ALMOST AUTOMORPHIC DELAYED DIFFERENTIAL EQUATIONS AND LASOTA-WAZEWSKA MODEL

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1 ALMOST AUTOMOPHIC DELAYED DIFFEENTIAL EQUATIONS AND LASOTA-WAZEWSKA MODEL ANÍBAL COONEL, CHISTOPHE MAULÉN, MANUEL PINTO, DANIEL SEPULVEDA arxiv: v2 [mah.ds] 28 Mar 216 Abrac. Exience of almo auomorphic oluion for abrac delayed differenial equaion i eablihed. Uing ergodiciy, exponenial dichoomy and Bi-almo auomorphiciy on he homogeneou par, ufficien condiion for he exience and uniquene of almo auomorphic oluion are given. 1. Inroducion The developmen of he heory of almo periodic ype funcion ha been rongly imulaed by problem ariing in differenial equaion, abiliy heory, dynamical yem and many oher area of cience. Nowaday, here exi alo a wide range of applicaion aring from he baic mahemaical model baed on linear ordinary differenial equaion, including nonlinear linear ordinary differenial equaion, differenial equaion in Banach pace and alo parial differenial equaion. Moreover, here exi everal relaed concep which arie a generalizaion of he almo periodic concep. For inance he noion of almo auomorphic, aympoically almo periodic, aympoically almo auomorphic and peudo almo periodic. Since here are pleny of reul in lieraure, le u ju quoe, for heir applicaion in engineering and life cience, for example aympoically almo periodic funcion [17, 18, 19, 2, 27, 29, 3, 31, 32, 33, 4], and peudo almo periodic funcion [9, 11, 28]. Moreover, we recall ha N Guérékaa ha given a huge impule o he udy of almo auomorphic oluion of differenial equaion [1, 7, 12, 14, 15, 21, 23, 25, 26]. For ome recen reul of almo auomorphic differenial equaion conul alo [5, 6]. In hi paper, we are iniially moivaed by a biological-mahemaical model [13, 16, 22, 37, 38, 41] which i a delayed differenial equaion of he following ype y ( = δ(y(+p(g(y( τ, wih τ >, δ and p almo auomorphic funcion and g a Lipchiz funcion. Then, we focu our aenion on he exience and uniquene of oluion of he following delayed differenial equaion y = A(y +f(+g(,y( τ wih τ, (1.1 under everal aumpion on A,f and g. Naurally, he aumpion on A and f are relaed o he almo auomorphic behavior and he aumpion on g are mainly relaed wih a Lipchiz requiremen. We noe ha (1.1 naurally, include a paricular cae he following equaion y = A(y, (1.2 y = A(y +f(, (1.3 y = A(y +f(+g(,y(. (1.4 Thu, following a naural equence of he claical yemic udy of ordinary differenial equaion we ar by analyzing he homogeneou linear equaion (1.2. Then, we develop he heory for he non-homogeneou linear equaion (1.3 by applying he mehod of variaion of parameer. In a Dae: Augu 1, 218. Key word and phrae. Abrac delay differenial equaion, Almo auomorphic, Exponenial dichoomy, Ergodiciy, Evoluion operaor. Parially uppored by FONDECYT CONICYT-PCHA/Magíer Nacional/ Aníbal Coronel hank for he uppor of reearch projec /, F/E and GI/C a Univeridad del Bío-Bío, Chile. 1

2 2 A. COONEL, CH. MAULÉN, M. PINTO, D. SEPULVEDA hird place, we analyze he nonlinear equaion (1.4 by uing he fixed poin argumen. Finally, by a compoiion reul of auomorphic funcion we ge an exenion of he reul for (1.4 o he delay equaion (1.1. The main conribuion and he organizaion of he paper are given a follow. In ecion 2 we inroduce he general aumpion, recall he concep of almo auomorphiciy and ergodic funcion, define a convoluion operaor and ge he reul for (1.2. To be a lile more precie in hi ecion, we obain ome condiion for he exponenial dichoomy uing ergodic funcion and we prove he Bi-almo periodiciy and Bi-almo auomorphiciy of he Green funcion when he evoluion operaor commue wih he projecion. We noe ha, he inegral Bi-almo auomorphiciy propery of he Green funcion i fundamenal o obain he main reul. In ecion 3, almo auomorphiciy of oluion of nonauonomou yem (1.3, (1.4 and (1.1 are obained. Here, he reul of almo auomorphiciy of he differenial equaion oluion are obained by auming ha A and f are almo auhomorphic and g aifie (2.1. Finally, in ecion 4 we udy a biological model eablihing an explici condiion under which here exi a unique almo auomorphic oluion of he Laoa-Wazewka equaion. 2. Preliminarie In hi ecion we preen ome general aumpion, precie he concep relaed wih he almo auomorphic noion, we recall he noion of ergodic funcion, we define a convoluion operaor and he α-exponenial dichoomy and inroduce everal reul for he homogeneou equaion (1.2 in he calar, yem and abrac cae General aumpion. Here we preen wo general aumpion. Firly, hroughou of he paper (V, V will be a Banach pace and le (BC(,V, will be ued o denoe he Banach pace of bounded coninuou funcion from ino V endowed wih he up norm ϕ = up ϕ( V. Second, concerning o he aumpion on he coefficien A,f and g for equaion (1.1-(1.4 we commen ha i will be pecifically done on he hypohei of each reul. However, in order o give a unified preenaion, we inroduce ome noaion relaed o he aumpion of he local Lipchiz behavior of g. Indeed, given a funcion g, i i aumed ha: (g g(, = for all ; (g 1 The funcion g(,y i coninuou on (ϕ,ρ wih (ϕ,ρ he open ball cenred in a given {(fix funcion ϕ : V and } wih radiu ρ +, i.e., (ϕ,ρ = ϕ : V ϕ ϕ ρ. In paricular, in ubecion will be aumed ha ϕ i of he form (2.1 ϕ ( = G(,f(d wih G he Green funcion defined on (3.17; (g 2 There exi a poiive conan L uch ha he inequaliy g(,y 1 g(,y 2 L y 1 y 2 hold for all (,y 1,y 2 (ϕ,ρ 2. Thi e of condiion (g -(g 2 appear in everal par of he paper and eenially when we udy he nonlinear equaion in ubecion Almo auomorphic noion and relaed concep. We recall ha he almo auomorphic funcion have been developed by Bochner [2, 3] a a generalizaion of almo periodic funcion. We recall ha a funcion f BC(,V i called Bohr almo periodic [8] if for each ǫ >, hereexi l ǫ > uchha everyinervalof lengh l ǫ conainanumber ξ wih he propery: f(+ξ f( V ǫ for. The e of Bohr almo periodic will be denoed by BC(,V. Then, we precie he concep of almo auomorphic funcion and marice. Definiion 2.1. Conider V a Banach pace. Then, (i A coninuou funcion ψ : V i called an almo auomorphic funcion if for any equence of real number { τ n } n=1, here exi a ubequence {τ n} n=1 of { τ n} n=1 uch ha he limi of he equence {ψ(+ τ n } n=1, denoed by ψ(, i well defined for all and he equence { ψ( τ n } n=1 converge poinwie on o ψ(, or equivalenly ψ( = lim ψ(+τ n and ψ( = lim ψ( τ n (2.2 n n

