ISRN Applied Mathematics

VolumeΒ 2011, Article IDΒ 696917, 13 pages

http://dx.doi.org/10.5402/2011/696917

## Some Results for the Family *KKM(X,Y)* and the -Mapping in *FC*-Spaces

Department of Mathematics, Chengdu University of Information Technology, Chengdu, Sichuan 610103, China

Received 15 July 2011; Accepted 24 August 2011

Academic Editor: A.Β Bellouquid

Copyright Β© 2011 Rong-Hua He. 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 first establish a fixed point theorem for a -set contraction map on the family , which does not need to be a compact map. Next, we present the type theorems, matching theorems, coincidence theorems, and minimax theorems on the family and the -mapping in *FC*-spaces. Our results improve and generalize some recent results.

#### 1. Introduction

In 1929, Knaster et al. [1] first established the well-known theorem in finite-dimensional spaces. In 1961, Fan [2] generalized the theorem to infinite dimensional topological vector spaces and gave some applications in several directions. Later, Chang and Yen [3] introduced the family and got some results about fixed point theorems, coincidence theorems, and its applications on this family. Recently, Lin and Chen [4] studied the coincidence theorems for two families of multivalued functions, Chen and Chang [5] obtained some results for the family and the -mapping in Hausdorff topological vector spaces. For the related results, the reader may consult [4β10]. In this paper, we first establish a fixed point theorem for a -set contraction map on the family , which does not need to be a compact map. Next, we present the type theorems, matching theorems, coincidence theorems, and minimax theorems on the family and the -mapping in -spaces. Our results improve and generalize the corresponding results in [5, 8, 9].

#### 2. Preliminaries

Let be a nonempty set. We denote by and the family of all subsets of and the family of all nonempty finite subsets of , respectively. For each , we denote by the cardinality of . Let denote the standard *n*-dimensional simplex with the vertices . If is a nonempty subset of , we will denote by the convex hull of the vertices .

Let and be two sets, and be a set-valued mapping. We will use the following notations in the following material: (i),(ii),(iii),(iv),(v) denote the family of single-valued continuous mappings from to .

For topological spaces and , is said to be closed if its graph is closed. is said to be compact if the image of under is contained in a compact subset of . A subset of is said to be compactly open (resp., compactly closed) if for each nonempty compact subset of , is open (resp., closed) in . The compact closure of and the compact interior of (see [5]) are defined, respectively, by

It is easy to see that , is compactly open (resp., compactly closed) in if and only if (resp., ). For each nonempty compact subset of , and , where (resp., )) denotes the closure (resp., interior) of in .

A set-valued mapping is said to be transfer compactly closed valued on (see [5]) if for each and , there exists such that . is said to be transfer compactly open valued on if for each and , there exists such that . is said to have the compactly local intersection property on if for each nonempty compact subset of and for each with , there exists an open neighborhood of in such .

Let is transfer compactly open valued, then . It is clear that each transfer open valued correspondence is transfer compactly open valued. The inverse is not true in general.

Throughout this paper, all topological spaces are assumed to be Hausdorff. The following notion of a finitely continuous topological space (in short, -space) was introduced by Ding in [11].

*Definition 2.1. * is said to be an -space if is a topological space and for each where some elements in may be same, there exists a continuous mapping . If and are two subsets of , is said to be an -subspace of relative to if for each and for any , , where .

If , then is called an -subspace of . Clearly, each -subspace of is also an -space.

For a subset of , we can define the -hull of (see [12]) as follows: .

*Definition 2.2 (see [8]). *An -space is said to be a locally -space, denoted by , if is a uniform topological space with a uniform structure having an open base of symmetric entourages such that for each and for each , is an -subspace of .

Lemma 2.3 (see [5]). *Let and be two topological spaces, and let be a set-valued mapping. Then the following conditions are equivalent:*(1)* has the compactly local intersection property,*(2)*for each compact subset of and for each , there exists an open subset of such that and ,*(3)*for any compact subset of , there exists a set-valued mapping such that for each , is open in and for each , and ,*(4)*for each compact subset of and for each , there exists such that and ,*(5)* is transfer compactly open valued on ,*(6)*.*

Now, we introduce the following Definitions 2.4 and 2.6.

