DSpace at VNU: Massera-type theorem for the existence of C (n)-almost-periodic solutions for partial functional differential equationswith infinite delay

Nonlinear Analysis 69 (2008) 1413–1424 www.elsevier.com/locate/na Massera-type theorem for the existence of C (n)-almost-periodic solutions for partial functional differential equations with infinite delay Khalil Ezzinbi a , Samir Fatajou a , Gaston Mandata N’gu´er´ekata b,∗ a Universit´e Cadi Ayyad, Facult´e des Sciences Semlalia, D´epartement de Math´ematiques, B.P 2390 Marrakech, Morocco b Department of Mathematics, Morgan State University, 1700 East Cold Spring Lane, Baltimore, MD 21251, USA Received 23 May 2007; accepted 29 June 2007 Abstract In this paper, we study the existence of C (n) -almost-periodic solutions for partial functional differential equations with infinite delay We assume that the undelayed part is not necessarily densely defined and satisfies the Hille–Yosida condition We use the reduction principle developed recently in [M Adimy, K Ezzinbi, A Ouhinou, Variation of constants formula and almost-periodic solutions for some partial functional differential equations with infinite delay, Journal of Mathematical Analysis and Applications 317 (2006) 668–689] to prove the existence of a C (n) -almost- periodic solution when there is at least one bounded solution in R+ We give an application to the Lotka–Volterra model with diffusion c 2007 Elsevier Ltd All rights reserved MSC: 34C27; 34K14; 35R10 Keywords: Hille–Yosida condition; Infinite delay; C0 -semigroup; Integral solution; Fading memory space; Reduction principle; C (n) -almostperiodic solution; Exponential dichotomy Introduction The aim of this work is to study the existence of C (n) -almost-periodic solutions for the following partial functional differential equation with infinite delay d x(t) = Ax(t) + L(xt ) + f (t) dt for t ∈ R, (1.1) where A : D(A) → X is a not necessarily densely defined linear operator on a Banach space X , for every t ∈ R, the history function xt ∈ B is defined by xt (θ ) = x(t + θ ) for θ ∈ (−∞, 0], ∗ Corresponding author Tel.: +1 443 885 3964; fax: +1 443 885 8216 E-mail addresses: ezzinbi@ucam.ac.ma (K Ezzinbi), gnguerek@morgan.edu (G.M N’gu´er´ekata) 0362-546X/$ - see front matter c 2007 Elsevier Ltd All rights reserved doi:10.1016/j.na.2007.06.041 1414 K Ezzinbi et al / Nonlinear Analysis 69 (2008) 1413–1424 where B is a normed linear space of functions mapping (−∞, 0] to X and satisfying some fundamental axioms given by Hale and Kato in [7] L is a bounded linear operator from B to X and f is an almost-periodic X -valued function on R They assume that the undelayed part A satisfies the Hille–Yosida condition (H0 ) there exist M0 ≥ 1, ω0 ∈ R such that (ω0 , +∞) ⊂ ρ(A) and |(λI − A)−n | ≤ M0 (λ − ω0 )n for n ∈ N and λ > ω0 , where ρ(A) is the resolvent set of A Recall in [10], Massera proposed a new approach to show the existence of periodic solutions of ordinary differential equations in finite dimensional spaces For some kind of ordinary differential equations, the author proved the existence of periodic solutions under a minimal condition, namely, the existence of a bounded solution on R+ is enough to get periodic solutions Many authors used Massera’s approach to prove the existence of periodic, almost periodic or C (n) -almost-periodic solutions in the context of differential equations In [2], the authors proved that the existence of a bounded solution on R+ implies the existence of an almost-periodic solution Firstly, they established