Does Kirk’s theorem hold for multivalued nonexpansive mappings?

Hindawi Publishing Corporation Fixed Point Theory and Applications Volume 2010, Article ID 546761, 20 pages doi:10.1155/2010/546761 Research Article Does Kirk’s Theorem Hold for Multivalued Nonexpansive Mappings? T. Dom´ ınguez Benavides and B. Gavira Facultad de Matem´ aticas, Universidad de Sevilla, P.O. Box 1160, 41080 Sevilla, Spain Correspondence should be addressed to T. Dom´ ınguez Benavides, [email protected] Received 25 September 2009; Accepted 29 December 2009 Academic Editor: Mohamed A. Khamsi Copyright q2010 T. Dom´ ınguez Benavides and B. Gavira. 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. Fixed Point Theory for multivalued mappings has many useful applications in Applied Sciences, in particular, in Game Theory and Mathematical Economics. Thus, it is natural to try of extending the known fixed point results for single-valued mappings to the setting of multivalued mappings. Some theorems of existence of fixed points of single-valued mappings have already been extended to the multivalued case. However, many other questions remain still open, for instance, the possibility of extending the well-known Kirk’s Theorem, that is: do Banach spaces with weak normal structure have the fixed point property FPPfor multivalued nonexpansive mappings? There are many properties of Banach spaces which imply weak normal structure and consequently the FPP for single-valued mappings for example, uniform convexity, nearly uniform convexity, uniform smoothness,.... Thus, it is natural to consider the following problem: do these properties also imply the FPP for multivalued mappings? In this way, some partial answers to the problem of extending Kirk’s Theorem have appeared, proving that those properties imply the existence of fixed point for multivalued nonexpansive mappings. Here we present the main known results and current research directions in this subject. This paper can be considered as a survey, but some new results are also shown. 1. Introduction The presence or absence of a fixed point i.e., a point which remains invariant under a map is an intrinsic property of a map. However, many necessary or sufficient conditions for the existence of such points involve a mixture of algebraic, topological, or metric properties of the mapping or its domain. By Metric Fixed Point Theory, we understand the branch of Fixed Point Theory concerning those results which depend on a metric and which are not preserved when this metric is replaced by another equivalent metric. The first metric fixed point theorem was given by Banach in 1922. 2 Fixed Point Theory and Applications Theorem 1.1 Banach Contraction Principle, 1.Let Xbe a complete metric space and T:X→ Xa contractive mapping, that is, there exists k∈0,1such that dTx,Ty≤kdx,yfor every x,y ∈X.ThenThas a (unique) fixed point x0. Moreover, x0limnTnxfor every x∈X. Banach Theorem is a basic tool in Functional Analysis, Nonlinear Analysis and Differential Equations. Thus, it is natural to look for some generalizations under weaker assumptions. For many years Metric Fixed Point Theory just studied some extensions of Banach Theorem relaxing the contractiveness condition, and the extension of this result for multivalued mappings. In the 1960s, Metric Fixed Point Theory received a strong boost when Kirk 2proved that every singlevaluednonexpansive mapping T:C→C, defined from a convex closed bounded subset Cof a reflexive Banach space with normal structure, has a fixed point. The celebrated Kirk’s theorem had a profound impact in the development of Fixed Point Theory and iniciated the search of more general conditions for a Banach space and for asubsetCwhich assure the existence of fixed points. The result obtained by Kirk is, in some sense, surprising because it uses geometric properties of Banach spaces commonly used in Linear Functional Analysis, but