Research Article | Open Access

# On System of Generalized Vector Quasiequilibrium Problems with Applications

**Academic Editor:**Marco Squassina

#### Abstract

We introduce a new system of generalized vector quasiequilibrium problems which includes system of vector quasiequilibrium problems, system of vector equilibrium problems, and vector equilibrium problems, and so forth in literature as special cases. We prove the existence of solutions for this system of generalized vector quasi-equilibrium problems. Consequently, we derive some existence results of a solution for the system of generalized quasi-equilibrium problems and the generalized Debreu-type equilibrium problem for both vector-valued functions and scalar-valued functions.

#### 1. Introduction and Formulations

In the resent years, the vector equilibrium problems have been studied in [1–7] and the references therein which is a unified model of several problems, for instance, vector variational inequality, vector variational-like inequality, vector complementarity problems, vector optimization problems. A comprehensive bibliography on vector equilibrium problems, vector variational inequalities, vector variational-like inequalities and their generalizations can be found in a recent volume [1]. Ansari and Yao [8] and Chiang et al. [9] introduced and studied some vector quasi-equilibrium problems which generalized those quasi-equilibrium problems in [10–17] to the case of vector-valued function. Very recently, the system of vector equilibrium problems was introduced by Ansari et al. [18] with applications in Nash-type equilibrium problem for vector-valued functions. The system of vector quasi-equilibrium problems was introduced by Ansari et al. [19] with applications in Debreu-type equilibrium problem for vector-valued functions. As a generalization of the above models, we introduce a new system of generalized vector quasi-equilibrium problems, that is, a family of generalized quasi-equilibrium problems for vector-valued maps defined on a product set.

Throughout this paper, for a set in a topological space, we denote by , , the convex hull, interior, and the convex closure of , respectively.

Let be an index set. For each , let , and let be topological vector spaces. Consider two family of nonempty convex subsets with and with . Let

An element of the set will be denoted by , therefore, will be written as . Similarly, an element of the set will be denoted by . For each , let , and be set-valued maps with nonempty values, and let be a the vector-valued function. Then the system of generalized vector quasi-equilibrium problems (in Short, SGVQEP) is to find in such that for each ,

Here are some special cases of the (SGVQEP).

(i) For each , let be a vector-valued function. We define a trifunction as , . Then the (SGVQEP) reduces to the generalized Debreu-type equilibrium problem for vector-valued functions (in short, G-Debreu VEP), which is to find in such that for each ,

(ii) We denote by and the set of real numbers and the set of real nonnegative numbers, respectively. For each , if and for all , then the (SGVQEP) reduces to the system of generalized quasi-equilibrium problems (in short, SGQEP), which is to find in such that for each ,

And the G-Debreu VEP reduces to the generalized Debreu-type equilibrium problem for scalar-valued functions (in short, G-Debreu EP), which is to find in such that for each ,

(iii) Let . For each , let , for all , where is a set-valued map. We define a function and a function as , and , then (SGVQEP) and (G-Debreu VEP), respectively, reduce to the system of vector quasi-equilibrium problems and the (Debreu VEP) introduced by Ansari et al. [19] which contain those mathematical in [18, 20] as special cases. The (SGQEP) reduces to the mathematical models in [21, page 286] and [22, pages 152-153] and the (G-Debreu EP) reduces to the abstract economy in [23, page 345] which contains the noncooperative game in [24] as a special case.

(iv) If the index set is singleton, , and , then the (SGVQEP) becomes the implicit vector variational inequality in [9] and the (SGQEP) reduces to the quasi-equilibrium problem investigated in [14–17].

The rest of this paper is arranged in the following manner. The following section deals with some preliminary definitions, notations and results which will be used in the sequel. In Section 3, we establish existence results for a solution to the (SGVQEP) and the (SGQEP) with or without involving -condensing maps by using similar techniques in [19]. In Section 4, as applications of the results of Section 3, we derive some existence results of a solution for the (G-Debreu VEP) and the the (G-Debreu EP).

#### 2. Preliminaries

In order to prove the main results, we need the following definitions.

*Definition 2.1 ([19, 25]). *Let be a nonempty convex subset of a topological vector space and a real topological space with a closed and convex cone with apex at the origin. A vector-valued function is called(i)-quasifunction if and only if, for all , the set is convex,(ii)natural -quasifunction if and only if, , and , .

*Definition 2.2 ([13]). *Let and be two topological spaces. be a set-valued map. Then is said to be upper semicontinuous if the set is open in for every open subset of . Also is said to be lower semicontinuous if the set is open in for every open subset of is said to have open lower sections if the set is open in for each .

