Volume 2010, Article ID 340631,12pages doi:10.1155/2010/340631
Research Article
Measures of Noncircularity and Fixed Points of
Contractive Multifunctions
Isabel Marrero
Departamento de An´alisis Matem´atico, Universidad de La Laguna, 38271 La Laguna (Tenerife), Spain
Correspondence should be addressed to Isabel Marrero,[email protected]
Received 24 October 2010; Accepted 8 December 2010
Academic Editor: N. J. Huang
Copyrightq2010 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 Hausdorffmeasure 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
LetX, · be a Banach space over the fieldK∈ {R,C}. In what follows, we writeBX{x∈ X :x ≤1}for the closed unit ball ofX. Denote by 2Xthe collection of all subsets ofXand consider
bX:C∈2X\ {∅}:Cbounded. 1.1
ForC, D∈bX, define their nonsymmetric Hausdorffdistance by
hC, D:sup c∈C
inf
and their symmetric Hausdorffdistanceor Hausdorff-Pompeiu distanceby
HC, D:max{hC, D, hD, C}. 1.3
ThisHis a pseudometric onbX, since
HC, D HC, DHC, DHC, D, 1.4
whereAdenotes the closure ofA∈2X.
Around 1955, Darbo1ensured 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 ifM∈ bX is closed and convex,κis a measure of noncompactness, andf : M → Mis continuous andκ-contractive, that is,κfA≤rκA A∈bMfor somer∈0,1, thenfhas a fixed point. Below we recall the axiomatic definition of a regular measure of noncompactness on X; we refer to2for details.
Definition 1.1. A function κ : bX → 0,∞ will be called a regular measure of noncompactness ifκsatisfies the following axioms, forA, B∈bX, andλ∈K:
1κA 0 if, and only if,Ais compact.
2κcoA κA κA, where coAdenotes the convex hull ofA.
3 monotonicityA⊂BimpliesκA≤κB.
4 maximum propertyκA∪B max{κA, κB}.
5 homogeneityκλA |λ|κA.
6 subadditivityκAB≤κA κB.
A regular measure of noncompactnessκpossesses the following properties:
1κA≤κBXδA, where
δA: sup x,y∈A
x−y 1.5
is the diameter ofA∈bX cf.2, Theorem 3.2.1.
2 Hausdorffcontinuity|κA−κB| ≤κBXHA, B A, B∈bX 2, page 12.
3 Cantor propertyIf{An}n∞0 ⊂ bXis a decreasing sequence of closed sets with
limn→ ∞κAn 0, thenA∞∞n0An/∅, andκA∞ 03, Lemma 2.1.
while inR2 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, inSection 4 we prove a result of Darbo type forγ-contractive multimapsseeSection 4for precise definitions. It is shown that the origin is a fixed point of everyγ-contractive multimapFwith closed values defined on a closed setM∈bXsuch thatFM⊂M.
2. The E-L Measure of Noncircularity
The definition of the Eisenfeld-Lakshmikantham measure of nonconvexity4motivates the following.
Definition 2.1. ForC∈bX, set
αC:HbaC, C hbaC, C, 2.1
where baCdenotes the balanced hull ofC, that is,
baC:μc:μ≤1, c∈C . 2.2
By analogy with the Eisenfeld-Lakshmikantham measure of nonconvexity, we shall refer toα as the E-L measure of noncircularity.
Next we gather some properties ofαwhich 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, forC, D∈bX, andλ∈K, the following hold:
1αC 0if, and only if,Cis balanced.
2αcoC≤αC αC.
3αC∪D≤max{αC, αD}.
4αλC |λ|αC.
5αCD≤αC αD.
6αC≤2C, where
C:sup c∈C
c 2.3
is the norm ofC. In particular, if0∈CthenαC≤2δC, where
δC: sup x,y∈C
x−y 2.4
is the diameter ofC.
Proof. Let baCdenote the closed balanced hull ofC. The identity
baCbaC 2.5
holds. Indeed,C⊂baCimplies baC⊂baC. Conversely,C⊂baCimplies baC⊂baC.
1By definition, αC HbaC, C hbaC, C 0 if, and only if, baC ⊂ C or, equivalently, baC ⊂ C. This means that baC C, which by2.5occurs if, and only if,Cis balanced.
2In view of1.4and2.5,
αCHbaC, CHbaC, C
HbaC, CHbaC, C αC.