rarely considered in Nonlinear Analysis until then. Thus, it is the starting point for a new mathematical field: the application of the Geometric Theory of Banach Spaces to Fixed Point Theory. From that moment on, many researchers have tried to exploit this connection, essentially considering some other geometric properties of Banach spaces which can be applied to prove the existence of fixed points for different types of nonlinear operators e.g., uniform smoothness, Opial property, nearly uniform convexity, nearly uniform smoothness, etc.. Fixed Point Theory for multivalued mappings has useful applications in Applied Sciences, in particular, in Game Theory and Mathematical Economics. Thus, it is natural to study the problem of the extension of the known fixed point results for singlevalued mappings to the setting of multivalued mappings. Some theorems of existence of fixed points of single-valued mappings have already been extended to the multivalued case. For example, in 1969 Nadler 3extended the Banach Contraction Principle to multivalued contractive mappings in complete metric spaces. However, many other questions remain open, for instance, the possibility of extending the well-known Kirk’s Theorem 2, that is, do Banach spaces with weak normal structure have the fixed point property FPPfor multivalued nonexpansive mappings? There are many properties of Banach spaces which imply weak normal structure and consequently the FPP for singlevalued mappings e.g., uniform convexity, nearly uniform convexity, uniform smoothness, .... Thus, it is natural to consider the following problem: Do these properties also imply the FPP for multivalued mappings? As a consequence, some partial answers to the problem of extending Kirk’s Theorem have appeared, which are directed to prove that those properties imply the existence of fixed point for multivalued nonexpansive mappings. Here we present the main known results and current research directions in this subject. This paper can be considered as a survey, but some new results are also included. 2. Preliminaries In this section we recall the notion of normal structure and some properties of Banach spaces which imply normal structure. Fixed Point Theory and Applications 3 Normal structure plays an essential role in some problems of Metric Fixed Point Theory, especially those concerning nonexpansive mappings. The notion of normal structure was introduced by Brodski˘ ıand Mil’man 4in 1948 in order to study fixed points of isometries. Later, the notion of normal structure was generalized for the weak topology. Definition 2.1. A Banach space Xis said to have normal structure NSresp., weak normal structure w-NS if for every bounded closed resp., weakly compactconvex subset Cof X with diamC:sup{x−y:x, y ∈C}>0,there exists x∈Csuch that sup{x−y:y∈ C}<diamC. In 1965 Kirk 2obtained a strong connection between normal structure and the FPP for nonexpansive mappings. Theorem 2.2. Let Cbe a bounded closed (resp., weakly compact) convex subset of a Banach space X and let T:C→Cbe a nonexpansive mapping (i.e., Tx−Ty≤x−yfor every x, y ∈C). If Xis a reflexive Banach space with normal structure (resp., a Banach space with w-NS), then Thas a fixed point. Bynum 5defined two coefficients related to normal structure and weak normal structure. Definition 2.3. The normal structure coefficient of a Banach space Xis defined by NXinfdiamA rA:A⊂Xconvex closed and bounded with diamA>0,2.1 where diamAdenotes the diameter of Adefined by diamAsup{x−y:x, y ∈A}and rAdenotes the Chebyshev radius of Adefined by rAinf{sup{x−y:y∈A}:x∈A}. The weakly convergent sequence coefficient of Xis defined by WCSXinfdiama{xn} ra{xn},2.2 where the infimum is taken over all weakly convergent sequences {xn}which are not norm convergent, where, diama{xn}lim k→∞sup{xn−xm:n, m ≥k}, ra{xn}inflim sup nxn−x:x∈co{xn}2.3 denote the asymptotic diameter and radius of {xn},respectively. We recall that Xis said to have uniform normal structure UNSresp., weak uniform normal structure w-UNS if NX>1resp., WCSX>1. Notice that this is not the common definition of weak uniform normal structure and is often known as Bynum’s condition. It is known that if Xhas uniform normal structure, then Xis reflexive 6. 