Asymptotically nonexpansive mappings in modular function spaces

ASYMPTOTICALLY NONEXPANSIVE MAPPINGS IN MODULAR FUNCTION SPACES T. DOMINGUEZ-BENAVIDES, M.A. KHAMSI AND S. SAMADI ABSTRACT In his pape , we p o e ha i ρis a con ex, σ- ini e modula unc ion sa is ying a ∆2- ype condi ion, Ca con ex, ρ-bounded, ρ-a.e. compac subse o Lρand T:C→Caρ-asymp o ically nonexpansi e mapping, hen Thas a ixed poin . In pa icula , any asymp o ically nonexpansi e sel -map de ined on a con ex subse o L1(Ω, µ) which is compac o he opology o local con e gence in measu e has a ixed poin . 1991 Ma hema ics subjec classi ica ion : P ima y 46E30; Seconda y 47H09, 47H10. Key Wo ds: asymp o ically nonexpansi e mappings, ixed poin , modula unc- ions. The i s au ho is pa ially suppo ed by PB-96-1338-C01-C02 and PAI-FMQ-0127. 1 2 T. DOMINGUEZ-BENAVIDES, M.A. KHAMSI AND S. SAMADI INTRODUCTION Le (M, d) be a me ic space. A mapping, T:M→Mis said o be asymp- o ically nonexpansi e i he e exis s a sequence {kn}o eal numbe s wi h lim n→∞ kn= 1 such ha d(Tnx, Tny)≤knd(x, y) o any x, y ∈Mand n∈N.In 1970 Goebel and Ki k [5] p o ed ha Thas a ixed poin whene e Mis a con ex bounded closed subse o a Banach space X. Fu he gene aliza ions o his esul we e p o ed by Yu and Dai [14] when Xis 2-uni o mly o und, by Ma ´ınez Ya˜nez [10] and Xu [12] when Xis k-uni o mly o und o some k≥1,by Xu [13] when Xis nea ly uni o mly con ex and by Kim and Xu [9] when Xhas uni o m no mal s uc u e. Some special s udies on he heo y o he ixed poin o asymp o ically nonexpansi e mappings we e made by many o he au ho s (see, o example, [2,11]). The i s ixed poin esul s in modula unc ion spaces we e gi en by Khamsi, Koz lowski and Reich [7]. E en hough a me ic is no de ined, many p oblems in me ic ixed poin heo y can be e o mula ed in modula spaces. Fo ins ance, ixed poin heo ems a e p o ed in [6,7] o nonexpansi e mappings, in [3] o asymp o ically egula mappings and in [4] o uni o mly Lipschi zian mappings. In his pape we will p o e he exis ence o ixed poin s o asymp o ically nonex- pansi e mappings in modula unc ion spaces when he modula ρsa is ies some con exi y and ∆2- ype p ope ies. Ou esul s can be, in pa icula , applied o L1(Ω, µ), showing ha asymp o i- cally nonexpansi e mappings ha e a ixed poin when hey a e de ined on a con ex subse o L1(Ω, µ) which is compac wi h espec o he opology o con e gence local in measu e. 1. PRELIMINARIES We s a by e iewing some basic ac s abou modula spaces as o mula ed by Koz lowski [8]. Fo mo e de ails he eade may consul [6,7]. Le Ω be a nonemp y se and Σ be a non i ial σ-algeb a o subse s