Volume 2007, Article ID 19745,7pages doi:10.1155/2007/19745
Research Article
Diametrically Contractive Multivalued Mappings
S. Dhompongsa and H. Yingtaweesittikul
Received 1 March 2007; Accepted 2 May 2007
Recommended by Tomas Dominguez Benavides
Diametrically contractive mappings on a complete metric space are introduced by V. I. Istratescu. We extend and generalize this idea to multivalued mappings. An easy example shows that our fixed point theorem is more applicable than a former one obtained by H. K. Xu. A convergence theorem of Picard iteratives is also provided for multivalued mappings on hyperconvex spaces, thereby extending a Proinov’s result.
Copyright © 2007 S. Dhompongsa and H. Yingtaweesittikul. This is an open access arti-cle distributed under the Creative Commons Attribution License, which permits unre-stricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
1. Introduction
Let (X,d) be a complete metric space. A mappingT:X→Xis acontractionif for some
α∈(0, 1),
d(Tx,T y)≤αd(x,y) ∀x,y∈X. (1.1)
The mappingTis said to becontractiveif
d(Tx,T y)< d(x,y) ∀x,y∈X,x=y. (1.2)
By the well-known Banach’s contraction principle, every contraction has a unique fixed pointx0and the Picard iteration{Tnx}converges tox0for everyx∈X. Examples in [1,2] show that a contractive mapping may fail to have a fixed point. However, a question of the existence of a fixed point is of interest. In fact, it has been left open the following question.
Question 1.1[3]. LetMbe a weakly compact subset of a Banach space and letT:M→M
Istratescu [4] introduced a proper subclass of the class of the contractive mappings, whose elements are called the diametrically contractive mappings. Xu [2] proved, in the framework of Banach spaces, the following theorem.
Theorem1.2 [2, Theorem 2.3]. LetM be a weakly compact subset of a Banach spaceX and letT:M→Mbe a diametrically contractive mapping. ThenThas a fixed point.
The following problems raised in [2] had been answered in the negative way in [5].
Problem 1.3. Can we substitute “weakly compact” subset with “closed convex bounded” one inTheorem 1.2?
Problem 1.4. IfTis diametrically contractive andx∗is the fixed point ofT, do we have
Tnx→x∗for all (or at least for some)x∈M?
In this paper, we weaken the condition in the definition of diametrically contractive mappings and obtain a corresponding fixed point theorem for nonself multivalued map-pings. Moreover, we also apply a Proinov’s fixed point theorem to a selection of a multi-valued mapping with externally hyperconvex values and obtain its unique fixed point on a hyperconvex metric space.
2. Diametrically contractive mappings
In [4], Istratescu introduced a new class of mappings strictly lying between contractions and contractive mappings.
Definition 2.1. A mappingTon a complete metric space (X,d) is said to bediametrically contractiveifδ(TA)< δ(A) for all closed subsetsAwith 0< δ(A)<∞.
(Hereδ(A) :=sup{d(x,y) :x,y∈A}is the diameter ofA⊂X.)
In the following, we consider a multivalued version of mappings inTheorem 1.2. We also can weaken the condition required inDefinition 2.1.
LetᏲ(X) be the collection of nonempty closed subsets ofX and let FixT denote the set of fixed points ofT. Recall thatTA=a∈ATa.
Theorem2.2. Let M be a weakly compact subset of a Banach spaceX and letT:M→
Ᏺ(X),Tx∩M= ∅for allx∈Mandδ(TA∩A)< δ(A)for all closed setsAwithδ(A)>0. ThenThas a unique fixed point.
