- Research Article
- Open Access

# Boundedness and Nonemptiness of Solution Sets for Set-Valued Vector Equilibrium Problems with an Application

- Ren-You Zhong
^{1}, - Nan-Jing Huang
^{1}and - YeolJe Cho
^{2}Email author

**2011**:936428

https://doi.org/10.1155/2011/936428

© Ren-You Zhong et al. 2011

**Received: **25 October 2010

**Accepted: **19 January 2011

**Published: **26 January 2011

## Abstract

This paper is devoted to the characterizations of the boundedness and nonemptiness of solution sets for set-valued vector equilibrium problems in reflexive Banach spaces, when both the mapping and the constraint set are perturbed by different parameters. By using the properties of recession cones, several equivalent characterizations are given for the set-valued vector equilibrium problems to have nonempty and bounded solution sets. As an application, the stability of solution set for the set-valued vector equilibrium problem in a reflexive Banach space is also given. The results presented in this paper generalize and extend some known results in Fan and Zhong (2008), He (2007), and Zhong and Huang (2010).

## Keywords

- Variational Inequality
- Nonempty Closed Convex Subset
- Reflexive Banach Space
- Vector Variational Inequality
- Vector Equilibrium Problem

## 1. Introduction

We denote the solution sets of SVEP and DSVEP by and , respectively.

We denote the solution sets of SVEP and DSVEP by and , respectively.

In 1980, Giannessi [1] extended classical variational inequalities to the case of vector-valued functions. Meanwhile, vector variational inequalities have been researched quite extensively (see, e.g., [2]). Inspired by the study of vector variational inequalities, more general equilibrium problems [3] have been extended to the case of vector-valued bifunctions, known as vector equilibrium problems. It is well known that the vector equilibrium problem provides a unified model of several problems, for example, vector optimization, vector variational inequality, vector complementarity problem, and vector saddle point problem (see [4–9]). In recent years, the vector equilibrium problem has been intensively studied by many authors (see, e.g., [1–3, 10–26] and the references therein).

Among many desirable properties of the solution sets for vector equilibrium problems, stability analysis of solution set is of considerable interest (see, e.g, [27–33] and the references therein). Assuming that the barrier cone of has nonempty interior, McLinden [34] presented a comprehensive study of the stability of the solution set of the variational inequality, when the mapping is a maximal monotone set-valued mapping. Adly [35], Adly et al. [36], and Addi et al. [37] discussed the stability of the solution set of a so-called semicoercive variational inequality. He [38] studied the stability of variational inequality problem with either the mapping or the constraint set perturbed in reflexive Banach spaces. Recently, Fan and Zhong [39] extended the corresponding results of He [38] to the case that the perturbation was imposed on the mapping and the constraint set simultaneously. Very recently, Zhong and Huang [40] studied the stability analysis for a class of Minty mixed variational inequalities in reflexive Banach spaces, when both the mapping and the constraint set are perturbed. They got a stability result for the Minty mixed variational inequality with -pseudomonotone mapping in a reflexive Banach space, when both the mapping and the constraint set are perturbed by different parameters, which generalized and extended some known results in [38, 39].

Inspired and motivated by the works mentioned above, in this paper, we further study the characterizations of the boundedness and nonemptiness of solution sets for set-valued vector equilibrium problems in reflexive Banach spaces, when both the mapping and the constraint set are perturbed. We present several equivalent characterizations for the vector equilibrium problem to have nonempty and bounded solution set by using the properties of recession cones. As an application, we show the stability of the solution set for the set-valued vector equilibrium problem in a reflexive Banach space, when both the mapping and the constraint set are perturbed by different parameters. The results presented in this paper extend some corresponding results of Fan and Zhong [39], He [38], Zhong and Huang [40] from the variational inequality to the vector equilibrium problem.

The rest of the paper is organized as follows. In Section 2, we recall some concepts in convex analysis and present some basic results. In Section 3, we present several equivalent characterizations for the set-valued vector equilibrium problems to have nonempty and bounded solution sets. In Section 4, we give an application to the stability of the solution sets for the set-valued vector equilibrium problem.