o Ω. Le Pbe a δ- ing o subse s o Σ, such ha E∩A∈ P o any E∈ P and A∈Σ. Le us assume ha he e exis s an inc easing sequence o se s Kn∈ P such ha ASYMPTOTICALLY NONEXPANSIVE MAPPINGS IN MODULAR FUNCTION SPACES 3 Ω = SKn. By Ewe deno e he linea space o all simple unc ions wi h suppo s om P. By Mwe will deno e he space o all measu able unc ions, i.e. all unc ions : Ω →Rsuch ha he e exis s a sequence {gn}∈E,|gn|≤| |and gn(ω)→ (ω) o all ω∈Ω. By 1Awe deno e he cha ac e is ic unc ion o he se A. De ini ion 1.1. A unc ional ρ:E × Σ→[0,∞] is called a unc ion modu- la i : (P1)ρ(0, E) = 0 o any E∈Σ, (P2)ρ( , E)≤ρ(g, E) whene e | (ω)| ≤ |g(ω)| o any ω∈Ω, , g ∈ E and E∈Σ, (P3)ρ( , .) : Σ →[0,∞] is a σ-subaddi i e measu e o e e y ∈ E, (P4)ρ(α, A)→0 as αdec eases o 0 o e e y A∈ P, whe e ρ(α, A) = ρ(α1A, A), (P5) i he e exis s α > 0 such ha ρ(α, A) = 0, hen ρ(β, A) = 0 o e e y β > 0, (P6) o any α > 0ρ(α, .) is o de con inuous on P, ha is ρ(α, An)→0 i {An} ∈ P and dec eases o ∅. The de ini ion o ρis hen ex ended o ∈ M by ρ( , E) = sup{ρ(g, E); g∈ E,|g(ω)| ≤ | (ω)| o e e y ω∈Ω}. De ini ion 1.2. A se Eis said o be ρ-null i and only i ρ(α, E) = 0 o α > 0. A p ope y p(ω) is said o hold ρ-almos e e ywhe e (ρ-a.e.) i he se {ω∈Ω; p(ω) does no hold}is ρ-null. Fo example we will say equen ly n→ ρ-a.e. Fo he sake o simplici y we w i e ρ( ) ins ead o ρ( , Ω). De ini ion 1.3. A modula unc ion ρis called σ- ini e i he e exis s an in- c easing sequence o se s Kn∈ P such ha 0 < ρ(Kn)<∞and Ω = SKn. I is easy o see ha he unc ional ρ:M → [0,∞] is a modula and sa is ies he ollowing p ope ies: (i) ρ( ) = 0 i = 0 ρ-a.e. 4 T. DOMINGUEZ-BENAVIDES, M.A. KHAMSI AND S. SAMADI (ii) ρ(α ) = ρ( ) o e e y scala αwi h |α|= 1 and ∈ M. (iii) ρ(α +βg)≤ρ( ) + ρ(g) i α+β= 1, α≥0, β ≥0 and , g ∈ M. In addi ion, i he ollowing p ope y is sa is ied (iii)’ ρ(α +βg)≤αρ( ) + βρ(g) i α+β= 1 ; α≥0, β ≥0 and , g ∈ M, we say ha ρis a con ex modula . The modula ρde ines a co esponding modula space, i.e he ec o space Lρ gi en by Lρ={ ∈ M;ρ(λ )→0 as λ→0}. When ρis con ex, he o mula || ||ρ= in nα > 0; ρ α≤1o de ines a no m in he modula space Lρwhich is equen ly called he Luxembu g no m. We can also conside he space Eρ={ ∈ M;ρ(α , An)→0 as n→ ∞ o e e y An∈ Σ ha dec eases o ∅and α > 0}. De ini ion 1.4. A unc ion modula is said o sa is y he ∆2-condi ion i sup n≥1 ρ(2 n, Dk)→0 as k→ ∞ whene e { n}n≥1⊂ M, Dk∈ Σ dec eases o ∅and sup n≥1 ρ( n, Dk)→0 as k→ ∞. We know om [8] ha Eρ=Lρwhen ρsa is ies he ∆2-condi ion. De ini ion 1.5. A unc ion modula is said o sa is y he ∆2- ype condi ion i he e exis s K > 0 such ha o any ∈Lρwe ha e ρ(2 )≤Kρ( ). In gene al, ∆2- ype condi ion and ∆2-condi ion a e no equi alen , e en hough i is ob ious ha ∆2- ype condi ion implies ∆2-condi ion on he modula space Lρ. De ini ion 1.6. Le Lρbe a modula space. ASYMPTOTICALLY NONEXPANSIVE MAPPINGS IN MODULAR FUNCTION SPACES 5 (1) The sequence { n}n⊂Lρis said o be ρ-con e gen o ∈Lρi ρ( n− )→0 as n→ ∞. (2) The sequence { n}n⊂Lρis said o be ρ-a.e. con e gen o ∈Lρi he se {ω∈Ω; n(ω)6→ (ω)}is ρ-null. (3) The sequence { n}n⊂Lρis said o be ρ-Cauchy i ρ( n− m)→0 as n and mgo o ∞. (4) A subse Co Lρis called ρ-closed i he ρ-limi o a ρ-con e gen sequence o Calways belongs o C. (5) A subse Co Lρis called ρ-a.e. closed i he ρ-a.e. limi o a ρ-a.e. con e gen sequence o Calways belongs o C. (6) A subse Co Lρis called ρ-a.e. compac i e e y sequence in Chas a ρ-a.e. con e gen subsequence in C. (7) A subse Co Lρis called ρ-bounded i δρ(C) = sup{ρ( −g); , g ∈C}<∞. We ecall wo basic esul s (see [7]) in he heo y o modula spaces. (i) I he e exis s a numbe α > 0 such ha ρ(α( n− )) →0, hen he e exis s a subsequence {gn}no { n}nsuch ha gn→ ρ-a.e. (ii) (Lebesgue’s Theo em) I n, ∈ M, n→ ρ-a.e. and he e exis s a unc ion g∈Eρsuch ha | n| ≤ |g|ρ-a.e. o all n, hen || n− ||ρ→0. We know, by [6,7] ha unde ∆2-condi ion he no m con e gence and modula con e gence a e equi alen , which implies ha he no m and modula con e - gence a e also he same when we deal wi h he ∆2- ype condi ion. In he sequel we will assume ha he modula unc ion ρis con ex and sa is ies he ∆2- ype condi ion. De ini ion 1.7. Le ρbe as abo e. We de ine a g ow h unc ion ωby: ω( ) = sup ρ( ) ρ( ), ∈Lρ {0} o all 0 ≤ < ∞. We ha e he ollowing: Lemma 1.1. [3] Le ρbe as abo e. Then he g ow h unc ion ωhas he ollowing p ope ies: 6 T. DOMINGUEZ-BENAVIDES, M.A. KHAMSI AND S. SAMADI (1) ω( )<∞,∀ ∈[0,∞) (2) ω: [0,∞)→[0,∞)is a con ex, s ic ly inc easing unc ion. So, i is con inuous. (3) ω(αβ)≤ω(α)ω(β); ∀α, β ∈[0,∞) (4) ω−1(α)ω−1(β)≤ω−1(αβ);∀α, β ∈[0,∞),whe e ω−1is he unc ion in- e se o ω. The ollowing lemma shows ha he g ow h unc ion can be used o gi e an uppe bound o he no m o a unc ion. Lemma 1.2. [3] Le ρbe a con ex unc ion modula sa is ying he ∆2- ype con- di ion. Then || ||ρ≤1 ω−11 ρ( )whene e ∈Lρ. The nex lemma will be o majo in e es h oughou his wo k. Lemma 1.3. [6] Le ρbe a unc ion modula sa is ying he ∆2-condi ion and { n}nbe a sequence in Lρsuch ha n ρ−a.e → ∈Lρand he e exis s k > 1such ha sup n ρ(k( n− )) <∞. Then, lim in n→∞ ρ( n−g) = lim in n→∞ ρ( n− ) + ρ( −g) o all g∈Lρ. Mo eo e , we ha e ρ( )≤lim in n→∞ ρ( n). 