2.6
It only remains to prove that αcoC ≤ αC. SupposeαC < ε, so that baC ⊂ CεBX. The set coCεBXbeing convex, it follows that bacoC⊂cobaC⊂coCεBX, whenceαcoC≤ε. From the arbitrariness ofεwe conclude thatαcoC≤αC.
3Assume max{αC, αD}< ε, that is,αC< εandαD< ε. Then baC⊂CεBX, baD⊂DεBX, and the fact that baC∪baDis a balanced set containingC∪D, imply
baC∪D⊂baC∪baD⊂C∪D εBX, 2.7
whenceαC∪D≤ε. The arbitrariness ofεyieldsαC∪D≤max{αC, αD}.
4Forλ0, this is obvious. Supposeλ /0. If|λ|αC< εthen baC⊂C ε/|λ|BX C ε/λBX, whence baλC λbaC ⊂ λCεBX. Thus αλC ≤ ε, and from the arbitrariness ofε we infer thatαλC ≤ |λ|αC. Conversely, assume αλC < ε. Then baλC⊂λCεBX, whence baC 1/λbaλC⊂C ε/λBX C ε/|λ|BX. ThereforeαC≤ε/|λ|, and from the arbitrariness ofεwe conclude that|λ|αC≤ αλC.
5Let αC αD < ε and chooseε1, ε2 > 0 such thatε ε1ε2,αC < ε1 and
αD< ε2. Then baC⊂Cε1BX, baD ⊂Dε2BXand the fact that baCbaDis a
balanced set containingCD, imply baCD⊂baCbaD⊂CDεBX, so that αCD≤ε. The arbitrariness ofεyieldsαCD≤αC αD.
6Pickxμu∈baC, with|μ| ≤1 andu∈C, and letc∈C. As
x−cμu−c ≤μuc ≤2C, 2.8
we obtain
αC sup x∈baC
inf
c∈Cx−c ≤2C ≤2δC, 2.9
7It 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
hbaC,baD αD hD, C.
2.12
To complete the proof we will establish that hbaC,baD ≤ hC, D. Indeed, suppose hC, D < ε, and letx μc ∈ baC, with|μ| ≤ 1 andc ∈ C. Then there existsd ∈ D such thatc−d< ε. Consequently, foryμd∈baDwe have
x−yμc−μdμc−d< ε. 2.13
This means that baC ⊂ baDεBX, so thathbaC,baD ≤ ε. From the arbitrariness ofεwe conclude thathbaC,baD≤hC, D.
Remark 2.3. The identityαcoC αC C∈bXmay not hold, as can be seen by choosing C {−1,1} ∈ 2R. In fact, coC −1,1is balanced, whileCis not. Therefore,αcoC 0 < αC.
In general, the identity αC∪D max{αC, αD}C, D ∈ bXdoes not hold either. To show this, chooseCandD, respectively, as the upper and lower closed half unit disks of the complex plane. ThenC∪Dequals the closed unit disk, which is balanced, while C,Dare not. Thus,αC∪D 0<max{αC, αD}.
Note thatαis not monotone: fromC, D ∈ bXand C ⊂ D, it does not necessarily follow thatαC≤αD. Otherwise,αD 0 would implyαC 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 Hausdorffmeasure of noncompactness
cf.2, Theorem 2.1.
Definition 3.1. We define the Hausdorffmeasure of noncircularity ofC∈bXby
βC:HC, bbX inf
B∈bbXHC, B, 3.1
In general,αC/βC, as the next example shows.
Example 3.2. LetC{1} ∈2R. Then baC −1,1, and
αC sup |x|≤1
|x−1|2. 3.2
IfBr −r, r r ≥0is any closed bounded balanced set inR, we have
hC, Br inf
|x|≤r|x−1|, hBr, C sup|x|≤r|x−1|, 3.3
so that
HC, Br max{hC, Br, hBr, C}hBr, C. 3.4
Since
hBr, C sup |x|≤r
|x−1|1r, 3.5
we obtain
βC inf
r≥0HC, Br infr≥01r 1. 3.6
Thus, 2βC 2αC.
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, forC, D∈bX, andλ∈K, the following hold:
1βC≤αC≤2βC, and the estimates are sharp.
2βC 0if, and only if,Cis balanced.
3βcoC≤βC βC.
4βC∪D≤max{βC, βD}.
5βλC |λ|βC.
6βCD≤βC βD.
