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.

Academic Editor:Mohamed-Aziz Taoudi

Department of Mathematics, Sichuan University, Chengdu, Sichuan 610064, China

Received 11 December 2014; Revised 16 March 2015; Accepted 25 March 2015; 18 May 2015

This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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 [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] be four normed vector spaces. Let [figure omitted; refer to PDF] be a nonempty closed subset of [figure omitted; refer to PDF] . Let [figure omitted; refer to PDF] and [figure omitted; refer to PDF] be two set-valued mappings. For [figure omitted; refer to PDF] , we consider that the following parametric generalized vector quasiequilibrium problems consist of finding [figure omitted; refer to PDF] such that [figure omitted; refer to PDF]

We define a solution mapping [figure omitted; refer to PDF] to [figure omitted; refer to PDF] by [figure omitted; refer to PDF]

Definition 1 (see [29]).

Let [figure omitted; refer to PDF] and [figure omitted; refer to PDF] be two topological vector spaces. A set-valued mapping [figure omitted; refer to PDF] is said to be

(i) upper semicontinuous (u.s.c.) at [figure omitted; refer to PDF] if, for any neighborhood [figure omitted; refer to PDF] of [figure omitted; refer to PDF] , there exists a neighborhood [figure omitted; refer to PDF] of [figure omitted; refer to PDF] such that, for every [figure omitted; refer to PDF] , [figure omitted; refer to PDF] ;

(ii) lower semicontinuous (l.s.c.) at [figure omitted; refer to PDF] if, for any [figure omitted; refer to PDF] and any neighborhood [figure omitted; refer to PDF] of [figure omitted; refer to PDF] , there exists a neighborhood [figure omitted; refer to PDF] of [figure omitted; refer to PDF] such that, for every [figure omitted; refer to PDF] , [figure omitted; refer to PDF] .

A set-valued mapping [figure omitted; refer to PDF] is said to be u.s.c. and l.s.c. on [figure omitted; refer to PDF] , if it is u.s.c. and l.s.c. at each [figure omitted; refer to PDF] , respectively. We say that [figure omitted; refer to PDF] is continuous on [figure omitted; refer to PDF] , if it is both u.s.c and l.s.c on [figure omitted; refer to PDF] .

Definition 2 (see [15]).

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

Remark 3.

It is easy to see that if [figure omitted; refer to PDF] is l.s.c. at [figure omitted; refer to PDF] , then it is [figure omitted; refer to PDF] -l.s.c. at [figure omitted; refer to PDF] . In fact, since [figure omitted; refer to PDF] , we have [figure omitted; refer to PDF] . It follows from [figure omitted; refer to PDF] that [figure omitted; refer to PDF] The following example shows that the reverse is not true in general.

Example 4.

Let [figure omitted; refer to PDF] . We define a set-valued mapping [figure omitted; refer to PDF] as follows: [figure omitted; refer to PDF] It is easy to see that [figure omitted; refer to PDF] is [figure omitted; refer to PDF] -l.s.c. at [figure omitted; refer to PDF] , but [figure omitted; refer to PDF] is not l.s.c. at [figure omitted; refer to PDF] .

Lemma 5 (see [29]).

A set-valued mapping [figure omitted; refer to PDF] is l.s.c. at [figure omitted; refer to PDF] if and only if, for any sequence [figure omitted; refer to PDF] with [figure omitted; refer to PDF] and for any [figure omitted; refer to PDF] , there exists [figure omitted; refer to PDF] such that [figure omitted; refer to PDF] .

Lemma 6 (see [30]).

Assume that [figure omitted; refer to PDF] is a set-valued mapping. If [figure omitted; refer to PDF] is compact for some [figure omitted; refer to PDF] , then [figure omitted; refer to PDF] is u.s.c. at [figure omitted; refer to PDF] if and only if, for any sequence [figure omitted; refer to PDF] with [figure omitted; refer to PDF] and for any [figure omitted; refer to PDF] , there exist [figure omitted; refer to PDF] and a subsequence [figure omitted; refer to PDF] of [figure omitted; refer to PDF] such that [figure omitted; refer to PDF] .

3. Upper Semicontinuity of Solution Mapping to [figure omitted; refer to PDF]

In this section, we establish the upper semicontinuity of [figure omitted; refer to PDF] at [figure omitted; refer to PDF] .

Theorem 7.

