Measures of Noncircularity and Fixed Points of Contractive Multifunctions

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.

1. Introduction

Let be a Banach space over the field . In what follows, we write for the closed unit ball of . Denote by the collection of all subsets of and consider

(1.1)

For , define their nonsymmetric Hausdorff distance by

(1.2)

and their symmetric Hausdorff distance (or Hausdorff-Pompeiu distance) by

(1.3)

This is a pseudometric on , since

(1.4)

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:

(1), where

(1.5)

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

(2)(Hausdorff continuity) [2, page 12].

• (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.

For , set

(2.1)

where denotes the balanced hull of , that is,

(2.2)

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).

(6), where

(2.3)

is the norm of . In particular, if then , where

(2.4)

is the diameter of .

(7).

Proof.

Let denote the closed balanced hull of . The identity

(2.5)

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.

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

(2.6)

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

(3)Assume , that is, and . Then , , and the fact that is a balanced set containing , imply

(2.7)

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 .

(6)Pick , with and , and let . As

(2.8)

we obtain

(2.9)

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

(7)It is enough to show that

(2.10)

since then, by symmetry,

(2.11)

whence the desired result. Now

(2.12)

To complete the proof we will establish that . Indeed, suppose , and let , with and . Then there exists such that . Consequently, for we have

(2.13)

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.

We define the Hausdorff measure of noncircularity of by

(3.1)

where denotes the class of all balanced sets in .

In general, , as the next example shows.

Example 3.2.

Let . Then , and

(3.2)

If is any closed bounded balanced set in , we have

(3.3)

so that

(3.4)

Since

(3.5)

we obtain

(3.6)

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).

(7), where

(3.7)

is the norm of . In particular, if then , where

(3.8)

is the diameter of .

(8).

Proof.

• (1) That follows immediately from the definitions of and . Let and choose satisfying , so that and . Then and , thus proving that . Now

  

  (3.9)

and the arbitrariness of yields . Example 3.2 shows that this estimate is sharp. In order to exhibit a set such that , let . Then , and

(3.10)

On the other hand, let be any closed bounded balanced subset of . For a fixed , there holds

(3.11)

Therefore,

(3.12)

so that

(3.13)

• (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) By (1.4), there holds

  

  (3.14)

Now we only need to show that . Assuming , choose for which , so that

(3.15)

The sum of convex sets being convex, we infer

(3.16)

Since is balanced we obtain and, as is arbitrary, we conclude that .

• (4) Suppose , that is, and . Pick satisfying and . Then

  

  (3.17)

Thus we get

(3.18)

whence and, being balanced, also . From the arbitrariness of we conclude that .

• (5) If , the property is obvious. Assume . Given, there exists such that

  

  (3.19)

Then

(3.20)

so that . Since is balanced, it follows that and, being arbitrary, we obtain . Conversely, let . Then there exists such that

(3.21)

Hence,

(3.22)

Therefore, . Since is balanced we conclude that , or . The arbitrariness of finally yields .

• (6) Let and let satisfy , and . Choose such that and . Then

  

  (3.23)

Thus we obtain

(3.24)

whence and, being balanced, also . From the arbitrariness of we conclude that .

• (7) This follows from Proposition 2.2.

• (8) For there holds , whence . Therefore, . By symmetry, , thus yielding , as claimed.

Remark 3.4.

By the same reasons as , the measure fails to be monotone and, in general, the identities and

(3.25)

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 .

If is a multifunction and , then

(4.1)

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

(i)a -contraction (of constant ), if

(4.2)

for some ;

(ii)a -contraction, if

(4.3)

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.

Let be a Banach space and a decreasing sequence of closed sets such that , where denotes either or . Then the set

(4.4)

satisfies

(4.5)

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 .

Since , necessarily

(4.6)

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.

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 of nonempty, closed, bounded, convex sets with empty intersection. To construct such a sequence, just take a unitary continuous linear functional in a real, nonreflexive Banach space which fails to be norm-attaining on the closed unit ball of (the existence of such an is guaranteed by a classical, well-known theorem of James, cf. [8]), and define

(4.7)

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.

Our hypotheses imply

(4.8)

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

(4.9)

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 .