Gap functions and Hausdorff continuity of solution mappings to parametric strong vector quasiequilibrium problems

Lam Quoc Anh,Nguyen Van Hung

doi:10.3934/jimo.2017037

Abstract

In this paper, we consider parametric strong vector quasiequilibrium problems in Hausdorff topological vector spaces. Firstly, we introduce parametric gap functions for these problems, and study the continuity property of these functions. Next, we present two key hypotheses related to the gap functions for the considered problems and also study characterizations of these hypotheses. Then, afterwards, we prove that these hypotheses are not only sufficient but also necessary for the Hausdorff lower semicontinuity and Hausdorff continuity of solution mappings to these problems. Finally, as applications, we derive several results on Hausdorff (lower) continuity properties of the solution mappings in the special cases of variational inequalities of the Minty type and the Stampacchia type.

Highlights

One of the classes of problems in optimization, which has attracted attention of mathematicians all over the world, is the class of equilibrium problem
Answering an open question put forward in [21], Zhong and Huang [51] proved that the hypothesis (Hg) was a sufficient and necessary condition for Hausdorff lower semicontinuity of solution mapping to the set-valued weak vector variational inequality in Banach spaces
Similar to the above mentioned papers, we present two key hypotheses related to the gap functions of the problems and study some characterizations of these hypotheses

Summary

Introduction

One of the classes of problems in optimization, which has attracted attention of mathematicians all over the world, is the class of equilibrium problem. In 1997, Zhao [50] introduced the key hypothesis (H1) related to a gap function of optimization problem and showed that (H1) was a sufficient condition for the Hausdorff lower semicontinuity of the solution mapping to the parametric nonlinear optimization problem. Answering an open question put forward in [21], Zhong and Huang [51] proved that the hypothesis (Hg) was a sufficient and necessary condition for Hausdorff lower semicontinuity of solution mapping to the set-valued weak vector variational inequality in Banach spaces. We establish that conditions (Hp(γ0)) and (Hh(γ0)) are sufficient and necessary for the Hausdorff lower semicontinuity and Hausdorff continuity of the solution mappings to (QEP1) and (QEP2).

It follows from the direct computations that

Let y