Fixed Point Theory and Applications

, 2010:340631 | Cite as

Measures of Noncircularity and Fixed Points of Contractive Multifunctions

Open Access
Research Article

Abstract

In analogy to the Eisenfeld-Lakshmikantham measure of nonconvexity and the Hausdorff measure of noncompactness, we introduce two mutually equivalent measures of noncircularity for Banach spaces satisfying a Cantor type property, and apply them to establish a fixed point theorem of Darbo type for multifunctions. Namely, we prove that every multifunction with closed values, defined on a closed set and contractive with respect to any one of these measures, has the origin as a fixed point.

Keywords

Banach Space Fixed Point Theorem Hausdorff Distance Hausdorff Measure Continuous Linear 

Open image in new window

1. Introduction

Let Open image in new window be a Banach space over the field Open image in new window . In what follows, we write Open image in new window for the closed unit ball of Open image in new window . Denote by Open image in new window the collection of all subsets of Open image in new window and consider
For Open image in new window , define their nonsymmetric Hausdorff distance by
and their symmetric Hausdorff distance (or Hausdorff-Pompeiu distance) by

where Open image in new window denotes the closure of Open image in new window .

Around 1955, Darbo [1] ensured the existence of fixed points for so-called condensing operators on Banach spaces, a result which generalizes both Schauder fixed point theorem and Banach contractive mapping principle. More precisely, Darbo proved that if Open image in new window is closed and convex, Open image in new window is a measure of noncompactness, and Open image in new window is continuous and Open image in new window -contractive, that is, Open image in new window for some Open image in new window , then Open image in new window has a fixed point. Below we recall the axiomatic definition of a regular measure of noncompactness on Open image in new window ; we refer to [2] for details.

Definition 1.1.

A function Open image in new window will be called a regular measure of noncompactness if Open image in new window satisfies the following axioms, for Open image in new window , and Open image in new window :

(1) Open image in new window if, and only if, Open image in new window is compact.

(2) Open image in new window , where Open image in new window denotes the convex hull of Open image in new window .

(3)(monotonicity) Open image in new window implies Open image in new window .

(4)(maximum property) Open image in new window .

(5)(homogeneity) Open image in new window .

(6)(subadditivity) Open image in new window .

A regular measure of noncompactness Open image in new window possesses the following properties:

is the diameter of Open image in new window (cf. [2, Theorem  3.2.1]).

(2)(Hausdorff continuity) Open image in new window [2, page 12].
  1. (3)

    (Cantor property) If Open image in new window is a decreasing sequence of closed sets with Open image in new window , then Open image in new window , and Open image in new window [3, Lemma  2.1].

     

In Sections 2 and 3 of this paper we introduce two mutually equivalent measures of noncircularity, the kernel (that is, the class of sets which are mapped to 0) of any of them consisting of all those Open image in new window such that Open image in new window is balanced. Recall that Open image in new window is balanced provided that Open image in new window for all Open image in new window with Open image in new window . For example, in Open image in new window the only bounded balanced sets are the open or closed intervals centered at the origin. Similarly, in Open image in new window as a complex vector space the only bounded balanced sets are the open or closed disks centered at the origin, while in Open image in new window as a real vector space there are many more bounded balanced sets, namely all those bounded sets which are symmetric with respect to the origin.

Denoting by Open image in new window either one of the two measures introduced, in Section 4 we prove a result of Darbo type for Open image in new window -contractive multimaps (see Section 4 for precise definitions). It is shown that the origin is a fixed point of every Open image in new window -contractive multimap Open image in new window with closed values defined on a closed set Open image in new window such that Open image in new window .

2. The E-L Measure of Noncircularity

The definition of the Eisenfeld-Lakshmikantham measure of nonconvexity [4] motivates the following.

Definition 2.1.

where Open image in new window denotes the balanced hull of Open image in new window , that is,

By analogy with the Eisenfeld-Lakshmikantham measure of nonconvexity, we shall refer to Open image in new window as the E-L measure of noncircularity.

Next we gather some properties of Open image in new window which justify such a denomination. Their proofs are fairly direct, but we include them for the sake of completeness.

Proposition 2.2.

In the above notation, for Open image in new window , and Open image in new window , the following hold:

(1) Open image in new window if, and only if, Open image in new window is balanced.

(2) Open image in new window .

(3) Open image in new window .

(4) Open image in new window .

(5) Open image in new window .

is the diameter of Open image in new window .

(7) Open image in new window .

Proof.

Let Open image in new window denote the closed balanced hull of Open image in new window . The identity

holds. Indeed, Open image in new window implies Open image in new window . Conversely, Open image in new window implies Open image in new window .

(1)By definition, Open image in new window if, and only if, Open image in new window or, equivalently, Open image in new window . This means that Open image in new window , which by (2.5) occurs if, and only if, Open image in new window is balanced.

