International Journal of Mathematics and Mathematical Sciences

Volume 2012 (2012), Article ID 862791, 12 pages

doi:10.1155/2012/862791

Research Article

Fixed Point Theorems for Asymptotically Contractive Multimappings

M. Djedidi¹ and K. Nachi²

¹Department of Mathematics, El Oued University, Alger, Algeria

²Department of Mathematics, Oran University, Oran, Algeria

Received 19 February 2012; Accepted 4 April 2012

Academic Editor: Yonghong Yao

Copyright © 2012 M. Djedidi and K. Nachi. 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 present fixed point theorems for a nonexpansive set-valued mapping from a closed convex subset of a reflexive Banach space into itself under some asymptotic contraction assumptions. Some existence results of coincidence points and eigenvalues for multimappings are given.

1. Introduction

In this paper, we investigate fixed point theorems for nonexpansive multifunctions (relations, multimaps, set-valued mappings, or correspondences) satisfying some asymptotic condition. This study has been the subject of numerous works [1–6] for an asymptotically contractive mapping. Our aim here is to obtain some generalization by using the notion of (semi-) asymptotically contractive multimappings which is introduced below. For doing so, we need to fix some notations and conventions. Given a normed vector space (n.v.s.) , the open ball with center and radius in is denoted by ; the closed, unit ball is denoted by . For any subsets , , we set

Recall that a multifunction is a contractive (resp., nonexpansive) multifunction on if there exists such that for any , one has

Note that when , where is a mapping, is a contraction with rate (resp., nonexpansive) on if and only if is a contraction with rate (resp., nonexpansive) mapping on for any

The existence theorem of fixed points for contractive multifunction is well known (see [7]). More generally, a generalization of Picard-Banach theorem to pseudo-contractive multifunction is given in ([8, 9], [10, Lemma 1, page 31] and [11, Proposition 2.5]). Let us recall that result for the sake of clarity.

Proposition 1.1 (see [8, 10, 11]). Let be a complete metric space, and let be a multifunction with closed, nonempty values. Suppose that is pseudo--contractive with respect to some ball for some (i.e., for and . Then the fixed point set of is nonempty and

In this work, the reflexivity of Banach spaces and the property of demiclosedness of multifunctions play an important role to have fixed points results. Let us recall that is said to be demiclosed if its graph is sequentially closed in the product of the weak topology on with the norm topology on a Banach space , that is,

where .

It is well known that if is nonexpansive on , a closed convex subset of a uniformly convex Banach space , then is demi-closed ([6], [12, Proposition 10.9, page 476]), where a Banach space is uniformly convex if and only if for any , there exists such that for any , one has

As examples, every Hilbert space is uniformly convex, the spaces and are uniformly convex for ( is a domain in ), which is not the case for . It is also well known that every uniformly convex Banach space is reflexive ([12, Proposition 10.7, page 475]).

2. Fixed Point Theorem under Asymptotical Conditions

The following definition generalizes the notion of asymptotically contractive mapping to set-valued mappings. Note that the meaning of the word “asymptotic” is not related to the iterations of the multimapping as in [13] but bears on the behavior of the set-valued mapping at infinity. This behavior can be studied using concepts of asymptotic cones and asymptotic compactness as in [14–19].

Definition 2.1. Let be a subset of a Banach space , and let be a multimapping with nonempty values. We say that is asymptotically contractive on if there exists such that

Let us note that when , where is a mapping, we get the definition of the asymptotically contractive mapping on given in [6] as a variant of the notion introduced in [17].

If for any (particularly, if is a multimapping with bounded values), then the condition (2.1) is independent of the choice of : indeed, let (). Since , we have

Proposition 2.2 is a multivalued version of the main result of [6].

Proposition 2.2. Let be a reflexive Banach space and a (nonempty) closed convex subset of . Let be a multifunction with closed and nonempty values such that is nonexpansive on . Assume that is asymptotically contractive on at with bounded. If and is demi-closed, then admits a fixed point.