a new variation of constants formula for Eq (1.1) Secondly, they used the spectral decomposition of the phase space to get a (new) reduction principle of Eq (1.1) to a finite dimensional space when B is a uniform fading memory space More precisely, they established a relationship between the bounded solutions of Eq (1.1) on R+ with the bounded solutions on R for an ordinary differential equation in a finite dimensional space In this work, we propose to use the reduction principle established in [2] and the Massera’s approach to show the existence of C (n) -almost-periodic solutions of Eq (1.1) C (n) -almost-periodic functions are functions such that the ith derivative are almost periodic for i = 1, , n; they have many applications in dynamical systems For more details, we refer to [1] and references therein In [4], the authors discussed some properties of C (n) -almost-periodic functions taking values in Banach spaces In [3], the authors proved the existence of a C (n) -almost-periodic solution for the following nonautonomous differential equation d x(t) = A(t)x(t) + θ (t) dt for t ∈ R, (1.2) where A(t) generates an exponentially stable family in a Banach space, they showed that when θ is C (n) -almost periodic, then the only bounded solution of Eq (1.2) is also C n -almost periodic In [9], the authors proved the existence of C (n) -almost-periodic solutions for some ordinary differential equations by using the exponential dichotomy approach In this work, we first discuss the existence of C (n) -almost- periodic solution for the following ordinary differential equation d x(t) = Gx(t) + e(t) dt for t ∈ R (1.3) where G is a constant n × n-matrix and e : R → Rn is C (n) -almost periodic We then prove the Massera-type theorem for the existence of C (n) -almost-periodic solution, more precisely, we prove that the existence of a bounded solution in R+ implies the existence of a C (n) -almost-periodic solution Moreover, we show that every bounded solution in R is C (n+1) -almost periodic We use the reduction principle developed recently in [2] to obtain a Massera-type theorem for the existence of C (n) -almost-periodic solution of Eq (1.1) This work is organized as follows, in Section 2, we recall some results on the existence of solutions of Eq (1.1) and we give the variation of constants formula that will be used in this work In Section 3, we give the reduction principle of Eq (1.1) to a finite dimensional ordinary differential equation Section is devoted to state some results on C (n) almost-periodic functions In Section 5, we use the reduction principle to show that the existence of a bounded solution on R+ implies the existence of an C (n) -almost-periodic solution of Eq (1.1) In Section 6, we prove the existence and uniqueness of an C (n) -almost-periodic solution of Eq (1.1) where the solution semigroup of Eq (1.1) with f = has an exponential dichotomy Finally, for illustration, we propose to study the existence of a C (n) -almost-periodic solution for a Lotka–Volterra model with diffusion K Ezzinbi et al / Nonlinear Analysis 69 (2008) 1413–1424 1415 Integral solutions and variation of constants formula We use the (classical) axiomatic approach of Hale and Kato [7] for the phase space B We assume that (B, · ) is a normed space of functions mapping (−∞, 0] into a Banach space X and satisfying the following fundamental axioms: (A) there exist a positive constant N , a locally bounded function M(·) on [0, +∞) and a continuous function K (·) on [0, +∞), such that if x : (−∞, a] → X is continuous