Almost periodic and asymptotically almost periodic functions: part I

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1 Lassoued e a. Advances in Difference Equaions (28) 28:47 hps://doi.org/.86/s R E S E A R C H Open Access Amos periodic and asympoicay amos periodic funcions: par I Dhaou Lassoued,2, Rahim Shah 3 and Tonging Li 4,5* * Correspondence: 4 LinDa Insiue of Shandong Provincia Key Laboraory of Nework Based Ineigen Compuing, Linyi Universiy, Linyi, P.R. China 5 Schoo of Informaion Science and Engineering, Linyi Universiy, Linyi, P.R. China Fu is of auhor informaion is avaiabe a he end of he arice Absrac In his paper, we review indispensabe properies and characerizaions of amos periodic funcions and asympoicay amos periodic funcions in Banach spaces. Specia accen is pu on he Sepanov generaizaions of amos periodic funcions and asympoicay amos periodic funcions. We aso recoec some basic resus regarding equi-wey-amos periodic funcions and Wey-amos periodic funcions. The cass of asympoicay Wey-amos periodic funcions, inroduced in his work, seems o be no considered esewhere even in he scaar-vaued case. We acuay inroduce eigh new casses of asympoicay amos periodic funcions and anayze reaions beween hem. In order o make a picure as compee and cear as possibe, severa iusraing eampes and couner-eampes are given. I is worh noing ha he opics dea wih in his paper seem o be of an inrinsic connecion wih he probem of eisence and uniqueness of souions of differenia and difference equaions, in boh deerminis and sochasic cases. MSC: 42A75 Keywords: amos periodic funcion; asympoicay amos periodic funcion Inroducion The heory of amos periodic funcions has graduay been increased o a comprehensive and eensive heory by he conribuions of numerous mahemaicians. Indeed, he prehisory of amos periodiciy begins wih Escangon and Boh. The heory of amos periodic funcions was deveoped in is main feaures by Bohr as a generaizaion of pure periodiciy in hree raher ong papers 3, under he common ie ZurTheoriederfasperiodischen Funkionen in 925 and 926. The firs of hese papers dea wih he amos periodic funcions of a rea variabe, whie he hird one ook up he case of a compe variabe. Aferwards, he heory of amos periodic funcions was coninuousy geing esabished by severa mahemaicians ike Amerio and Prouse 4, Levian 5, Besicovich,Bochner,vonNeumann,Fréche,Ponryagin,Lusernik,Sepanov,Wey,ec.;wih respec o his maer, we cie 6 and he references herein. In 962, Bochner defined and sudied he amos periodic funcions wih vaues in Banach spaces. He showed ha hese funcions incude cerain earier generaizaions of he noion of amos periodic funcions. Some eensions of Bohr s concep have been inroduced, mos noaby by Besicovich, Sepanov, Wey and Eberein. One can remark ha speaking abou Sepanov, The Auhor(s) 28. This arice is disribued under he erms of he Creaive Commons Aribuion 4. Inernaiona License (hp://creaivecommons.org/icenses/by/4./), which permis unresriced use, disribuion, and reproducion in any medium, provided you give appropriae credi o he origina auhor(s) and he source, provide a ink o he Creaive Commons icense, and indicae if changes were made.

2 Lassoued e a. Advances in Difference Equaions (28) 28:47 Page 2 of 9 Wey or Besicovich merics impiciy means deaing wih he reaed quoien spaces, because oherwise we shoud raher speak abou Sepanov, Wey or Besicovich. In fac, he firs moivaion for he sudy of amos periodic funcions is he se of various ways o combine periodic funcions wih differen periods. For insance, he funcion cos + cos(5) is periodic, and his remains rue when 5 is repaced by any oher raiona number. However, he sum of he periodic funcions e i and e i 2 is no periodic. Hence, when such funcions, obained by using a combinaion of periodic funcions, are no periodic, hey are no wihou properies: hey are amos periodic funcions. In he courses of mechanics, we usuay encouner some wo-dimensiona differenia sysems of he form = A + e(), where A is a 2 2 mari wih purey imaginary eigenvaue and e( ) is a periodic eerior force. I is we known ha when hese forced sysems possess a periodic osciaion, hen he period of his osciaion is eacy he period of he eerior force. I is no menioned in hese courses, bu hese forced sysems possess amos periodic souions. More generay, we know ha when a he souions of an auonomous inear finie dimensiona sysem are bounded, hen a hese souions are amos periodic. In more physica erms, he amos periodic rajecories are rajecories wih a discree specrum. Besides, among he acua ieraure abou he chaos heory, a famous mode of ransiion owards he chaos is he Landau-Hopf mode 2 where he invoved poenia is an amos periodic poenia. Maurice Aais (Nobe Price of Economics) has wrien a wide work abou he foundaions of he heory of probabiiies 3. The major concusion of his work is he foowing: many naura phenomena are considered as sochasic phenomena, bu, in fac, hey are amos periodic phenomena which are bady undersood. In suppor of his viewpoin, Aais has esabished (wih rigorous proofs 3) a mahemaica heorem which says ha he sampings of an amos periodic funcion converge o he Lapace-Gauss disribuion. Ever since heir inroducion by Bohr in he mid-wenies, amos-periodic (a.p.) funcions have payed an imporan roe in various branches of mahemaics. Aso, in he course of ime, various varians and eensions of Bohr s concep have been inroduced, mos noaby by Besicovich, Sepanov and Wey. Accordingy, here are a number of monographs and papers covering a wide specrum of noions of amos periodiciy and appicaions (see, for insance, he arge is of references 4, Chapers and 2). An eension of Bohr s origina (scaar) concep of a differen kind is he generaizaion o vecorvaued amos-periodic funcions, saring wih Bochner s work in he hiries. Here, oo, are a number of monographs on he subjec, mos noaby by Amerio and Prouse 4and Levian and Zhikov 5. This vecor-vaued (Banach space-vaued) case is paricuary imporan for appicaions o (he asympoic behavior of souions o) differenia equaions and dynamica sysems. As aforemenioned, he noion and properies of amos periodic funcions, eiher in heir iniia or in generaized form, urned ou o be of grea imporance in various fieds of anaysis, funcion heory, opoogy and appied mahemaics. The necessiy of a manuscrip giving a concise and sysemaic eposiion of he fundamenas of he heory of amos periodic funcions was becoming more and more obvious. The ask of wriing such a manuscrip in a jus ony one par was an arduous one. Therefore, i is no asonishing ha he presen arice wi ead o oher fuure works running in he same aim. In he presen arice, we sudy he basic properies of amos periodic funcions and asympoicay amos periodic funcions. These opics are inrinsicay conneced wih

