Upper Semicontinuity of Solution Mappings to Parametric Generalized Vector Quasiequilibrium Problems

Abstract

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.

1. Introduction

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 [1, 2]. In recent years, the existence of solutions for various types of vector equilibrium problems has been investigated intensively by many authors under different conditions (see, e.g., [3–8] and the references therein).

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 [9–17]. Recently, Anh and Khanh [18] obtained the semicontinuity of the solution mapping to parametric vector quasiequilibrium problems. Khanh and Luu [19] discussed the upper semicontinuity of solution mapping to parametric vector quasivariational inequalities involving multifunctions without monotonicity assumptions. Fang and Huang [20] established upper semicontinuity of the solution maps to the vector homogeneous quasiequilibrium problems. By using a scalarization method, Cheng and Zhu [21] investigated the upper semicontinuity and lower semicontinuity of the solution mapping to a parametric weak vector variational inequality in finite-dimension Euclidean spaces. Li and Fang [22] studied the lower semicontinuity of the solution mappings to a parametric generalized Ky Fan inequality by using a key assumption that includes the information about the solutions set. By virtue of a density result and scalarization technique, Gong and Yao [23] first discussed the lower semicontinuity of the set of efficient solutions to parametric vector equilibrium problems. Li et al. [24] investigated the upper semicontinuity and lower semicontinuity of the solution mappings to a parametric generalized strong vector equilibrium problem. By using the Hölder relation, Zhang et al. [25] obtained the lower semicontinuity of the efficient solution mapping to a parametric vector equilibrium problem. Fan et al. [26] studied the continuity of the solution mapping concerned with a class of vector quasiequilibrium problems with an application to traffic network problems. Xu and Li [27] established the lower semicontinuity of solution mappings to a parametric generalized strong vector equilibrium problem by using a scalarization method. Very recently, by using a new proof method which is different from the ones used in the literature, Han and Gong [28] established the lower semicontinuity of the solution mappings to parametric generalized strong vector equilibrium problems without the assumptions of monotonicity and compactness.

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 2, we recall some basic concepts and known lemmas. In Section 3, we show a main result in connection with the upper semicontinuity of the solution mapping for the parametric generalized vector quasiequilibrium problem. Some applications of the main result are given in Section 4.

2. Preliminaries

Throughout this paper, unless otherwise specified, let , , , and be four normed vector spaces. Let be a nonempty closed subset of . Let and be two set-valued mappings. For , we consider that the following parametric generalized vector quasiequilibrium problems consist of finding such that

We define a solution mapping to by

Definition 1 (see [29]). Let and be two topological vector spaces. A set-valued mapping is said to be (i)upper semicontinuous (u.s.c.) at if, for any neighborhood of , there exists a neighborhood of such that, for every , ;(ii)lower semicontinuous (l.s.c.) at if, for any and any neighborhood of , there exists a neighborhood of such that, for every , .A set-valued mapping is said to be u.s.c. and l.s.c. on , if it is u.s.c. and l.s.c. at each , respectively. We say that is continuous on , if it is both u.s.c and l.s.c on .

Definition 2 (see [15]). Let and be two topological vector spaces and let be a cone. A set-valued mapping is said to be -lower semicontinuous (-l.s.c.) at if, for any and any neighborhood of , there exists a neighborhood of such that A set-valued mapping is said to be -l.s.c. on , if it is -l.s.c. at each .

Remark 3. It is easy to see that if is l.s.c. at , then it is -l.s.c. at . In fact, since , we have . It follows from that The following example shows that the reverse is not true in general.

Example 4. Let . We define a set-valued mapping as follows: It is easy to see that is -l.s.c. at , but is not l.s.c. at .

Lemma 5 (see [29]). A set-valued mapping is l.s.c. at if and only if, for any sequence with and for any , there exists such that .

Lemma 6 (see [30]). Assume that is a set-valued mapping. If is compact for some , then is u.s.c. at if and only if, for any sequence with and for any , there exist and a subsequence of such that .

3. Upper Semicontinuity of Solution Mapping to

In this section, we establish the upper semicontinuity of at .

Theorem 7. Let . Assume that is nonempty compact, is continuous at , and is l.s.c. on . Then is u.s.c. at .

Proof. Suppose to the contrary that is not u.s.c. at . Then there exists a neighborhood of ; for any neighborhood of , there exists such that . Hence, there exists a sequence with such that Then there exists such that Since is u.s.c. at and , by Lemma 6, there exist and a subsequence of such that . Without loss of generality, we can assume that .
We claim that . In fact, suppose that . Then there exists such that Hence there exists such that Since is l.s.c. at and , by Lemma 5, there exists such that . Since is l.s.c. at and , by Lemma 5, there exists such that . Noting (9) and that is closed, we know that for large enough, which contradicts with (6). Therefore, . It is easy to see that , which contradicts with (7).

Remark 8. We would like to point out that the assumptions of Theorem 7 are quite natural and easy to be verified.

We give an example to illustrate Theorem 7.

Example 9. Let , , and . Let be the closed unit ball of . Assume that is defined by where We define a set-valued mapping as follows: Then it is easy to see that and so . Moreover, it is easy to check that all the assumptions of Theorem 7 are satisfied. Thus, it follows from Theorem 7 that is u.s.c. at .

