ASYMPTOTIC EQUIVALENCE OF ALMOST PERIODIC SOLUTIONS FOR A CLASS OF PERTURBED ALMOST PERIODIC SYSTEMS

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1 Glasgow Mah. J. 52 (2010) C Glasgow Mahemaical Journal Trus doi: /s ASYMPTOTIC EQUIVALENCE OF ALMOST PERIODIC SOLUTIONS FOR A CLASS OF PERTURBED ALMOST PERIODIC SYSTEMS MANUEL PINTO Deparameno de Maemáicas, Faculad de Ciencias, Universidad de Chile, Casilla 653, Saniago , Chile pinoj@uchile.cl VICTOR TORRES Deparameno de Ciencias Físicas y Maemáicas, Universidad Aruro Pra, Avda. Aruro Pra 2120, Iquique , Chile vorres@unap.cl and GONZALO ROBLEDO Deparameno de Maemáicas, Faculad de Ciencias, Universidad de Chile, Casilla 653, Saniago , Chile grobledo@uchile.cl (Received 9 June 2009; revised 30 December 2009; acceped 17 June 2010) Absrac. The soluions of a perurbed linear ordinary differenial equaion (ODE) sysem are sudied. Provided ha some inegrabiliy and oddness condiions are saisfied, we show ha hey are asympoically equivalen a =± o he soluions of he unperurbed one. This fac is used o deermine he exisence of almos periodic or pseudo-almos periodic soluions of he perurbed sysem Mahemaics Subjec Classificaion. 34E10, 34C27, 34C Inroducion. In his noe, we consider he perurbed sysems of ordinary differenial equaions and is relaion wih he linear sysem y = A()y + f (, y), (1.1) z = A()z + B()z + h() (1.2) x = A()x. (1.3) By following an idea developed by Akhme e al. [1] for he linear sysem (1.2), we obain sufficien condiions ensuring he asympoic equivalence beween (1.1) (1.2) and (1.3). The resuls are used o prove he exisence of asympoically almos periodic soluions and pseudo-almos periodic soluions of (1.1) (1.2) provided ha (1.3) has non-rivial almos periodic soluions. Throughou his noe, f : n n, h: n are coninuous. A( ),B( ) are n n coninuous marices. A soluion of (1.1) wih iniial condiion y( 0 ) = b n will be denoed as y(, 0, b), he fundamenal marix of (1.3) is denoed by () and Downloaded from hps:// IP address: , on 15 Sep 2018 a 02:42:09, subjec o he Cambridge Core erms of use, available a hps:// hps://doi.org/ /s

2 584 MANUEL PINTO, VICTOR TORRES AND GONZALO ROBLEDO (, s) = () 1 (s), he marix and vecorial norm are denoed by and,he Landau s noaion: p 1 () = o(1) and p 1 () = O(p 2 ()) means, respecively, ha lim + p 1 () = 0 and ha here exis a consan M verifying p 1 () < M p 2 () for big values of. Finally, hroughou his noe, we assume ha (L) There exiss a measurable funcion η : + such ha f (, y 1 ) f (, y 2 ) η() y 1 y 2 for any and y 1, y 2 n. Our main resuls are: THEOREM 1. If he following condiions are saisfied: (H1) 1 () () η() and 1 ()f (, 0) are inegrable on (0, + ), (H2) (, s) (s) η(s) ds = o(1) and lim + (, s)f (s, 0) ds = 0, hen all he soluions of (1.1) are defined on + and here exiss a homeomorphism beween he soluions x() = x(, 0, b) of sysem (1.3) and y() = y(, 0, H(b)) of (1.1), where H is a n homeomorphism and b is arbirary. Moreover, hese soluions saisfy: lim x() y() =0. (1.4) + In addiion, if he assumpions (H3) 1 ()f (, ()x) is odd for any x n, (H4) lim (, s) (s) η(s) ds = 0 and lim are saisfied, hen he soluions saisfy: (, s)f (s, 0) ds = 0 lim x() y() =0. (1.5) THEOREM 2. If he following condiions are saisfied: (L1) 1 ()B() () and 1 ()h() are inegrable on (0, + ), (L2) (, s)b(s) (s) ds = o(1) and lim + (, s)h(s) ds = 0, hen he soluions of (1.2) are defined on + and here exiss a homeomorphism beween he soluions x() = x(, 0, b) of (1.3) and z() = z(, 0, H(b)) of (1.2), whereh is a n