The τ-fixed point property for nonexpansive mappings

THE τ-FIXED POINT PROPERTY FOR NONEXPANSIVE MAPPINGS TOM´ AS DOM´ INGUEZ BENAVIDES, JES´ US GARC´ IA FALSET AND MARIA A. JAP´ ON PINEDA Abstract. Let Xbe a Banach space and τa topology on X. We say that Xhas the τ-fixed point property (τ-FPP) if every nonexpansive mapping T defined from a bounded convex τ-sequentially compact subset Cof Xinto Chas a fixed point. When τsatisfies certain regularity conditions, we show that normal structure assures the τ-FPP and Goebel-Karlovitz’s Lemma still holds. We use this results to study two geometrical properties which imply the τ-FPP: the τ-GGLD and M(τ) properties. We show several examples of spaces and topologies where these results can be applied, specially the topology of convergence locally in measure in Lebesgue spaces. In the second part we study the preservence of the τ-FPP under isomorphisms. In order to do that we study some geometric constants for a Banach space Xsuch that the τ-FPP is shared by any isomorphic Banach space Ysatisfying that the Banach-Mazur distance between Xand Yis less than some of these constants. 1. Introduction Let (M,d) be a metric space. We recall that a mapping T:M→M is said to be nonexpansive if d(Tx,Ty)≤d(x, y) for every x, y ∈M.We say that a Banach space Xhas the fixed point property (FPP) if every nonexpansive mapping Tdefined from a convex bounded closed subset Cof Xinto Chas a fixed point. In 1965 Browder [5,6] and Kirk [33] proved that Xhas the FPP if Xis either a Hilbert space, or a uniformly convex space 1991 Mathematics Subject Classification. 47H09, 47H10. Key words and phrases. Fixed point, nonexpansive mapping, stability of the fixed point property, uniformly Lipschitzian mappings, asymptotically regular mappings, local convergence in measure, Lebesgue spaces. This research is partially supported by D.G.I.C.Y.T. PB 96-1338-C01-C02 and J.A. FQM 0127 Received: March 4, 1998 c 1996 Mancorp Publishing, Inc. 343 344 DOM´ INGUEZ BENAVIDES, GARC´ IA FALSET AND JAP ´ ON PINEDA or a reflexive space with normal structure. From this starting point, many authors have studied different geometric conditions on X, assuring the FPP. (See, for instance, [19] and references therein). However, c0is an easy example of a Banach space without the FPP [25]. The failure of the FPP in c0is a consequence of the noncompactness of its unit ball in the weak topology. This fact has originated that many authors study the existence of fixed points for a nonexpansive mapping, under the stronger assumption: C is a weakly compact convex subset of X. (See, for instance, [20] and references therein). If every nonexpansive mapping T:C→C,Cas above, has a fixed point, we say that Xhas the weak fixed point property (w-FPP). In several papers, the weak topology is replaced by some other topologies. For instance, in [1,13,14,28,29,38,40,42] the existence of fixed points is studied when Cis a weakcompact convex set of a dual space. In [3,36,37] the same problem is studied for the topology of convergence in measure. More general topologies are considered in [7,15,16,17,31,32,34,35]. In this paper we consider that τis any topology on Xand Cis a convex subset of Xwhich is sequentially compact for the topology τ. If every nonexpansive mapping T:C→Chas a fixed point, we say that Xhas the τ-FPP. Normal structure and Goebel-Karlovitz’s lemma are the most usual tools to prove the existence of fixed points for nonexpansive mappings. When τsatisfies certain regularity conditions, we show that normal structure still assures the τ-FPP and Goebel-Karlovitz’s lemma still holds. In Section 2, we use these tools to study two geometrical