Proof. The uniqueness of the fixed point is obvious. To prove the existence we consider the familyᐁ:= {A⊂M:Ais a nonempty weakly compact subset ofM, TA∩M⊂A}. Clearly,ᐁ= ∅. Partially orderᐁby saying thatA1A2ifA1⊃A2forA1,A2∈ᐁ. Every chainᏯinᐁhas a finite intersection property, thus it has a nonempty intersection, that is,B:=A∈ᏯA= ∅. SinceTB∩M⊂TA∩M⊂Afor allA∈Ꮿ,TB∩M⊂B, that is,
Put A0=TA∩Aw. ThereforeA0=TA∩Aw⊂TA∩Mw⊂Aw=Aand so A0⊂A. Moreover, we haveTA0∩M⊂TA∩M⊂A. ThereforeTA0∩M⊂TA∩A⊂TA∩Aw= A0. ThusA0∈ᐁand by maximality ofA, we haveA=A0=TA∩Aw. By lower semicon-tinuity of the norm ofXwe see thatδ(A)=δ(TA∩Aw)=δ(TA∩A). SinceTis diamet-rically contractive we must haveδ(A)=0 andAconsists of exactly one point, sayξ. By the condition∅ =TA∩M⊂Awe see thatξ∈Tξ, and we have a fixed point.
The proof given above is a modification of the proof ofTheorem 1.2. The following example shows thatTheorem 2.2is strictly stronger thanTheorem 1.2.
Example 2.3. M=[0, 5],T:M→Rdefined byTx=x+ 1 ifx≤3, andTx=4 ifx >3. Now, ifAis a closed subset ofM, then there will bea,b inMsuch thatA⊂[a,b] and
δ(A)=b−a. If [a,b]⊂[0, 3], thenTA⊂[a+ 1,b+ 1] and TA∩A⊂[a+ 1,b]. Thus
δ(TA∩A)≤b−a−1< δ(A). Ifa≤3≤b, thenTA⊂[a+ 1, 4] and thereforeδ(TA∩
A)≤3−a < b−a=δ(A). The case when [a,b]⊂[3, 5],T clearly satisfiesδ(TA∩A)= 0< δ(A). ThusThas a fixed point byTheorem 2.2. Note that 4 is the unique fixed point of
T. We observe thatTdoes not satisfy the condition inTheorem 1.2becauseδ(T[0, 1])= 1=δ([0, 1]).
Example 2.4. LetTx=[0,x−log(x+ 1)] forx∈[0, 100]. IfAis a bounded closed sub-set of [0, 100], then for somea,b >0 we haveA⊂[a,b], andδ(A)=b−a. ClearlyTA⊂
x∈A[0,x−log(x+ 1)]⊂[0,b−log(b+ 1)], and soTA∩A⊂[a,b−log(b+ 1)]. There-foreδ(TA∩A)< δ(A). 0 is the unique fixed point ofT.
Next we will replace the diameterδ(A) of a setAbyα(A), whereαis the Kuratowski measure of noncompactness:
α(A)=inf{>0 :Acan be covered by finitely many sets with diameters ≤}. (2.1)
Definition 2.5. LetMbe a nonempty subset of a metric space (X,d). A mappingT:M→ 2Xis said to beak-set contractionif, for eachA⊂MwithAbounded,TAis bounded and
α(TA)≤kα(A). Ifα(TA)< α(A) for all suchA, thenTis said to beα-condensing.
Suppose thatMis a bounded subset of a metric space (X,d). Then: (i) co(M)={B⊂X:Bis a closed ball inXsuch thatM⊂B}, and
(ii)M is said to besubadmissible[6], if for eachA∈ M, co(A)⊂M, whereM denotes the class of all nonempty finite subsets ofM.
For a nonempty subsetMofXand a topological spaceY, if two set-valued mappings
T,F:M→2Y satisfy the conditionT(co(A)∩M)⊂F(A), for anyA∈ M, thenF is called a generalized KKM mapping with respect toT.
