Asymptotically regular mappings in modular function spaces

ASYMPTOTICALLY REGULAR MAPPINGS IN MODULAR FUNCTION SPACES T. DOMINGUEZ-BENAVIDES, M.A. KHAMSI AND S. SAMADI 1. ABSTRACT Le ρbe a modula unc ion sa is ying a ∆2- ype condi ion and Lρ he co - esponding modula space. The main esul in his pape s a es ha i Cis a ρ-bounded and ρ-a.e sequen ially compac subse o Lρand T:C→Cis an asymp o ically egula mapping such ha lim in n→∞ |Tn|<2, whe e |S|deno es he Lipschi z cons an o S, hen Thas a ixed poin . We show ha he es ima e lim in n→∞ |Tn|<2 canno be, in gene al, imp o ed. 1991 Ma hema ics subjec classi ica ion: P ima y 46E30; Seconda y 47H09, 47H10. Key Wo ds: asymp o ically egula mappings, ixed poin , modula unc ions, Opial p ope y. 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 2. INTRODUCTION Le Xbe a me ic space. A mapping T:X→Xis said o be asymp o ically egula i lim nd(Tn+1x, Tnx) = 0 o each x∈X. This no ion was de ined by B owde and Pe yshyn [2]. The exis ence o ixed poin s o asymp o ically egula mappings has been widely s udied [3, 4, 5, 6, 7, 8, 9, 10, 11, 16, 17]. When Xis a con ex, closed, subse o a Banach space i is known [12] ha he p oblem o he exis ence o ixed poin o a nonexpansi e mapping is equi alen o he same p oblem o a nonexpansi e asymp o ically egula mapping. On he o he hand, he heo y o modula spaces was ini ia ed by Nakano [21] 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 [20] in 1959. Besides he idea o de ining a no m and conside ing pa icula Banach spaces o unc ions, ano he di ec ion is based on conside ing an abs ac ly gi en un ional de ined on a linea space o unc ions which con ols he g ow h o membe s o he space. 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 (see, o ins ance [13] and e e ences he ein). In his pape , we s udy he exis ence o ixed poin s o asymp o ically egula mapping de ined om a ρ-bounded and ρ-a.e sequen ially compac se Co a modula space Lρ in o C. We ac ually p o e ha Thas a ixed poin i lim in n|Tn|<2, |T|being he exac Lipschi z cons an o T. I is wo hed o no ice he simplici y o his s a emen in compa ison wi h simila esul s in Banach spaces, whe e a di e en uppe bound o lim in n|Tn|mus be conside ed o each space. E en in Hilbe spaces (see [1, chap e IX]) a bes es ima e is unknown. We also gi e an example showing ha his esul can no be, in gene al, imp o ed. 3. PRELIMINARIES We s a by ecalling some basic concep s and ac s o modula spaces as o - mula ed by Kozlowski. Fo mo e de ails he eade is e e ed o [13], [14], [15] and [19]. 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 REGULAR MAPPINGS IN MODULAR FUNCTION SPACES 3 Ω = SKn. In o he wo ds, he amily Pplays he ole o he δ- ing o subse s o ini e measu e. 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 → < such 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 3.1. 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(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)|ω∈Ω}. This will enable us o de ine ρ(α, E) o se s Eno in P; o he sake o simplici y, we w i e ρ( ) ins ead o ρ( , Ω). De ini ion 3.2. 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. 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 . 