properties which imply the τ-FPP: the τ-GGLD and the M(τ) properties. We show several examples of spaces and topologies where these results can be applied. When Xhas the τ-FPP the following question can be interesting: If Yis a Banach space isomorphic to Xand the Banach-Mazur distance d(X,Y ) is “small”, does Yhave the τ-FPP? This problem, originated by Bynum [8], has been widely studied for the weak topology (see, for instance, [10,11,18,23,30,39]) and, occasionally, for the weaktopology [42]. However we do not know any previous result for a general topology. According to the assumptions on the topology τ, we can obtain a scale of stability bounds. When τis an arbitrary topology we can use the fixed point theory for uniformly Lipschitzian and asymptotically regular mappings to obtain stability bounds for nonexpansive mappings by means of the coefficient κτ(X). We show some classes of spaces and topologies where this coefficient is computed. In particular, for L1(Ω) and the topology of locally convergence in measure we obtain the best possible stability bound. When Xis a separable Banach space, τis a linear topology and the norm is a τ-sequentially lower semicontinuous (τ-slsc) function we define the coefficient τCS(X), which becomes Bynum’s weakly convergent sequences coefficient when τis the weak topology. However, we show that we cannot use a similar definition as that given by Bynum. This coefficient is used, in two different ways, to obtain stability bounds of the τ-FPP. In special classes of spaces, we compare these bounds and those obtained using κτ(X). Finally, assuming stronger regu- THE τ-FIXED POINT PROPERTY 345 larity properties of the norm we obtain better stability bounds using the coefficient Mτ(X), inspired in [10]. In particular, if X=Lp(Ω), 1 ≤p<∞ and τis the topology of convergence locally in measure, we obtain stability bounds which are identical to those which appear in [10] for the FPP in p-spaces. 2. Geometric Conditions Implying the τ-FPP Throughout this paper Xwill be a Banach space and τan arbitrary topology on X. In some theorems additional conditions will be assumed on τ. Definition 1. We say that Xhas the τ-FPP if every nonexpansive mapping Tdefined from a convex norm-bounded τ-sequentially compact subset Cof Xinto Chas a fixed point. From the Eberlein-Smulian Theorem , it is clear that the τ-FPP is the w-FPP when τis the weak topology. Definition 2. We say that Xhas normal structure with respect to τ(τ- NS) if for every convex norm-bounded τ-sequentially compact C⊂Xwith diamC>0, there exists an element x∈Cwhich is not diametral, i.e. sup{x−y:y∈C}<diam (C). Theorem 1. Suppose that ·is a τ-slsc function. If Xhas τ-NS then X has the τ-FPP. Proof. It is an easy consequence of Theorem 1 in [34]. Indeed, let Cbe a convex norm-bounded τ-sequentially compact subset of Xand T:C→Ca nonexpansive mapping. Denote by Sthe family of all convex τ-sequentially closed subsets of C. Then Sis countably compact, stable under arbitrary intersections, and normal (see definitions in [34]). Furthermore it is easy to prove that the τ-sequential lower semicontinuity of the norm implies that every closed ball Bis τ-sequentially closed. Thus, Scontains B∩Cfor every closed ball B. Theorem 1 in [34] implies that Thas a fixed point. Definition 3. We shall say that Xhas the τ-GGLD property if lim nxn<lim n,m;n=mxn−xm for every norm-bounded τ-null sequence such that both limits exist and limnxn=0. Remark 1. If τis the weak topology, the τ-GGLD property is implied by the GGLD property [22], which implies weakly normal structure and hence the w-FPP. Our main goal in this section is to prove an equivalent result