Abstract

We consider a system of operator quasi equilibrium problems and system of generalized quasi operator equilibrium problems in topological vector spaces. Using a maximal element theorem for a family of set-valued mappings as basic tool, we derive some existence theorems for solutions to these problems with and without involving -condensing mappings.

1. Introduction

In 2002, Domokos and Kolumbán [1] gave an interesting interpretation of variational inequality and vector variational inequalities (for short, VVI) in Banach space settings in terms of variational inequalities with operator solutions (for short, OVVI). The notion and viewpoint of OVVI due to Domokos and Kolumbán [1] look new and interesting even though it has a limitation in application to VVI. Recently, Kazmi and Raouf [2] introduced the operator equilibrium problem which generalizes the notion of OVVI to operator vector equilibrium problems (for short, OVEP) using the operator solution. They derived some existence theorems of solution of OVEP with pseudomonotonicity, without pseudomonotonicity, and with -pseudomonotonicity. However, they dealt with only the single-valued case of the bioperator. It is very natural and useful to extend a single-valued case to a corresponding set-valued one from both theoretical and practical points of view.

The system of vector equilibrium problems and the system of vector quasi equilibrium problems were introduced and studied by Ansari et al. [3, 4]. Inspired by above cited work, in this paper, we consider a system of operator quasi equilibrium problems (for short, SOQEP) in topological vector spaces. Using a maximal element theorem for a family of set-valued mappings according to [5] as basic tool, we derive some existence theorems for solutions to SOQEP with and without involving -condensing mappings.

Further, we consider a system of generalized quasi operator equilibrium problems (for short, SGQOEP) in topological vector spaces and give some of its special cases and derive some existence theorems for solutions to SOQEP with and without involving -condensing mappings by using well-known maximal element theorem [5] for a family of set-valued mappings, and, consequently, we also get some existence theorems for solutions to a system of operator equilibrium problems.

2. Preliminaries

Let be an index set, for each , and let be a Hausdorff topological vector space. We denote , the space of all continuous linear operators from into , where is topological vector space for each . Consider a family of nonempty convex subsets with in .

Let Let be a set-valued mapping such that, for each , is solid, open, and convex cone such that and .

For each , let be a bifunction and let be a set-valued mapping with nonempty values. We consider the following system of operator quasi equilibrium problems (for short, SOQEP). Find such that, for each ,

We remarked that, for the suitable choices of , and , SOQEP (2) reduces to the problems considered and studied by [3–6] and the references therein.

Now, we will give the following concepts and results which are used in the sequel.

Definition 1. Let be a nonempty and convex subset of a topological vector space, and let be a topological vector space with a closed and convex cone with apex at the origin. A vector-valued function is said to be as follows: (i)P-function if and only if and : (ii)natural P-quasifunction if and only if and : where denotes the convex hull of ;(iii)P-quasifunction if and only if and the set is convex.

Definition 2 (see [7]). Let be a topological vector space and let be a lattice with a minimal element, denoted by . A mapping is called a measure of noncompactness provided that the following conditions hold for any : (i), where denotes the closed convex hull of ;(ii) if and only if is precompact;(iii).

Definition 3 (see [7]). Let be a topological vector space, , and let be a measure of noncompactness on . A set-valued mapping is called -condensing provided that with ; then is relative compact; that is, is compact.

Remark 4. Note that every set-valued mapping defined on a compact set is -condensing for any measure of noncompactness . If is locally convex, then a compact set-valued mapping (i.e., is precompact) is -condensing for any measure of noncompactness . Obviously, if is -condensing and satisfies , for all , then is also -condensing.

The following maximal element theorems will play key role in establishing existence results.

Theorem 5 (see [8]). For each , let be a nonempty convex subset of a topological vector space and let be the two set-valued mappings. For each , assume that the following conditions hold: (a)for all , ;(b)for all , ;(c)for all , is compactly open ;(d)there exist a nonempty compact subset of and a nonempty compact convex subset , for each , such that, for all , there exists such that . Then, there exists such that for each .