7βC≤2C, where
C:sup c∈C
is the norm ofC. In particular, if0∈CthenβC≤2δC, where
δC: sup x,y∈C
x−y 3.8
is the diameter ofC.
8|βC−βD| ≤HC, D.
Proof. 1ThatβC≤αCfollows immediately from the definitions ofβandα. Letε >2βC and chooseB∈bbXsatisfyingHC, B< ε/2, so thatC⊂Bε/2BXandB⊂Cε/2BX. Then baC⊂B ε/2BXandB⊂baC ε/2BX, thus proving thatHbaC, B≤ε/2. Now
αC HbaC, C≤HbaC, B HB, C< ε, 3.9
and the arbitrariness ofεyieldsαC≤2βC.Example 3.2shows that this estimate is sharp. In order to exhibit a setC∈2Rsuch thatβC αC, letC{−1,1}. Then baC −1,1, and
αC sup |x|≤1
inf
c∈C|x−c|1. 3.10
On the other hand, letBr −r, r r≥0be any closed bounded balanced subset ofR. For a fixedr≥0, there holds
hBr, C sup |x|≤r
inf
c∈C|x−c|
⎧ ⎨ ⎩
1, r ≤1 max{1, r−1}, r >1,
hC, Br sup c∈C
inf
|x|≤r|x−c|
⎧ ⎨ ⎩
1−r, r ≤1 0, r >1.
3.11
Therefore,
HBr, C max{hBr, C, hC, Br}
⎧ ⎨ ⎩
1, r ≤1
max{1, r−1}, r >1, 3.12
so that
βC inf
r≥0HBr, C 1αC. 3.13
2Let 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,Cis balanced.
3By1.4, there holds
βC inf
B∈bbXHC, B B∈infbbXH
Now we only need to show thatβcoC≤ βC. AssumingβC< ε, chooseB∈ bbXfor whichHC, B< ε, so that
C⊂BεBX, B⊂CεBX. 3.15
The sum of convex sets being convex, we infer
coC⊂coBεBX, coB⊂coCεBX. 3.16
Since coBis balanced we obtainβcoC≤εand, asεis arbitrary, we conclude thatβcoC≤ βC.
4Suppose max{βC, βD}< ε, that is,βC< εandβD< ε. PickB1, B2∈bbX
satisfyingHC, B1< εandHD, B2< ε. Then
C⊂B1εBX, B1⊂CεBX,
D⊂B2εBX, B2⊂DεBX.
3.17
Thus we get
C∪D⊂B1∪B2 εBX, B1∪B2⊂C∪D εBX, 3.18
whenceHC∪D, B1∪B2 ≤ ε and,B1∪B2 being balanced, alsoβC∪D ≤ ε. From the
arbitrariness ofεwe conclude thatβC∪D≤max{βC, βD}.
5 Ifλ 0, the property is obvious. Assume λ /0. Given ε > |λ|βC, there exists B∈bbXsuch that
C⊂B
ε
|λ|
BXBε λ
BX,
B⊂C
ε
|λ|
BX Cε λ
BX.
3.19
Then
λC⊂λBεBX, λB⊂λCεBX, 3.20
so thatHλC, λB≤ε. SinceλBis balanced, it follows thatβλC≤εand,εbeing arbitrary, we obtainβλC≤ |λ|βC. Conversely, letε > βλC. Then there existsB∈bbXsuch that
Hence, C⊂ 1 λ
Bε λ BX 1 λ B ε
|λ|
BX, 1 λ
B⊂Cε λ
BXC
ε
|λ|
BX.
3.22
Therefore,HC,1/λB≤ε/|λ|. Since1/λBis balanced we conclude thatβC≤ε/|λ|, or
|λ|βC≤ε. The arbitrariness ofεfinally yields|λ|βC≤βλC.
6LetβC βD< εand letε1, ε2 >0 satisfyε ε1ε2,βC < ε1andβD< ε2.
ChooseB1, B2∈bbXsuch thatHC, B1< ε1andHD, B2< ε2. Then
C⊂B1ε1BX, B1⊂Cε1BX,
D⊂B2ε2BX, B2⊂Dε2BX.
3.23
Thus we obtain
CD⊂B1B2εBX, B1B2⊂CDεBX, 3.24
whenceHCD, B1B2 ≤ ε and,B1B2 being balanced, alsoβCD ≤ ε. From the
arbitrariness ofεwe conclude thatβCD≤βC βD.