when τis an arbitrary topology. The standard arguments to prove that GGLD implies w-NS [22] are strongly based upon a well known fact: Convex norm-closed subsets of Xare weakly closed. This fact is not true 346 DOM´ INGUEZ BENAVIDES, GARC´ IA FALSET AND JAP ´ ON PINEDA for arbitrary topologies on X. For instance, we consider X=L1([0,1]) endowed with the topology of convergence in measure and the sequence fn=nχ[0,1 n]which converges to the null function in measure. However, 0/∈¯co{fn}because g1= 1 for every g∈co{fn}. Since the standard approach does not work when τis not the weak topology, we need to use different arguments. Theorem 2. Let Xbe a separable Banach space and τa linear topology on X.IfXhas the τ-GGLD property then Xhas τ-NS. Proof. Suppose that Xfails to have τ-NS. Then, there exists a convex, normbounded, τ-sequentially compact subset Cwith diam(C) = 1, which is a diametral set. We shall find a sequence {xn}⊂Csuch that limnxn−x=1 for every x∈C. Let {yk}be a dense sequence in C. Inspired by [37] we are going to construct, by induction, a sequence {xn}⊂Csuch that limnxn−yk=1 for every k∈N. Let x1=y1and suppose that we have found x2,... ,x n−1∈Csuch that xm−yk≥1−1 mfor k≤m,m=1,... ,n−1. Denote by bthe geometric center of y1,y 2,... ,y n, that is: b= n  i=1 yi n. Since Cis a diametral set, there exists xn∈Csuch that xn−b≥1−1 n2. Then if k≤nwe have 1−1 n2≤xn−b=     n  i=1 xn n−yi n     ≤1 nxn−yk+ n  i=1;i=k 1 nxn−yi≤1 nxn−yk+n−1 n, which implies xn−yk≥1−1 nfor k=1,... ,n. Consequently, for every k∈Nwe get lim n1−1 n≤lim inf nxn−yk≤lim sup nxn−yk≤1, and therefore, {xn}satisfies the required condition. Notice that if we define the functions ϕ1(x) = lim infnxn−x,ϕ2(x)= lim supnxn−x, both are continuous and constant equal to 1 in a dense subset of C. Thus, limnxn−x= 1 for every x∈C. On the other hand, Cis a τ-sequentially compact set. Hence, there exist a subsequence {xnk}⊂{xn}and a vector x∈Csuch that {xnk}converges THE τ-FIXED POINT PROPERTY 347 to xin the topology τ. We can also assume that limk,j;k=jxnk−xnj exists (see [2, Theorem III.1.5]). Notice that the condition satisfied by {xn} implies that this limit must be equal to 1. Finally, consider the sequence yk=xnk−xwhich converges to 0 in τbecause τis a linear topology. Moreover, limkyk= 1 and limk,j;k=jyk−yj=1. ThusXdoes not satisfy the τ-GGLD property. The converse of Theorem 2 is not true even if τis the weak topology. Example 1. In the space L1([0,1]) we introduce the equivalent norm |||x|||2=x2 1+∞  k=1 |x(k)| 2k2 where x= ∞  k=1 x(k)ek,{ek}is a Schauder basis in L1([0,1]) (for instance, the Haar system) and ·1is the usual norm on L1([0,1]). Since (L1([0,1]),· 1) fails to have the w-FPP, it is known [22] that L1[0,1] does not have the w-GGLD property. Then, there exists a weakly null sequence {xn}such that limnxn1= limn,m;n=mxn−xm1. It is easy to check that if {xn}is a weakly null sequence then lim n ∞  k=1 |xn(k)| 2k=0. Thus limn,m;n=m|||xn−xm|||2 limn|||xn|||2=limn,m;n=mxn−xm2 1 limnxn2 1 =1. But (L1([0,1]),||| · |||) is U.C.E.D. [9, Corollary 6.9, page 66] and consequently this space has the w-NS. From Theorem 1 and 2 the following result is clear Theorem 3. Let Xbe a separable Banach space and τa linear topology on X.IfXhas the τ-GGLD property and the norm is a τ-slsc function, then Xhas the τ-FPP. Let {xn}beaτ-null sequence which is norm-bounded. The function φ(xn)(x) = lim sup nx−xn will be called a function of τ-null type. A basic tool in Fixed Point Theory for nonexpansive mappings is Goebel-Karlovitz’s Lemma. Its proof is based upon the weak lower semicontinuity of the functions of w-null type and the following fact: if Xdoes not have the