We will use the following particular form of a maximal element theorem for a family of set-valued mappings due to Deguire et al. [5].

Theorem 6 (see [5]). Let be any index set, for each , let be a nonempty convex subset of a Hausdorff topological vector space , and let be a set-valued mapping. Assume that the following conditions hold: (i) and ; is convex;(ii) and ; , where is the th component of ;(iii) and ; is open ;(iv)there exist a nonempty compact subset of and a nonempty compact convex subset such that and there exists such that . Then, there exists such that for each .

Remark 7. If is nonempty, closed, and convex subset of a locally convex Hausdorff topological vector space , then condition (iv) of Theorem 6 can be replaced by the following condition:
the set-valued mapping is defined as , -condensing.

3. Main Result

Throughout this paper, unless otherwise stated, for any index set and for each , let be a topological vector space and let be a set-valued mapping such that, for each , is proper, solid, open, and convex cone such that and . We denote , the space of all continuous linear operators from into . We also assume that is a set-valued mapping such that is nonempty and convex, is open in , and the set is closed in , where is the th component of .

Now, we have the following existence result for SOQEP (2).

Theorem 8. For each , let be nonempty and convex subset of a Hausdorff topological vector space and let be a bifunction. Suppose that the following conditions hold: (i) and , where is the th component of ;(ii) and ; the vector-valued function is natural -quasifunction;(iii) and ; the set is closed in ;(iv)there exist a nonempty compact subset of and a nonempty compact convex subset of , for each such that ; there exists and such that and . Then SOQEP (2) has a solution.

Proof. Let us define, for each given , a set-valued mapping by First, we claim that and is convex. Fix an arbitrary and . Let and ; then we have Since is natural -quasifunction, there exists such that From the inclusion of and , we get Hence, and therefore is convex. Since and are arbitrary, is convex, and .
Hence, our claim is then verified.
Now and ; the complement of in can be defined as From condition (iii) of the above theorem, will be closed in .
Suppose that and ; we define another set-valued mapping by
Then, it is clear that and is convex, because and are both convex. Now, by condition (i), . Since and , is open in , because and are open in .
Condition (iv) of Theorem 6 is followed from condition (iv). Hence, by fixed point Theorem 6, there exists such that . Since and is nonempty, we have . Therefore, and .
This completes the proof.

Now, we establish an existence result for SOQEP (2) involving -condensing maps.

Theorem 9. For each , let be a nonempty, closed, and convex subset of a locally convex Hausdorff topological vector space , suppose that is a bifunction, and let the set-valued mapping defined as be -condensing. Assume that conditions (i), (ii), and (iii) of Theorem 8 hold. Then SOQEP (2) has a solution.

Proof. In view of Remark 7, it is sufficient to show that the set-valued mapping defined as , is -condensing, where s are the same as defined in the proof of Theorem 8. By the definition of and and therefore . Since is -condensing, by Remark 7, we have being also -condensing.
This completes the proof.

4. System of Generalized Quasi Operator Equilibrium Problem

Throughout this section, unless otherwise stated, let be any index set. For each , let be a Hausdorff topological vector space. We denote , the space of all continuous linear operators from into , where is topological vector space for each and for each ; let be a closed, pointed, and convex cone with , where denotes the interior of set . Consider a family of nonempty convex subsets with in . Let, for each , a bifunction and two set-valued mappings be with nonempty values.

Let be the unit vector in , for each , and also such that , where are two real numbers such that .

Now, we consider the system of generalized quasi operator equilibrium problems (for short, SGQOEP). Find such that, for each ,

4.1. Special Cases

(I)If , then SGQOEP (10) reduces to finding of such that, for each , (II)If, in Case (I), we take , then and ; then problem (10) reduces to the system of generalized quasi operator equilibrium problems with lower and upper bounds (for short, SGQOEPLUB). Find such that, for each ,

Now, we establish the existence result for SGQOEP (10).