## 2. Preliminaries

In this section, we introduce some basic notations and preliminary results.

Let be a reflexive Banach space and be a nonempty closed convex subset of . The symbols " " and " " are used to denote strong and weak convergence, respectively.

Definition 2.1.

We say is continuous at if it is both upper and lower semicontinuous at , and we say is continuous on if it is both upper and lower semicontinuous at every point of .

It is evident that is lower semicontinuous at if and only if, for any sequence with and , there exists a sequence with such that .

Definition 2.2.

A set-valued mapping is said to be weakly lower semicontinuous at if, for any and for any sequence with , there exists a sequence such that .

We say is weakly lower semicontinuous on if it is weakly lower semicontinuous at every point of . By Definition 2.2, we know that a weakly lower semicontinuous mapping is lower semicontinuous.

Definition 2.3.

We say that is -convex if is both upper -convex and lower -convex.

Definition 2.4.

Lemma 2.5 (see [36]).

Let be a nonempty closed convex subset of with . Then there exists no sequence such that and .

Lemma 2.6 (see [39]).

Let be a nonempty closed convex subset of with . Then there exists no sequence with each such that .

Lemma 2.7 (see [39]).

Let be a metric space and be a given point. Let be a set-valued mapping with nonempty values and let be upper semicontinuous at . Then there exists a neighborhood of such that for all .

Lemma 2.8 (see [41]).

## 3. Boundedness and Nonemptiness of Solution Sets

In this section, we present several equivalent characterizations for the set-valued vector equilibrium problem to have nonempty and bounded solution set. First of all, we give some assumptions which will be used for next theorems.

Let be a nonempty convex and closed subset of . Assume that is a set-valued mapping satisfying the following conditions:

( )for each , is weakly lower semicontinuous on ;

( )for each , the set is closed, here stands for the closed line segment joining and .

Remark 3.1.

Remark 3.2.

If, for each , the mapping is lower semicontinuous in , then condition is fulfilled. Indeed, for each and for any sequence with , we have and . By the lower semicontinuity of , for any , there exists such that . Since , we have and so by the closedness of . This implies that and the set is closed.

The following example shows that conditions can be satisfied.

Example 3.3.

which shows that is -convex on and so holds. Thus, satisfies all conditions .

Theorem 3.4.

Let be a nonempty closed convex subset of and be a set-valued mapping satisfying assumptions - . Then .

Proof.

This implies that and so . Letting , by assumption , we have . Thus, and . This completes the proof.

Theorem 3.5.

Proof.

Then this completes the proof.

Remark 3.6.

Thus, we know that Theorem 3.5 is a generalization of [40, Theorem 3.1]. Moreover, by [40, Remark 3.1], Theorem 3.5 is also a generalization of [38, Lemma 3.1].

Theorem 3.7.

- (i)
- (ii)
- (iii)
- (iv)

Proof.

The implications (i) (ii) and (ii) (iii) follow immediately from Theorems 3.4 and 3.5 and the definition of recession cone.

and is weakly lower semicontinuous, we know that and so . However, it contradicts the assumption that . Thus (iv) holds.

We first prove that is a closed subset of . Indeed, for any with , we have . It follows from the weakly lower semicontinuity of that . This shows that and so is closed.

which is a contradiction with (3.17). Thus we know that is a KKM mapping.

Indeed, by Lemma 2.8, intersection in (3.19) is nonempty. Moreover, if there exists some but , then by (iv), we have for some . Thus, and so , which is a contradiction to the choice of .

Let . Then by (3.19) and so . This shows that the collection has finite intersection property. For each , it follows from the weak compactness of that is nonempty, which coincides with the solution set of DSVEP .

Remark 3.8.

which was considered by Zhong and Huang [40]. Therefore, Theorem 3.7 is a generalization of [40, Theorem 3.2]. Moreover, by [40, Remark 3.2], Theorem 3.7 is also a generalization of Theorem 3.4 due to He [38].