*Definition 2.4. *Let be a topological space and be an -space. A set-valued mapping is called a -mapping if there exists a set-valued mapping such that(i)for each , is a nonempty -subspace of relative to ,(ii) satisfies one of the conditions (1)β(6) in Lemma 2.3.

The mapping is said to be a companion mapping of .

*Remark 2.5. *If be a -mapping, then for each nonempty subset of , is also a -mapping.

The class was introduced by Ding [8]. Let be an -space and let be a topological space. If are two set-valued mappings such that for each and for each , then is said to be a generalized - mapping with respect to . Let be a set-valued mapping such that if is a generalized - mapping with respect to , then the family has the finite intersection property, where denotes the closure of , then is said to have the property. Write

Let be an -space and be the family of nonempty bouned subsets. Let = is a family of seminorms which determines the topology on . Let be the set of all nonnegative real numbers. If is a family of seminorms which determines the topology on , then for each and , we define the set-measure of noncompactness by where -diameter of a set .

*Definition 2.6. *Let be an -space, and a mapping is said to be a -set contraction map, if there exists such that for each , with for each bounded subset of and is bounded.

#### 3. Main Results

In order to prove our main results, we need the following Lemmas. The following results are [8, Lemma 3.1(i) and Theorem 3.1].

Lemma 3.1. *Let be an -space and let be a topological space. Then we have if and only if for each nonempty subset of .*

Lemma 3.2. *Let be a locally -space. If is a compact mapping, then for each open entourage there exists such that .*

Lemma 3.3. *Let be a compact topological space and be an -space. Let be a -mapping. Then there exists a continuous function such that for each , ; that is, has a continuous selection.*

*Proof. *Since is a -mapping, there exists a companion mapping such that(i)for each , for each , and for any ;(ii).

Since be a compact, there exists such that . Let be the continuous partition of unity subordinated to the open covering of , then for each and , we have
Define a mapping by , then is continuous and for each , , where . By (3.1), we have . From the definition of -mapping, we obtain for each , . It follows that
This shows that for each , , that is, has a continuous selection. This completes the proof.

Lemma 3.4. *Let be an -space and let be a topological space. If is a -mapping, then .*

*Proof. *Since is a -mapping, we have that for each and in , is also a -mapping. By Lemma 3.3, has a continuous selection function. Then . It follows from Lemma 3.1 that . This completes the proof.

Lemma 3.5. *Let and be two topological spaces and be an -space. If and , then .*

*Proof. *Let be a generalized - mapping with respect to such that is closed in for each . Take and for any , we have , and hence . So is a generalized - mapping with respect to . Since , then has the finite intersection property, and hence has also the finite intersection property and . This completes the proof.

*Remark 3.6. *Lemmas 3.3, 3.4, and 3.5 generalize Lemmas 2, 3, and 5 in [5], respectively.

Corollary 3.7. *Let be a locally -space and is compact and closed. Then has a fixed point in .*

*Proof. *Since is compact, we have is compact in . Without loss of generality, assume that be a basis of symmetric open entourages for the uniformity . It follows from the Lemma 3.2 that for each there exists such that . Hence, for each there exists and . Since and is compact, we may assume that converges to some , and then also converges to . Since is closed, we have . This completes the proof.

Theorem 3.8. *Let be a bounded -space. Assume that be a -set contraction map, . Then contains a precompact -subspace.*

*Proof. *Since is a -set contraction map, , there exists such that for each , we have for each subset of . Take . Let
Then(1) is -subspace of for each ,(2) for each ,(3) for each ,(4) for each .

Thus , as , and hence is a nonempty precompact -subspace of . This completes the proof.

*Remark 3.9. *Theorem 3.8 generalized [5, Theorem 1] from a nonempty bounded convex subset of a Hausdorff topological vector space to a bounded -space. In the process of the proof Theorem 3.8, we call the set a precompact-inducing -subspace of .

The following result is a fixed point theorem for a -set contraction map on the family , which does not need to be a compact map.

