Journal of Function Spaces and Applications

Volume 2013 (2013), Article ID 517469, 10 pages

http://dx.doi.org/10.1155/2013/517469

Research Article

A Collectively Fixed Point Theorem in Abstract Convex Spaces and Its Applications

Haishu Lu¹ and Qingwen Hu²

¹School of Business, Jiangsu University of Technology, Changzhou, Jiangsu 213001, China

²Department of Mathematical Sciences, University of Texas at Dallas, Richardson, TX 75080, USA

Received 1 April 2013; Revised 11 June 2013; Accepted 16 June 2013

Academic Editor: Stanislav Hencl

Copyright © 2013 Haishu Lu and Qingwen Hu. 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.

The main purpose of this paper is to establish a new collectively fixed point theorem in noncompact abstract convex spaces. As applications of this theorem, we obtain some new existence theorems of equilibria for generalized abstract economies in noncompact abstract convex spaces.

1. Introduction

Collectively fixed point theorems for a family of set-valued mappings play a vital role in studying various nonlinear problems. In 1991, Tarafdar [1] established a collectively fixed point theorem in topological vector spaces and gave applications to mathematical economies, game theory, and problems of social sciences. Since then, a lot of generalizations and applications of collectively fixed point theorem under different assumptions and different underlying spaces have been studied by many authors (see [2–7] and the references therein).

Inspired and motivated by the above results, in this paper, we establish a new collectively fixed point theorem in noncompact abstract convex spaces. As applications of this fixed point theorem, some new existence theorems of equilibria for generalized abstract economies are proved under the setting of noncompact abstract convex spaces.

2. Preliminaries

Let be a set. We will denote by the family of all subsets of , by the family of nonempty finite subsets of . Let be a subset of a topological space ; we will denote the interior and the closure of by and , respectively. Let and be two nonempty sets and a set-valued mapping. Then the set-valued mapping is defined by for each .

Definition 1 (see [8]). An abstract convex space consists of a topological space , a nonempty set , and a set-valued mapping with nonempty values. One may denote for each .

Let be an abstract convex space. For any , the -convex hull of is denoted and defined by

( is reserved for the convex hull in vector spaces). A subset of is called a -convex subset of relative to if, for each , we have ; that is, . This means that itself is an abstract convex space called a subspace of . When , the space is denoted by . In such case, a subset of is said to be -convex if ; in other words, is -convex relative to . In case , let .

Remark 2. There are a lot of examples of abstract convex spaces; see [7–13] and references therein. Here, for convenience, we give the following three examples of abstract convex spaces which are cited in this paper.

(a)Let be a topological space and be a given family of nonempty contractible subsets of indexed by such that whenever . The couple is called an -space (see [14]). A set is said -convex if for each . A set is said -compact if, for each , there is a compact -convex set such that .(b)A generalized convex space or a -convex space (see [15]) consists of a topological space and a nonempty set such that, for each with the cardinality , there exist a subset of and a continuous function such that implies . Here, is the standard -simplex with vertices and the face of corresponding to ; that is, if and , then .(c)A semilattice (see [16]) is a partially ordered set , with the partial ordering denoted by , for which, any pair of elements has a least upper bound, denoted by . A topological semilattice is a topological space with a partial ordering for which it is a semilattice with a continuous sup operation; that is, the function , is continuous.

It is evident that each nonempty finite subset of a semilattice has a least upper bound, denoted by . In a partially ordered set , two arbitrary elements and do not have to be comparable. In case , the set is called an order interval. Now assume that is a semilattice and is a nonempty finite subset of . Then the set is well defined. A subset of is called -convex if, for any , we have .

Definition 3 (see [13]). Let be an abstract convex space and a set. For a set-valued mapping with nonempty values, if a set-valued mapping satisfies for  each , then is called a KKM map with respect to . A KKM map is a KKM map with respect to the identity map .

A set-valued mapping is said to have the KKM property and called a -map, if, for any KKM map with respect to , the family has the finite intersection property. We denote

Let be a topological space. A set-valued mapping is called a -map (resp., -map) if, for any closed-valued (resp., open-valued) KKM map with respect to , the family has the finite intersection property. In this case, we denote (resp., ). When , we will write (resp., ) instead of (resp., ). Note that if is a discrete space, then three classes , , and are identical. For more details, we refer to [12, 13, 17, 18] and the references therein.

Definition 4 (see [17, 18]). The partial KKM principle for an abstract convex space is the statement ; that is, for any closed-valued KKM map , the family has the finite intersection property. The KKM principle is the statement that the same property also holds for any open-valued KKM map (i.e., ).