w-FPP, there exists a convex 348 DOM´ INGUEZ BENAVIDES, GARC´ IA FALSET AND JAP ´ ON PINEDA weakly compact subset Cof Xand a fixed point free nonexpansive mapping T:C→Csuch that Cis minimal, i.e., if Kis a nonempty weakly compact convex subset of Cwhich is invariant under T, then K=C. In order to assure the existence of a minimal set in the setting of the τ- FPP we need that τ-sequentially compact sets are τ-compact. This property is not only satisfied by either the weak topology or a metric topology. In the case that the Banach space Xis separable, any topology τwhich is weaker than the norm topology also verifies that τ-sequentially compact sets are τ-compact. Indeed, Xis Lindel¨of for the norm topology because there is a countable basis of open sets and the same is true for the topology τbecause it is weaker than the norm topology. Thus τ-sequentially compact sets are countably compact and Lindel¨of, which implies that they are τ-compact. On the other hand, if Tis a nonexpansive mapping defined from a convex norm-bounded subset Cof Xinto Cwe can always find an approximated fixed point sequence (a.f.p.s.) for Tin C(i.e. lim xn−Txn= 0). For that, we extend the mapping T, in a continuous way, to the set ¯ C·. This extension is also a nonexpansive mapping. Hence, Banach’s Contractive Mapping Principle let us find an a.f.p.s. for the extension of Tin ¯ C·. Finally, by an approximation argument, we can construct an a.f.p.s. for T in C. If we replace the weak topology by a linear topology τon Xsuch that τ-sequentially compact sets are τ-compact and the functions of τ-null type are τ-slsc, it is easy to prove that Goebel-Karlovitz’s Lemma still holds (see [32,41]). Thus, we can state: Lemma 1. Let Xbe a Banach space and τa linear topology on Xsuch that τ-sequentially compact sets are τ-compact and the functions of τ-null type are τ-slsc. Let Cbe a τ-sequentially compact convex set, T:C→C a nonexpansive mapping and assume that Cis minimal, i. e. if Kis a nonempty τ-sequentially compact convex subset of Cwhich invariant under T, then K=C.If{xn}is an approximated fixed point sequence for Tin C then limnx−xn=diam (C), for all x∈C. Definition 4. A Banach space Xis said to have the property M(τ)ifthe functions of τ-null type are constant on the spheres, i.e. φ(xn)(x)=φ(xn)(y) whenever x=y. Remark 2. If τis the weak topology, M(τ) is known as property Mof Kalton (see [26,27]), which implies the w-FPP [18]. We are going to prove that property M(τ) implies the τ-FPP. Lemma 2. Let Xbe a Banach space with the property M(τ)where τis a linear topology on X. Then the norm and the functions of τ-null type are τ-slsc. THE τ-FIXED POINT PROPERTY 349 Proof. We are going to prove that the unit ball BXis τ-sequentially closed which easily implies that the norm is τ-slsc. Let {xn}be a sequence in BXwhich is τ-convergent to x0. Since {xn−x0}is τ-null we obtain from property M(τ) 1≥lim sup nxn= lim sup nxn−x0+x0= lim sup nxn−2x0. Hence, 2x0≤lim sup nxn−2x0+ lim sup nxn≤2 and we obtain that x0∈BX. To prove the second assertion, we shall first prove that functions of τnull type are nondecreasing with respect to x. Bearing in mind that X satisfies property M(τ) we only need prove that φ(xn)(tx) is a nondecreasing function with respect to tin [0,∞). Indeed, assume 0 <t 1<t 2. Then, there exists β∈(0,1) such that t1x=β(−t2)x+(1−β)t2x. The convexity of φ(xn)implies φ(xn)(t1x)≤βφ(xn)(−t2x)+(1−β)φ(xn)(t2x)(∗) From property M(τ) we derive φ(xn)(−t2x)=φ(xn)(t2x)(∗∗) and (*) and (**) imply that φ(xn)(t1x)≤φ(xn)(t2x). Now, we are going to prove that for every α>0 the set Aα:= {x∈X:φ(xn)(x)≤α} is τ-sequentially closed, which will imply that functions of