Theorem 3.10. *Let be a bounded -subspace of a locally -space , and let be a -set contraction map, and closed with . Then has a fixed point in .*

*Proof. *By the same process of the proof Theorem 3.8, we get a precompact-inducing -subspace of . Since and for each , we have for each . Since as , we have is a nonempty compact subset of .

Since and is a nonempty -subspace of , by Lemma 3.1, we have .

Let be a basis of symmetric open entourages for the uniformity , then there exists such that . We now claim that for there exists such that . If it is false, then there exists such that for all . Let . Define by
then is a compact for each . Next, let , then , we prove that is a generalized - mapping with respect to .

Suppose, on contrary, is not a generalized - mapping with respect to . Then there exists and such that
where . Hence there exist and such that . From the definition of it follows that for all . Noting that is symmetric, we have . Since is -subspace of , we have . By the symmetry of , we obtain and which contradicts the fact for all . Therefore is a generalized - mapping with respect to .

Since and is a generalized - mapping with respect to , the family has the finite intersection property, and so we conclude that . Choose , then for all . Since , hence there exists such that . But , a contradiction. Therefore, we have prove that for each , there exists such that . Let . Since and is compact, we may assume that converges to some , and then also converges to . Since is closed, we have . This completes the proof.

By Lemma 3.4 and Theorem 3.10, we can get the following result immediately.

Corollary 3.11. *Let be a bounded -subspace of a locally -subspace , and let be a -mapping, -set contraction, and closed with . Then has a fixed point in .*

*Remark 3.12. *Theorem 3.10 generalizes [5, Theorem 2] from a nonempty bounded convex subset of a locally convex space to a bounded -subspace of a locally -space and [8, Theorem 3.10]. Corollary 3.11 generalizes [5, Corollary 1] in several aspects.

#### 4. Applications

By Definition 4.3 of Ding [13], we have the following definition.

*Definition 4.1. *Let be an -space. be topological space. is said to be -quasiconvex (resp., -quasiconcave) if for each and for each , the set (resp., ) is an -subspace of .

*Definition 4.2 (see [5]). *Let and be two topological spaces, and let be a function. Then is said to be transfer compactly lower semicontinuous (in short, transfer compactly l.s.c) in if for each and with , there exists such that . is said to be transfer compactly u.s.c in if is transfer compactly l.s.c in .

Now, we establish the following -type theorem for a -set contraction map.

Theorem 4.3. *Let be a bounded -space and let be a topological space. If are two set-valued mappings satisfying the following:*(i)* is a -set contraction map, , with ,*(ii)*for any , is compactly closed in ,*(iii)* is a generalized - mapping with respect to .**Then
**
where is the precompact-inducing -subspace of .*

*Proof. *Let be a basis of symmetric open entourages for the uniformity . By the same process of the proof Theorem 3.10, we get a compact subset of , and , since .

Define by
By condition (ii), is compact in , for each . We now claim that is a generalized - mapping with respect to . Let and for each . By condition (iii), . Thus, we have shown that is a generalized - mapping with respect to . Since , the family has the finite intersection property. And, since is compact, , that is, . This completes the proof.

Theorem 4.4. *Let be a bounded -space and let be a topological space. If are two set-valued mappings satisfying the following:*(i)* is a -set contraction map, , with ,*(ii)*for any , is transfer compactly closed in ,*(iii)* is a generalized - mapping with respect to .**Then
**
where is the precompact-inducing -subspace of .*

*Proof. *Define a mapping by for each , it is easy to see that is also a generalized - mapping with respect to with compactly closed values. By the same process of the proof Theorem 3.10, we get a compact subset of . By Theorem 4.3, . And since for any , is transfer compactly closed in , by Lemma 2.2 [9], we have . This completes the proof.

*Remark 4.5. *Theorem 4.3 generalizes [5, Theorem 3] from a nonempty bounded convex subset of a Hausdorff topological vector space to a bounded -space and [9, Theorem 3.1]. Theorem 4.4 generalizes [5, Theorem 3] in several aspects and [9, Theorem 3.2].

The following results are the generalization of the Ky Fan matching theorem and coincidence theorems.

