1. Introduction

Let be a Hausdorff topological vector space, a nonempty subset of , and a function. Then, the scalar equilibrium problem consists in finding such that

(1.1)

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.

Now, let be another Hausdorff topological vector space and a closed convex cone with , where denotes the topological interior of . Let be a given map. Recently, Ansari et al. [2] studied the following vector equilibrium problems to find such that

(1.2)

or to find such that

(1.3)

In case that the map is multivalued, Ansari et al. [2] also studied the following multivalued vector equilibrium problem (for short, MVEP) to find such that

(1.4)

or to find such that

(1.5)

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

Very recently, in [3, 4], the authors studied the following implicit vector equilibrium problems (for short, IVEP): to find such that

(1.6)

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.

In this paper, we consider the following implicit multivalued vector equilibrium problem (for short, IMVEP) to find such that

(2.1)

We call this a solution for IMVEP.

Some special cases of IMVEP.

(1)If is a single-valued map, then IMVEP reduces to the problem of finding such that

(2.2)

which has been studied in [5].

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

(3)If is a single-valued map, , then IMVEP reduces to the problem of finding such that

(2.3)

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.

Let and be topological vector spaces, a nonempty convex subset of and a nonempty convex cone of . A multivalued map is said to be -convex if, for any and , one has

(2.4)

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

(i)-pseudomonotone with respect to and on if, for any and any , , one has

(2.5)

(ii)weakly -pseudomonotone with respect to and on if, for any and any , one has

(2.6)

(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

(2.7)

Remark 2.4.

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

Example 2.5.

Let , , and for all . Let , , and be defined as follows:

(2.8)

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 .

Indeed, for any , , and , it is obvious that . If

(2.9)

then, , that is, . It follows that

(2.10)

Hence is -pseudomonotone with respect to and on .

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

Indeed, let . Taking , for any open set with , that is,

(2.11)

there exists such that

(2.12)

Let , for all . Clearly, .

If , then for any and any , we have

(2.13)

And so

(2.14)

If , taking , then for any and , we have

(2.15)

It follows that

(2.16)

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).

Proof.

  1. (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

(3.1)

Let , for all . Since is convex, we have . Then, there exists such that

(3.2)

Since for any and any , is -convex, we have

(3.3)

Notice that for all and any , . Then, we can obtain that

(3.4)

Indeed, suppose to the contrary, that for some , then

(3.5)

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

We claim that there exists such that

(3.6)

and so is a solution of Problem (I).

In fact, if it is not the case, then we have, for any , . And then it follows from the fact that is -hemicontinuous with respect to and on that there exists and such that and

(3.7)

(3.7) contradicts (3.4), and so (3.6) holds.

  1. (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 ;

  1. (v)

    there exists a nonempty, compact and closed subset and such that, for all , one has

(3.8)

Then, IMVEP is solvable, that is, there exists such that

(3.9)

Proof.

Define two multivalued maps as follows, for any ,

(3.10)

The proof is divided into the following steps.

  1. (I)

    For all , .

Indeed, let , then there exists such that

(3.11)

If , then there exists such that

(3.12)

Since is -pseudomonotone with respect to and on , then, by (3.12), we have for all ,

(3.13)

which contradicts (3.11). Thus, , and so .

  1. (II)

    For all , is convex.

In fact, for any and , by the definition of , we have, for each ,

(3.14)

Let . Since is convex, we have . Noting that is -convex, we have

(3.15)

By the arbitrary of , we have , and so is convex.

  1. (III)

    For any , is compactly open.

Indeed, for any given compact subset , let . We will show that is open in by proving that is closed in . Let be an arbitrary net such that . Then, for each , , that is, , thus, for any ,

(3.16)

It follows that, for each , there exists such that , that is,

(3.17)

Since is continuous in the first variable and is u.s.c. and compact-valued, it follows from Lemma 2.6 that there exists and a subnet of , we still denote this subnet by , such that . Notice that is closed. We have . It follows that , and so

(3.18)

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

(3.19)

This implies that , that is, . And thus .

  1. (V)

    has no fixed point in .

Suppose that it is not the case, then there exists such that , that is,

(3.20)

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.

Since has no fixed point in , it follows from Lemma 2.7 that there exists such that , that is,

(3.21)

From Lemma 3.1, we have such that

(3.22)

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 map**.

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

Let , for any set with .

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

Taking , then for any and , we have

(3.23)

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

(3.24)

Taking and , we have

(3.25)

It follows that for any and any , then we have . Thus,

(3.26)

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

Hence, is u.s.c. on .

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

For any and . Let for each and . Since

(3.27)

it follows that

(3.28)

Thus, we have

(3.29)

This shows that is -convex.

(IV)Obviously, for any and any ,

(3.30)

that is, for any and , .

For any and , we have

(3.31)

thus, .

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

Indeed, let , then for any , there exists such that

(3.32)

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 .

Then, IMVEP is solvable, that is, there exists such that

(3.33)

Proof.

Define two multivalued maps as follows, for any ,

(3.34)

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.

Indeed, for any given compact subset , let . We will show that is open in by proving that is closed in . Let be an arbitrary net such that . Then, for each , , that is, . Thus, for each , there exists such that

(3.35)

Since is compact valued, there exists a subnet of such that . Then, by virtue of (3.35), for each , there exists such that , that is,

(3.36)

Since is continuous in the second variable and is continuous in the first variable, we have

(3.37)

In addition, is u.s.c. and compact valued, it follows from Lemma 2.6 that there exist and a subnet of , we still denote this subnet by , such that . Notice that is closed. We have . It follows that , and so

(3.38)

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

Hence, as in the proof of Theorem 3.2, there exists such that , that is,

(3.39)

From Lemma 3.1, we have such that

(3.40)

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.

Then, IMVEP is solvable, that is, there exists such that

(3.41)

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.

Indeed, for any given compact subset , let . We will show that is open in by proving that is closed in . Let be an arbitrary net such that . Then, for each , , that is, . Thus, for each , there exists such that

(3.42)

Since is u.s.c. and compact-valued, it follow from Lemma 2.6 that there exists and a subnet such that . Then, by the assumption (i), we have

(3.43)

Furthermore, by virtue of (3.42), for each , there exists some such that . It follows that

(3.44)

Since is u.s.c. and compact-valued, it follows from Lemma 2.6 that there exist and a subnet of , we still denote this subnet by such that . Notice that is closed. We have , that is, , and so

(3.45)

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

Thus, as in the proof of Theorem 3.2, there exists such that , that is,

(3.46)

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 .

Let

(4.1)

Let be two compact sets in a normed space . Recall the Hausdorff metric defined by

(4.2)

For any given , let

(4.3)

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

Assume that the multivalued maps and satisfy all the conditions of Theorem 3.2. Then, it follows from Theorem 3.8 that for any , IMVEP has a solution, that is, there exists such that

(4.4)

Let

(4.5)

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 .

Indeed, for any and , since , there exists such that

(4.6)

Notice that and are compact-valued. Then, for any open set with , there exists such that

(4.7)

where . Since and is u.s.c., there exists such that, for all ,

(4.8)
(4.9)

It follows from (4.8) that

(4.10)

By (4.10), (4.9), and (4.7), for all , we have

(4.11)

Since and for all , , by Lemma 2.9, there exists a subsequence of such that

(4.12)

Since is continuous, is continuous in the first variable, and , we have

(4.13)

Then, it follows from (4.6) that

(4.14)

Thus, there exist such that , that is, . Since is u.s.c. and compact valued, there exist and a subsequence of , we still denote this subsequence by , such that . Notice that is closed. We have , that is, , and so

(4.15)

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.