Fixed point theorems for multivalued mappings in modular function spaces

FIXED POINT THEOREMS FOR MULTIVALUED MAPPINGS IN MODULAR FUNCTION SPACES* S. DHOMPONGSA, T. DOMINGUEZ BENAVIDES, A. KAEWCHAROEN, AND B. PANYANAK Abs ac . The pu pose o his pape is o s udy he exis ence o ixed poin s o mul i alued nonexpansi e mappings in modula unc ion spaces. We apply ou main esul o ob ain ixed poin heo ems o mul i alued mappings in he Banach spaces L1and l1. 1. In oduc ion The heo y o modula spaces was ini ia ed by Nakano [?] in 1950 in connec- ion wi h he heo y o o de spaces and ede ined and gene alized by Musielak and O licz [?] in 1959. 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 and sol ed in modula spaces (see, o ins ance, [?,?,?,?]). In pa icula , some ixed poin heo ems o (single alued) nonexpansi e mappings in modula unc ion spaces a e gi en in [?]. In 1969, Nadle [?] es ablished he mul i alued e sion o Banach’s con ac ion p inciple in me ic spaces. Since hen he me ic ixed poin heo y o mul i alued mappings has been apidly de eloped and many o pape s ha e appea ed p o ing he exis ence o ixed poin s o mul i alued nonexpansi e mappings in special classes o Banach spaces (see, o ins ance, [?,?,?,?]). In his pape , we s udy simila p oblems in he se ing o modula unc ion spaces. Namely, we p o e ha e e y ρ−con ac ion T:C→Fρ(C) has a ixed poin whe e ρis a con ex unc ion modula sa is y- ing he ∆2− ype condi ion and Cis a nonemp y ρ−bounded ρ−closed subse o Lρ.By using his esul , we can asse he exis ence o ixed poin s o mul i al- ued ρ−nonexpansi e mappings. Finally, we apply ou main esul o ob ain ixed poin heo ems in he Banach space L1( esp. l1) o mul i alued mappings whose domains a e compac in he opology o he con e gence locally in measu e ( esp. w∗− opology). Key wo ds and ph ases : Mul i alued mappings, ixed poin s, Modula unc ion spaces. * Suppo ed by Thailand Resea ch Fund unde g an BRG4780013. The second au ho was pa ially suppo ed by DGES, G an D.G.E.S. REF. PBMF2003-03893-C02-C01 and Jun a de Andalucia, G an FQM-127. The hi d and ou h au ho s we e suppo ed by he Royal Golden Jubilee p og am unde g an PHD/0250/2545 and PHD/0251/2545, espec i ely. 1 2 S. DHOMPONGSA, T. DOMINGUEZ BENAVIDES, A. KAEWCHAROEN, AND B. PANYANAK 2. P elimina ies We s a by ecalling some basic concep abou modula unc ion spaces. Fo mo e de ails he eade is e e ed o [?,?]. 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 Ω, 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 Ω = SKn( o ins ance, Pcan be he se s o ini e measu e in a σ− ini e measu e space). By Ewe deno e he linea space o all simple unc ions wi h suppo 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. Le us ecall ha a se unc ion µ: Σ →[0,∞] is called a σ−subaddi i e measu e i µ(∅)=0, µ(A)≤µ(B) o any A⊂Band µ(SAn)≤Pµ(An) o any sequence o se s {An} ⊂ Σ.By χAwe deno e he cha ac e is ic unc ion o he se A. De ini ion 2.1. A unc ional ρ:E × Σ→[0,∞] is