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Abstract and Applied Analysis

Volume 2013 (2013), Article ID 310796, 6 pages

http://dx.doi.org/10.1155/2013/310796

## 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.

#### Abstract

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 [1]. 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 [15]. In a recent paper [16], 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 [16]). *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 [16]). *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 [12]). * If is an .-space and is an -convex subset of , then is also -convex. *

Lemma 2 (see [12]). * 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 [7]). * 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.

We remark that Lemma 4 extends [7, Theorem 2] from topological vector spaces to general -spaces. When and has open lower sections, Lemma 4 reduces to [18, Theorem 3.1].

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 [21], 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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