7This follows fromProposition 2.2.
8ForB∈bbXthere holdsHC, B≤HC, DHD, B, whenceβC≤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 asα, the measureβfails to be monotone and, in general, the identitiesβcoC βCand
βC∪D maxβC, βD 3.25
do not holdcf.Remark 2.3.
4. A Fixed Point Theorem for Multimaps
The study of fixed points for multivalued mappings was initiated by Kakutani5in 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 spaceX. We begin by recalling some definitions.
IfFis a multifunction andA∈2M, then
FA: x∈A
Fx. 4.1
Definition 4.2. GivenM∈2X\{∅}, letF:MX, and letγrepresent any of the two measures of noncircularity introduced above. A fixed point ofF is a pointx ∈Msuch thatx∈Fx. The multifunctionFwill be called
iaγ-contractionof constantk, if
γFB≤kγB B∈bX∩2M 4.2
for somek∈0,1;
iiaγ, φ-contraction, if
γFB≤φγB B∈bX∩2M, 4.3
whereφ:0,∞→ 0,∞is a comparison function, that is,φis increasing,φ0 0, andφnr → 0 asn → ∞for eachr >0.
Note that aγ-contraction of constantkcorresponds to aγ, φ-contraction withφr kr r≥0.
In order to establish our main result, we prove a property of Cantor type for the E-L and Hausdorffmeasures of noncircularity.
Proposition 4.3. LetXbe a Banach space and{Ak}k∞0⊂bXa decreasing sequence of closed sets such thatlimk→ ∞γAk 0, whereγdenotes eitherαorβ. Then the set
A∞: ∞
k0
Ak 4.4
satisfies
A∞ ∞
k0
baAk. 4.5
HenceA∞belongs tobXand is closed and balanced.
Proof. ByProposition 3.3we have limk→ ∞αAk 0 if, and only if, limk→ ∞βAk 0. Thus for the proof it suffices to setγα.
SinceAk⊂baAk k∈N, necessarily
A∞ ∞
k0
Ak⊂ ∞
k0
Conversely, letx∈∞k0baAk. As limk→ ∞αAk 0, to everyε >0 there correspondsN ∈N
such thatn∈N,n≥Nimplies baAn⊂AnεBX. This yields an increasing sequence{nm}∞m1 of positive integers and vectorsanm ∈Anmwhich satisfyx−anm ≤1/mm∈N, m≥1. Thus the sequence{anm}∞m1converges toxasm → ∞. Moreover, sinceanm ∈Anm ⊂ Ak m, k ∈
N, m≥ 1, nm ≥kandAkis closed, we find thatx ∈Ak k ∈ N. In other words,x ∈A∞. This proves4.5.
Note that∅/An ⊂ baAn implies 0 ∈ baAn n ∈ N, whence 0 ∈ A∞/∅. Since the intersection of closed, bounded and balanced sets preserves those properties, so doesA∞.
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 functionalfin a real, nonreflexive Banach spaceXwhich fails to be norm-attaining 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
x∈BX:fx≥1− 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 noncircularityαorβ.
Theorem 4.5. LetX be a Banach space, and letM ∈bXbe closed. IfF : M Mis aγ, φ -contraction with closed values, then0∈Mand 0 is a fixed point ofF.
Proof. Our hypotheses imply
Fn1M⊂FnM n∈N,
lim n→ ∞γF
nM≤ lim n→ ∞φ
nγM0. 4.8
SettingAn FnM n∈N, from Propositions2.2and3.3we find that{An}∞
n0 ⊂bXis a
decreasing sequence of closed sets with limn→ ∞γAn 0.Proposition 4.3shows thatA∞is a nonempty, balanced subset ofM; in particular, 0∈A∞⊂M. Now,{0}being balanced, we have
γF0≤φγ{0}0, 4.9
whenceγF0 0. This shows that the nonempty setF0 F0is balanced and forces 0∈F0, as asserted.
Corollary 4.6. Let X be a Banach space, and letM ∈ bXbe closed. If F : M M is a γ -contraction with closed values, then0∈Mand 0 is a fixed point ofF.
Acknowledgments
This paper has been partially supported by ULLMGC grantsand MEC-FEDER MTM2007-65604, MTM2007-68114. It is dedicated to Professor A. Martin ´on on the occasion of his 60th birthday. The author is grateful to Professor J. Bana´s for his interest in this work.
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