3 ALMOST AUTOMOPHIC DELAYED EQUATIONS 3 are well defined for all. The collecion of all almo auomorphic funcion from o V i denoed by AA(,V. (ii A marix valued funcion A : C d1 d2 i called an almo auomorphic marix valued funcion or equivalenly (mo of he ime by briefne A( C d1 d2 i called an almo auomorphic marix if for any equence {ξ n } n=1, here exi a ubequence {ξ n} n=1 of {ξ n} n=1 and a marix B( C d1 d2 uch ha he equence {A( + ξ n } n=1 and {B( ξ n } n=1 converge poinwie o B( and A(, repecively. We noe ha he convergence in (2.2 i poinwie. Then, he funcion ψ in (2.2 i meaurable, bu no necearily coninuou. Moreover, we noe if we conider ha convergence in definiion 2.1 i uniform on inead of poinwie convergence, we ge ha he funcion ψ i Bochner almo periodic. I i well known ha boh definiion of almo periodiciy (Bohr and Bochner are equivalen, ee for inance [8]. Now, we noe ha AP(,V and AA(,V are vecorial pace and AP(,V i a proper ubpace of AA(,V, ince for inance ψ( = co ( [2+in(+in( 2] 1 i an almo periodic funcion bu no almo auhomorphic. Similarly, i i proven ha he incluion AP(,V BC(,V, for an exenive dicuion conul [4, 3, 15, 14, 21, 23, 24, 25, 26, 34, 35, 42, 44, 1, 1, 39, 43, 12]. To cloe hi ubecion we inroduce wo addiional fac. Firly, we noe ha he impler equaion (1.3 wih A, i.e., y ( = f(, wih f AA(,V ha no necearily a oluion y AA(,V. However, hi fac i rue in a uniformly convex Banach pace V and hence in every Hilber pace, ee Theorem 2.1. In he econd place we need a compoiion reul [7], which will be fundamenal for he analyi of (1.4 and (1.1, ee Propoiion 2.2. Theorem 2.1. Denoe by C he vecorial pace formed by he funcion which vanihe a infiniy. Conider ha V i a Banach pace which doe no conain C a an iomorphic ubpace and le f AA(,V. Then, he funcion F( = f(d i in AA(,V if and only if i i bounded. Such Banach pace wih hi propery for F will be called a Banach pace wih he Bohl-Bohr propery. Propoiion 2.2. Le g = g(,y AA( V,V uniformly in for y in a compac e conained in V and g aifie he aumpion given on (2.1. Then, g(,ϕ( AA(,V for all ϕ AA(, (ϕ,ρ Ergodic funcion. Here we inroduce he concep of ergodic funcion and deduce ha hee ype of funcion implie naurally an exponenial behavior (or α-exponenial dichoomy o be more precie. Definiion 2.2. A funcion f BC(,V i called an ergodic funcion if he limi 1 M(f = lim T 2T T+ξ T+ξ f(d exi uniformly wih repec o ξ and i value i independen of ξ. The complex number M(f i called he mean of he funcion f. The mean of an ergodic funcion ha everal properie, a complee li of properie may be conuled in [45]. Among hee ueful baic properie, we only recall he ranlaion invariance propery, ince i will be ued frequenly in he proof given below in hi paper. Indeed, he ranlaion invariance propery of M(f, e ha M(f aifie he following ideniy M(f = M(f ξ, (2.3 where f ξ denoe a ξ-ranlaion of f, i.e. f ξ ( = f(+ξ for all and any arbirarily given ξ (bu fixed. Lemma 2.3. Conider µ BC(,C an ergodic funcion wih e(m(µ and alo conider α +. Then, here exi wo poiive conan T (big enough and c uch ha he wo aerion given below are valid:

