Stable Analysis of Solution Set for System of Quasivariational Relations with Applications

Shi, Lefeng; Yang, Zhe

doi:https://doi.org/10.1155/2014/109616

Journal of Applied Mathematics

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Research Article | Open Access

Volume 2014 | Article ID 109616 | https://doi.org/10.1155/2014/109616

Stable Analysis of Solution Set for System of Quasivariational Relations with Applications

Lefeng Shi^1,2and Zhe Yang^3,4

Academic Editor: Naseer Shahzad

Received09 Apr 2014

Revised22 Jul 2014

Accepted22 Jul 2014

Published12 Aug 2014

Abstract

The essential stability of solutions for system of quasivariational relations is studied. We show that most of systems of quasivariational relations are essential (in the sense of Baire category) and that, for any system of quasivariational relations, there exists at least one essential component of its solution set. As applications, the existence of essential components of solution set for systems of KKM problems and systems of quasivariational inclusions is obtained.

1. Introduction

It is well known that the equilibrium problem is a unified model of several problems, namely, optimization problem, saddle point problem, variational inequality problem, fixed point problem, Nash equilibrium problem, and so forth. Recently, several people focused on the study of equilibrium problems and their generalizations. Luc [1] introduced a more general model of equilibrium problems which is called a variational relation problem (in short, VR). The stability of the solution set of variational relation problems was studied in [2, 3]. Various types of sufficient conditions for the existence of solutions of variational relation problems have been investigated in many recent papers (see [4–12]). Furthermore, Agarwal et al. [13] presented a unified approach for studying the existence of solutions for two types of variational relation problems, and Balaj and Lin [14] established the existence criteria for the solutions of two very general types of variational relation problems.

However, One could also argue that a sensible equilibrium should be stable against slight perturbations in the payoffs of the game (van Damme [15]). The notation of an essential solution for fixed points was firstly introduced by Fort [16], which means that, for a fixed point of a mapping , if each mapping sufficiently near to has a fixed point arbitrarily near to , is said to be essential. The method of essential solution has been widely used in various fields recently. It plays a crucial role in the study of stability of solutions including optimal solutions, Nash equilibria, and fixed points (see [17–30]).

Motivated and inspired by research works mentioned above, in this paper, we study the notions of essential stability of solutions for system of quasivariational relations. The results of this paper improve and generalize several known results on the stability of solution set for variational relation problems.

2. Definitions and Preliminaries

Lin and Ansari [8] introduced a system of quasivariational relations (SQVP) and established the existence of solutions of SQVP by means of maximal element theorem for a family of multivalued mappings. Let be any index set. For each , let be nonempty set, let , be multivalued mappings with nonempty values, and let be a relation linking and . A system of quasivariational relations consists in finding such that, for each , and holds for any .

Definition 1. Let be a finite set, and . Let be the set of all systems of quasivariational relations such that the following hold: (i) for each , is nonempty convex compact subset of a normed linear space; (ii) for each , is upper semicontinuous with nonempty convex compact values; (iii) for each , is open in for any ; (iv) for any and any ; (v) for each , any finite set , and any , there is such that holds, where ; (vi) is closed for any .

Clearly, for any , we have that (1) is closed; (2) for all ; (3) since is convex and , then . By Theorem 3.1 of [8], the problem has at least one solution. Here, for any and each , define two closed-valued mappings by and . Denote by the solution set of . Thus, a correspondence is well defined. For each , define the distance on by where is the Hausdorff distance on and is the Hausdorff distance on . Clearly, is a metric space.

In what follows, the notions of essential solutions, essential problems are introduced (see [25]). Let , be two metric spaces, and let be the Hausdorff distance defined on . A set-valued mapping is said to be (1) upper semicontinuous at if, for any open subset of with , there exists an open neighborhood of such that for any ; (2) upper semicontinuous on if is upper semicontinuous on each ; (3) an mapping if is upper semicontinuous on and is compact for each ; (4) lower semicontinuous at if, for any open subset of with , there exists an open neighborhood of such that for any ; (5) lower semicontinuous on if is lower semicontinuous on each ; (6) closed if is closed. Let . A point is said to be an essential point of if, for any open neighborhood of in , there is a such that for any with . If all are essential, then is said to be essential. A nonempty closed subset of is said to be an essential set of if, for any open set , , there is a such that for any with . An essential subset is said to be a minimal essential set of if it is a minimal element of the family of essential sets ordered by set inclusion. A component is called an essential component of if is essential.

Remark 2 (see [25]). (1) It is easy to see that the problem is essential if and only if the mapping is lower semicontinuous at . (2) Let two nonempty closed sets be , of , if , and is essential, so is .

Lemma 3 (see [31]). If is a complete metric space and is , then the set of points, where is lower semicontinuous, is a dense residual set in .