Proof. Let be a sequence in such that . For any , we define a multifunction by setting

It is clear that for any and . On the other hand, for and , from (2.3), there exists such that . Applying (1.2) since is nonexpansive, there exists satisfying . Thus, for , one has

Then is a contraction with rate on . The Nadler's theorem [7] ensures that each multivalued admits a fixed point in . So, from (2.3) and for some , one has

Observe that if the sequence has a bounded subsequence, the proof is finished. Indeed, taking a subsequence if necessary, admits a weak limit ( is closed, convex in the reflexive space ). As is bounded (by equality (2.5)), the sequence converges to 0. We conclude that , that is, is a fixed point of .

Thus, to complete the proof of the proposition, let us show that the sequence is bounded. If this is not the case, taking a subsequence if necessary, we may assume that . As condition (2.1) is satisfied, there exist and such that

For large , we have and , so that

There exists then a sequence in such that

On the other hand, from equalities (2.5) and (2.6), we get

Dividing by , we obtain

Passing to the limit and using the fact that is bounded, and , a contradiction follows. So the sequence has a bounded subsequence and the proposition is proved.

The preceding results can be applied to coincidence properties between two multifunctions. Let us give first a precise definition.

Definition 2.3. Let be a set, let be a linear space, and let be two multimappings. We say that and present a coincidence on if there exists such that

The point is called a coincidence point of and .

Note that if and for all , we obtain the definition of a fixed point of the multifunction . Also observe that the relation can be written , so that two mappings present a coincidence on if and only if there exists such that .

The following corollary is an immediate consequence giving the existence of a fixed point of a sum (resp., a coincidence point of two multifunctions).

Corollary 2.4. Let be a nonempty closed convex cone of a reflexive Banach space . Let be a -contraction (resp., be a -contraction) set-valued mapping on with closed and nonempty values. Assume that is demi-closed and there exists such that , are bounded and one has

Then the multifunction admits a fixed point on , which is a coincidence point of and .

Proof. Since for any subsets , , , of one has

the multimapping is nonexpansive and

Since is a convex cone, is contained in , the result is a consequence of Proposition 2.2.

Observe that if is a fixed point of such that and if is a convex cone, then is a common fixed point of and .

Corollary 2.5. Let be a nonempty closed convex cone of a Banach uniformly convex space . Let , be a -contraction (resp., be a -contraction) set-valued mapping on . Assume that

Then the multifunction admits a fixed point on , which is a coincidence point of and .

The notion of eigenvalue is very important in nonlinear analysis. It has many applications as the notion of fixed point. We present now some results related to eigenvalues. We obtain in particular an existence result for eigenvalues of nonexpansive mappings.

Let us recall that a real number is said to be an eigenvalue for a set-valued mapping if there exists an element , such that . When , where is a mapping, we obtain the usual definition of an eigenvalue for a mapping.

The next proposition gives an existence result.

Proposition 2.6. Let be a closed convex cone of a reflexive Banach space . Let and let be a nonexpansive set-valued mapping on whose values are nonempty, closed and . Assume that is demi-closed and that there exists such that is bounded and one has

Then is an eigenvalue for associated to an eigenvector . And if is a cone, then is a fixed point of .

Proof. By taking with , we have ( a convex cone), bounded, and so that is demi-closed. Moreover, using the inequality (2.14), we get

Therefore, there exists a fixed point of , that is, we have

so that and (). Remark that if is a cone, we get from (2.19) that , that is, is a fixed point of .

Corollary 2.7. Let be a closed convex cone of an uniformly convex Banach space . Let , and let be a nonexpansive mapping on such that . Assume that there exists such that

Then is an eigenvalue for associated to an eigenvector .

2.1. Asymptotic Contraction Condition with Respect to Semi-Inner Product

In this section, we will present some fixed points results for multimappings under another asymptotic condition. This study is inspired by the work [20]. For this aim, let us introduce some definitions. Recall that a semi-inner product on a vector space is a function satisfying the following properties for any and :

It is proved in [21, 22] that a semi-inner-product space is a normed linear space with the norm and every Banach space can be endowed with different semi-inner-products unless for Hilbert spaces where is the inner product. We say that the semi-inner-product on an n.v.s. is compatible with the norm if .

Let us introduce the following definition of asymptotically contractive multimappings with respect to (w.r.t) a semi-inner product on a Banach .