3 Lassoued e a. Advances in Difference Equaions (28) 28:47 Page 3 of 9 he probem of eisence and uniqueness of souions of differenia equaions. To give a compee and cear picure for he differen spaces sudied in our work, we iusrae hem by severa eampes and couner-eampes. Throughou his paper, we use he usua noaion N, Z, R and C for he ses of naura, ineger, rea and compe numbers, respecivey. For any rea number s R,wedenoe s = sup{ Z : s } and s = inf{ Z : s }. Uness saed oherwise, we assume ha X is an infinie-dimensiona compe Banach space, and he norm of an eemen X is denoed by. Assuming ha Y is anoher compe Banach space, we denoe by L(X, Y ) he space consising of a coninuous inear mappings from X ino Y and L(X) L(X, X). The norm on L(X, Y ) sha be denoed by he same noaion.theopoogyonl(x, Y )andx := L(X, C), he dua space of X, are inroduced in he usua way. The symbo I denoes he ideniy operaor on X. Insome paces, we need o have wo differen pivo spaces, hus we someimes use he symbos Y, Z,...,E in pace of X. Le I = R or I =, ). The space of a Bochner inegrabe funcions from I ino X is denoed by L (I : X), equipped wih he norm f = I f () d. For p < and (, R, μ) ameasurespace,byl p ( : X) we denoe he space consising of a srongy μ- measurabe funcions f : X such ha f p := ( f ( ) p dμ) /p is finie. By C b (I : X) we denoe he space consising of a bounded coninuous funcions from I ino X. The symbo C (, ):X) denoes he cosed subspace of C b (I : X) consising of funcions vanishing as he modue of he argumen ends o infiniy. By BUC(I : X) we denoe he space consising of a bounded uniformy coninuous funcions from I o X. The sup-norm urns hese spaces ino Banach s. The noaion c wi be deserved o he space of a compe sequences (a n ) n ha converge o zero a infiniy, ha is, such ha im n a n =.Someimes, we use he noaion X I for he se of a appicaions from I ino X. For an appicaion f X I, R(f ) denoes is range (or is image). Our paper is organized in hree big secions. In he firs secion, we sudy genera amos periodic funcions and asympoicay amos periodic ones. In he second secion, we dea wih he Sepanov generaizaion for amos periodiciy and asympoic amos periodiciy. Then, Wey amos periodic funcions and asympoicay amos periodic funcions are considered. These secions are muuay cosey conneced since we anayze some comparison reaions inking he differen funciona spaces defined in each paragraph. 2 Amos periodic funcions and asympoicay amos periodic funcions As underined above, he concep of amos periodiciy was inroduced by Danish mahemaician Bohr around and aer generaized by many oher auhors (cf., 5 2 for more deais on he subjec). Le I = R or I =, ), and e f : I R be a coninuous funcion. Given ε >,weca τ >anε-period for f ( ) if and ony if, for a I, f ( + τ) f () ε. (2.) The se consising of a ε-periods for f ( )isdenoedbyv(f, ε). I is said ha he funcion f ( ) is amos periodic, a.p. for shor, if and ony if, for each ε >,hesev(f, ε) is reaivey dense in I, which means ha here eiss a consan > such ha any subinerva of I of engh mees V(f, ε).

4 Lassoued e a. Advances in Difference Equaions (28) 28:47 Page 4 of 9 Since for each ε >wehavev(f, ε) V( f, ε), which is a consequence of he inequaiy y y,, y X, i immediaey foows from he definiion ha he amos periodiciy of funcion f : I X impies he amos periodiciy of scaar-vaued funcion f : I R. Furhermore, i can be easiy seen ha he amos periodiciy of funcion f : I X impies he amos periodiciy of vecor-vaued funcions f ( +a)andf(a ), where a I. We ca f ( ) weaky amos periodic, w.a.p. for shor, if and ony if for each X he funcion (f ( )) is amos periodic (i is we known ha any funcion f BUC(I : X)which has a reaivey compac range in X and which is w.a.p., needs o be a.p., cf. 2, Proposiion 4.5.2). A famiy of funcions F X I is said o be uniformy amos periodic if and ony if, for each ε >, here eiss a consan > such ha any subinerva of I of engh conains a number τ >suchha(2.)hodsforaf F. The space consising of a amos periodic funcions from he inerva I ino X wi be denoed by AP(I : X). Equipped wih he sup-norm, AP(I : X) becomes a Banach space. For he seque, we need some preiminaries from he pioneering paper 22byBarand Godberg. The ransaion semigroup (W()) on AP(, ):X), given by W()f (s):= f ( + s),, s, f AP(, ):X), consiss soey of surjecive isomeries W()( ) and can be eended o a C -group (W()) R of isomeries on AP(, ) :X), where W( ):=W() for >. Furhermore, he mapping E : AP(, ):X) AP(R : X), defined by Ef ():= W()f (), R, f AP (, ):X ), is a inear surjecive isomery and Ef is he unique coninuous amos periodic eension of a funcion f from AP(, ):X) o he whoe rea ine. We have ha E(Bf ) = B(Ef )for a B L(X)andf AP(, ):X). The mos inriguing properies of amos periodic vecor-vaued funcions are coeced in he foowing wo heorems (in he case ha I = R hese asserions are we known in he eising ieraure, in he case ha I =, ), hen hese asserions can be deduced by using heir vaidiy in he case I = R and he properies of eension mapping E( )). Theorem 2. Le f AP(I : X). Then he foowing asserions hod: () f BUC(I : X); (2) if g AP(I : X), h AP(I : C), α, β C, hen αf + βg and hf beong o AP(I : X); (3) Bohr s ransform of f ( ) P r (f ):= im eiss for a r R and e irs f (s) ds P r (f )= im +α α e irs f (s) ds for a α I and r R; (4) if P r (f )=for a r R, hen f ()=for a I; (5) σ (f ):={r R : P r (f ) } is a mos counabe;