Lemma 4 (see [22]). Let be two nonempty, convex, and compact subsets of linear normed space . Then , where , .

Lemma 5 (see [24]). Let be a metric space, let and be two nonempty compact subsets of , and let and be two nonempty disjoint open subsets of . If , then

Lemma 6 (see [32]). Let and be two Hausdorff topological spaces with compact. If is a closed set-valued mapping from to , then is upper semicontinuous.

Lemma 7 (see [23]). Let , , be three metric spaces, and let and be two set-valued mappings. Suppose that there exists at least one essential component of for each and there exists a continuous single-valued mapping such that for each . Then, there exists at least one essential component of for each .

3. Main Results

Theorem 8. is a complete metric space.

Proof. Let be any Cauchy sequence in , and then, for any , there is such that for any ; that is, for any . (1)Easily, for each , there exists a set-valued mapping such that for any and is upper semicontinuous with nonempty compact convex values. (2)For each , there exist two closed-valued mappings such that, for any , Further, for each , define the mapping and relation by the following: We need to prove that . (3)Easily, for each and any , is open in . And, since is closed, it follows that is closed. (4)If there exist and such that , then . It follows from that there exists a sequence of such that and , which implies that . It is a contradiction. Thus for any and any . (5)Suppose the existence of and such that , and then there exist such that . From , that is, , it follows that for enough large ; that is, . Moreover, since , then for enough large . Thus, for enough large , and , which implies that . It is a contradiction. Thus for any and any . (6)Suppose that there exist , finite set , for all , and with , for all such that does not hold for any , and then for any . Since for any and any , then, for enough large , for any , which implies that does not hold for any . It is a contradiction. Thus, for each , any finite set , and any , there is such that holds, where . Hence . This completes the proof.

Theorem 9. The mapping is upper semicontinuous with compact values.

Proof. By Lemma 6, we only prove that is closed. For any with and any with , we will prove .
It follows from for any that, for each , and holds for any . Since and , then for any . Next, if there exist and such that does not hold, then and . Then, for enough large , and , which implies that and does not hold for enough large . It is a contradiction. Thus, for each , and holds for any ; that is, . This completes the proof.

Theorem 10. There exists a dense residual subset of such that, for each , is essential.

Proof. Since is complete by Theorem 8, and the mapping is upper semicontinuous with nonempty compact values by Theorem 9, by Lemma 3, there is a dense residual subset of , where is lower semicontinuous; thus, is essential for each by Remark 2(1).

Theorem 11. For each , there exists at least one minimal essential subset of .

Proof. By Theorem 9, is upper semicontinuous with nonempty compact values; that is, for each open set , there exists such that, for any with , . Hence is an essential set of itself. Let denote the family of all essential sets of ordered by set inclusion. Then is nonempty and every decreasing chain of elements in has a lower bound (because by the compactness the intersection is in ); therefore, by Zorn’s lemma, has a minimal element and this minimal element is a minimal essential set of .

Theorem 12. For each , there exists at least one connected minimal essential subset of .

Proof. For each , let be a minimal essential subset of . Suppose that was not connected, then there exist two nonempty compact subsets , with , and there exist two disjoint open subsets , in such that and . Since is a minimal essential set of , neither nor is essential. There exist two open sets and such that, for any , there exist with Now, we choose two open sets , such that and .
Since is essential, then, for , there exists such that for any with . Since is a minimal essential set of , then neither nor is essential. Thus, for , there exist two such that Thus .
Now, define the following system of quasivariational relations by where
Easily, we can check the following.(i) is upper semicontinuous with nonempty compact convex values for each .(ii) is open for any and any .(iii)For any and any , since is closed, then is closed.(iv)For each , since and for any , then and for any , which implies that for any ; that is, for any .(v)Next, we show for any and any . For any fixed , we have three cases. If , then and . It follows from that If , then and . It follows from that If , then and . For any , we have and . Then Thus . (vi)Suppose that there exist , finite set , for all , and with , for all such that does not hold for any , and then for any . Since , without loss of generality, we assume . For any , since , then , for all . It follows from that , ; that is, does not hold for any . It is a contradiction. (vii)By Lemmas 4 and 5, we have Thus and .
Since and , then we assume without loss of generality. Then there exists such that, for each , , , and holds for any . When , we have that, for each , that is, for each , and holds for any , which implies that . Hence , which contradicts . Thus is connected.

Theorem 13. For each , there exists at least one essential component of .

Proof. By Theorem 12, there exists at least one connected minimal essential subset of . Thus, there is a component of such that . It is obvious that is essential by Remark 2(2). Thus is an essential component.

