- Research Article
- Open Access

# Existence and Stability of Solutions for Implicit Multivalued Vector Equilibrium Problems

- Sanhua Wang
^{1}Email author and - Qiuying Li
^{2}

**2011**:381218

https://doi.org/10.1155/2011/381218

© S.Wang and Q. Li. 2011

**Received:**31 October 2010**Accepted:**14 January 2011**Published:**24 January 2011

## Abstract

A class of implicit multivalued vector equilibrium problems is studied. By using the generalized Fan-Browder fixed point theorem, some existence results of solutions for the implicit multivalued vector equilibrium problems are obtained under some suitable assumptions. Moreover, a stability result of solutions for the implicit multivalued vector equilibrium problems is derived. These results extend and unify some recent results for implicit vector equilibrium problems, multivalued vector variational inequality problems, and vector variational inequality problems.

## Keywords

- Equilibrium Problem
- Existence Result
- Convex Cone
- Topological Vector Space
- Variational Inequality Problem

## 1. Introduction

This problem provides a unifying framework for many important problems, such as, optimization problems, variational inequality problems, complementary problems, minimal inequality problems, and fixed point problems, and has been widely applied to study the problems arising in economics, mechanics, and engineering science (see [1]). In recent years, lots of existence results concerning equilibrium problems and variational inequality problems have been established by many authors in different ways. For details, we refer the reader to [1–26] and the references therein.

By using an abstract monotonicity condition, they gave some existence theorems of solutions for MVEP.

where is a vector-valued map and is a multivalued map such that, for all , is a closed convex cone in with . By using the famous FKKM theorem and section theorem, they gave some existence results of solutions for IVEP.

Inspired and motivated by the research work mentioned above, in this paper, we consider a class of implicit multivalued vector equilibrium problems and introduce the concepts of -pseudomonotonicity and -hemicontinuity for multivalued maps. By using the fixed point theorem of Chowdhury and Tan [27], we obtain some existence results of solutions for the implicit multivalued vector equilibrium problems in the setting of topological vector spaces. Furthermore, we derive a stability result of solutions for the implicit multivalued vector equilibrium problems. These results extend and unify some recent results for implicit vector equilibrium problems, multivalued vector variational inequality problems, and vector variational inequality problems.

## 2. Preliminaries

Throughout this paper, unless otherwise specified, we suppose that , , and are topological vector spaces, and and are nonempty subsets. We also suppose that is a multivalued map such that, for any , is a proper, closed, and convex cone in with , , are vector-valued maps, and , are multivalued maps.

We call this a solution for IMVEP.

Some special cases of IMVEP.

which has been studied in [5].

(2)If is a single-valued map, , then IMVEP reduces to IVEP.

which has been studied in [6].

(4)If , ( is a closed convex cone in with ), then IMVEP reduces to MVEP.

Definition 2.1 (see[28]).

Let and be two topological spaces. A multivalued map is said to be

(i)upper semicontinuous (for short, u.s.c.) at if, for each open set in with , there exists an open neighborhood of such that for all ;

(ii)lower semicontinuous (for short, l.s.c.) at if, for each open set in with , there exists an open neighborhood of such that for all ;

(iii)closed if the graph is a closed subset of .

(iv)compact-valued if, for each , is a nonempty compact subset of .

Definition 2.2.

Definition 2.3.

Let , , and be topological vector spaces, a nonempty convex subset of , and a nonempty subset of . Let be a multivalued map such that, for any , is a proper, closed, and convex cone in with . Given two vector-valued maps , , and two multivalued maps , . Then, is said to be

(iii) -hemicontinuous with respect to and on if, for any , , and any , there exists such that, for any open set with , there exists such that

Remark 2.4.

The above -hemicontinuity for multivalued map is a generalization of -hemicontinuity for continuous linear operator.

Example 2.5.

Then, is -pseudomonotone with respect to and on . Moreover, is -hemicontinuous with respect to and on .

Proof.

Firstly, we show that is -pseudomonotone with respect to and on .

Hence is -pseudomonotone with respect to and on .

Secondly, we show that is -hemicontinuous with respect to and on .