2. AN EQUIVALENT TOPOLOGY The concep o ρ-a.e. closed, compac se s ha e been s udied ex ensi ely in he sequen ial case. One o he p oblem ha many au ho s ha e ound ha d o ci cum en is whe he hese no ions a e ela ed o a opology. In his sec ion we will discuss his p oblem. In pa icula , we will cons uc a opology τ o which he ρ-a.e. compac ness is equi alen o he usual compac ness o τ. This is c ucial when we y o use Zo n’s lemma. ASYMPTOTICALLY NONEXPANSIVE MAPPINGS IN MODULAR FUNCTION SPACES 7 F om now on, we assume ha he modula unc ion ρis, in addi ion, σ- ini e. Se d( , g) = ∞ X k=1 1 2k 1 ρ(1Kk)ρ| −g| 1 + | −g|1Kk o any , g ∈Lρ. Some basic p ope ies sa is ied by da e discussed in he ollowing p oposi ion. P oposi ion 2.1. The unc ional dsa is ies he ollowing: (1) d( , g) = 0 i and only i =g ρ-a.e.; (2) d( , g) = d(g, ); (3) d( , g)≤ω(2) 2d( , h) + d(h, g); o any , g and hin Lρ. P oo . (1) and (2) a e ob ious. To p o e (3) we only need o ecall he inequali y |a+b| 1 + |a+b|≤|a| 1 + |a|+|b| 1 + |b| o all posi i e numbe s a, b and use he de ini ion o he g ow h unc ion ω.  Rema k 2.1. The unc ional dis no a dis ance because o (3). Bu he e a e many ma hema ical objec s which ail he iangle inequali y bu a e e y use ul ools. Tha is he case wi h d. In he nex p oposi ion, we discuss he ela ionship be ween ρ-a.e. con e gence and he con e gence o he unc ional d. P oposi ion 2.2. Le ρbe a con ex, σ- ini e modula sa is ying he ∆2- ype condi ion and { n}nbe a sequence o measu able unc ions. I { n}nis ρ-a.e. con e gen o , hen lim n→∞ d( n, ) = 0. Mo eo e , i lim n→∞ d( n, ) = 0, 8 T. DOMINGUEZ-BENAVIDES, M.A. KHAMSI AND S. SAMADI hen he e exis s a subsequence { nk}kwhich con e ges ρ-a.e. o . P oo . Assume ha { n}nρ-a.e. con e ges o . We will show ha lim n→∞ d( n, ) = 0.Le ε > 0,and choose N∈Nsuch ha ∞ X k=N+1 1 2k< ε. We ha e lim n→∞ d( n, )≤lim n→∞ N X k=1 1 2k 1 ρ(1Kk)ρ| n− | 1 + | n− |1Kk+ε = N X k=1 lim n→∞ 1 2k 1 ρ(1Kk)ρ| n− | 1 + | n− |1Kk+ε. Since | n− | 1 + | n− |1Kk ρ−a.e −→ 0 as n→ ∞ o any k∈Nand | n− | 1 + | n− |1Kk≤1Kk, om Lebesgue’s Theo em we ob ain lim n→∞ ρ| n− | 1 + | n− |1Kk= 0 o e e y non null in ege k. Thus lim n→∞ d( n, )≤ε o each ε > 0 which means ha lim n→∞ d( n, ) = 0. Assume now ha lim n→∞ d( n, ) = 0.Fo e e y non null in ege kwe ha e lim n→∞ ρ| n− | 1 + | n− |1Kk= 0. Thus, he e exis s a subsequence { 1 n}no { n}nsuch ha | 1 n− | 1 + | 1 n− |1K1 ρ−a.e −→ 0 and so 1 n ρ−a.e −→ in K1i.e. lim n→∞ 1 n(x) = (x) whene e x∈K1 A1whe e A1⊂K1and ρ(1A1) = 0. By induc ion and using a diagonal a gumen we ob ain a subsequence o { n}n which con e ges ρ-a.e. o .  