Let X be a nearly uniformly convex Banach space, C a convex closed bounded subset of X and T : C → 2 C a multivalued nonexpansive
mapping with convex compact values. We prove that T has a fixed point. This result improves former results in Domínguez Benavides, T., P. Lorenzo, Fixed point theorems for multivalued nonexpansive mappings without uniform convexity, Abstr. Appl. Anal. 2003:6 (2003), 375–386 and solves an open problem appearing in  Xu, H.K., Metric fixed point theory for multivalued mappings, Dissertationes Math. (Rozprawy Mat.) 389 (2000), 39 pp.Dirección General de Enseñanza SuperiorJunta de Andalucí

Domínguez Benavides, Tomás

Lorenzo Ramírez, Josefa

idUS. Depósito de Investigación Universidad de Sevilla

ANNALESUNIVERS ITAT I S MARIAE CUR IE - SKŁODOWSKALUBL IN – POLONIAVOL. LVIII, 2004 SECTIO A 37–45T. DOMI´NGUEZ BENAVIDES and P. LORENZOAsymptotic centers and fixed pointsfor multivalued nonexpansive mappingsDedicated to W.A. Kirk on the occasion ofHis Honorary Doctorate ofMaria Curie-Skłodowska UniversityAbstract. Let X be a nearly uniformly convex Banach space, C a convexclosed bounded subset of X and T : C → 2C a multivalued nonexpansivemapping with convex compact values. We prove that T has a fixed point. Thisresult improves former results in [4] and solves an open problem appearing in[17].1. Introduction. In 1969 Nadler [15] extended the Banach ContractionPrinciple to multivalued contractive mappings in complete metric spaces.Namely, he proved: Let X be a complete metric space and T : X → 2X acontraction with closed bounded values. Then T has a fixed point. Sincethen, many authors have studied the possibility of extending classical fixedpoint theorems for single-valued nonexpansive mappings to the setting ofmultivalued nonexpansive mappings. Even though several authors have2000 Mathematics Subject Classification. 47H09, 47H10, 46B20.Key words and phrases. Multivalued nonexpansive mappings, asymptotic centers, fixedpoints, normal structure, nearly uniform convexity.The authors are partially supported by DGES, Grant D.G.E.S. REF. PBMF2003-03893-C02-C01 and Junta de Andaluc´ıa, Grant FQM-127.38 T. Domı´nguez Benavides and P. Lorenzoobtained fixed point results for Banach spaces satisfying some strong geo-metric restriction (see [14], [2], [12] and the survey [17]) many problemsremain open in this theory. For instance the following very general problemis still open [16]: Let X be a Banach space satisfying the fixed point prop-erty (FPP), i.e. every nonexpansive single valued mapping defined from aconvex bounded closed subset of X into itself has a fixed point. Does X sat-isfy the same property for multivalued nonexpansive mappings with closedbounded values? The answer to this question could be strongly connectedwith the problem of obtaining a nonexpansive selection for any nonexpansivemultivalued mapping. In spite of the well known Michael selection theoremwhich gives a continuous selection for multivalued upper semicontinuousmappings, almost nothing is known about obtaining a nonexpansive selec-tion. A positive result in this direction was obtained by W.A. Kirk, M.A.Khamsi and C. Mart´ınez Yan˜ez [6] for a class of nonexpansive multivaluedmappings in hyperconvex metric spaces. However these problems are toogeneral and we cannot expect a positive answer for them. Thus, it seemsto be more convenient to study particular problems. For instance, the cele-brated Kirk’s theorem [7] which states the FPP for reflexive Banach spaceswith normal structure yields to a very natural question: Do reflexive Banachspaces with normal structure have the FPP for multivalued nonexpansivemappings ? The answer is unknown, either. Since normal structure is im-plied by different geometrical properties of Banach spaces, it is natural toconsider the following problem: Do these properties imply the FPP for mul-tivalued mappings? Let us list some of the properties implying reflexivityand normal structure:(1) X is uniformly