Abstract and Applied Analysis

Volume 2010 (2010), Article ID 314616, 9 pages

http://dx.doi.org/10.1155/2010/314616

## A Class of Fan-Browder Type Fixed-Point Theorem and Its Applications In Topological Space

College of Mathematics and Statistics, Chongqing Technology and Business University, Chongqing 400067, China

Received 9 March 2010; Accepted 26 April 2010

Academic Editor: W. A. Kirk

Copyright © 2010 Yi-An Chen and Yi-Ping Zhang. 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

A fixed-point theorem is proved under noncompact setting of general topological spaces. By applying the fixed-point theorem, several new existence theorems of solutions for equilibrium problems are proved under noncompact setting of topological spaces. These theorems improve and generalize the corresponding results in related literature.

#### 1. Introduction

Let and be nonempty sets, let be a single-valued mapping, let be a set-valued mapping, let and be functions. The quasi-equilibrium problem is to find such that The was introduced and studied by Noor and Oettli [1]. Cubiotti [2] and Ding [3] proved some existence theorems of solutions for the in finite-dimensional space and topological vector spaces, respectively.

The quasi-equilibrium problem is to find such that The has been studied by many authors; for example, see [4–8] and others.

The and include many optimization problems, Nash-type equilibrium problems, quasivariational inequality problems, quasi-complementary problems, and others as special cases; see [1–8] and the references therein.

In this paper, we first prove a new Fan-Browder-type fixed-point theorem under noncompact setting of general topological spaces. Next, by applying the fixed-point theorem, some new existence theorems of solutions for the and are proved in noncompact setting of general topological spaces. These results include a number of important known results in the fields as special cases.

#### 2. Preliminaries

For a set , we will denote by and the family of all subsets of and the family of all nonempty finite subsets of . For any , let denote the cardinality of . Let denote the standard -dimensional simplex with vertices . If is a nonempty subset of , we denote by the convex hull of the vertices . A set-valued mapping is a function from a set into the power set of , that is, a function with the value for each and the fiber for each . For , let .

A subset of is said to be compactly open (resp., compactly closed) in if, for any nonempty compact subset of , is open (resp., closed) in . The following notions were introduced by Ding [9]. For any given nonempty subset of , we define the compact closure and the compact interior of , denoted by ccl and cint, as If is a subset of a vector space, then co denotes the convex hull of .

If and are two topological spaces and is a set-valued mapping, then is said to be transfer compactly open-valued (resp., transfer compactly closed-valued) on if, for each and for each compact subset of with , (resp., ) implies that there exists a point such that (resp., ). A set-valued mapping 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 that ; see [10]. A multivalued map is said to be transfer open-valued [11], if for any , there exists a such that .

In [12], Deng and Xia introduced the following concept which is crucial to the study of KKM theory in general topological spaces.

Let be a nonempty set and let be a topological space. is said to be a generalized relatively KKM (-KKM) mapping if, for each with , there exists a continuous mapping such that, for each , where is the standard subsimplex of corresponding to .

Throughout this paper, all topological spaces are assumed to be Hausdorff.

In order to prove our main theorems, we need the following results.

Lemma 2.1. *Let and be two topological spaces and a set-valued mapping with nonempty values. Then the following conditions are equivalent:*(I)* has the compactly local intersection property,*(II)*for each nonempty compact subset of and for each , there exists an open subset of (which may be empty) such that and ,*(III)*for each nonempty compact subset of , there exists a set-valued mapping such that for each , is open or empty in and for each , and ,*(IV)*for each nonempty compact subset of and for each , there exists such that and ,*(V)* is transfer compactly open-valued on ,*(VI)*,*(VII)*for each , contains a relatively open subset of ( could be empty set for some ) such that ,*(VIII)*let be two multivalued maps, and is nonempty, and is open in .*

*Proof. *By Lemma of Ding in [10], (I), (II), (III), (IV), and (V) are equivalent. By Lemma of Lin and Ansari in [13], (V) and (VI) are equivalent, and by Ansari in [14], (VI), (VII), and (VIII) are equivalent. This completes our proof.

*Remark 2.2. *Lemma 2.1 includes Lemma of Ding in [10] and Lemma of Ansari in [13] as special cases.

Lemma 2.3 (see [15]). *Let and be topological spaces, let be a nonempty closed subset of , and let be two set-valued mappings such that for each . Suppose that are both transfer compactly open-valued on . Then the mapping defined by
**
is such that is also transfer compactly open-valued on .*

#### 3. Fan-Browder Type Fixed-Point Theorem

Theorem 3.1. *Let be a topological space, let be a nonempty compact subset of , and let be such that*(i)* has nonempty values and satisfies one of the conditions (I)–(VIII) in Lemma 2.1,*(ii)*either,(a) for each , there is a nonempty compact subset of containing such that there is a generalized -KKM mapping satisfying, for each , the fact that implies that , having, moreover, that , or(b) if is a nonempty convex set of a topological vector space, for each , there is a nonempty compact subset of containing such that there is a mapping satisfying, for each , the fact that is convex and, for each , the fact that implies that .*

*Then there exists a point such that .*

*Proof. *By (i) and Lemma 2.1, we have . Since is a nonempty compact subset of , there exists a finite set such that
For the , consider the compact subset in condition (ii) satisfying
By (3.1), we have
Noting that , it follows from (3.2) and (3.3) that . Since is compact, there exists a finite set such that
Since is a generalized -KKM mapping, there exists a continuous mapping such that for each , where is the face of corresponding to . Let be a continuous partition of unity subordinated to the open covering , that is, for each , is continuous, and
such that for all . Define by
Then for all , where . Therefore, we have
It is easy to see that is continuous. By Browder's fixed-point theorem, there exists such that . Let , then we have . This completes our proof.

