Near-infinity concentrated norms and the fixed point property for nonexpansive maps on closed, bounded, convex sets

NEAR-INFINITY CONCENTRATED NORMS AND THE FIXED POINT PROPERTY FOR NONEXPANSIVE MAPS ON CLOSED, BOUNDED, CONVEX SETS F.E. CASTILLO-S´ ANTOS, P.N. DOWLING, H. FETTER, M. JAP ´ ON, C.J. LENNARD, B. SIMS, B. TURETT Abs ac . In his pape we de ine he concep o a nea -in ini y concen a ed no m on a Banach space Xwi h a boundedly comple e Schaude basis. When k·k is such a no m, we p o e ha (X, k·k) has he ixed poin p ope y (FPP); ha is, e e y nonexpansi e sel -mapping de ined on a closed, bounded, con ex subse has a ixed poin . In pa icula , P.K. Lin’s no m in `1[14] and he no m νp(·) (wi h p= (pn) and limnpn= 1) in oduced in [3] a e examples o nea -in ini y concen a ed no ms. When νp(·) is equi alen o he `1-no m, i was an open p oblem as o whe he (`1, νp(·)) had he FPP. We p o e ha he no m νp(·) always gene a es a non e lexi e Banach space X=R⊕p1(R⊕p2(R⊕p3...)) sa is ying he FPP, ega dless o whe he νp(·) is equi alen o he `1-no m. We also ob ain some s abili y esul s. 1. In oduc ion and P elimina ies Le (X, k · k) be a Banach space and Ca subse o X. A mapping T:C→Cis said o be nonexpansi e i kT x −Tyk ≤ kx−yk o e e y x, y ∈C. The Banach space Xendowed wi h he no m k·khas he ixed poin p ope y (FPP) i e e y nonexpansi e mapping de ined om a closed bounded con ex subse Co Xin o i sel has a ixed poin . This p ope y is no p ese ed by isomo phism, ha is, i s ongly depends on he unde lying no m [14] . The e is a wide li e a u e ela ing geome ic p ope ies o e lexi e Banach spaces wi h he ul ilmen o he ixed poin p ope y (see, o ins ance, he monog aphs [9], [13] and he e e ences he ein). The Banach space `1endowed wi h i s s anda d no m k·k1is a classical example o a non e lexi e Banach space ha ails o ha e he FPP. I is possible o “pe u b” his (`1,k·k1)-example o ob ain o he Banach spaces ha ail o ha e he FPP. One such class o Banach spaces a e hose ha con ain asymp o ically isome ic copies o `1. Recall ha a Banach space (X, k · k) con ains an asymp o ically isome ic copy (a.i.c.) o `1i he e a e a sequence (xn)⊂Xand a dec easing sequence (n)⊂(0,1) wi h limnn= 0 such ha ∞ X n=1 (1 −n)| n| ≤      ∞ X n=1 nxn     ≤ ∞ X n=1 | n| 1991 Ma hema ics Subjec Classi ica ion. 46B03, 47H09, 47H10. Key wo ds and ph ases. ixed poin p ope y; nonexpansi e mappings; eno ming heo y. The hi d au ho is pa ially unded by CONACYT g an s 243722 and 613207. The ou h au ho is pa ially suppo ed by MCIN, G an MTM2015-65242-C2-1-P and Andalusian Regional Go e nmen G an FQM-127. She is also g a e ul o he Depa men o Ma hema ics o he Uni e si y o Pi sbu gh o i s suppo while p epa ing his manusc ip . The i h au ho hanks he Uni e si y o Newcas le o i s inancial suppo