An abstract convex space is called a (partial) KKM space if it satisfies the (partial) KKM principle. A lot of examples of (partial) KKM spaces can be found in [9, 17, 18] and the references therein.

Definition 5 (see [19]). Let be a topological space, a nonempty set, and a set-valued mapping. is said to have local intersection property if, for each with , there exists an open neighborhood of such that .

Remark 6. By Proposition 1 in Lin [19] and Lemma 3.1 in Llinares [20], we can see that the following conditions are equivalent.

(i) has the local intersection property and, for all , is nonempty;(ii);(iii)for each ,   contains an open subset of and .

Lemma 7 (see [8]). Let be a family of abstract convex spaces, where is an index set. Let be equipped with the product topology and . For each , let be the projection. Define by for each . Then is an abstract convex space.

Lemma 8 (see [13]). Let be an abstract convex space, a subspace of , and a topological space. If , then .

3. A Collectively Fixed Point Theorem

The following lemma is a special case of Theorem 11 in Park [12].

Lemma 9. Let be an abstract convex space with (i.e., satisfies the partial KKM principle). Let be two set-valued mappings satisfying the following conditions:

(i)for each , (i.e., for each , );(ii) has open values;(iii)there exists such that .

Then has a fixed point ; that is, .

By Lemma 9, we can obtain the following collectively fixed point theorem which is the main result of our paper.

Theorem 10. Let be a finite index set; let be a family of abstract convex spaces such that is an abstract convex space defined as in Lemma 7. Let be nonempty compact subset . For each , let , be set-valued mappings such that,

(i)for each , ;(ii)for each , is open in ;(iii);(iv)for each , there exists a compact -convex subset of containing such that, for , we have

If satisfies , then there exists such that for each .

Proof. Since is compact subset of , by (ii) and (iii), for each , there exists such that

Then by (iv), for each , there exists a compact -convex subset of containing such that, for , we have

By (4), we have

Then it follows from (5) and (6) that

For each , since is -convex subset of , it follows from Lemma 1 in Park [9] that is an abstract convex space, which is a subspace of . Then by Lemma 7, is an abstract convex space, which is a subspace of . Now, for each , define two set-valued mappings by

Furthermore, we define two set-valued mappings by

Next, we prove that and satisfy all the conditions of Lemma 9 as follows.

(a) For each ,   implies that . In fact, implies that for each . Then it follows that for each . By (i) and the fact that is a subspace of for each , we have

 Then it follows from (10) that

 which implies that for each .(b) For each , is relatively open in . In fact, by (8), for each and each , we have

 Therefore, for each , we obtain

 By (ii) and the fact that is a finite index set, we know that is relatively open in for each .(c) There exists such that . In fact, it follows from (7) and (12) that

 Let be given. Then by (14), for each , there exists such that

 Therefore, from the arbitrary of , we have that . Since is compact, there exists such that .(d) Since , it follows from Lemma 8 that .

Hence, by Lemma 9, there exists such that ; that is, for each . This completes the proof.

Remark 11. Theorem 10 is a new result and completely different from the corresponding collectively fixed point theorems in [1–6], the proofs of which are mainly based on the unity partition theorem. Therefore, the topological spaces in these fixed point theorems satisfy Hausdorff property.

Theorem 12. Let be a finite index set; let be a family of abstract convex spaces such that is an abstract convex space defined as in Lemma 7. Let be nonempty compact subset . For each , let be set-valued mappings such that,

(i)for each , ;(ii);(iii)for each , there exists a compact -convex subset of containing such that, for , we have

If satisfies , then there exists such that for each .

Proof. For each , define a set-valued mapping as follows:

Then by (i), we have for each and each . By (17), for each and each , we have

which is open in . By (ii) and (18), we have for each . By (iii) and (18), there exists a nonempty compact subset of such that, for each and each , there exists a compact -convex subset of containing such that, for , we have

Since satisfies , it follows that all the conditions of Theorem 10 for and hold. Therefore, by Theorem 10, there exists such that for each .

Remark 13. We have shown that Theorem 10 implies Theorem 12. It is clear that Theorem 12 implies Theorem 10. Therefore, Theorem 10 is equivalent to Theorem 12.

4. Particular Fixed Point Theorems

In this section, we give simple consequences of Theorems 10 and 12 and their applications obtained by other authors. We omit their proofs.

Proposition 14 (see [21]). Let () be nonempty compact convex sets, each in a topological vector space (not necessary Hausdorff). Let each have the local intersection property and nonempty convex values. Then there exists such that for each .

