Uniformly Lipschitzian mappings in modular function spaces

UNIFORMLY LIPSCHITZIAN MAPPINGS IN MODULAR FUNCTION SPACES T. DOMINGUEZ BENAVIDES, M.A. KHAMSI AND S. SAMADI ABSTRACT Le ρbe a con ex modula unc ion sa is ying a ∆2- ype condi ion and Lρ he co esponding modula space. Assume ha Cis a ρ-bounded and ρ-a.e compac subse o Lρand T:C→Cis a k-uni o mly Lipschi zian mapping. We p o e ha Thas a ixed poin i k < (˜ N(Lρ))−1/2whe e ˜ N(Lρ) is a geome ical coe i- cien o no mal s uc u e. We also show ha ˜ N(Lρ)<1 in modula O licz spaces o uni o mly con ex O licz unc ions. 1991 Ma hema ics subjec classi ica ion: P ima y 46E30; Seconda y 47H09, 47H10. Key Wo ds: uni o mly Lipschi zian mappings, ixed poin , modula unc ions, uni o m no mal s uc u e, uni o m con ex O licz unc ion, modulus o con exi 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 INTRODUCTION The heo y o modula spaces was ini ia ed by Nakano [14] 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 [13] in 1959. De ining a no m, pa icula Banach spaces o unc ions can be conside ed. Me ic ixed heo y o hese Banach spaces o unc ions has been widely s udied (see, o ins ance, [15]). Ano he di ec ion is based on conside ing an abs ac ly gi en unc ional which con ols he g ow h o he unc ions. E en hough a me ic is no de ined, many p oblems in ixed poin heo y o nonexpansi e mappings can be e o mula ed in modula spaces (see, o ins ance, [8] and e e ences he ein). In his pape , we s udy he exis ence o ixed poin s o a mo e gene al class o mappings: uni o mly Lipschi zian mappings. Fixed poin heo ems o his class o mappings in Banach spaces ha e been s udied in [3,4] and in me ic spaces in [11,12] ( o u he in o ma ion abou his subjec , see [2, chap e VIII] and e e ences he ein). The main ool in ou app oach is he coe icien o no mal s uc u e ˜ N(Lρ). We p o e ha unde sui able condi ions a k-uni o mly Lipschi zian mapping has a ixed poin i k < (˜ N(Lρ))−1/2.In he las sec ion we show a class o modula spaces whe e ˜ N(Lρ)<1 and so, he abo e heo em can be success ully applied. 1. PRELIMINARIES We s a by eco ding a b ie collec ion o 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 [7], [8], [10] and [13]. 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 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. UNIFORMLY LIPSCHITZIAN MAPPINGS IN MODULAR FUNCTION SPACES 3 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)|ω∈Ω}. A se Eis said o be ρ-null i ρ(α, E) = 0 o e e y α > 0.Fo he sake o simplici y we w i e ρ( ) ins ead o ρ( , Ω). 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 Lρ gi en by Lρ={ ∈ M;ρ(λ )→0 as λ→0}. 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}. 