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 Abs ac . We s udy he ixed poin p ope y wi h espec o gene al ec o opologies in L-embedded Banach spaces. Conside ing a class o opologies in l1such ha he s anda d basis is con e gen , we cha ac e ize hose o hem o which he ixed poin p ope y holds. We show ha in c0-sums o some Banach spaces he weak opology is in a sense he coa ses opology o which he ixed poin p ope y holds. 1. In oduc ion E e y Banach space is equipped wi h he no m opology and he weak opology. Bo h o hem play impo an oles in he ixed poin heo y. In pa icula , i is possible o cha ac e ize se s wi h he ixed poin p ope y o some class o mappings in e ms o he weak opology. Such cha ac e iza ion o he class o con inuous a ine mappings on bounded con ex se s in a bi a y Banach spaces can be ound in [8] and o he class o nonexpansi e mappings on bounded con ex subse s o c0can be ound in [1] and [2]. In he case o L1spaces ano he opology was success ully applied. This is he opology o con e gence locally in measu e (see [13]). As a gene aliza ion o his opology an abs ac measu e opology in L-embedded Banach spaces was in oduced (see [17]). I s applica ions o he me ic ixed poin heo y can be ound in [10] and [11]. In his pape we gi e ano he one. Ou esul conce ns exis ence o ixed poin s o mappings o asymp o ically nonexpansi e ype in an L-embedded Banach space Xendowed wi h a ec o opology sa is ying he Kadec-Klee p ope y. We s udy in de ails he special case when X=l1. Fo a pa icula amily o se s in l1a cha ac e iza ion o he ixed poin p ope y o nonexpansi e mappings was ound in [5]. Mo eo e , i is well known ha l1has he ixed poin p ope y o nonexpansi e mappings on con ex se s which a e compac wi h espec o he weak∗ opology gene a ed by he p edual c0and lacks his p ope y i we eplace c0by c. This leads o he p oblem o cha ac e izing locally con ex opologies τin l1 o which he ixed poin p ope y holds. We ind a solu ion o his p oblem in he case when he s anda d basis o l1is τ-con e gen . In he las sec ion o his pape we deal wi h c0-sums o e lexi e spaces. Using an idea om [1] we show ha o some such spaces he weak opology is in a sense he coa ses opology o which he ixed poin p ope y holds. 2. P elimina ies Le Xbe a no med space. I s closed uni ball will be deno ed by BX. By a ec o opology in Xwe mean a Hausdo opology τsuch ha he ec o ope a ions a e con inuous wi h espec o τ. Gi en a subspace Yo he dual space X∗, by σ(X, Y ) we deno e he coa ses opology in X o which all unc ionals ∈Ya e con inuous. Recall 2000 Ma hema ics Subjec Classi ica ion. 47H09, 47H10. 1 2 M. A. JAP ´ ON PINEDA AND S. PRUS ha Yis o al i o e e y x∈X {0} he e exis s ∈Ysuch ha (x)6= 0. In his case σ(X, Y ) is a ec o opology. O cou se X∗is o al and σ(X, X∗) is jus he weak opology which we deno e also by w. I X=Y∗, hen Yconside ed as a subspace o X∗is o al and σ(X, Y ) is he weak∗ opology. Ano he examples can be ob ained o Banach spaces Xwhich a e no e lexi e. Then ke Fis o al o e e y F∈X∗∗ X. We will also deal wi h he so-called abs ac measu e opologies. Le us ecall ha a sequence (xn) in a Banach space Xspans an asymp o ically isome ic copy o l1i he e exis s a noninc easing sequence (δn) in [0,1) ending o 0 such ha ∞ X n=1 (1 −δn)|αn| ≤      ∞ X n=1 αnxn     ≤ ∞ X n=1 |αn| o e e y sequence (αn)∈l1. In his case we w i e (xn)∼(asy)l1. We say ha a opology τin a Banach space Xis an abs ac measu e opology p o ided ha a no m bounded sequence (xn) in Xcon e ges o xwi h espec o τi and only i e e y subsequence (yn) o (xn−x) has a subsequence (ynk) such ha ei he (ynk/kynkk)∼(asy)l1o limk→∞ kynkk= 0. Some ec o opologies a e abs ac measu e opologies. Le X=L1(Ω, µ) whe e µ is a σ- ini e measu e on a σ- ield o subse s o Ω. Then he opology o con e gence locally in measu e is an abs ac measu e opology (see [17]). In he pa icula case when X=l1 his opology coincides wi h he opology o coo dina ewise con e gence. On BX his is jus he weak∗ opology σ(l1, c0). The Be gman space A1p o ides ano he such example. To ecall he de ini ion o A1we pu D={z∈C:|z|<1}and conside he no malized Lebesgue measu e µon D.A1is he subspace o L1(D, µ) consis ing o all analy ic unc ions on D. I is a dual space and o bounded sequences weak∗ con e gence is equi alen o uni o m con e gence on compac se s (see [16]). This shows ha he weak∗ opology is ine han he opology o con e gence in measu e on BA1 and consequen ly, hese wo opologies coincide on BA1. The weak∗ opology in A1is he e o e an abs ac measu e opology. Le τbe a ec o opology in a Banach space X. A unc ion :X→Ris sequen ially lowe semicon inuous wi h espec o τ(τ-SLSC o sho ) i (x)≤lim in n→∞ (xn) o e e y sequence (xn) in Xwhich con e ges o xwi h espec o τ. Obse e ha ‘lim in ’ may be eplaced by ‘lim sup’ in his de ini ion. The space Xhas he Kadec-Klee p ope y wi h espec o τ(KK(τ) o sho ) p o ided ha i (xn) is a sequence in X wi hou a no m con e gen subsequence and (xn) con e ges o xwi h espec o τ, hen kxk<lim sup n→∞ kxnk. I τis coa se han he no m opology, hen he KK(τ) p ope y implies ha he no m k·k is τ-SLSC. Le Cbe a nonemp y subse o X. A mapping T:C→Cis nonexpansi e i kT(x)−T(y)k ≤ kx−yk o all x, y ∈C. In he case when s ic inequali y holds in he abo e condi ion whene e x6=y, we say ha Tis con ac i e. A mapping T:C→Cis o asymp o ically FIXED POINT PROPERTY FOR GENERAL TOPOLOGIES 3 nonexpansi e ype i TNis con inuous o some N∈Nand lim sup n→∞ (sup {kTn(x)−Tn(y)k−kx−yk:y∈C})≤0 o e e y x∈C. The space Xhas he τ- ixed poin p ope y (τ-FPP o sho ) p o ided ha i Cis a nonemp y bounded con ex and τ-sequen ially compac subse o Xand T:C→Cis nonexpansi e, hen Thas a ixed poin . A mapping T:C→Cis said o sa is y he (P)τ- ixed poin p ope y i Thas a ixed poin in e e y nonemp y con ex τ- sequen ially closed subse Do Csuch ha i x∈D, hen each τ-limi o a subsequence o (Tn(x)) belongs o D. Le τbe a ec o opology in a space X. In he sequel τBXwill deno e he es ic ion o he opology τ o he ball BX. We say ha τis coa se han he weak opology on he uni ball i τBXis coa se han wBX. We will conside mainly locally con ex opologies, i.e. ec o opologies which admi local bases consis ing o con ex se s. Le τbe such a opology in a space Xand Ebe he space dual o (X, τ). Gi en nonemp y se s A⊂X,D⊂E, we conside he pola se s A◦= ∈E: sup x∈A | (x)| ≤ 1 and D◦=x∈X: sup ∈D | (x)| ≤ 1. P oposi ion 1. Le Xbe a no med space and Y= (X, τ)whe e τis a