Existence Result of Generalized Vector Quasiequilibrium Problems in Locally -Convex Spaces

This paper deals with the generalized strong vector quasiequilibrium problems without convexity in locally -convex spaces. Using the Kakutani-Fan-Glicksberg fixed point theorem for upper semicontinuous set-valued mapping with nonempty closed acyclic values, the existence theorems for them are established. Moreover, we also discuss the closedness of strong solution set for the generalized strong vector quasiequilibrium problems.

Let be real topological vector space, and let be a nonempty closed convex subset of . Let be a bifunction, where is the set of real numbers. The equilibrium problem for is to find such that

Problem (1.1) was studied by Blum and Oettli [1]. The set of solution of (1.1) is denoted by . The equilibrium problem contains many important problems as special cases, including optimization, Nash equilibrium, complementarity, and fixed point problems (see [1–3] and the references therein). Recently, there has been an increasing interest in the study of vector equilibrium problems. Many results on the existence of solutions for vector variational inequalities and vector equilibrium problems have been established (see, e.g., [4–16]).

Let and be real topological vector spaces and a nonempty subset of . Let be a closed and convex cone in with , where denotes the topological interior of . For a bifunction , the vector equilibrium problem (for short, VEP) is to find such that

which is a unified model of several known problems, for instance, vector variational and variational-like inequality problems, vector complementarity problem, vector optimization problem, and vector saddle point problem; see, for example, [3, 8, 17, 18] and references therein. In 2003, Ansari and Yao [19] introduced vector quasiequilibrium problem (for short, VQEP) to find such that

where is a multivalued map with nonempty values.

Recently, Ansari et al. [4] considered a more general problem which contains VEP and generalized vector variational inequality problems as special cases. Let and be real locally convex Hausdorff space, a nonempty subset and a closed convex pointed cone. Let be a given set-valued mapping. Ansari et al. [4] introduced the following problems, to find such that

or to find such that

It is called generalized vector equilibrium problem (for short, GVEP), and it has been studied by many authors; see, for example, [20–22] and references therein. For other possible ways to generalize VEP, we refer to [23–25]. If is nonempty and satisfies (1.4), then we call a weak efficient solution for VEP, and if satisfies (1.5), then we call a strong solution for VEP. Moreover, they also proved an existence theorem for a strong vector equilibrium problem (1.5) (see [4]).

On the other hand, it is well known that a strong solution of vector equilibrium problem is an ideal solution; it is better than other solutions such as efficient solution, weak efficient solution, proper efficient solution, and supper efficient solution (see [12]). Thus, it is important to study the existence of strong solution and properties of the strong solution set. In 2003, Ansari and Flores-Bazán [26] considered the following generalized vector quasiequilibrium problem (for short, GVQEP): to find such that

Very recently, the generalized strong vector quasiequilibrium problem (in short, GSVQEP) is introduced by Hou et al. [27] and Long et al. [16]. Let , and be real locally convex Hausdorff topological vector spaces, and nonempty compact convex subsets, and a nonempty closed convex cone. Let , and be three set-valued mappings. They considered the GSVQEP, finding such that and

Moreover, they gave an existence theorem for a generalized strong vector quasiequilibrium problem without assuming that the dual of the ordering cone has a weak compact base.

Throughout this paper, motivated and inspired by Hou et al. [27], Long et al. [16], and Yuan [28], we will introduce and study the generalized vector quasiequilibrium problem on locally -convex Hausdorff topological vector spaces. Let , , and be real locally -convex Hausdorff topological vector spaces, and nonempty compact subsets, and a nonempty closed convex cone. We also suppose that , and are set-valued mappings.

The generalized vector quasiequilibrium problem of type (I) (GSVQEP I) is to find such that

The generalized vector quasiequilibrium problem of type (II) (GSVQEP II) is to find such that

We denote the set of all solution to the (GSVQEP I) and (GSVQEP II) by and , respectively. The main motivation of this paper is to prove the existence theorems of the generalized strong vector quasiequilibrium problems in locally -convex spaces, by using Kakutani-Fan-Glicksberg fixed point theorem for upper semicontinuous set-valued mapping with nonempty closed acyclic values, and the closedness of and . The results in this paper generalize, extend, and unify some well-known some existence theorems in the literature.