convex (UC),(2) X is nearly uniformly convex (NUC),(3) ε0(X) < 1 where ε0(X) is the characteristic of convexity,(4) εα(X) < 1 where εα(X) is the characteristic of noncompact convexityfor the Kuratowski measure of noncompactness,(5) εβ(X) < 1 where εβ(X) is the characteristic of noncompact convexityfor the separation measure of noncompactness.Furthermore, we have the following relationships between these notions:UC ⇒ NUC⇓ ⇓ε0(X) < 1⇒ εα(X) < 1⇒ εβ(X) < 1Hence the following question arises: Does any of the above propertiesimply the FPP for multivalued nonexpansive mappings? Of course, thesequestions are scaled. A positive answer for the case εβ(X) < 1 solves allcases.Asymptotic centers and fixed points... 39T.C. Lim [13] obtained a fixed point theorem for a multivalued nonex-pansive self-mapping in a uniformly convex Banach space. W.A. Kirk [8]gave an extension of Lim’s theorem proving the existence of a fixed pointin a Banach space for which the asymptotic center of a bounded sequencein a closed bounded convex subset is nonempty and compact (note thatthe asymptotic center is a singleton in UC spaces). First, he proved thefollowing result.Theorem 1.1. Let C be a nonempty weakly compact and separable subsetof a Banach space X, T : C → 2C a nonexpansive mapping with compactvalues and {xn} a sequence in C such that limn d(xn, Txn) = 0. Then, thereexists a subsequence {zn} of {xn} such thatTx ∩A 6= ∅, ∀x ∈ A := A(C, {zn}).Noting that we can reduce to a separable setting and using a fixed pointtheorem for compact operators he proved that Banach spaces such thatasymptotic centers of bounded sequences are compact enjoy the FPP formultivalued nonexpansive mappings with convex bounded closed values.However, the asymptotic center of bounded sequences in NUC spaces canbe noncompact [11]. Thus, Kirk’s result does not solve the other questionsarisen above. Using some inequalities relating characteristic of noncompactconvexity, Chebyshev centers and asymptotic centers, the following partialextension of Kirk’s result is obtained in [4].Theorem 1.2. Let X be a Banach space such that εβ(X) < 1 and Xsatisfies the nonstrict Opial condition, C a convex bounded subset of X andT : C → 2C a nonexpansive mapping with convex compact values. Then Thas a fixed point.An open question in [4] is the possibility of removing the nonstrict Opialproperty from the assumptions. We will prove in this paper that the aboveresult holds without any assumption on Opial conditions.2. Notation. Let us fix the notation which will be used. Let C be anonempty bounded closed subset of a Banach space X and {xn} a boundedsequence in X, we use r(C, {xn}) and A(C, {xn}) to denote the asymptoticradius and the asymptotic center of {xn} in C, respectively, i.e.r(C, {xn}) = inf{lim supn‖xn − x‖ : x ∈ C},A(C, {xn}) ={x ∈ C : lim supn‖xn − x‖ = r(C, {xn})}.It is known that A(C, {xn}) is a nonempty weakly compact convex set as Cis.40 T. Domı´nguez Benavides and P. LorenzoIf D is a bounded subset of X, the Chebyshev radius of D relative to Cis defined byrC(D) := inf {sup{‖x− y‖ : y ∈ D} : x ∈ C} .Let X be a Banach space. We denote by CB(X) the family of allnonempty closed bounded subsets of X and by K(X) (resp. KC(X)) thefamily of all nonempty compact (resp. compact convex) subsets of X. OnCB(X) we have the Hausdorff metric H given byH(A,B) := max{supa∈Ad(a,B), supb∈Bd(b, A)}, A,B ∈ CB(X)where for x ∈ X and E ⊂ X, d(x,E) := inf{‖x−y‖ : y ∈ E} is the distancefrom the point x to the subset E.If C is a closed convex subset of X and k ∈ [0, 1), then a multivaluedmapping T : C → CB(X) is said to be k-contractive ifH(Tx, Ty) ≤ k‖x− y‖, x, y ∈ C,and T is said to be nonexpansive ifH(Tx, Ty) ≤ ‖x− y‖, x, y ∈ C.A multivalued mapping T : C → 2X is called k-φ-contractive where φ isa measure of noncompactness if, for each bounded subset B of C, we haveφ(T (B)) ≤ kφ(B).Let us recall the definition of a nearly uniformly convex space.Definition 2.1. X is said to be nearly uniformly