Remark 3.9.

By using a asymptotic analysis methods, many authors studied the necessary and sufficient conditions for the nonemptiness and boundedness of the solution sets to variational inequalities, optimization problems, and equilibrium problems, we refer the reader to references [42–49] for more details.

## 4. An Application

As an application, in this section, we will establish the stability of solution set for the set-valued vector equilibrium problem when the mapping and the constraint set are perturbed by different parameters.

Let and be two metric spaces. is a set-valued mapping satisfying the following assumptions:

( )for each , , , implies that ;

( )for each , , , is -convex on ;

( )for each , and , for any sequences , and with , and , there exists a sequence with such that .

The following Theorem 4.1 plays an important role in proving our results.

Theorem 4.1.

Proof.

Assume that the conclusion does not hold, then there exist a sequence in with such that .

Since is cone, we can select a sequence with such that for every . As is reflexive, without loss of generality, we can assume that , as . Since is a continuous set-valued mapping, hence, is upper semicontinuous and lower semicontinuous at . From the upper semicontinuity of , by Lemma 2.7, we have as large enough and hence as large enough. Since is a closed convex cone and hence weakly closed. This implies that . Moreover, it follows from Lemma 2.6 that .

For any , and , from the lower semicontinuity of , there exists such that . Since , it follows that . Together with , from assumption , there exists such that . Since , we have and . Letting , we obtain that . Since and are arbitrary, from the above discussion, we obtain with . This contradicts our assumption that . This completes the proof.

Remark 4.2.

where is a set-valued mapping, is a proper, convex, lower semicontinuous function and , from Remark 3.6, we know that (4.1) and (4.2) in Theorem 4.1 reduce to (4.1) and (4.2) in [40, Theorem 4.1], respectively. Therefore, Theorem 4.1 is a generalization of [40, Theorem 4.1]. Moreover, by [40, Remark 4.1], Theorem 4.1 is also a generalization of [39, Theorem 3.1].

From Theorem 4.1, we derive the following stability result of the solution set for the vector equilibrium problem.

Theorem 4.3.

- (i)
- (ii)

Proof.

If is nonempty and bounded, then by Theorem 3.7 we have . It follows from Theorem 4.1 that there exists a neighborhood of , such that for every . By using Theorem 3.7 again, we have is nonempty and bounded for every . This verifies the first assertion.

Next, we prove the second assertion - . For any given sequence with , we need to prove that - . Let - . Then there exists a sequence with each such that weakly converges to . We claim that there exists such that . Indeed, if the claim does hold, then there exist that a subsequence of and some , such that , for all . This implies that and so , which contradicts with the upper semicontinuity of . Thus, we have the claim. Moreover, we obtain as is a closed convex subset of and hence weakly closed.

Now we prove for all and hence . For any and , from the lower semicontinuity of , there exist such that . Moreover, from assumption , there exists a sequence of elements such that . Since , we have and so . Letting , we obtain that . Since is arbitrary, we have . This yields that . Thus, have the second assertion. This completes the proof.

Remark 4.4.

which was considered by Zhong and Huang [40]. Therefore, Theorem 4.3 is a generalization of [40, Theorem 4.2]. Moreover, by [40, Remark 4.2], Theorem 4.3 ia also a generalizationof Theorems 4.1 and 4.4 due to He [38] and Theorem 3.5 due to Fan and Zhong [39].

The following examples show the necessity of the conditions of Theorem 4.3.

Example 4.5.

Example 4.6.

Example 4.7.

## Declarations

### Acknowledgments

The authors are grateful to the editor and reviewers for their valuable comments and suggestions. This work was supported by the Key Program of NSFC (Grant no. 70831005), the National Natural Science Foundation of China (10671135) and the Korea Research Foundation Grant funded by the Korean Government (KRF-2008-313-C00050).

## Authors’ Affiliations

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