Let , for all . Clearly, .

Thus, is -hemicontinuous with respect to and on .

Lemma 2.6 (see [29]).

Let and be two topological spaces and a multivalued map.

(i) is closed if and only if for any net with and any net such that with , one has .

(ii)If is compact valued, then is u.s.c. at if and only if for any net with and any net with , there exists and a subnet such that .

The following lemma, which is a generalized form of Fan-Browder fixed piont theorem [30, 31], is very important to establish our existence results of solutions for IMVEP.

Lemma 2.7 (see [27]).

Let be a nonempty convex subset of a topological vector space and be two multivalued maps such that

(i)for any , ;

(ii)for any , is convex;

(iii)for any , is compactly open (i.e., is open in for each nonempty compact subset of );

(iv)there exists a nonempty, closed, and compact subset and such that ;

(v)for any , .

Then, there exists such that .

Lemma 2.8 (see [28]).

Let and be two Hausdorff topological vector spaces and a multivalued map. If is closed and is compact, then is u.s.c., where and denotes the closure of the set .

Lemma 2.9 (see [32]).

Let be a metric space and be compact subsets. If, for any open set with , there exists such that for all , then any sequence , satisfying , for , has some subsequence which converges to some point of A.

## 3. Existence of Solutions for IMVEP

In this section, we will apply the generalized Fan-Browder fixed point theorem to establish some existence results of solutions for IMVEP. First of all, we have the following lemma.

Lemma 3.1.

Let , , and be topological vector spaces, and a nonempty convex subset of , and a nonempty subset of . Let be a multivalued map such that, for any , is a proper, closed, and convex cone in with . Given two vector-valued maps , , and two multivalued maps , . Consider the following problems.

(I)Find such that, , ;

(II)Find such that, , ;

(III)Find such that, , ;

Then,

(i)Problem (I) implies Problem (II) if is weakly -pseudomonotone with respect to and on , moreover, implies Problem (III) if is -pseudomonotone with respect to and on ;

(ii)Problem (II) implies Problem (I) if is -hemicontinuous with respect to and on and, for any and any , is -convex and ;

(iii)Problem (III) implies Problem (II).

- (i)
It follows from the weakly -pseudomonotone with respect to and on and -pseudomonotone with respect to and on , respectively.

(ii)Let be a solution of (II). Then, , such that

which contradicts (3.2), and so (3.4) holds.

and so is a solution of Problem (I).

- (iii)
is obvious.

This completes the proof.

Now, we are ready to prove some existence theorems for IMVEP under suitable pseudomonotonicity assumptions.

Theorem 3.2.

Let , , and be topological vector spaces, and a nonempty convex subset of and a nonempty subset of . Let be a multivalued map such that, for any , is a proper, closed, and convex cone in with . Given two maps , , and two multivalued maps , . Suppose the following conditions are satisfied:

(i) is continuous in the first variable;

(ii) is -pseudomonotone and -hemicontinuous with respect to and on ;

(iii)the multivalued map , defined by is closed;

(iv) is u.s.c. and compact-valued, and it satisfies the following conditions:

(a)for any and , is -convex and ;

(b)for any , there exists such that ;

- (v)
there exists a nonempty, compact and closed subset and such that, for all , one has

Proof.

- (I)
For all , .

- (II)
For all , is convex.

- (III)
For any , is compactly open.

Then, by the arbitrary of , we have , that is, . Since , we know that , and so is closed in .

(IV)By the assumption (v), there exists a nonempty, compact, and closed subset and such that, for all , we have

- (V)
has no fixed point in .

By the assumption (iv), we have for some , and it follows that . This implies that is an absorbing set in , which contradicts the assumption that is proper in . Therefore, has no fixed point.

This completes the proof.

Remark 3.3.

Theorem 3.2 is a multivalued extension of [7, Theorem 3].

Remark 3.4.

The condition (v) of Theorem 3.2 is satisfied automatically if is compact.

We now give an example to illustrate Theorem 3.2.

Example 3.5.

Let , and be as in Example 2.5. We will show that all conditions of Theorem 3.2 are satisfied.

