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Abstract and Applied Analysis
Volume 2013 (2013), Article ID 310796, 6 pages
On Abstract Economies and Their Applications
Department of Mathematics, National Taiwan Normal University, No. 88, Section 4, Ting-Chou Road, Taipei 116, Taiwan
Received 10 July 2013; Accepted 22 August 2013
Academic Editor: Wei-Shih Du
Copyright © 2013 Chien-Hao Huang and Liang-Ju Chu. 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.
We establish a new equilibrium existence theorem of generalized abstract economies with general preference correspondences. As an application, we derive an existence theorem of generalized quasi-variational inequalities in the general setting of -spaces without any linear structure.
1. Introduction and Preliminary
Let be any (finite or infinite) set of agents. A generalized abstract economy is defined as a family of order quintuples with such that for each , is a topological space, are constraint correspondences, is a fuzzy constraint correspondence, and is a preference correspondence. In a real market, any preference of a real agent would be unstable by the fuzziness of consumers’ behavior or market situations. Thus, it is reasonable to introduce fuzzy constraint correspondences in defining an abstract economy. An equilibrium point of is a point such that for each , , , and , where and denote the projections of and from to , respectively.
In case for each and is independent of the second variable, that is, , the above generalized abstract economy reduces to the standard abstract economy , in which an equilibrium point of is a point such that for each , and . When and each is a topological vector space, the standard abstract economy coincides with the classical definition of Shafer and Sonnenschein . For more details on abstract economies, see, for example, [2–14] and the references therein.
Throughout this paper, all topological spaces are assumed to be Hausdorff. In order to establish our main results, we first give some basic notations. For a nonempty set of a topological space , we denote the set of all subsets of by , the set of all nonempty finite subsets of by , the interior of by , and the closure of by .
Let be a family of some nonempty contractible subsets of a topological space indexed by such that whenever . The pair is called an -space. Given an -space , a nonempty subset of is said to be -convex if for all . For a nonempty subset of , we define the -convex hull of as
It is known that if , then there exists a finit subset of such that . Moreover, for any , is called a polytope. We will say that is an -space with precompact polytopes if any polytope of is precompact. For example, a locally convex topological vector space is an -space with precompact polytopes, by setting for all .
An -space is called an .-space if is a uniform space whose topology is induced by its uniformity , and there is a base consisting of symmetric entourages in such that for each , the set for some is -convex whenever is -convex. We will use the notation to stand for an .-space. For details of uniform spaces, we refer to . In a recent paper , we introduce a new measure of precompactness of a subset in an .-space by
Let be a family of .-spaces with precompact polytopes, where is a finite or infinite index set and . For each , let be the projection of onto and a measure of precompactness in . We say that a set-valued mapping is -condensing if for every satisfying is a nonprecompact subset of . It is clear that for any set-valued mapping and any measure in , is -condensing whenever is compact.
Let be a topological space, let be an -space, and let be two set-valued mappings.(1) is said to be upper semicontinuous if for each and each open subset of with , there exists a neighborhood of such that for all . (2) is said to be transfer open valued on if for each , for each , there exists some such that . (3) is said to be transfer open inverse valued in if is transfer open valued on , where is defined by (4) The set-valued mappings and are defined by
Further, we denote by the class of all . set-valued mappings with nonempty closed -convex values.
2. Main Results
The following fundamental theorems will play an important role in proving our main theorem.
Theorem A (see ). Let be a family of .-spaces with precompact polytopes, , and let be -condensing. Then there exists a nonempty compact -convex subset of such that .
Theorem B (see ). Let be a family of .-spaces with precompact polytopes and . If is an -condensing mapping with closed -convex values for each , then has a fixed point.
Next, we list and establish some essential lemmas as follows.
Lemma 1 (see ). If is an .-space and is an -convex subset of , then is also -convex.
Lemma 2 (see ). Let be a topological space and let be a compact .-space. If is an . set-valued mapping, then the mapping is also . with compact -convex values.
Lemma 3 (see ). Let and be topological spaces and let be a transfer open valued mapping. Then and hence is open in .
Lemma 4. Let be paracompact, an -space, and be two set-valued mappings such that (1) and for each , (2) is transfer open inverse valued in . Then has a continuous selection; that is, there exists a continuous function such that for each .
Proof. Since for each , , it follows that for some . Since is transfer open inverse valued in , there exists some such that . This yields that forms an open cover of . Since is paracompact, there exists a locally finite open cover such that for each . By [17, Theorem 3.1], there exists a continuous function such that for all . Note that for any , there exist finitely many such that . This implies , and hence . It follows that for each , we get Thus, the proof is complete.
Lemma 5. Let be a compact -space, and let be a set-valued mapping such that for each , is open: then so is .
Proof. For each , we fix an . Since , there is a finite set in such that . Since each is open, it follows that the set is also open and . To complete the proof, we will show that . For any , we have for all . Accordingly, for all . Hence,
That is, . Consequently, .
Theorem 6. Let be a generalized abstract economy, where is a set of agents and such that for each , (1) is an .-space with precompact polytopes, (2) for each , (3) both and are -condensing mappings in , (4) for each , (5) is transfer open inverse valued in , (6) is paracompact. Then has an equilibrium point .