(2)In view of (1.4) and (2.5),

It only remains to prove that Open image in new window . Suppose Open image in new window , so that Open image in new window . The set Open image in new window being convex, it follows that Open image in new window , whence Open image in new window . From the arbitrariness of Open image in new window we conclude that Open image in new window .

whence Open image in new window . The arbitrariness of Open image in new window yields Open image in new window .

(4)For Open image in new window , this is obvious. Suppose Open image in new window . If Open image in new window then Open image in new window , whence Open image in new window . Thus Open image in new window , and from the arbitrariness of Open image in new window we infer that Open image in new window . Conversely, assume Open image in new window . Then Open image in new window , whence Open image in new window . Therefore Open image in new window , and from the arbitrariness of Open image in new window we conclude that Open image in new window .

(5)Let Open image in new window and choose Open image in new window such that Open image in new window , Open image in new window and Open image in new window . Then Open image in new window , Open image in new window and the fact that Open image in new window is a balanced set containing Open image in new window , imply Open image in new window , so that Open image in new window . The arbitrariness of Open image in new window yields Open image in new window .

we obtain

where for the validity of the latter estimate we have assumed Open image in new window .

(7)It is enough to show that
since then, by symmetry,
whence the desired result. Now
To complete the proof we will establish that Open image in new window . Indeed, suppose Open image in new window , and let Open image in new window , with Open image in new window and Open image in new window . Then there exists Open image in new window such that Open image in new window . Consequently, for Open image in new window we have

This means that Open image in new window , so that Open image in new window . From the arbitrariness of Open image in new window we conclude that Open image in new window .

Remark 2.3.

The identity Open image in new window may not hold, as can be seen by choosing Open image in new window . In fact, Open image in new window is balanced, while Open image in new window is not. Therefore, Open image in new window .

In general, the identity Open image in new window does not hold either. To show this, choose Open image in new window and Open image in new window , respectively, as the upper and lower closed half unit disks of the complex plane. Then Open image in new window equals the closed unit disk, which is balanced, while Open image in new window , Open image in new window are not. Thus, Open image in new window .

Note that Open image in new window is not monotone: from Open image in new window and Open image in new window , it does not necessarily follow that Open image in new window . Otherwise, Open image in new window would imply Open image in new window , which is plainly false since not every subset of a balanced set is balanced.

3. The Hausdorff Measure of Noncircularity

The following definition is motivated by that of the Hausdorff measure of noncompactness (cf. [2, Theorem  2.1]).

Definition 3.1.

We define the Hausdorff measure of noncircularity of Open image in new window by

where Open image in new window denotes the class of all balanced sets in Open image in new window .

In general, Open image in new window , as the next example shows.

Example 3.2.

If Open image in new window is any closed bounded balanced set in Open image in new window , we have
we obtain

Thus, Open image in new window .

Next we compare the measures Open image in new window and Open image in new window and establish some properties for the latter. Again, most proofs derive directly from the definitions, but we include them for completeness.

Proposition 3.3.

In the above notation, for Open image in new window , and Open image in new window , the following hold:

(1) Open image in new window , and the estimates are sharp.

(2) Open image in new window if, and only if, Open image in new window is balanced.

(3) Open image in new window .

(4) Open image in new window .

(5) Open image in new window .

(6) Open image in new window .

is the diameter of Open image in new window .

(8) Open image in new window .

and the arbitrariness of Open image in new window yields Open image in new window . Example 3.2 shows that this estimate is sharp. In order to exhibit a set Open image in new window such that Open image in new window , let Open image in new window . Then Open image in new window , and
On the other hand, let Open image in new window be any closed bounded balanced subset of Open image in new window . For a fixed Open image in new window , there holds
Therefore,
  1. (2)

    Let Open image in new window . As we just proved, Open image in new window if, and only if, Open image in new window . In view of Proposition 2.2, this occurs if, and only if, Open image in new window is balanced.

     
  2. (3)
    By (1.4), there holds
     
Now we only need to show that Open image in new window . Assuming Open image in new window , choose Open image in new window for which Open image in new window , so that
The sum of convex sets being convex, we infer
Thus we get
whence Open image in new window and, Open image in new window being balanced, also Open image in new window . From the arbitrariness of Open image in new window we conclude that Open image in new window .
  1. (5)
    If Open image in new window , the property is obvious. Assume Open image in new window . Given Open image in new window , there exists Open image in new window such that
     
so that Open image in new window . Since Open image in new window is balanced, it follows that Open image in new window and, Open image in new window being arbitrary, we obtain Open image in new window . Conversely, let Open image in new window . Then there exists Open image in new window such that
Thus we obtain
whence Open image in new window and, Open image in new window being balanced, also Open image in new window . From the arbitrariness of Open image in new window we conclude that Open image in new window .
  1. (7)

    This follows from Proposition 2.2.

     
  2. (8)
     

Remark 3.4.

