Hindawi Publishing Corporation
Fixed Point Theory and Applications
Volume 2010, Article ID 340631, 12 pages
doi:10.1155/2010/340631
Research Article
Measures of Noncircularity and Fixed Points of
Contractive Multifunctions
Isabel Marrero
Departamento de Análisis Matemático, Universidad de La Laguna, 38271 La Laguna (Tenerife), Spain
Correspondence should be addressed to Isabel Marrero, imarrero@ull.es
Received 24 October 2010; Accepted 8 December 2010
Academic Editor: N. J. Huang
Copyright q 2010 Isabel Marrero. This is an open access article distributed under the Creative
Commons Attribution License, which permits unrestricted use, distribution, and reproduction in
any medium, provided the original work is properly cited.
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 X, · be a Banach space over the field K ∈ {R, C}. In what follows, we write BX {x ∈
X : x ≤ 1} for the closed unit ball of X. Denote by 2X the collection of all subsets of X and
consider
bX : C ∈ 2X \ {∅} : C bounded .
1.1
For C, D ∈ bX, define their nonsymmetric Hausdorff distance by
hC, D : sup inf c − d
c∈C d∈D
1.2
2
Fixed Point Theory and Applications
and their symmetric Hausdorff distance or Hausdorff-Pompeiu distance by
HC, D : max{hC, D, hD, C}.
1.3
This H is a pseudometric on bX, since
HC, D H C, D H C, D H C, D ,
1.4
where A denotes the closure of A ∈ 2X .
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 M ∈ bX
is closed and convex, κ is a measure of noncompactness, and f : M → M is continuous
and κ-contractive, that is, κfA ≤ rκA A ∈ bM for some r ∈0, 1, then f has a fixed
point. Below we recall the axiomatic definition of a regular measure of noncompactness on
X; we refer to 2 for details.
Definition 1.1. A function κ : bX → 0, ∞ will be called a regular measure of
noncompactness if κ satisfies the following axioms, for A, B ∈ bX, and λ ∈ K:
1 κA 0 if, and only if, A is compact.
2 κco A κA κA, where co A denotes the convex hull of A.
3 monotonicity A ⊂ B implies κA ≤ κB.
4 maximum property κA ∪ B max{κA, κB}.
5 homogeneity κλA |λ|κA.
6 subadditivity κA B ≤ κA κB.
A regular measure of noncompactness κ possesses the following properties:
1 κA ≤ κBX δA, where
δA : sup x − y
x,y∈A
1.5
is the diameter of A ∈ bX cf. 2, Theorem 3.2.1.
2 Hausdorff continuity |κA − κB| ≤ κBX HA, B A, B ∈ bX 2, page 12.
⊂ bX is a decreasing sequence of closed sets with
3 Cantor property If {An }∞
n0 ∅, and κA∞ 0 3, Lemma 2.1.
limn → ∞ κAn 0, then A∞ ∞
n0 An /
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 C ∈ bX such that C is balanced. Recall that A ∈ 2X \ {∅} is balanced
provided that μA ⊂ A for all μ ∈ K with |μ| ≤ 1. For example, in R the only bounded balanced
sets are the open or closed intervals centered at the origin. Similarly, in C as a complex vector
space the only bounded balanced sets are the open or closed disks centered at the origin,
Fixed Point Theory and Applications
3
while in R2 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 γ either one of the two measures introduced, in Section 4 we prove a
result of Darbo type for γ-contractive multimaps see Section 4 for precise definitions. It is
shown that the origin is a fixed point of every γ-contractive multimap F with closed values
defined on a closed set M ∈ bX such that FM ⊂ M.
2. The E-L Measure of Noncircularity
The definition of the Eisenfeld-Lakshmikantham measure of nonconvexity 4 motivates the
following.
Definition 2.1. For C ∈ bX, set
αC : HbaC, C hbaC, C,
2.1
where baC denotes the balanced hull of C, that is,
baC : μc : μ ≤ 1, c ∈ C .
2.2
By analogy with the Eisenfeld-Lakshmikantham measure of nonconvexity, we shall refer to α
as the E-L measure of noncircularity.
