Fixed point property for general topologies in some Banach spaces

FIXED POINT PROPERTY FOR GENERAL TOPOLOGIES IN SOME BANACH SPACES MARIA A. JAP´ ON PINEDA AND STANIS LAW PRUS Abstract. We study the fixed point property with respect to general vector topologies in L-embedded Banach spaces. Considering a class of topologies in l1such that the standard basis is convergent, we characterize those of them for which the fixed point property holds. We show that in c0-sums of some Banach spaces the weak topology is in a sense the coarsest topology for which the fixed point property holds. 1. Introduction Every Banach space is equipped with the norm topology and the weak topology. Both of them play important roles in the fixed point theory. In particular, it is possible to characterize sets with the fixed point property for some class of mappings in terms of the weak topology. Such characterization for the class of continuous affine mappings on bounded convex sets in arbitrary Banach spaces can be found in [8] and for the class of nonexpansive mappings on bounded convex subsets of c0can be found in [1] and [2]. In the case of L1spaces another topology was successfully applied. This is the topology of convergence locally in measure (see [13]). As a generalization of this topology an abstract measure topology in L-embedded Banach spaces was introduced (see [17]). Its applications to the metric fixed point theory can be found in [10] and [11]. In this paper we give another one. Our result concerns existence of fixed points of mappings of asymptotically nonexpansive type in an L-embedded Banach space Xendowed with a vector topology satisfying the Kadec-Klee property. We study in details the special case when X=l1. For a particular family of sets in l1a characterization of the fixed point property for nonexpansive mappings was found in [5]. Moreover, it is well known that l1has the fixed point property for nonexpansive mappings on convex sets which are compact with respect to the weak∗topology generated by the predual c0and lacks this property if we replace c0by c. This leads to the problem of characterizing locally convex topologies τin l1for which the fixed point property holds. We find a solution to this problem in the case when the standard basis of l1is τ-convergent. In the last section of this paper we deal with c0-sums of reflexive spaces. Using an idea from [1] we show that for some such spaces the weak topology is in a sense the coarsest topology for which the fixed point property holds. 2. Preliminaries Let Xbe a normed space. Its closed unit ball will be denoted by BX. By a vector topology in Xwe mean a Hausdorff topology τsuch that the vector operations are continuous with respect to τ. Given a subspace Yof the dual space X∗, by σ(X, Y ) we denote the coarsest topology in Xfor which all functionals f∈Yare continuous. Recall 2000 Mathematics Subject Classification. 47H09, 47H10. 1 2 M. A. JAP ´ ON PINEDA AND S. PRUS that Yis total if for every x∈X\ {0}there exists f∈Ysuch that f(x)6= 0. In this case σ(X, Y ) is a vector topology. Of course X∗is total and σ(X, X∗) is just the weak topology which we denote also by w. If X=Y∗, then Yconsidered as a subspace of X∗is total and σ(X, Y ) is the weak∗topology. Another examples can be obtained for Banach spaces Xwhich are not reflexive. Then ker Fis total for every F∈X∗∗ \X. We will also deal with the so-called abstract measure topologies. Let us recall that a sequence (xn) in a Banach space Xspans an