Proof. For each , we define by
Assume that . Then for each , we have some . Equivalently, . It follows that . Since each is transfer open inverse valued in by (5), it follows from Lemma 3 that is open in .
For , if , by using (5), we have some such that . Thus, the restriction is transfer open inverse valued in . Moreover, by (3), each is nonempty and -convex. Therefore, by Lemma 4, there exists a continuous function such that for each .
Since and are -condensing, applying Theorem A, we have two nonempty compact -convex subsets and of such that and . Using these notations, we define a set-valued mapping by
We will show that . Let be an open subset of . Since for each , we have
It follows from Lemma 2 and the upper semicontinuity of that is open in . Hence, is . Further, by (3) and Lemma 1, each is nonempty, closed, and -convex. Therefore, .
Next, we define a set-valued mapping by
Since is compact, each is -condensing in . Hence, by Theorem B, the set-valued mapping has a fixed point ; that is, for each . If , then
Thus, , which contradicts with (4). Therefore, and hence , , and for each . That is, is an equilibrium of .
Remark that condition (4) of Theorem 6 can be replaced by a milder condition for each . Further, when each .-space satisfies , condition (4) can be modified by without affecting the conclusion.
Corollary 7. Let be a generalized abstract economy, where is a set of agents and such that for each , (1) is an .-space with precompact polytopes, and for each , (2) for each , (3) both and are -condensing mappings in , (4) for each , (5) is transfer open inverse valued in , (6) is paracompact. Then has an equilibrium point .
Proof. According to the proof of Theorem 6 and by virtue of the condition for each , we obtain . It follows that the set-valued mapping can be defined by
Thus, by an analogue proof to Theorem 6, we may conclude that has an equilibrium point.
Following the proof of Theorem 6 by taking , we may obtain a new version of equilibrium existence theorem as follows.
Corollary 8. Let be a generalized abstract economy, where is a set of agents and such that for each , (1) is an .-space with precompact polytopes, (2) for each , (3) both and are -condensing mappings in , (4) for each , (5) is transfer open inverse valued in , (6) is paracompact. Then has an equilibrium point .
Notice that Theorem 6 generalizes [7, Kim-Tan, Theorem 2], in which they deal with the case of locally convex topological vector spaces under some compactness conditions, and it also improves [19, Wu-Yuan, Theorem 3] in the setting of locally -convex spaces. We also note that if is metrizable, the set is also metrizable and hence is paracompact. Therefore, the assumption (6) of Theorem 6 is automatically satisfied. Furthermore, if each is compact, then both and are obviously -condensing. Thus, we have an immediate consequence, which is a generalization of [7, Kim-Tan, Corollary 1] to -spaces.
Corollary 9. Let be a generalized abstract economy, where is a set of agents such that for each , (1) is a metrizable compact .-space, and for each , (2) for each , (3), and , (4) for each , (5) is transfer open inverse valued in . Then has an equilibrium point .
We note that our main results focus on the setting of general .-spaces without any linear structure; further, the correspondences are not necessarily lower semicontinuous and do not require the usual open lower section assumption, such as the earlier works [3, Theorem 4], [13, Theorem 3 and its Corollary], [19, Theorems 1 and 3], and [18, Theorem 6.1]. In fact, we can give a simple example applicable for Corollary 9, while previous results do not.
Example 10. Consider the set of agents is singleton. Let and the correspondences be defined by , and for each . The preference correspondence is defined as follows: Then is transfer open inverse valued in . Indeed, is open in , and for any and , we always have . However, the lower section is not open. Indeed, let and let ; then and converges to , which does not belong to . This means that the set is not closed, and hence is not open. Further, for each , . Thus, all hypotheses of Corollary 9 are satisfied so that the generalized abstract economy has an equilibrium point in . In fact, all the equilibria of are the points , where .
Let and be two topological spaces. Given three set-valued mappings , , , and a function , a generalized quasi-variational inequality is defined as follows:
In particular, if for each , then . Therefore, the () reduces to the usual quasi-variational inequality as follows:
Theorem 11. Let be an .-space with precompact polytopes, for each , and let be a topological space. The set-valued mappings and satisfy , , and , and is open for all . Suppose that is a function such that (1) for all and , (2) for each fixed , the mapping is lower semicontinuous, (3) for each fixed , the mapping is -quasiconvex in the following sense that for any finite set in , Then there is a solution to ().
Proof. Define a set-valued mapping by
By [20, Proposition 23, page 121], for each fixed , the mapping is lower semicontinuous. Thus, the set is open for each . It follows that is open. By Lemma 5, is also open. Next, we show that for all . Assume that there are and satisfying . Then there is a finite subset of such that . For each fixed , since the mapping is -quasiconvex, it follows that
By Kneser’s minimax theorem , together with for all , we have
This is a contradiction. Thus, all hypotheses of Corollary 7 are satisfied. Therefore, there exist such that , , and , It follows that
Since is compact, there is such that for all . That is, is a solution to ().
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