du ing pa o he p epa a ion o his pape ; and B ailey, Gail and he Ma hema ics Depa men o hei hospi ali y. 1 2 CASTILLO-S´ ANTOS, DOWLING, FETTER, JAP ´ ON, LENNARD, SIMS, TURETT o e e y ( n)∈`1. I was p o ed in [4] ha i a Banach space con ains an a.i.c. o `1 hen i ails o ha e he FPP. I u ns ou ha he e exis equi alen no ms on `1which ail o con ain an a.i.c. o `1. Le us s a e some examples: •The so-called P.K. Lin no m, de ined as |||x|||L:= sup k≥1 γk ∞ X n=k |x(n)|;x= ∞ X n=1 x(n)en whe e (γk) is a nondec easing sequence in (0,1) wi h limkγk= 1. In [5] i was p o ed ha (`1,||| · |||L) ails o con ain an a.i.c. o `1. La e on, P.K. Lin [14] p o ed ha (`1,|||·|||L) has he FPP o γk:= 8k 1+8k. This condi ion was ex ended o e e y sequence (γk) wi h limkγk= 1 (see [7] and [11]). P. K. Lin’s esul opened new a enues o esea ch in he ixed poin heo y o nonexpansi e mappings, since he se led nega i ely he long-s anding open ques ion: “Does he ixed poin p ope y imply e lexi i y?” Since hen, many o he a icles ha e appea ed ob aining su icien condi ions ha imply he FPP o equi alen no ms on `1(see o ins ance [2, 6, 7, 8, 10, 11, 12, 15]). •Fix a noninc easing sequence p= (pn)n⊂(1,+∞) wi h limnpn= 1. In he sequence space c00 o all eal sequences wi h ini ely many non-null coo dina es, we de ine he no m νp(x) = limnνn(p, x) whe e ν1(p, x) := |x1|, νn+1(p, x) := (|x1|p1+νn(Sp, Sx)p1)1/p1, wi h x= (x1, x2, ...) and Sz := (z2, z3, ...) when z= (z1, z2, ...). The comple ion o c00 wi h he νp(·) no m gi es us a Banach space Xwi h aboundedly comple e Schaude basis (en). Also, Xis he se o all eal sequences x= (xn) o which νp(x) := supnνn(p, x) = limnνn(p, x)<∞; which we summa ize by w i ing X=R⊕p1(R⊕p2(R⊕p3. . . )). Le q= (qn) be he sequence sa is ying 1 pn+1 qn= 1 o e e y n∈N. Whene e he sequence (pn) con e ges o 1 quickly enough, he no m νp(·) p o ides an equi alen no m in `1; ha is, (X, νp(·)) and `1a e isomo phic Banach spaces. In ac , i was p o ed in [3, P oposi ion 1] ha νp(·) is equi alen o he `1no m i and only i he e exis s some δ > 0 so ha qn≥δlog n o all n∈N. I is also known ha (`1, νp(·)) ails o con ain asymp o ically isome ic copies o `1[3, Theo em 1]. Howe e , unlike P.K. Lin’s no m, i was unknown whe he `1wi h he no m νp(·) had he ixed poin p ope y. In wha ollows, we enla ge he class o no ms on `1sa is ying he FPP and we include, as a pa icula case, he no m νp(·) de ined in [3]. We will ex end ou esul o a mo e gene al amewo k. Fo ins ance, we will p o e he ul ilmen o he FPP o (X, νp(·)) e en when his no m ails o be an `1-no m. Fu he mo e, we ob ain s abili y o he ixed poin p ope y o ce ain no ms along ays emana ing om nea -in ini y concen a ed no ms. 2. Nea -in ini y concen a ed no ms and he FPP Th oughou his pape , le Xdeno e a Banach space wi h a