Let be the standard -dimensional simplex in with vertices . For any nonempty subset of , we denote by the convex hull of the vertices . The following definition was essentially given by Park and Kim [29].

Definition 2.1.

A generalised convex space, or say, a -convex space consists of a topological space , a nonempty subset of and a function such that

(i)for each if ,

(ii)for each with , there exists a continuous function such that for each , where and denotes the face of corresponding to the subindex of in .

A subset of the -convex space is said to be -convex if for each for all . For the convenience of our discussion, we also denote by or if there is no confusion for , where is the set of all indices for the set ; that is, . A space is said to have a -convex structure if and only if is a -convex space.

In order to cover general economic models without linear convex structures, Park and Kim [29] introduced another abstract convexity notion called a -convex space, which includes many abstract convexity notions such as -convex spaces as special cases. For the details on G-convex spaces, see [30–34], where basic theory was extensively developed.

Definition 2.2.

A -convex is said to be a locally -convex space if is a uniform topological space with uniformity , which has an open base of symmetric entourages such that for each , the set is a -convex set for each .

We recall that a nonempty space is said to be acyclic if all of its reduced Čech homology groups over the rationals vanish.

Definition 2.3 (see [35]).

Let be a topological space. A subset of is called contractible at , if there is a continuous mapping such that for all and for all .

In particular, each contractible space is acyclic and thus any nonempty convex or star-shaped set is acyclic. Moreover, by the definition of contractible set, we see that each convex space is contractible.

Definition 2.4.

Let and be two topological vector spaces and a nonempty subset of , and let be a set-valued mapping.

(i) is called upper -continuous at if, for any neighbourhood of the origin in , there is a neighbourhood of such that, for all ,

(ii) is called lower -continuous at if, for any neighbourhood of the origin in , there is a neighbourhood of such that for all ,

Definition 2.5.

Let and be two topological vector spaces and a nonempty convex subset of . A set-valued mapping is said to be properly -quasiconvex if, for any and , we have

Definition 2.6.

Let and be two topological vector spaces and a set-valued mapping.

(i) is said to be upper semicontinuous at if, for any open set containing , there exists an open set containing such that for all , ; is said to be upper semicontinuous on if it is upper semicontinuous at all .

(ii) is said to be lower semicontinuous at if, for any open set with , there exists an open set containing such that for all , ; is said to be lower semicontinuous on if it is lower semicontinuous at all .

(iii) is said to be continuous on if it is at the same time upper semicontinuous and lower semicontinuous on .

(iv) is said to be closed if the graph, , of , that is, , is a closed set in .

Lemma 2.7 (see [36]).

Let and be two Hausdorff topological vector spaces and a set-valued mapping. Then, the following properties hold:

(i)if is closed and is compact, then is upper semicontinuous, where and denotes the closure of the set ,

(ii)if is upper semicontinuous and for any is closed, then is closed,

(iii) is lower semicontinuous at if and only if for any and any net , there exists a net such that and .

We now have the following fixed point theorem in locally -convex spaces given by Yuan [28] which is a generalization of the Fan-Glickberg-type fixed point theorems for upper semicontinuous set-valued mapping with nonempty closed acyclic values given in several places (e.g., see Kirk and Shin [37], Park and Kim [29], and others in locally convex spaces).

Lemma 2.8 (see [28]).

Let be a compact locally -convex space and an upper semicontinuous set-valued mappings with nonempty closed acyclic values. Then, has a fixed point; that is, there exists an such that .

In this section, we apply the Kakutani-Fan-Glicksberg fixed point theorem for upper semicontinuous set-valued mapping with nonempty closed acyclic values to establish two existence theorems of strong solutions and obtain the closedness of the strong solutions set for generalized strong vector quasiequilibrium problem.

Theorem 3.1.