τ-null type are τslsc. Assume that {ym}is a sequence in Aαwhich is τ-convergent to y∈X. Since ym−φ(xn)(0) ≤φ(xn)(ym)≤α we know that {ym}is bounded by α+φ(xn)(0). The τ-sequential lower semicontinuity of the norm implies y≤lim infmym.Thus, given ε>1 there exists m0∈Nsuch that y≤εym0, which implies φ(xn)(y)≤φ(xn)(εym0). Hence, we have φ(xn)(εym0)≤lim sup nxn−ym0+(ε−1)ym0=φ(xn)(ym0)+(ε−1)ym0 and φ(xn)(y)≤α+(ε−1)(α+φ(xn)(0)). Taking limits as εgoes to 1 we obtain the result. From Lemma 1 we can prove the following Proposition. For the weak topology the proof can be found in [24, Proof of Theorem 1]. 350 DOM´ INGUEZ BENAVIDES, GARC´ IA FALSET AND JAP ´ ON PINEDA Proposition 1. Assume that X,Tand Care as in Lemma 1 and that T has no fixed point in C.Ifdiam(C)=1,{xn}is a τ-null approximated fixed point sequence in C, for every ε>0and t∈[0,1] there exists a sequence {zn}in Csuch that (a.-) {zn}is τ−convergent, say to z∈C. (b.-) zn>1−εfor any positive integer n. (c.-) lim supnlim supmzn−zm≤t. (d.-) lim supnzn−xn≤1−t. Proof. From Lemma 1 we know that limnwn=1if{wn}is an a.f.p.s. in C. Then, for every ε>0 there exists δ(ε)>0 such that x>1−εif x∈Cand Tx−x<δ(ε). Indeed, otherwise, there exists ε0>0 such that we can find xn∈Cwith Txn−xn<1 nand xn≤1−ε0for every n∈N. Then, the sequence {xn}is an approximated fixed point sequence in C, but lim supnxn≤1−ε0. Let ε>0 and t∈[0,1]. Choose γ>0 such that γ<min{1,δ(ε)}and a null decreasing sequence {ηn}with γ+ηn<min{1,δ(ε)}for every n∈N. For every n∈Nwe define the contractive mapping Sn:¯ C· →¯ C· by Sn(x)=(1−γ)¯ Tx+γtxnfor every x∈¯ C· where ¯ Tis the unique nonexpansive extension of Tto the set ¯ C·. Using Banach’s Contractive Mapping Principle, there exists yn∈¯ C· such that yn=(1−γ)¯ Tyn+γtxn. Since ¯ Tynis defined as the limit of the images under Tof any sequence in Cwhich converges to the vector ynwe can find zn∈Cwith zn−(1 −γ)Tzn−γtxn<η n. In addition, from the τ-sequential compactness of Cwe can assume that the sequence {zn}τ-converges to a vector z∈C. Following now the arguments in the proof of Theorem 1 in [24] we obtain the proposition. Theorem 4. Let Xbe a Banach space which satisfies the property M(τ) where τis a linear topology on Xsuch that τ-sequentially compact sets are τ-compact and the functions of τ-null type are τ-slsc. Then Xhas the τ−FPP. Proof. Assume that Xdoes not satisfy the τ-FPP. Using standard arguments we can prove that there exist a τ-sequentially compact norm-bounded convex subset Kof Xwith diam(K) = 1 and a fixed point free nonexpansive mapping T:K→Ksuch that Kis minimal and contains a τ-null approximated fixed point sequence {xn}.Forε∈(0,1/3) and t=1/2+ε/2,we can construct {zn}as in Proposition 1. Notice that the choice of εand t implies t<1−εand 1 −t=t−ε. Since the norm is τ-slsc (Lemma 2) we have z≤lim inf nzn−xn≤1−t. THE τ-FIXED POINT PROPERTY 351 On the other hand, let ηbe a positive number such that t+η<1−ε. Since lim supnlim supmzn−zm≤twe can assume lim supnzn−zm≤t+η for nlarge enough. Again, using that the norm is τ-slsc we obtain lim sup nzn−z≤lim sup n lim inf mzn−zm≤t+η<1−ε. This implies that lim supnzn−z≥t−εbecause otherwise we obtain the contradiction 1−ε≤lim sup nzn≤lim sup nzn−z+z<t−ε+1−t=1−ε Let d= lim supnzn−z. Then z≤1−t=t−ε≤lim sup nzn−z=d Consider the sequence yn:= (z−zn)/d. This sequence is τ-null and lim supnyn= 1. Since z/d≤1, we have for any m φ(yn)(z/d)≤φ(yn)(ym/ym)≤φ(yn)(ym)+|ym−1| which implies φ(yn)(z/d)≤lim sup m φ(yn)(ym) = lim sup m lim sup nym−yn. Thus, we obtain the contradiction 1−ε≤lim sup zn≤dlim sup (zn−z)/d +z/d =dφ(yn)(z/d)≤lim sup m lim sup nzn−zm≤t. We are going to show some examples, inspired from [31], where the above results can be applied. In fact, these examples satisfy the following stronger property than M(τ). Definition 5. Let Xbe a Banach space and τa topology on X. We say that Xsatisfies the property L(τ) if there exists a function δ:[0,∞)×[0,∞)→ [0,∞) satisfying (1).