Next we gather some properties of α 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 C, D ∈ bX, and λ ∈ K, the following hold:
1 αC 0 if, and only if, C is balanced.
2 αco C ≤ αC αC.
3 αC ∪ D ≤ max{αC, αD}.
4 αλC |λ|αC.
5 αC D ≤ αC αD.
6 αC ≤ 2C, where
C : supc
c∈C
2.3
is the norm of C. In particular, if 0 ∈ C then αC ≤ 2δC, where
δC : sup x − y
x,y∈C
is the diameter of C.
7 |αC − αD| ≤ 2HC, D.
2.4
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Fixed Point Theory and Applications
Proof. Let baC denote the closed balanced hull of C. The identity
ba C baC
2.5
holds. Indeed, C ⊂ baC implies ba C ⊂ baC. Conversely, C ⊂ ba C implies baC ⊂ ba C.
1 By definition, αC HbaC, C hbaC, C 0 if, and only if, baC ⊂ C or,
equivalently, baC ⊂ C. This means that baC C, which by 2.5 occurs if, and
only if, C is balanced.
2 In view of 1.4 and 2.5,
α C H baC, C H ba C, C
H baC, C HbaC, C αC.
2.6
It only remains to prove that αco C ≤ αC. Suppose αC < ε, so that baC ⊂
CεBX . The set co CεBX being convex, it follows that bacoC ⊂ cobaC ⊂ co CεBX ,
whence αco C ≤ ε. From the arbitrariness of ε we conclude that αco C ≤ αC.
3 Assume max{αC, αD} < ε, that is, αC < ε and αD < ε. Then baC ⊂ C εBX ,
baD ⊂ D εBX , and the fact that baC ∪baD is a balanced set containing C ∪D, imply
baC ∪ D ⊂ baC ∪ baD ⊂ C ∪ D εBX ,
2.7
whence αC ∪ D ≤ ε. The arbitrariness of ε yields αC ∪ D ≤ max{αC, αD}.
4 For λ 0, this is obvious. Suppose λ /
0. If |λ|αC < ε then baC ⊂ C ε/|λ|BX C ε/λBX , whence baλC λbaC ⊂ λC εBX . Thus αλC ≤ ε, and from the
arbitrariness of ε we infer that αλC ≤ |λ|αC. Conversely, assume αλC < ε.
Then baλC ⊂ λC εBX , whence baC 1/λbaλC ⊂ C ε/λBX C ε/|λ|BX .
Therefore αC ≤ ε/|λ|, and from the arbitrariness of ε we conclude that |λ|αC ≤
αλC.
5 Let αC αD < ε and choose ε1 , ε2 > 0 such that ε ε1 ε2 , αC < ε1 and
αD < ε2 . Then baC ⊂ C ε1 BX , baD ⊂ D ε2 BX and the fact that baC baD is a
balanced set containing C D, imply baC D ⊂ baC baD ⊂ C D εBX , so that
αC D ≤ ε. The arbitrariness of ε yields αC D ≤ αC αD.
6 Pick x μu ∈ baC, with |μ| ≤ 1 and u ∈ C, and let c ∈ C. As
x − c μu − c ≤ μu c ≤ 2C,
2.8
αC sup infx − c ≤ 2C ≤ 2δC,
2.9
we obtain
x∈baC c∈C
where for the validity of the latter estimate we have assumed 0 ∈ C.
Fixed Point Theory and Applications
5
7 It is enough to show that
αC ≤ αD hC, D hD, C,
2.10
since then, by symmetry,
αD ≤ αC hC, D hD, C,
2.11
whence the desired result. Now
αC HbaC, C hbaC, C
≤ hbaC, baD hbaD, D hD, C
2.12
hbaC, baD αD hD, C.
To complete the proof we will establish that hbaC, baD ≤ hC, D. Indeed, suppose
hC, D < ε, and let x μc ∈ baC, with |μ| ≤ 1 and c ∈ C. Then there exists d ∈ D such
that c − d < ε. Consequently, for y μd ∈ baD we have
x − y μc − μd μc − d < ε.
2.13
This means that baC ⊂ baD εBX , so that hbaC, baD ≤ ε. From the arbitrariness of ε we
conclude that hbaC, baD ≤ hC, D.