Schaude basis {en}n. Gi en x= ∞ X n=1 x(n)en∈X, we deno e by supp(x) = {n∈N:x(n)6= 0}, Qk(x) = ∞ X n=k x(n)enand Pk(x) = k−1 X n=1 x(n)en(P1= 0). The basis is said o be p emono one o he no m k·kwhen kQkk ≤ 1 o e e y k∈N. NEAR-INFINITY CONCENTRATED NORMS AND THE FPP 3 Gi en k∈Nand x∈X, we w i e k≤xwhene e k≤min{supp(x)}and k < x whene e k < min{supp(x)}. We say ha (yn) is a block basic sequence o {en}n i i is bounded and he e exis posi i e in ege s p1≤q1< p2≤q2< ... such ha ynbelongs o he span o {epn,· · · , eqn} o e e y n∈N. The Schaude basis is said o be boundedly comple e i sup n     n X i=1 iei     <+∞im- plies ha ∞ X i=1 iei∈X. When he Schaude basis (en) is boundedly comple e, he Banach space Xis isomo phic o a dual space Z∗, whe e Zis he closed subspace spanned by he bio hogonal unc ionals (e∗ n) in X∗. In his case, we can conside in X he weak∗ opology σ(X, Z), o which he con e gence coincides wi h he coo dina e- o-coo dina e con e gence o no m-bounded sequences. Mo eo e , he closed uni ball is σ(X, Z)-sequen ially compac and he e o e e e y bounded se- quence in Xhas a subsequence which con e ges coo dina ewise (see o ins ance Theo em 3.2.10 in [1]). In wha ollows he weak∗ opology always e e s o he σ(X, Z) opology o Banach spaces wi h boundedly comple e Schaude basis. In he case whe e X=`1endowed wi h he s anda d Schaude basis, his w∗- opology coincides wi h he σ(`1, c0) opology. De ini ion 2.1. [2] A no m ||| · ||| on a Banach space Xwi h a Schaude basis {en}is said o be a sequen ially sepa a ing no m i o e e y  > 0 he e exis s some k∈Nsuch ha |||x||| + lim sup n |||xn||| ≤ (1 + ) lim sup n |||x+xn||| whene e k≤xand (xn)nis a block basic sequence o {en}nin X. De ini ion 2.2. Le Xbe a Banach space wi h a Schaude basis {en}nand le ||| · ||| be a no m on X. This no m is called nea -in ini y concen a ed (n.i.c.) i i has he ollowing p ope ies: (1) I is a sequen ially sepa a ing no m. (2) I is p emono one. (3) The e exis R0>5and M∈[0,1) such ha o e e y k∈N, he e exis s a unc ion Fk: (0,+∞)→[0,+∞)sa is ying he ollowing condi ions: (a) limλ→0+ Fk(λ) λ≤M R0 . (b) Fo e e y bounded poin wise-null sequence (xn)wi h lim in n|||xn||| ≥ 1, o all λ∈(0,+∞), and o e e y z∈Xwi h Qk(z)=0and |||z||| ≤ R0, lim sup n |||xn+λz||| ≤ lim sup n |||xn||| +Fk(λ)|||z||| . Rema k 2.3. Obse e ha P ope y (3) can be e-w i en as: The e exis s K≥0 such ha o e e y k∈N, he e exis s a unc ion Fk: (0,+∞)→[0,+∞)sa is ying (a)’ and (b); whe e condi ion (a)’ is: limλ→0+ Fk(λ) λ≤K < 1 5. Gi en K, we may ake M:= 1 −1−5K K+1 =6K K+1 and R0:= 5 + 1−5K K+1 =6 K+1 . No e ha i ||| · ||| is an equi alen no m on `1sa is ying akkQk(x)k1≤ |||Qk(x)||| ≤ bkkQk(x)k1, o all x∈`1, 4 CASTILLO-S´ ANTOS, DOWLING, FETTER, JAP ´ ON, LENNARD, SIMS, TURETT o e e y k∈N, wi h 0 < ak≤bkand