Let , , and be real locally -convex topological vector spaces, and nonempty compact subsets, and a nonempty closed convex cone. Let be a continuous set-valued mapping such that for any , the set is a nonempty closed contractible subset of . Let be an upper semicontinuous set-valued mapping with nonempty closed acyclic values and a set-valued mapping satisfy the following conditions:

(i)for all ,

(ii)for all are properly -quasiconvex,

(iii) are upper -continuous,

(iv)for all are lower -continuous.

Then, the solutions set is nonempty and closed subset of .

Proof.

For any , we define a set-valued mapping by

Since for any is nonempty. So, by assumption (i), we have that is nonempty. Next, we divide the proof into five steps.

Step 1 (to show that is acyclic).

Since every contractible set is acyclic, it is enough to show that is contractible. Let , thus and . Since is contractible, there exists a continuous mapping such that and . Now, we set for all . Then, is a continuous mapping, and we see that and for all . Let . We claim that . In fact, if , then there exists such that

Since is properly -quasiconvex, we can assume that

It follows that

which contradicts . Therefore, , and hence is contractible.

Step 2 (to show that is a closed subset of ).

Let be a sequence in such that . Then, . Since is a closed subset of , . Since is a lower semicontinuous, it follows by Lemma 2.7(iii) that for any and any net , there exists a net such that and . This implies that

Since are lower -continuous, we note that for any neighbourhood of the origin in , there exists a subnet of such that

From (3.5) and (3.6), we have

We claim that . Assume that there exists and . Then, we note that , and the set is closed. Thus, is open, and . Since is a locally -convex space, there exists a neighbourhood of the origin such that and . Thus, we note that , and hence , which contradicts to (3.7). Hence, , and therefore, . Then, is a closed subset of .

Step 3 (to show that is upper semicontinuous).

Let be given such that , and let such that . Since and is upper semicontinuous, it follows by Lemma 2.7(ii) that . We claim that . Assume that . Then, there exists such that

which implies that there is a neighbourhood of the origin in such that

Since is upper -continuous, it follows that for any neighbourhood of the origin in , there exists a neighbourhood of such that

Without loss of generality, we can assume that . This implies that

Thus, there is such that

it is a contradiction to . Hence, , and therefore, is a closed mapping. Since is a compact set and is a closed subset of , is compact. This implies that is compact. Then, by Lemma 2.7(i), we have is upper semicontinuous.

Step 4 (to show that the solutions set is nonempty).

Define the set-valued mapping by

Then, is an upper semicontinuous mpping. Moreover, we note that is a nonempty closed acyclic subset of for all . By Lemma 2.8, there exists a point such that . Thus, we have , . It follows that there exists and such that and

Hence, the solutions set .

Step 5 (to show that the solutions set is closed).

Let be a net in such that . By definition of the solutions set , we note that , and there exist satisfying

Since is a continuous closed valued mapping, . From the compactness of , we can assume that . Since is an upper semicontinuous closed valued mapping, it follows by Lemma 2.7(ii) that is closed. Thus, we have . Since is a lower -continuous, we have

This means that belongs to . Therefore, the solutions set is closed. This completes the proof.

Theorem 3.1 extends Theorem 3.1 of Long et al. [16] to locally -convex which includes locally convex Hausdorff topological vector spaces.

Corollary 3.2.

Let , and be real locally convex Hausdorff topological vector spaces, and two nonempty compact convex subsets, and a nonempty closed convex cone. Let be a continuous set-valued mapping such that for any , is a nonempty closed convex subset of . Let be an upper semicontinuous set-valued mapping such that for any , is a nonempty closed convex subset of . Let be a set-valued mapping satisfying the following conditions:

(i)for all ,

(ii)for all are properly -quasiconvex,

(iii) are upper -continuous,

(iv)for all are lower -continuous.

Then, the solutions set is nonempty and closed subset of .

Theorem 3.3.

Let , and be real locally -convex topological vector spaces, and nonempty compact subsets, and a nonempty closed convex cone. Let be a continuous set-valued mapping such that for any , the set is a nonempty closed contractible subset of . Let be an upper semicontinuous set-valued mapping with nonempty closed acyclic values and a set-valued mapping satisfying the following conditions:

(i)for all ,

(ii)for all are properly -quasiconvex,

(iii) are upper -continuous,

(iv)for all are lower -continuous.