locally con ex opology in Xcoa se han he weak opology on he uni ball. Then a bounded sequence (xn)con e ges o xwi h espec o he opology σ(X, Y ∗)i and only i (xn)con e ges o xwi h espec o τ. P oo . Ou assump ion gua an ees ha τis coa se han he no m opology. This shows in pa icula ha Y∗⊂X∗. Le Ube a con ex, balanced and τ-closed neighbo hood o ze o. The e is > 0 such ha BX⊂U. I ollows ha he pola U◦o Uin Y∗is bounded. By ou assump ion o each  > 0 he e is a ini e se F⊂X∗such ha F◦∩BX⊂U. We pu Z=Tx∗∈Fke x∗. Then Z⊂F◦which yields 1 U◦= (U)◦⊂(Z∩BX)◦= (BZ)◦. Mo eo e , i g∈(BZ)◦, hen we can ind y∗∈X∗so ha ky∗k= g|Z ≤1 and y∗ |Z=g|Z. Thus h|Z= 0 whe e h=g−y∗. I ollows ha h∈span(F). We he e o e see ha (BZ)◦⊂span(F) + BX∗. Hence U◦⊂span(F) + BX∗. Using his ac , one can easily show ha he se U◦is ela i ely compac in he no m opology. Le now (xn) be a bounded sequence in Xcon e ging o xwi h espec o σ(X, Y ∗). We can assume ha x= 0 and (xn) is con ained in BX. Fo e e y con ex balanced τ-closed neighbo hood Uo ze o we ind a ini e 1/2-ne { 1, . . . , m}in U◦. The e exis s 4 M. A. JAP ´ ON PINEDA AND S. PRUS n0∈Nsuch ha max1≤k≤m| k(xn)| ≤ 1/2 o e e y n≥n0. Gi en ∈U◦, we choose k o which k − kk ≤ 1/2. Then | (xn)| ≤ | k(xn)|+k − kk ≤ 1 which shows ha xn∈(U◦)◦ o e e y n≥n0. Bu by he bipola heo em (see [12]), (U◦)◦=U. We he e o e see ha (xn) con e ges o xwi h espec o τ. The emaining pa o he conclusion is ob ious.  Co olla y 2. Le Xbe a no med space and Y= (X, τ)whe e τis a locally con ex opology in Xcoa se han he weak opology on he uni ball. I (xn)is a bounded sequence in Xcon e ging o xwi h espec o σ(X, Y ∗), hen he se C=( 0x+ ∞ X n=1 nxn: ∞ X n=0 n= 1, n≥0, n = 0,1,2, . . . ) is τ-sequen ially compac . P oo . Conside he mapping Φ : l1→Xgi en by he o mula Φ(λ1, λ2, . . . ) = λ1x+ ∞ X j=2 λjxj. Using P oposi ion 1, one can easily show ha Φ is σ(l1, c) o τsequen ially con inuous. I su ices now o obse e ha C= Φ(K) whe e K=((λ1, λ2, . . . )∈l1: ∞ X n=1 λn= 1, λn≥0, n = 1,2, . . . ) is sequen ially compac wi h espec o σ(l1, c).  Le τbe a locally con ex opology in a space X. A modi ica ion o he easoning used in he p oo o P oposi ion 1 shows ha i a con ex balanced τ-closed neighbo hood Uo ze o con ains an open weak neighbo hood o ze o, hen span(U◦) is a ini e dimensional subspace o Y∗. Since U◦is bounded, i is con ained in an absolu e con ex hull o a ini e se A⊂Y∗. Consequen ly, A◦⊂(U◦)◦=U. This shows ha i a locally con ex opology τin Xis coa se han he weak opology, hen τ=σ(X, Y ∗). The assump ion o P oposi ion 1 does no gua an ee his conclusion. Indeed, le X be an in ini e dimensional no med space and Bbe he amily o all pola se s A◦whe e Ais a nonemp y compac subse o X∗. Then Bis a local basis a ze o o a locally con ex opology τin Xwhich is ine han he weak opology. Consequen ly, X∗is he dual space o (X, τ). I A⊂X∗is a compac se which is no con ained in any ini e dimensional subspace o X∗, hen A◦does no con ain any open weak neighbo hood o ze o. This shows ha τdoes no coincide wi h he weak opology. On he o he hand, i is easy o see ha τBXcoincides wi h wBX. 3. L-embedded spaces Le us ecall ha Xis an L-embedded Banach space i he e exis s a closed subspace Zo X∗∗ such