# Measures of Noncircularity and Fixed Points of Contractive Multifunctions

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## 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## 1. Introduction

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]).

- (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.

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.

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.

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 .

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

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.

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.

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 .

- (1)That Open image in new window follows immediately from the definitions of Open image in new window and Open image in new window . Let Open image in new window and choose Open image in new window satisfying Open image in new window , so that Open image in new window and Open image in new window . Then Open image in new window and Open image in new window , thus proving that Open image in new window . Now(3.9)

- (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.

- (3)By (1.4), there holds(3.14)

- (4)Suppose Open image in new window , that is, Open image in new window and Open image in new window . Pick Open image in new window satisfying Open image in new window and Open image in new window . Then(3.17)

- (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(3.19)

- (6)Let Open image in new window and let Open image in new window satisfy Open image in new window , Open image in new window and Open image in new window . Choose Open image in new window such that Open image in new window and Open image in new window . Then(3.23)

- (7)
This follows from Proposition 2.2.

- (8)
For Open image in new window there holds Open image in new window , whence Open image in new window . Therefore, Open image in new window . By symmetry, Open image in new window , thus yielding Open image in new window , as claimed.

Remark 3.4.

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.

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.

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.

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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