Remark 2.3. The identity αco C αC C ∈ bX may not hold, as can be seen by choosing
C {−1, 1} ∈ 2R . In fact, co C −1, 1 is balanced, while C is not. Therefore, αco C 0 <
αC.
In general, the identity αC ∪ D max{αC, αD} C, D ∈ bX does not hold
either. To show this, choose C and D, respectively, as the upper and lower closed half unit
disks of the complex plane. Then C ∪ D equals the closed unit disk, which is balanced, while
C, D are not. Thus, αC ∪ D 0 < max{αC, αD}.
Note that α is not monotone: from C, D ∈ bX and C ⊂ D, it does not necessarily
follow that αC ≤ αD. Otherwise, αD 0 would imply αC 0, 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 C ∈ bX by
βC : HC, bbX inf HC, B,
B∈bbX
where bbX denotes the class of all balanced sets in bX.
3.1
6
Fixed Point Theory and Applications
In general, αC /
βC, as the next example shows.
Example 3.2. Let C {1} ∈ 2R . Then baC −1, 1, and
αC sup|x − 1| 2.
3.2
|x|≤1
If Br −r, r r ≥ 0 is any closed bounded balanced set in R, we have
hBr , C sup|x − 1|,
3.3
HC, Br max{hC, Br , hBr , C} hBr , C.
3.4
hBr , C sup|x − 1| 1 r,
3.5
βC inf HC, Br inf1 r 1.
3.6
hC, Br inf |x − 1|,
|x|≤r
|x|≤r
so that
Since
|x|≤r
we obtain
r≥0
r≥0
Thus, 2βC 2 αC.
Next we compare the measures α and β 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 C, D ∈ bX, and λ ∈ K, the following hold:
1 βC ≤ αC ≤ 2βC, and the estimates are sharp.
2 βC 0 if, and only if, C is balanced.
3 βco C ≤ βC βC.
4 βC ∪ D ≤ max{βC, βD}.
5 βλC |λ|βC.
6 βC D ≤ βC βD.
7 βC ≤ 2C, where
C : supc
c∈C
3.7
Fixed Point Theory and Applications
7
is the norm of C. In particular, if 0 ∈ C then βC ≤ 2δC, where
δC : sup x − y
3.8
x,y∈C
is the diameter of C.
8 |βC − βD| ≤ HC, D.
Proof. 1 That βC ≤ αC follows immediately from the definitions of β and α. Let ε > 2βC
and choose B ∈ bbX satisfying HC, B < ε/2, so that C ⊂ B ε/2BX and B ⊂ C ε/2BX .
Then baC ⊂ B ε/2BX and B ⊂ baC ε/2BX , thus proving that HbaC, B ≤ ε/2. Now
αC HbaC, C ≤ HbaC, B HB, C < ε,
3.9
and the arbitrariness of ε yields αC ≤ 2βC. Example 3.2 shows that this estimate is sharp.
In order to exhibit a set C ∈ 2R such that βC αC, let C {−1, 1}. Then baC −1, 1, and
αC sup inf|x − c| 1.
3.10
|x|≤1 c∈C
On the other hand, let Br −r, r r ≥ 0 be any closed bounded balanced subset of R. For a
fixed r ≥ 0, there holds
hBr , C sup inf|x − c| |x|≤r c∈C
hC, Br sup inf |x − c| c∈C |x|≤r
⎧
⎨1,
r≤1
⎩max{1, r − 1}, r > 1,
⎧
⎨1 − r, r ≤ 1
⎩0,
3.11
r > 1.
Therefore,
HBr , C max{hBr , C, hC, Br } ⎧
⎨1,
r≤1
⎩max{1, r − 1},
r > 1,
3.12
so that
βC inf HBr , C 1 αC.
r≥0
3.13
2 Let C ∈ bX. As we just proved, βC 0 if, and only if, αC 0. In view of
Proposition 2.2, this occurs if, and only if, C is balanced.
3 By 1.4, there holds
βC inf HC, B B∈bbX
inf H C, B β C .
