- Research Article
- Open Access

# Measures of Noncircularity and Fixed Points of Contractive Multifunctions

- Isabel Marrero
^{1}Email author

**2010**:340631

https://doi.org/10.1155/2010/340631

© Isabel Marrero. 2010

**Received:**24 October 2010**Accepted:**8 December 2010**Published:**14 December 2010

## 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 denotes the closure of .

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 is closed and convex, is a measure of noncompactness, and is continuous and -contractive, that is, for some , then has a fixed point. Below we recall the axiomatic definition of a regular measure of noncompactness on ; we refer to [2] for details.

Definition 1.1.

A function will be called a regular measure of noncompactness if satisfies the following axioms, for , and :

(1) if, and only if, is compact.

(2) , where denotes the convex hull of .

(3)(monotonicity) implies .

(4)(maximum property) .

(5)(homogeneity) .

(6)(subadditivity) .

A regular measure of noncompactness possesses the following properties:

is the diameter of (cf. [2, Theorem 3.2.1]).

- (3)
(Cantor property) If is a decreasing sequence of closed sets with , then , and [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 such that is balanced. Recall that is balanced provided that for all with . For example, in the only bounded balanced sets are the open or closed intervals centered at the origin. Similarly, in as a complex vector space the only bounded balanced sets are the open or closed disks centered at the origin, while in 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 either one of the two measures introduced, in Section 4 we prove a result of Darbo type for -contractive multimaps (see Section 4 for precise definitions). It is shown that the origin is a fixed point of every -contractive multimap with closed values defined on a closed set such that .

## 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 as the E-L measure of noncircularity.

Next we gather some properties of 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 , and , the following hold:

(1) if, and only if, is balanced.

(2) .

(3) .

(4) .

(5) .

is the diameter of .

(7) .

Proof.

holds. Indeed, implies . Conversely, implies .

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

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

whence . The arbitrariness of yields .

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

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

where for the validity of the latter estimate we have assumed .

This means that , so that . From the arbitrariness of we conclude that .

Remark 2.3.

The identity may not hold, as can be seen by choosing . In fact, is balanced, while is not. Therefore, .

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

Note that is not monotone: from and , it does not necessarily follow that . Otherwise, would imply , 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 denotes the class of all balanced sets in .

In general, , as the next example shows.

Example 3.2.

Thus, .

Next we compare the measures and 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 , and , the following hold:

(1) , and the estimates are sharp.

(2) if, and only if, is balanced.

(3) .

(4) .

(5) .

(6) .

is the diameter of .

(8) .

- (1)

- (2)
Let . As we just proved, if, and only if, . In view of Proposition 2.2, this occurs if, and only if, is balanced.

- (3)

- (4)

- (5)

- (6)

- (7)
This follows from Proposition 2.2.

- (8)
For there holds , whence . Therefore, . By symmetry, , thus yielding , 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 . We begin by recalling some definitions.

Definition 4.1.

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

Definition 4.2.

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

for some ;

where is a comparison function, that is, is increasing, , and as for each .

Note that a -contraction of constant corresponds to a -contraction with .

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 belongs to and is closed and balanced.

Proof.

By Proposition 3.3 we have if, and only if, . Thus for the proof it suffices to set .

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

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

Remark 4.4.

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

Theorem 4.5.

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

Proof.

whence . This shows that the nonempty set is balanced and forces , as asserted.

Corollary 4.6.

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

Proof.

It suffices to apply Theorem 4.5, with , for .

## Declarations

### 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.

## Authors’ Affiliations

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