5 Lassoued e a. Advances in Difference Equaions (28) 28:47 Page 5 of 9 (6) if c X, which means ha X does no conain an isomorphic copy of c, I = R and g()= f (s) ds ( R) is bounded, hen g AP(R : X); (7) if (g n ) n N is a sequence in AP(I : X) and (g n ) n N converges uniformy o g, hen g AP(I : X); (8) if I = R and f BUC(R : X), hen f AP(R : X); (9) (specra synhesis) f span{e iμ : μ σ (f ), R(f )}; () R(f ) is reaivey compac in X; () we have f = sup f (), I; (2.2) (2) if I = R and g L (R), hen g f AP(R : X), where (g f )()= g( s)f (s) ds, R. Theorem 2.2 (Bochner s crierion) Le f BUC(R : X).Then f ( ) is amos periodic if and ony if, for any sequence (b n ) n of numbers from R, here eiss a subsequence (a n ) n of (b n ) n such ha (f ( + a n )) n converges in BUC(R : X). Remark 2.3 I is worh noing ha asserion (8) in Theorem 2. hods in he case ha I =, ). More precisey, eing f AP(, ) :X) andf BUC(, ) :X), hen f AP(, ) : X). To see his, i suffices o appy asserion (7) from he same heorem by noicing ha he sequence defined by f n ():=nf( +/n) f (), of amos periodic funcions converges uniformy o f ()for, because fn () f() +/n = n f (s) f () ds +/n n f (s) f () ds, and f ( ) is bounded uniformy coninuous on, ). Beforeproceedinganyfurher,wewoudikeomenionhahenecessaryandsufficien condiion for X o conain c is given in 2, Theorem : c X if and ony if here eiss a divergen series n= n in X which is uncondiionay bounded, i.e., here eiss a consan M >suchha m j= n j M, whenevern j N (j =,2,...,m) suchha n < n 2 < < n m. The imporance of condiion c X has been recognized aready by Bohrand aerempoyedby many ohers (see, e.g., Kade sheorem 2, Theorem 4.6.). By eiher AP( : X) orap (I : X), where is a nonempy subse of I, wedenoehe vecor subspace of AP(I : X) consising of a funcions f AP(I : X) for which he incusion σ (f ) hods good. I can be easiy seen ha AP( : X)isacosedsubspaceofAP(I : X) and herefore Banach space isef. The reaive compacness of subses in AP(I : X) has been eamined by Corduneanu 23 (seeaso7, Theorem 3.). A funcion f BUC(I : X) issaidobeweakyamos periodic in he sense of Eberein if and ony if {f ( + s):s I} is reaivey weaky compac in X. This imporan cass of funcions wi no be considered in he seque (for furher deais concerning his inriguing opic and connecions beween amos periodiciy and

6 Lassoued e a. Advances in Difference Equaions (28) 28:47 Page 6 of 9 Caremanspecrumof funcions, one mayrefero he monograph 2 and he references cied herein). 2. Asympoicay amos periodic funcions The noion of an asympoicay amos periodic funcion was inroduced by Fréche in 94 (for more deais concerning he vecor-vaued asympoicay amos periodic funcions and asympoicay amos periodic differenia equaions, see, e.g., 7, 8, 24 3). Afuncionf C b (, ):X) is said o be asympoicay amos periodic if and ony if, for every ε >,wecanfindnumbers >andm > such ha every subinerva of, ) of engh conains,aeas,onenumberτ such ha f ( + τ) f () ε for a M.The space consising of a asympoicay amos periodic funcions from, ) inox wi be denoed by AAP(, ): X). I is we known ha (see Ruess, Summers, and Vũ QuôcPhóng27, 3 33), for any funcion f C(, ): X), he foowing saemens are equivaen: (i) f AAP(, ):X); (ii) here eis uniquey deermined funcions g AP(, ): X) and C (, ):X) such ha f = g + ; (iii) he se H(f ):={f( + s):s >} is reaivey compac in C b (, ):X),whichmeans ha for any sequence (b n ) n of nonnegaive rea numbers here eiss a subsequence (a n ) n of (b n ) n such ha (f ( + a n )) n converges in C b (, ):X). The funcions g and from (ii) are caed he principa and correcive erms of he funcion f, respecivey. ThenweknowhaR(g) R(f ) (see, e.g., 7, Lemma3.43). By C (, ) Y : X), we denoe he space of a coninuous funcions h :, ) Y X such ha im h(, y) = uniformy for y in any compac subse of Y.Aconinuousfuncion f : I Y X is caed uniformy coninuous on bounded ses, uniformy for I if and ony if, for every ε > and every bounded subse K of Y,hereeissanumberδ ε,k > such ha f (, ) f (, y) ε for a I and a, y K saisfying ha y δ ε,k.if f : I Y X, henwedefineˆf : I Y L p (, : X) byf (, ˆ y):=f ( +, y),, y Y. For he purpose of research of (asympoicay) amos periodic properies of souions o semiinear Cauchy incusions, we need o remind ourseves of he foowing we-known definiions and resus (see, e.g., Zhang 34,LongandDing35 and Proposiion 2.6 beow). Definiion 2.4 Le p <. () A funcion f : I Y X is caed amos periodic if and ony if f (, ) is bounded, coninuous as we as, for every ε >and every compac K Y,hereeissan (ε, K) >such ha every subinerva J I of engh (ε, K) conains a number τ wih he propery ha f ( + τ, y) f (, y) ε for a I, y K. The coecion of such funcions wi be denoed by AP(I Y : X). (2) A funcion f :, ) Y X is said o be asympoicay amos periodic if and ony if i is bounded coninuous and admis a decomposiion f = g + q,where g AP(, ) Y : X) and q C (, ) Y : X). Denoe by AAP(, ) Y : X) he vecor space consising of a such funcions. The foowing composiion principes are we known in he eising ieraure (see, e.g., 34).

