We establish the upper semicontinuity of solution mappings for a class of parametric generalized vector quasiequilibrium problems. As applications, we obtain the upper semicontinuity of solution mappings to several problems, such as parametric optimization problem, parametric saddle point problem, parametric Nash equilibria problem, parametric variational inequality, and parametric equilibrium problem.

It is well known that the vector equilibrium problem provides a unified model of several problems, such as the vector optimization problem, the vector saddle point problem, the vector complementarity problem, and the vector variational inequality problem [

On the other hand, the stability analysis of the solution mapping to vector equilibrium problems is an important topic in vector optimization theory. In recent years, the lower semicontinuity and the upper semicontinuity of of the solution mappings to parametric optimization problems, parametric vector variational inequalities, and parametric vector equilibrium problems have been intensively studied in the literature; for instance, we refer the reader to [

The aim of this paper is to establish the upper semicontinuity of solution mappings for a class of parametric generalized vector quasiequilibrium problems under some suitable conditions. We provide a uniform method to deal with the upper semicontinuity of solution mappings for several problems, such as parametric optimization problem, parametric saddle point problem, parametric Nash equilibria problem, parametric variational inequality, and parametric equilibrium problem. The rest of the paper is organized as follows. In Section

Throughout this paper, unless otherwise specified, let

We define a solution mapping

Let

upper semicontinuous (u.s.c.) at

lower semicontinuous (l.s.c.) at

Let

It is easy to see that if

Let

A set-valued mapping

Assume that

In this section, we establish the upper semicontinuity of

Let

Suppose to the contrary that

We claim that

We would like to point out that the assumptions of Theorem

We give an example to illustrate Theorem

Let

We define a set-valued mapping

In this section, we give some applications of Theorem

Define a solution mapping

From Theorem

Let

Define a solution mapping

From Theorem

Let

For every

Define a solution mapping

From Theorem

Let

Define a solution mapping

From Theorem

Let

Define a solution mapping

The proof of the following corollary is similar to the proof of Theorem

Let

Suppose to the contrary that

By (

For any

Define a solution mapping

From Theorem

Let

Define a solution mapping

From Theorem

Let

Define a solution mapping

From Theorem

Let

Corollary

From Corollary

Let

In the proof of upper semicontinuity of solution mapping, Corollary

The authors declare that there is no conflict of interests regarding the publication of this paper.

The authors are grateful to the editor and the referees for their valuable comments and suggestions. This work was supported by the National Natural Science Foundation of China (11171237, 11471230).