LetT:M→2Y be a set-valued mapping such that the family{Fx:x∈M}has the finite intersection property (whereFx denotes the closure of Fx) for each generalized KKM mappingF:M→2Y with respect toT, then we say thatT has the KKM property. Denote
Theorem2.6 [7, Theorem 1]. Let(X,d)be a complete metric space and letM be a non-empty bounded nearly subadmissible subset ofX. IfT∈KKM(M,M)is ak-set contraction,
0< k <1, and closed withTM⊂M, thenThas a fixed point inM.
The next result shows that we can replacek-set contractions inTheorem 2.6byα -condensing mappings.
Theorem2.7. Let(X,d)be a complete metric space and letM be a nonempty bounded nearly subadmissible subset of X. If T∈KKM(M,M)is α-condensing, and closed with TM⊂M, thenThas a fixed point inM.
In the course of the proof, we will apply the technique in the proof of the following lemma.
Lemma2.8 [8, Lemma 2.2]. LetFbe a selfmapping of an arbitrary setYand let f :Y→R+ be a nonnegative valued function defined onY. Suppose that the following conditions hold:
(i)there exists a functionϕ∈Φ1(i.e.,ϕ:R+→R+satisfying: for any>0, there exists δ >such that< t < δimpliesϕ(t)≤) such that f(F y)≤ϕ(f(y))for ally∈Y;
(ii) f(y)>0implies f(F y)< f(y)and f(y)=0implies f(F y)=0. Thenlimf(Fny)=0for eachy∈Y.
Proof ofTheorem 2.7. We follow the proof ofTheorem 2.6. Lety∈Mbe any point, and letM0=M. DefineM1=co(T(M0)∪ {y})∩M, andMn+1=co(T(Mn)∪ {y})∩M, for eachn. Then
αMn+1
≤αTMn
< αMn
≤ ···< α(M), (2.3)
for eachn(see [7]). If we can prove that
limαMn=0, (2.4)
then the rest of the proof will follow the same lines as ofTheorem 2.6. To achieve (2.4), we will apply the proof ofLemma 2.8. For eacht∈R+, letAt= {Mn:α(Mn)≤t}and
Bt= {α(Mn+1) :Mn∈At}. From (2.3), it is seen thatBt= ∅ and bounded ifAt= ∅. Defineϕ(t)=supBtifAt= ∅. Otherwise, putϕ(t)=0. We claim thatϕ∈Φ1. Consider the setAfor>0. Ifα(Mn)≤for somen, letn0be the smallest suchn. Ifn0=0, then by (2.3) it is seen thatϕ(t)≤for allt >. Otherwise, letδ=α(Mn0−1). Thusδ >, and
if< t < δ, then by (2.3) we haveϕ(t)≤α(Mn0+1)< α(Mn0)≤. Thereforeϕ∈Φ1. We
now prove (2.4). Clearly, we have
αMn+1
≤ϕαMn
, αMn+1
< αMn
by (2.3) for eachn. (2.5)
It follows from (2.3) that{α(Mn)}is strictly decreasing, hence it converges to some≥0. Suppose>0. Sinceϕ∈Φ1, we have for someδ >,ϕ(t)≤for allt∈(,δ). Choose n0so that< α(Mn0)< δ. Thusϕ(α(Mn0))≤. But then (2.5) impliesα(Mn)≤for all
3. Picard iteratives for multivalued mappings on hyperconvex metric spaces
A metric space (X,d) ishyperconvexif for any family of points{xα}inXand any family of positive numbers{rα}satisfyingd(xα,xβ)≤rα+rβ, we haveαB(xα,rα)= ∅whereB(x,r) is the closed ball with center atx and radiusr. A subsetEofX is said to beexternally hyperconvexif for any of those families{xα},{rα}withd(xα,xβ)≤rα+rβand dist(xα,E)≤
rα, we haveαB(xα,rα)∩E= ∅. The class of all externally hyperconvex subsets ofXwill be denoted byᏱ(X). LetHbe the Hausdorffmetric.