Let [figure omitted; refer to PDF] . Assume that [figure omitted; refer to PDF] is nonempty compact, [figure omitted; refer to PDF] is continuous at [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] is l.s.c. on [figure omitted; refer to PDF] . Then [figure omitted; refer to PDF] is u.s.c. at [figure omitted; refer to PDF] .

Proof.

Suppose to the contrary that [figure omitted; refer to PDF] is not u.s.c. at [figure omitted; refer to PDF] . Then there exists a neighborhood [figure omitted; refer to PDF] of [figure omitted; refer to PDF] ; for any neighborhood [figure omitted; refer to PDF] of [figure omitted; refer to PDF] , there exists [figure omitted; refer to PDF] such that [figure omitted; refer to PDF] . Hence, there exists a sequence [figure omitted; refer to PDF] with [figure omitted; refer to PDF] such that [figure omitted; refer to PDF] Then there exists [figure omitted; refer to PDF] such that [figure omitted; refer to PDF] Since [figure omitted; refer to PDF] is u.s.c. at [figure omitted; refer to PDF] and [figure omitted; refer to PDF] , by Lemma 6, there exist [figure omitted; refer to PDF] and a subsequence [figure omitted; refer to PDF] of [figure omitted; refer to PDF] such that [figure omitted; refer to PDF] . Without loss of generality, we can assume that [figure omitted; refer to PDF] .

We claim that [figure omitted; refer to PDF] . In fact, suppose that [figure omitted; refer to PDF] . Then there exists [figure omitted; refer to PDF] such that [figure omitted; refer to PDF] Hence there exists [figure omitted; refer to PDF] such that [figure omitted; refer to PDF] Since [figure omitted; refer to PDF] is l.s.c. at [figure omitted; refer to PDF] and [figure omitted; refer to PDF] , by Lemma 5, there exists [figure omitted; refer to PDF] such that [figure omitted; refer to PDF] . Since [figure omitted; refer to PDF] is l.s.c. at [figure omitted; refer to PDF] and [figure omitted; refer to PDF] , by Lemma 5, there exists [figure omitted; refer to PDF] such that [figure omitted; refer to PDF] . Noting (9) and that [figure omitted; refer to PDF] is closed, we know that [figure omitted; refer to PDF] for [figure omitted; refer to PDF] large enough, which contradicts with (6). Therefore, [figure omitted; refer to PDF] . It is easy to see that [figure omitted; refer to PDF] , 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 [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] . Let [figure omitted; refer to PDF] be the closed unit ball of [figure omitted; refer to PDF] . Assume that [figure omitted; refer to PDF] is defined by [figure omitted; refer to PDF] where [figure omitted; refer to PDF]

We define a set-valued mapping [figure omitted; refer to PDF] as follows: [figure omitted; refer to PDF] Then it is easy to see that [figure omitted; refer to PDF] and so [figure omitted; refer to PDF] . Moreover, it is easy to check that all the assumptions of Theorem 7 are satisfied. Thus, it follows from Theorem 7 that [figure omitted; refer to PDF] is u.s.c. at [figure omitted; refer to PDF] .

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 [figure omitted; refer to PDF] be a mapping. Let [figure omitted; refer to PDF] be a nonempty subset of [figure omitted; refer to PDF] . A point [figure omitted; refer to PDF] is called a solution of optimization problem if and only if [figure omitted; refer to PDF] Let [figure omitted; refer to PDF] be a mapping and let [figure omitted; refer to PDF] be a set-valued mapping. For [figure omitted; refer to PDF] , we consider that the following parametric optimization problem consists of finding [figure omitted; refer to PDF] such that [figure omitted; refer to PDF]

Define a solution mapping [figure omitted; refer to PDF] to [figure omitted; refer to PDF] by [figure omitted; refer to PDF]

From Theorem 7, we can get the following corollary.

Corollary 10.

Let [figure omitted; refer to PDF] . Assume that [figure omitted; refer to PDF] is nonempty compact, [figure omitted; refer to PDF] is continuous at [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] is continuous on [figure omitted; refer to PDF] . Then the mapping [figure omitted; refer to PDF] is u.s.c. at [figure omitted; refer to PDF] .

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

Define a solution mapping [figure omitted; refer to PDF] to [figure omitted; refer to PDF] by [figure omitted; refer to PDF]

From Theorem 7, we can get the following corollary.

Corollary 11.