homeomorphism and b is arbirary. Moreover, he soluions saisfy lim x() z() =0. (1.6) + Moreover, if he condiions (L3) 1 ()B() () and 1 ()h() are odd. (L4) lim (, s)b(s) (s) ds = 0 and lim are saisfied, hese soluions saisfy lim x() z() =0. (, s)h(s) ds = 0, The exisence of an homeomorphism beween x(, 0, b)and y(, 0, H(b)) combined wih (1.4) is known as asympoic equivalence beween (1.1) and (1.3) (see [1, 3, 4]and references herein). In addiion, a bi-asympoic equivalence exiss when (1.5) is also verified (see [1] for deails). Noice ha (L3) (L4) can be saisfied in several cases. For example, if ( ) is eiher even or odd and B( ) is odd, hen 1 ()B() () is odd. Noice ha if A( ) is odd, hen ( ) is even. Finally, 1 ()B() () is also odd in oher cases: i) when B( ) isoddand Downloaded from hps:// IP address: , on 15 Sep 2018 a 02:42:09, subjec o he Cambridge Core erms of use, available a hps:// hps://doi.org/ /s

3 ASYMPTOTIC EQUIVALENCE OF ALMOST PERIODIC SOLUTIONS 585 commues wih ( ). ii) when A() commues wih 0 A(s) ds and B(). A number of concree resuls and examples appear in he las secion. An ineresing applicaion of hese resuls is a crierion o deermine he exisence of pseudo-almos periodic soluions for (1.1) (1.2). The pseudo-almos periodic funcions were inroduced by Zhang [22] and generalizes he concep of almos periodic funcion: DEFINITION 1. [1, 7, 10, 17, 21, 22] A coninuous funcion r: n is (i) Almos periodic i.e. r AP( ), if for each ε>0here exiss l ε > 0 such ha every inerval of lengh l ε conains a number τ wih r( + τ) r() <ε. (ii) Asympoically almos periodic i.e. r AAP( ), if r = g + ϕ wih g AP( ) and ϕ = o(1) as +. In addiion, if ϕ = o(1) also for, we will say ha r AAP 0 ( ). (iii) Pseudo-almos periodic i.e. r PAP( ), if r = g + ϕ wih g AP( ) and he ergodic componen ϕ PAP 0 ( ) is bounded coninuous and saisfies lim τ + 1 2τ τ τ ϕ() d = 0. REMARK 1. Noice ha AP( ) AAP 0 ( ) PAP( ) (seee.g.[1, 2, p. 548]). The exisence of AP( ),AAP 0 ( ) andpap( ) soluions is among he mos aracive opics in qualiaive heory of ordinary differenial equaions due o heir applicaions, especially in biology [14, 20], number heory [8, 11], neural neworks [13, 15] and physics [16]. A well known resul relaes exponenial dichoomy (see e.g. [6, 18]) wih he exisence of a unique AP( )orpap( ) soluion of perurbed sysems: PROPOSITION 1. [10, Theorem 8.1], [21, Theorem 3.1] Le us consider he perurbed sysem y = A()y + h(, y), where A( ) AP( ) and h: n n is such ha (i) h(, y 0 ) AP( )(resp. PAP( )) for any y 0 n. (ii) There exiss L > 0 such ha h(, x) h(, y) L x y x, y n. If he sysem (1.3) has an exponenial dichoomy, hen for L sufficienly small i follows ha he perurbed sysem has a unique AP( ) (resp. PAP( )) soluion. Wihou considering exponenial dichoomy, he resuls abou exisence of AP( ) soluions are limied in number [10, 12, 22]: Van Vleck [19] and Buron [5] consider A( ) ω periodic and skew symmeric (i.e. A() = A( + ω) anda T () = A()) and obain condiions for he exisence of AP( ) soluions for (1.3). Akhme e al. [1] recenly sudied A( ) odd,b( ) even and obain he exisence of AAP 0 ( ) soluions for (1.2), hey also define AAP 0 ( ) as bi-asympoically almos periodic funcions. Our objecive is o deermine when he non-zero AP( ) soluions of he linear sysem (1.3) arise AP( ) soluions for he perurbed sysems (1.1) (1.2). To work in his framework has wo consequences: firsly, exponenial dichoomy is no verified because i would imply ha zero is he unique AP( ) soluion of (1.3). Secondly, he condiions (i) (ii) of Proposiion 1 are no sufficien o ensure exisence of almos periodic soluions for he perurbed sysems and new condiions are needed. We will propose a se of inegrabiliy and oddness condiions ensuring he exisence of a Downloaded from hps:// IP address: , on 15 Sep 2018 a 02:42:09, subjec o he Cambridge Core erms of use, available a hps:// hps://doi.org/ /s