Remark 15. By Remarks 6 and 13, we know that Proposition 14 is a particular form of Theorems 10 and 12.

By using Proposition 14, Prokopovych [21] proved a theorem on the existence of a pure strategy -Nash equilibrium in every compact, quasiconcave, and payoff secure game; meanwhile, by means of Proposition 14, he also proved an approximate equilibrium existence theorem that covers a number of known game models.

By Proposition 14, we can obtain the following famous Fan-Browder fixed theorem.

The Fan-Browder Fixed Point Theorem (see [22, 23]).  Let be a nonempty compact convex subset of a topological vector space. Let be a set-valued mapping such that it has nonempty convex values and open inverse values (i.e., each is open in ). Then has a fixed point.

By using the Fan-Browder fixed point theorem, Yu and Yuan [24] obtained the existence results of weight Nash-equilibria and Pareto equilibria for multiobjective games. Kim and Yuan [25] applied the Fan-Browder fixed point theorem to prove a maximal element theorem for -majorized mappings in topological vector spaces from which they obtained an existence theorem of maximal elements for the family of -majorized mappings in which domains are not compact. Balaj and Muresan [26] obtained two minimax inequalities by using the Fan-Browder fixed point theorem. Recently, Luo [27] applied the Fan-Browder fixed point theorem to establish some generalized Ky Fan minimax inequalities for vector-valued mappings.

In 1992, Park [28] generalized the Fan-Browder fixed point theorem to noncompact setting and obtained the following result, which is a particular case of Theorems 10 and 12 with .

Proposition 16 (see [28]). Let be a nonempty convex subset of a topological vector space, nonempty convex, and a nonempty compact subset. Let and be two set-valued mappings. Assume that,

(a)for each , is convex and ;(b)for each , ;(c)for each , is open in ;(d)for each finite subset of , there is a compact convex subset such that and for all .

Then has a fixed point.

Remark 17. The coercivity condition (d) in Proposition 16 can be replaced by the following condition:

(d)′ for each finite subset of , there is a compact convex subset such that and .

Many authors applied Proposition 16 or particular forms of Proposition 16 to study various nonlinear problems in topological vector spaces, for example, generalized Minty vector variational inequality problems, generalized variational inequality problems, generalized vector equilibrium problems, and the constrained or the competitive Nash type equilibrium problems. For more details, see Park [29] and the references therein.

The following particular form of Proposition 16 can be found in Ding and Tan [30].

Proposition 18 (see [30]). Let be a nonempty convex subset of a topological vector space. Let be a set-valued mapping. Assume that,

(a)for each ,   ;(b)for each , is compactly open; that is, for each nonempty compact subset of , is open in ; (c)there exist a nonempty compact convex subset of and a nonempty compact subset of (not necessarily convex) such that, for each , we have .

Then there exists such that .

Ding and Tan [30] applied Proposition 18 to obtain an existence theorem of equilibria for one person games. By using Proposition 18, Ding and Yuan [31] proved a maximal element theorem from which they obtained some existence theorems of equilibria for generalized games without lower semicontinuity for both constraint and preference correspondences.

Remark 19. In condition (b) of Proposition 18, “compactly” can be removed; see Park [11].

The following result (see Corollary 2.3 in Tarafdar [2]) is known in the setting of -spaces without linear structure.

Proposition 20 (see [2]). Let be a compact -space and a set-valued mapping such that,

(a)for each , is a nonempty -convex subset of ;(b)for each , contains an open set ( may be empty for some );(c).

Then has a fixed point.

By using Proposition 20, Chang et al. [32] proved an existence theorem of solutions for the quasi-variational inequality problem in the setting of -spaces. After that, Chang et al. [33] applied Proposition 20 with each being open to prove some existence theorems of loose saddle point, saddle point, and minimax problems for vector-valued multifunctions in the framework of -spaces. On the basis of an equivalent form of Proposition 20, Wu [34] obtained two existence theorems for maximal elements in -spaces from which he proved the existence of solutions of Fan-Yen minimax inequalities, qualitative games, and abstract economies.

The following result is a noncompact generalization of Proposition 20.

Proposition 21 (see [35]). Let be an -space, an -compact set, and a set-valued mapping such that,

(a)for each , is a nonempty -convex subset of ;(b)for each , is open in ;(c) for  all .

Then has a fixed point.

By using Proposition 21, Cubiotti and Nordo [36] obtained an existence result for the Nash equilibria of generalized games with strategy sets in -spaces.