Definition 2.8. Let be a subset of a Banach space , and let a multimapping with nonempty values. We say that is asymptotically contractive on with respect to if there exists such that

Note that when , where is a mapping, we get a definition of the asymptotically contractive mapping on as a variant of the notion introduced in [20]. Indeed, condition (2.22) becomes in this case as follows: there exists so that

Observe that if is a Hilbert space endowed with the scalar product noted by , the above inequality is then written

and a map is said to be scalarly asymptotically contractive on if (2.24) is satisfied for some .

In the sequel, we consider only semi-inner products on Banach spaces which are compatible with the norm . The next theorem is a multivalued version of the main result of [20, Theorem 3.2] for correspondances.

Theorem 2.9. Let be a reflexive Banach space and a (nonempty) closed convex subset of . Let be a nonexpansive multifunction on with closed and nonempty values. Assume that is asymptotically contractive on with respect to . If and is demi-closed, then admits a fixed point on .

Proof. Let such that (2.22) is satisfied, and let be a sequence in such that . For any , we define a multifunction by setting

It is clear that for any and . On the other hand, for and , from (2.25), there exists such that . Applying (1.2) since is nonexpansive and is closed, convex in the reflexive space , there exists satisfying . Thus, for , one has

Then is a contraction with rate on . The Nadler's theorem [7] ensures that each multivalued admits a fixed point in . So, from (2.25) and for some , one has

As in the proof of Proposition 2.2, it suffices to show that is bounded. Suppose on the contrary, by taking a subsequence if necessary, that . As condition (2.24) is satisfied, there exist and such that

For large , we have and with so that

From the properties of the semi-inner product, we get

Dividing by and taking the limit, we obtain , which leads to a contradiction and the conclusion of the proposition follows.

Let us remark that when , where is a mapping, we get the following corollary, which is a variant of [6, Proposition 1].

Corollary 2.10. Let be a reflexive Banach space and a (nonempty) closed convex subset of . Let be a nonexpansive function on . Assume that for some , one has

If and is demi-closed, then admits a fixed point.

We introduce now the following concept, which generalizes the definition of -asymptotically bounded maps to multimaps.

Definition 2.11. Let be a Banach space, a (nonempty) closed convex subset of , a multifunction with nonempty values, and let . We say that is -symptotically bounded on if for some , there exist such that for any and one has

We want to give a fixed point result for a nonexpansive multimapping when satisfies some asymptotic contraction condition under the assumption that is -asymptotically bounded multifunction. More precisely, we have the following proposition.

Proposition 2.12. Let be a reflexive Banach space and a (nonempty) closed convex subset of . Let be a nonexpansive multifunction with (nonempty) closed values, and let be a -asymptotically bounded multifunction on at with . Assume that there exist , such that for some one has

If and is demi-closed, then admits a fixed point on .

Proof. It is enough to prove thatthe condition (2.22) is satisfied. Let , , such that (2.32) and(2.33) are hold. Consider and . We have then, for some ,

We conclude that

Hence by Theorem 2.9, admits a fixed point.

Corollary 2.13. Let be a reflexive Banach space and a (nonempty) closed convex cone of . Let be a -contraction mapping with and a -contraction multifunction with closed and nonempty values. Assume that is -asymptotically bounded multifunction at with .

If and is demi-closed, then admits a fixed point on .

Proof. Let us verify the assumptions of Proposition 2.12 with a semi-inner-product compatible with the norm in . It is clear that is nonexpansive on and as is a convex cone, . Consider now and . There exists then some such that . By the properties of , we get the following inequalities:

Thus for all such that , we obtain

Hence the property (2.33) is satisfied with the constant and the corollary follows.

The following result is close to Corollary 2.7 giving the existence of an eigenvalue of a nonexpansive and -asymptotically bounded multifunction.

Corollary 2.14. Let be a nonempty closed convex cone of a real reflexive Banach space . Let , , and let be a nonexpansive multifunction with closed and nonempty values on such that and . Assume that is demiclosed and that is -asymptotically bounded multifunction with . Then is an eigenvalue of the multifunction associated to an eigenvector .

Proof. Since all assumptions of the above corollary are satisfied, there exists or . Thus and as . And the conclusion follows.

Acknowledgment

This paper was supported by the National Research Program of the Ministry of Higher Education and Scientific Research (Algeria), under Grant no. 50/03.

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