Theorem 10. For each , let be a nonempty convex subset of a topological vector space and are the bifunctions, is a set-valued mapping such that the set is compactly closed, is a set-valued mapping with nonempty values such that, for each is compactly open in , and are the unit vector such that , where are two real numbers such that . For each , assume that the following conditions hold: (i)for all , ;(ii)for all , or ;(iii)for all and for every nonempty finite subset , we have (iv)for all , the set is compactly closed in ;(v)there exist a nonempty compact subset of and a nonempty compact convex subset , for each , such that, for all , there exists such that satisfying and either or . Then the problem SGQOEP (10) has a solution.

Proof. For each and for all , define two set-valued mappings by Condition (iii) implies that, for each and for all , .
From condition (ii), we have for all and for each .
Thus, for each and for all , We have complement of in : which is compactly closed by virtue of condition (iv). Therefore, for each and for all is compactly open in .
For each , define two set-valued mappings by Thus, for each and for all and in view of condition (i), we obtain . It is easy to see that for each and for all . Thus, for each and for all and are compactly open in . We have being compactly open in . Also for all and for each .
Then, by Theorem 5, there exists such that for each . If , then , which contradicts the fact that is nonempty for each and for all . Hence, , for each . Therefore, and , for all . Thus, for each and for all . This completes the proof.

Now, we establish an existence result for SGQOEP (10) involving -condensing maps.

Theorem 11. For each , assume that conditions (i)–(iv) of Theorem 10. hold. Let be a measure of noncompactness on . Further, assume that the set-valued mapping defined as is a nonempty, closed, and convex subset of a locally convex Hausdorff topological vector space and is a bifunction and let the set-valued mapping defined as be -condensing. Then, there exists a solution of SGQOEP (10).

Proof. In view of Remark 7, it is sufficient to show that the set-valued mapping defined as , is -condensing, where s are the same as defined in the proof of Theorem 10. By the definition of and and therefore . Since is -condensing, by Remark 7, we have being also -condensing.
This completes the proof.

Next, we derive the existence result for the solution of SGQOEPLUB (12).

Corollary 12. For each , let be a nonempty convex subset of a topological vector space and are the bifunctions, is a set-valued mapping such that the set is compactly closed, is a set-valued mapping with nonempty values such that, for each is compactly open in , and are two real numbers such that . For each , assume that the following conditions hold:(i)for all , ;(ii)for all , or ;(iii)for all and for every nonempty finite subset , we have (iv)for all , the set is compactly closed in ;(v)there exist a nonempty compact subset of and a nonempty compact convex subset , for each , such that, for all , there exists such that satisfying and either or . Then the problem SGQOEPLUB (12) has a solution.

Proof. For each and for all , define two set-valued mappings by Condition (iii) implies that, for each and for all , .
From condition (ii), we have for all and for each .
Thus, for each and for all , We have complement of in : which is compactly closed by virtue of condition (iv). Therefore, for each and for all is compactly open in .
For each , define two set-valued mappings by Thus, for each and for all and in view of condition (i), we obtain . It is easy to see that for each and for all . Thus, for each and for all and are compactly open in . We have being compactly open in . Also for all and for each .
Then, by Theorem 5, there exists such that for each . If , then , which contradicts the fact that is nonempty for each and for all . Hence, , for each . Therefore, and , for all . Thus, for each and for all . This completes the proof.

Now, we establish an existence result for SGQOEPLUB (12) involving -condensing maps.

Theorem 13. For each , assume that conditions (i)–(iv) of Corollary 12 hold. Let be a measure of noncompactness on . Further, assume that the set-valued mapping defined as is a nonempty, closed, and convex subset of a locally convex Hausdorff topological vector space and is a bifunction and let the set-valued mapping defined as , be -condensing. Then, there exists a solution of SGQOEPLUB (12).

Proof. In view of Remark 7, it is sufficient to show that the set-valued mapping defined as , is -condensing, where are the same as defined in the proof of Theorem 10. By the definition of and and therefore . Since is -condensing, by Remark 7, we have being also -condensing.
This completes the proof.

Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.