*Definition 2.3 ([26]). *Let be a Hausdorff topological space and a lattice with least element, denoted by . A map is a measure of noncompactness provided that the following conditions hold :(i) iff is precompact ( i.e., it is relatively compact),(ii),(iii)max.

*Definition 2.4 ([26]). *Let be a measure of noncompactness on and . A set-valued map is called -condensing provided that, if with , then is relatively compact.

*Remark 2.5. *Note that every set-valued map defined on a compact set is -condensing for any measure of noncompactness . If is locally convex and is a compact set-valued map (i.e., is precompact), then is -condensing for any measure of noncompactness . It is clear that if is -condensing and satisfies , then is also -condensing.

We will use the following particular forms of two maximal element theorems for a family of set-valued maps due to Deguire et al. [27, Theorem 7] and Chebbi and Florenzano [28, Corollary 4].

Lemma 2.6 ([19, 27]). *Let be a family of nonempty convex subsets where each is contained in a Hausdorff topological vector space , For each , let be a set-valued map such that *(i)*for each , is convex,*(ii)*for each , ,*(iii)*for each , is open in .*(iv)*there exist a nonempty compact subset of and a nonempty compact convex subset of for each such that for each there exists satisfying . Then there exists such that for all .*

Lemma 2.7 ([19, 28]). *Let be any index set and be a family of nonempty, closed and convex subsets where each is contained in a locally convex Hausdorff topological vector space . For each , let be a set-valued map. Assume that the set-valued map defined as , , is -condensing and the conditions (i), (ii), (iii) of Lemma 2.6 hold. Then there exists such that for all .*

#### 3. Existence Results

An existence result of a solution for the system of generalized vector quasi-equilibrium problems with or without -condensing maps are will shown in this section.

Theorem 3.1. *Let be any index set. For each , let be a topological vector space, let and be two Hausdorff topological vector spaces, let and be nonempty and convex subsets, let and be set-valued maps with nonempty convex values and open lower sections, and the set be closed in and let be a vector-valued function. For each , let be a set-valued map such that be a proper closed and convex cone with apex at the origin and for all and . Assume that*(i)*for all , for all , ;*(ii)*for each , is natural -quasifunction;*(iii)*for all , the set is closed in ;*(iv)*there exist nonempty and compact subsets and and nonempty, compact and convex subsets , for each such that and , satisfying , and .**Then, there exists in such that for each ,
**
That is, the solution set of the (SGVQEP) is nonempty.*

*Proof. *For each , let us define a set-valued map by
Then, and , is a convex set.

To prove it, let us fix arbitrary and . Let and , then we have

Since is natural -quasifunction, such that
From (3.3) and (3.4), we get

Hence and, therefore, is convex.

It follows from condition (i) that, for each and for all ,

It follows from condition (iii) that for each and each , the set

is open in . That is, has open lower sections on . For each , we also define another set-valued map by
Then, it is clear that and , is convex, and . Since and ,
and , , and are open in , we have is open in .

From condition (iv), there exist a nonempty and compact subset and a nonempty, compact, and convex subset for each such that and . Hence, by Lemma 2.6, such that . Since and , and are nonempty, we have and , . This implies and , . Therefore, ,

That is, the solution set of the (SGVQEP) is nonempty.

*Remark 3.2. *(1) The condition (iii) of Theorem 3.1 is satisfied if the following conditions hold :

(a) is a set-valued map such that for each and the set-valued map is upper semicontinuous;(b)for all , the map is continuous on ;

(2) If , and , , a (fixed) proper, closed and convex cone in , then the condition (ii) and (iii) of Theorem 3.1 can be replaced, respectively, by the following conditions:

(c), the vector-valued function , is -quasifunction;(d), , the map is -upper semicontinuous on ;

(3) Theorem 3.1 extends and generalizes in [19, Theorem 2], [20, Theorem 2.1] and [18, Theorem 2.1] in several ways.

(4) If , is a nonempty, compact and convex subset of a Hausdorff topological vector space , then the conclusion of Theorem 3.1 holds without condition (iv).

Theorem 3.3. *Let be any index set. For each , let be a topological vector space, let and be two locally convex Hausdorff topological vector spaces, let and be nonempty, closed and convex subsets, let and be set-valued maps with nonempty convex values and open lower sections, the set be closed in and be a vector-valued function. For each , let be a set-valued map such that be a proper closed and convex cone with apex at the origin and for all and . Assume that the set-valued map defined as , , is -condensing and for each , the conditions (i), (ii) and (iii) of Theorem 3.1 hold. Then the solution set of the (SGVQEP) is nonempty.*

*Proof. *In view of Lemma 2.7 and the proof of Theorem 3.1, it is sufficient to show that the set-valued map defined as , is -condensing, where 's are the same as in the proof of Theorem 3.1. By the definition of , for all and for each , and therefore for all . Since is -condensing, by Remark 2.5, we have is also -condensing.