4 T. DOMINGUEZ BENAVIDES, M.A. KHAMSI AND S. SAMADI 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 (see [10]) ha Eρ=Lρwhen ρsa is ies he ∆2-condi ion. When ρis con ex, he o mula || ||ρ= in nα > 0; ρ α≤1o de inies a no m in he modula space Lρwhich is equen ly called he Luxem- bu g no m. De ini ion 1.2. (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 sequen ially 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 sequen ially 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}<∞. Le Bbe a bounded subse o Lρ.We de ine he ρ-ball o cen e ∈Lρand adius > 0 by B( , ) = {g∈Lρ, ρ(g− )≤ }.We will deno e ( , B) = sup{ρ( − g), g ∈B}, δ(B) = sup{ ( , B), ∈B}, R(B) = in { ( , B), ∈B}.We de ine he admissible hull o Bas he in e sec ion o all ρ-ball con aining B, i.e: ad(B) = {A:B⊂A⊂Lρ,whe e Ais a ρ-ball}. UNIFORMLY LIPSCHITZIAN MAPPINGS IN MODULAR FUNCTION SPACES 5 Bis said admissible i ad(B) = B. We de ine he no mal s uc u e coe icien ˜ N(Lρ) o Lρby ˜ N(Lρ) = sup R(B) δ(B), B is admissible, ρ-bounded and ρ-a.e sequen ially compac . The use ul ollowing p oposi ion is easily seen: P oposi ion 1.1. Le Bbe a ρ-bounded subse o Lρand ∈Lρ.Then (1) ( , ad(B)) = ( , B). (2) δ(ad(B)) = δ(B). 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.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. Assume ha ρis con ex and sa is ies he ∆2- ype condi ion. We de ine a g ow h unc ion ωby ω( ) = sup ρ( ) ρ( ),0< ρ( )<∞ o all 0 ≤ < ∞. The ollowing p ope ies o he g ow h unc ion can be easily seen. Lemma 1.1. Le ρbe a con ex unc ion modula sa is ying he ∆2- ype condi ion. Then he g ow h un 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. 6 T. DOMINGUEZ BENAVIDES, M.A. KHAMSI AND S. SAMADI Lemma 1.2. [5] 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 ollowing lemma can be ound in [7]. Lemma 1.3. Le ρbe a unc ion modula sa is ying he ∆2-condi ion and { n}n be a sequence in Lρsuch ha n ρ−a.e → ∈Lρand he e exis s k > 1such ha supnρ(k( n− )) <∞. Then, lim in n→∞ ρ( n−g) = lim in n→∞ ρ( n− ) + ρ( −g) o all g∈Lρ. Lemma 1.4. Le ρbe a modula unc ion sa is ying he ∆2- ype condi ion. Le B be a ρ-a.e sequen ially closed and ρ-bounded subse o Lρ. Le {gn}nbe a sequence in Bsuch ha gn ρ−a.e →g. Then, (1) ρ(g)≤lim in n→∞ ρ(gn). (2) B(0, )∩Bis ρ-a.e sequen ially closed. (3) ad(A)∩Bis ρ-a.e sequen ially closed, o all A⊂Lρ. P oo . Condi ion (1) is a s aigh o wa d consequence o Lemma 1.3 applied o he sequence gn ρ−a.e →gand he null unc ion. Condi ion (2) and (3) can be easily deduced om (1). 2. FIXED POINT FOR UNIFORMLY LIPSCHITZIAN MAPPINGS The ollowing lemma is he key o ou ixed poin esul . Lemma 2.1. Le ρbe a modula unc ion sa is ying he ∆2- ype condi ion and Baρ-bounded and ρ-a.e sequen ially compac subse o Lρ. Le { n}nand {gn}n be sequences in B. Then, he e exis s g∈ ∩∞ n=1ad(gj, j ≥n)∩Bsuch ha lim sup n→∞ ρ(g− n)≤lim sup j→∞ lim sup n→∞ ρ(gj− n) P oo . Le { n}nand {gn}nbe sequences in B. We de ine θ(h) = lim supn→∞ ρ(h− n) o all h∈B. Since Bis ρ-sequen ially compac and ρ-bounded, he e ex- is a subsequence {gφ(n)}n⊂ {gn}nsuch ha gφ(n) ρ−a.e →gand a subsequence { ψ(n)}n⊂ { n}nsuch ha limn→∞ ρ( ψ(n)−g) = lim supn→∞ ρ( n−g) and UNIFORMLY LIPSCHITZIAN MAPPINGS IN MODULAR FUNCTION SPACES 7 ψ(n) ρ−a.e → ∈B. Since gφ(n)∈ad(gj, j ≥n)∩Bwhich is ρ-a.e sequen ially closed (by p ope y (3) o Lemma 1.4) and gφ(n) ρ−a.e →g, we ob ain g∈ad(gj, j ≥ n)∩B o all n≥1.We will see ha θ(g)≤lim sup j→∞ θ(gj).Indeed, om Lemma 2.3 we