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

Journal of Inequalities and Applications volume 2011, Article number: 936428 (2011) Cite this article

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).

1. Introduction

Let and be reflexive Banach spaces. Let be a nonempty closed convex subset of . Let be a set-valued mapping with nonempty values. Let be a closed convex pointed cone in with . The cone induces a partial ordering in , which was defined by . We consider the following set-valued vector equilibrium problem, denoted by SVEP, which consists in finding such that

(1.1)

It is well known that (1.1) is closely related to the following dual set-valued vector equilibrium problem, denoted by DSVEP, which consists in finding such that

(1.2)

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

Let and be two metric spaces. Suppose that a nonempty closed convex set is perturbed by a parameter , which varies over , that is, is a set-valued mapping with nonempty closed convex values. Assume that a set-valued mapping is perturbed by a parameter , which varies over , that is, . We consider a parametric set-valued vector equilibrium problem, denoted by SVEP, which consists in finding such that

(1.3)

Similarly, we consider the parameterized dual set-valued vector equilibrium problem, denoted by DSVEP, which consists in finding such that

(1.4)

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.

The barrier cone of , denoted by , is defined by

(2.1)

The recession cone of , denoted by , is defined by

(2.2)

It is known that for any given ,

(2.3)

We give some basic properties of recession cones in the following result which will be used in the sequel. Let be any family of nonempty sets in . Then

(2.4)

If, in addition, and each set is closed and convex, then we obtain an equality in the previous inclusion, that is,

(2.5)

Let be a proper convex and lower semicontinuous function. The recession function of is defined by

(2.6)

where is any point in . Then it follows that

(2.7)

The function turns out to be proper convex, lower semicontinuous and so weakly lower semicontinuous with the property that

(2.8)

Definition 2.1.

A set-valued mapping is said to be

  1. (i)

    upper semicontinuous at if, for any neighborhood of , there exists a neighborhood of such that

    (2.9)
  2. (ii)

    lower semicontinuous at if, for any and any neighborhood of , there exists a neighborhood of such that

    (2.10)

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.

A set-valued mapping is said to be

  1. (i)

    upper -convex on if for any and , ,

    (2.11)
  2. (ii)

    lower -convex on if for any and , ,

    (2.12)

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

Definition 2.4.

Let be a sequence of sets in . We define

(2.13)

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]).

Let be a nonempty convex subset of a Hausdorff topological vector space and be a set-valued mapping from into satisfying the following properties:

  1. (i)

    is a KKM mapping, that is, for every finite subset of , ;

  2. (ii)

    is closed in for every ;

  3. (iii)

    is compact in E for some .

Then .

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 , ;

()for each , implies that ;

()for each , is -convex on ;

()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.

If

(3.1)

where is a set-valued mapping, is a proper, convex, lower semicontinuous function and , then condition reduces to the following -pseudomonotonicity assumption which was used in [40]. (See [40, Definition  2.2(iii)] of [40]): for all in the graph,

(3.2)

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.

Let , , and . Let

(3.3)

It is obvious that holds. Since for each , and are lower semicontinuous on , by Remark 3.2, we known that conditions and hold. For each , if , then we have . This implies that

(3.4)

and so holds. Moreover, for each , and with , it is easy to verify that

(3.5)

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.

From the assumption , it is easy to see that . We now prove that . Let . Then for all , . Set , where . Clearly, . From the upper -convexity of , we have

(3.6)

Since , we obtain

(3.7)

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

Theorem 3.5.

Let be a nonempty closed convex subset of and be a set-valued mapping satisfying assumptions . If the solution set is nonempty, then

(3.8)

Proof.

From the proof of Theorem 3.4, we know that

(3.9)

Let . Then . By the assumptions and , we know that the set is nonempty closed and convex. It follows from (2.5) and Theorem 3.4 that

(3.10)

Then this completes the proof.

Remark 3.6.

If

(3.11)

where is a set-valued mapping, is a proper, convex, lower semicontinuous function and , then it follows from (3.8) and (2.8) that

(3.12)

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.

Let be a nonempty closed convex subset of and be a set-valued mapping satisfying assumptions . Suppose that . Then the following statements are equivalent:

  1. (i)

    the solution set of SVEP is nonempty and bounded;

  2. (ii)

    the solution set of DSVEP is nonempty and bounded;

  3. (iii)

    ;

  4. (iv)

    there exists a bounded set such that for every , there exists some such that .

Proof.

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

Now we prove that (iii) implies (iv). If (iv) does not hold, then there exists a sequence such that for each , and for every with . Without loss of generality, we may assume that weakly converges to . Then by the definition of the recession cone. Since , by Lemma 2.5, we know that . Let and be any fixed points. For sufficiently large, by the lower -convexity of ,

(3.13)

Since

(3.14)

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

Since (i) and (ii) are equivalent, it remains to prove that (iv) implies (ii). Let be a set-valued mapping defined by

(3.15)

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.

We next prove that is a KKM mapping from to . Suppose to the contrary that there exist with , and such that . Then

(3.16)

By assumption , we have

(3.17)

It follows from the upper -convexity of that

(3.18)

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

We may assume that is a bounded closed convex set (otherwise, consider the closed convex hull of instead of ). Let be finite number of points in and let . Then the reflexivity of the space yields that is weakly compact convex. Consider the set-valued mapping defined by for all . Then each is a weakly compact convex subset of and is a KKM mapping. We claim that

(3.19)

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.

Theorem 3.7 establishes the necessary and sufficient conditions for the vector equilibrium problem to have nonempty and bounded solution sets. If

(3.20)

where is a set-valued mapping, is a proper, convex, lower semicontinuous function and , then problem (1.2) reduces to the following Minty mixed variational inequality: finding such that

(3.21)

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 , , , ;

()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.

Let and be two metric spaces, and be given points. Let be a continuous set-valued mapping with nonempty closed convex values and . Suppose that is a set-valued mapping satisfying the assumptions . If

(4.1)

then there exists a neighborhood of such that

(4.2)

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.

If

(4.3)

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.

Let and be two metric spaces, and be given points. Let be a continuous set-valued mapping with nonempty closed convex values and . Suppose that is a set-valued mapping satisfying the assumptions -. If is nonempty and bounded, then

  1. (i)

    there exists a neighborhood of such that for every , is nonempty and bounded;

  2. (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.

If

(4.4)

where is a set-valued mapping, is a proper, convex, lower semicontinuous function and , then problem (1.4) reduces to the following parametric Minty mixed variational inequality: finding such that

(4.5)

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.

Let , , and ,

(4.6)

Note that is continuous on . However, is not lower semicontinuous at . Clearly, we have and for any . Thus,

(4.7)

Example 4.6.

Let , , and ,

(4.8)

Note that satisfies the assumptions , and is upper semicontinuous. However, is not lower semicontinuous at . Clearly, we have and for any . Thus,

(4.9)

Example 4.7.

Let , , , ,

(4.10)

Note that satisfies the assumptions and is lower semicontinuous. However, is not upper semicontinuous at . Clearly, we have and for any . Thus,

(4.11)

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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).

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Authors and Affiliations

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

    Ren-You Zhong & Nan-Jing Huang

  2. Department of Mathematics Education and the RINS, Gyeongsang National University, Chinju, 660-701, Republic of Korea

    YeolJe Cho

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  1. Ren-You Zhong
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Correspondence to YeolJe Cho.

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Zhong, RY., Huang, NJ. & Cho, Y. Boundedness and Nonemptiness of Solution Sets for Set-Valued Vector Equilibrium Problems with an Application. J Inequal Appl 2011, 936428 (2011). https://doi.org/10.1155/2011/936428

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