By Theorem 3.1 and Remark 3.2, we can easily get the following result.

Corollary 3.4. *Let be any index set. For each , let and be two Hausdorff topological vector spaces, let and be nonempty and convex subsets, let and be set-valued maps with nonempty convex values and open lower sections, let the set be closed in and be a function. Assume that*(i)*for all , for all , ;*(ii)*for each , is quasiconvex;*(iii)*for all , the set is closed in ;*(iv)*there exist nonempty and compact subsets and and nonempty, compact and convex subsets , for each such that and , satisfying , and .**Then, there exists in such that for each ,
**
That is, the solution set of the (SGQEP) is nonempty.*

By Theorem 3.3, we can easily get the following result.

Corollary 3.5. *Let be any index set. For each , let be a topological vector space, let and be two locally convex Hausdorff topological vector spaces, let and be nonempty, closed and convex subsets, let and be set-valued maps with nonempty convex values and open lower sections, the set be closed in and be a function. Assume that the set-valued map defined as , , is -condensing and for each , the conditions (i), (ii) and (iii) of Corollary 3.4 hold. Then the solution set of the (SGQEP) is nonempty.*

*Remark 3.6. *Theorem 3.3 is a generalization of [19, Theorem 3]. Corollaries 3.4 and 3.5 extend and generalize the main results in [10–17].

#### 4. Applications

In this section, we present some existence of a solution for the (G-Debreu VEP) and the (G-Debreu EP).

Theorem 4.1. *Let be any index set. For each , let be a topological vector space, let and be two Hausdorff topological vector spaces, let and be nonempty and convex subsets, let be a set-valued map such that is a proper, closed and convex cone with apex at the origin and for each and , and be set-valued maps with nonempty convex values and open lower sections, the set be closed in and be a bifunction from into . For each , assume that*(i)* is upper semicontinuous;*(ii)*For all and , is natural -quasifunction, where ;*(iii)* is continuous on ;*(iv)*there exist nonempty and compact subsets and and nonempty, compact and convex subsets , for each such that and , satisfying , and .**Then, there exists in such that for each ,
**
That is, the solution set of the (G-Debreu VEP) is nonempty.*

*Proof. *For each , we define a trifunction as
Since is natural quasi-function, by [19, Remark 2], for all and for all such that
Hence
Hence, for all , is natural quasifunction.

By condition (iii), we know that for all , the map is continuous on . So it follows from Remark 3.2 that condition (iii) of Theorem 3.1 holds. It is easy to verify that the other conditions of Theorem 3.1 are satisfied. By Theorem 3.1, we know that the conclusion holds.

Similarly, by Theorem 3.3, Corollaries 3.4 and 3.5, respectively, we have the following results.

Theorem 4.2. *Let be any index set. For each , let be a topological vector space, let and be two locally convex Hausdorff topological vector spaces, let and be nonempty, closed and convex subsets, let be a set-valued map such that is a proper, closed and convex cone with apex at the origin and for each and , and be set-valued maps with nonempty convex values and open lower sections, the set be closed in and be a vector-valued function. Assume that the set-valued map defined as , , is -condensing and (i), (ii), and (iii) of Theorem 4.1 hold. Then, the solution set of the (G-Debreu VEP) is nonempty.*

Theorem 4.3. *Let be any index set. For each , let and be nonempty and convex subsets, let and be set-valued maps with nonempty convex values and open lower sections, the set be closed in and be a bifunction from into . For each , assume that*(i)*for all and , is quasiconvex;*(ii)* is continuous on ;*(iii)*Then, there exists in such that for each ,
**
That is, the solution set of the (G-Debreu EP) is nonempty.*

Theorem 4.4. *Let be any index set. For each , let and be two locally convex Hausdorff topological vector spaces, and be nonempty, closed and convex subsets, let and be set-valued maps with nonempty convex values and open lower sections, the set be closed in and be a function. Assume that the set-valued map defined as , , is -condensing and (i), and (ii) of Theorem 4.3 hold. Then, the solution set of the (G-Debreu EP) is nonempty.*

*Remark 4.5. *Theorem 4.1 extends and generalizes [19, Theorem 5] and [20, Theorems 3.1, 3.6 and Corollaries 3.2, 3.3, and 3.5]. Theorem 4.2 extends and generalizes [19, Theorem 6]. Theorems 4.3 and 4.4 are generalizations of [20, Corollaries 3.5 and 3.7] and the corresponding results in [21–24].

#### Acknowledgment

This research was supported by the National Natural Science Foundation of China (Grant no. 10771228 and Grant no. 10831009).

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#### Copyright

Copyright © 2009 Jian-Wen Peng and Lun Wan. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.