By the same reasons as Open image in new window , the measure Open image in new window fails to be monotone and, in general, the identities Open image in new window and

do not hold (cf. Remark 2.3).

4. A Fixed Point Theorem for Multimaps

The study of fixed points for multivalued mappings was initiated by Kakutani [5] in 1941 in finite dimensional spaces and extended to infinite dimensional Banach spaces by Bohnenblust and Karlin [6] in 1950 and to locally convex spaces by Fan [7] in 1952. Since then, it has become a very active area of research, both from the theoretical point of view and in applications. In this section we use the previous theory to obtain a fixed point theorem for multifunctions in the Banach space Open image in new window . We begin by recalling some definitions.

Definition 4.1.

Let Open image in new window . A multimap or multifunction Open image in new window from Open image in new window to the class Open image in new window of all nonempty subsets of a given set Open image in new window , written Open image in new window , is any map from Open image in new window to Open image in new window .

Definition 4.2.

Given Open image in new window , let Open image in new window , and let Open image in new window represent any of the two measures of noncircularity introduced above. A fixed point of Open image in new window is a point Open image in new window such that Open image in new window . The multifunction Open image in new window will be called

for some Open image in new window ;

where Open image in new window is a comparison function, that is, Open image in new window is increasing, Open image in new window , and Open image in new window as Open image in new window for each Open image in new window .

Note that a Open image in new window -contraction of constant Open image in new window corresponds to a Open image in new window -contraction with Open image in new window .

In order to establish our main result, we prove a property of Cantor type for the E-L and Hausdorff measures of noncircularity.

Proposition 4.3.

Let Open image in new window be a Banach space and Open image in new window a decreasing sequence of closed sets such that Open image in new window , where Open image in new window denotes either Open image in new window or Open image in new window . Then the set
satisfies

Hence Open image in new window belongs to Open image in new window and is closed and balanced.

Proof.

By Proposition 3.3 we have Open image in new window if, and only if, Open image in new window . Thus for the proof it suffices to set Open image in new window .

Conversely, let Open image in new window . As Open image in new window , to every Open image in new window there corresponds Open image in new window such that Open image in new window , Open image in new window implies Open image in new window . This yields an increasing sequence Open image in new window of positive integers and vectors Open image in new window which satisfy Open image in new window . Thus the sequence Open image in new window converges to Open image in new window as Open image in new window . Moreover, since Open image in new window and Open image in new window is closed, we find that Open image in new window . In other words, Open image in new window . This proves (4.5).

Note that Open image in new window implies Open image in new window , whence Open image in new window . Since the intersection of closed, bounded and balanced sets preserves those properties, so does Open image in new window .

Remark 4.4.

In contrast to Proposition 4.3, the Eisenfeld-Lakshmikantham measure of nonconvexity does not necessarily satisfy a Cantor property. Indeed, in real, nonreflexive Banach spaces one can find a decreasing sequence Open image in new window of nonempty, closed, bounded, convex sets with empty intersection. To construct such a sequence, just take a unitary continuous linear functional Open image in new window in a real, nonreflexive Banach space Open image in new window which fails to be norm-attaining on the closed unit ball Open image in new window of Open image in new window (the existence of such an Open image in new window is guaranteed by a classical, well-known theorem of James, cf. [8]), and define

Now we are in a position to derive the announced result. Here, and in the sequel, Open image in new window will stand for any one of the measures of noncircularity Open image in new window or Open image in new window .

Theorem 4.5.

Let Open image in new window be a Banach space, and let Open image in new window be closed. If Open image in new window is a Open image in new window -contraction with closed values, then Open image in new window and 0 is a fixed point of Open image in new window .

Proof.

Our hypotheses imply
Setting Open image in new window , from Propositions 2.2 and 3.3 we find that Open image in new window is a decreasing sequence of closed sets with Open image in new window . Proposition 4.3 shows that Open image in new window is a nonempty, balanced subset of Open image in new window ; in particular, Open image in new window . Now, Open image in new window being balanced, we have

whence Open image in new window . This shows that the nonempty set Open image in new window is balanced and forces Open image in new window , as asserted.

Corollary 4.6.

Let Open image in new window be a Banach space, and let Open image in new window be closed. If Open image in new window is a Open image in new window -contraction with closed values, then Open image in new window and 0 is a fixed point of Open image in new window .

Proof.

It suffices to apply Theorem 4.5, with Open image in new window , for Open image in new window .

Notes

Acknowledgments

This paper has been partially supported by ULL (MGC grants) and MEC-FEDER (MTM2007-65604, MTM2007-68114). It is dedicated to Professor A. Martinón on the occasion of his 60th birthday. The author is grateful to Professor J. Banaś for his interest in this work.

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Copyright information

© Isabel Marrero. 2010

This article is published under license to BioMed Central Ltd. 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.

Authors and Affiliations

  1. 1.Departamento de Análisis MatemáticoUniversidad de La LagunaLa LagunaSpain

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