Then, the solutions set is nonempty and closed subset of .

Proof.

For any , define a set-valued mapping by

Proceeding as in the proof of Theorem 3.1, we need to prove that is closed acyclic subset of for all . We divide the remainder of the proof into three steps.

Step 1 (to show that is a closed subset of ).

Let be a sequence in such that . Then, and . Since is a closed subset of , we have . By the lower semicontinuity of and Lemma 2.7(iii), we note that for any and any net , there exists a net such that and . Thus, we have

which implies that there exists a neighbourhood of the origin in such that

Since are lower -continuous, it follows that for any neighbourhood of the origin in , there exists a subnet of such that

Without loss of generality, we can assume that . Then, by (3.18), (3.19), and (3.20), we have

This means that and so is a closed subset of .

Step 2 (to show that is upper semicontinuous).

Let be given such that , and let such that . Then, and . Since is upper semicontinuous closed valued mapping, it follows by Lemma 2.7(ii) that . We claim that . Indeed, if , then there exists a such that

Since is upper -continuous, we note that for any neighbourhood of the origin in , there exists a neighbourhood of such that

From (3.22) and (3.23), we obtain

As in the proof of Step 2 in Theorem 3.1, we can show that for all . Hence, there is such that

it is a contradiction to . Hence, , and therefore, is a closed mapping. Since is a compact set and is a closed subset of , is compact. This implies that is compact. Then, by Lemma 2.7(i), we have that is upper semicontinuous.

Step 3 (to show that the solutions set is nonempty and closed).

Define the set-valued mapping by

Then, is an upper semicontinuous mapping. Moreover, we note that is a nonempty closed acyclic subset of for all . Hence, by Lemma 2.8, there exists a point such that . Thus, we have and . This implies that there exists and such that and

Hence, . Similarly, by the proof of Step 5 in Theorem 3.1, we have is closed. This completes the proof.

In this section, we discuss the stability of the solutions for the generalized strong vector quasiequilibrium problem (GSVQEP II).

Throughout this section, let , be Banach spaces, and let be a real locally -convex Hausdorff topological vector space. Let and be nonempty compact subsets, and let be a nonempty closed convex cone. Let is a continuous set-valued mapping with nonempty closed contractible values, and is an upper semicontinuous set-valued mapping with nonempty closed acyclic values}.

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

where .

For , we define

where being the appropriate Hausdorff metrics. Obviously, is a metric space. Now, we assume that satisfies the assumptions of Theorem 3.3. Then, for each , (GSVQEP II) has a solution . Let

Thus, , which conclude that defines a set-valued mapping from into .

We also need the following lemma in the sequel.

Lemma 4.1 (see [8, 38]).

Let be a metric space, and let be compact sets in . Suppose that for any open set , there exists such that for all . Then, any sequence satisfying has a convergent subsequence with limit in .

In the following theorem, we replaced the convex set by the contractible set and acyclic set in Theorem 4.1 in [16]. The following theorem can acquire the same result appearing on the Theorem 4.1 by utilized Lemma 4.1. Now, we need only to present stability theorem for the solution set mapping for (GSVQEP II).

Theorem 4.2.

is an upper semicontinuous mapping with compact values.

Proof.

Since is compact, we need only to show that is a closed mapping. In fact, let be such that . Since , we have , and there exists such that

By the same argument as in the proof of Theorem 4.1 in [16], we can show that and .

Since is lower semicontinuous at and , it follows by Lemma 2.7(iii) that for any , there exists such that . To finish the proof of the theorem, we need to show that for all . Since , it follows by the same argument as in the proof of Theorem 4.1 in [16] that there exists a subsequence of such that , , , and

From the upper -continuous of , we have

Then, , and so is closed. The theorem is proved.

S. Plubtieng would like to thank the Thailand Research Fund for financial support under Grant no. BRG5280016. Moreover, K. Sitthithakerngkiet would like to thank the Office of the Higher Education Commission, Thailand, for supporting by grant fund under Grant no. CHE-Ph.D-SW-RG/41/2550, Thailand.

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