4 586 MANUEL PINTO, VICTOR TORRES AND GONZALO ROBLEDO manifold of AAP 0 ( ) (resp.pap( )) soluions of (1.1) and (1.2). These resuls are complemenary o he previous ones and are saed as follows: THEOREM 3. Assume ha he sysem (1.3) has a k parameer (k n) family S 1 of AP( ) (resp. PAP( )) soluions. If he assumpions (H1) (H4) are saisfied, hen (1.1) has a k parameer family S 2 of AAP 0 ( )(resp. PAP( )) soluions homeomorphic o S 1. THEOREM 4. Assume ha he sysem (1.3) has a k parameer (k n) family S 1 of AP( ) (resp. PAP( )) soluions. If B() and h() saisfies he assumpions (L1) (L4) hen, (1.2) has a k parameer family S 2 of AAP 0 ( ) (resp. PAP( )) soluions homeomorphic o S 1. The paper is organized as follows: Secion 2 gives a se of preliminary resuls and he proofs of he Theorems 1 and 2. Theorems 3 and 4 are proved in Secion 3, where some illusraive examples and consequences are also included. 2. Exisence of soluions and asympoic equivalence. The following resul will be used a several seps of he paper: THEOREM 5. Le us consider he non-linear sysem w = g(,w). (2.1) If g: n n saisfies he condiions: (G1) There exiss a measurable and inegrable funcion ξ : + such ha g(,w 1 ) g(,w 2 ) ξ() w 1 w 2 for any and w i n, (G2) The funcion g(, 0) is inegrable, (G3) g(,w) = g(,w) for any and w n, hen here exiss a Lipschiz homeomorphism beween he soluions of (2.1) and n given by he correspondence beween b n and he unique even soluion of (2.1) saisfying lim ± w() = b. More precisely, w() = b g(s, 0) ds + ϕ() wih ϕ() = ( ( O exp ξ(s) ds ( ) ( O exp ξ(s) ds ) ) 1 ) 1 Proof. Noice ha (2.1) can be wrien as follows: 0, < 0. ẇ = g(,w) + g(, 0) wih g(,w) = g(,w) g(, 0). (2.2) Firsly, we will sudy he soluions defined on 0, he exisence is proved by using successive approximaions: le w 0 () = b g(s, 0) ds and w k () = b g(s, 0) ds g ( s,w k 1 (s) ) ds k 1. Downloaded from hps:// IP address: , on 15 Sep 2018 a 02:42:09, subjec o he Cambridge Core erms of use, available a hps:// hps://doi.org/ /s

5 ASYMPTOTIC EQUIVALENCE OF ALMOST PERIODIC SOLUTIONS 587 By inducion and (G1) (G2), i is sraighforward o prove ha for 0: ( g(s, 0) ds, w k () w k 1 () b +ρ k! ξ(s) ds) k, ρ = sup 0 combining his fac wih w k () = w 0 () + k 1 j=0 {w j+1() w j ()}, i follows ha k 1 w k () b +ρ + j=1 b +ρ j! ( j. ξ(s) ds) (2.3) Assumpion (G1) implies ha w k is uniformly convergen o a bounded and coninuous funcion w(). By leing k, i follows ha w() = b g(s,w(s)) ds = w 0 () g(s,w(s)) ds, 0 (2.4) and he exisence of a soluion w( ) convergen oward b when + follows. Noice ha on, w 0 () is even since g(, 0) is odd. Moreover, i can be proved by inducion ha on, w k () is even and we conclude ha w()isevenon. The asympoic esimaion (2.2) can be proved by leing k + in (2.3) combined wih (G1) and (2.4). Given wo soluions w i () (i = 1, 2) saisfying lim + w i () = b i, le us denoe u() = w 1 () w 2 (). By using (G1) combined wih (2.4), we can prove ha u() b 1 b 2 + ξ(s) u(s) ds. By following Gronwall s Lemma argumens, we can deduce ha w 1 () w 2 () e ξ(s) ds b 1 b 2, 0 and he uniqueness of he soluion convergen o b follows. Moreover, here exiss a homeomorphism H: n n defined by