De ini ion 2.1. Le Cbe a subse o Lρ. ASYMPTOTICALLY NONEXPANSIVE MAPPINGS IN MODULAR FUNCTION SPACES 9 (a) Cis said o be d-closed i o any sequence { n}nin Cwhich d-con e ges o , hen we ha e ∈C. (b) Cis d-open i Lρ Cis d-closed. (c) Cis said o be d-sequen ially compac i o each sequence { n}n he e exis s a subsequence { nk}kwhich d- con e ges o a poin in C. I is easily seen ha he amily o all d-open subse s o Lρ o m a opology on Lρ.Fu he mo e, om p oposi ion (2.2) d-sequen ially compac se s and ρ-a.e. compac se s a e iden ical. On he o he hand, e en hough dsa is ies (3) ins ead o he iangula inequali y, he usual a gumen s which p o e ha sequen ial compac ness and compac ness a e iden ical in me ic spaces hold in his se ing. We also ha e d-sequen ial compac ness and d-compac ness a e iden ical. 3. TECHNICAL LEMMAS In he sequel we assume ha ρis a con ex, σ- ini e modula unc ion sa is ying he ∆2- ype condi ion, Cis a con ex, ρ-bounded and ρ-a.e. compac subse o he modula unc ion space Lρand T:C→Cis a ρ-asymp o ically nonexpansi e mapping, i.e. he e exis s a sequence o posi i e in ege s {kn}nwhich con e ge o 1 such ha o e e y n∈Nand , g ∈Cwe ha e ρ(Tn −Tng)≤knρ( −g). Lemma 3.1. Unde he abo e assump ions, le { n}nbe a sequence o elemen s o C. Conside he unc ional Φ : C→Rde ined by Φ(g) = lim sup n→∞ ρ( n−g). Then, o any sequence {gm}min Cwhich ρ-a.e. con e ges o g∈Cwe ha e Φ(g)≤lim in m→∞ Φ(gm). P oo . Since Cis ρ-a.e. compac , he e exis s a subsequence { φ(n)}no { n}n such ha φ(n) ρ−a.e −→ ∈Cand lim n→∞ ρ( φ(n)−g) = lim sup n→∞ ρ( n−g).Hence Φ(gm) = lim sup n→∞ ρ( n−gm) ≥lim sup n→∞ ρ( φ(n)−gm) ≥lim in n→∞ ρ( φ(n)−gm). 16 T. DOMINGUEZ-BENAVIDES, M.A. KHAMSI AND S. SAMADI Fix n≥n0. The e exis s k0≥1 such ha o all k≥k0, we ha e n(k)≥n+n0 and ρ(Tn −Tn(k) ) = ρTn −Tn+(n(k)−n) =ρTn −Tn(Tn(k)−n ) ≤knρ −Tn(k)−n < kn( +η). No e ha i n ρ−a.e −→ and Sep{ n}n≥ε, hen by Lemma (1.3), we ha e ε≤lim in m→∞ lim in n→∞ ρ( n− m)≤2 lim in n→∞ ρ( n− ). Combined wi h Lemma (1.3), we ge lim in n→∞ ρ( n) = lim in n→∞ ρ( n− ) + ρ( )≥ 2+ρ( ). In pa icula , since {Tn(k) −Tn }kis ρ-a.e. con e gen o ∞−Tn as k→ ∞ and sa is ies Sep({Tn(k) −Tn }k)≥, we ge ρ(Tn − ∞)≤lim in k→∞ ρ(Tn(k) −Tn )− 2. Hence ρ( ∞−Tn )≤ +η− 2 which implies = lim sup n→∞ ρ( ∞−Tn )≤ +η− 2< . This con adic ion comple es he p oo o Theo em 4.2.  