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Measures of Noncircularity and Fixed Points of Contractive Multifunctions
Fixed Point Theory and Applications volume 2010, Article number: 340631 (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.
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F340631/MediaObjects/13663_2010_Article_1261_IEq1_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F340631/MediaObjects/13663_2010_Article_1261_Equ1_HTML.gif)
For , define their nonsymmetric Hausdorff distance by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F340631/MediaObjects/13663_2010_Article_1261_Equ2_HTML.gif)
and their symmetric Hausdorff distance (or Hausdorff-Pompeiu distance) by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F340631/MediaObjects/13663_2010_Article_1261_Equ3_HTML.gif)
This is a pseudometric on
, since
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F340631/MediaObjects/13663_2010_Article_1261_Equ4_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F340631/MediaObjects/13663_2010_Article_1261_Equ5_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F340631/MediaObjects/13663_2010_Article_1261_Equ6_HTML.gif)
where denotes the balanced hull of
, that is,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F340631/MediaObjects/13663_2010_Article_1261_Equ7_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F340631/MediaObjects/13663_2010_Article_1261_Equ8_HTML.gif)
is the norm of . In particular, if
then
, where
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F340631/MediaObjects/13663_2010_Article_1261_Equ9_HTML.gif)
is the diameter of .
(7).
Proof.
Let denote the closed balanced hull of
. The identity
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F340631/MediaObjects/13663_2010_Article_1261_Equ10_HTML.gif)
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),
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F340631/MediaObjects/13663_2010_Article_1261_Equ11_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F340631/MediaObjects/13663_2010_Article_1261_Equ12_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F340631/MediaObjects/13663_2010_Article_1261_Equ13_HTML.gif)
we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F340631/MediaObjects/13663_2010_Article_1261_Equ14_HTML.gif)
where for the validity of the latter estimate we have assumed .
(7)It is enough to show that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F340631/MediaObjects/13663_2010_Article_1261_Equ15_HTML.gif)
since then, by symmetry,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F340631/MediaObjects/13663_2010_Article_1261_Equ16_HTML.gif)
whence the desired result. Now
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F340631/MediaObjects/13663_2010_Article_1261_Equ17_HTML.gif)
To complete the proof we will establish that . Indeed, suppose
, and let
, with
and
. Then there exists
such that
. Consequently, for
we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F340631/MediaObjects/13663_2010_Article_1261_Equ18_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F340631/MediaObjects/13663_2010_Article_1261_Equ19_HTML.gif)
where denotes the class of all balanced sets in
.
In general, , as the next example shows.
Example 3.2.
Let . Then
, and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F340631/MediaObjects/13663_2010_Article_1261_Equ20_HTML.gif)
If is any closed bounded balanced set in
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F340631/MediaObjects/13663_2010_Article_1261_Equ21_HTML.gif)
so that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F340631/MediaObjects/13663_2010_Article_1261_Equ22_HTML.gif)
Since
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F340631/MediaObjects/13663_2010_Article_1261_Equ23_HTML.gif)
we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F340631/MediaObjects/13663_2010_Article_1261_Equ24_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F340631/MediaObjects/13663_2010_Article_1261_Equ25_HTML.gif)
is the norm of . In particular, if
then
, where
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F340631/MediaObjects/13663_2010_Article_1261_Equ26_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F340631/MediaObjects/13663_2010_Article_1261_Equ28_HTML.gif)
On the other hand, let be any closed bounded balanced subset of
. For a fixed
, there holds
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F340631/MediaObjects/13663_2010_Article_1261_Equ29_HTML.gif)
Therefore,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F340631/MediaObjects/13663_2010_Article_1261_Equ30_HTML.gif)
so that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F340631/MediaObjects/13663_2010_Article_1261_Equ31_HTML.gif)
-
(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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F340631/MediaObjects/13663_2010_Article_1261_Equ33_HTML.gif)
The sum of convex sets being convex, we infer
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F340631/MediaObjects/13663_2010_Article_1261_Equ34_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F340631/MediaObjects/13663_2010_Article_1261_Equ36_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F340631/MediaObjects/13663_2010_Article_1261_Equ38_HTML.gif)
so that . Since
is balanced, it follows that
and,
being arbitrary, we obtain
. Conversely, let
. Then there exists
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F340631/MediaObjects/13663_2010_Article_1261_Equ39_HTML.gif)
Hence,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F340631/MediaObjects/13663_2010_Article_1261_Equ40_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F340631/MediaObjects/13663_2010_Article_1261_Equ42_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F340631/MediaObjects/13663_2010_Article_1261_Equ43_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F340631/MediaObjects/13663_2010_Article_1261_Equ44_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F340631/MediaObjects/13663_2010_Article_1261_Equ45_HTML.gif)
for some ;
(ii)a -contraction, if
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F340631/MediaObjects/13663_2010_Article_1261_Equ46_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F340631/MediaObjects/13663_2010_Article_1261_Equ47_HTML.gif)
satisfies
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F340631/MediaObjects/13663_2010_Article_1261_Equ48_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F340631/MediaObjects/13663_2010_Article_1261_Equ49_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F340631/MediaObjects/13663_2010_Article_1261_Equ50_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F340631/MediaObjects/13663_2010_Article_1261_Equ51_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F340631/MediaObjects/13663_2010_Article_1261_Equ52_HTML.gif)
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
.
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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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Marrero, I. Measures of Noncircularity and Fixed Points of Contractive Multifunctions. Fixed Point Theory Appl 2010, 340631 (2010). https://doi.org/10.1155/2010/340631
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DOI: https://doi.org/10.1155/2010/340631