ha X∗∗ =X⊕Zand kx+zk=kxk+kzk o all x∈Xand z∈Z. In pa icula e e y space L1(Ω, µ) is an L-embedded space. Fo ho ough s udy o L-embedded spaces he eade may consul he monog aph [6]. In [11], he ollowing p ope y o L-embedded spaces was es ablished. FIXED POINT PROPERTY FOR GENERAL TOPOLOGIES 5 P oposi ion 3. Le Xbe an L-embedded Banach space. I a bounded sequence (xn) con e ges o 0 in an abs ac measu e opology, hen lim sup n→∞ kx+xnk=kxk+ lim sup n→∞ kxnk o e e y x∈X. P oposi ion 3 will be used many imes in his pape . As he i s applica ion we ob ain he ollowing lemma. Lemma 4. Le τbe a ec o opology in an L-embedded Banach X,(xn)be a bounded sequence in Xsuch ha he se {xn}is ela i ely sequen ially compac in an abs ac measu e opology and (x) = lim sup n→∞ kxn−xk whe e x∈X. (i) I he no m o Xis τ-SLSC, hen he unc ion is τ-SLSC. (ii) I Xhas he KK(τ)p ope y and (zn)is a sequence in Xsuch ha (zn)con e ges o zwi h espec o τand (zn)does no ha e a no m con e gen subsequence, hen (z)<lim sup n→∞ (zn). P oo . Le a sequence (zn) con e ge o zwi h espec o τ. We ind a sequence (nk) so ha (z) = limk→∞ kxnk−zkand (xnk) con e ges o some ywi h espec o he abs ac measu e opology. Then using P oposi ion 3, we ob ain (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→∞ (zm). This comple es he p oo o (i) and he p oo o (ii) is simila .  We can now p o e ou gene al ixed poin esul s o L-embedded spaces. Theo em 5. Le Xbe an L-embedded Banach space and τbe a ec o opology in X coa se han he no m opology such ha e e y τ-sequen ially compac subse o Xis τ- compac and Xhas he KK(τ)p ope y. Le a nonemp y bounded con ex se C⊂Xbe τ- sequen ially compac and ela i ely sequen ially compac in an abs ac measu e opology. Then e e y mapping T:C→Co asymp o ically nonexpansi e ype has he (P)τ- ixed poin p ope y. P oo . We ollow a easoning o m [18]. Le T:C→Cbe a mapping o asymp o ically nonexpansi e ype. We pu x(y) = lim sup n→∞ kTn(x)−yk whe e x, y ∈C. Clea ly, 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( )k−ky− k: ∈C}+kTn−m(x)−yk 6 M. A. JAP ´ ON PINEDA AND S. PRUS o all x, y ∈Cand n>m. Hence (1) lim sup m→∞ lim sup n→∞ kTn(x)−Tm(y)k ≤ x(y). Le Fbe he amily o all nonemp y con ex τ-sequen ially closed subse s Ko Csuch ha i y∈Kand zis a limi wi h espec o τo a subsequence o (Tn(y)), hen z∈K. We ix D∈ F. F om he Zo n lemma i ollows ha he e exis s K0∈ F which is minimal wi h espec o inclusion in he amily {K∈ F :K⊂D}. Le x∈K0. We will show ha he se {Tn(x)}is ela i ely compac in he no m opology. Le K1be he se o all z∈K0a which he unc ion xa ains i s in imum on K0. Lemma 4 shows ha K1is nonemp y and τ-sequen ially closed. Ob iously i is also con ex. Le z∈K1and (nk) be an inc easing sequence such ha (Tnk(z)) con e ges o some uwi h espec o τ. By Lemma 4 and (1) x(u)≤lim sup k→∞ x(Tnk(z)) ≤ x(z). This shows ha u∈K1and we see ha K1∈ F. Consequen ly, K1=K0and in pa icula xa ains a xi s in imum on K0. Suppose ha he e exis s an inc easing sequence (nk) such ha (Tnk(x)) does no ha e a no m con e gen subsequence. We can assume ha (Tnk(x)) con e ges o some u∈K0wi h espec o τ. Then Lemma 4 and (1) show ha x(u)<lim sup k→∞ x(Tnk(x)) ≤ x(x) which is a con adic ion. Now i su ices o use he easoning om he p oo o Lemma 2 in [18].  