limkbk/ak= 1, hen i is clea ha ||| · ||| is a sequen ially sepa a ing no m. Ne e heless, he e exis some equi alen no ms on `1which do no sa isi y his condi ion bu hey a e s ill sequen ially sepa a ing [2, Example 3.2]. Fu he mo e, he e exis Banach spaces wi h sequen ially sepa a ing no ms ha a e no isomo phic o `1, al hough he exis ence o such a no m implies ha he Banach space Xis “simila ” o `1, in he sense ha i has he Schu p ope y, and so is he edi a ily `1[2, Co olla y 7.4]. Recall ha a Banach space X is he edi a ily `1i each in ini e dimensional closed subspace o Xcon ains a u he subspace isomo phic o `1. This implies, in pa icula , ha i a Banach space wi h an uncondi ional Schaude basis has a sequen ially sepa a ing no m, hen he basis is boundedly comple e, since o he wise Xwould con ain an isomo phic copy o c0 (see o ins ance [1, Theo em 3.3.2]). Also no e ha in De ini ion 2.2, P ope y (3)(b), i (xn) is an a bi a y sequence o “bump unc ions sliding owa ds in ini y”, each wi h hei ||| · |||-no m asymp- o ically no less han 1, hen lim supn|||xn+λz||| − lim supn|||xn||| λ is smalle han one would expec om jus he iangle inequali y: o all z=Pk(z) wi h |||z||| ≤ R0, o all λposi i e and e y small, he “uppe asymp o ic alue” o he no m o xnis changed less han expec ed when we pe u b each, xnby λz, since Fk(λ)R0/λ is app oxima ely bounded by M < 1. In his sense, |||·||| is “nea - in ini y concen a ed”. Mo eo e , his hi d p ope y p e en s X om con aining an asymp o ically isome ic copy o `1, which we will now p o e. Lemma 2.4. Le Xbe a Banach space wi h a boundedly comple e Schaude basis. I ||| · ||| is an equi alen no m in Xsa is ying p ope y (3) in De ini ion 2.2, hen (X, ||| · |||) ails o ha e an a.i.c. o `1. P oo . Assume o he con a y ha he e exis s a basic sequence (xn) in Xgen- e a ing an a.i.c. o `1, ha is, he e is a dec easing sequence (n)⊂(0,1) wi h limnn= 0 such ha ∞ X n=1 (1 −n)| n| ≤  ∞ X n=1 nxn ≤ ∞ X n=1 | n|. By ex ac ing a subsequence, we can assume ha (xn) is w∗-con e gen and, by eplacing (xn) by ((x2n−x2n−1)/2)), ha i is w∗-con e gen o he null ec o . Finally, using he sliding hump me hod and he ac ha asymp o ically isome ic copies a e s able by adding no m-null sequences, we can assume ha he sequence (xn) gene a ing he a.i.c. o `1is a disjoin ly suppo ed w∗-null sequence. Take R0>5 and M∈[0,1) as in (3) o De ini ion 2.2. By omi ing he i s ew e ms o he sequence (xn)n, we can also assume ha 1<(R0−M)/R0. F om he p e ious inequali ies |||xn||| ≤ 1 o e e y n∈Nand limn|||xn||| = 1. Le k:= 1 + max{supp(x1)}. Since ||| · ||| sa is ies p ope y (3) o a nea -in ini y concen a ed no m, he e exis s a unc ion Fk(λ) such ha limλ→0+Fk(λ)/λ ≤M R0, and o e e y λ > 0 lim sup n |||xn+λR0x1||| ≤ lim sup n |||xn||| +Fk(λ)R0|||x1||| ≤ 1 + Fk(λ)R0. On he o he hand, o e e y n≥2, 1−n+λR0(1 −1)≤ |||xn+λR0x1|||. Le ing n end o in ini y, we see ha 1 + λR0(1 −1)≤1 + Fk(λ)R0 NEAR-INFINITY CONCENTRATED NORMS AND THE FPP 5 and so λ(1 −1)≤Fk(λ) o e e y λ > 0. Le ing λ→0, we ge ha (1 − 1)≤limλ→0+Fk(λ) λ≤M R0, which implies ha R0(1 −1)≤M, and his is a con adic ion.  