4 Fixed Point Theory and Applications In the latest fifty years, some geometrical properties implying normal structure have been studied. Here we are going to recall some of these properties and some results which prove that these properties imply the existence of fixed point for multivalued mappings. First we consider the Opial property. Opial 7was the first who studied such a property giving applications to Fixed Point Theory. The uniform Opial property was defined in 8by Prus, and the Opial modulus was introduced in 9by Lin et al. Definition 2.4. We will say that a Banach space Xsatisfies the Opial property if for every weakly null sequence {xn}and every x/ 0inX, we have lim inf n→∞ xn<lim inf n→∞ xnx.2.4 We will say that Xsatisfies the nonstrict Opial property if lim inf n→∞ xn≤lim inf n→∞ xnx2.5 under the same conditions. The Opial modulus of Xis defined for c≥0as rXcinflim inf nxnx−1,2.6 where the infimum is taken over all x∈Xwith x≥cand all weakly null sequences {xn} in Xwith lim infnxn≥1. We will say that Xsatisfies the uniform Opial property if rXc>0 for all c>0. There are some relationships between the notions of Opial property and normal structure. If Xis a Banach space which satisfies the Opial property, then Xhas w-NS 10. On the other hand, WCSX≥1rX19, Theorem 3.2. Consequently, Xhas w-UNS if rX1>0. Next we study the uniform convexity of the space, which is another geometrical property related with normal structure. We recall that a Banach space Xis uniformly convex UCif and only if δX:inf1−    xy 2   :x,y ∈Bx, x−y ≥>02.7 for each ∈0,2, or equivalently ε0X:sup{≥0:δX0}0.2.8 The Clarkson modulus δXand the coefficient of normal structure NXare related by the following inequality: NX≥1−δX1−1. Consequently, the condition δX1>0 implies that Xis reflexive and has uniform normal structure. In particular, notice that not only do uniformly convex spaces have normal structure, but so do all those spaces which do not have segments of length greater than or equal to 1 near the unit sphere. Fixed Point Theory and Applications 5 In 1980 Huff11initiated the study of nearly uniform convexity which is an infinite-dimensional generalization of uniform convexity. Independently of Huff,Goebeland Se¸kowski 12also introduced a property which is equivalent to nearly uniform convexity under the name of noncompact uniform convexity. It is known that a Banach space Xis nearly uniformly convex NUCif and only if ΔX,φ:inf1−d0,A :A⊂BXconvex,φ A> >02.9 for each >0, or equivalently εφX:sup≥0:ΔX,φ00,2.10 where φis a measure of noncompactness. Also we are going to use the following equivalent definition: Xis NUC if and only if Xis reflexive and ΔX:inf1−x:{xn}⊂BX,x nx,lim inf nxn−x≥>02.11 for each >0, or equivalently Δ0X:sup{>0:ΔX0}0.2.12 When Xis a reflexive Banach space, βis the separation measure and χis the Hausdorff measure for definitions see, for instance, 13or 14, we have the following relationships among the different moduli: ΔX,β≤ΔX≤ΔX,χ,2.13 and consequently, εβX≥Δ0X≥εχX.2.14 If the space Xsatisfies the nonstrict Opial property, then Δ0Xcoincides with εχX. On the other hand, if εβX<1in particular, if Xis NUC, then Xis reflexive and has weak uniform normal structure see 13, page 125. The dual concept of uniform convexity is uniform smoothness which is also related to normal structure. A Banach space Xis said to be uniformly smooth USif ρ X0lim t→0 ρXt t0,2.15 6 Fixed Point Theory and Applications where ρXis the modulus of smoothness of X, defined by ρXtsup1 2 xty  x−ty −1:x≤1, y ≤12.16 for t≥0. It is known that ρ X0<1/2 implies that Xis reflexive and has uniform normal structure 15–17. However, the infinite-dimensional generalization of uniform smoothness, nearly uniform smoothness, does not imply normal structure 13, Example VI.2. 