(I)It follows from Example 2.5 that the condition (ii) of Theorem 3.2 is satisfied. And it is obvious that is compact valued and is a closed mapping.

(II)We will show that is u.s.c. on .

Let , for any set with .

(1)If , then , and then there exists such that .

(2)If , then , and then there exists such that

(3)If , the argument is similar to (2).

Hence, is u.s.c. on .

(III)We will show that for any and , is -convex.

This shows that is -convex.

that is, for any and , .

thus, .

By the above arguments, we know that all the conditions of Theorem 3.2 are satisfied. By Theorem 3.2, IMVEP is solvable.

Thus, is a solution of IMVEP.

We now obtain an existence theorem for IMVEP for weakly -pseudomonotone maps with respect to and under additional assumptions.

Theorem 3.6.

Let , , , , , , , , , and be as in Theorem 3.2. Assume that the conditions (iii)–(v) of Theorem 3.2 and the following conditions are satisfied:

is continuous in the second variable and is continuous in the first variable;

is compact-valued, weakly -pseudomonotone and -hemicontinuous with respect to and on .

Proof.

By using the same arguments as in the proof (I) of Theorem 3.2 and weakly -pseudomonotonicity with respect to and on of , we see that for any , .

We have already seen in the proof of Theorem 3.2 that for each , is convex and the multivalued map has no fixed point. Moreover, there exists a nonempty, compact, and closed subset and such that .

Next, we will show that, for each , is compactly open.

Thus, , that is, . Since , we know that , and so is closed in .

This completes the proof.

Remark 3.7.

Theorem 3.6 is a multivalued extension of [7, Theorem 4].

Next, we will prove an existence result for IMVEP without any kind of pseudomonotonicity assumption.

Theorem 3.8.

Let , , , , , , , , , and be as in Theorem 3.2. Assume that the conditions (iii)–(v) of Theorem 3.2 and the following conditions are satisfied:

is continuous in both variables and is continuous in the first variable;

is u.s.c. and compact-valued.

Proof.

Define a multivalued map as in the proof of Theorem 3.2. As we have seen in the proof in Theorem 3.2 that, for each , is convex and the multivalued map has no fixed point. Moreover, there exists a nonempty, compact, and closed subset and such that .

Now, we have only to show that, for any , is compactly open.

Thus, , that is, . Since , we know that , and so is closed in .

This completes the proof.

Remark 3.9.

From the proof of Theorem 3.8, we can see that, if is compact, then the condition can be replaced by the following condition is closed.

## 4. Stability of Solution Sets for IMVEP

In this section, we discuss the stability of solutions for IMVEP.

Throughout this section, let and be Banach spaces, and a topological vector space, a nonempty compact convex subset and a nonempty subset, and a multivalued map such that, for all , is a proper, closed convex cone in with .

where is the Hausdorff metric. Then, it is easy to verify that is a metric space.

Then , which implies that is a multivalued map from into .

Theorem 4.1.

is an upper semicontinuous map with nonempty compact values.

Proof.

Since is compact, it follows from Lemma 2.8 that we need only to show that is closed. Let and . We will show that .

This implies that , and so is closed.

This completes the proof.

## 5. Conclusions

In this paper, the existence and stability of solutions for a class of implicit multivalued vector equilibrium problems are studied. By using the generalized Fan-Browder fixed point theorem [27], some existence results of solutions for the implicit multivalued vector equilibrium problems are obtained under some suitable assumptions. These results generalize and extend some corresponding results of Ansari et al. [7]. Also, in Section 4 of this paper, a stability result of solutions for the implicit multivalued vector equilibrium problems is obtained. It is worth mentioning that, up till now, there is no paper to consider the stability of solutions for the implicit multivalued vector equilibrium problems. So, the stability result obtained in Section 4 of this paper is new and interesting.

## Declarations

### Acknowledgments

This paper was supported by the National Natural Science Foundation of China (11061023, 11071108), the Natural Science Foundation of Jiangxi Province (2010GZS0145), and the Youth Foundation of Jiangxi Educational Committee (GJJ10086).

## Authors’ Affiliations

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