Let tbe a single-valued selfmapping on a metric space (X,d). A fixed point oft is said to be contractive(cf. [9]) if all Picard iteratives oft converge to this fixed point. A selfmappingt on a metric space (X,d) is said to be asymptotically regular(cf. [10]) if limd(tn(x),tn+1(x))=0 for eachxinX. Extend the concept naturally to multivalued mappings with the Hausdorffmetric taken into action.
Theorem3.1 [8, Theorem 4.1]. Lettbe a continuous and asymptotically regular selfmap-ping on a complete metric space satisfying the following conditions:
(i)there existsϕ∈Φ1such thatd(t(x),t(y))≤ϕ(D(x,y))for allx,y∈X; (ii)d(t(x),t(y))< D(x,y)for allx,y∈Xwithx=y.
Then t has a contractive fixed point. Here D(x,y)=d(x,y) +r[d(x,t(x)) +d(y,t(y))], r≥0.
ReplacingDbyd, we present a multivalued version ofTheorem 3.1on a special setting, namely, on the class of hyperconvex metric spaces.
Theorem3.2. Let(X,d)be a bounded hyperconvex metric space, and letT:X→Ᏹ(X)be asymptotically regular satisfying the following conditions:
(i)there existsϕ∈Φ1such thatϕ(x)≤x,ϕ(x+y)≤ϕ(x) +ϕ(y),ϕ(x)=0if and only ifx=0, andH(Tx,T y)≤ϕ(d(x,y))for allx,yinX;
(ii)H(Tx,T y)< d(x,y)for allx,y∈Xwithx=y.
Then, ifδ(Tnx)→0for eachx∈X,T has a contractive fixed point. That is, there exists a
unique pointξinXsuch that, for eachx∈X,Tnx→ {ξ} =FixT.
Proof. The uniqueness of the fixed point is evident. We are going to find a selectiont:
X→Xso thatt(x)∈Txfor allx∈Xandtsatisfies the conditions inTheorem 3.1. Thus, there is a pointξin Fixtsatisfyingtn(x)→ξfor allx∈X. To find a selectiont, we apply Zorn’s lemma to the family Ᏺ= {(A,t) :∅ =A⊂X, t:A→Aasymptotically regular,
t(a)∈Tafor alla∈A, andtsatisfies (i) and (ii) inTheorem 3.1}. Partially orderᏲby (A1,t1)(A2,t2) ifA1⊂A2andt2|A1=t1.
SupposeA= ∅or (A,t)∈Ᏺandx0∈X\A. We will define a countable set{x0,x1, x2,...}, possibly finite, and an extension functiont∗oftoverA∪ {x0,x1,x2,...}so that (A∪ {x0,x1,x2,...},t∗)∈Ᏺ. Let x0,x1,...,xn∈X\A have been defined for some n≥ 0 so that, for 1≤k≤n, xk∈Txk−1, and when n≥2, d(xi+1,xj+1)≤ϕ(d(xi,xj)) and
d(xi+1,xj+1)< d(xi,xj) fori < jin{1,...,n−1}. Moreover,d(t(x),xi+1)≤ϕ(d(x,xi)) and
Putrk=ϕ(d(xk−1,xn)), andrt(x)=ϕ(d(x,xn)) for each 1≤k≤n, and for allx∈A. Thus, for 1≤k≤n,x∈A, and fori < jin{1,...,n−1},
(i) dist(xk,Txn)≤H(Txk−1,Txn)≤ϕ(d(xk−1,xn))=rk, (ii) dist(t(x),Txn)≤H(Tx,Txn)≤rt(x),
(iii)d(t(x),xk)≤ϕ(d(x,xk−1))≤ϕ(d(x,xn)) +ϕ(d(xk−1,xn))=rt(x)+rk, and (iv)d(xi,xj)≤ϕ(d(xi−1,xj−1))≤ϕ(d(xi−1,xn)) +ϕ(d(xj−1,xn))=ri+rj.