ha e θ(gj) = lim supn→∞ ρ( n−gj)≥lim in n→∞ ρ( ψ(n)−gj) = lim in n→∞ ρ( ψ(n)− ) + ρ( −gj).Thus, again using Lemma 2.3, we ob ain lim sup j→∞ θ(gj)≥lim in n→∞ ρ( ψ(n)− ) + lim sup j→∞ ρ( −gj) ≥lim in n→∞ ρ( ψ(n)− ) + lim in j→∞ ρ( −gφ(j)) = lim in n→∞ ρ( ψ(n)− ) + lim in j→∞ ρ(gφ(j)−g) + ρ( −g). On he o he hand θ(g) = lim supn→∞ ρ( n−g) = lim in n→∞ ρ( ψ(n)−g) = lim in n→∞ ρ( ψ(n)− ) + ρ( −g). The e o e, θ(g)≤lim supj→∞ θ(gj). The ollowing lemma is inspi ed on [3] whe e a simila lemma is p o ed in e lexi e Banach spaces (see also [12, Lemma 6] o a e sion in me ic spaces wi h addi ional p ope ies). Lemma 2.2. Le ρbe a unc ion modula sa is ying he ∆2- ype condi ion and Ban be admissible, ρ-a.e sequen ially compac and ρ-bounded subse o Lρ.Le { n}be a sequence in Band ca cons an such ha c > ˜ N(Lρ). Then he e exis s ∈Bsuch ha (1) lim sup n→∞ ρ( − n)≤c δ({ n}n). (2) ρ( −g)≤lim sup n→∞ ρ( n−g) o all g∈B. P oo . Le { n}nbe a sequence o B. Deno e Am=ad( j:j≥m)⊂Band A=T∞ m=1 Am.Since Bis ρ-a.e sequen ially compac , he e exis s a subsequence o { n}nρ−a.e con e gen , say o h. I is clea ha h∈Aand so A6=∅. Fu he mo e, om P oposi ion 1.1 (2), we ha e δ(An)≤δ({ n}n).On he o he hand, o any ∈Aand g∈Bwe ha e ρ(g− )≤ (g, A)≤ (g, An) = (g, { j: j≥n}) = supj≥nρ(g− j).The e o e, ρ(g− )≤lim supn→∞ ρ(g− n) and (2) holds o any ∈A. We will p o e ha he e exis s ∈Asa is ying (1). Wi hou loss o gene ali y we may assume ha δ({ n}n)>0.Choose ε > 0 such ha 8 T. DOMINGUEZ BENAVIDES, M.A. KHAMSI AND S. SAMADI ˜ N(Lρ)δ({ n}n)+ε≤c δ({ n}n).By de ini ion o R(An), he e exis s gn∈Ansuch ha (gn, An)< R(An)+ε≤˜ N(Lρ)δ(An)+ε≤˜ N(Lρ)δ({ n}n)+ε≤c δ({ n}n). Since (gn, An) = (gn,{ j}j≥n) = supj≥nρ(gn− j),we ha e, lim sup j→∞ ρ(gn− j)≤c δ({ n}n) (A). Using Lemma 2.5, he e exis s ∈ ∩∞ n=1ad(gi, i ≥n) such ha lim sup j→∞ ρ( − j)≤lim sup n→∞ lim sup j→∞ ρ(gn− j) (B). We will check ha ∈A. Indeed, o all i, n in ege s such ha i≥nwe ha e gi∈Ai⊂An.Thus, {gi}i≥n⊂Anwhich implies ad(gi, i ≥n)⊂Anand ∈A. Using (B) and (A) i is clea ha lim supj→∞ ρ( − j)≤c δ({ n}n). Theo em 2.1. Le ρbe a con ex unc ion modula sa is ying he ∆2-condi ion and B an admissible, ρ-a.e sequen ially compac and ρ-bounded subse o Lρ.Sup- pose ha ˜ N(Lρ)<1and le T:B→Bbe a k-uni o mly lipschi zian mapping sa is ying k < (˜ N(Lρ))−1/2.Then, Thas a ixed poin . P oo . We can assume ha k > 1; o he wise Twill be nonexpansi e and he exis ence o a ixed poin is a consequence o [8, Theo em 3.5]. Choose a cons an c,˜ N(Lρ)< c < 1 such ha 1 < k < c−1/2.Fix 0∈B. By Lemma 2.6, we can induc i ely cons uc a sequence { j}j≥0⊂Bsuch ha o each j≥0 (1) lim supn→∞ ρ(Tn( j)− j+1)≤c δ({Tn( j)}n). (2) ρ( j+1 −g)≤lim supn→∞ ρ(Tn( j)−g) o all g∈B. Deno e Dj= lim supn→∞ ρ(Tn( j)− j+1) and h=ck2<1.Fo n≥m≥0,we ha e ρ(Tm j−Tn j)≤kρ( j−Tn−m j) ≤klim sup i→∞ ρ(Ti j−1−Tn−m j) ≤k2lim sup i→∞ ρ(Ti−(n−m) j−1− j) ≤k2Dj−1. UNIFORMLY LIPSCHITZIAN MAPPINGS IN MODULAR FUNCTION SPACES 9 Since Dj= lim supn→∞ ρ(Tn( j)− j+1)≤c δ({Tn( j)}n),we ob ain Dj≤ c k2Dj−1=hDj−1.Thus, Dj≤hjD0and we ha e ρ( j+1 − j)≤ω(2)ρ( j+1 −Tn j) + ρ( j−Tn j) ≤ω(2)ρ( j+1 −Tn j) + lim sup m→∞ ρ(Tm j−1−Tn j) ≤ω(2)ρ( j+1 −Tn j) + klim sup m→∞ ρ(Tm−n j−1− j) ≤ω(2)ρ( j+1 −Tn j) + kDj−1. Taking limsup as n→ ∞,we ob ain ρ( j+1 − j)≤ω(2)(Dj+kDj−1) ≤ω(2)(hj+khj−1)D0 ≤ω(2)(h+k)hj−1D0 ≤Ahj,whe e A=ω(2)h+k hD0. Hence, he e exis s an in ege Nand some β < 1 such ha o j > N we ha e ρ( j+1 − j)≤βj,which implies 1 βj≤1 ρ( j+1 − j).Using p ope ies (2) and (3) o Lemma 1.1 we ob ain ω−11 βj≤ω−11 ρ( j+1 − j) and ω−11 βj ≤ω−11 ρ( j+1 − j). The e o e, by Lemma 2.2 we ha e || j+1 − j||ρ≤1 ω−11 ρ( j+1− j)≤1 ω−1(1 β)j. Hence { j}is a Cauchy sequence in (Lρ,||.||ρ), he e exis s ∈Lρsuch ha || j− ||ρ→0, because (Lρ,||.||ρ) is comple e. Since unde ∆2-condi ion no m- con e gence and modula -con e gence a e iden ical, { j}is modula con e gen o . Thus, he e exis s a subsequence o { j}jρ-a.e con e gen o [1, Theo em 16 T. DOMINGUEZ BENAVIDES, M.A. KHAMSI AND S. SAMADI ACKNOWLEDGEMENTS The i s and hi d au ho s a e e y g a e ul o he Depa men o Ma hema - ical Sciences a he Uni e si y o Texas a El Paso o hei hospi ali y while comple ing his wo k. Re e ences [1] A.Kaminska. On uni o m con exi y o O licz spaces. Ma hema ics, P oceedings. Konink. Nede l, Ak. We . Ams e dam, A 85 (1) (1982), 27- 36. [2] J.M.Aye be, T.Dominguez Bena ides, G.Lopez Acedo. Measu es o Noncompac ness in Me ic Fixed Poin Theo y. Bikhause : Basel (1997). [3] E.Casini, E.Malu a. Fixed poin s o uni o mly Lipchi zian mappins in spaces wi h uni o mly no mal s uc u e. Nonlinea Anal. 9(1985), 103-108. [4] T.Dominguez Bena ides. Fixed poin heo ems o uni o mly Lipschi zian mappings and asymp o ically egula mappings. Nonlinea Anal. 32 (1) (1998), 15-27. [5] T.Dominguez Bena ides, M.A.Khamsi, S.Samadi. Asymp o ically egula mappings in mod- ula unc ion spaces, (p ep in ). [6] H.Hudzik, A.Kaminska, J.Musielak. On he con exi y coe icien o O licz spaces. Ma h. Z. 197 (1988), 291-295. [7] M.A.Khamsi. Fixed poin heo y in modula unc ion spaces. Recen Ad ances on Me ic Fixed Poin Theo y. 31-58. Uni e sidad de Se illa, Se illa (1996). [8] M.A.Khamsi, W.M.Kozlowski, S.Reich. Fixed poin heo y in modula unc ion spaces. Non- linea Anal. 14 (1990), 935-953. [9] M.A.Khamsi, W.M.Kozlowski, C.Shu ao. Some geome ical p ope ies and ixed poin he- o ems in O licz Spaces. J. Ma h. Anal. Appl. 155 (2) (1991), 393-412. [10] W.M.Koslowski. Modula unc ion spaces. Dekke : New Yo k, Basel (1988). [11] E.A.Li schi z. Fixed poin heo ems o ope a o s in s ongly con ex spaces (Russian) Vo onez. Gos. Uni . T udy Ma . Fak. 16 (1975), 23-28. [12] T.C.Lim, H.K.Xu. Uni o mly Lipschi zian mappings in me ic spaces wi h uni o m no mal s uc u e. Nonlinea Anal. 25 (11) (1995), 1231-1235. [13] J.Musielak, W.O licz. On modula spaces. S udia Ma h. 18 (1959), 591-597. [14] H. Nakano. Modula ed semi-o de ed spaces, Tokyo (1950). [15] D.Van Dul s , V.de Valk. (KK)-p ope ies, no mal s uc u e and ixed poin s o nonexpan- si e mappings in O licz spaces. Canad. J. Ma h. 38 (1986), 728-750. UNIFORMLY LIPSCHITZIAN MAPPINGS IN MODULAR FUNCTION SPACES 17 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]