B∈bbX
3.14
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Fixed Point Theory and Applications
Now we only need to show that βco C ≤ βC. Assuming βC < ε, choose B ∈ bbX for
which HC, B < ε, so that
C ⊂ B εBX ,
B ⊂ C εBX .
3.15
co B ⊂ co C εBX .
3.16
The sum of convex sets being convex, we infer
co C ⊂ co B εBX ,
Since co B is balanced we obtain βco C ≤ ε and, as ε is arbitrary, we conclude that βco C ≤
βC.
4 Suppose max{βC, βD} < ε, that is, βC < ε and βD < ε. Pick B1 , B2 ∈ bbX
satisfying HC, B1 < ε and HD, B2 < ε. Then
C ⊂ B1 εBX ,
B1 ⊂ C εBX ,
D ⊂ B2 εBX ,
B2 ⊂ D εBX .
3.17
Thus we get
C ∪ D ⊂ B1 ∪ B2 εBX ,
B1 ∪ B2 ⊂ C ∪ D εBX ,
3.18
whence HC ∪ D, B1 ∪ B2 ≤ ε and, B1 ∪ B2 being balanced, also βC ∪ D ≤ ε. From the
arbitrariness of ε we conclude that βC ∪ D ≤ max{βC, βD}.
5 If λ 0, the property is obvious. Assume λ /
0. Given ε > |λ|βC, there exists
B ∈ bbX such that
ε
ε
BX ,
BX B λ
|λ|
ε
ε
BX .
BX C B⊂C
λ
|λ|
C⊂B
3.19
Then
λC ⊂ λB εBX ,
λB ⊂ λC εBX ,
3.20
so that HλC, λB ≤ ε. Since λB is balanced, it follows that βλC ≤ ε and, ε being arbitrary,
we obtain βλC ≤ |λ|βC. Conversely, let ε > βλC. Then there exists B ∈ bbX such that
λC ⊂ B εBX ,
B ⊂ λC εBX .
3.21
Fixed Point Theory and Applications
9
Hence,
ε
ε
1
1
B
BX B
BX ,
λ
λ
λ
|λ|
ε
ε
1
BX C BX .
B⊂C
λ
λ
|λ|
C⊂
3.22
Therefore, HC, 1/λB ≤ ε/|λ|. Since 1/λB is balanced we conclude that βC ≤ ε/|λ|, or
|λ|βC ≤ ε. The arbitrariness of ε finally yields |λ|βC ≤ βλC.
6 Let βC βD < ε and let ε1 , ε2 > 0 satisfy ε ε1 ε2 , βC < ε1 and βD < ε2 .
Choose B1 , B2 ∈ bbX such that HC, B1 < ε1 and HD, B2 < ε2 . Then
C ⊂ B1 ε1 BX ,
B1 ⊂ C ε1 BX ,
D ⊂ B2 ε2 BX ,
B2 ⊂ D ε2 BX .
3.23
Thus we obtain
C D ⊂ B1 B2 εBX ,
B1 B2 ⊂ C D εBX ,
3.24
whence HC D, B1 B2 ≤ ε and, B1 B2 being balanced, also βC D ≤ ε. From the
arbitrariness of ε we conclude that βC D ≤ βC βD.
7 This follows from Proposition 2.2.
8 For B ∈ bbX there holds HC, B ≤ HC, D HD, B, whence βC ≤ HC, D βD. Therefore, βC − βD ≤ HC, D. By symmetry, βD − βC ≤ HC, D, thus yielding
|βC − βD| ≤ HC, D, as claimed.
Remark 3.4. By the same reasons as α, the measure β fails to be monotone and, in general, the
identities βco C βC and
βC ∪ D max βC, βD
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 X. We begin by recalling some definitions.
Definition 4.1. Let M ∈ 2X \ {∅}. A multimap or multifunction F from M to the class 2Y \ {∅}
of all nonempty subsets of a given set Y , written F : M Y , is any map from M to 2Y \ {∅}.
10
Fixed Point Theory and Applications
If F is a multifunction and A ∈ 2M , then
FA :
Fx.
4.1
x∈A
Definition 4.2. Given M ∈ 2X \{∅}, let F : M X, and let γ represent any of the two measures
of noncircularity introduced above. A fixed point of F is a point x ∈ M such that x ∈ Fx.