Finally, for x,y∈A, d(t(x),t(y))≤ϕ(d(x,y))≤ϕ(d(x,xn)) +ϕ(d(y,xn))=rt(x)+rt(y). Therefore there exists a point xn+1∈x∈AB(t(x),rt(x))nk=1B(xk,rk)∩Txn. The point
xn+1has the following property: fork≤n,d(xk,xn+1)≤rk=ϕ(d(xk−1,xn)), and for each
x∈A,d(t(x),xn+1)≤rt(x)=ϕ(d(x,xn)). Clearly,d(xk,xn+1)<d(xk−1,xn) andd(t(x),xn+1)< d(x,xn).
Ifxn+1∈A, the process terminates. Otherwise, we obtain a subset{x0,x1,...,xn,...} ofX\Asatisfying the conditions (i) and (ii) inTheorem 3.1where we extendttot∗by definingt∗(xn)=xn+1forn≥0. Thus (A∪ {x0,x1,...},t∗)∈Ᏺ.
In summary, the above argument shows that, ifA= ∅, then ({x0,x1,...},t∗)∈Ᏺ, that is,Ᏺ= ∅. On the other hand if (A,t) is a maximal element inᏲ(by Zorn’s lemma), we must haveA=X, that is, (X,t) belongs toᏲfor somet. ApplyTheorem 3.1, to conclude that there exists a fixed pointξ oftsuch that tn(x)→ξ for each x∈X. Consequently,
Tnx→ {ξ}for eachx∈Xand FixT= {ξ}.
Acknowledgments
The authors would like to thank the referees for valuable comments. They also would like to thank the Thailand Research Fund (Grant BRG4780013) and the Development and Promotion of Science and Technology Talent Project (DPST) for their support.
References
[1] W. A. Kirk, “A fixed point theorem for mappings which do not increase distances,”The American Mathematical Monthly, vol. 72, no. 9, pp. 1004–1006, 1965.
[2] H.-K. Xu, “Diametrically contractive mappings,”Bulletin of the Australian Mathematical Society, vol. 70, no. 3, pp. 463–468, 2004.
[3] K. Deimling,Multivalued Differential Equations, vol. 1 ofde Gruyter Series in Nonlinear Analysis and Applications, Walter de Gruyter, Berlin, Germany, 1992.
[4] V. I. Istratescu, “Some fixed theorems for convex contraction mappings and mappings with con-vex diminishing diameters. IV nonexpansive diameter mappings in uniformly concon-vex spaces,” preliminary report. Abstract of the American Mathematiccal Society, 82T-46-316, 1982. [5] M. Baronti, E. Casini, and P. L. Papini, “Diametrically contractive maps and fixed points,”Fixed
Point Theory and Applications, vol. 2006, Article ID 79075, 8 pages, 2006.
[6] A. Amini, M. Fakhar, and J. Zafarani, “KKM mappings in metric spaces,”Nonlinear Analysis: Theory, Methods & Applications, vol. 60, no. 6, pp. 1045–1052, 2005.
[7] C.-M. Chen, “KKM property and fixed point theorems in metric spaces,”Journal of Mathemat-ical Analysis and Applications, vol. 323, no. 2, pp. 1231–1237, 2006.
[8] P. D. Proinov, “Fixed point theorems in metric spaces,”Nonlinear Analysis: Theory, Methods & Applications, vol. 64, no. 3, pp. 546–557, 2006.
[10] F. E. Browder and W. V. Petryshyn, “The solution by iteration of nonlinear functional equations in Banach spaces,”Bulletin of the American Mathematical Society, vol. 72, pp. 571–575, 1966.
S. Dhompongsa: Department of Mathematics, Faculty of Science, Chiang Mai University, Chiang Mai 50200, Thailand
Email address:[email protected]
H. Yingtaweesittikul: Department of Mathematics, Faculty of Science, Chiang Mai University, Chiang Mai 50200, Thailand