4 4 A. COONEL, CH. MAULÉN, M. PINTO, D. SEPULVEDA (i If e(m(µ ], α[, hen he following inequaliie hold rue: ( e µ(rdr < α( for > T, (2.4 ( µ(rdr exp cexp( α( for all (, 2 uch ha. (2.5 (ii If e(m(µ ]α, [ +, hen he following inequaliie hold rue: ( e µ(rdr < α( for > T, (2.6 ( µ(rdr exp cexp(α( for all (, 2 uch ha. (2.7 Proof. Le u aume ha µ BC(,C i an ergodic funcion. Then, by he Definiion 2.2 and he ranlaion invariance propery of M(f (ee (2.3 we have ha T µ(+τdτ = [M(µ+o(1]T when T. (2.8 Here and hroughou of he paper o(1 correpond o he well known Bachmann-Landau noaion, i.e. f = o(g if and only if (f/g(x when x. Then, when T =, we ge µ(rdr = T µ( +τdτ (2.9 and he proof of (2.4 follow immediaely. The proof of (i i a conequence of he exponenial funcion increaing behavior. Thu, he proof of iem (i i compleed. Now, he proof of iem (ii i imilar and we omi i The convoluion operaor. Le u denoe by L 1 ( and L ( he pace of Lebegue inegrable funcion on and eenially bounded funcion on, repecively. Then, he convoluion operaor on L : L ( L ( i defined a he operaor uch ha L(ϕ( = h( ϕ(d, h L 1 (,, (2.1 for all ϕ L (. Some properie of L, which will be needed in he proof he main reul, are ummarized in he following lemma. Lemma 2.4. Conider L he convoluion operaor defined by (2.1. Then, he pace BC(, AP( and AA( are invarian under he operaor L. Moreover, he inequaliie Lϕ L ( ϕ L ( h L 1 ( for ϕ BC(, (2.11 (Lϕ ξ Lϕ L ( (ϕ ξ ϕ L ( h L 1 ( for ξ and ϕ AP(, (2.12 hold. Here (Lϕ ξ and (ϕ ξ are he ξ-ranlaion funcion for Lϕ and ϕ, repecively. Proof. Le u elec ϕ AA(. Then, by Definiion 2.1, given an arbirary equence of real number { τ n } n=1, here exi a ubequence {τ n } n=1 of { τ n } n=1 uch ha (2.2 i aified. Now, if we conider ψ = L(ϕ, we have ha (2.2 i equivalenly rewrien a follow ψ( = lim ψ(+τ n and ψ( = lim ψ( τ n. (2.13 n n Indeed, hi fac can be proved by applicaion of Lebegue dominaed convergence heorem, ince ψ( = lim L(ϕ τ n ( = lim h(rϕ τn ( rdr n n = h(r ϕ( rdr = L( ϕ(. Le u conider ϕ BC(, hen we deduce (2.11 by applicaion of he Hölder inequaliy. Now, from (2.11 we follow he invariance of BC(.

5 ALMOST AUTOMOPHIC DELAYED EQUATIONS 5 We noe ha, if we define { exp( αx, x >, h 1 (x =, oherwie, and h 2 (x = { exp(αx, x <,, oherwie, for ome α + and denoe by L 1 and L 2 he correponding convoluion operaor aociaed wih h 1 and h 2, repecively. Then, we ge an inereing reul by applicaion of Lemma 2.4. More preciely, we have he following Corollary. Corollary 2.5. Conider α + and he operaor L i, i = 1,2 defined by L 1 (ϕ( = e α( ϕ(d and L 2 (ϕ( = e α( ϕ(d, (2.14 repecively. Then, he pace BC(, AP( and AA( are invarian under he operaor L i, i = 1,2. Moreover, he inequaliie (2.11 and (2.12 are aified wih L i,i = 1,2, inead of L Some concep and properie relaed o equaion (1.2. In hi ubecion we udy he equaion (1.2. In order o inroduce he concep and reul, we recall he andard noaion of fundamenal marix and flow aociaed wih (1.2, which are denoed by Φ A and Ψ A, repecively. More preciely Given a marix A(, hen he noaion Φ A = Φ A ( and Ψ A are ued for a fundamenal marix of he yem (1.2 and for he applicaion defined a follow Ψ A (, = Φ A (Φ 1 A (. Lemma 2.6. Conider he noaion (2.15. Then, he ideniie Ψ A (, Ψ B (, = Ψ A (+ξ,+ξ Ψ B (, = are aified for all (,,ξ 3. (2.15 Ψ B (,r[a(r B(r]Ψ A (r,dr, (2.16 Ψ B (,r[a(r +ξ B(r]Ψ A (r+ξ,+ξdr, (2.17 Proof. Le u denoe by H he funcion defined by he following correpondence rule H(, = Ψ A (, Ψ B (,. Then, by parial differeniaion of H wih repec o he fir variable and making ome rearrangemen, we ge H(, = Ψ A(, Ψ B(, = (A( B(Ψ A (,+B((Ψ A (, Ψ B (, = (A( B(Ψ A (,+B(H(,. Now, by muliplying o he lef by Ψ B (,r and implifying, we deduce ha Ψ B (,r(a(r B(rΨ A (r, = Ψ B (,r r H(r, Ψ B(,rB(rH(r, = Ψ B (,r ( r H(r, r Ψ B(,r H(r, = r (Ψ B(,rH(r,. Thu, by inegraion over he inerval [,], we have ha Ψ B (,r[a(r B(r]Ψ A (r,dr = Ψ B (,H(, Ψ B (,H(,, which implie (2.16 by noicing ha Ψ B (, = I and H(, =. Now, he proof (2.17 follow by imilar argumen or by direc applicaion of (2.16.

6 6 A. COONEL, CH. MAULÉN, M. PINTO, D. SEPULVEDA Lemma 2.7. Conider he noaion (2.15 and he e 2 and 2 defined a follow { } 2 = (, 2 { } : > and 2 = (, 2 : <, repecively. Aume ha he following hree aemen are rue: A( i an almo periodic marix (ee definiion 2.1, P i a conan projecion marix ha commue wih Φ A and for ome given poiive conan c and α he inequaliy Ψ A (,P c exp( α, (2.18 i aified for all (, ( 2 or for all (,. 2 Then, for all (,,ξ ( 2 repecively (, 2, here exi wo real conan c 1 > and α ],α[, uch ha Ψ A (+ξ,+ξp Ψ A (,P c 1 A( +ξ A( exp( α. (2.19 In paricular, if ξ i an ǫ-almo period of A he inequaliy Ψ A (+ξ,+ξp Ψ A (,P c 1 ǫexp( α, (2.2 i aified for all (,,ξ 2 (repecively (, 2 or equivalenly Ψ A (,P i Bi-almo periodic. Proof. Since he proof wih (,,ξ 2 or wih (,,ξ 2 are analogou, we conider only one of he cae. Then, in order o fix idea, le u conider (,,ξ 2 and recall he noaion ξ defined by ξ F(x = F(x+ξ F(x for any funcion F. (2.21 In paricular, for inance, we have ha ξ Φ(, = Φ( + ξ, + ξ Φ(, and ξ A( = A( + ξ A(. From (2.16 and he hypohei ha P i a conan projecion marix, i.e. P 2 = P, which commue wih Φ( for every, we have ha ξ Ψ A (,P = Ψ A (,rp ξ A(rΨ Aξ (r,pdr. (2.22 Now, by he aumpion (2.18 we follow ha (2.22 implie he following eimae ξ Ψ A (,P c exp( α ξ A(r dr c exp( α ξ A, which implie (2.19. The inequaliy (2.2 follow immediaely from (2.19 uing he fac ha ξ i an ǫ-almo period of A. Definiion 2.3. Conider he noaion (2.15. If he fac ha A( i an almo auomorphic marix (ee definiion 2.1-(ii implie he following convergence (Ψ Aξn Ψ B (, Pd = lim (Ψ B ξn Ψ A (, Pd =, (2.23 n lim n lim n he applicaion Ψ A i called inegrally Bi-almo auomorphic on ],]. Similarly, if he almo auomorphic behavior of A( implie he following convergence (Ψ Aξn Ψ B (, Pd = lim (Ψ B ξn Ψ A (, Pd =, (2.24 n he applicaion Ψ A i called inegrally Bi-almo auomorphic on [, [. Here {ξ n } n=1 and B( denoe he ubequence and he marix wih he properie given on definiion 2.1-(ii. Lemma 2.8. Conider he noaion (2.15. If he aumpion of Lemma 2.7 hold, hen Ψ A i inegrally Bi-almo auomorphic on ],] and on [, [.