on [σ, a] with x ∈ B, for some σ < a, then for all t ∈ [σ, a], (i) xt ∈ B, (ii) t → xt is continuous with respect to · on [σ, a], (iii) N |x(t)| ≤ xt ≤ K (t − σ ) sup |x(s)| + M(t − σ ) xσ σ ≤s≤t (B) B is a Banach space The following lemma is well known Lemma 2.1 ([2, p 140]) Assume that (H0 ) holds Let A0 be the part of the operator A in D(A), which is defined by D(A0 ) = {x ∈ D(A) : Ax ∈ D(A)} A0 x = Ax Then A0 generates a C0 -semigroup (T0 (t))t≥0 on D(A) To Eq (1.1), we associate the following Cauchy problem d x(t) = Ax(t) + L(xt ) + f (t) dt xσ = φ ∈ B for t ≥ σ, (2.1) The following results are taken from [2] Definition 2.2 ([2]) Let φ ∈ B A function u : R → X is called an integral solution of Eq (2.1) on R if the following conditions hold: (i) u is continuous on [σ, ∞), (ii) u σ = φ, t (iii) σ u(s)ds ∈ D(A) for t ≥ σ , t (iv) u(t) = φ(0) + A σ u(s)ds + t σ L(u s )ds + t σ f (s)ds for t ≥ σ For simplicity, integral solutions will be called solutions in this work Theorem 2.3 ([2]) Assume that (H0 ), (A) and (B) hold Then for all φ ∈ B such that φ(0) ∈ D(A), Eq (2.1) has a unique solution u = u(·, φ, L , f ) on R which is given by  t  T0 (t − σ )φ(0) + lim T0 (t − s)λR(λ, A) [L(u s ) + f (s)] ds, for t ≥ σ, u(t) = λ→+∞ σ  φ(t) for t ≤ σ Let B A = {φ ∈ B : φ(0) ∈ D(A)} be the phase space corresponding to Eq (2.1) For t ≥ 0, we define the operator U (t) for φ ∈ B A , by U (t)φ = u t (·, φ, L , 0), where u(·, φ, L , 0) is the solution of Eq (2.1) with f = and σ = Theorem 2.4 ([2]) Assume that (H0 ), (A) and (B) hold Then (U (t))t≥0 is a C0 -semigroup on B A That is (i) U (0) = I d, (ii) U (t + s) = U (t)U (s) for t, s ≥ 0, (iii) for all φ ∈ B A , t → U (t)φ is continuous from [0, ∞) to B A 1416 K Ezzinbi et al / Nonlinear Analysis 69 (2008) 1413–1424 Moreover, (U (t))t≥0 satisfies, for t ≥ 0, φ ∈ B A , the translation property (U (t + θ )φ) (0), for t + θ ≥ φ(t + θ ), for t + θ ≤ (U (t)φ) (θ ) = Due to the relationship between the semigroup and its generator, it is fundamental to compute the infinitesimal generator of (U (t))t≥0 ; to this end, we assume furthermore that (D1 ) If (φn )n is a sequence in B such that φn → in B as n → +∞, then for all θ ≤ 0, φn (θ ) → in X as n → +∞ (D2 ) B ⊂ C((−∞, 0]; X ), where C((−∞, 0]; X ) is the space of continuous functions from (−∞, 0] into X (D3 ) There exists λ0 ∈ R such that for all λ ∈ C with Re λ > λ0 and x ∈ X , we have eλ· x ∈ B and K0 = sup Re λ>λ0 ,x∈X x=0 eλ· x |x| < ∞, where (eλ· x)(θ ) = eλθ x for θ ∈ (−∞, 0] and x ∈ X The aim of the following result is the computation of the infinitesimal generator of (U (t))t≥0 Lemma 2.5 ([2]) Assume that (H0 ), (A), (B), (D1 ) and (D2 ) hold Then the infinitesimal generator AU of (U (t))t≥0 is given by D(AU ) = {φ ∈ C ((−∞, 0]; X ) ∩ B A : φ ∈ B A ,φ(0) ∈ D(A) and φ (0) = Aφ(0) + L(φ)}, AU φ = φ Recently in [2], a variation of constants formula for Eq (2.1) has been established In order to recall this formula, we need to give some preliminary results Firstly, we consider the space X := B A ⊕ X , where X = {X x : x ∈ X } and X x is the discontinuous function defined by (X x) (θ ) = x for θ ∈ (−∞, 0) for θ = The space X endowed with the norm φ + X x = φ + |x| is a Banach space According to Axiom (D3 ), we define for each complex number λ such that Re λ > λ0 , the linear operator (λ) : D(A) → X by (λ) = λI − A − L(eλ I ), where L(eλ I ) is a bounded linear operator on X , which is defined by L(eλ I )(x) = L(eλ x) for x ∈ X Theorem 2.6 ([2]) Assume that (H0 ), (A), (B), (D1 )–(D3 ) hold Then the extension