Let [figure omitted; refer to PDF] . Assume that [figure omitted; refer to PDF] is nonempty compact, [figure omitted; refer to PDF] is continuous at [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] is continuous on [figure omitted; refer to PDF] . Then the mapping [figure omitted; refer to PDF] is u.s.c. at [figure omitted; refer to PDF] .

Nash Equilibria Problem. Let [figure omitted; refer to PDF] be a finite index set. For every [figure omitted; refer to PDF] , let [figure omitted; refer to PDF] be a nonempty subset of [figure omitted; refer to PDF] and let [figure omitted; refer to PDF] be a mapping. Let [figure omitted; refer to PDF] . For [figure omitted; refer to PDF] , we define [figure omitted; refer to PDF] . A point [figure omitted; refer to PDF] is called a Nash equilibrium if and only if, for any [figure omitted; refer to PDF] , we have [figure omitted; refer to PDF]

For every [figure omitted; refer to PDF] , let [figure omitted; refer to PDF] be a mapping and let [figure omitted; refer to PDF] be a set-valued mapping. Let [figure omitted; refer to PDF] for [figure omitted; refer to PDF] . For [figure omitted; refer to PDF] , we consider that the following parametric Nash equilibria problem consists of finding [figure omitted; refer to PDF] such that [figure omitted; refer to PDF]

Define a solution mapping [figure omitted; refer to PDF] to [figure omitted; refer to PDF] by [figure omitted; refer to PDF]

From Theorem 7, we can get the following corollary.

Corollary 12.

Let [figure omitted; refer to PDF] . For every [figure omitted; refer to PDF] , assume that [figure omitted; refer to PDF] is nonempty compact, [figure omitted; refer to PDF] is continuous at [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] is continuous on [figure omitted; refer to PDF] . Then the mapping [figure omitted; refer to PDF] is u.s.c. at [figure omitted; refer to PDF] .

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

Define a solution mapping [figure omitted; refer to PDF] to [figure omitted; refer to PDF] by [figure omitted; refer to PDF]

From Theorem 7, we can get the following corollary.

Corollary 13.

Let [figure omitted; refer to PDF] . Assume that [figure omitted; refer to PDF] is nonempty compact, [figure omitted; refer to PDF] is continuous at [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] is continuous on [figure omitted; refer to PDF] . Then the mapping [figure omitted; refer to PDF] is u.s.c. at [figure omitted; refer to PDF] .

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

Define a solution mapping [figure omitted; refer to PDF] to [figure omitted; refer to PDF] by [figure omitted; refer to PDF]

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 [figure omitted; refer to PDF] . Assume that [figure omitted; refer to PDF] is nonempty compact, [figure omitted; refer to PDF] is continuous at [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] is u.s.c. on [figure omitted; refer to PDF] with nonempty compact values. Then the mapping [figure omitted; refer to PDF] is u.s.c. at [figure omitted; refer to PDF] .

Proof.

Suppose to the contrary that [figure omitted; refer to PDF] is not u.s.c. at [figure omitted; refer to PDF] . Then there exists a neighborhood [figure omitted; refer to PDF] of [figure omitted; refer to PDF] ; for any neighborhood [figure omitted; refer to PDF] of [figure omitted; refer to PDF] , there exists [figure omitted; refer to PDF] such that [figure omitted; refer to PDF] . Hence, there exists a sequence [figure omitted; refer to PDF] with [figure omitted; refer to PDF] such that [figure omitted; refer to PDF] Then there exists [figure omitted; refer to PDF] such that [figure omitted; refer to PDF] Since [figure omitted; refer to PDF] is u.s.c. at [figure omitted; refer to PDF] and [figure omitted; refer to PDF] , by Lemma 6, there exist [figure omitted; refer to PDF] and a subsequence [figure omitted; refer to PDF] of [figure omitted; refer to PDF] such that [figure omitted; refer to PDF] . Without loss of generality, we can assume that [figure omitted; refer to PDF] .

By (24), there exists [figure omitted; refer to PDF] such that [figure omitted; refer to PDF] Since [figure omitted; refer to PDF] , by Lemma 6, there exist [figure omitted; refer to PDF] and a subsequence [figure omitted; refer to PDF] of [figure omitted; refer to PDF] such that [figure omitted; refer to PDF] . Without loss of generality, we can assume that [figure omitted; refer to PDF] .