4. Some Applications

In this section, we give some applications of Theorem 7 to the optimization problem, the saddle point problem, the Nash equilibria problem, the variational inequality, the variational inequality with set-valued mappings, the equilibrium problem, the generalized strong vector equilibrium problem, and the generalized weak vector equilibrium problem.

Optimization Problem. Let be a mapping. Let be a nonempty subset of . A point is called a solution of optimization problem if and only if Let be a mapping and let be a set-valued mapping. For , we consider that the following parametric optimization problem consists of finding such that

Define a solution mapping to by

From Theorem 7, we can get the following corollary.

Corollary 10. Let . Assume that is nonempty compact, is continuous at , and is continuous on . Then the mapping is u.s.c. at .

Saddle Point Problem. Let be a mapping. Let be a nonempty subset of . is called a saddle point on if and only if Let be a mapping and let be a set-valued mapping. For , we consider that the following parametric saddle point problem consists of finding such that

Define a solution mapping to by

From Theorem 7, we can get the following corollary.

Corollary 11. Let . Assume that is nonempty compact, is continuous at , and is continuous on . Then the mapping is u.s.c. at .

Nash Equilibria Problem. Let be a finite index set. For every , let be a nonempty subset of and let be a mapping. Let . For , we define . A point is called a Nash equilibrium if and only if, for any , we have

For every , let be a mapping and let be a set-valued mapping. Let for . For , we consider that the following parametric Nash equilibria problem consists of finding such that

Define a solution mapping to by

From Theorem 7, we can get the following corollary.

Corollary 12. Let . For every , assume that is nonempty compact, is continuous at , and is continuous on . Then the mapping is u.s.c. at .

Variational Inequality. Let be the topological dual space of . Let be a mapping. Let be a nonempty subset of . A point is called a solution of variational inequality if and only if Let be a mapping and let be a set-valued mapping. For , we consider that the following parametric variational inequality consists of finding such that

Define a solution mapping to by

From Theorem 7, we can get the following corollary.

Corollary 13. Let . Assume that is nonempty compact, is continuous at , and is continuous on . Then the mapping is u.s.c. at .

Variational Inequality with Set-Valued Mappings. Let be the topological dual space of . Let be a set-valued mapping. Let be a nonempty subset of . A point is called a solution of variational inequality with set-valued mappings if and only if Let and be two set-valued mappings. For , we consider that the following parametric variational inequality with set-valued mappings consists of finding such that

Define a solution mapping to by

The proof of the following corollary is similar to the proof of Theorem 7. For the convenience of the readers, we also give the proof.

Corollary 14. Let . Assume that is nonempty compact, is continuous at , and is u.s.c. on with nonempty compact values. Then the mapping is u.s.c. at .

Proof. Suppose to the contrary that is not u.s.c. at . Then there exists a neighborhood of ; for any neighborhood of , there exists such that . Hence, there exists a sequence with such that Then there exists such that Since is u.s.c. at and , by Lemma 6, there exist and a subsequence of such that . Without loss of generality, we can assume that .
By (24), there exists such that Since , by Lemma 6, there exist and a subsequence of such that . Without loss of generality, we can assume that .
For any , by Lemma 5, there exists such that . Noting (26), we have It follows from that Therefore, . We can see that , which contradicts with (25).

Equilibrium Problem. Let be a mapping. Let be a nonempty subset of . A point is called a solution of equilibrium problem if and only if Let and be two set-valued mappings. For , we consider that the following parametric equilibrium problem consists of finding such that

Define a solution mapping to by

From Theorem 7, we can get the following corollary.

Corollary 15. Let . Assume that is nonempty compact, is continuous at , and is continuous on . Then the mapping is u.s.c. at .

Generalized Strong Vector Equilibrium Problem. Assume that is a closed cone. Let be a set-valued mapping. Let be a nonempty subset of . A point is called a solution of generalized strong vector equilibrium problem if and only if Let and be two set-valued mappings. For , we consider that the following parametric generalized strong vector equilibrium problem consists of finding such that

Define a solution mapping to by

From Theorem 7, we can get the following corollary.

Corollary 16. Let . Assume that is nonempty compact, is continuous at , and is l.s.c. on . Then the mapping is u.s.c. at .

Generalized Weak Vector Equilibrium Problem. Assume that is a cone with nonempty interior. Let be a set-valued mapping. Let be a nonempty subset of . A point is called a solution of generalized weak vector equilibrium problem if and only if Let and be two set-valued mappings. For , we consider that the following parametric generalized weak vector equilibrium problem consists of finding such that

Define a solution mapping to by

From Theorem 7, we can get the following corollary.

Corollary 17. Let . Assume that is nonempty compact, is continuous at , and is l.s.c. on . Then the mapping is u.s.c. at .

Remark 18. Corollary 17 is similar to Theorem 3.4 of [18].

From Corollary 17, it is easy to get the following corollary.

Corollary 19. Let . Assume that is nonempty compact, is continuous at , and is -l.s.c. on . Then is u.s.c. at .

Remark 20. In the proof of upper semicontinuity of solution mapping, Corollary 19 improves Theorem 3.1 of [15].

Conflict of Interests

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

Acknowledgments

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).