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Abstract and Applied Analysis
Volume 2013 (2013), Article ID 346094, 4 pages
http://dx.doi.org/10.1155/2013/346094
Research Article

Set Contractions and KKM Mappings in Banach Spaces

Department of Mathematics, Karaj Branch, Islamic Azad University, Karaj, Iran

Received 10 April 2013; Accepted 17 June 2013

Academic Editor: Chengming Huang

Copyright © 2013 A. Razani and N. Karamikabir. 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

Some fixed point theorems for generalized set contraction maps and KKM type ones in Banach spaces are presented. Moreover, a new generalized set contraction is introduced. As an application, some coincidence theorems for KKM type set contractions are obtained.

1. Introduction

Let be a Banach space and Thus , , , , , , , and denote the classes of all bounded, closed, convex, compact, connected, closed-bounded, compact-convex, and relatively compact subsets of , respectively [1]. Let and be two Banach spaces. A multivalued mapping is said to be (i)upper semicontinuous if and only if for every closed subset of , the set is a closed subset of ,(ii)closed if its graph is a closed subset of , (iii)compact if is a compact subset of . The first fixed point theorem for multivalued mappings is due to Kakutani in Banach spaces, in 1941 [2]. He proved a generalization of Brouwer’s fixed point theorem to the multivalued mappings.

Theorem 1 (see [2]). Let be a compact subset of a Banach space and let be an upper semicontinuous multivalued operator. Then has a fixed point.

The above theorem has been extended in the literature by generalizing or modifying the domain space , domain set , and the nature of the multivalued operator . Here, compactness plays an essential role. The following definition of measure of noncompactness on a bounded subset of the Banach space is given by Dhage in 2010.

Definition 2 (see [1]). A function is called a measure of noncompactness if it satisfies (i), (ii)if then , (iii), where denotes the closure of , (iv), where denotes the convex hull of , (v)if is a decreasing sequence of sets in satisfying , then the limiting set is nonempty.

Definition 3 (see [1]). A multivalued mapping is called -set Lipschitz if there exists a continuous nondecreasing function such that for all with , where . Sometimes we call the function to be a -function of on . In the spatial case, when , , is called a -set Lipschitz mapping and if , then is called a -set contraction on . Further, if for , then is called a nonlinear -set contraction on .

Dhage proved a generalization of Theorem 1 under weaker upper semicontinuity conditions in 2010 [1].

Theorem 4. Let be a nonempty, closed, convex, and bounded subset of a Banach space and let be a closed and nonlinear -set contraction. Then has a fixed point.

Lemma 5 (see [3]). If is a -function with for , then for all .

Recall that a function is called a comparison function if is increasing and for all [4]. As a consequence, for any , is continuous at , and .

A function is called a (c)-comparison function if is increasing and there exist , and a convergent series of non-negative terms such that for and any [5]. If is a (c)-comparison function, then is a comparison function [5]. So, we can define -set contraction as follows.

Definition 6. A multivalued mapping is called -set contraction if there exists a continuous (c)-comparison function such that for all with .

Let be a metric space. The Hausdorff metric induced by the metric and defined as follows

Definition 7 (see [6]). Let . A multivalued contractive (-contractive) map is a map such that for all with (and for some , , resp.), .

Theorem 8 (see [7]). Let be a nonempty compact and connected metric space and let be a multivalued -contractive map, then has a fixed point.

We need the following definitions of KKM theory in the sequel [8].

Assume that is a convex subset of a topological vector space and is a topological space. If and are two multivalued maps and for each , , then is called generalized KKM mapping with respect to , where denote the family of all nonempty finite subsets of . More generally, if satisfies the requirement that for any generalized KKM mapping with respect to and the family has the finite intersection property, then is said to have the KKM property. Let

Lemma 9 (see [8]). Let be a nonempty convex subset of Hausdorff topological vector space . Then whenever and is a nonempty convex subset of .

Chen and Chang obtained some fixed point theorems for KKM type set contraction mappings in various spaces [912]. In 2010, Amini-Harandi et al. introduced generalized set contraction on topological spaces [13].

In Section 2, we present some fixed point theorems for generalized set contractions which are -contractive (KKM -contractive) multivalued maps. In the first step of Section 3, we introduce a new type of generalized set contraction and then prove that the results of Section 2 hold for them. Section 4 is devoted to some KKM coincidence theorems as applications of these results.

2. Generalized Set Contractions

In this section by applying Theorem 8, we obtain some fixed point theorems for -contractive multivalued maps which are either generalized set contraction or KKM type ones. In all cases, the multivalued maps are not necessarily compact values. We consider measurement of noncompactness in Definition 2.

Definition 10 (see [13]). A multivalued mapping is said to be a generalized set contraction, if for each there exists such that for with , there exists such that .

Lemma 11 (see [13]). Let be a topological space and let be a measure of noncompactness on . Suppose that is a generalized set contraction on . Then for every subset of for which and , one has .

Proposition 12. Let be a Hausdorff topological space. If is a decreasing sequence of closed and connected sets in such that , then is nonempty, compact, and connected.

Proof. Clearly, is a nonempty, closed, and compact subset of . Let and be two nonempty, disjoint, and closed sets so that . We can find disjoint open sets and around and , respectively. For every , is nonempty. If not, then and are nonempty and , which cannot happen. The collection of is also a decreasing sequence of nonempty closed sets. Since then as . Hence, is nonempty, that is, , which is a contradiction.

