Measures of Noncircularity and Fixed Points of Contractive Multifunctions
© Isabel Marrero. 2010
Received: 24 October 2010
Accepted: 8 December 2010
Published: 14 December 2010
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.
Around 1955, Darbo  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  for details.
is the diameter of (cf. [2, Theorem 3.2.1]).
(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  motivates the following.
(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 .
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, .
3. The Hausdorff Measure of Noncircularity
The following definition is motivated by that of the Hausdorff measure of noncompactness (cf. [2, Theorem 2.1]).
This follows from Proposition 2.2.
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  in 1941 in finite dimensional spaces and extended to infinite dimensional Banach spaces by Bohnenblust and Karlin  in 1950 and to locally convex spaces by Fan  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.
In order to establish our main result, we prove a property of Cantor type for the E-L and Hausdorff measures of noncircularity.
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).
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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