7 Lassoued e a. Advances in Difference Equaions (28) 28:47 Page 7 of 9 Theorem 2.5 () Le f AP(I Y : X) and h AP(I : Y ). Then he mapping f (, h()), I, beongs o he space AP(I : X). (2) Le f AAP(, ) Y : X) and h AAP(, ):Y ). Then he mapping f (, h()),, beongs o he space AAP(, ):X). In Definiion 2.4(2), a grea number of auhors assume a priori ha g AP(R Y : X). This is sighy redundan on accoun of he foowing proposiion. Proposiion 2.6 Le f :, ) Y XandeS Y. Suppose ha, for every ε >,here eiss an (ε, S)>such ha every subinerva J, ) of engh (ε, S) conains a number τ wih he propery ha f ( +τ, y) f (, y) ε for a, y S (his, in paricuar, hods provided ha f AP(I Y : X)). Denoe by F(, y) he unique amos periodic eension of funcion f (, y) from he inerva, ) o he whoe rea ine for fied y S. Then, for every ε >,wih he same (ε, S)>chosen as above, we have ha every subinerva J R of engh (ε, S) conains a number τ wih he propery ha F( + τ, y) F(, y) ε for a R, y S. Proof Le ε >begiveninadvance,(ε, S) >beasabove,andej =a, b R. The asserion is cear provided ha a >.Supposenowhaa <.Wechooseanumberτ > arbirariy. Then here eiss a τ J =τ, τ + b a, ) suchha f ( + τ, y) f (, y) ε for a, y S. Since τ := τ τ a J, i suffices o show ha F( + τ, y) F(, y) ε for a R, y S. To his end, fi a number R and an eemen y S. Since he mapping s F(s + τ τ a, y) F(s τ a, y), s R is amos periodic, equaion (2.2)showsha F ( s + τ τ a, y ) F ( s τ a, y ) ( F s + τ τ a, y ) F ( s τ a, y ) ( F s + τ τ a, y ) F ( s τ a, y ) = sup s τ + a = sup s τ + a = sup s ε. f ( s + τ τ a, y ) f ( s τ a, y ) f ( s + τ, y ) f (s, y) This ends he proof of he proposiion. 3 Sepanov amos periodic funcions and asympoicay Sepanov amos periodic funcions Le p <, >,andf, g L p oc (I : X), where I = R or I =, ). We define he Sepanov meric by D p S f ( ), g( ) = sup I + f () g() /p d p. (3.)

8 Lassoued e a. Advances in Difference Equaions (28) 28:47 Page 8 of 9 Then we know ha, for every wo numbers, 2 >, here eis wo posiive rea consans k, k 2 > independen of f, g such ha k D p S f ( ), g( ) D p S 2 f ( ), g( ) k2 D p S f ( ), g( ), (3.2) aswe as ha (see, e.g., 6, pp ) in he scaar-vaued case here eiss D p W f ( ), g( ) = im D p S f ( ), g( ) (3.3) in, ). Thedisanceappearingin(3.3) is caed he Wey disance of f ( )andg( ). The Sepanov and Wey norms of f ( ) are now respecivey defined by and f S p = D p S f ( ), f W p = D p W f ( ),. Taking ino accoun (3.2), in he seque of his secion i wi be appropriae o assume ha = 2 =.Wesayhaafuncionf L p oc (I : X) issepanovp-bounded, Sp -bounded shory, if and ony if f S p = sup I + f () /p d p <. The space L p S (I : X)consisingofaSp -bounded funcions becomes a Banach space when equipped wih he above norm. A funcion f L p S (I : X) issaidobesepanovp-amos periodic, S p -amos periodic shory, if and ony if he funcion ˆf : I L p (, : X)defined by ˆf ()(s)=f ( + s), I, s, is amos periodic (cf. 4 for more deais). 3. Asympoicay Sepanov amos periodic funcions I is said ha f L p S (, ):X) is asympoicay Sepanov p-amos periodic, asympoicay S p -amos periodic shory, if and ony if ˆf :, ) L p (, : X) is asympoicay amos periodic. I is a we-known fac ha if f ( ) is an amos periodic (respecivey, a.p.) funcion, hen f ( )isasos p -amos periodic (respecivey, S p -a.p.) for p <. The converse saemen is fase, however, as he foowing eampe from he book of Levian 5shows. Eampe 3. Assume ha α, β R and αβ is a we-defined irraiona number. Then he funcions f ()=sin 2+cos(α)+cos(β), R

9 Lassoued e a. Advances in Difference Equaions (28) 28:47 Page 9 of 9 and g()=cos 2+cos(α)+cos(β), R are Sepanov p-amos periodic bu no amos periodic ( p < ). Denoe by APS p (I : X) he space consising of a S p -amos periodic funcions f : I X. For any S p -amos periodic funcion f ( ) andforanyreanumberδ (, ), we define he funcion f δ ()= δ +δ f (s) ds, I. Arguing as in he scaar-vaued case, we can prove ha he funcion f δ ( ) isamos periodic ( < δ <)asweasha f δ f S p converges o as δ +. Hereafer, we wi aso use he Bochner heorem, which assers ha any BUC funcion ha is Sepanov p-amos periodic needs o be amos periodic ( p < ). The noion of a scaar S p -amos periodic funcion, sighy differen from he noion of usuay considered weaky S p -amos periodic funcion, is given as foows: a funcion f L p S (I : X)issaidobescaarySepanovp-amos periodic if and ony if, for each X,we have ha he funcion (f ):, ) C defined by (f )():= (f ()),, is Sepanov p-amos periodic. Definiion 3.2 Afuncionf : I Y X is caed Sepanov p-amos periodic, S p -amos periodic shory, if and ony if ˆf : I Y L p (, : X) is amos periodic. By 34, Theorem 2.6, we have ha a bounded coninuous funcion f :, ) Y X is asympoicay amos periodic if and ony if, for every ε > and every compac K Y, here eis (ε, K) >andm(ε, K) >suchhaeverysubinervaj, ) ofengh (ε, K) conainsanumberτ wih he propery ha f ( + τ, y) f (, y) ε for a > M(ε, K), y K. We inroduce he noion of an asympoicay Sepanov p-amos periodic funcion f (, ) as foows. Definiion 3.3 Le p <. A funcion f :, ) Y X is said o be asympoicay S p -amos periodic if and ony if ˆf :, ) Y L p (, : X) is asympoicay amos periodic. The coecion of such funcions wi be denoed by AAPS p (, ) Y : X). I is very eemenary o prove ha any asympoicay amos periodic funcion is aso asympoicay Sepanov p-amos periodic ( p < ). We need he asserion of 36, Lemma. Lemma 3.4 Suppose ha f :, ) X is an asympoicay S p -amos periodic funcion. Then here are wo ocay p-inegrabe funcions g : R Xandq:, ) Xsaisfying he foowing condiions: () g is S p -amos periodic; (2) ˆq beongs o he cass C (, ):L P (, : X));