convex (NUC) if it isreflexive and its norm is uniformly Kadec-Klee, that is, for any positivenumber  there exists a corresponding number δ = δ() > 0 such that forany sequence {xn}‖xn‖ ≤ 1 n = 1, 2, . . .w- limnxn = xsep({xn}) := inf{‖xn − xm‖ : n 6= m} ≥  =⇒ ‖x‖ ≤ 1− δ.Assume that C is a subset of a Banach space X. Looking at C as ametric space we can consider the Hausdorff measure of noncompactness χCdefined for any bounded subset A of C by χC(A) = inf{ > 0 : A can becovered by finitely many balls centered at points in C with radii less than}. It must be noted that this measure depends on C and it is, in general,different from χ := χX . Furthermore, if C is a convex closed set, it is easyto check that the usual arguments to prove χX(A) = χX(co(A)) (see, forAsymptotic centers and fixed points... 41instance [1, Theorem 2.4]) equally well apply to prove χC(A) = χC(co(A))for any bounded subset A ⊂ C. Furthermore, if C is separable, for anybounded subset A of C there exists B ⊂ A such that χC(B) = χC(A) andB is χC-minimal, i.e. χC(B) = χC(D) for any infinite subset D of B (fordefinition and properties of χC-minimal sets, see [10], Chapter 8). Apartfrom χ and χC , we shall consider in this paper the separation measure ofnoncompactness defined byβ(B)= sup{ : there exists a sequence {xn} in B such that sep({xn})≥ }for any bounded subset B of a Banach space X.Definition 2.2. Let X be a Banach space. The modulus of noncompactconvexity associated to β is defined in the following way:∆X,β() = inf{1− d(0, A) : A ⊂ BX is convex, β(A) ≥ }(BX is the unit ball of X).The characteristic of noncompact convexity of X associated with themeasure of noncompactness β is defined byβ(X) = sup{ ≥ 0 : ∆X,β() = 0}.When X is a reflexive Banach space we have the following alternativeexpression for the modulus of noncompact convexity associated with β,∆X,β() = inf{1− ‖x‖ : {xn} ⊂ BX , x = w- limnxn, sep({xn}) ≥ }.It is known that X is NUC if and only if β(X) = 0. The above-mentioneddefinitions and properties can be found in [1].3. Fixed point results. In the sequel we are going to use the followingresult.Theorem 3.1 ([3, Lemma 11.5]). Let X be a Banach space and ∅ 6= D ⊂ Xbe compact convex. Let F : D → 2X be upper semicontinuous with closedconvex values. If Fx ∩ ID(x) 6= ∅ on D then F has a fixed point. (HereID(x) is called the inward set at x defined by ID(x) := {x+ λ(y − x) : λ ≥0, y ∈ D}.)The following lemma is the key of this paper, stating a relationship be-tween k-contractive and k-χC-contractive mappings.Theorem 3.2. Let C be a weakly compact convex separable subset of aBanach space X. Assume that T : C → 2C is a multivalued k-contractivemapping with compact values. Then, T is k-χC-condensing.42 T. Domı´nguez Benavides and P. LorenzoProof. Let A be a bounded subset of C. Since C is separable there existsa χC-minimal subset B ⊂ T (A) such that χC(B) = χC(T (A)). We canassume that B is countable, i.e. B = {yn : n ∈ N}. Since C is separable,taking a subsequence, we can assume that lim ‖yn−x‖ exists for any x ∈ C.Then χC({yn : n ∈ N}) = r(C, {yn}). Indeed, denote h = χC({yn : n ∈N}). For any  > 0 there exist a1, . . . , aN ∈ C such that {yn : n ∈ N} ⊂⋃Ni=1B(ai, h + ) where B(a, r) denotes the open ball centered at a withradius r. Thus, for a subsequence {zn} of {yn} and some i ∈ {1, . . . , N} wehave lim supn ‖zn − ai‖ = limn ‖yn − ai‖ ≤ h+ . Hence r(C, {yn}) ≤ h+ and, since  is arbitrary, we have r(C, {yn}) ≤ h. On the other hand,denote r = r(C, {yn}). We know that the asymptotic center A(C, {yn})is nonempty. Take a ∈ A(C, {yn}) and  > 0. There exists n0 such that‖yn− a‖ < r+  for n > n0. Thus χC({yn : n ∈ N}) = χC({yn : n > n0}) ≤r +  and we obtain the opposite inequality.Choose xn ∈ A such that yn ∈ Txn. Taking again a subsequence we canassume that the set {xn : n ∈ N} is χC-minimal and limn ‖xn−x‖ exists forevery x ∈ C. The same argument as above proves that χC({xn : n ∈ N}) =r(C, {xn}). Let u ∈ A(C, {xn}), i.e. limn ‖xn − u‖ = χC({xn : n ∈ N}).Since T is compact valued, we can take un ∈ Tu such that ‖yn − un‖ =d(yn, Tu). Using the compactness of Tu and taking again a subsequence wecan assume that {un} converges strongly to a point v ∈ Tu. Hence, we haveχC(T (A)) = r(C, {yn}) ≤ limn‖yn − v‖ = lim supn‖yn − un‖= lim supnd(yn, Tu) ≤ lim supnH(Txn, Tu)≤ k limn‖xn − u‖ = kχC({xn : n ∈ N})≤ kχC(A).Theorem 3.3. Let C be a weakly compact convex separable subset of aBanach space X. Assume that T : C → 2C is a multivalued k-contractive(k < 1) mapping with convex compact values. Assume that A is a convexclosed subset of C such that Tx ∩ A 6= ∅ for every x ∈ A. Then, T has afixed point in A.Proof. According to Theorem 3.2, the mapping T is k-χC-condensing. Wecould check that the arguments in the proof of [3, Theorem 11.5] equally wellwork for the measure χC when we assume that T (C) ⊂ C. However, we aregoing to prove that Theorem 3.1 can be directly applied to obtain the fixedpoint. To do that, we follow an induction argument. Denote A1 = A andassume that we have defined a finite decreasing sequence of convex closedsets An ⊂ An−1 ⊂ . . . ⊂ A1 such that Tx ∩ Ak 6= ∅ for every x ∈ Ak andfor all k = 1, . . . , n. Define An+1 = [coT (An)]∩An. Then, An+1 is a closedAsymptotic centers and fixed points... 43convex subset of An. Furthermore, for every x ∈ An+1 we have Tx∩An 6= ∅.Since Tx ⊂ T (An) we obtain that Tx ∩An+1 is nonempty. FurthermoreχC(An+1) = χC([coT (An)] ∩An) ≤ χC(T (An)) ≤ kχC(An).Then, A∞ :=⋂∞n=1An is a nonempty compact convex subset of A. Letx ∈ A∞ and take an ∈ Tx ∩ An which is nonempty. The sequence {an},which lies in a weakly compact set, has some cluster points for the weaktopology. Assume that a is a weak cluster point. Then a ∈ A∞. Since Txis weakly closed and the sequence {an} lies in Tx we have that a ∈ Tx anda ∈ A∞ which implies that Tx ∩ A∞ 6= ∅. Since A∞ is compact we obtainfrom Theorem 3.1 that T has a fixed point in A∞ ⊂ A. Next, we present a theorem which gives a connection between the asymp-totic center of a sequence and the modulus of noncompact convexity.Theorem 3.4 ([4, Theorem 3.1]). Let C be a closed convex separable subsetof a reflexive Banach space X and let {yn} be a bounded sequence in C.Then, there exists a subsequence {xn} such thatrC(A(C, {xn})) ≤ (1−∆X,β(1−))r(C, {xn}).We can prove now that all conditions (1)–(5), assuring normal structure,in the introduction, imply the FPP for multivalued nonexpansive mappingswith compact convex values. In particular, NUC spaces enjoy this propertysolving an open problem appearing in [17].Theorem 3.5. Let C be a nonempty closed bounded convex subset of a Ba-nach space X such that β(X) < 1, and T : C → KC(C) be a nonexpansivemapping. Then T has a fixed point.Proof. From [11] we can assume that C is separable. We claim that forany bounded closed subset A of C such that Tx ∩ A 6= ∅ for every x ∈ A,there exists an approximated fixed point sequence of T in A, i.e. there exists{xn} ⊂ A such that d(xn, Txn) → 0. Indeed, let x0 ∈ A be fixed and, foreach n ≥ 1, defineTnx :=1nx0 +(1− 1n)Tx, x ∈ C.Then, Tn is (1 − 1/n)-contractive and from Theorem 3.3 has a fixed pointxn. It is easily seen that limn d(xn, Txn) = 0. Using this fact, we can followthe proof as in [4, Theorem 4.1]. To do that we consider A1 = C. Assumethat sets A1, . . . , Am and approximated fixed point sequences {xkn} ⊂ Ak areconstructed where Ak = A(C, {xk−1n }) and rC(Ak) ≤ (1−∆X,β(1−))krC(A1)for k = 2, . . . ,m. Defining Am+1 = A(C, {xmn }) and choosing a suitableapproximated fixed point sequence {xm+1n } in Am+1 we obtain rC(Am+1) ≤44 T. Domı´nguez Benavides and P. Lorenzo(1−∆X,β(1−))rC(Am) and we can continue the induction process. As in [4,Theorem 4.1] it can be proved that the diagonal sequence {xnn} convergesto a fixed point of T . Remark. According to [13], every uniformly convex space has the FPP formultivalued nonexpansive mappings with compact values. In Theorem 3.5,we need to assume, in addition, that T