For any [figure omitted; refer to PDF] , by Lemma 5, there exists [figure omitted; refer to PDF] such that [figure omitted; refer to PDF] . Noting (26), we have [figure omitted; refer to PDF] It follows from [figure omitted; refer to PDF] that [figure omitted; refer to PDF] Therefore, [figure omitted; refer to PDF] . We can see that [figure omitted; refer to PDF] , which contradicts with (25).

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

Define a solution mapping [figure omitted; refer to PDF] to [figure omitted; refer to PDF] by [figure omitted; refer to PDF]

From Theorem 7, we can get the following corollary.

Corollary 15.

Let [figure omitted; refer to PDF] . Assume that [figure omitted; refer to PDF] is nonempty compact, [figure omitted; refer to PDF] is continuous at [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] is continuous on [figure omitted; refer to PDF] . Then the mapping [figure omitted; refer to PDF] is u.s.c. at [figure omitted; refer to PDF] .

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

Define a solution mapping [figure omitted; refer to PDF] to [figure omitted; refer to PDF] by [figure omitted; refer to PDF]

From Theorem 7, we can get the following corollary.

Corollary 16.

Let [figure omitted; refer to PDF] . Assume that [figure omitted; refer to PDF] is nonempty compact, [figure omitted; refer to PDF] is continuous at [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] is l.s.c. on [figure omitted; refer to PDF] . Then the mapping [figure omitted; refer to PDF] is u.s.c. at [figure omitted; refer to PDF] .

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

Define a solution mapping [figure omitted; refer to PDF] to [figure omitted; refer to PDF] by [figure omitted; refer to PDF]

From Theorem 7, we can get the following corollary.

Corollary 17.

Let [figure omitted; refer to PDF] . Assume that [figure omitted; refer to PDF] is nonempty compact, [figure omitted; refer to PDF] is continuous at [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] is l.s.c. on [figure omitted; refer to PDF] . Then the mapping [figure omitted; refer to PDF] is u.s.c. at [figure omitted; refer to PDF] .

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 [figure omitted; refer to PDF] . Assume that [figure omitted; refer to PDF] is nonempty compact, [figure omitted; refer to PDF] is continuous at [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] is [figure omitted; refer to PDF] -l.s.c. on [figure omitted; refer to PDF] . Then [figure omitted; refer to PDF] is u.s.c. at [figure omitted; refer to PDF] .

Remark 20.

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

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).

Conflict of Interests

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

References

[1] G. Y. Chen, X. X. Huang, X. Q. Yang Vector Optimization: Set-valued and Variational Analysis , vol. 541, of Lecture Notes in Economics and Mathematical Systems, Springer, Berlin, Germany, 2005.

[2] F. Giannessi Vector Variational Inequalities and Vector Equilibria , Kluwer Academic Publishers, Dordrecht, The Netherlands, 2000.

[3] G. Y. Chen, "Existence of solutions for a vector variational inequality: an extension of the Hartmann-Stampacchia theorem," Journal of Optimization Theory and Applications , vol. 74, no. 3, pp. 445-456, 1992.

[4] Q. H. Ansari, I. V. Konnov, J. C. Yao, "Existence of a solution and variational principles for vector equilibrium problems," Journal of Optimization Theory and Applications , vol. 110, no. 3, pp. 481-492, 2001.

[5] Q. H. Ansari, "Existence of solutions of systems of generalized implicit vector quasi-equilibrium problems," Journal of Mathematical Analysis and Applications , vol. 341, no. 2, pp. 1271-1283, 2008.

[6] X. H. Gong, "Strong vector equilibrium problems," Journal of Global Optimization , vol. 36, no. 3, pp. 339-349, 2006.

[7] N.-J. Huang, Y.-P. Fang, "On vector variational inequalities in reflexive Banach spaces," Journal of Global Optimization , vol. 32, no. 4, pp. 495-505, 2005.

[8] N. J. Huang, J. Li, J. C. Yao, "Gap functions and existence of solutions for a system of vector equilibrium problems," Journal of Optimization Theory and Applications , vol. 133, no. 2, pp. 201-212, 2007.

[9] L. Q. Anh, P. Q. Khanh, "On the stability of the solution sets of general multivalued vector quasiequilibrium problems," Journal of Optimization Theory and Applications , vol. 135, no. 2, pp. 271-284, 2007.