*Remark 3.2. *Theorem 3.1 generalized Theorem of Ding in [15] by dropping all the contractible conditions. Theorem 3.1 is also a noncompact variant of [8, Theorem ] in general topological spaces.

#### 4. Equilibrium Existence of and

First, we prove the following equilibrium existence theorems of .

Theorem 4.1. *Let be a topological space, let be a nonempty compact subset of , and let be a nonempty set. Let , , and be such that*(i)* has nonempty values on and satisfies one of the conditions (I)–(VIII) in Lemma 2.1,*(ii)*the set is nonempty closed in ,*(iii)*the mapping is transfer compactly open-valued, where is defined by
*(iv)*for each , there is a nonempty compact subset of containing such that there is a generalized -KKM mapping satisfying, for each , the fact that implies that and, for each , the fact that implies that . Moreover, for each , if , then there exists such that ; if , then there exists such that .**Then there exists such that**
that is, is a solution of the .*

*Proof. *Define a mapping by
From conditions (i), (iii) and Lemma 2.1, it follows that the mappings are both transfer compactly open-valued on . Note that for each and is nonempty closed by (ii); by Lemma 2.3, is also transfer compactly open-valued on . From condition (iv), it follows that, for each and for each , implies that . Moreover, . Hence, condition (ii) of Theorem 3.1 holds. Now assume that, for each , . Then for each , by (i). It is easy to see that all conditions of Theorem 3.1 are satisfied. By Theorem 3.1, there exists such that . By the definition of and , we must have . It follows that . In particular, we obtain which is impossible. Therefore, there exists such that , that is, and for all . This completes the proof.

*Remark 4.2. *Theorem 4.1 generalized Theorem .1 of Ding in [15] by dropping all the contractible conditions. Theorem 4.1 also improved and generalized [3, Theorem ] and [2, Theorem ] from topological vector spaces to general noncompact topological spaces without linear structure under much weaker assumptions.

Theorem 4.3. *Let and be two topological spaces and let be a nonempty compact subset of . Let , , and be such that*(i)* has nonempty values on such that is compactly open-valued,*(ii)*the set is nonempty closed in ,*(iii)* and are continuous,*(iv)*for each , there is a nonempty compact subset of containing such that there is a generalized -KKM mapping satisfying, for each , the fact that implies that , where is defined by , and, for each , the fact that implies that . Moreover, for each , if , then ; if , then there exists satisfying .**Then there exists such that
**
that is, is a solution of the .*

*Proof. *Since and are both continuous, we have that, for each , = is open in , and hence, has compactly open values. It follows that the mapping defined by is such that the mapping also has compactly open values on , and hence, it is also transfer compactly open-valued on . By condition (iv), for each , if , we have , and hence, there exists such that ; if , we have and . Hence, condition (iv) implies that condition (iv) of Theorem 4.1 is satisfied. It is easy to see that all conditions of Theorem 4.1 are satisfied. By Theorem 4.1, the conclusion of Theorem 4.3 holds. This completes the proof.

Second, we prove the following equilibrium existence theorem of .

Theorem 4.4. *Let be a topological space, let be a nonempty compact subset of . Let and be such that*(i)* has nonempty values on such that has compactly open values and the mapping defined by (the closure of ) is upper semicontinuous,*(ii)* is a continuous function,*(iii)*for each , there is a nonempty compact subset of containing such that there is a generalized -KKM mapping satisfying, for each , the fact that implies that , where and , and for each , the fact that implies that . Moreover, for each , if , then ; if , then there exists satisfying .**Then there exists such that
**
If one we further assumes that for all , then one has that
**
that is, is a solution of the .*

*Proof. *Since is upper semicontinuous with closed values, the set must be closed in . By letting , with being the identity mapping and being in place of , it is easy to see that all conditions of Theorem 4.3 are satisfied. By Theorem 4.3, there exists such that and , for all . If for all , then we must have and , for all , that is, is a solution of the .

*Remark 4.5. *Theorem 4.4 generalized Theorem of Ding in [15] by dropping all the contractible conditions. Theorem 4.4 is also a noncompact variant of [4, Theorem ], [5, Theorem ], Theorem of [6, 7], and [8, Theorem ] in general topological spaces.

From Theorem 4.4, we obtain the following existence result for generalized quasi-equilibrium problems.

Theorem 4.6. *Let and be two topological spaces and let be a nonempty compact subset of . Let have a continuous selection . Let and be such that*(i)* satisfies condition (i) of Theorem 4.4,*(ii)* is a continuous function,*(iii)*for each , there is a nonempty compact subset of containing such that there is a generalized -KKM mapping satisfying, for each , the fact that implies that , where and , and, for each , the fact that implies that . Moreover, for each , if , then ; if , then there exists satisfying .**Then there exists and such that
**
If one further assumes that for all , then one has that
*

*Proof. *Define by
Then the conclusion of Theorem 4.6 holds from Theorem 4.4.

*Remark 4.7. *Theorem 4.6 generalized Theorem of Ding in [15] by dropping all the contractible conditions. Theorem 4.6 is a noncompact variant of [8, Corollary ] and [16, Theorem ] in general topological spaces.

#### Acknowledgment

The authors would like to express their thanks to the Editor-in-Chief and the anonymous referees for their valuable comments and suggestions on a previous draft, which resulted in the present version of the paper.

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