Remark 14. If is a singleton, and for any and any , the system of quasivariational relations coincides with the variational relation problem considered in [3]. Thus, the results of [3] are obtained as a special case of this paper’s results. There, is the set of variational relations linking such that (i) is nonempty convex compact subset of a normed linear space; (ii) for any finite set , and any , there is such that holds; (iii) is closed for any . Then we have the following results, which are Lemma 3.2, Theorems 3.1–3.4 in [3].

Theorem 15. The mapping is upper semicontinuous with compact values.

Theorem 16. There exists a dense residual subset of such that, for each , is essential.

Theorem 17. For each , there exists at least one minimal essential subset of .

Theorem 18. For each , there exists at least one minimal essential subset of .

Theorem 19. For each , there exists at least one essential component of .

4. Applications (I): Systems of KKM Problems

Let be any index set. For each , let be a nonempty convex compact subset of topological vector space and , and let and be multivalued mappings with nonempty values. A system of KKM problems consists in finding such that, for each , and .

Definition 20. Let be a finite set, and . Let be the set of systems of KKM problems such that, for each , (1) is a nonempty convex compact subset of a normed linear space ; (2) for any and any , and is open in for all ; (3) is upper semicontinuous with nonempty compact convex values; (4) the multivalued mapping holds nonempty closed values; (5) for any finite set , and any , there is such that holds, where .

For each , denote by the solution set of and by , where Easily is nonempty. Furthermore, for any , we define the distance on by . Thus there exists an isometric mapping such that and .

Theorem 21. For each , there exists at least one essential component of .

Proof. Since is an isometric mapping such that , it is continuous. By Theorem 13, there exists at least one essential component of for each , and by Lemma 7, there exists at least one essential component of for each .

5. Applications (II): Systems of Variational Inclusions

Let be any index set. For each , let be a nonempty subset of Hausdorff topological space, let be a Hausdorff topological vector space, and let , be multivalued mappings with nonempty values.

A system of quasivariational inclusions of type (I) consists in finding such that, for each , and for all .

A system of quasivariational inclusions of type (II) consists in finding such that, for each , and for all .

A system of quasivariational inclusions of type (III) consists in finding such that, for each , and for all .

Definition 22. Let be a finite set, and . Let be the set of systems of quasivariational inclusions of type (I) such that, for each , (1) is a nonempty convex compact subset of a normed linear space ; (2) for any and any , and is open in for all ; (3) is upper semicontinuous with nonempty compact convex values; (4) for any , is lower semicontinuous, and is closed; (5) for any of and any , there is such that , where .
For each , denote by the solution set of . For any , define by , where By Theorem 3.5 of [8], . Easily, is nonempty. Further, for any , we define the distance on by . Thus there exists an isometric mapping such that and .

Theorem 23. For each , there exists at least one essential component of .

Proof. Since is an isometric mapping such that , it is continuous. By Theorem 13, there exists at least one essential component of for each , and by Lemma 7, there exists at least one essential component of for each .

It is similar to Definition 22 and Theorem 23, and the following results are obtained. Here, we do not repeat the process.

Definition 24. Let be a finite set, and . Let be the set of systems of quasivariational inclusions of type (II) such that, for each , (1) is a nonempty convex compact subset of a normed linear space ; (2) for any and any , and is open in for all ; (3) is upper semicontinuous with nonempty compact convex values; (4) for any , is upper semicontinuous with nonempty compact values and is closed; (5) for any of and any , there is such that , where .

Definition 25. Let be a finite set, and . Let be the set of systems of quasivariational inclusions of type (III) such that, for each , (1) is a nonempty convex compact subset of a normed linear space ; (2) for any and any , and is open in for all ; (3) is upper semicontinuous with nonempty compact convex values; (4) for any , is closed; (5) for any of and any , there is such that , where .

For each and each , denote the solution set of by .

Theorem 26. For and each , there exists at least one essential component of .

6. Conclusions

In this paper, we study the notions of essential stability of solutions for system of quasivariational relations. We show that most of systems of quasivariational relations (in the sense of Baire category) are essential and that, for any system of quasivariational relations, there exists at least one essential component of its solution set. As applications, the existence of essential components of solution set for systems of KKM problems and systems of quasivariational inclusions is obtained.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Authors’ Contribution

All authors contributed equally and significantly in writing this paper. All authors read and approved the final paper.

Acknowledgments

This research was supported by project supported by the China’s Post-Doctoral Science Fund (no. 2014M552325), the Chen Guang Project sponsored by the Shanghai Municipal Education Commission and Shanghai Education Development Foundation (no. 13CG35), the open project of Key Laboratory of Mathematical Economics (SUFE), Ministry of Education (no. 201309KF02), and the Chongqing Foundation and Cutting-Edge Research Program (no. cstc2014jcyjA90016).

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Copyright © 2014 Lefeng Shi and Zhe Yang. 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.

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