asymptotically isometric copy of l1if there exists a nonincreasing sequence (δn) in [0,1) tending to 0 such that ∞ X n=1 (1 −δn)|αn| ≤      ∞ X n=1 αnxn     ≤ ∞ X n=1 |αn| for every sequence (αn)∈l1. In this case we write (xn)∼(asy)l1. We say that a topology τin a Banach space Xis an abstract measure topology provided that a norm bounded sequence (xn) in Xconverges to xwith respect to τif and only if every subsequence (yn) of (xn−x) has a subsequence (ynk) such that either (ynk/kynkk)∼(asy)l1or limk→∞ kynkk= 0. Some vector topologies are abstract measure topologies. Let X=L1(Ω, µ) where µ is a σ-finite measure on a σ-field of subsets of Ω. Then the topology of convergence locally in measure is an abstract measure topology (see [17]). In the particular case when X=l1this topology coincides with the topology of coordinatewise convergence. On BXthis is just the weak∗topology σ(l1, c0). The Bergman space A1provides another such example. To recall the definition of A1we put D={z∈C:|z|<1}and consider the normalized Lebesgue measure µon D.A1is the subspace of L1(D, µ) consisting of all analytic functions on D. It is a dual space and for bounded sequences weak∗ convergence is equivalent to uniform convergence on compact sets (see [16]). This shows that the weak∗topology is finer than the topology of convergence in measure on BA1 and consequently, these two topologies coincide on BA1. The weak∗topology in A1is therefore an abstract measure topology. Let τbe a vector topology in a Banach space X. A function f:X→Ris sequentially lower semicontinuous with respect to τ(τ-SLSC for short) if f(x)≤lim inf n→∞ f(xn) for every sequence (xn) in Xwhich converges to xwith respect to τ. Observe that ‘lim inf’ may be replaced by ‘lim sup’ in this definition. The space Xhas the Kadec-Klee property with respect to τ(KK(τ) for short) provided that if (xn) is a sequence in X without a norm convergent subsequence and (xn) converges to xwith respect to τ, then kxk<lim sup n→∞ kxnk. If τis coarser than the norm topology, then the KK(τ) property implies that the norm k·k is τ-SLSC. Let Cbe a nonempty subset of X. A mapping T:C→Cis nonexpansive if kT(x)−T(y)k ≤ kx−yk for all x, y ∈C. In the case when strict inequality holds in the above condition whenever x6=y, we say that Tis contractive. A mapping T:C→Cis of asymptotically FIXED POINT PROPERTY FOR GENERAL TOPOLOGIES 3 nonexpansive type if TNis continuous for some N∈Nand lim sup n→∞ (sup {kTn(x)−Tn(y)k−kx−yk:y∈C})≤0 for every x∈C. The space Xhas the τ-fixed point property (τ-FPP for short) provided that if Cis a nonempty bounded convex and τ-sequentially compact subset of Xand T:C→Cis nonexpansive, then Thas a fixed point. A mapping T:C→Cis said to satisfy the (P)τ-fixed point property if Thas a fixed point in every nonempty convex τsequentially closed subset Dof Csuch that if x∈D, then each τ-limit of a subsequence of (Tn(x)) belongs to D. Let τbe a vector topology in a space X. In the sequel τBXwill denote the restriction of the topology τto the ball BX. We say that τis coarser than the weak topology on the unit ball if τBXis coarser than wBX. We will consider mainly locally convex topologies, i.e. vector topologies which admit local bases consisting of convex sets. Let τbe such a topology in a space Xand Ebe the space dual to (X, τ). Given nonempty sets A⊂X,D⊂E, we consider the polar sets A◦=f∈E: sup x∈A |f(x)| ≤ 1 and D◦=x∈X: sup f∈D |f(x)| ≤ 1. Proposition 1. Let Xbe a normed space and Y= (X, τ)where τis a locally