Be o e s a ing ou main esul , we ecall some s anda d a gumen s used o p o e he FPP (see o ins ance [14] o [11]): Le Cbe a closed bounded con ex subse o a Banach space (X, k·k) and T:C→Cbe a nonexpansi e mapping. Using Banach’s Con ac ion Mapping Theo em, we can always ind a sequence (xn)⊂Csuch ha limnkxn−Txnk= 0. Such sequences a e called app oxima e ixed poin sequences (a. .p.s.). In ac , i (xn) is an a. .p.s. and > 0, he se {x∈C: lim sup n kxn−xk ≤ } is ei he emp y, o a non-emp y closed con ex T-in a ian subse o C, in which we can ind new app oxima e ixed poin sequences. In a dual Banach space Xwi h sepa able p edual Y, e e y a. .p.s. has a subse- quence which is w∗-con e gen . Fo example, i Xis a Banach space wi h a bound- edly comple e Schaude basis {en}n, and co esponding bio hogonal unc ionals { n}n⊂X∗, hen o Y:= he closed linea span o { n}nin X∗,Y∗is isomo - phic o Xand e e y a. .p.s. in Xhas a subsequence ha is σ(X, Y )-con e gen . The e o e, we will subsequen ly assume ha app oxima e ixed poin sequences in bounded subse s o Xa e w∗-con e gen . Using Can o ’s heo em (see [14] o [11, Lemma 1]), he abo e a gumen le s us deduce ha i T:C→Cis a ixed poin ee nonexpansi e mapping, he e exis some a > 0 and a closed con ex T-in a ian subse , deno ed again by C, such ha lim supnkyn−yk> a whene e (yn)⊂Cis an a. .p.s. and y=w∗-lim yn. No e ha om De ini ion 2.1 i is no di icul o check he ollowing [2]: Lemma 2.5. A no m ||| · ||| in a Banach space Xwi h a Schaude basis {en}nis sequen ially sepa a ing i and only i limkSk(X, ||| · |||)=1, wi h Sk(X, ||| · |||) := sup |||x||| + lim supn|||xn||| lim supn|||x+xn||| , whe e he sup emum is aken o e all ec o s x∈Xwi h k≤xand all block basic sequences o {en}n. I we ix he no m ||| · ||| in he Banach space X, we will use Sk o deno e Sk(X, ||| · |||). Lemma 2.6. Le (X, |||·|||)be a Banach space wi h a boundedly comple e Schaude basis {en}nsuch ha ||| · ||| is p emono one and sequen ially sepa a ing. The ol- lowing holds: i (xn),(yn)a e wo sequences in X ha a e w∗-con e gen o xand y espec i ely, hen lim sup m lim sup n |||xn−ym||| ≥ lim sup n |||xn−x||| + lim sup m |||ym−y|||. P oo . Le k∈Nand δ > 0 be gi en. Choose a subsequence (xn`) o (xn) such ha lim sup n |||Qk(xn−x)||| = lim `|||Qk(xn`−x)||| . Fix m∈N. Then lim sup n |||xn−ym||| ≥ lim sup n |||Qk(xn−ym)||| = lim sup n |||Qk(xn−x)−Qk(ym−x)||| ≥lim sup ` |||Qk(xn`−x)−Qk(ym−x)||| . 