called a unc ion modula 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(αχA, 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 > 0, ρ(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∈Ωª. De ini ion 2.2. A se Eis said o be ρ−null i ρ(α, E) = 0 o e e y 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. No e ha a coun able union o ρ−null se s is s ill ρ−null. In he sequel we will iden i y se s Aand Bwhose symme ic di e ence A∆Bis ρ−null, simila ly we will iden i y measu able unc ions which di e only on a ρ−null se . In he abo e condi ion, we de ine he unc ion ρ:M → [0,∞] by ρ( ) = ρ( , Ω).We know om [?] ha ρsa is ies he ollowing p ope ies : (i) ρ( ) = 0 i and only i = 0 ρ−a.e. (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, β ≥0 and , g ∈ M. In addi ion, i he ollowing p ope y is sa is ied FIXED POINT THEOREMS FOR MULTIVALUED MAPPINGS IN MODULAR FUNCTION SPACES3 (iii)’ ρ(α +βg)≤αρ( ) + βρ(g) i α+β= 1, α, β ≥0 and , g ∈ M, we say ha ρis con ex modula . The modula ρde ines a co esponding modula space Lρ,which is gi en by Lρ={ ∈ M :ρ(λ )→0 as λ→0}. In gene al he modula ρis no subaddi i e and he e o e does no beha e as a no m o a dis ance. Bu one can associa e o a modula an F−no m. Recall ha a unc ional k·k:X→[0,∞] de ines an F−no m i and only i (1) kxk= 0 i and only i x= 0, (2) kαxk=kxkwhene e |α|= 1, (3) kx+yk ≤ kxk+kyk, (4) kαnxn−αxk → 0 i αn→αand kxn−xk → 0. The modula space Lρcan be equipped wi h an F−no m de ined by k kρ= in nα > 0 : ρµ α¶≤αo. We know om [?] ha he linea space (Lρ,k·kρ) is a comple e me ic space. I ρis con ex he o mula k kρ= in nα > 0 : ρµ α¶≤1o de ines a no m which is equen ly called he Luxembu g no m. The o mula k ka= in ½1 k(1 + ρ(k )) : k > 0¾ de ines a di e en no m which is called Amemiya no m. Mo eo e , k · kρand k · ka a e equi alen no ms. We can also conside he space Eρ={ ∈ M :ρ(α , ·) is o de con inuous o all α > 0}. De ini ion 2.3. A unc ion modula ρis said o sa is y he ∆2−condi ion i sup n≥1 ρ(2 n, Dk)→0 as k→ ∞ whene e { n} ⊂ M, Dk∈Σ dec eases o ∅and sup n≥1 ρ( n, Dk)→0 as k→ ∞. I is known ha he ∆2−condi ion is equi alen o Eρ=Lρ. De ini ion 2.4. A unc ion modula ρis 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, he ∆2− ype condi ion and ∆2−condi ion a e no equi alen , e en hough i is ob ious ha he ∆2− ype condi ion implies he ∆2−condi ion. De ini ion 2.5. Le Lρbe a modula space. 4 S. DHOMPONGSA, T. DOMINGUEZ BENAVIDES, A. KAEWCHAROEN, AND B. PANYANAK (1) The sequence { n} ⊂ Lρis said o be ρ−con e gen o ∈Lρi ρ( n− )→ 0 as n→ ∞. (2) The sequence { n} ⊂ Lρis said o be ρ−a.e. con e gen o ∈Lρi he se {ω∈Ω : n(ω)9 (ω)}is ρ−null. (3) A subse Co Lρis called ρ−closed i he ρ−limi o a ρ−con e gen se- quence o Calways belongs o C. (4) 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. (5) A subse Co Lρis called ρ−compac i e e y sequence in Chas a ρ−con e gen subsequence in 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 diamρ(C) = sup{ρ( −g) : , g ∈C}<∞. We know by [?] ha unde he ∆2−condi ion he no m con e gence and mod- ula 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 2.6. Le ρbe as abo e. We de ine a g ow h unc ion ωby ω( ) = sup nρ( ) ρ( ): ∈Lρ,0< ρ( )<∞o o all 0 ≤ < ∞. The ollowing p ope ies o he g ow h unc ion can be ound in [?]