Luo [37] proved the following fixed point theorem in topological ordered spaces.

Proposition 22 (see [37]). Let be a nonempty compact -convex subset of a topological semilattice with path-connected intervals and a set-valued mapping such that

(a)for each , is nonempty and -convex;(b)for each , is open in .

Then has a fixed point.

Luo [37, 38] applied Proposition 22 to prove a saddle-point theorem, existence theorems of solutions for some generalized quasi-Ky Fan inequalities, and Nash equilibrium points for a game system in the setting of topological ordered spaces. By using Proposition 22, Vinh [39] proved a coincidence theorem from which he obtained a Sion-Neumann type minimax theorem.

As it is well known, -convex spaces are typical example of abstract convex spaces. The following extension of the Fan-Browder fixed point theorem to -convex spaces is a particular form of Theorem 3.3 in Park [29], and it includes the fixed point theorems mentioned previously as special cases.

Proposition 23. Let be a -convex space, and let , be two set-valued mappings such that,

(a)for each , implies ;(b);(c)there exists a nonempty compact subset of such that, for each , there exists a compact -convex subset of containing such that

Then there exists such that .

Remark 24. The coercivity condition (c) in Proposition 23 can be replaced by the following equivalent condition:

(c)′ there exists a nonempty compact subset of such that, for each , there exists a compact -convex subset of containing such that

Lin [19] applied the equivalent form of Proposition 23 to obtain some minimax inequalities, existence of maximal element, intersection theorems, and KKM type theorems. At the same year, Lin and Yu [40] applied special cases of Proposition 23 to study scalar equilibrium problems and vectorial equilibrium problems in the setting of -convex spaces. Ding and Park [41] applied Proposition 23 to a class of abstract generalized vector equilibrium problems in -convex spaces. Recently, by using Proposition 23, Balaj and Lin [42] proved a new fixed point theorem for set-valued mappings in -convex spaces from which they obtained some coincidence theorems and existence theorems for maximal elements. Applications of these results to generalized equilibrium and minimax theory were also given.

In 2010, Park [7] established the following generalized Fan-Browder fixed point theorem in abstract convex spaces.

Proposition 25 (see [7]). Let be a finite index set; let be a family of compact abstract convex spaces such that is an abstract convex space as defined as in Lemma 7, and satisfies the partial KKM principle. For each , let be a set-valued mapping such that,

(a)for each , ;(b).

Then there exists such that for each .

Park [7] applied Proposition 25 to establish the von Neumann-Fan type intersection theorem under the setting of abstract convex spaces satisfying the partial KKM principle. By using Proposition 25 with being a singleton and , Yang et al. [43] established some minimax theorems for vector-valued mappings in abstract convex spaces. They also gave some examples to illustrate their results.

5. Equilibria for Generalized Abstract Economies

Considering any preference of a real agent could be unstable because of the fuzziness of consumers’ behavior or market situations, Kim and Tan [44] introduced the fuzzy constraint correspondences in defining the following generalized abstract economy.

Let be any set of agents. For each , let be the strategy set or commodity space of the agent , and let . Following the method of Kim and Tan [44], let be a generalized abstract economy, where are two constrained correspondences such that and are the states attainable for the agent at ; is a fuzzy constrained correspondence such that is the unstable state for the agent at , and is a preference correspondence such that is the state preference by the agent at . An equilibrium for is a point such that for each ,  ,  , and .

As an application of Theorem 10, we derive the following equilibrium existence theorem for generalized abstract economies in noncompact abstract convex spaces.

Theorem 26. Let be a finite index set, a family of abstract convex spaces such that and are two abstract convex spaces as defined in Lemma 7. Let be a generalized abstract economy, and let be a nonempty compact subset of . For each , assume that,

(i)for each , and ;(ii)for each , is nonempty -convex;(iii)for each , ;(iv)for each , is open in ;(v)for each , there exist compact -convex subsets , of containing , respectively, such that, for , we have

If satisfies , then there exists such that, for each , , and .

Proof. By Lemma 7, for each , is an abstract convex space. For each , define two set-valued mappings , by

By (i), (ii), and the definition of , we have and for each and for all . For each and for all , we see from

and (iv) that is open in . Since for each and for all , we have the following:

and so, we have

By (v), for each and each , there exists a compact -convex subset of containing , such that, for , we have

Since satisfies , we can see that all the conditions of Theorem 10 are satisfied. So, by Theorem 10, there exists such that for each . If for some , then we have

And, hence, and so , which contradicts (iii). Therefore, we must have for all . It follows from the definitions of and that, for each , , , and .