convex topology in Xcoarser than the weak topology on the unit ball. Then a bounded sequence (xn)converges to xwith respect to the topology σ(X, Y ∗)if and only if (xn)converges to xwith respect to τ. Proof. Our assumption guarantees that τis coarser than the norm topology. This shows in particular that Y∗⊂X∗. Let Ube a convex, balanced and τ-closed neighborhood of zero. There is r > 0 such that rBX⊂U. It follows that the polar U◦of Uin Y∗is bounded. By our assumption for each  > 0 there is a finite set F⊂X∗such that F◦∩BX⊂U. We put Z=Tx∗∈Fker x∗. Then Z⊂F◦which yields 1 U◦= (U)◦⊂(Z∩BX)◦= (BZ)◦. Moreover, if g∈(BZ)◦, then we can find y∗∈X∗so that ky∗k= g|Z ≤1 and y∗ |Z=g|Z. Thus h|Z= 0 where h=g−y∗. It follows that h∈span(F). We therefore see that (BZ)◦⊂span(F) + BX∗. Hence U◦⊂span(F) + BX∗. Using this fact, one can easily show that the set U◦is relatively compact in the norm topology. Let now (xn) be a bounded sequence in Xconverging to xwith respect to σ(X, Y ∗). We can assume that x= 0 and (xn) is contained in BX. For every convex balanced τ-closed neighborhood Uof zero we find a finite 1/2-net {f1, . . . , fm}in U◦. There exists 4 M. A. JAP ´ ON PINEDA AND S. PRUS n0∈Nsuch that max1≤k≤m|fk(xn)| ≤ 1/2 for every n≥n0. Given f∈U◦, we choose k for which kf−fkk ≤ 1/2. Then |f(xn)| ≤ |fk(xn)|+kf−fkk ≤ 1 which shows that xn∈(U◦)◦for every n≥n0. But by the bipolar theorem (see [12]), (U◦)◦=U. We therefore see that (xn) converges to xwith respect to τ. The remaining part of the conclusion is obvious.  Corollary 2. Let Xbe a normed space and Y= (X, τ)where τis a locally convex topology in Xcoarser than the weak topology on the unit ball. If (xn)is a bounded sequence in Xconverging to xwith respect to σ(X, Y ∗), then the set C=(t0x+ ∞ X n=1 tnxn: ∞ X n=0 tn= 1, tn≥0, n = 0,1,2, . . . ) is τ-sequentially compact. Proof. Consider the mapping Φ : l1→Xgiven by the formula Φ(λ1, λ2, . . . ) = λ1x+ ∞ X j=2 λjxj. Using Proposition 1, one can easily show that Φ is σ(l1, c) to τsequentially continuous. It suffices now to observe that C= Φ(K) where K=((λ1, λ2, . . . )∈l1: ∞ X n=1 λn= 1, λn≥0, n = 1,2, . . . ) is sequentially compact with respect to σ(l1, c).  Let τbe a locally convex topology in a space X. A modification of the reasoning used in the proof of Proposition 1 shows that if a convex balanced τ-closed neighborhood Uof zero contains an open weak neighborhood of zero, then span(U◦) is a finite dimensional subspace of Y∗. Since U◦is bounded, it is contained in an absolute convex hull of a finite set A⊂Y∗. Consequently, A◦⊂(U◦)◦=U. This shows that if a locally convex topology τin Xis coarser than the weak topology, then τ=σ(X, Y ∗). The assumption of Proposition 1 does not guarantee this conclusion. Indeed, let X be an infinite dimensional normed space and Bbe the family of all polar sets A◦where Ais a nonempty compact subset of X∗. Then Bis a local basis at zero of a locally convex topology τin Xwhich is finer than the weak topology. Consequently, X∗is the dual space of (X, τ). If A⊂X∗is a compact set which is not contained in any finite dimensional subspace of X∗, then A◦does not contain any open weak neighborhood of zero. This shows that τdoes not coincide with the weak topology. On the other hand, it is easy to see that τBXcoincides with wBX. 3. L-embedded spaces Let us recall that Xis an L-embedded Banach space if there exists a closed subspace Zof X∗∗ such that X∗∗ =X⊕Zand kx+zk=kxk+kzkfor all x∈Xand z∈Z. In particular every space L1(Ω, µ) is an