has convex values. We do not knowif this assumption can be removed, but our method in the proof does notwork without convexity. Indeed, the main tool in our proof is to obtain anapproximated fixed point sequence for T in a set A such that Tx∩A 6= ∅. Todo that, we cannot use fixed point results for contractive mappings, becausethese results do not hold for mappings which are not self-mappings. Thefollowing example illustrates this fact: Assume X = R, A = [0, 1] and defineT : A→ 2X byTx = [−1, 2] \(3x4− 12,3x4+12).Then, T is 3/4-contractive and satisfies Tx ∩ A 6= ∅ for every x ∈ A.However, T is fixed point free. Thus, in the proof of Theorem 3.5 we needto use a fixed point result for compact mappings. On the other hand,we cannot expect such a result without convexity assumptions. Indeed,consider the following easy example. Assume that X = R2 and D is theclosed unit disk. Define T (0) = ∂D and T (x) = ∂D \ B(x/‖x‖, ‖x‖) forx 6= 0. Then T is a continuous and fixed point free mapping.References[1] Ayerbe, J.M., T. Domı´nguez Benavides and G. López Acedo, Measures of Non-compactness in Metric Fixed Point Theory, Birkha¨user-Verlag, Basel–Boston–Berlin,1997.[2] Browder, F.E., Nonlinear operators and nonlinear equations of evolution in Banachspaces, Nonlinear Functional Analysis, Proc. Sympos. Pure Math., vol. 18, pt. 2,Amer. Math. Soc., Providence, RI, 1976, 1–308.[3] Deimling, K., Multivalued Differential Equations, Walter de Gruyter, Berlin–NewYork, 1992.[4] Domı´nguez Benavides, T., P. Lorenzo, Fixed point theorems for multivalued non-expansive mappings without uniform convexity, Abstr. Appl. Anal. 2003:6 (2003),375–386.[5] Goebel, K., W.A. Kirk, Topics in Metric Fixed Point Theory, Cambridge UniversityPress, Cambridge, 1990.[6] Khamsi, M.A., W.A. Kirk and C. Mart´ınez-Ya´n˜ez, Fixed point and selection theoremsin hyperconvex spaces, Proc. Amer. Math. Soc. 128 (11) (2000), 3275–3283.[7] Kirk, W.A., A fixed point theorem for mappings which do not increase the distances,Amer. Math. Monthly 72 (1965), 1004–1006.Asymptotic centers and fixed points... 45[8] Kirk, W.A., Nonexpansive mappings in product spaces, set-valued mappings and k-uniform rotundity, Nonlinear Functional Analysis and Its Applications, (Ed. F. Brow-der) Proc. Sympos. Pure Math. 45, pt. 2, Amer. Math. Soc., Providence, RI, 1986,51–64.[9] Kirk, W.A., S. Massa, Remarks on asymptotic and Chebyshev centers, Houston J.Math. 16 (3) (1990), 357–364.[10] Kirk, W.A., B. Sims (eds.), Handbook of Metric Fixed Point Theory, Kluwer Aca-demic Publishers, Dordrecht–Boston–London, 2001.[11] Kuczumow, T., S. Prus, Asymptotic centers and fixed points of multivalued nonex-pansive mappings, Houston J. Math. 16 (1990), 465–468.[12] Lami Dozo, E., Multivalued nonexpansive mappings and Opial’s condition, Proc.Amer. Math. Soc. 38 (1973), 286–292.[13] Lim, T.C., A fixed point theorem for multivalued nonexpansive mappings in a uni-formly convex Banach space, Bull. Amer. Math. Soc. 80 (1974), 1123–1126.[14] Markin, J., A fixed point theorem for set valued mappings, Bull. Amer. Math. Soc.74 (1968), 639–640.[15] Nadler, S.B., Jr., Multi-valued contraction mappings, Pacific J. Math. 30 (1969),475–488.[16] Reich, S., Some problems and results in fixed point theory, Contemporary Math. 21(1983), 179–187.[17] Xu, H.K., Metric fixed point theory for multivalued mappings, Dissertationes Math.(Rozprawy Mat.) 389 (2000), 39 pp.T. Domı´nguez Benavides P. LorenzoFacultad de Matema´ticas Facultad de Matema´ticasUniversidad de Sevilla Universidad de SevillaP.O. Box 1160, 41080-Sevilla P.O. Box 1160, 41080-SevillaSpain Spaine-mail: tomasd@us.es e-mail: ploren@us.esReceived July 1, 2004

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Asymptotic centers and fixed points for multivalued nonexpansive mappings

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Asymptotic centers and fixed points for multivalued nonexpansive mappings

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