I he no m o Xis no only τ-SLSC, bu τ-lowe semicon inuous, hen he assump ions o Theo em 5 ac ually gua an ee ha he e exis s a nonexpansi e e ac ion R om C on o he se Fix(T) o all ixed poin s o Tsuch ha R◦T=Rand e e y con ex τ-sequen ially closed T-in a ian subse o Cis also R-in a ian (see [15] o [14] whe e only he case o τ=wis conside ed). The o mula ion o Theo em 5 can be simpli ied i he space Xadmi s an abs ac measu e opology such ha bounded se s a e ela i ely sequen ially compac . Fu he simpli ica ion is possible i Xis sepa able. Then τ-sequen ially compac se s a e τ- compac (see [10]). Bo h ema ks apply o ins ance o he spaces A1and l1. Co olla y 6. Le τbe a ec o opology in l1coa se han he no m opology such ha l1has he KK(τ)p ope y. I a nonemp y bounded con ex se C⊂Xis τ-sequen ially compac , hen e e y mapping T:C→Co asymp o ically nonexpansi e ype has he (P)τ- ixed poin p ope y. In pa icula l1has he τ-FPP. In Co olla y 6 we ob ained a condi ion su icien o he τ-FPP in l1. Ou nex esul gi es a necessa y condi ion. Be o e passing o his heo em we es ablish some no a ion. Le Γ be a nonemp y se . Gi en x∈l1(Γ), we w i e x= (x(i))i∈Γwhe e x(i) a e scala s. I x6= 0, we se supp x={i∈Γ : x(i)6= 0}. E en i Γ is uncoun able, his se is a mos coun able. Theo em 7. Le Γbe an in ini e se and τbe a locally con ex opology in l1(Γ) coa se han he weak opology on he uni ball. I he s anda d no m o l1(Γ) is no τ-SLSC, hen he e exis a bounded con ex τ-sequen ially compac se C⊂l1(Γ) and a con ac i e mapping T:C→Cwhich does no ha e a ixed poin . FIXED POINT PROPERTY FOR GENERAL TOPOLOGIES 7 P oo . By he assump ion he e exis s a sequence (xn) in l1(Γ) such ha (xn) con e ges o xwi h espec o τand kxk>lim n→∞ kxnk. We can assume ha (xn) con e ges coo dina ewise o some y. Se ing un=xn−y and u0=x−y, we ob ain a sequence (un) which con e ges o u0wi h espec o τ and con e ges coo dina ewise o 0. Then (un) does no con e ge o 0 in no m and by P oposi ion 3 ku0k ≥ kxk−kyk>lim n→∞ kxnk−kyk= lim n→∞ kunk. We can assume ha An= supp unis ini e, kunk= 1 o e e y n≥1 and he se s An a e pai wise disjoin . Then ku0k>1+ o some  > 0 and he e a e ec o s u0 0,u00 0such ha u0=u0 0+u00 0,ku0 0k − ku00 0k>1 + ,A= supp u0 0is ini e and A∩supp u00 0=∅. We can also assume ha A∩An=∅ o e e y n≥1. We pu k= (1 + /k)uk o k≥1 and C=(λ0u0+ ∞ X j=1 λj j: ∞ X j=0 λj= 1, λj≥0, j = 0,1,2, . . . ). Clea ly, Cis bounded and con ex. Co olla y 2 shows ha Cis τ-sequen ially compac . We now se T λ0u0+ ∞ X j=1 λj j!= ∞ X j=0 λj j+1. This o mula de ines a mapping T:C→Cwi hou a ixed poin . Mo eo e ,      ∞ X j=0 γj j+1     = ∞ X j=0 |γj|1 +  j+ 1 <|γ0|(ku0 0k−ku00 0k) +      ∞ X j=1 γj j     =     γ0u0 0+ ∞ X j=1 γj j     − kγ0u00 0k ≤     γ0u0+ ∞ X j=1 γj j     o e e y nonze o (γn)∈l1, which shows ha Tis con ac i e.  