7 ALMOST AUTOMOPHIC DELAYED EQUATIONS 7 Proof. Le u conider ha A( i an almo auomorphic marix. Then by definiion 2.1-(ii we follow ha for any equence {ξ n } n=1, here exi a ubequence {ξ n} n=1 of {ξ n } n=1 and a marix B( uch ha lim A ξ n n ( = B( and lim B ξ n n ( = A( for all. (2.25 Now, from he ideniy (2.16, we deduce ha he aumpion (2.18 implie he following bound (Ψ Ψ Aξn B (, P ce α (A ξn B(rdr c 1 e α ( A + B, (2.26 (Ψ Ψ B ξn A (, P ce α (B ξn A(rdr c 1 e α ( A + B, (2.27 for all (,,n 2 N and ome real conan c 1 > and α ],α[. Then, by applying four ime he Lebegue dominaed convergence heorem we deduce he inegrally Bi-almo auomorphic propery of Ψ A. Indeed, firly by he fir limi given in (2.25 we deduce ha for each (, 2 he inegral (A ξ n B(r dr converge o when n. Then, wih hi convergence in mind, in a econd applicaion, from (2.26 we ge ha for each he inegral (ΨAξn Ψ B (,P d converge o when n. Similarly, applying wice more he Lebegue heorem (in he econd inegral (2.25 and in he inequaliy (2.27 we deduce ha for each (, 2 he inegral (B ξ n A(r dr converge o when n and for each he inegral (ΨB ξn Ψ A (,P d converge o when n. Thu, (2.23 hold and Ψ A i inegrally Bi-almo auomorphic. The proof of (2.24 can be obained by imilar argumen. Definiion 2.4. Conider he noaion (2.15. The linear yem (1.2 ha an α-exponenial dichoomy if here exi a projecion P and wo poiive conan c and α uch ha for all (, 2 he eimae G A (, cexp( α wih G A (, = { Φ A (PΦ 1 A (,, Φ A ((I PΦ 1 A (, oherwie, (2.28 i aified. The marix G A i called he Green marix aociaed wih he dichoomy. Lemma 2.9. Conider he noaion (2.15 and define he Green operaor Γ a follow (Γϕ( = G(,ϕ(d,. Aume ha A( i an almo auomorphic marix and (1.2 ha an exponenially dichoomy uch he i projecion commue wih he fundamenal marix Φ A. Then, he following aerion are aified: (i The Green marix G A i inegrally Bi-almo auomorphic. (ii The pace BC(,V, AP(,V and AA(,V are invarian under he operaor Γ. Moreover, here exi wo poiive conan c 1 and c 2 uch ha he following inequaliie Γϕ ϕ α for ϕ BC(,V, 2 2 ( ξ Γϕ( c 1 ϕ L i ( ξ A +c 2 L i ( ξ ϕ for ϕ AP(,V, i=1 are aified. Here L i and ξ denoe he operaor defined on (2.14 and (2.21, repecively. i=1

8 8 A. COONEL, CH. MAULÉN, M. PINTO, D. SEPULVEDA Proof. The proof of (i and (ii are raighforward. Indeed, for (i, le u conider A(, B( and {ξ n } n=1 a given in he proof of Lemma 2.8. Now, by he aumpion we can deduce ha (G Aξn G B (, d and (G B ξn G A (, d when n where G A i he Green marix defined on (2.28. Thu, we can follow ha G A i inegrally Bialmo auomorphic. Meanwhile, we follow he proof of (ii by applicaion of Lemma 2.3, 2.4 and 2.9-(i. 3. Main eul In hi ecion we preen everal reul of Maera ype for (1.3 and (1.1 and relaed wih he almo auhomorphic behavior of he A and f eul for (1.3. Here we preen a reul for he calar abrac cae, ee Theorem 3.1. Then, we exend hi reul can o linear riangular yem and general linear conan yem, ee Theorem 3.2. We alo, preen imple and ueful relaion beween finie and infinie dimenion i deduced from Theorem 3.3. Finally, we preen o reul on he general cae, ee Theorem 3.4 and 3.5. Theorem 3.1. Conider he equaion (1.3 wih A = µ : C and denoe by g he applicaion defined by ( g(, = exp µ(rdr. (3.1 Then, he following aerion are valid: (i Aume ha µ i a funcion belong o AA(,C aifying M(e(µ. Alo, aume ha f i belong o AA(,V. Then, a oluion y of equaion (1.3 i bounded if and only if y AA(,V or equivalenly he unique oluion of equaion (1.3 belong AA(,V i given by g(,f(d, M(e(µ <, y( = (3.2 g(,f(d, M(e(µ >. (ii Aume ha µ( = ia( wih a(d bounded and V a Bohl-Bohr Banach pace. Then, a oluion y of equaion (1.3 i bounded if and only if y i belong AA(,V and i given by ( y( = exp i a(rdr v + ( exp i a(rdr f(d for all v V. (3.3 Proof. (i Before of prove he iem we deduce wo eimae (ee (3.5 and (3.6 and inroduce ome noaion (ee (3.7 o (3.8. Firly, by Lemma 2.3 we can deduce ha he calar equaion x = µ(x ha an α-exponenial dichoomy. Indeed, we noe ha by he hypohei M(e(µ we can alway elec α aifying M(e(µ α >. Then, by applicaion of Lemma 2.3, we have ha here exi a poiive conan c uch ha g(, ce α wih g defined in (3.1. (3.4 Moreover, by applicaion of Lemma 2.7 and inegraion on, we deduce ha here exi c 1 + and α ],α[ uch he following inequaliie ξ g(, f ξ ( d c 1 α f ξ µ for all, (3.5 ξ g(, f ξ ( d c 1 α f ξ µ for all, (3.6