L-embedded space. For thorough study of L-embedded spaces the reader may consult the monograph [6]. In [11], the following property of L-embedded spaces was established. FIXED POINT PROPERTY FOR GENERAL TOPOLOGIES 5 Proposition 3. Let Xbe an L-embedded Banach space. If a bounded sequence (xn) converges to 0 in an abstract measure topology, then lim sup n→∞ kx+xnk=kxk+ lim sup n→∞ kxnk for every x∈X. Proposition 3 will be used many times in this paper. As the first application we obtain the following lemma. Lemma 4. Let τbe a vector topology in an L-embedded Banach X,(xn)be a bounded sequence in Xsuch that the set {xn}is relatively sequentially compact in an abstract measure topology and r(x) = lim sup n→∞ kxn−xk where x∈X. (i) If the norm of Xis τ-SLSC, then the function ris τ-SLSC. (ii) If Xhas the KK(τ)property and (zn)is a sequence in Xsuch that (zn)converges to zwith respect to τand (zn)does not have a norm convergent subsequence, then r(z)<lim sup n→∞ r(zn). Proof. Let a sequence (zn) converge to zwith respect to τ. We find a sequence (nk) so that r(z) = limk→∞ kxnk−zkand (xnk) converges to some ywith respect to the abstract measure topology. Then using Proposition 3, we obtain r(z) = ky−zk+ lim sup k→∞ kxnk−yk ≤lim sup m→∞ ky−zmk+ lim sup k→∞ kxnk−yk = lim sup m→∞ lim sup k→∞ kxnk−zmk ≤lim sup m→∞ r(zm). This completes the proof of (i) and the proof of (ii) is similar.  We can now prove our general fixed point results for L-embedded spaces. Theorem 5. Let Xbe an L-embedded Banach space and τbe a vector topology in X coarser than the norm topology such that every τ-sequentially compact subset of Xis τcompact and Xhas the KK(τ)property. Let a nonempty bounded convex set C⊂Xbe τsequentially compact and relatively sequentially compact in an abstract measure topology. Then every mapping T:C→Cof asymptotically nonexpansive type has the (P)τ-fixed point property. Proof. We follow a reasoning form [18]. Let T:C→Cbe a mapping of asymptotically nonexpansive type. We put rx(y) = lim sup n→∞ kTn(x)−yk where x, y ∈C. Clearly, kTn(x)−Tm(y)k= Tm(y)−TmTn−m(x) − ky−Tn−m(x)k+ky−Tn−m(x)k ≤sup{kTm(y)−Tm(v)k−ky−vk:v∈C}+kTn−m(x)−yk 6 M. A. JAP ´ ON PINEDA AND S. PRUS for all x, y ∈Cand n>m. Hence (1) lim sup m→∞ lim sup n→∞ kTn(x)−Tm(y)k ≤ rx(y). Let Fbe the family of all nonempty convex τ-sequentially closed subsets Kof Csuch that if y∈Kand zis a limit with respect to τof a subsequence of (Tn(y)), then z∈K. We fix D∈ F. From the Zorn lemma it follows that there exists K0∈ F which is minimal with respect to inclusion in the family {K∈ F :K⊂D}. Let x∈K0. We will show that the set {Tn(x)}is relatively compact in the norm topology. Let K1be the set of all z∈K0at which the function rxattains its infimum on K0. Lemma 4 shows that K1is nonempty and τ-sequentially closed. Obviously it is also convex. Let z∈K1and (nk) be an increasing sequence such that (Tnk(z)) converges to some uwith respect to τ. By Lemma 4 and (1) rx(u)≤lim sup k→∞ rx(Tnk(z)) ≤rx(z). This shows that u∈K1and we see that K1∈ F. Consequently, K1=K0and in particular rxattains at xits infimum on K0. Suppose that there exists an increasing sequence (nk) such that (Tnk(x)) does not have a norm convergent subsequence. We can assume that (Tnk(x)) converges to some u∈K0with respect to τ. Then Lemma 4 and (1) show that rx(u)<lim sup k→∞ rx(Tnk(x)) ≤rx(x) which is a contradiction. Now it suffices to use the reasoning from the proof of Lemma 2 in [18].  