Remark 27. Theorem 26 is a new result, which can be compared with Theorem in [45], Theorems 4.1-4.2 in [46], and Theorem 3.1 in [47] in several aspects.

Corollary 28. Let be a finite index set; let be a family of abstract convex spaces such that and are two abstract convex spaces as defined in Lemma 7. Let be a generalized abstract economy, and let be a nonempty compact subset of . For each , assume that,

(i)for each , and ;(ii)for each , is nonempty -convex;(iii) for each , ;(iv)for each , , , and are open sets;(v)the set is closed in ;(vi)for each , there exist compact -convex subsets of containing , respectively, such that, for , we have

If satisfies , then there exists such that, for each , , , and .

Proof. By (iv) and (v), for each , the set is open in . Hence, the conclusion of Corollary 28 follows from Theorem 26.

Corollary 29. Let be a finite index set; let be a family of abstract convex spaces such that and are two abstract convex spaces as defined in Lemma 7. Let be a generalized abstract economy, and let be a nonempty compact subset of . For each , assume that,

(i)for each , and ;(ii)for each , is nonempty -convex;(iii)for each , ;(iv)for each , , , and are open sets;(v)the set is closed in ;(vi)for each , there exist compact -convex subsets of containing , respectively, such that, for , we have

If satisfies , then there exists such that, for each , , , and .

Proof. By (iii), for each , . Hence, the conclusion of Corollary 29 follows from Corollary 28.

If for each and each in Theorem 26, then we have the following theorem.

Theorem 30. Let be a finite index set; let be a family of abstract convex spaces such that and are two abstract convex spaces as defined as in Lemma 7. Let be a generalized qualitative game, and let be a nonempty compact subset of . For each , assume that,

(i)for each , ;(ii)for each , is open in ;(iii)for each , there exist compact -convex subsets of containing , respectively, such that, for , we have

If satisfies , then the generalized qualitative game has a constrained maximal element; that is, there exists such that, for each , and .

By Theorem 30, we can obtain the following equilibrium existence theorem for generalized abstract economies.

Theorem 31. Let be a finite index set; let be a family of abstract convex spaces such that and are two abstract convex spaces as defined in Lemma 7. Let be a generalized abstract economy, and let be a nonempty compact subset of . For each , assume that,

(i) the set is open in ;(ii)if with , we have and ; if with and , we have ;(iii)for each , , , and are open sets;(iv)the set = is closed in ;(v)for each , there exist compact -convex subsets of containing , respectively, such that, for each , there exists satisfying

 where .

If satisfies , then there exists such that, for each , , , and .

Proof. For each , define a set-valued mapping by

Then for each and each , we have

By (i), (iii), and (iv), for each and each , the set is open in . By (ii), for each and each with , we have . By (v), for each and each , there exist compact -convex subsets of containing , respectively, such that, for , we have

So, all the conditions of Theorem 30 are satisfied. Hence, by Theorem 30, there exists such that

If for some , then, by the definition of , we have

which contradicts the first part of (ii). Thus, we have

which implies that, for each , , , and ; that is, is an equilibrium point of generalized abstract economy .

In Theorems 26 and 31, when and for each and for all , we can derive the following equilibrium existence results for abstract economies.

Corollary 32. Let be a finite index set; let be a family of abstract convex spaces such that is an abstract convex space as defined in Lemma 7. Let be an abstract economy, and let be a nonempty compact subset of . For each , assume that,

(i)for each , and ;(ii)for each , ;(iii)for each , is open in ;(iv)for each , there exists a compact -convex subset of containing , such that, for , we have

If satisfies , then there exists such that, for each , and .

Corollary 33. Let be a finite index set; let be a family of abstract convex spaces such that is an abstract convex space as defined in Lemma 7. Let be an abstract economy, and let be a nonempty compact subset of . For each , assume that,

(i)the set is open in ;(ii)if , we have and ; if with , then we have ;(iii)for each , the sets and are open in ;(iv)the set is closed in ;(v)for each , there exists a compact -convex subset of containing such that, for each , there exists satisfying

where .

If satisfies , then there exists such that, for each , and .

Acknowledgments

The authors thank the referee for his many valuable suggestions and helpful comments which improve the exposition of this paper. This work was partially supported by the Planning Foundation for Humanities and Social Sciences of Ministry of Education of China (Research on utilizing conflict of water resource and initial water right allocation in a river basin—on the basis of game theory, no. 12YJAZH084) and Jiangsu Overseas Research & Training Program for University Prominent Young & Middle-aged Teachers and Presidents.

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