- δis continuous (2).- δ(., s) is nondecreasing. (3).- δ(r, .) is nondecreasing (4).- δ(lim supnxn,y) = lim supnxn−yfor every y∈Xand every τ-null sequence. It is clear that L(τ) implies M(τ). 358 DOM´ INGUEZ BENAVIDES, GARC´ IA FALSET AND JAP ´ ON PINEDA with |||f||| =(1/2 0|f|,1 1/2|f|)mand (·,·)mis the norm on R2whose unit ball is the balanced convex hull of the set {(0,1),(1 + a, a)}. Trivially, the norm ·Xis τ-slsc, however, we shall check that ·Yis not τ-slsc. For that, we consider the sequence {(0,f n)}⊂Ywith fn=2χ[0,1/2]+anχ[n−1 n,1]. Then (0,f n)→n(0,2χ[0,1 2]) in the topology τbut (0,f n)Y=(1,a)m=a2+1 1+a<1=(1,0)m=(0,2χ[0,1/2])Y. It is not difficult to prove that d(X,Y )=1+2a. Then, when atends to 0, d(X,Y ) tends to 1. Therefore, there does not exist, in general, an upper bound of the Banach-Mazur distance between two isomorphic Banach spaces such that the norm on Yis τ-slsc if the norm on Xis τ-slsc and d(X,Y )is less than that bound. Since ·Yis not τ-slsc we cannot apply Theorem 6 to assure the τ-FPP in Y. However, we can deduce from Theorem 7 that Y has the τ-FPP if 1+2a<√2. It must be noted that we cannot apply Theorem 6 either if 21/p ≤1+2a. For topologies such that the functions of τ-null type are τ-slsc we can obtain better stability results using a coefficient inspired in the definition of M(X) in [10]. Definition 8. Let Xbe a Banach space and τa topology on X. For any nonnegative number awe define the coefficient Rτ(a, X) = sup{lim inf xn+x} where the supremum is taken over all x∈Xwith x≤aand all τ-null sequences in the unit ball such that limn,m;n=mxn−xm≤1. We define the coefficient Mτ(X)as sup 1+a Rτ(a, X):a≥0. Theorem 8. Let Xbe a Banach space and τa linear topology such that τ-sequentially compact sets are τ-compact. Let |·|be an equivalent norm on Xsuch that |x|≤x≤d|x| for every x∈X, and let Y=(X,|·|).Ifd<M τ(X)and the functions of τ-null type are τ-slsc in Y, then Yhas the τ-FPP. Proof. We shall use Proposition 1 to give an elemental proof. Since d< Mτ(X) there exists a≥0 such that dRτ(a, X)<1+a. Since Rτ(·,X)isa continuous function we can assume that a>0. Let t=(1+a)−1∈(0,1) and choose a positive ε<1−(1 + a)−1dRτ(a, X). If Ydoes not have THE τ-FIXED POINT PROPERTY 359 the τ-FPP, a standard argument proves that there exist Tand Cas in Proposition 1. Consider the sequence there defined. Taking a subsequence, if necessary, we can assume that limn,m , n=mzn−zmexists and {zn}is τ-convergent, say to z. Furthermore, limn,m , n=mzn−zm≤dt because lim supnlim supm|zn−zm|≤tand z≤d(1 −t) because |z|≤1−t. Let 0<η<(1 −ε)/Rτ(a, X)−dt. Then we can assume that the sequence {(zn−z)/(dt +η)}belongs to the unit ball in Y. Thus, we obtain the contradiction 1−ε dt +η<|zn| dt +η≤ zn−z dt +η+z dt +η≤Rτ(a, X). Theorem 8 is of special interest when every τ-convergent sequence is weakly convergent, because in this case the functions of τ-null type are τ-slsc and this condition is preserved under isomorphisms. That happens for Lp(Ω), p>1, if τis the clm topology (see, for instance, [21], page 207). If p>1 we know that Lp(Ω) has the clm-FPP, because every convex, norm-bounded, clm-compact subset of Lp(Ω) is weakly compact and it is well known that Lp(Ω) has the w-FPP if p>1. In fact, it is known that WCS(Lp(Ω)) ≥min{21/p,21−1/p}. (The equality holds if either p>2orµ is not purely atomic [2, Theorem VI.6.3]). However, better stability bounds can be