10 Lassoued e a. Advances in Difference Equaions (28) 28:47 Page of 9 (3) f ()=g()+q() for a. Moreover, hereeissanincreasingsequence( n ) n N of posiive reas such ha im n n = and g()=im n f ( + n ) a.e.. Now we sae he foowing wo-variabe anaogue of Lemma 3.4. Lemma 3.5 Suppose ha f :, ) Y X is an asympoicay S p -amos periodic funcion. Then here are wo funcions g : R Y Xandq:, ) Y Xsaisfyingha, for each y Y, he funcions g(, y) and q(, y) are ocay p-inegrabe as we as ha he foowing hod: () ĝ : R Y L p (, : X) is amos periodic; (2) ˆq beongs o he cass C (, ) Y : L P (, : X)); (3) f (, y)=g(, y)+q(, y) for a and y Y. Moreover, for every compac se K Y, hereeissanincreasingsequence( n ) n N of posiive reas such ha im n n = and g(, y)=im n f ( + n, y) for a y Kanda.e.. Proof By he foregoing, we have ha ˆf :, ) Y X is bounded coninuous and admis a decomposiion ˆf = G + Q, whereg AP(, ) Y : L p (, : X)) and Q C (, ) Y : L p (, : X)). Moreover, he proof of 34, Theorem 2.6 shows ha, for every compac se K Y,hereeissanincreasingsequence( n ) n N of posiive reas such ha im n n = and G(, y)=im n ˆf ( + n, y) for a y Y and. The remaining par of proof foows by appying Lemma 3.4 o he funcion ˆf (, y)forafiedeemeny Y and he uniqueness of decomposiion g( )+q( ) in his emma. In he case ha he vaue of p is irreevan, we simpy say ha he funcion under our consideraion is (asympoicay, scaary) Sepanov amos periodic. Hereafer, we wi use he foowing emma (see, e.g., 6, p. 7 for he scaar-vaued case). Lemma 3.6 Le < a < b <, p < p <, and f L p (a, b :X). Then f L p (a, b:x) and b a b a f (s) p ds /p b a b a f (s) p ds 4 Wey amos periodic funcions and asympoicay Wey amos periodic funcions Uness specified oherwise, in his secion i wi be aways assumed ha I = R or I =, ). The pivo Banach space wi be denoed by X. The noion of an (equi-)wey amos periodic funcion is given as foows (cf. aso (3.)). Definiion 4. Le p < and f L p oc (I : X). () We say ha he funcion f ( ) is equi-wey-p-amos periodic, f e-wap(i p : X) for shor, if and ony if, for each ε >, we can find wo rea numbers >and L >such ha any inerva I I of engh L conains a poin τ I such ha /p. sup I + f ( + τ) f () /p d p ε,

11 Lassoued e a. Advances in Difference Equaions (28) 28:47 Page of 9 ha is, D p S f ( + τ), f ( ) ε. (2) We say ha he funcion f ( ) is Wey-p-amos periodic, f Wap(I p : X) for shor, if and ony if, for each ε >, we can find a rea number L >such ha any inerva I I of engh L conains a poin τ I such ha ha is, im sup I + /p f ( + τ) f () p d ε, Le us reca ha im Dp S f ( + τ), f ( ) ε. APS p (I : X) e-wap p (I : X) W ap p (I : X) in he se heoreica sense and ha any of hese wo incusions can be sric (see 37). For insance, he scaar-vaued funcion f : R C defined by f ():=χ (,/2) (), R is no Sepanov -amos periodic, bu i is equi-wey-amos--periodic (see, e.g., 37, Eampe 4.27); and he scaar-vaued funcion f : R C defined by f ():=χ (, ) (), R is no equi-wey-amos--periodic, bu i is Wey-amos--periodic (see, e.g., 38, Eampe ). Here, χ( ) denoes he characerisic funcion. We aso wan o poin ou ha he space of scaar-vaued funcions Wap(R p : R) seems o be defined and anayzed for he firs ime by Kovanko 39 in 944 (according o he informaion given in he survey paper 37). I is we known ha for any funcion f L p oc (I : X) is Sepanov boundedness is equivaen o is Wey boundedness, i.e., f S p < f W p <. Inheseque,weuseabbreviaionse-W ap (I : X) andw ap (I : X) o denoe he spaces e-wap (I : X) andw ap (I : X), respecivey (he case p =wibemosimporaninour furher anaysis). Simiary, we say ha a funcion is (equi)-wey-amos periodic if and ony if i is (equi)-wey--amos periodic. I is very imporan o sae he foowing characerisic of he space e-wap(i p : X), see, e.g., 37 for he scaar-vaued case. Theorem 4.2 Le p < and f L p oc (I : X). Then f e-w ap(i p : X) if and ony if, for every ε >,here eiss a rigonomeric X-vaued poynomia P ε ( ) such ha D p W Pε ( ), f ( ) ε. A Bochner ype heorem hods for Wey amos periodic funcions; see 5, 4.

