Well‐Posedness by Perturbations of Generalized Mixed Variational Inequalities in Banach Spaces

Lu-Chuan Ceng,Ching-Feng Wen

doi:10.1155/2012/194509

Abstract

We consider an extension of the notion of well‐posedness by perturbations, introduced by Zolezzi (1995, 1996) for a minimization problem, to a class of generalized mixed variational inequalities in Banach spaces, which includes as a special case the class of mixed variational inequalities. We establish some metric characterizations of the well‐posedness by perturbations. On the other hand, it is also proven that, under suitable conditions, the well‐posedness by perturbations of a generalized mixed variational inequality is equivalent to the well‐posedness by perturbations of the corresponding inclusion problem and corresponding fixed point problem. Furthermore, we derive some conditions under which the well‐posedness by perturbations of a generalized mixed variational inequality is equivalent to the existence and uniqueness of its solution.

Highlights

Let X be a real Banach space and f : X → R ∪ { ∞} a real-valued functional on X
We consider an extension of the notion of well-posedness by perturbations, introduced by Zolezzi 1995, 1996 for a minimization problem, to a class of generalized mixed variational inequalities in Banach spaces, which includes as a special case the class of mixed variational inequalities
A minimization problem is said to be Tikhonov well-posed if it has a unique solution toward which every minimizing sequence of the problem converges

Summary

Introduction

Let X be a real Banach space and f : X → R ∪ { ∞} a real-valued functional on X. In the setting of Banach spaces, they established some metric characterizations and showed that the well-posedness by perturbations of a mixed variational inequality is closely related to the well-posedness by perturbations of the corresponding inclusion problem and corresponding fixed point problem. They derived some conditions under which the well-posedness by perturbations of the mixed variational inequality is equivalent to the existence and uniqueness of its solution. We derive some conditions under which the well-posedness by perturbations of the generalized mixed variational inequality is equivalent to the existence and uniqueness of its solution

Preliminaries

Well-Posedness by Perturbations and Metric Characterizations

Links with the Well-Posedness by Perturbations of Inclusion Problems

Links with the Well-Posedness by Perturbations of Fixed Point Problems