. Lemma 2.7. Le ρbe as abo e. Then he g ow h unc ion ωhas he ollowing p ope ies : (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 2.8 (T. Dominguez Bena ides e al. [?]).Le ρbe as abo e. Then k kρ≤1 ω−1(1/ρ( )) whene e ∈Lρ {0}. The ollowing lemma is a echnical lemma which will be need because o lack o he iangula inequali y. FIXED POINT THEOREMS FOR MULTIVALUED MAPPINGS IN MODULAR FUNCTION SPACES5 Lemma 2.9 (T. Dominguez Bena ides e al. [?]).Le ρbe as abo e, { n}and {gn}be wo sequences in Lρ.Then lim n→∞ ρ(gn) = 0 =⇒lim sup n→∞ ρ( n+gn) = lim sup n→∞ ρ( n) and lim n→∞ ρ(gn) = 0 =⇒lim in n→∞ ρ( n+gn) = lim in n→∞ ρ( n) In he same way as he Hausdo dis ance de ined on he amily o bounded closed subse s o a me ic space, we can de ine he analogue o he Hausdo dis ance o modula unc ion spaces. We will call ρ−Hausdo dis ance e en hough i is no a me ic. De ini ion 2.10. Le Cbe a nonemp y subse o Lρ.We shall deno e by Fρ(C) he amily o nonemp y ρ−closed subse s o Cand by Kρ(C) he amily o nonemp y ρ−compac subse s o C. Le Hρ(·,·) be he ρ−Hausdo dis ance on Fρ(Lρ),i.e., Hρ(A, B) = max nsup ∈A dis ρ( , B),sup g∈B dis ρ(g, A)o, A, B ∈Fρ(Lρ). whe e dis ρ( , B) = in {ρ( −g) : g∈B}is he ρ−dis ance be ween and B. A mul i alued mapping T:C→Fρ(Lρ) is said o be a ρ−con ac ion i he e exis s a cons an k∈[0,1) such ha (2.1) Hρ(T , T g)≤kρ( −g), , g ∈C. I (??) is alid when k= 1, hen Tis called ρ−nonexpansi e. A unc ion ∈Cis called a ixed poin o a mul i alued mapping Ti ∈T . 3. Main esul s We begin s a ing he Banach Con ac ion P inciple o mul i alued mappings in modula unc ion spaces. Theo em 3.1. Le ρbe a con ex unc ion modula sa is ying he ∆2− ype con- di ion, Ca nonemp y ρ−bounded ρ−closed subse o Lρ,and T:C→Fρ(C)a ρ−con ac ion mapping, i.e., he e exis s a cons an k∈[0,1) such ha (3.1) Hρ(T , T g)≤kρ( −g), , g ∈C. Then Thas a ixed poin . P oo . Le 0∈Cand α∈(k, 1).Since T 0is nonemp y, he e exis s 1∈T 0 such ha ρ( 0− 1)>0 (o he wise 0is a ixed poin o T). In iew o (??), we ha e dis ρ( 1, T 1)≤Hρ(T 0, T 1)≤kρ( 0− 1)< αρ( 0− 1). Since dis ρ( 1, T 1) = in {ρ( 1−g) : g∈T 1},i ollows ha he e exis s 2∈T 1 such ha ρ( 1− 2)< αρ( 0− 1). 6 S. DHOMPONGSA, T. DOMINGUEZ BENAVIDES, A. KAEWCHAROEN, AND B. PANYANAK Simila ly, he e exis s 3∈T 2such ha ρ( 2− 3)< αρ( 1− 2). Con inuing in his way, he e exis s a sequence { n}in Csa is ying n+1 ∈T n and ρ( n− n+1)< αρ( n−1− n) < α2(ρ( n−2− n−1)) < ... < αn−1(ρ( 1− 2)) < αn(ρ( 0− 1)) ≤αndiamρ(C), Le M= diamρ(C), hen 1 αnM<1 ρ( n− n+1). By Lemma ??, we ha e ³ω−1¡1 α¢´n ω−1³1 M´< ω−1³1 ρ( n− n+1)´, I ollows ha 1 ω−1³1 ρ( n− n+1)´<1 ³ω−1¡1 α¢´n ω−1³1 M´. By Lemma ??, we ob ain k