Theorem 4.6. *Let be a bounded -space. If are two set-valued mappings satisfying the following:*(i)* is a -set contraction map, , with ,*(ii)*for any , is compactly open in ,*(iii)*for the precompact-inducing -subspace of , .**Then for the precompact-inducing -subspace of satisfying the following condition:
*

*Proof. *Let be a basis of symmetric open entourages for the uniformity . By the same process of the proof Theorem 3.10, we get a compact subset of , and , since .

We claim that there exists such that . On the contrary, assume that for any , then . Since is -subspace of , and for any , we have , where . This implies that is a generalized - mapping with respect to . By condition (ii), for any , is compactly closed in . Follows Theorem 4.3, we have , which implies , a contradiction to condition (iii). This completes the proof.

Theorem 4.7. *Let be a locally -space. Assume that*(i)* is a -set contraction map, , with ,*(ii)* is a -mapping.**Then there exists such that and .*

*Proof. *By the same process of the proof Theorem 3.10, we get a compact subset of , and , since .

Let is compact. Then is a -mapping, and by Lemma 3.3, has a continuous selection . So, by Lemma 3.5, we have and so, it follows from Theorem 3.10 that there exists such that ; that is, there exists such that . This completes the proof

Theorem 4.8. *Let be a locally -space, and let be a topological space. Assume that*(i)* is compact and closed,*(ii)* is a -mapping.**Then there exists such that and .*

*Proof. *Since is compact, we have is compact in . By condition (ii), we have is also a -mapping. By Lemma 3.3, has a continuous selection . So, by Lemma 3.5, we have , and so, it follows from Corollary 3.7 that there exists such that ; that is, there exists such that . This completes the proof.

As a consequence of the above Theorem 4.6, we have the following generalized variational inequality.

Corollary 4.9. *Let be a bounded -space, and let be a -set contraction map, , with . If are two real-valued mappings satisfying the following:*(i)* for each ,*(ii)*for fixed , the mapping is lower semicontinuous on for each compact subset of ,*(iii)*for fixed , for each and , such that implies that there exists such that .**Then for the precompact-inducing -subspace of , there exists such that for each .*

*Proof. *Define by
By condition (i), we have , and by condition (ii), is compactly closed for each . The condition (iii) implies that for each and , , and then ; that is, is a generalized -*KKM* mapping with respect to .

Thus, all conditions of Theorem 4.3 are satisfied. By Theorem 4.3, and for the precompact-inducing -subspace of , we have that . Let , and hence we have for each . This completes the proof.

Applying Theorem 4.8 and Lemma 3.4, we have the following result.

Corollary 4.10. *Let be a locally -space, and let be a topological space. If , are two -mappings, and is compact and closed, then there exists such that and .*

From Corollary 4.10, we have the following result.

Corollary 4.11. *Let be a locally -space, and let be a compact -space. If are four real-valued functions, and be two real numbers. Suppose the following conditions hold:*(i)* and for all ,*(ii)*for each , be -quasiconcave on and for each be -quasiconvex on ,*(iii)*for each , be transfer compactly lower semicontinuous in and for each , be transfer compactly upper semicontinuous in ,*(iv)* is upper semicontinuous on .**Then one of the following statements holds:*(1)*there exists such that for each ,*(2)*there exists such that for each ,*(3)*there exists such that and .*

*Proof. *Let and be defined by
By the condition (iii), is transfer compactly open valued on and is transfer compactly open valued on . By the condition (i), we have for each and for each . By the condition (ii), is -subspace of for each , and so is -subspace of for each .

Suppose that the Statements (1) and (2) are false. Then for each and for each . So, we conclude that is a -mapping with a companion mapping and is a -mapping with a companion mapping . By the Condition (iv), is closed. Thus, all conditions of Corollary 4.9 are satisfied. By Corollary 4.9, there exists such that and ; that is, and . This completes the proof.

From Corollary 4.11, we have the following minimax theorem.