6 CASTILLO-S´ ANTOS, DOWLING, FETTER, JAP ´ ON, LENNARD, SIMS, TURETT By he Bessaga-Pe lczynski Selec ion P inciple (see, o example, [1, p. 14]), passing o a u he subsequence i necessa y, we may choose a block basic sequence (u`) o (en) such ha Qk(u`) = u`and |||u`−Qk(xn`−x)||| < δ. Then lim sup n |||xn−ym||| ≥ lim sup ` |||u`−Qk(ym−x)||| − δ ≥1 Sk|||Qk(ym−x)||| + lim sup ` |||u`|||−δ ≥1 Sk|||Qk(ym−x)||| + lim `|||Qk(xn`−x)|||−2δ =1 Sk|||Qk(ym−x)||| + lim sup n |||Qk(xn−x)|||−2δ . Le ing m end o ∞; no ing ha Sk≥1; and using a pe u ba ion a gumen simila o he one abo e hen yields: lim sup m lim sup n |||xn−ym||| ≥ 1 Sklim sup m |||Qk(ym−x)||| + lim sup n |||Qk(xn−x)|||−2δ =1 Sklim sup n |||xn−x||| + lim sup m |||Qk(ym−y) + Qk(y−x)|||−2δ ≥1 Sk lim sup n |||xn−x||| +1 S2 k|||Qk(y−x)||| + lim sup m |||Qk(ym−y)|||−4δ ≥1 Sk lim sup n |||xn−x||| +1 S2 k lim sup m |||Qk(ym−y)||| − 4δ =1 Sk lim sup n |||xn−x||| +1 S2 k lim sup m |||ym−y||| − 4δ . In he abo e calcula ion, we used he ac ha lim supn|||Qk(xn−x)||| = lim supn|||xn− x||| and lim supm|||Qk(ym−y)||| = lim supm|||ym−y||| o e e y k∈N. Since he abo e inequali ies hold o e e y k∈N, le ing k end o in ini y gi es lim sup m lim sup n |||xn−ym||| ≥ lim sup n |||xn−x||| + lim sup m |||ym−y||| − 4δ , o e e y δ > 0. Since δ > 0 is a bi a y, we ob ain he desi ed inequali y.  Theo em 2.7. Le Xbe a Banach space wi h a boundedly comple e Schaude basis and le ||| · ||| be a nea -in ini y concen a ed (n.i.c.) no m on X. Then (X, ||| · |||) has he FPP, ha is, e e y nonexpansi e sel -map on a closed bounded con ex subse o Xhas a ixed poin . P oo . Assume, o he con a y, ha he e exis s a closed bounded con ex subse Co Xand T:C→Ca nonexpansi e mapping such ha b= in {lim sup n |||yn−y||| : (yn)⊂Cis an a. .p.s. and yn w∗ →y}>0. Wi hou loss o gene ali y we can assume ha b= 1. We p oceed as ollows. Fix some 0 < 1<min{1 4(1 −M+1 2),1 10 (R0−5)}, whe e M∈[0,1) and R0>5 a e he cons an s gi en by condi ion (3) in De ini ion 2.2. Conside an a. .p.s. (xn) in Csuch ha xn w∗ →x0∈Xand lim supn|||xn−x0||| < 1 + 1. Again, wi hou loss o gene ali y, we can assume ha x0= 0 so ha lim supn|||xn||| <1 + 1. De ine he se D:= z∈C: lim sup n |||xn−z||| ≤ 2(1 + 1). NEAR-INFINITY CONCENTRATED NORMS AND THE FPP 7 Then Dis a closed con ex T-in a ian subse o C. Mo eo e , using he iangle inequali y, lim sup m lim sup n |||xn−xm||| ≤ 2 lim sup n |||xn||| <2(1 + 1) ; so Dis no emp y and we can assume ha xn∈D o nla ge enough. De ine c:= in lim sup n |||yn−y||| : (yn)⊂Dis an a. .p.s. and yn w∗ →y. No ice ha 1 ≤c. To simpli y he no a ion, we de ine A∗(D) := ny∈X:∃(yn)⊂Dan a. .p.s. such ha w∗- lim nyn=yo. We now p o e ha sup y∈A∗(D) |||y||| ≤ 4+41: Indeed, le (yn)⊂Dbe an a. .p.s. wi h w∗-lim yn=y. In pa icula , lim supn|||xn− ym||| ≤ 2(1 + 1) o e e y m∈N. Using he iangle inequali y and Lemma 2.6 (wi h x= 0): |||y||| ≤ lim sup m |||y−ym|||+lim sup n |||xn|||+lim sup m lim sup n |||xn−ym||| ≤ 4(1+1). Nex we show ha sup y∈A∗(D) |||Qk(y)||| ≤ µk:= 2S2 k(1 + 1)−Sk−1 o e e y k∈N. Using he p oo o Lemma 2.6 we deduce ha : 2(1 + 1)≥lim sup m lim sup n |||xn−ym||| ≥ 1 Sk lim sup n |||xn|||+1 S2 klim sup m |||ym−y||| +|||Qk(y)|||≥1 Sk +1 S2 k [1 + |||Qk(y)|||], which implies ha |||Qk(y)||| ≤ 2S2 k(1 + 1)−Sk−1 o all y∈A∗(D). Choose x:= xνwi h ν∈Nla ge enough so ha x∈Dand |||x||| <1 + 1. Since he no m sa is ies condi ion (1) in De ini ion 2.2, we know ha limkSk= 1 and he e o e limkµk= 21. Take k1∈Nso ha |||Qk(x)||| < 1,and µk<31 i k≥k1. In pa icula , his implies ha |||Qk1(y−x)||| ≤ |||Qk1(y)||| +|||Qk1(x)||| ≤ 31+1= 41<1, and |||Pk1(y−x)||| ≤ |||x−y||| +|||Qk1(y−x)||| ≤ |||x||| +|||y||| + 41 ≤1 + 1+ 4 + 41+ 41= 5 + 91< R0 o e e y y∈A∗(D). Gi en k1∈Nas be o e, he e exis s a co esponding unc ion F(λ) := Fk1(λ) sa is ying p ope y (3) in De ini ion 2.2. Since limλ→0+F(λ) λ≤M R0, ake λ∈(0,1) such ha F(λ) λ<M+ 1 2R0 <1−41 R0 ≤c−41 R0 , which implies ha (2 −λ)c+F(λ)R0+λ41<2c. Now, choose 2>0 wi h (2 −λ)(c+2) + F(λ)R0+λ41<2c. 8 CASTILLO-S´ ANTOS, DOWLING, FETTER, JAP ´ ON, LENNARD, SIMS, TURETT Choose (yn)⊂Dan a. .p.s. wi h w∗-limnyn=yand such ha lim sup n |||yn−y||| ≤ c+2. By passing o a subsequence, we may also suppose ha lim in n|||yn−y||| = lim sup n |||yn−y||| ≥ c≥1. No ice ha , o e e y m∈N, he ec o s (1 −λ)ym+λx ∈D. We claim ha (∗∗) lim sup m lim sup n |||yn−[(1 −λ)ym+λx]||| <2c. Assume ha (∗∗) holds. Then we can ind some m∈Nsuch ha lim sup n |||yn−[(1 −λ)ym+λx]||| <2c. This implies ha o some ∈(0,2c) he se G:= {z∈D: lim sup n |||yn−z||| ≤ } is a nonemp y closed con ex T-in a ian subse o D, and he e o e i con ains an a. .p.s. (zs), which ends o some z∈Xwi h espec o he w∗- opology. In his case, using he de ini ion o c, Lemma 2.6, and ha each zs∈G, we ha e 2c≤lim sup s |||zs−z||| + lim sup n |||yn−y||| ≤ lim sup s lim sup n |||yn−zs||| ≤ , which is a con adic ion. We inish by p o ing he claim (∗∗). No ing ha lim in n|||yn−y||| ≥ 1 and by p ope y (3) in De ini ion 2.2, we ha e: lim supn|||(yn−y) + λPk1(y−x)||| ≤ lim supn|||yn−y||| +F(λ)|||Pk1(y−x)||| ≤c+2+F(λ)R0. The e o e, lim supmlim supn|||yn−[(1 −λ)ym+λx]||| = lim supmlim supn|||yn−y+y−(1 −λ)ym−λx||| = lim supmlim supn|||yn−y+ (1 −λ)y+λy −(1 −λ)ym−λx||| ≤lim supmlim supn[(1 −λ)|||ym−y||| +|||(yn−y) + λ(y−x)|||] ≤(1 −λ) lim supm|||ym−y||| + lim supn|||(yn−y) + λPk1(y−x)||| +λ|||Qk1(y−x)||| ≤(1 −λ)(c+2) + c+2+F(λ)R0+λ41 ≤(2 −λ)(c+2) + F(λ)R0+λ41 <2c , which p o es (∗∗), and comple es he p oo o he heo em.  3. No ms wi h he Fixed Poin P ope y Th oughou his sec ion, we will s udy se e al examples o no ms which a e nea -in ini y concen a ed no ms and he e o e hey sa is y he FPP acco ding o Theo