n− n+1kρ<³1 ω−1¡1 α¢´n·1 ω−1³1 M´. Since ω−1is s ic ly inc easing, we ha e 1 ω−1¡1 α¢<1.This implies ha { n}is a Cauchy sequence in (Lρ,k · kρ).Since (Lρ,k·kρ) is a comple e me ic space, he e exis s ∈Lρsuch ha { n}is k · kρ−con e gen o . Since unde he ∆2− ype condi ion, no m con e gence and modula con e gence a e iden ical, { n} is ρ−con e gen o and ∈Cbecause Cis ρ−closed. Since n∈T n−1, we ha e (3.2) dis ρ( n, T )≤Hρ(T n−1, T )≤kρ( n−1− )−→ 0. We obse e ha , o each n, he e exis s gn∈T such ha (3.3) ρ( n−gn)≤dis ρ( n, T ) + 1 n. Thus, (??) and (??) imply ha lim n→∞ ρ( n−gn) = 0.By Lemma ??, lim sup n→∞ ρ(gn− ) = lim sup n→∞ ρ(gn− n+ n− ) = lim sup n→∞ ρ( n− ) = 0. Since T is ρ−closed, we can conclude ha ∈T . ¤ FIXED POINT THEOREMS FOR MULTIVALUED MAPPINGS IN MODULAR FUNCTION SPACES7 The ollowing esul s will be e y use ul in he p oo o ou main heo em. Theo em 3.2 (M. A. Khamsi [?]).Le { n} ⊂ Lρbe ρ−a.e. con e gen o 0. Assume he e exis s k > 1such ha sup n≥1 ρ(k n) = M < ∞. Le g∈Eρ, hen we ha e lim in n→∞ ρ( n+g) = lim in n→∞ ρ( n) + ρ(g). The ollowing lemma gua an ee ha e e y nonemp y ρ−compac subse o Lρ a ains a nea es poin . Lemma 3.3. Le ρbe a con ex unc ion modula sa is ying he ∆2− ype condi ion, ∈Lρ,and Ka nonemp y ρ−compac subse o Lρ.Then he e exis s g0∈Ksuch ha ρ( −g0) = dis ρ( , K). P oo . Le m= dis ρ( , K).Fo each n∈N, he e exis s gn∈Ksuch ha m−1 n≤ρ( −gn)≤m+1 n. By he ρ−compac ness o K, we can assume, by passing h ough a subsequence, ha gn ρ −→ g0∈K. By Lemma ??, we ob ain m= lim sup n→∞ ρ(gn− ) = lim sup n→∞ ρ(gn−g0+g0− ) = lim sup n→∞ ρ(g0− ) =ρ(g0− ). ¤ We can now s a e ou main heo em. Theo em 3.4. Le ρbe a con ex unc ion modula sa is ying he ∆2− ype condi- ion, Ca nonemp y ρ−a.e. compac ρ−bounded con ex subse o Lρ,and T:C→ Kρ(C)aρ−nonexpansi e mapping. Then Thas a ixed poin . P oo . Fix 0∈C. Fo each n∈N, he ρ−con ac ion Tn:C→Fρ(C) is de ined by Tn( ) = 1 n 0+ (1 −1 n)T , ∈C. By Theo em ??, we can conclude ha Tnhas a ixed poin , say n.I is easy o see ha dis ρ( n, T n)≤1 ndiamρ(C)−→ 0. 8 S. DHOMPONGSA, T. DOMINGUEZ BENAVIDES, A. KAEWCHAROEN, AND B. PANYANAK Because o ρ−a.e. compac ness o C, we can assume, by passing h ough a subse- quence, ha n ρ−a.e. −→ o some ∈C. By Lemma ??, o each n∈N, he e exis s gn∈T nand hn∈T such ha ρ( n−gn) = dis ρ( n, T n) and ρ(gn−hn) = dis ρ(gn, T )≤Hρ(T n, T )≤ρ( n− ). Because o ρ−compac ness o T , we can assume, by passing h ough a subsequence, ha hn ρ −→ h∈T . Since ρsa is ies he ∆2− ype condi ion, he e exis s K > 0 such ha ρ(2( n− )) ≤Kρ( n− ) o all n∈N. This implies ha sup n≥1 ρ(2( n− )) ≤Ksup n≥1 ρ( n− )<∞. By Theo em ?? and Lemma ??, we ob ain lim in n→∞ ρ( n− ) + ρ( −h) = lim in n→∞ ρ( n− + −h) = lim in n→∞ ρ( n−h) = lim in n→∞ ρ( n−gn+gn−hn+hn−h) = lim in n→∞ ρ(gn−hn) ≤lim in n→∞ ρ( n− ). I ollows ha ρ( −h) = 0 and hen we ha e =h∈T . ¤ Conside he space Lp(Ω, µ) o a σ− ini e measu e µwi h he usual no m. Le Cbe a bounded closed con ex subse o Lp o 1 < p < ∞and T:C→K(C) a mul i alued nonexpansi e mapping. Because o uni o m con exi y o Lp,i is known ha Thas a ixed poin . Fo p= 1, T can ail o ha e a ixed poin e en in he single alued case o a weakly compac con ex se C(see [?]). Howe e , since L1is a modula space whe e ρ( ) = RΩ| |dµ =k k o all ∈L1,Theo em ?? implies he exis ence o a ixed poin when we de ine mappings on a ρ−a.e. compac ρ−bounded con ex subse o L1.Thus we can s a e : Co olla y 3.5. Le (Ω, µ)be as abo e, C⊂L1(Ω, µ)a nonemp y bounded con ex se which is compac o he opology o he con e gence locally in measu e, and T:C→K(C)a nonexpansi e mapping. Then Thas a ixed poin . P oo . Unde he abo e hypo hesis ρ−a.e. compac se s and compac se s in he opology o he con e gence locally in measu e a e iden ical (see [?]). Consequen ly, Theo em ?? can be applied o ob ain a ixed poin o T. ¤ In he case o he space l1we also can ob ain a bounded closed con ex se C and a nonexpansi e mapping T:C→Cwhich is ixed poin ee. Indeed, conside FIXED POINT THEOREMS FOR MULTIVALUED MAPPINGS IN MODULAR FUNCTION SPACES9 he ollowing easy and well known example : Le C=n{xn} ∈ l1: 0 ≤xn≤1 and ∞ X n=1 xn= 1o. De ine a nonexpansi e mapping T:C→Cby T(x) = (0, x1, x2, x3, ...) whe e x={xn}. Then Tis a ixed poin ee. Howe e , i we conside Lρ=l1whe e ρ(x) = kxk,∀x∈l1.Then ρ−a.e. con e gence and w∗−con e gence a e iden ical on bounded subse s o l1(see [?]). This ac leads us o ob ain he ollowing co ol- la y : Co olla y 3.6. Le Cbe a nonemp y w∗−compac con ex subse o l1and T:C→ K(C)a nonexpansi e mapping. Then Thas a ixed poin . P oo . By he abo e a gumen , we know ha ρ−a.e. compac bounded se s and w∗−compac se s a e iden ical. Then we can apply Theo em ?? o asse he exis ence o a ixed poin o T. ¤ In ac ?? and ?? a e consequences o a gene al esul : Assume ha Xis a linea no med space and τis a Hausdo opology on X. We say ha Xsa is ies he s ic τ-Opial p ope y i lim sup n→∞ kxn−xk<lim sup n→∞ kxn−yk o each sequence {xn}in Xwhich con e ges o x o he opology τand each y6=x. Following he same a gumen as in [?] i is easy o p o e he ollowing heo em: Theo em 3.7. Le Xbe a Banach space, Ca con ex bounded sequen ially τ- compac subse o X, and T:C→K(C)a nonexpansi e mapping. I Xsa is ies he s ic τ-Opial p ope y, hen Thas a ixed poin . When Xis a modula unc ion space equipped wi h ei he Luxembu g o Amemiya no m, we can conside he opology τo con e gence ρ-a.e. In his case, ?? yields o he ollowing: Theo em 3.8. Le ρbe a con ex unc ion modula sa is ying he ∆2− ype condi- ion. Assume ha Lρis equipped ei he wi h Luxembu g o Amemiya no m. Le C be a nonemp y ρ−a.e. compac ρ−bounded con ex subse o Lρ,and T:C→K(C) a nonexpansi e mapping. Then Thas a ixed poin . P oo . F om [?] (Theo em 4.1 and 4.3), Xsa is ies he uni o m Opial p ope y wi h espec o he opology o ρ-a.e. con e gence. Since ρ-a.e. compac se s and ρ-a.e. sequen ially compac se s a e iden ical o his opology (see [?]), we can deduce he esul om ?? ¤