Assume ha Lρ=Lp(Ω, µ) o a σ- ini e measu e µ. I Cis a con ex, bounded and closed subse o Lp o 1 < p < ∞and T:C→Cis asymp o ically non- expansi e, i is known ha Chas a ixed poin because Lpis uni o mly con ex. Howe e he esul does no hold o p= 1 (e en o nonexpansi e mappings, see [1]). Since L1is a modula space, Theo em (4.1) implies he exis ence o ixed poin i p= 1 when Cis ρ-a.e. compac . Thus we can s a e. Co olla y 4.1. Le (Ω, µ) be as abo e, C⊂L1(Ω, µ) a con ex bounded se which is compac o he opology o local con e gence in measu e and T:C→C asymp o ically nonexpansi e. Then, Thas a ixed poin . ASYMPTOTICALLY NONEXPANSIVE MAPPINGS IN MODULAR FUNCTION SPACES 17 P oo . Unde he abo e hypo hesis ρ-a.e. compac se s and compac se s in he opology o local con e gence in measu e a e iden ical.  Re e ences [1] D.E. Alspach. A ixed poin ee nonexpansi e map. P oc. Am. Ma h. Soc., (1981), 82, 423-424. [2] S.C. Bose. Weak con e gence o he ixed poin o an asymp o ically nonexpansi e map. P oc. Am. Ma h. Soc., (1978), 68, 305-308. [3] T. Dominguez Bena ides, M.A. Khamsi, S. Samadi. Asymp o ically egula mappings in modula unc ion spaces. P ep in . [4] T.Dominguez Bena ides, M.A. Khamsi, S. Samadi. Uni o mly Lipschi zian mappings in modula unc ion spaces. P ep in . [5] K. Goebel and W.A. Ki k. A ixed poin heo em o asymp o ically nonexpansi e mappings. P oc. Am. Ma h. Soc., (1972), 35, 171-174. [6] M.A. Khamsi. Fixed poin heo y in modula unc ion spaces. Recen Ad ances on Me ic Fixed Poin Theo y. Uni e sidad de Se illa, Se illa, (1996), 31-58. [7] M.A. Khamsi, W.M. Koz lowski, S. Reich. Fixed poin heo y in modula unc ion spaces. Nonlinea Anal., (1990), 14, 935-953. [8] W.M. Kos lowski. Modula unc ion spaces. Dekke : New Yo k, Basel, (1988). [9] T.-H. Kim and H.-K. Xu. Rema ks on asymp o ically nonexpansi e mappings. Nonlinea Anal., o appea . [10] C. Ma ´ınez Ya˜nez. A ixed poin heo em on k-uni o mly o und spaces. Nonlinea Anal., (1988), 13 857-861. [11] G. Pass y. Cons uc ion o ixed poin s o asymp o ically nonexpansi e mappings. P oc. Am. Ma h. Soc., (1982), 84, 213-216. [12] H.-K. Xu. k-Uni o m o undi y and ixed poin s o mappings o asymp o ically nonexpansi e ype. To appea in Chinese. [13] H.-K. Xu. Exis ence and con e gence o ixed poin s o mappings o asymp o ically non- expansi e ype. Nonlinea Anal., (1991) 16(12), 1139-1146. [14] X.T. Yu and X. Dai. A ixed poin heo em o asymp o ically nonexpansi e mappings. J. Ma h. (PRC), (1986), 6, 255-262. 18 T. DOMINGUEZ-BENAVIDES, M.A. KHAMSI AND S. SAMADI Tomas Dominguez-Bena ides, Depa men o Ma hema ical Analysis, Uni e - si y o Se ille, P.O.Box 1160. 41080. Se ille (Spain). E-mail add ess:[email p o ec ed] Mohamed Amine Khamsi, Depa men o Ma hema ical Science, The Uni e - si y o Texas a El Paso, El Paso, TX 79968, (U.S.A). E-mail add ess:[email p o ec ed] Sedki Samadi, Depa men o Ma hema ical Analysis, Uni e si y o Se ille, P.O.Box 1160. 41080. Se ille (Spain). E-mail add ess:[email p o ec ed]