9 ALMOST AUTOMOPHIC DELAYED EQUATIONS 9 hold. Now, le u conider{ξ n } n=1 aequence in. Then, by he aumpion µ belongaa(,v, here exi a ubequence {ξ n } n=1 of {ξ n} n=1 and he funcion µ uch ha lim µ ξ n n ( = µ(, lim µ ξ n n ( = µ(, for all. (3.7 Similarly, given he equence {ξ n } n=1 by he hypohei f AA(,V, here exi a ubequence {ξ n} n=1 of {ξ n} n=1 and he funcion f uch ha lim f ξ n n ( = f(, lim n f ξ ( = f(, for all. (3.8 n Now we develop he proof of he iem. Indeed, we conider ha y i defined by (3.2 and we prove ha y i belong o AA(,V. Le u ar by conidering he noaion y ± and ỹ ± for he funcion defined a follow y + ( = y ( = g(,f(d, ỹ + ( = g(,f(d and ỹ ( = repecively. Then, by algebraic rearrangemen we deduce ha (y + ξ n ( ỹ +( = (y ξ n ( ỹ ( = ξ n g(,f ξ n (d+ ξ n g(,f ξ n (d g(, f(d, (3.9 g(, f(d, (3.1 g(,(f ξ n f(d, (3.11 g(,(f ξ n f(d, (3.12 Now, by uing Lebegue dominaed convergence heorem we ge ha he four inegral on (3.11- (3.12 converge o when n. Indeed, by (3.5 and (3.7 we follow ha he fir inegral in (3.11 converge o when n. We ee ha he econd inegral in (3.11 vanihe when n by conequence of (3.8. Meanwhile, we noe ha boh inegral in (3.12 converge o when n by applicaion of (3.6, (3.7 and (3.8. Conequenly, we have ha lim (y ± ξn ( = ỹ ± ( for all. (3.13 n Similarly, we can prove ha (ỹ ± ξn ( y( for all and when n. Hence y AA(,V. (ii Noicing ha h( = exp(i af( AA(,V and he Banach pace V ha he Bohl-Bohr propery we follow he proof by applicaion of Theorem 2.1. Theorem 3.2. Conider ha A( AA(,C p p i an upper riangular of order p p. Then he following aerion are valid: (i Aume ha A( aifie he condiion e(m(a ii for all i = 1,...,p and f AA(,V p. Then, a oluion y of (1.3 i bounded if and only if y AA(,V p (ii Aume ha A( aifie he condiion a kk ( = iβ k ( wih β k (d bounded for all k = 1,,p. (3.14 Aume ha he Banach pace V ha he Bohl-Bohr propery. Then any oluion y of yem (1.3 i bounded if and only if y i belong AA(,V p. Proof. (i If we conider ha A( i riangular marix, we have ha he yem (1.3 i of he following ype y 1 = a 11(y 1 + a 12 (y 2 + a 13 (y a 1p (y p + f 1 ( y 2 = a 22 (y 2 + a 23 (y a 2p (y p + f 2 (. y p =.. a pp(y p + f p (. (3.15

10 1 A. COONEL, CH. MAULÉN, M. PINTO, D. SEPULVEDA We noe ha he p-h equaion in (3.15 can be analyzed by applicaion of Theorem 3.1. Indeed, by Theorem 3.1-(i, we have ha here exi y p AA(,V given for g p (,f(d, M(e(µ p <, ( y( = wih g p (, = exp a pp (rdr. g p (,f(d, M(e(µ p >. Similarly, by ubiuing y p AA(,V in (p 1 h equaion of (3.15 and by a new applicaion of Theorem 3.1-(i we can find an explici expreion for y p 1 (. Thi argumen can be repeaed o conruc y p 2 (,y p 3 (,...,y 2 ( and y 1 ( by backward ubiuion and applicaion of Theorem 3.1-(i in he yem (3.15. Hence, we can conruc y( an alo ge ha he concluion of he heorem 3.2-(i i valid. (ii The proof of hi iem i imilar o he proof of he preceden iem (i of he Theorem 3.2. In a broad ene, in hi cae we apply Theorem 3.1-(ii inead of Theorem 3.1-(i and imilarly we ue backward ubiuion. Theorem 3.3. Le V be a Banach pace having he Bohl-Bohr propery. Le {µ i } p i=1 be he eigenvalue of he p p conan marix A aifying µ i = 1. Then any bounded oluion of (1.3 y AA(,V p. When all he eigenvalue µ i are diinc, hee oluion have he form [ ] y( = exp(a v + exp( Af(d for all v V p. (3.16 In he general cae, a formula for he bounded oluion can be alo obained. Proof. If {µ i } p i=1 are diinc, he conan yem (1.2 i imilar o a diagonal yem. Then, wihou lo of generaliy, we can uppoe ha A i an upper riangular marix. Hence, he reul (3.16 follow by applicaion of he Theorem 3.2-(i. Theorem 3.4. Conider A : V V an infinieimal generaor of a C group of bounded linear operaor T( wih and define he Green funcion { T( P, G(, = (3.17 T( (I P,. Aume ha A ha an α-exponenial dichoomy, i.e., here exi wo conan c and α uch ha G(, ce α for all (, 2. (3.18 Then if f AA(,V, equaion (1.3 ha a unique oluion y AA(,V given by y( = G(,f(d for all (3.19 and aifying he following eimae Proof. From (3.17 and (3.18 and y BC(,V y 2c α f. (3.2 lim G(,y( = (3.21 ± Indeed, for we have ha G(,y( ce α( y, which implie ha G(,y( when. Similarly, we ge ha G(,y( vanihe when. We noe ha, for (fix, applying T( on he ideniy y ( = Ay( + f( and uing he fac ha A commue wih T( on he domain of A, we ge T( y ( = T( Ay(+T( f( = AT( y(+t( f(. (3.22

