strict pseudocontraction in a real q-uniformly smooth Banach space. Some strong convergence theorems are obtained under appropriate conditions. As an application, we prove some strong convergence theorems for fixed point problems and variational inequality problems or equilibrium problems in Hilbert spaces. These results improve and extend the corresponding results announced by many others.

Highlights

Let C be a subset of a real Banach space X and T be a mapping from C into itself
We recall that the generalized duality mapping Jq : X → X∗ is defined by
We introduce a viscosity iterative algorithm for finding a common element of the set of solutions of a general variational inequality ( . ) and the set of fixed points of a δ-strict pseudocontraction in a real q-uniformly smooth Banach space

Summary

Introduction

Let C be a subset of a real Banach space X and T be a mapping from C into itself. In what follows, we use F(T) to denote the set of fixed points of T. ([ ]) Let C be a nonempty convex subset of a real q-uniformly smooth Banach space X and T : C → C be a λ-strict pseudocontraction.

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