The multifunction F will be called
i a γ-contraction of constant k, if
B ∈ bX ∩ 2M
γFB ≤ kγB
4.2
for some k ∈0, 1;
ii a γ, φ-contraction, if
B ∈ bX ∩ 2M ,
γFB ≤ φ γB
4.3
where φ : 0, ∞ → 0, ∞ is a comparison function, that is, φ is increasing, φ0 0,
and φn r → 0 as n → ∞ for each r > 0.
Note that a γ-contraction of constant k corresponds to a γ, φ-contraction with φr kr r ≥ 0.
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 X be a Banach space and {Ak }∞
k0 ⊂ bX a decreasing sequence of closed sets
such that limk → ∞ γAk 0, where γ denotes either α or β. Then the set
A∞ :
∞
Ak
4.4
baAk .
4.5
k0
satisfies
A∞ ∞
k0
Hence A∞ belongs to bX and is closed and balanced.
Proof. By Proposition 3.3 we have limk → ∞ αAk 0 if, and only if, limk → ∞ βAk 0. Thus
for the proof it suffices to set γ α.
Since Ak ⊂ baAk k ∈ N, necessarily
A∞ ∞
Ak ⊂
k0
∞
baAk .
k0
4.6
Fixed Point Theory and Applications
11
Conversely, let x ∈ ∞
k0 baAk . As limk → ∞ αAk 0, to every ε > 0 there corresponds N ∈ N
such that n ∈ N, n ≥ N implies baAn ⊂ An εBX . This yields an increasing sequence {nm }∞
m1
of positive integers and vectors anm ∈ Anm which satisfy x−anm ≤ 1/m m ∈ N, m ≥ 1. Thus
the sequence {anm }∞
m1 converges to x as m → ∞. Moreover, since anm ∈ Anm ⊂ Ak m, k ∈
N, m ≥ 1, nm ≥ k and Ak is closed, we find that x ∈ Ak k ∈ N. In other words, x ∈ A∞ .
This proves 4.5.
∅. Since the
Note that ∅ /
An ⊂ baAn implies 0 ∈ baAn n ∈ N, whence 0 ∈ A∞ /
intersection of closed, bounded and balanced sets preserves those properties, so does A∞ .
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 {An }∞
n1 of nonempty, closed, bounded,
convex sets with empty intersection. To construct such a sequence, just take a unitary
continuous linear functional f in a real, nonreflexive Banach space X which fails to be normattaining on the closed unit ball BX of X the existence of such an f is guaranteed by a
classical, well-known theorem of James, cf. 8, and define
An 1
x ∈ BX : fx ≥ 1 −
n
n ∈ N, n ≥ 1.
4.7
Now we are in a position to derive the announced result. Here, and in the sequel, γ
will stand for any one of the measures of noncircularity α or β.
Theorem 4.5. Let X be a Banach space, and let M ∈ bX be closed. If F : M M is a γ, φcontraction with closed values, then 0 ∈ M and 0 is a fixed point of F.
Proof. Our hypotheses imply
F n1 M ⊂ F n M n ∈ N,
lim γF n M ≤ lim φn γM 0.
n→∞
4.8
n→∞
Setting An F n M n ∈ N, from Propositions 2.2 and 3.3 we find that {An }∞
n0 ⊂ bX is a
decreasing sequence of closed sets with limn → ∞ γAn 0. Proposition 4.3 shows that A∞ is
a nonempty, balanced subset of M; in particular, 0 ∈ A∞ ⊂ M. Now, {0} being balanced, we
have
γF0 ≤ φ γ{0} 0,
4.9
whence γF0 0. This shows that the nonempty set F0 F0 is balanced and forces
0 ∈ F0, as asserted.
Corollary 4.6. Let X be a Banach space, and let M ∈ bX be closed. If F : M M is a γcontraction with closed values, then 0 ∈ M and 0 is a fixed point of F.
Proof. It suffices to apply Theorem 4.5, with φr kr r ≥ 0, for k ∈0, 1.
12
Fixed Point Theory and Applications
Acknowledgments
This paper has been partially supported by ULL MGC grants and MEC-FEDER MTM200765604, 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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