11 ALMOST AUTOMOPHIC DELAYED EQUATIONS 11 Now, he proof coni of hree main par: (a we prove ha he oluion y of (1.3 i given by (3.19; (b we prove ha y AA(,V and (c we prove he uniquene. (a. Proof of ha he oluion y of (1.3 i given by (3.19. Firly, we noe ha a formal inegraion of (3.22 on (, give he following ideniy T( Py (d = AT( Py(d+ T( Pf(d. (3.23 Moreover we have ha d d T( y( = AT( y(+t( y (. (3.24 Then an inegraion on [r,] implie he following relaion Py( T( rpy(r = Now, by (3.21 leing r, we deduce ha Py( = r AT( Py(d+ AT( Py(d+ Here, we noe ha a inegraion of (3.22 on [, wih Q = I P give T( Qy ( = and a inegraion of (3.24 on [,r] yield T( rqy(r Qy( = r AT( Qy(d+ AT( Qy(d+ Now, by (3.21 and leing r in he la relaion we ge (I Py( = AT( (I Py(d+ The relaion (3.28 ogeher wih (3.27 yield (I Py( = Then, from (3.23, (3.26 and (3.29 we obain r T( Py (d. (3.25 T( Py (d. (3.26 r T( Qf(d. (3.27 T( Qy (d. T( (I Py (d. (3.28 T( (I Pf(d. (3.29 y( = Py(+(I Py( (3.3 = = T( Pf(d G(, f(d. Thu, we conclude he proof of (3.19. T( (I Pf(d (b. Proof y AA(,V. Theproofofhiproperyfollowby(3.3hehypoheif AA(,V and applicaion of Lemma 2.9. (c. Proof of uniquene of bounded oluion. The uniquene of he bounded oluion for (1.3 follow by he fac ha x i he unique bounded oluion on of he linear equaion (1.2. Indeed, ifx BC(,V iaoluionofhelinearyemwehaveha x( = Px(+(I Px( = x 1 ( + x 2 (. Noe ha x 1 a and x 2 a, by he exponenial dichoomy. Finally, we noe ha (3.2 i a conequence of (3.3.

12 12 A. COONEL, CH. MAULÉN, M. PINTO, D. SEPULVEDA Theorem 3.5. Aume ha A AA(,C p p and (1.2 ha an exponenial dichoomy wih a projecion P ha commue wih he fundamenal marix Φ A (. Aume ha f AA(,V p. Then, he linear non-homogeneou equaion (1.3 ha a unique AA(,V p oluion given by y( = G(, f(d, aifying (3.2. Proof. By applicaion of Lemma 2.9. Theorem 3.6. Conider V be a Hilber pace and A a linear compac operaor on V. Suppoe ha V = k=1 V k i a Hilber um uch ha V k i a finie dimenional ubpace of V for each k N. Suppoe ha each orhogonal projecion P k on V k commue wih A. If f AA(,V, hen every bounded oluion y of (1.3 i belong AA(,V. Proof. Noicing ha for any y V, we have ha y = k=1 P ky = k=1 y k. Then, by he fac ha A i bounded on V, we deduce ha Ay = Ay k = AP k y = P k Ay. (3.31 k=1 k=1 Now, from he hypohei ha f AA(,V, we have ha for any ubequence { τ n } n=1, here exi a ubequence {τ n } n=1 { τ n } n=1 and a funcion f uch ha f τn ( f( and f τn ( f( poinwie on when n. Then, by compacne of A we deduce ha Af τn ( A f( and A f τn ( Af( poinwie on when n. Now, chooing y k ( = P k y( and auming ha y i oluion of equaion (1.3 we can deduce ha k=1 y k = P k y = P k (Ay(+f( = AP k y(+p k f( = Ay k (+P k f( or equivalenly y k aifie he equaion (1.3 in he finie dimenional pace V k wih P k f( AA(,V k incep k iaboundedlinearoperaor. { } Thu, y k iboundedifandonlyify k AA(,V k. Now, if y( i bounded he e Ay( i relaively compac in V. Hence k=1 P kay( = Ay( uniformly on. On he oher hand P k Ay( AA(,V k ince P k Ay( = AP k y( = Ay k (. Then, Ay( AA(,V and y ( AA(,V ince y( aifie he equaion (1.3. Therefore, uing he Theorem 2.1, y AA(,V ince y BC(,V and V i a Hilber pace eul for (1.4. Before ar we recall he noaion (ϕ,ρ and ϕ given on (2.1. Here, in hi ubecion, we preen wo reul for (1.4 auming fundamenally ha g aifie he aumpion given on (2.1 and f if a funcion uch ha he inequaliy f αρ 2c, (3.32 i aified for ome poiive conan α and c, hen (ϕ,ρ or equivalenly ϕ ρ. Theorem 3.7. Conider A( AA(,C p p uch ha (1.2 ha an α-exponenial dichoomy wih a projecion P ha commue wih Φ A. Aume ha g aifie he aumpion given on (2.1 and f i eleced uch ha he (3.32 hold. Then, if 4cL < α he equaion (1.4 ha a unique oluion belong AA(,V p. Proof. Le = AA(,V p (ϕ,ρ and G = G(, he Green marix aociaed o α- exponenial dichoomy, i.e G(, ce α for ome c,α + and for all (, 2.