Theorem 13. Let be a nonempty, bounded, closed, and connected subset of Banach space . If is an -contractive and generalized set contraction, then has a fixed point and the set of fixed points of is compact.

Proof. Let and for all . Since is an -contractive map, then is continuous and . On the other hand, , so we have and . Since is continuous with closed and connected values, then by [14, Lemma 1.6], is connected. Hence, which is closure of connected set for all , is connected. But , and by Lemma 11 we have . Therefore, by Proposition 12, is nonempty compact and connected. Since , then the desired conclusion followed by an application of Theorem 8 to the multivalued map .
Let . We claim that . If then . Since is a generalized set contraction, then for , there exists such that for with , there exists such that , which is a contradiction. So is relatively compact, and since is continuous, then is a compact subset of .

If is a -set contraction, then is a generalized set contraction, but the converse is not true [13]. Therefore, we have the following result.

Corollary 14. Let be a nonempty, bounded, closed, and connected subset of Banach space . If is an -contractive and -set contraction, then has a fixed point.

Corollary 15. Let be a nonempty, closed, and bounded subset of Banach space . If is an -contractive and generalized set contraction, then there exists a compact subset of such that .

Theorem 16. Let be a nonempty, closed, and bounded subset of Banach space . If is an -contractive and generalized set contraction with nonempty closed and bounded values, then has a fixed point.

Proof. By Corollary 15, there exists a compact subset of such that . Let . Hence there exists a finite subset of such that , where . Define a map by for all ; then is closed for each and . Since and , then by Lemma 9, . Thus, is not a generalized KKM map with respect to . Hence, there exists of such that . Thus there exists such that , that is, for some and , so for . Since , then . Since is a compact subset of , then converges to some as . Consequently, converges to as . Since is continuous, then by [14, Lemma 1.6] we have .

3. Asymptotic Generalized Set Contractions

In this section, we define a new type of set contraction in Banach spaces. Then we prove that the results of Section 2 hold for them. Also, we conclude some fixed point theorems for nonlinear -set contractions.

Definition 17. Let be a nonempty, closed, and bounded subset of a Banach space . A multivalued mapping is said to be an asymptotic generalized set contraction, if there exists a sequence of functions from in to itself satisfies (i)for each , there exists and such that for all , (ii).

Theorem 18. Let be a nonempty, bounded, closed, and connected subset of Banach space . If is an -contractive and asymptotic generalized set contraction, then has a fixed point.

Proof. Define a sequence of sets in such that and for all . As the proof of Theorem 13, , , and is connected for all . If there exists an integer such that , then is a compact and connected set and invariant under . Thus Theorem 8 implies that has a fixed point. So we assume that for all . Define and . If , by Definition 17, there exists , and such that for all and , so which is a contradiction. Hence as . Now by Proposition 12, is nonempty, compact, and connected. Moreover . So by Theorem 8, the multivalued map has a fixed point.

Corollary 19. Let be a nonempty, closed, and bounded subset of Banach space . If is an -contractive and asymptotic generalized set contraction, then there exists a compact subset of such that .

The proof of following theorem is similar to that of Theorem 16; hence it is omitted.

Theorem 20. Let be a nonempty, closed, and bounded subset of Banach space . If is an -contractive and asymptotic generalized set contraction with nonempty closed and bounded values, then has a fixed point.

Proposition 21. Let be a nonempty, closed, and bounded subset of Banach space . If is a nonlinear -set contraction, then is an asymptotic generalized set contraction.

Proof. Let be a nonlinear -set contraction with -function . Define for all . Clearly, , on the other hand, by Lemma 5 we have as . Thus is an asymptotic generalized set contraction.

Applying Proposition 21, Theorems 18 and 20, it is easy to conclude the following results.

Corollary 22. Let be a nonempty, bounded, closed, and connected subset of Banach space . If is an -contractive and nonlinear -set contraction, then has a fixed point.

Corollary 23. Let be a nonempty, closed, and bounded subset of Banach space . If is an -contractive and nonlinear -set contraction with nonempty closed and bounded values, then has a fixed point.

Remark 24. Since every -set contraction is a nonlinear -set contraction, then Theorem 4, Corollaries 22 and 23 hold for these mappings.

4. Some Applications in KKM Theory

In this section we obtain two coincidence theorems for KKM type set contractions.

Theorem 25. Let be a nonempty, closed, bounded, and convex subset of Banach space . If and are two multivalued mappings satisfying (i), (ii) is a generalized (an asymptotic) set contraction and -contractive map, (iii)for each compact subset of and any , is open in , then, there exists such that and .

Proof. By Corollary 15 (Corollary 19), there exists a compact subset of such that . Since and is compact, then and for some . Define a map by for each , then . Therefore, is not a generalized KKM map with respect to . So there exists a finite subset of such that . Hence, there exist and such that . Thus and so for . Since is convex, then and so .

By Proposition 21, Corollary 19, and slight modification of the proof of Theorem 25, we have the following theorem.

Theorem 26. Let be a nonempty, closed, bounded, and convex subset of Banach space . If and are two multivalued mappings satisfying (i) is a nonlinear -set contraction. (ii)for each compact subset of and any , is open in , then, there exists such that and .

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