Corollary 4.12. *Let be a locally -space, and let be a compact -space. If are four real-valued functions, and let be two real numbers. Suppose the following conditions hold*(i)* for all ,*(ii)*for each , let be -quasiconcave on and for each , let be -quasiconvex on ,*(iii)*for each , let be transfer compactly lower semicontinuous in and for each , let be transfer compactly upper semicontinuous in , *(iv)* is upper semicontinuous on .**Then
*

*Proof. *Let and let
Then for each , there exists such that , and for each , there exists such that . Therefore, Conclusions (1) and (2) of Corollary 4.11 are false. So there exist such that and ; that is
So, by Condition (i), we have
Since is arbitrary positive number, by letting , we get
This completes the proof.

*Remark 4.13. *Theorem 4.6 (resp., Theorems 4.7 and 4.8, Corollaries 4.9, 4.10, 4.11, and 4.12) generalize Theorem 4 (resp., Theorems 7, 6, 5, 8, 9, and 10) in [5] in several aspects.

#### Acknowledgment

The authorβs research was supported by the Scientific Research Foundation of CUIT under Grant KYTZ201114.

#### References

- H. Knaster, C. Kuratowski, and S. Mazurkiewicz, βEin Beweis des Fixpunktsatzes fΓΌr n-dimensional simplexe,β
*Fundamenta Mathematicae*, vol. 14, pp. 132β137, 1929. View at Google Scholar - K. Fan, βA generalization of Tychonoff's fixed point theorem,β
*Mathematische Annalen*, vol. 142, pp. 305β310, 1961. View at Google Scholar - T. H. Chang and C. L. Yen, β
*KKM*property and fixed point theorems,β*Journal of Mathematical Analysis and Applications*, vol. 203, no. 1, pp. 224β235, 1996. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH - L. J. Lin and H. I. Chen, βCoincidence theorems for families of multimaps and their applications to equilibrium problems,β
*Abstract and Applied Analysis*, vol. 2003, no. 5, pp. 295β309, 2003. View at Publisher Β· View at Google Scholar - C. M. Chen and T. H. Chang, βSome results for the family
*KKM(X, Y)*and the $\mathrm{\Xi \xa6}$-mapping,β*Journal of Mathematical Analysis and Applications*, vol. 329, no. 1, pp. 92β101, 2007. View at Publisher Β· View at Google Scholar - A. Amini, M. Fakhar, and J. Zafarani, β
*KKM*mappings in metric spaces,β*Nonlinear Analysis: Theory, Methods & Applications*, vol. 60, no. 6, pp. 1045β1052, 2005. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH - X. P. Ding, βAbstract convexity and generalizations of Himmelberg type fixed-point theorems,β
*Computers & Mathematics with Applications*, vol. 41, no. 3-4, pp. 497β504, 2001. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH - X. P. Ding, βGeneralizations of Himmelberg type fixed point theorems in locally
*FC*-spaces,β*Journal of Sichuan Normal University*, vol. 29, no. 1, pp. 1β6, 2006. View at Google Scholar Β· View at Zentralblatt MATH - X. P. Ding and T. M. Ding, β
*KKM*type theorems and generalized vector equilibrium problems in noncompact*FC*-spaces,β*Journal of Mathematical Analysis and Applications*, vol. 331, no. 2, pp. 1230β1245, 2007. View at Publisher Β· View at Google Scholar - J. C. Jeng, H. C. Hsu, and Y. Y. Huang, βFixed point theorems for multifunctions having
*KKM*property on almost convex sets,β*Journal of Mathematical Analysis and Applications*, vol. 319, no. 1, pp. 187β198, 2006. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH - X. P. Ding, βSystem of generalized vector quasi-equilibrium problems in locally
*FC*-space,β*Acta Mathematica Sinica*. In press. - X. P. Ding, βMaximal element theorems in product
*FC*-spaces and generalized games,β*Journal of Mathematical Analysis and Applications*, vol. 305, no. 1, pp. 29β42, 2005. View at Publisher Β· View at Google Scholar - X. P. Ding, βCollective fixed points and a system of coincidence theorems in product
*FC*-spaces and applications,β*Nonlinear Analysis: Theory, Methods & Applications*, vol. 66, no. 11, pp. 2604β2617, 2007. View at Publisher Β· View at Google Scholar