12 Lassoued e a. Advances in Difference Equaions (28) 28:47 Page 2 of 9 Theorem 4.3 Le p < and e f W p ap(i : X) be uniformy coninuous. Then f AP(I : X). I is we known ha he funcions beonging o he space e-w p ap(i : X) need o be Wey uniformy coninuous in he foowing sense (see 6,p.84). Theorem 4.4 Le p < and f W p ap(i : X). Then, for every ε >,hereeiswofinie numbers L >and δ >such ha D p S L f ( + δ), f ( ) ε for δ δ. For some oher noions of Wey-amos periodiciy, ike equi-w p -normaiy and W p - normaiy, we refer he reader o 37, Secion Asympoicay Wey amos periodic funcions For he beginning, we need o inroduce he foowing noion. If q L p oc (, ):X), hen we define he funcion q(, ):, ), ) X by q(, s):=q( + s),, s. Definiion 4.5 I is said ha q L p oc (, ):X)isWey-p-vanishing if and ony if im q(, ) W p =, i.e., im im sup + /p q( + s) p ds =. (4.) I is cear ha, for any funcion q L p oc (, ):X), we can repace he imis in (4.). We say ha q L p oc (, ):X)isequi-Wey-p-vanishing if and ony if im im sup + q( + s) /p ds p =. (4.2) Since he second imi in (4.) awayseissin, ) (on accoun of (3.3)) and he second imi in (4.2) awayseissin, ) (aking ino accoun he fac ha he mapping sup + q( + s) p ds/ /p, is monoonicay decreasing), condiion (4.) is equivaen o ε >, (ε)>, (ε), >, > : + /p q( + s) p ds ε, (4.3) sup whie condiion (4.2)is equivaen o ε >, (ε)>, (ε), >, > : + /p q( + s) p ds ε. (4.4) sup

13 Lassoued e a. Advances in Difference Equaions (28) 28:47 Page 3 of 9 Now, assume ha q L p (, ):X). Then, for each ε >,hereeissa (ε)>such ha q(s) p ds ε p, (ε). In paricuar, + q(s) p ds ε p, (ε), and he funcion ˆq :, ) L p (, : X) beongs o he cass C (, ):L p (, : X)). The converse saemen is no rue, however, since he scaar-vaued funcion q()= /(2p), >saisfies ha ˆq C (, ):L p (, : X)) and q / L p (, ):X). If q L p oc (, ):X)andˆq C (, ):L p (, : X)), hen he compuaion sup + q( + s) /p ( ds p ) + + q(s) /p ds p + ++ ( ε p ) /p 2ε, hoding for any, shows ha he funcion q( )isequi-wey-p-vanishing, wih (ε)= and = (ε) chosen so ha + q(s) p ds ε p, (ε)( > (ε)). As he foowing simpe couner-eampe shows, he converse saemen does no hod in genera. Eampe 4.6 Define q():= χ n 2,n 2 +(),. n= Since n 2 + n 2 q(s) p ds =,n N,iiscearhaˆq / C (, ):L p (, : X)). On he oher hand, he inerva, + conainsamos squares of nonnegaive inegers, so ha + q( + s) p ds sup + q(s) p ds ( + +2 ) ( 2+ + ),,, so ha (4.4)hodswih (ε) > sufficieny arge and = ( > (ε)). If q L p oc (, ):X) andq( ) isequi-wey-p-vanishing, hen q( ) iswey-p-vanishing. To see his, assume ha (4.4)hodswih (ε)>andpuaferha (ε):= (ε). Therefore, sup (ε) + (ε) q( + s) /p ds p ε, (ε). (4.5) For any fied > (ε), we se := (ε). Then i suffices o show ha, for any >,we have sup + /p q( + s) p ds 2ε.

14 Lassoued e a. Advances in Difference Equaions (28) 28:47 Page 4 of 9 This foows from (4.5) and a simpe anaysis invoving he second inequaiy in par (i) of 38,Proposiion: sup + 2 /p sup /p q( + s) p ds (ε) + (ε) q( + s) /p ds p, > = (ε). Again, he converse saemen does no hod in genera and a Wey-p-vanishing funcion need no be equi-wey-p-vanishing. Eampe 4.7 Define q():= nχn 2,n 2 +(),. n= Then i is cear ha + q( + s) p ds sup + + q(s) p ds,,, 2 so ha (4.3)hodswih (ε) > chosen so ha /( + ) ε p and = +.Hence,q( )is Wey-p-vanishing. On he oher hand, q( ) canno be equi-wey-p-vanishing because, for each number >, here does no eis a finie imi + im q( + s) /p ds p. To see his, i suffices o observe ha, for each >andn N such ha n 2,wehave + sup q( + s) p n ds. Before proceeding furher, we woud ike o noe ha an equi-wey-p-vanishing funcion q( ) need no be bounded as. Eampe 4.8 Define q():= n /(4p) χ n 4,n 4 +(),. n= Then, simiary as in Eampe 4.6, we can prove ha + q( + s) p ds sup ( 2+ + q(s) p ds + ),,, which yieds he required concusions.

15 Lassoued e a. Advances in Difference Equaions (28) 28:47 Page 5 of 9 Denoe by W p (, ) :X) ande-w p (, ) :X) he ses consising of a Wey-pvanishing funcions and equi-wey-p-vanishing funcions, respecivey. The symbo S p (, ):X) wi be used o denoe he se of a funcions q Lp oc (, ):X) suchha ˆq C (, ):L p (, : X)). By our previous consideraions, Eampes 4.6 and 4.7, we have he foowing resu. Theorem 4.9 The foowing incusions hod: L p(, ):X ) S p and any of hem can be sric. and ( ) p( ) p( ), ):X e-w, ):X W, ):X Now, we inroduce he foowing funcion spaces: AAPW p(, ):X ) := AP (, ):X ) + W p (, ):X ), e- AAPW p(, ):X ) := AP (, ):X ) + e-w p (, ):X ), AAPSW p(, ):X ) := APS p(, ):X ) + W p (, ):X ), e- AAPSW p(, ):X ) := APS p(, ):X ) + e-w p ( ), ):X, e-waap p ( ) ( ), ):X := e-w p p( ) ap, ):X + W, ):X, ee-waap p ( ) ( ), ):X := e-w p p( ) ap, ):X + e-w, ):X, Waap p ( ) ( ), ):X := W p p( ) ap, ):X + W, ):X, ( ) ( ), ):X := W p p( ) ap, ):X + e-w, ):X. e-w p aap Then i is cear ha AAPW p(, ):X ) AAPSW p(, ):X ) e-waap p ( ) ( ), ):X W p aap, ):X e- AAPW p(, ):X ) e- AAPSW p(, ):X ) ee-w p aap(, ):X ) e-w p aap (, ):X ), and ha any of hese incusions can be sric. By he anaysis conained in 37, Eampe 4.27, he funcion f :, ) C defined by f () :=χ (,/2) (), > is equi-wey-amos periodic. Since his funcion is aso in cass e-w p (, ) :X), we have ha he sums defining e-w p aap(, ) :X), ee-w p aap(, ) : X), W p aap(, ) :X) ande-w p aap(, ) :X) are no direc. For he firs four spaces AAPW p (, ):X), e- AAPW p (, ):X), AAPSW p (, ):X) ande- AAPSW p (, ): X), he sums in heir definiions are direc, which foow from he foowing proposiion. Proposiion 4. Le p <. Then W p (, ):X ) APS p (, ):X ) = {}.