13 ALMOST AUTOMOPHIC DELAYED EQUATIONS 13 Now, for ϕ, by Propoiion 2.2 we follow ha he funcion g(,ϕ( AA(,V p. Then, by Theorem 3.5, we have ha (Γϕ( = G(, [f( + g(, ϕ(]d (3.33 i he unique AA(,V p oluion of he yem (1.4. Moreover, (Γϕ( aifie he inequaliy Γϕ ϕ 2cL 2cL ϕ α α ( ϕ ϕ + ϕ 4cL α ρ ρ. Thu, i invarian under Γ. Furhermore, we noe noe ha Γ i a conracion, ince (Γϕ 1 ( (Γϕ 2 ( L G(, ϕ 1 ( ϕ 2 ( d 2cL α ϕ 1 ϕ 2. Hence by he Banach fixed poin argumen, we deduce ha Γ ha a unique fixed poin ϕ. Conequenly, he funcion ϕ i he unique oluion of (1.4. Theorem 3.8. Conider ha A i a linear operaor aifying he aumpion given on Theorem 3.4. Aume ha g aifie he aumpion given on (2.1 and f i eleced uch ha he (3.32 hold. Then, if 4cL < α he equaion (1.4 ha a unique mild oluion belong AA(,V p. Proof. Le = AA(,V (ϕ,ρ and ϕ AA(,V. Then ϕ τ AA(,V. Now, by he compoiion Propoiion 2.2, we have ha ψ ϕ ( = g(,ϕ τ ( AA(,V. Moreover, applying imilar argumen o ha ued in Theorem (3.7, we deduce ha he Green operaor (Γϕ( = ϕ (+ G(,ψ ϕ (d map ino and addiionally we can prove ha Γ : i a ric conracion. Thu, he Banach principle inure he exience of a unique ϕ aifying (3.32. The proof i now complee eul for (1.1. In hi ubecion we generalize he Theorem 3.7 and 3.8 o he cae of he delay equaion (1.1. Theorem 3.9. Conider ha τ > i a conan delay. Then, we have ha he following aerion are valid: (i If A, f and g aify he hypohei of Theorem 3.7. Then, he concluion of Theorem 3.7 hold for he delayed equaion (1.1. (ii If A, f and g aify he hypohei of Theorem 3.8. Then, he concluion of Theorem 3.8 are rue for he delayed equaion 1.1 Proof. The proof follow by applicaion of Theorem 3.7 and 3.8 ince by Propoiion 2.2 we have ha g(,φ( τ AA(,V for every φ AA(, (ϕ,ρ. Indeed, hi fac i a conequence of he following fac: φ AA(,V implie ha he ranlaion φ τ ( = φ( τ i belong AA(,V. 4. Applicaion o he Delayed Laoa-Wazewka Model The Laoa-Wazewka model i an auonomou differenial equaion of he form y ( = δy(+pe γy( τ,. (4.1 I wa occupied by Wazewka-Czyzewka and Laoa [36] o decribe he urvival of red blood cell in he blood of an animal. In hi equaion, y( decribe he number of red cell blood in he ime,δ > i he probabiliy of deah of a red blood cell; p,γ are poiive conan relaed wih he producion of red blood cell by uniy of ime and τ i he ime required o produce a red blood cell. In hi ecion, we udy he following delayed model: y ( = δ(y(+p(g(y( τ, (4.2

14 14 A. COONEL, CH. MAULÉN, M. PINTO, D. SEPULVEDA where τ >, δ(, p( are poiive almo auomorphic funcion and g( i a poiive Lipchiz funcion wih Lipchiz conan γ. Equaion (4.2 model everal iuaion in he real life, ee [22]. We will aume he following condiion (D The mean of δ aifie M(δ > δ >. In hi ecion, he principal goal i he following Theorem: Theorem 4.1. In he above condiion, for γ ufficienly mall, he equaion (4.1 ha a unique almo auomorphic oluion. By Lemma 2.3, he linear par of equaion (4.1 ha an exponenial dichoomy. Le ψ( be a real almo auomorphic funcion and conider he equaion y ( = δ(y(+p(g(ψ( τ. (4.3 Then, he bounded oluion for he equaion (4.3 aifie ( y( = exp δ(d p(ug(ψ(u τdu. u The homogeneou par of equaion of (4.3 ha an exponenial dichoomy and ince δ i almo auomorphic funcion, by Lemma 2.8, i i inegrally Bi-almo auomorphic. Therefore, Theorem 4.1 follow from Theorem 3.7. Taking g(x = e γx, α >, we have he Laoa-Wazewka model: y ( = δ(y(+p(e γy( τ,. (4.4 Corollary 4.2. For γ mall enough, he delayed Laoa-Wazewka model (4.4 ha a unique aympoically able almo auomorphic oluion. eference [1] J. Blo, G. Mophou, G. M. N Guérékaa and D. Pennequin. Weighed peudo almo auomorphic funcion and applicaion o abrac differenial equaion. Nonlinear Analyi, 71 (29, [2] S. Bochner. A new approach o almo periodiciy. Proceeding of he Naional Academic Science of he Unied Sae of America, 48 ( [3] S. Bochner. Coninuou mapping of almo auomorphic and almo periodic funcion, Proceeding of he Naional Academic Science of he Unied Sae of America, 52 ( [4] T. Caraballo, D. N. Cheban, Almo periodic and almo auomorphic oluion of linear differenial/difference equaion wihou Favard eparaion condiion. I and II, J. Differenial Equaion 246(1 (29, and [5] S. Caillo, M. Pino, Dichoomy and almo auomorphic oluion of difference yem. Elecron. J. Qual. Theory Differ. Equ. 213, No. 32, 17 pp. [6] A. Chávez, S. Caillo, M. Pino, Diconinuou almo auomorphic funcion and almo auomorphic oluion of differenial equaion wih piecewie conan argumen. Elecron. J. Differenial Equaion 214, No. 56, 13 pp. [7] P. Cieua, S. Faajou and G. M. N Guérékaa. Compoiion of peudo almo periodic and peudo almo auomorphic funcion and applicaion o evoluion equaion. Applicable Analyi, 89 1 (21, [8] C. Corduneanu, Almo Periodic Funcion. John Wiley and Son, New York, [9] C. Cueva, M. Pino, Exience and uniquene of peudo-almo periodic oluion of emilinear Cauchy problem wih non dene domain. Nonlinear Anal., 45 (21, no.1, [1] H.-S. Ding, T.-J. Xiao, J. Liang, Aympoically almo auomorphic oluion for ome inegrodifferenial equaion wih nonlocal iniial condiion. J. Mah. Anal. Appl. 338 ( [11] T. Diagana, Peudo Almo Periodic Funcion in Banach Space. Nova Science Publiher Inc., New York. 27 [12] S. Faajou, N. V. Minh, G. N Guérékaa and A. Pankov. Sepanov-like almo auomorphic oluion for nonauonomou evoluion equaion. Elecronic Journal of Differenial Equaion, 121 ( [13] C. Feng, On he exience and uniquene of almo periodic oluion for delay logiic equaion. Appl. Mah. Compu., 136(2-3 ( [14] S. G. Gal and G. M. N Guérékaa. Almo auomorphic fuzzy-number-valued funcion. Journal of Fuzzy Mahemaic, 13 1 ( [15] J. A. Goldein, G. M. N Guérékaa. Almo auomorphic oluion of emilinear evoluion equaion. Proc. Amer. Mah. Soc., 133(8 (