If the norm of Xis not only τ-SLSC, but τ-lower semicontinuous, then the assumptions of Theorem 5 actually guarantee that there exists a nonexpansive retraction Rfrom C onto the set Fix(T) of all fixed points of Tsuch that R◦T=Rand every convex τ-sequentially closed T-invariant subset of Cis also R-invariant (see [15] or [14] where only the case of τ=wis considered). The formulation of Theorem 5 can be simplified if the space Xadmits an abstract measure topology such that bounded sets are relatively sequentially compact. Further simplification is possible if Xis separable. Then τ-sequentially compact sets are τcompact (see [10]). Both remarks apply for instance to the spaces A1and l1. Corollary 6. Let τbe a vector topology in l1coarser than the norm topology such that l1has the KK(τ)property. If a nonempty bounded convex set C⊂Xis τ-sequentially compact, then every mapping T:C→Cof asymptotically nonexpansive type has the (P)τ-fixed point property. In particular l1has the τ-FPP. In Corollary 6 we obtained a condition sufficient for the τ-FPP in l1. Our next result gives a necessary condition. Before passing to this theorem we establish some notation. Let Γ be a nonempty set. Given x∈l1(Γ), we write x= (x(i))i∈Γwhere x(i) are scalars. If x6= 0, we set supp x={i∈Γ : x(i)6= 0}. Even if Γ is uncountable, this set is at most countable. Theorem 7. Let Γbe an infinite set and τbe a locally convex topology in l1(Γ) coarser than the weak topology on the unit ball. If the standard norm of l1(Γ) is not τ-SLSC, then there exist a bounded convex τ-sequentially compact set C⊂l1(Γ) and a contractive mapping T:C→Cwhich does not have a fixed point. FIXED POINT PROPERTY FOR GENERAL TOPOLOGIES 7 Proof. By the assumption there exists a sequence (xn) in l1(Γ) such that (xn) converges to xwith respect to τand kxk>lim n→∞ kxnk. We can assume that (xn) converges coordinatewise to some y. Setting un=xn−y and u0=x−y, we obtain a sequence (un) which converges to u0with respect to τ and converges coordinatewise to 0. Then (un) does not converge to 0 in norm and by Proposition 3 ku0k ≥ kxk−kyk>lim n→∞ kxnk−kyk= lim n→∞ kunk. We can assume that An= supp unis finite, kunk= 1 for every n≥1 and the sets An are pairwise disjoint. Then ku0k>1+for some  > 0 and there are vectors u0 0,u00 0such that u0=u0 0+u00 0,ku0 0k − ku00 0k>1 + ,A= supp u0 0is finite and A∩supp u00 0=∅. We can also assume that A∩An=∅for every n≥1. We put vk= (1 + /k)ukfor k≥1 and C=(λ0u0+ ∞ X j=1 λjvj: ∞ X j=0 λj= 1, λj≥0, j = 0,1,2, . . . ). Clearly, Cis bounded and convex. Corollary 2 shows that Cis τ-sequentially compact. We now set T λ0u0+ ∞ X j=1 λjvj!= ∞ X j=0 λjvj+1. This formula defines a mapping T:C→Cwithout a fixed point. Moreover,      ∞ X j=0 γjvj+1     = ∞ X j=0 |γj|1 +  j+ 1 <|γ0|(ku0 0k−ku00 0k) +      ∞ X j=1 γjvj     =     γ0u0 0+ ∞ X j=1 γjvj     − kγ0u00 0k ≤     γ0u0+ ∞ X j=1 γjvj     for every nonzero (γn)∈l1, which shows that Tis contractive.  Theorem 7 may be extended to spaces of the form Pi∈ΓXil1(Γ) where Xiare finite dimensional. Assume that τis a vector topology in l1such that if (yn) converges to ywith respect to τand converges to 0 coordinatewise, then kyk ≤ lim sup n→∞ kynk. Then the norm k·k is τ-SLSC. Indeed, let a bounded sequence (xn) converge to xwith respect to τ. Passing to a subsequence, we can assume that (xn) converges coordinatewise 8 M. A. JAP ´ ON PINEDA AND S. PRUS to some z. By our assumption and Proposition 3 kxk ≤ kx−zk+kzk ≤ lim sup n→∞ kxn−zk+kzk= lim sup n→∞ kxnk. A similar remark applies to the KK(τ) property. Under