3. Some Properties Implying Weak Normal Structure and the FPP for Multivalued Mappings In this section we are going to show some results which prove that some properties implying weak normal structure also imply the existence of fixed point for multivalued nonexpansive mappings. As a consequence these results give some partial answers to the problem of extending Kirk’s Theorem. Throughout this section KXresp., KCX will denote the family of all nonempty compact resp., compact convexsubsets of X. We recall that a multivalued mapping T: X→KXis said to be nonexpansive if HTx,Ty≤x−yfor every x, y ∈X, where H·,·denotes the Hausdorffmetric given by HA, B:maxsup a∈A da, B,sup b∈B db,A3.1 for every bounded subsets Aand Bof X. In 1973 Lami Dozo gave the following result of existence of fixed point for those spaces which satisfy the Opial property. Theorem 3.1 Lami Dozo 18, Theorem 3.2.Let Xbe a Banach space which satisfies the Opial property, let Cbe a weakly compact convex subset of X, and let T:C→KCbe a nonexpansive mapping. Then Thas a fixed point, that is, there exists x∈Csuch that x∈Tx. In 1974 Lim 19gave a similar result for uniformly convex spaces using Edelstein’s method of asymptotic centers 20. Theorem 3.2 Lim 19.Let Xbe a uniformly convex Banach space, let Cbe a closed bounded convex subset of Xand T:C→KCbe a nonexpansive mapping. Then Thas a fixed point. In 1990 Kirk and Massa proved the following partial generalization of Lim’s Theorem using asymptotic centers of sequences and nets. We recall that, given a bounded sequence {xn}in a Banach space Xand a subset Cof X, the asymptotic center of {xn}with respect to Cis defined by AC, {xn}:x∈C: lim sup nxn−xrC, {xn},3.2 Fixed Point Theory and Applications 7 where rC, {xn}denotes the asymptotic radius of {xn}with respect to Cdefined by rC, {xn}:inflim sup nxn−x:x∈C.3.3 Theorem 3.3 Kirk and Massa 21.Let Cbe a closed bounded convex subset of a Banach space Xand T:C→KCCa nonexpansive mapping. If the asymptotic center in Cof each bounded sequence of Xis nonempty and compact, then Thas a fixed point. We do not know a complete characterization of those spaces in which asymptotic centers of bounded sequences are compact. Nevertheless, there are some partial answers, for example, k-uniformly convex Banach spaces satisfy that condition 22. However, an example given by Kuczumov and Prus 23shows that in nearly uniformly convex spaces, the asymptotic center of a bounded sequence with respect to a closed bounded convex subset is not necessarily compact. Therefore, the problem of obtaining fixed point results in nearly uniformly convex spaces remained open. This question together with the same question for uniformly smooth spacesexplicitly appeared in a survey about Metric Fixed Point Theory for multivalued mappings published by Xu 24in 2000. The analysis of the importance of the asymptotic center in Kirk-Massa Theorem led Dom´ ınguez Benavides and Lorenzo to study some connections between asymptotic centers and the geometry of certain spaces, including nearly uniformly convex spaces. Thus, in 25 Dom´ ınguez and Lorenzo obtained the following relationship between the Chebyshev radius of the asymptotic center of a bounded sequence and the modulus of noncompact convexity with respect to the measures βand χ. Theorem 3.4 see 25, Theorem 3.4.Let Cbe a closed convex subset of a reflexive Banach space Xand {xn}a bounded sequence in Cwhich is regular with respect to C(i.e., the asymptotic radius is invariant for all subsequences of {xn}). Then rCAC, {xn} ≤1−ΔX,β1−rC, {xn},3.4 where the Chebyshev radius of a bounded subset Dof Xrelative to Cis defined by rCD:infsup x−y :y∈D:x∈C.3.5 Moreover, if Xsatisfies the nonstrict Opial property, then rCAC, {xn} ≤1−ΔX,χ1−rC, {xn}.3.6 The previous inequalities give an iterative method which reduces at each step the value of the Chebyshev radius for a chain of asymptotic centers. Consequently, Dom´ ınguez and Lorenzo deduced in 26the following partial extension of Kirk’s Theorem which, in particular, assures that nearly uniformly convex spaces have the fixed point property for multivalued nonexpansive mappings. 