Theo em 7 may be ex ended o spaces o he o m Pi∈ΓXil1(Γ) whe e Xia e ini e dimensional. Assume ha τis a ec o opology in l1such ha i (yn) con e ges o ywi h espec o τand con e ges o 0 coo dina ewise, hen kyk ≤ lim sup n→∞ kynk. Then he no m k·k is τ-SLSC. Indeed, le a bounded sequence (xn) con e ge o xwi h espec o τ. Passing o a subsequence, we can assume ha (xn) con e ges coo dina ewise 8 M. A. JAP ´ ON PINEDA AND S. PRUS o some z. By ou assump ion and P oposi ion 3 kxk ≤ kx−zk+kzk ≤ lim sup n→∞ kxn−zk+kzk= lim sup n→∞ kxnk. A simila ema k applies o he KK(τ) p ope y. Unde an addi ional assump ion we can gi e a simple cha ac e iza ion o opologies τ o which l1has he τ-FPP. By (en) we deno e he s anda d basis o l1. Theo em 8. Le τbe a locally con ex opology in he eal space l1coa se han he weak opology on he uni ball. Assume ha (en)con e ges o some e∈l1wi h espec o τ. Then l1has he τ-FPP i and only i one o he ollowing condi ions holds (i) kek<1 (ii) kek= 1 and he se N+={n∈N:e(n)≥0}is ini e. P oo . Gi en z= (z(k))k∈N∈l1we se s(z) = ∞ X k=1 z(k). Mo eo e , we pu Pn(z) = n X k=1 z(k)ek whe e n∈N. Conside a bounded sequence (xn) in l1which con e ges o xwi h espec o τand con e ges o 0 coo dina ewise. I addi ionally he limi s= limn→∞ s(xn) exis s, hen x=se. Indeed, le Ybe he space dual o (l1, τ). I x∗∈Y, hen |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 o e e y m. I ollows ha |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 o al, so we ob ain he desi ed o mula x=se. Assume now ha (i) o (ii) holds. Then kek ≤ 1. Le (xn) be a bounded sequence in l1which con e ges o xwi h espec o τand con e ges o 0 coo dina ewise. Passing o a subsequence, we can assume ha he limi s= limn→∞ s(xn) exis s. Then kxk ≤ |s|= lim n→∞ |s(xn)| ≤ lim sup n→∞ kxnk. This shows ha he no m o l1is τ-SLSC. Assume ha l1does no ha e he τ-FPP. Lemma 4 enables us o use a gene alized Goebel-Ka lo i z lemma (see [7, Lemma 1] and [9, Lemma 2.6]) and ob ain a sequence (xn) such ha i con e ges o x0wi h espec o τand limn→∞ ku−xnk= 2 o e e y u∈con {xn:n≥0}. We can assume ha (xn) con e ges coo dina ewise o some y∈l1. Then he ec o s yn=xn−y end o z=x0−ywi h espec o τand end o 0 FIXED POINT PROPERTY FOR GENERAL TOPOLOGIES 9 coo dina ewise. We can also assume ha he limi s limn→∞ kynkand s= limn→∞ s(yn) exis . Using P oposi ion 3, we see ha 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 ha limn→∞ kynk= 1 = kzkand consequen ly, |s| ≤ 1. Bu z=se. I ollows ha kek= 1 and |s|= 1. We he e o e see ha (i) does no hold, so by ou assump ion he se N+is ini e. Conside he case when s= 1. We choose n o which s(yn)>1/2 and kPm(yn)k<1/4 whe e m= max N+. Then he se B={k∈N:e(k)yn(k)<0}is nonemp y. I is easy o see ha |a+b|=|a|+|b| − 2 min{|a|,|b|} whene e a, b ∈R,ab < 0. Consequen ly, ke+ynk=kek+kynk − 2c whe e c=Pk∈Bmin{|e(k)|,|yn(k)|} >0. Applying P oposi ion 3, we he e o e ob ain 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 con adic ion. The case when s=−1 is simila . We ha e p o ed ha i (i) o (ii) holds, hen l1has he τ-FPP and om Theo em 7 we know ha i kek>1, hen l1does no ha e his p ope y. To comple e he p oo i he e o e emains o show ha i kek= 1 and he se N+is in ini e, hen l1lacks he τ-FPP. Le (nk) be an in ini e sequence in N+such ha w0=e−u06= 0 whe e u0=P∞ k=1 e(nk)enk. We se w=1 kw0kw0and C=(µ1e+µ2w+ ∞ X k=1 µk+2enk: ∞ X j=1 µj= 1, µj≥0, j = 1,2, . . . ). The se Cis bounded and con ex. By Co olla y 2 i is also τ-sequen ially compac . Mo eo e , i P∞ j=1 µj= 1 and µj≥0 o all j∈N hen 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.