15 ALMOST AUTOMOPHIC DELAYED EQUATIONS 15 [16]. C. Grimmer, eolven operaor for inegral equaion in a Banach pace. Tran. Amer. Mah. Soc., 273( [17] E. Hernández, J. P. C. do Sano, Aympoically almo periodic oluion for a cla of parial inegrodifferenial equaion. Elecron. J. Differenial Equaion, 38 (26, 1-8. [18] H.. Henríquez, M. Pierri, P. Táboa, On S-aympoically ω-periodic funcion on Banach pace and applicaion, J. Mah. Anal. Appl., 343(2 ( [19] H.. Henríquez, M. Pierri, P. Táboa, Exience of S-aympoically ω-periodic oluion for abrac neural equaion. Bull. Aural. Mah. Soc., 78 (3 (28, [2] Z. C. Liang, Aympoically periodic oluion of a cla of econd order nonlinear differenial equaion. Proc. Amer. Mah. Soc., 99(4 (1987, [21] J. Liu, G. M. N Guérékaa and N. V. Minh. A Maera ype heorem for almo auomorphic oluion of differenial equaion. Journal of Mahemaical Analyi and Applicaion, 299 ( [22] E. Liz, C. Marínez, S. Trofimchuk, Araciviy properie of infinie delay Mackey-Gla ype equaion, Differenial and Inegral Equaion, Vol. 15 (22, [23] N. V. Minh, T. Naio and G. N Guérékaa. A pecral counabiliy condiion for almo auomorphy of oluion of differenial equaion. Proceeding of he American Mahemaical Sociey, ( [24] N. V. Minh and T. T. Da. On he almo auomorphy of bounded oluion of differenial equaion wih piecewie conan argumen. Journal of Mahemaical Analyi and Applicaion, ( [25] G. N Guérékaa. Topic in Almo Auomorphy. Springer-Verlag, New York, 25. [26] G. N Guérékaa. Almo Auomorphic and Almo Periodic Funcion in Abrac Space. Kluwer Academic/Plenum Publiher, New York, USA, 21. [27] S. Nicola, M. Pierre, A noe on S-aympoically periodic funcion. Nonlinear Analyi, eal World Applicaion, 1 (5(28, [28] M. Pino, Peudo-almo periodic oluion of neural inegral and differenial equaion and applicaion. Nonlinear Analyi, 72 ( [29] M. Pino, G. obledo, Cauchy marix for linear almo periodic yem and ome conequence. Nonlinear Anal. 74 (211, no. 16, [3] M. Pino, G. obledo, Diagonalizabiliy of nonauonomou linear yem wih bounded coninuou coefficien. J. Mah. Anal. Appl. 47 (213, no. 2, [31] M. Pino,G. obledo, Exience and abiliy of almo periodic oluion in impulive neural nework model. Appl. Mah. Compu. 217 (21, no. 8, [32] M. Pino, V. Torre, G. obledo, Aympoic equivalence of almo periodic oluion for a cla of perurbed almo periodic yem. Glag. Mah. J. 52 (21, no. 3, [33] W.. Uz, P. Walman, Aympoically almo periodiciy of oluion of a yem of differenial equaion. Proc. Amer. Mah. Soc., (18(1967, [34] W. A. Veech. Almo auomorphic funcion. Proceeding of he Naional Academy of Science of he Unied ae of America, 49 ( [35] W. A. Veech. Almo auomorphic funcion on group. American Journal of Mahemaic, 87 ( [36] M. Wazewka-Czyzewka and A. Laoa, Mahemaical problem of he red-blood cell yem, Ann. Polih Mah. Soc. Ser. III, Appl. Mah. 6 ( [37] F. Wei, K. Wang, Global abiliy and aympoically periodic oluion for non auonomou cooperaive Loka-Volerra diffuion yem. Applied Mah. and Compuaion, 182(26, [38] F. Wei, K. Wang, Aympoically periodic oluion of N-pecie cooperaion yem wih ime delay. Nonlinear Analyi, eal World and Applicaion. 7(26, [39] T. Xiao, X. Zhu, J. Liang, Peudo-almo auomorphic mild oluion o nonauonomou differenial equaion and applicaion, Nonlinear Analyi, 7(29 11, [4] T. Yohizawa, Sabiliy Theory and he Exience of Periodic Soluion and Almo Periodic Soluion. Applied Mahemaical Science, 14, Springer-Verlag, New York- Heidelberg, [41] H. Zhao, Exience and global araciviy of almo periodic oluion for cellular neural nework wih diribued delay. Appl. Mah. Compu., 154(3(24, [42] S. Zaidman. Almo auomorphic oluion of ame abrac evoluion equaion. Iniuo Lombardo, Accademia di Science e Leer, 11 2 ( [43] S. Zaidman. Exience of aympoically almo-periodic and of almo auomorphic oluion for ame clae of abrac differenial equaion. Annale de Science Mahémaique du Québec, 131 ( [44] M. Zaki. Almo auomorphic oluion of cerain abrac differenial equaion. Annali di Mahemaica pura e Applicaa, 11 1 ( [45] C. Zhang. Almo Periodic Type Funcion and Ergodiciy. Science Pre, Beijing, 23. Aníbal Coronel GMA, Deparameno de Ciencia Báica, Faculad de Ciencia, Univeridad del Bío-Bío, Campu Fernando May, Chillán, Chile. addre: acoronel@ubiobio.cl

16 16 A. COONEL, CH. MAULÉN, M. PINTO, D. SEPULVEDA Chriopher Maulén Deparameno de Maemáica, Faculad de Ciencia, Univeridad de Chile addre: Manuel Pino Deparameno de Maemáica, Faculad de Ciencia, Univeridad de Chile addre: Daniel Sepulveda Ecuela de Maemáica y Eadíica, Univeridad Cenral de Chile addre: daniel.ep.oe@gmail.com