[10] L. Q. Anh, P. Q. Khanh, "Continuity of solution maps of parametric quasiequilibrium problems," Journal of Global Optimization , vol. 46, no. 2, pp. 247-259, 2010.

[11] N. Dinh, M. A. Goberna, M. A. Lopez, "On the stability of the optimal value and the optimal set in optimization problems," Journal of Convex Analysis , vol. 19, no. 4, pp. 927-953, 2012.

[12] N. J. Huang, J. Li, H. B. Thompson, "Stability for parametric implicit vector equilibrium problems," Mathematical and Computer Modelling , vol. 43, no. 11-12, pp. 1267-1274, 2006.

[13] X. H. Gong, "Continuity of the solution set to parametric weak vector equilibrium problems," Journal of Optimization Theory and Applications , vol. 139, no. 1, pp. 35-46, 2008.

[14] C. R. Chen, S. J. Li, K. L. Teo, "Solution semicontinuity of parametric generalized vector equilibrium problems," Journal of Global Optimization , vol. 45, no. 2, pp. 309-318, 2009.

[15] B. Chen, N. J. Huang, "Continuity of the solution mapping to parametric generalized vector equilibrium problems," Journal of Global Optimization , vol. 56, no. 4, pp. 1515-1528, 2013.

[16] R. Lucchetti, F. Patrone, "Closure and upper semicontinuity results in mathematical programming, Nash and economic equilibria," Optimization , vol. 17, no. 5, pp. 619-628, 1986.

[17] T. Tanino, Y. Sawaragi, "Stability of nondominated solutions in multicriteria decision-making," Journal of Optimization Theory and Applications , vol. 30, no. 2, pp. 229-253, 1980.

[18] L. Q. Anh, P. Q. Khanh, "Semicontinuity of the solution set of parametric multivalued vector quasiequilibrium problems," Journal of Mathematical Analysis and Applications , vol. 294, no. 2, pp. 699-711, 2004.

[19] P. Q. Khanh, L. M. Luu, "Upper semicontinuity of the solution set to parametric vector quasivariational inequalities," Journal of Global Optimization , vol. 32, no. 4, pp. 569-580, 2005.

[20] Y.-P. Fang, N.-J. Huang, "Upper semicontinuity of the solution maps in homogeneous vector quasi-equilibrium problems," Optimization , vol. 55, no. 3, pp. 231-239, 2006.

[21] Y. H. Cheng, D. L. Zhu, "Global stability results for the weak vector variational inequality," Journal of Global Optimization , vol. 32, no. 4, pp. 543-550, 2005.

[22] S. J. Li, Z. M. Fang, "Lower semicontinuity of the solution mappings to a parametric generalized Ky Fan inequality," Journal of Optimization Theory and Applications , vol. 147, no. 3, pp. 507-515, 2010.

[23] X. H. Gong, J. C. Yao, "Lower semicontinuity of the set of efficient solutions for generalized systems," Journal of Optimization Theory and Applications , vol. 138, no. 2, pp. 197-205, 2008.

[24] S. J. Li, H. M. Liu, Y. Zhang, Z. M. Fang, "Continuity of the solution mappings to parametric generalized strong vector equilibrium problems," Journal of Global Optimization , vol. 55, no. 3, pp. 597-610, 2013.

[25] W. Y. Zhang, Z. M. Fang, Y. Zhang, "A note on the lower semicontinuity of efficient solutions for parametric vector equilibrium problems," Applied Mathematics Letters , vol. 26, no. 4, pp. 469-472, 2013.

[26] X. D. Fan, C. Z. Cheng, H. J. Wang, "Stability analysis for vector quasiequilibrium problems," Positivity , vol. 17, no. 3, pp. 365-379, 2013.

[27] Y. D. Xu, S. J. Li, "On the lower semicontinuity of the solution mappings to a parametric generalized strong vector equilibrium problem," Positivity , vol. 17, no. 2, pp. 341-353, 2013.

[28] Y. Han, X. H. Gong, "Lower semicontinuity of solution mapping to parametric generalized strong vector equilibrium problems," Applied Mathematics Letters , vol. 28, pp. 38-41, 2014.

[29] J.-P. Aubin, I. Ekeland Applied Nonlinear Analysis , John Wiley & Sons, New York, NY, USA, 1984.

[30] A. Göpfert, H. Riahi, C. Tammer, C. Zalinescu Variational methods in partially ordered spaces , Springer, New York, NY, USA, 2003.