4 T. DOMINGUEZ-BENAVIDES, M.A. KHAMSI AND S. SAMADI I is easy o see ha he unc ional ρ:M → [0,∞] is a modula because i sa is ies he ollowing p ope ies: (i) ρ( ) = 0 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≥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 Lρgi en by Lρ={ ∈ M;ρ(λ )→0 as λ→0}. The modula space Lρcan be equipped wi h an F-no m de ined by || ||ρ= in nα > 0; ρµ α¶≤αo. When ρis con ex he o mula || ||ρ= in nα > 0; ρµ α¶≤1o De ini ion 3.3. (a) The sequence { n} ⊂ Lρis said o be ρ-con e gen o ∈Lρi ρ( n− )→ 0 as n→ ∞, (a’) The sequence { n} ⊂ Lρis said o be ρ-a.e con e gen o ∈Lρi he se {ω∈Ω; n(ω)6→ (ω)}is ρ-null. (b) The sequence { n} ⊂ Lρis said o be ρ-Cauchy i ρ( n− m)→0 as n and mgo o ∞, (b’) The sequence { n} ⊂ Lρis said o be ρ-a.e Cauchy i {ω∈Ω; { n(ω)}is no a Cauchy sequence }is ρ-null. (c) A subse Co Lρis called ρ-closed i he ρ-limi o a ρ-con e gen sequence o Calways belongs o C. (c’) A subse Co Lρis called ρ-a.e sequen ially closed i he ρ-a.e limi o a ρ-a.e con e gen sequence o Calways belongs o C. ASYMPTOTICALLY REGULAR MAPPINGS IN MODULAR FUNCTION SPACES 5 (d) A subse Co Lρis called ρ-sequen ially compac i e e y sequence in C has a ρ-con e gen subsequence in C. (d’) A subse Co Lρis called ρ-a.e sequen ially compac i e e y sequence in Chas a ρ-a.e con e gen subsequence in C. (e) A subse Co Lρis called ρ-bounded i δρ(C) = sup{ρ( −g); , g ∈C}<∞. De ini ion 3.4. Le ρbe a unc ion modula , we de ine a g ow h un ion ωρ: [0,∞]→[0,∞] by ωρ( ) = sup ½ρ( ) ρ( ); ∈Lρ,0< ρ( )<∞¾, ≥0. The ollowing echnical esul [13] is undamen al o his wo k. Lemma 3.1. Le { n}nbe a sequence in Eρsuch ha n ρ−a.e → ∈Eρand he e exis s k > 1 such ha supnρ(k( n− )) <∞. Then, we ha e lim in n→∞ ρ( n−g) = lim in n→∞ ρ( n− ) + ρ( −g) o all g∈Eρ and, he e o e, lim in n→∞ ρ( n− )≤lim in n→∞ ρ( n−g) o all g∈Eρ. ¿F om his lemma a uni o m Opial p ope y- ype o Eρcan be de i ed. Re- call [22] ha a Banach space is said o sa is y he uni o m Opial p ope y wi h espec o an a bi a y opology τi o e e y c > 0 he e exis s > 0 such ha lim in n→∞ ||xn+x|| ≥ 1 + , i {xn}nis a τ-null sequence wi h lim in n→∞ ||xn|| ≥ 1 and ||x|| ≥ c. F om lemma 3.1 i is clea ha lim in n→∞ ρ( n+ )≥1+ci ρ( )≥cand { n}nis a ρ-a.e null sequence which sa is ies lim in n→∞ ρ( n)≥1 and he e exis s K > 1 such ha supnρ¡K( n− )¢<∞. 6 T. DOMINGUEZ-BENAVIDES, M.A. KHAMSI AND S. SAMADI 4. SOME TECHNICAL RESULTS De ini ion 4.1. Le ρbe a unc ion modula . We say ha ρsa is ies he ∆2- ype condi ion i he e exis s K > 0 such ha ρ(2 )≤Kρ( ) o all ∈Lρ. As examples o con ex un ion modula wi h ∆2- ype condi ion we men ion, he usual lpspaces and unc ion modula o O licz spaces, whe e he measu e space (Ω,Σ, µ) is σ- ini e, he measu e µis a omless and in ini e, and he O licz unc ion ψis con ex sa is ying ∆2- ype condi ion, i.e. lim sup u→∞ ψ(2u) ψ(u)<∞and lim sup u→0 ψ(2u) ψ(u)<∞. No e ha he ∆2- ype condi ion implies he ∆2-condi ion. Fo his condi ion we e e o [15] and [19]. In he sequel, we will assume ha ρis con ex and sa is ies he ∆2-condi ion. In his case we ha e Lρ=Eρ. The ollowing lemma can be easily p o ed. Lemma 4.1. The g ow h un ion ωρhas he ollowing p ope ies: (1) ωρ( )<∞,∀ ∈[0,∞) (2) ωρ: [0,∞)→[0,∞) is a con ex, s ic ly inc easing unc ion. So, i is con inuous. (3) ωρ(αβ)≤ωρ(α)ωρ(β); ∀α, β ∈[0,∞) (4) ω−1 ρ(α)ω−1 ρ(β)≤ω−1 ρ(αβ);∀α, β ∈[0,∞), whe e ω−1 ρis he in e se unc- ion o ωρ. The ollowing is a echnical lemma which will be needed because o lack o he iangula inequali y. Lemma 4.2. Le { 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). ASYMPTOTICALLY REGULAR MAPPINGS IN MODULAR FUNCTION SPACES 7 P oo . By p ope y (iii) o he modula ρ, we ha e ρ( n+gn)≤ρµ n 1−ε¶+ρ³gn ε´,∀ε∈(0,1) Thus, ρ( n+gn)≤ωµ1 1−ε¶ρ( n) + ωµ1 ε¶ρ(gn) and lim sup n→∞ ρ( n+gn)≤ωµ1 1−ε¶lim sup n→∞ ρ( n) Since εis a bi a y and ωµ1 1−ε¶→1 as ε→0+ we ob ain lim sup n→∞ ρ( n+gn)≤lim sup n→∞ ρ( n). Fu he mo e, he same a gumen p o es lim sup n→∞ ρ( n) = lim sup n→∞ ρ( n+gn−gn) ≤lim sup n→∞ ρ( n+gn). 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 4.3. Le Lρbe a unc ion modula space sa is ying he ∆2- ype condi- ion. Then || ||ρ≤1 ω−1³1 ρ( )´ P oo . Assume, α < || ||ρ.We ha e 1 < ρ( /α) which implies 1 ρ( )< ω µ1 α¶ 8 T. DOMINGUEZ-BENAVIDES, M.A. KHAMSI AND S. SAMADI and so ω−1µ1 ρ( )¶<1 α. Le ing α→ || ||ρ −, we ob ain || ||ρ≤1 ω−1³1 ρ( )´. 5. MAIN RESULT In his sec ion we p esen he main esul o his wo k. Indeed, we will p o e a ixed poin heo em o asymp o ically egula mappings. Ce ainly, one can also conside mappings which a e asymp o ically egula wi h espec o he F-no m induced by he modula unc ion. We should like o men ion ha , gene ally speaking, he e is no na u al ela ion be ween hese wo kinds o asymp o ically egula ness. Indeed all esul s exp essed in e ms o modula s a e mo e con e- nien in he sense ha hei assump ions a e much easie o e i y. Le Ca subse o Lρand T:C→C, we deno e by |T| he exac Lipschi z cons an o T, i.e. |T|= sup ½ρ(T −Tg) ρ( −g): 6=g, , g ∈C¾and s(T) = lim in n→∞ |Tn|. Theo em 5.1. Le ρbe a con ex modula unc ion sa is ying he ∆2- ype con- di ion, Caρ-bounded, ρ−a.e sequen ially compac subse o Lρ. Le T:C→C be an asymp o ically egula mapping such ha s(T)<2. Then, T has a ixed poin . P oo . Choose a sequence {nk}o posi i e in ege s such ha s(T) = lim k→∞ |Tnk|= lim in n→∞ |Tn|and de ine a unc ion on C by ( ) = in { > 0 : ∃g∈Csuch ha lim in k→∞ ρ( −Tnkg)≤ }. Since s(T)<2, he e exis s b∈(1,2) such ha s(T)< b < 2.Le ε∈(0,1) such ha ωρµ1 ε¶<1 b−1and choose γ∈(0,1) such ha ωρµ1 ε¶<γ b−1.Since γ+ (1 −b)ωρ¡1 ε¢>0, we can choose δ∈(0,1) such ha ASYMPTOTICALLY REGULAR MAPPINGS IN MODULAR FUNCTION SPACES 9 δ < γ + (1 −b)ωρ¡1 ε¢. Now, we choose a numbe µ∈(0,1) such ha µ < min (δ ωρ¡1 1−ε¢,γ−δ+ωρ¡1 ε¢(1 −b) ωρ¡1 ε¢b) Finally,deno e α= max (1 + µ−δ ωρ¡1 1−ε¢, b(1 + µ)−γ−δ ωρ¡1 ε¢). Then 0 < α < 1.Since γ < 1, we can ind a posi i e in ege k0such ha |Tnk0|< b and ρ( −Tnk0 )> γ ( ). Since µ > 0, we can also ind g∈Csuch ha lim in k→∞ ρ( −Tnkg)≤ ( )(1 + µ). Conside a subsequence {nk0}o {nk}such ha {Tnk0g}is ρ-a.e con e gen in C, say o h, and lim k0→∞ ρ( −Tnk0g) = lim in k→∞ ρ( −Tnkg). The e o e, using Lemma 4.2 and he asymp o ic egula i y o Twe ha e lim sup k0→∞ ρ(Tnk0 −Tnk0g)≤ |Tnk0|lim sup k0→∞ ρ( −Tnk0−nk0g) =|Tnk0|lim sup k0→∞ ρ( −Tnk0g+Tnk0g−Tnk0−nk0g) =|Tnk0|lim sup k0→∞ ρ( −Tnk0g). We spli he p oo in o wo cases: Case 1. Assume ha ρ( −h)≥δ ωρ¡1 1−ε¢ ( )