an additional assumption we can give a simple characterization of topologies τ for which l1has the τ-FPP. By (en) we denote the standard basis of l1. Theorem 8. Let τbe a locally convex topology in the real space l1coarser than the weak topology on the unit ball. Assume that (en)converges to some e∈l1with respect to τ. Then l1has the τ-FPP if and only if one of the following conditions holds (i) kek<1 (ii) kek= 1 and the set N+={n∈N:e(n)≥0}is finite. Proof. Given z= (z(k))k∈N∈l1we set s(z) = ∞ X k=1 z(k). Moreover, we put Pn(z) = n X k=1 z(k)ek where n∈N. Consider a bounded sequence (xn) in l1which converges to xwith respect to τand converges to 0 coordinatewise. If additionally the limit s= limn→∞ s(xn) exists, then x=se. Indeed, let Ybe the space dual to (l1, τ). If x∗∈Y, then |x∗(xn)−s(xn)x∗(e)|= ∞ X k=1 xn(k)(x∗(ek)−x∗(e)) ≤ kx∗k(1 + kek)kPm(xn)k+ sup k>m |x∗(ek)−x∗(e)|kxnk for every m. It follows that |x∗(x−se)|= lim n→∞ |x∗(xn)−s(xn)x∗(e)| ≤ lim m→∞ sup k>m |x∗(ek)−x∗(e)|lim sup n→∞ kxnk= 0. The subspace Yis total, so we obtain the desired formula x=se. Assume now that (i) or (ii) holds. Then kek ≤ 1. Let (xn) be a bounded sequence in l1which converges to xwith respect to τand converges to 0 coordinatewise. Passing to a subsequence, we can assume that the limit s= limn→∞ s(xn) exists. Then kxk ≤ |s|= lim n→∞ |s(xn)| ≤ lim sup n→∞ kxnk. This shows that the norm of l1is τ-SLSC. Assume that l1does not have the τ-FPP. Lemma 4 enables us to use a generalized Goebel-Karlovitz lemma (see [7, Lemma 1] and [9, Lemma 2.6]) and obtain a sequence (xn) such that it converges to x0with respect to τand limn→∞ ku−xnk= 2 for every u∈conv{xn:n≥0}. We can assume that (xn) converges coordinatewise to some y∈l1. Then the vectors yn=xn−ytend to z=x0−ywith respect to τand tend to 0 FIXED POINT PROPERTY FOR GENERAL TOPOLOGIES 9 coordinatewise. We can also assume that the limits limn→∞ kynkand s= limn→∞ s(yn) exist. Using Proposition 3, we see that 2 = lim n→∞ kx0−xnk= lim n→∞ kz−ynk =kzk+ lim n→∞ kynk ≤lim m→∞ kymk+ lim n→∞ kynk = lim m→∞ lim n→∞ kym−ynk= 2. This shows that limn→∞ kynk= 1 = kzkand consequently, |s| ≤ 1. But z=se. It follows that kek= 1 and |s|= 1. We therefore see that (i) does not hold, so by our assumption the set N+is finite. Consider the case when s= 1. We choose nfor which s(yn)>1/2 and kPm(yn)k<1/4 where m= max N+. Then the set B={k∈N:e(k)yn(k)<0}is nonempty. It is easy to see that |a+b|=|a|+|b| − 2 min{|a|,|b|} whenever a, b ∈R,ab < 0. Consequently, ke+ynk=kek+kynk − 2c where c=Pk∈Bmin{|e(k)|,|yn(k)|} >0. Applying Proposition 3, we therefore obtain lim m→∞     1 2(x0+xn)−xm    = lim m→∞     1 2(z+yn)−ym    =1 2kz+ynk+ lim m→∞ kymk =1 2lim m→∞ kz−ymk+ lim m→∞ kyn−ymk−c = 2 −c < 2 which is a contradiction. The case when s=−1 is similar. We have proved that if (i) or (ii) holds, then l1has the τ-FPP and from Theorem 7 we know that if kek>1, then l1does not have this property. To complete the proof it therefore remains to show that if kek= 1 and the set N+is infinite, then l1lacks the τ-FPP. Let (nk) be an infinite sequence in N+such that w0=e−u06= 0 where u0=P∞ k=1 e(nk)enk. We set w=1 kw0kw0and C=(µ1e+µ2w+ ∞ X k=1 µk+2enk: ∞ X j=1 µj= 1, µj≥0, j = 1,2, . . . ). The set Cis bounded and convex. By Corollary 2 it is also τ-sequentially compact. Moreover, if P∞ j=1 µj= 1 and µj≥0 for all j∈Nthen µ1e+µ2w+ ∞ X k=1 µk+2enk= (µ1kw0k+µ2)w+ ∞ X k=1 (µ1e(nk) + µk+2)enk and µ1kw0k+µ2+ ∞ X k=1 (µ1e(nk) + µk+2) = µ1(kw0k+ku0k) + ∞ X k=1 µk+1 = 1.