obtained when the clm-FPP is considered. Theorem 9. Let Xbe a Banach space. Assume that τis a linear topology such that τ-sequentially compact sets are τ-compact. Assume that τis sequentially stronger than the weak topology and Xsatisfies L(τ).IfYis an isomorphic Banach space and d(X,Y )<sup    1+a δ1 δ(1,1) ,a :a≥0   then Yhas the τ-FPP. In particular, Yhas the τ-FPP if d(X, Y )<(1 + √5)/2. Proof. We can assume that there exists a norm-bounded τ-null sequence which is not norm-convergent, because otherwise it is clear that Yhas the τ-FPP. Using property L(τ) we can easily deduce that Rτ(a, X)=δ1 τCS(X),a =δ1 δ(1,1),a . Since δis an homogeneous function, if we choose a=1/δ(1,1) we obtain Mτ(X)≥1+(1/δ(1,1)). For a= 0 we also know that Mτ(X)≥δ(1,1). Noting that max 1+ 1 δ(1,1),δ(1,1)≥1+√5 2 we obtain the result. From Theorem 9, we can obtain stability bounds in Lp(Ω) for the clm- FPP, which are identical to those obtained in [10] for the FPP in p-spaces. 360 DOM´ INGUEZ BENAVIDES, GARC´ IA FALSET AND JAP ´ ON PINEDA Corollary 3. Let (Ω,Σ,µ)be a positive σ-finite measure space and |·|a norm on Lp(Ω) ,p>1which satisfies |f|≤fp≤d|f| for every f∈Lp(Ω).LetX=(Lp(Ω),|·|).If d<1+2 1 p−1p−1 p, then Xhas the clm-FPP. Proof. A standard argument proves that sup (1 + a) (ap+1/2)1/p :a≥0=1+2 1 p−1p−1 p. References 1. G. Aksoy and M. A. Khamsi, Nonstandard Methods in Fixed Point Theory, Springer- Verlag, New York, 1990.. 2. J. M. Ayerbe, T. Dominguez Benavides and G. L´opez Acedo, Measures of Noncompactness in Metric Fixed Point Theory, Oper. Theory Adv. Appl. 99, Birkh¨auser, Basel, 1997.. 3. M. Besbes, Points fixes des contractions definies sur un convexe L0-ferm´edeL1,C. R. Acad. Sci. Paris S´er. I Math. 311 (1990), 243–246.. 4. H. Br´ezis and E. Lieb, A relation between pointwise convergence of functions and convergence of functionals, Proc. Amer. Math. Soc. 88 (1983), 486–490.. 5. F. E. Browder, Fixed point theorems for noncompact mappings in Hilbert spaces, Proc. Nat. Acad. Sci. U.S.A. 43 (1965), 1272–1276.. 6. F. E. Browder, Nonexpansive nonlinear operators in a Banach space, Proc. Nat. Acad. Sci. U.S.A. 54 (1965), 1041–1044.. 7. M. Budzynska, W. Kaczor, M. Koter-Morgowska and T. Kuczumow, Asymptotic normal structure and the semi-Opial property, Nonlinear Anal. 30 (1997), 3505– 3515.. 8. W.L. Bynum, Normal structure coefficients for Banach spaces, Pacific J. Math. 86 (1980), 427–435.. 9. R. Deville, G. Godefroy and V. Zizler, Smoothness and renorming in Banach spaces, Pitman Monographs and Surveys in Pure and Applied Mathematics, 64, Longman Scientific and Technical, Harlow, 1993.. 10. T. Dominguez Benavides, A geometrical coefficient implying the fixed point property and stability results, Houston J. Math. 22 (1996), 835–849.. 11. T. Dominguez Benavides and M. A. Jap´on Pineda, Stability of the fixed point property for nonexpansive mappings in some classes of spaces, omm. Appl. Nonlinear Anal. 5(1998), 37–46.. 12. T. Dominguez Benavides and H. K. Xu, A new geometrical coefficient for Banach spaces and its applications in fixed point theory, Nonlinear Anal. 25 1995, 311–325.. 13. D. van Dulst, Equivalent norms and the fixed point property for nonexpansive mappings, J. London Math. Soc. 25 (1982), 139–144.. 14. D. van Dulst and B. Sims, Fixed points of nonexpansive mappings and Chebyshev centers in Banach spaces with norms of type (KK), Lect. Notes Math. 991 (1993), 33–43.. THE τ-FIXED POINT PROPERTY 361 15. J. M. Dye, T. Kuczumow, P. K. Lin and S. Reich, Random products of nonexpansive mappings in spaces with the Opial property, Contemp. Math. 144 (1993), 87–93.. 