8 Fixed Point Theory and Applications Theorem 3.5 see 26, Theorem 3.5.Let Cbe a nonempty closed bounded convex subset of a Banach space Xsuch that εβX<1.LetT:C→KCCbe a nonexpansive mapping. Then Thas a fixed point. This result guarantees, in particular, the existence of fixed points in nearly uniformly convex spaces because εβX0ifXis NUC, giving a positive answer to one of the previous open problems proposed by Xu. Dhompongsa et al. 27observed that the main tool used in the proofs in 25,26,in order to obtain fixed point results for multivalued nonexpansive mappings, is a relationship between the Chebyshev radius of the asymptotic center of a bounded sequence and the asymptotic radius of the sequence. This relationship also gives an iterative method which reduces at each step the value of the Chebyshev radius for a chain of asymptotic centers. Consequently, in 27,28they introduced the Dom´ ınguez-Lorenzo condition DL- condition, in shortand property Din the following way. We recall that a sequence {xn}is regular with respect to Cif rC, {xn}rC, {xni} for all subsequences {xni}of {xn},and{xn}is asymptotically uniform with respect to Cif AC, {xn}AC, {xni}for all subsequences {xni}of {xn}. Definition 3.6. A Banach space Xis said to satisfy the DL-condition if there exists λ∈0,1 such that for every weakly compact convex subset Cof Xand for every bounded sequence {xn}in Cwhich is regular with respect to C rCAC, {xn} ≤λrC, {xn}.3.7 A Banach space Xis said to satisfy property Dif there exists λ∈0,1such that for any nonempty weakly compact convex subset Cof X, any bounded sequence {xn}in Cwhich is regular and asymptotically uniform with respect to C, and any sequence {yn}⊂AC, {xn} which is regular and asymptotically uniform with respect to C, we have rC, yn≤λrC, {xn}.3.8 From the definition it is easy to deduce that property Dis weaker than the DL- condition. Dhompongsa et al. proved in 28, Theorem 3.2and 28, Theorem 3.5that property Dimplies w-NS and the FPP for multivalued nonexpansive mappings. Theorem 3.7 see 28, Theorem 3.3.Let Xbe a Banach space satisfying property (D). Then Xhas w-NS. Theorem 3.8 see 28, Theorem 3.6.Let Cbe a nonempty weakly compact convex subset of a Banach space Xwhich satisfies property (D). Let T:C→KCCbe a nonexpansive mapping. Then Thas a fixed point. From Theorem 3.5 every Banach space with εβX<1satisfiestheDL-condition. In this paper we present some other properties concerning geometrical constants of Banach spaces which also imply the DL-condition or property D. Since our goal is to study if properties implying w-NS also imply the FPP for multivalued mappings, a possible approach to that problem is to study if these properties imply either the DL-condition or property D. These results will give only partial answers Fixed Point Theory and Applications 9 to the problem of extending Kirk’s Theorem for multivalued mappings because we know that uniform normal structure does not imply property D29,Proposition5; therefore, the problem of extending Kirk’s Theorem cannot be fully solved by this approach. In this setting the following results have been obtained. Theorem 3.9 Dhompongsa et al. 27, Theorem 3.4.Let Xbe a uniformly nonsquare Banach space with property WORTH. Then Xsatisfies the (DL)-condition. We recall that a Banach space Xis uniformly nonsquare if there exists δ>0such that x y∧x−y≤2−δfor every x, y ∈BXor equivalently JX<2,where JXdenotes the James constant of Xdefined by JXsup xy ∧ x−y :x,y ∈BX.3.9 Xis said to satisfy property WORTH if lim sup nxnxlim sup nxn−x3.10 for any x∈Xand any weakly null sequence {xn}in X. Theorem 3.10 Dhompongsa et al. 28, Theorem 3.7.Let Xbe Banach space such that CNJX<1WCSX2 4,3.11 where CNJXdenotes the Jordan-von Neumann constant of Xdefined by CNJXsup xy 2 x−y 2 2x22 y 2:x,y ∈Xnot both zero.3.12 Then Xsatisfies property (D). Theorem 3.11 Dom´ ınguez Benavides and Gavira 29, Corollary 1.Let Xbe a Banach space such that ρ X0<1 2.3.13 Then Xsatisfies the (DL)-condition. In particular, uniformly smooth Banach spaces (ρ X00) satisfy the (DL)-condition. Theorem 3.12 Dom´ ınguez Benavides and Gavira 29, Corollary 2.Let Xbe a Banach space such that one of the following two equivalent conditions is satisfied: 1rX1>0, 2Δ 0X<1. Then Xsatisfies the (DL)-condition. 16 Fixed Point Theory and Applications Theorem 4.7 see 40, Theorem 3.1.Let ρbe a convex function modular satisfying the Δ2-type condition, Ca nonempty ρ-bounded ρ-closed subset of Lρ, and T:C→FρCaρ-contraction mapping, that is, there exists a constant k∈0,1such that HρTf,Tg≤kρf−g,f,g∈C. 4.10 Then Thas a fixed point. By using that result, they proved the existence of fixed points for multivalued ρnonexpansive mappings. Theorem 4.8 see 40, Theorem 3.4.Let ρbe a convex function modular satisfying the Δ2-type condition, Ca nonempty ρ-a.e. compact ρ-bounded convex subset of Lρ, and T:C→KρCa ρ-nonexpansive mapping. Then Thas a fixed point. They also applied the above theorem to obtain fixed point results in the Banach space L1resp., 1for multivalued mappings whose domains are compact in the topology of the convergence locally in measure resp., w∗-topology. Consider the space LpΩ,μfor a σ-finite measure μwith the usual norm. Let Cbe a bounded closed convex subset of Lpfor 1 <p<∞and T:C→KCa multivalued nonexpansive mapping. Because of uniform convexity of Lp, it is known that Thas a fixed point. For p1, Tcan fail to have a fixed point even in the singlevalued case for a weakly compact convex set Csee 43. However, since L1is a modular space where ρfΩ|f|dμ ffor all f∈L1,Theorem 4.8 implies the existence of a fixed point when we define mappings on a ρ-a.e. compact ρ-bounded convex subset of L1. Thus the following can be stated. Corollary 4.9 see 40, Corollary 3.5.Let Ω,μbe as above, C⊂L1Ω,μa nonempty bounded convex set which is compact for the topology of the convergence locally in measure, and T:C→KC a nonexpansive mapping. Then Thas a fixed point. In the case of the space 1,we also can obtain a bounded closed convex set Cand a nonexpansive mapping T:C→Cwhich is fixed point free. Indeed, consider the following easy and well-known example. Let C{xn}∈1:0≤xn≤1,∞  n1 xn1.4.11 Define a nonexpansive mapping T:C→Cby Tx0,x 1,x 2,x 3,... ,where x{xn},4.12 then Tis a fixed point free map. However, if we consider Lρ1,where ρxx, for all x∈1, then ρ-a.e. convergence and ω∗-convergence are identical on bounded subsets of 1 see 36. This fact leads to the following corollary. Fixed Point Theory and Applications 17 Corollary 4.10 see 40, Corollary 3.6.Let Cbe a nonempty ω∗-compact convex subset of 1and T:C→KCa nonexpansive mapping. Then Thas a fixed point. Next we will give a property of closed convex bounded subsets of 1more general than weak star compactness which implies the fixed point property for nonexpansive mappings. Dom´ ınguez et al. introduced in 44some compactness conditions concerning proximinal subsets called Property P. Following this idea we will use the following similar notion for modular function spaces. Definition 4.11. Let Cbe a nonempty ρ-closed convex ρ-bounded subset of Lρ.ItissaidthatC has Property Pρif for every f∈Lρ,which is the ρ-a.e. limit of a sequence in C,thesetPρ,Cf is a nonempty and ρ-compact subset of C, where Pρ,Cf{g∈C:ρg−fdistρf,C}. Using that notion and the following two lemmas, we obtain a new fixed point result for multivalued ρ-nonexpansive