em 2.7. As a pa icula case o a mo e gene al esul , we will deduce ha (`1, νp(·)) has he FPP whene e νp(·) is a eno ming o `1. We will s a by p o ing ha P.K. Lin’s no m is an example o a nea -in ini y concen a ed no m. We will deduce his asse ion om he ollowing lemma. Lemma 3.1. Le (X, | · |)be a Banach space wi h a Schaude basis and assume ha |·| sa is ies p ope ies (1) and (2) in De ini ion 2.1. I (γk)is a nondec easing sequence in (0,1) con e ging o 1, hen he no m de ined as |x|1:= sup k γk|Qk(x)|, o all x∈X , NEAR-INFINITY CONCENTRATED NORMS AND THE FPP 9 is a nea -in ini y concen a ed no m on X ha is equi alen o |·|. P oo . No ice ha γk|Qk(x)|≤|Qk(x)|1≤ |Qk(x)| o e e y k∈Nand x∈X, which implies ha | · |1is a sequen ially sepa a ing no m whene e | · | sa is ies he same p ope y. I is also easy o check ha | · |1sa is ies (2) in De ini ion 2.2. I emains o p o e condi ion (3). Fix some k∈Nand R > 0. Le (xn) be a bounded poin wise-null sequence in Xwi h lim in n|x|1≥1. Wi hou loss o gene ali y we can assume ha Ql(xn) = Qk(xn) o e e y l≤k. Mo eo e , i is no di icul o check ha lim supn|xn|= lim supn|xn|1. Fo e e y z∈Xwi h Qk(z) = 0, |z|1≤Rand o e e y λ > 0 we ha e |xn+λz|1= suplγl|Ql(xn+λz)|= suplγl|Ql(xn) + λQl(z)| = max max 1≤l≤k−1γl|Ql(xn) + λQl(z)|,sup l≥k γl|Ql(xn)| = max max 1≤l≤k−1γl|Ql(xn) + λQl(z)|,|xn|1 ≤max{γk−1|xn|+λ|z|1,|xn|1} Taking limi s when ngoes o in ini y: lim sup n |xn+λz|1≤max{γk−1lim sup n |xn|1+λ|z|1,lim sup n |xn|1}. F om abo e, lim supn|xn|1≥1 and |z|1≤R; and so lim sup n |xn+λz|1≤lim sup n |xn|1+Fk(λ)|z|1, whe e Fk(λ) := 0 i λ≤(1 −γk−1)/R, and Fk(λ) := λo he wise. Taking M= 0 and any R > 5 in De ini ion 2.2(3), we see ha |·|1is a nea in ini y concen a ed no m.  I we le |·|:= k · k1in `1we ob ain ha | · |1coincides wi h P.K. Lin’s no m ||| · |||L. Mo eo e , gi en a no m |·|0:= |·|sa is ying (1) and (2) in De ini ion 2.2 and de ining in a ecu si e way he equi alen no ms |·|n= sup k γk|Qk(·)|n−1 o e e y n∈N, we can cons uc sequences o nea -in ini y concen a ed no ms. All o hese no ms |·|n(n≥1) sa is y he FPP when he basis is boundedly comple e, acco ding o Theo em 2.7. Lemma 3.2. Assume ha (pn)⊂(1,+∞)is a noninc easing sequence wi h limnpn= 1. Then he no m νp(·)is a nea -in ini y concen a ed no m in he Banach space X, de ined as he comple ion o c00 wi h he no m νp(·). P oo . Le us s a by p o ing ha νp(·) is a sequen ially sepa a ing no m, ha is, νp(·) sa is ies p ope y (1) in De ini ion 2.2. By Lemma 2.5, i su ices o check ha limkSk(X, νp(·)) = 1. Fix k∈N. Fi s no e ha i x= l X i=k x(i)eiwi h k≤land l < y hen νp(x+y) = νp l X i=k x(i)ei+νp(y)el+1!. On he o he hand, i is no di icul o check ha i 1 < p ≤qand a, b, c ≥0 hen: k(a, k(b, c)kp)kq≥ k(a, k(b, c)kq)kq=aq+k(b, c)kq q1/q = (aq+bq+cq)1/q = =k(k(a, b)kq, c)kq.