16. J. M. Dye, T. Kuczumow, P. K. Lin and S. Reich, Convergence of unrestricted products of nonexpansive mappings in spaces with the Opial property, Nonlinear Anal. 26 (1996), 767–773.. 17. J. M. Dye, T. Kuczumow and S. Reich, Random products of contractions, Proceedings of the First World Congress of Nonlinear Analysis, Walter de Gruyter, New York, II (1996), 1541–1548.. 18. J. Garcia-Falset, B. Sims, Property (M) and weak fixed point property, Proc. Amer. Math. Soc., to appear.. 19. K. Goebel and W.A. Kirk. Topics in Metric Fixed Point Theory, Cambridge Univ. Press, 1990.. 20. N. M. Gulevich, Fixed points of nonexpansive mappings, J. Math. Sci, 79 (1996), 755–815 .. 21. E. Hewitt, K. Stromberg, Real and Abstract Analysis, Springer-Verlag, New York, 1965.. 22. A. Jim´enez-Melado, Stability of weak normal structure in James quasi reflexive space, Bull. Austral. Math. Soc. 46 (1992), 367–372.. 23. A. Jim´enez-Melado and E. Llorens Fuster, Stability of the fixed point property for nonexpansive mappings, Houston J. Math.,917 (1992), 251–257.. 24. A. Jim´enez-Melado and E. Llorens Fuster, Opial modulus and stability of the fixed point property, Nonlinear Anal., to appear.. 25. S. Kakutani, Topological properties of the unit sphere of a Hilbert space, Proc. Imp. Acad. Tokyo,19 1943, 269–271.. 26. N. J. Kalton, M-ideals of compact operators, Illinois J. of Math. 37 (1993), 147–169.. 27. N. J. Kalton and D. Werner, Property (M), M-ideals and almost isometric structure of Banach spaces. J. Reine Angew. Math. 461 (1995), 137–178.. 28. L. A. Karlovitz. On nonexpansive mappings, Proc. Amer. Math. Soc. 55 (1976), 321-325.. 29. M. A. Khamsi, On the weak-fixed point property, Contemp. Math. 85 (1989), 325- 337.. 30. M. A. Khamsi, On the stability of the fixed point property in p,Rev. Colombiana Mat. 28 (1994), 1–6.. 31. M. A. Khamsi, On uniform Opial condition and Uniform Kadec-Klee property in Banach and metric spaces, Nonlinear Anal. 26 (1996), 1733–1748.. 32. M. A. Khamsi and Ph. Turpin, Fixed points of nonexpansive mappings in Banach lattices, Proc. Amer. Math. Soc. 105 (1989), 102–110.. 33. W.A. Kirk, A fixed point theorem for mappings which do not increase distances, Amer. Math. Monthly,72 (1965), 1004–1006.. 34. W.A. Kirk, An abstract fixed point theorem for nonexpansive mappings, Proc. Amer. Math. Soc. 82 (1981), 640–642.. 35. T. Kuczumow, S. Reich, M. Schmidt and A. Stachura, Strong asymptotic normal structure and fixed points in product spaces. Nonlinear Anal. 21 (1993), 501–515.. 36. E. Lami Dozo and Ph. Turpin, Nonexpansive mappings in generalized Orlicz spaces, Studia Math. 86 (1987), 155–188.. 37. C. Lennard, A new convexity property that implies a fixed point property for L1, Stud. Math. 100 (1991), 95–108.. 38. T. C. Lim, Asymptotic centers and nonexpansive mappings in conjugate Banach spaces, Pacific J. Math. 90 (1980), 135–143.. 39. S. Prus, On Bynum’s fixed point theorem, Atti Sem. Mat. Fis. Univ. Modena,38 (1990), 535–545.. 40. T. Sekowski, On normal structure, stability of fixed point property and the modulus of noncompact convexity, Rend. Sem. Mat. Fis. Milano,56 (1986), 147–153.. 362 DOM´ INGUEZ BENAVIDES, GARC´ IA FALSET AND JAP ´ ON PINEDA 41. B. Sims, The weakKarlovitz Lemma for dual lattices. Bull. Austral. Math. Soc. 45 (1992), 171–176.. 42. P. Soardi, Schauder basis and fixed points of nonexpansive mappings, Pacific J. Math. 101 (1982), 193–198.. Departamento de An´ alisis Matem´ atico University of Seville, SPAIN E-mail address: ayerb[email protected] Departamento de An´ alisis Matem´ atico University of Valencia, SPAIN E-mail address: [email protected] Departamento de An´ alisis Matem´ atico University of Seville, SPAIN E-mail address: jap[email protected]