H(b) = w(0), he iniial condiion a = 0 of he soluion w() b. Indeed, he bijeciviy is consequence of he exisence and uniqueness of he soluions of (2.1). Finally, i can be proved by Gronwall s inequaliy ha and we can conclude ha w 1 () w 2 () e 0 ξ(s) ds w 1 (0) w 2 (0), 0 e 0 ξ(s) ds w 1 (0) w 2 (0) b 1 b 2 e 0 ξ(s) ds w 1 (0) w 2 (0). By he evenness of w( ), his resul is valid on. Hence, H is a uniform homeomorphism and he Theorem follows. COROLLARY 1. Le us consider he linear sysem w = F()w + γ (), where F() is a coninuous and inegrable marix funcion and γ : n is inegrable. If F( ) and γ ( ) are odd funcions, hen for any b n here exiss a unique even soluion Downloaded from hps:// IP address: , on 15 Sep 2018 a 02:42:09, subjec o he Cambridge Core erms of use, available a hps:// hps://doi.org/ /s

6 588 MANUEL PINTO, VICTOR TORRES AND GONZALO ROBLEDO saisfying lim ± w() = b. More precisely, we have w() = b γ (s) ds + ϕ() wih ( ( ) ) O exp F(s) ds 1 0 ϕ() = ( ( ) ) O exp F(s) ds 1 < 0. The Theorem 5 and Corollary 1 will be applied in order o prove he Theorems 1 and Proof of Theorem 1. Le y() be a soluion of sysem (1.1) and u() = 1 ()y(). I is sraighforward o prove ha sysem (1.1) becomes u = 1 ()f (, ()u) = g(, u) (2.5) and he condiions (G1) (G2) of Theorem 1 can be verified by using (L) and (H1). Firsly, le us consider 0. By Theorem 5, for any b n here exiss a unique soluion u() convergen o b when +. Furhermore, here is a homeomorphism H: n n such ha H(b) = u(0) = y(0) and lim + 1 ()y() = lim + u() = b. For any b, we consider a soluion y(, 0, H(b)) of (1.1). By Theorem 3 applied o (2.5), here exiss a funcion y() solving he inegral equaion y() = (, 0)b (, s)f ( s, (s)u(s) ) ds, 0, (2.6) where u() = u(, 0, H(b)) is soluion of (2.5). I is clear ha a given iniial condiion H(b) resuls in a homeomorphism beween x(, 0, b) = (, 0)b and y(). Moreover, noice ha y() x() (, s) (s) u(s) η(s) ds + (, s)f (s, 0) ds. By (H2), we can see ha (1.4) follows. Moreover, (H3) implies ha (2.6) holds for 0 since (2.5) saisfies (G3) and Theorem 5 follows. Finally, (1.5) is a consequence of (H4) andhetheoremisproved Proof of Theorem 2. Noice ha u() = 1 ()z() ransforms (1.2) in u = 1 ()B() ()u + 1 ()h(). (2.7) As (L1) (L3) implies ha (2.7) saisfies assumpions of Corollary 1. Following he lines of he previous proof, we obain a represenaion y() = x() + ϕ() wih ϕ() = (, s)b(s) (s)u(s) ds and (L2) combined wih (L4) imply lim ± x() z() =0. (, s)h(s) ds 0 Downloaded from hps:// IP address: , on 15 Sep 2018 a 02:42:09, subjec o he Cambridge Core erms of use, available a hps:// hps://doi.org/ /s

7 ASYMPTOTIC EQUIVALENCE OF ALMOST PERIODIC SOLUTIONS Exisence of PAP( ) soluions and examples Proofs of Theorems 3 and 4. The proofs follow immediaely from Theorems 1 and 2 respecively. We will prove only Theorem 3: le x() = x(, 0, b) S 1. By Theorem 1, here exiss a soluion of (1.1) described by y() = y(, 0, H(b)) = x() + ϕ(), where ϕ is given by ϕ() = () 1 (s)f (s, y(s)) ds (3.1) such ha lim ± ϕ() = 0. Using his fac combined wih x( ) AP( ) (resp. x( ) PAP( )) implies ha y( ) AAP 0 ( )(resp.y( ) PAP( ))andhetheorem1 follows Consequences and illusraive examples. EXAMPLE 1. Assume ha (H1) (H4) are saisfied. If A( ) is ω periodic and skew symmeric (i.e. A() = A( + ω) anda T () = A()), hen all he soluions of (1.1) are AAP 0 ( ) soluion. Indeed, Theorem 1 from [5] says ha all he soluions of (1.3) are AP( ) and he resul follows from Theorem 3. THEOREM 6. Le S 1 be a k parameer family of AP( ) soluions of (1.3): (i) If (H1) (H2) are saisfied, hen here exiss a k parameer family S 2 of AAP( ) soluions of (1.1) homeomorphic o S 1. (ii) Moreover, if A() and f (, x) are odd for any x, hen here exiss a k parameer family S 2 of AAP 0 ( ) soluions of (1.1) homeomorphic o S 1. Proof. (i) is a sraighforward consequence of Theorems 1 and 3. Now we prove (ii): noice ha ( ) is even since A( ) is odd and we can conclude ha (H3) is saisfied. Following he lines of he proof of Theorem 1 combined wih Theorem 5, we can prove ha ϕ() is an even funcion. Finally, he asympoic behaviour (1.4) is obained by leing ± in (2.6) and using (H2) combined wih he evenness of ϕ(). Similarly, THEOREM 7. Le S 1 be a k parameer family of PAP( ) soluions of (1.3): (i) If (H1) (H2) are saisfied, hen here exiss a k parameer family S 2 of AAP( ) soluions of (1.1) homeomorphic o S 1. (ii) Moreover, if A() and f (, x) are odd for any x, hen here exiss a k parameer family S 2 of PAP( ) soluions of (1.1) homeomorphic o S 1. THEOREM 8. If (G1) (G2) are saisfied, hen every soluion of (2.1) is AAP( ). In addiion, if (G3) holds, hen every soluion is AAP 0 ( ). EXAMPLE 2. If (H1) (H2) are saisfied, f (, x) isoddforanyx n and A() is odd and ω periodic, hen all soluions of (1.1) are AAP 0 ( ): Indeed, Theorem 2 from [9] says ha all he soluions of (1.3) are ω periodic and he resul follows from Theorem 6. Observe ha in his las example, we are in presence of a bi-asympoically periodic equivalence. In addiion, all he soluions of examples 1 and 2 are AAP( )orpap( ). Downloaded from hps:// IP address: , on 15 Sep 2018 a 02:42:09, subjec o he Cambridge Core erms of use, available a hps:// hps://doi.org/ /s

8 590 MANUEL PINTO, VICTOR TORRES AND GONZALO ROBLEDO COROLLARY 2. If (H1) (H4) (resp. (L1) (L4)) are saisfied and moreover (i) A() is ω periodic, (ii) The linear sysem (1.3) has a unique bounded soluion, hen he sysems (1.1) (resp.(1.2)) have a soluion AAP 0 ( ). Proof. Le x() be he bounded soluion of sysem (1.3) and le x 1 () = x( + ω). Noice ha x 1 () is also a bounded soluion of (1.3). Hence x 1 () = x() andx() is ω periodic. The res of he proof follows from Theorem 1. COROLLARY 3. Assume ha (i) () and B() commue, (ii) B( ) is odd and B() is inegrable, (iii) lim ± () B(s) ds = 0 hen he conclusions of Theorem 2 holds for z = A()z + B()z. COROLLARY 4. Assume ha (i) A( ) and B( ) are odd, (ii) 1 ()B() () is inegrable, (iii) lim + (, s)b(s) (s) ds = 0 hen he conclusions of Theorem 2 holds for z = A()z + B()z. Observe ha Corollary 4 can be compared wih Theorem 4.1 from [1] inseveral ways: firsly, we work wih an odd perurbaion while in [1] an even perurbaion is considered. The inegrabiliy condiion (ii) is similar. Finally, our condiion (iii) is easier o verify in comparison wih [1] (See is assumpion [C2]). While Ráb Lemma plays a key role in he asympoic equivalence resuls from [1], in our case Theorem 5 makes possible o consider a non-linear version of Corollary 4.1 from [1], as done in Theorems 6 and 7. In addiion, his las one exend he resuls o a PAP( )case. COROLLARY 5. If f (, 0) and η() are inegrable, (H3) follows, he soluions of (1.3) are bounded and lim inf + 0 Tr A(s) ds > (3.2) hen, from each k parameer family of AP( ) (resp. PAP( )) soluions of (1.3) we obain an homeomorphic k parameer family of PAP( ) soluions of he sysem (1.1) such ha (1.4) follows. A corresponding resul holds for sysem (1.2). Proof. As () is bounded, (3.2) implies ha 1 () is bounded. The resul follows since he inegrabiliy of η()andf (, 0) implies (H1), (H2) and (H4). EXAMPLE 3. Le A be a 2n 2n marix wih simple purely imaginary eigenvalues λ =±iω k (k = 1,...,n), B() isa 2n 2n marix funcion, such ha A commues wih B()and B() is odd and inegrable. Consider he sysem: y = Ay + B()y, (3.3) where y 2n, hen Corollary 5 implies ha any soluion of (3.3) is AAP 0 ( ). Noice ha he classic resuls canno be applied o his example (see e.g. [10, 21]) and (1.3) has no an exponenial dichoomy. Downloaded from hps:// IP address: , on 15 Sep 2018 a 02:42:09, subjec o he Cambridge Core erms of use, available a hps:// hps://doi.org/ /s

9 ASYMPTOTIC EQUIVALENCE OF ALMOST PERIODIC SOLUTIONS 591 EXAMPLE 4. Le us consider he scalar sysem y ={cos() + cos(π)}y + e 2 y. Here A() is even and he soluions x() = e sin()+ 1 π sin(π) x(0) of (1.3) are AP( )bu neiher even nor odd. However, by Corollary 5, every soluion y( ) ispap( ). Noe ha his resul canno be obained from [1]. 4. Discussion. We exend some resuls and furnish alernaive mehods compared wih [1]: Firsly, we give an alernaive proof of asympoic equivalence ha employs Theorem 5 and can be applied also for non-linear sysems. On he oher hand, we poin ou ha (G1) could be relaxed by supposing ha he inequaliy is verified only locally (we are preparing his resul in a fuure aricle). Secondly, we can exend he mehods of [1] o prove he exisence of PAP( ) soluions. Finally, we exend some resuls from [1] by supposing B( )odd. We have given only sufficien condiions for he exisence of even PAP( ) soluions (resp. AAP 0 ( )). In his sense, a naural exension of our resuls would be o find sufficien condiions ensuring he exisence of soluions ha are no even (Example 4 shows us ha hese soluions exis). Neverheless, o prove he bi-asympoic equivalence by using condiions weaker han (G3) remain an open quesion, worh furher sudy. ACKNOWLEDGEMENT. The firs wo auhors were suppored by FONDECYT , DGI MATH UNAP The hird auhor was suppored by he Israel Council for Higher Educaion and UNESCO. REFERENCES 1. M. U. Akhme, M. A. Tleubergenova and A. Zafer, Asympoic equivalence of differenial equaions and asympoically almos periodic soluions, Nonlinear Anal. 67 (2007), A. Alonso, J. Hong and R. Obaya, Almos periodic ype soluions of differenial equaions wih piecewise consan argumen via almos periodic sequences, Appl. Mah. Le. 13 (2000), S. Bodine and D. A. Luz, On asympoic equivalence of perurbed linear sysems of differenial and difference equaions, J. Mah. Anal. Appl. 326 (2007), F. Brauer and J. S. W. Wong, On asympoic behavior of perurbed linear sysems, J. Differ. Equ. 6 (1969), T. A. Buron, Linear differenial equaions wih periodic coefficiens, Proc. Amer. Mah. Soc. 17 (1966), W. A. Coppel. Dichoomies in sabiliy heory. Lecure Noes in Mahemaics Vol. 629 (Springer Verlag, Berlin, New York, 1978). 7. C. Cuevas and M. Pino, Exisence and uniqueness of pseudo almos periodic soluions of semilinear Cauchy problems wih non dense domain, Nonlinear Anal. 45 (2001), J. Delsare, Essai sur l applicaion de la héorie des foncions presque périodiques à l arihméique, Ann. Sci. l École Normale Supérieure Ser. 3,62 (1945), I. J. Epsein, Periodic soluions of sysems of differenial equaions, Proc. Amer. Mah. Soc. 13 (1962), A. M. Fink, Almos periodic differenial equaions. Lecure Noes in Mahemaics (Springer-Verlag, Berlin, New York, 1974). Downloaded from hps:// IP address: , on 15 Sep 2018 a 02:42:09, subjec o he Cambridge Core erms of use, available a hps:// hps://doi.org/ /s

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