mappings. Lemma 4.12 see 40, Lemma 3.3.Let ρbe a convex function modular satisfying the Δ2-type condition, f∈Lρ, and Ka nonempty ρ-compact subset of Lρ. Then there exists g0∈Ksuch that ρf−g0distρf,K.4.13 Lemma 4.13 see 37, Lemma 1.3.Let ρbe a function modular satisfying the Δ2-type condition, and {fn}nbe a sequence in Lρsuch that fn ρ-a.e. →f∈Lρand there exists k>1such that supnρkfn− f <∞. Then, lim inf n→∞ ρfn−glim inf n→∞ ρfn−fρf−g∀g∈Lρ.4.14 Theorem 4.14. Let ρbe a convex function modular satisfying the Δ2-type condition, Ca nonempty ρ-closed ρ-bounded convex subset of Lρsatisfying Property Pρsuch that every sequence in Chas a ρ-a.e. convergent subsequence in Lρ, and T:C→KρCCaρ-nonexpansive mapping. Then Thas a fixed point. Proof. Fix f0∈C. For each n∈N,theρ-contraction Tn:C→FρCis defined by Tnf1 nf01−1 nTf, f ∈C. 4.15 By Theorem 4.7, we can conclude that Tnhas a fixed point, say fn.Itiseasytoseethat distρfn,Tf n≤1 ndiamρC−→ 0.4.16 By our assumptions, we can assume, by passing through a subsequence, that fn ρ-a.e. →ffor some f∈Lρ.ByLemma 4.12, for each n∈Nthere exists gn∈Tfnsuch that ρfn−gndistρfn,Tf n.4.17 18 Fixed Point Theory and Applications Now we are going to show that Pρ,Cf∩Th/ ∅for each h∈Pρ,Cf. Taking any h∈Pρ,Cf, from the ρ-compactness of Th and Lemma 4.12, we can find hn∈Th such that ρgn−hndistρgn,Th ≤HρTfn,Th ≤ρfn−h,4.18 and we can assume, by passing through a subsequence, that hn ρ →h0for some h0∈Th.From above and using Lemma 4.13, it follows that lim inf nρfn−h0lim inf nρgn−h0lim inf nρgn−hn≤lim inf nρfn−h lim inf nρfn−fρf−h. 4.19 On the other hand, by Lemma 4.13 we also have lim inf nρfn−h0lim inf nρfn−fρf−h0.4.20 Thus, we deduce ρf−h0≤ρf−h, which implies that h0∈Pρ,Cfand so Pρ,Cf∩Th/ ∅. Now we define the mapping  T:Pρ,Cf→KCPρ,Cf by  ThPρ,Cf∩Th. From 45,Proposition2.45we know that the mapping  Tis upper semicontinuous. Since Pρ,Cf∩Th is a nonempty ρ-compact convex set and the ρ-topology is a norm-topology, we can apply the Kakutani-Bohnenblust-Karlin Theorem see 14 to obtain a fixed point for  T and hence for T. If we apply the previous theorem in the particular case of the space L1Ω,μfor a σ-finite measure μwith the usual norm, we obtain the following result, which can be also deduced from 44, Theorem 4.9. Corollary 4.15. Let Ω,μbe as above, C⊂L1Ω,μa nonempty closed bounded convex set which satisfies Property (P). Suppose, in addition, that every sequence in Chas a convergent locally in measure subsequence in L1.IfT:C→KCCis a nonexpansive mapping, then Thas a fixed point. If we consider now the space 1, then the assumption of existence of a w∗-convergent subsequence for every sequence in Ccan be removed and we can state the following result. Corollary 4.16. Let Cbe a nonempty closed bounded convex subset of 1which satisfies Property (P). If T:C→KCCis a nonexpansive mapping, then Thas a fixed point. Notice that in 1there exists a subset with Property Pwhich is not w∗-compact. Example 4.17 see 44, Example 4.8.Letanbe a bounded sequence of nonnegative real numbers and let enbe the standard Schauder basis of 1. It is clear that the set C:convxn, where xn:1anen, is never weakly star compact. Nevertheless, by using 46, Example 1it is easy to show that Chas Property Pif and only if N0:{n∈N:aninfm∈Nam}is nonempty and finite. Fixed Point Theory and Applications 19 Acknowledgments The authors are very grateful to the anonymous referee for some useful suggestions to improve the presentation of this paper. This research was partially supported by DGES Grant no.BFM2006-13997-C02-01 and Junta de Andaluc´ ıa Grant no.FQM-127. This research is dedicated to W. A. Kirk celebrating his wide and deep contribution in Metric Fixed Point Theory. References 1S. Banach, “Sur les op´ erations dans les ensembles abstraits et leurs applications,” Fundamenta Mathematicae, vol. 3, pp. 133–181, 1922. 2W. A. 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