16 Lassoued e a. Advances in Difference Equaions (28) 28:47 Page 6 of 9 Proof Assume q W p (, ):X) APSp (, ):X).Inorderoprovehaq()=fora.e., i suffices o show ha ˆq()=,, in L p (, : X). Since ˆq( ) is amos periodic, we ony need o prove ha any Bohr-Fourier coefficien of ˆq( ) is equa o zero, i.e., ha im ( p /p e irs q(s + v) ds dv) =, r R. (4.6) To see ha (4.6) hods good, observe firs ha ( p ) /p e irs q(s + v) ds dv ( p /p q(s + v) ds dv), which can be furher majorized by using Lemma 3.6: ( ) /p ( p q(s + v) p ds dv = /p Hence, we need o prove ha im q(s + v) p ds dv = im s+ s ) /p q(s + v) p ds dv. q(r) p dr dv =. (4.7) Le ε > be given in advance. Since q W p (, ) :X), we know ha here eis wo finie numbers (ε)>and (ε)>suchha,forevery > (ε), sup + q ( (ε)+s ) /p ds p ε. (4.8) Le T (ε)>besuchha,foreach > T (ε), (ε) 2 and (ε). (4.9) The vaidiy of (4.9) ceary impies by (4.8)ha Since s+ s s+ s+ q(s) p ds ε, s,. (4.) q(r) p dr ( s+ + + s+2 s s+ ( s+ + s+ q S p + ε, s+ ) + s+ q(r) p dr s+ ) + + q(r) p dr by S p -boundedness of q( )and(4.), equaion (4.7)hodsrue.Theproofofheproposiion is hereby compee. s+

17 Lassoued e a. Advances in Difference Equaions (28) 28:47 Page 7 of 9 I is an easy ask o prove ha W p (, ) :X) ande-w p (, ) :X) are vecor spaces, so ha he inroduced eigh funcion spaces have a inear vecor srucure. Disregarding he erm (, ) : X) and aking ino consideraion he previousy defined spaces AAP and AAPS p, we have he foowing incusion diagram of asympoicay amos periodic funcion spaces (see Theorem 4.9): AAP e- AAPW p AAPW p AAPS p e- AAPSW p AAPSW p ee-w p aap e-w p aap e-w p aap W p aap By he foregoing, any incusion of his diagram can be sric. Furhermore, for any wo funcion spaces A and B beonging his diagram and saisfying addiionay ha here is no ransiive sequence of incusions connecing eiher A and B or B and A, wehaveha A \ B and B \ A (he diagram can be epanded by consrucing he sums of spaces of (equi-)wey amos periodic funcions wih S p (, ):X), which wi be no eamined here). We refer he reader o he paper 4 by Diagana e a. for more deais abou he noion of S p (n) -amos periodiciy (he noion of N-amos periodiciy is very we epored in he monograph 5 by Levian and Zhikov. Sepanov cases have no been inroduced so far, o he bes knowedge of he auhors). For an eceen survey of resus abou various casses of (Sepanov) amos periodic funcions and (Sepanov) asympoicay amos periodic funcions, we refer he reader o he review paper 37 by Andres e a. (cf. aso Andres e a. 38), aready cied muipe imes before. We round off our paper by inroducing he foowing imporan definiion. Definiion 4. Le I = R or I =, ), (R()) I L(X) be a srongy coninuous operaor famiy, and e denoe any of (asympoicay) amos periodic properies considered above. Then we say ha (R()) I is (asympoicay) amos periodic if and ony if he mapping R(), I is (asympoicay) amos periodic for a X.Iissaidha (R()) I is uniformy amos periodic if and ony if he famiy {R( ) : } is uniformy amos periodic. Acknowedgemens The auhors epress heir sincere graiude o he ediors and wo anonymous referees for he carefu reading of he origina manuscrip and usefu commens ha heped o improve he presenaion of he resus and accenuae imporan deais. This research is suppored by NNSF of P.R. China (Gran No. 6537), CPSF (Gran No. 25M5829), NSF of Shandong Province (Gran No. ZR26JL2), DSRF of Linyi Universiy (Gran No. LYDX25BS), and he AMEP of Linyi Universiy, P.R. China. Compeing ineress The auhors decare ha hey have no compeing ineress. Auhors conribuions A hree auhors conribued equay o his work. They a read and approved he fina version of he manuscrip.

18 Lassoued e a. Advances in Difference Equaions (28) 28:47 Page 8 of 9 Auhor deais Déparemen de Mahémaiques, Facué des Sciences de Gabès, Universié de Gabès, Cié Erriadh, Tunisia. 2 Laboraoire Équaions au dérivées pariees LEDP-LR3ES4, Facué des Sciences de Tunis, Universié de Tunis E Manar, Tunis, Tunisia. 3 Deparmen of Mahemaics, Universiy of Peshawar, Peshawar, Pakisan. 4 LinDa Insiue of Shandong Provincia Key Laboraory of Nework Based Ineigen Compuing, Linyi Universiy, Linyi, P.R. China. 5 Schoo of Informaion Science and Engineering, Linyi Universiy, Linyi, P.R. China. Pubisher s Noe Springer Naure remains neura wih regard o jurisdiciona caims in pubished maps and insiuiona affiiaions. Received: 8 Ocober 27 Acceped: 2 January 28 References. Bohr, H: Zur Theorie der fasperiodischen Funkionen I. Aca Mah. 45, (925) 2. Bohr, H: Zur Theorie der fasperiodischen Funkionen II. Aca Mah. 46, -24 (925) 3. Bohr, H: Zur Theorie der fasperiodischen Funkionen III. Aca Mah. 47, (926) 4. Amerio, L, Prouse, G: Amos-Periodic Funcions and Funciona Equaions. Springer, New York (97) 5. Levian, BM: Amos Periodic Funcions. Gos. Izda. Tekhn-Teor. Li., Moscow (953) (in Russian) 6. Besicovich, AS: Amos Periodic Funcions. Dover, New York (954) 7. Bochner, S: Curvaure and Bei numbers in rea and compe vecor bundes. Rend. Semin. Ma. (Torino) 5, ( ) 8. Corduneanu, C: Amos Periodic Funcions. Wiey, New York (968) 9. Favard, J: Leçons sur es foncions presque périodiques. Gauhier-Viars, Paris (933) (in French). Fink, AM: Amos Periodic Differenia Equaions. Lecure Noes in Mahemaics, vo Springer, Berin (974). Bochner, S: A new approach o amos periodiciy. Proc. Na. Acad. Sci. USA 48, (962) 2. Landau, L, Lifschiz, E: Cours de Physique héorique, ome 6, 2nd edn. Mir, Moscow (989) (in French) 3. Aais, M: Sur a disribuion normae des vaeurs à des insans réguièremen espacés d une somme des sinusoïdes. In: Compes Rendus de Académie des Sciences de Paris, ome 296, Série I, pp (983) (in French) 4. Lassoued, D: Foncions presque-périodiques e équaions différeniees. Ph.D. Disseraion, Universié Paris I, Panhéon-Sorbonne, Laboraoire SAMM (December 23) (in French) 5. Levian, BM, Zhikov, VV: Amos Periodic Funcions and Differenia Equaions. Cambridge Universiy Press, London (982) 6. Corduneanu, C: Amos Periodic Funcions, 2nd edn. Chesea, New York (989) 7. Diagana, T: Amos Auomorphic Type and Amos Periodic Type Funcions in Absrac Spaces. Springer, New York (23) 8. N Guérékaa, GM: Amos Auomorphic and Amos Periodic Funcions in Absrac Spaces. Kuwer Academic, Dordrech (2) 9. N Guérékaa, GM: Topics in Amos Auomorphy. Springer, New York (25) 2. Zaidman, S: Amos-Periodic Funcions in Absrac Spaces. Piman Research Noes in Mahemaics, vo. 26. Piman, Boson (985) 2. Arend, W, Bay, CJK, Hieber, M, Neubrander, F: Vecor-Vaued Lapace Transforms and Cauchy Probems. Birkhäuser, Base (2) 22. Bar, H, Godberg, S: Characerizaions of amos periodic srongy coninuous groups and semigroups. Mah. Ann. 236, 5-6 (978) 23. Corduneanu, C: Amos Periodic Osciaions and Waves. Springer, Berin (2) 24. Bay, CJK, van Neerven, J, Räbiger, F: Tauberian heorems and sabiiy of souions of he Cauchy probem. Trans. Am. Mah. Soc. 35, (998) 25. Cheban, DN: Asympoicay Amos Periodic Souions of Differenia Equaions. Hindawi Pubishing Corporaion, New York (29) 26. Deeeuw, K, Gicksberg, I: Appicaions of amos periodic compacificaions. Aca Mah. 5, (96) 27. Ruess, WM, Summers, WH: Compacness in spaces of vecor vaued coninuous funcions and asympoic amos periodiciy. Mah. Nachr. 35,7-33 (988) 28. Sahiyanahan, K, Nandha Gopa, T: Eisence of asympoicay amos periodic souions of inegrodifferenia equaions. App. Mah. Compu. Ine. 2, (23) 29. Xie, L, Li, M, Huang, F: Asympoic amos periodiciy of C-semigroups. In. J. Mah. Mah. Sci. 2, (23) 3. Zhang, C: Vecor-vaued pseudo amos periodic funcions. Czechosov. Mah. J. 47, (997) 3. Ruess, WM, Summers, WH: Asympoic amos periodiciy and moions of semigroups of operaors. Linear Agebra App. 84, (986) 32. Ruess, WM, Summers, WH: Inegraion of asympoicay amos periodic funcions and weak asympoic amos periodiciy. Diss. Mah. 279, -35 (989) 33. Ruess, WM, Vũ, QP: Asympoicay amos periodic souions of evouion equaions in Banach spaces. J. Differ. Equ. 22, (995) 34. Zhang, C: Ergodiciy and asympoicay amos periodic souions of some differenia equaions. In. J. Mah. Mah. Sci. 25(2), (2) 35. Long, W, Ding, H-S: Composiion heorems of Sepanov amos periodic funcions and Sepanov-ike pseudo-amos periodic funcions. Adv. Differ. Equ. 2, Arice ID (2).hps://doi.org/.55/2/ Henríquez, HR: On Sepanov-amos periodic semigroups and cosine funcions of operaors. J. Mah. Ana. App. 46, (99) 37. Andres, J, Bersani, AM, Grande, RF: Hierarchy of amos-periodic funcion spaces. Rend. Ma. App. (7) 26, 2-88 (26)

19 Lassoued e a. Advances in Difference Equaions (28) 28:47 Page 9 of Andres, J, Bersani, AM, Leśniak, K: On some amos-periodiciy probems in various merics. Aca App. Mah. 65, (2) 39. Kovanko, AS: Sur a compacié des sysèmes de foncions presque périodiques généraisées de H. Wey. C. R. (Dok.) Acad. Sci. URSS 43, (944) (in French) 4. Daniov, LI, Kudryavsev, LD, Levian, BM: Eemens of he Theory of Funcions. Pergamon, Oford (966) 4. Diagana, T, Neson, V, N Guérékaa, GM: Sepanov-ike C (n) -pseudo amos auomorphy and appicaions o some nonauonomous higher-order differenia equaions. Opusc. Mah. 32, (22)

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