AU of the operator AU defined on X by D( AU ) = {φ ∈ B A : φ ∈ B A and φ(0) ∈ D(A)}, AU φ = φ + X (Aφ(0) + Lφ − φ (0)), satisfies the Hille–Yosida condition on X More precisely, there exists ω ∈ R such that (ω, ∞) ⊂ ρ AU and for λ > ω, φ ∈ B A , x ∈ X , n ∈ N∗ , one has λ − AU −n (φ + X x) = (λ − AU )−n φ + (λ − AU )n−1 (eλ (λ)−1 x) K Ezzinbi et al / Nonlinear Analysis 69 (2008) 1413–1424 1417 Theorem 2.7 ([2]) Assume that (H0 ), (A), (B), (D1 )–(D3 ) hold Let φ ∈ B A Then the corresponding solution u of Eq (2.1) is given by the following variation of constants formula u t = U (t − σ )φ + lim n→+∞ σ t U (t − s) Bn (X f (s)) ds for t ≥ σ, (2.2) where Bn = n(n I − AU )−1 for n large enough Reduction principle in uniform fading memory spaces In this work, we assume that B satisfies Axioms (A), (B), (D1 )–(D3 ) Let C00 be the space of all X -valued continuous functions on (−∞, 0] with compact support We suppose the following axiom: (C) if a uniformly bounded sequence (ϕn )n in C00 converges to a function ϕ compactly in (−∞, 0], then ϕ is in B and ϕn − ϕ → as n → ∞ Let (S0 (t))t≥0 be the strongly continuous semigroup defined on the subspace B0 = {φ ∈ B : φ(0) = 0} by (S0 (t)φ) (θ ) = φ(t + θ) for t + θ ≤ 0 for t + θ ≥ Definition 3.1 Assume that the space B satisfies Axioms (A)–(C) B is said to be a fading memory space if for all φ ∈ B0 , S0 (t)φ −→ t→∞ in B Moreover, B is said to be a uniform fading memory space, if S0 (t) −→ t→∞ The following results give some properties of fading memory spaces Lemma 3.2 ([8, p 190]) The following statements hold (i) If B is a fading memory space, then the functions K (·) and M(·) in Axiom (A) can be chosen to be constants (ii) If B is a uniform fading memory space, then the functions K (·) and M(·) can be chosen such that K (·) is constant and M(t) → as t → ∞ Proposition 3.3 ([8]) If B is a fading memory space, then the space BC ((−∞, 0]; X ) of all bounded and continuous X -valued functions on (−∞, 0], endowed with the uniform norm topology, is continuously embedding in B By Proposition 3.3, one can observe that if B is a fading memory space then (D3 ) is satisfied with λ0 ≥ In order to study the qualitative behavior of the semigroup (U (t))t≥0 , we suppose the following assumption that (H1 ) T0 (t) is compact on D(A), for each t > Let V be a bounded subset of a Banach space Y The Kuratowskii measure of noncompactness α(V ) of V is defined by α(V ) = inf d > such that there exists a finite number of sets V1 , , Vn with n diam(Vi ) ≤ d such that V ⊆ ∪ Vi i=1 Moreover, for a bounded linear operator P on Y , we define |P|α by |P|α = inf{k > : α(P (V )) ≤ kα(V ) for any bounded set V of Y } 1418 K Ezzinbi et al / Nonlinear Analysis 69 (2008) 1413–1424 For the semigroup (U (t))t≥0 , we define the essential growth bound ωess (U ) by log |U (t)|α t We have the following fundamental result ωess (U ) = lim t→∞ Theorem 3.4 ([5]) Assume that (H0 ), (H1 ) hold and B is a uniform fading memory space Then ωess (U ) < Definition 3.5 Let C be a densely defined operator on Y The essential spectrum of C denoted by σess (C) is the set of λ ∈ σ (C) such that one of the following conditions holds: (i) Im(λI − C) is not closed, (ii) the generalized eigenspace Mλ (C) = (iii) λ is a limit point of σ (C) \ {λ} k≥1 Ker(λI − C)k is of infinite dimension, The essential radius of any bounded operator T is defined by ress (T ) = sup{|λ| : λ ∈ σess (C)} Theorem 3.6 ([2]) Assume that (H0 ), (H1 ) hold and B is a uniform fading memory space Then σ + (AU ) = {λ ∈ σ (AU ) : Re(λ) ≥ 0} is a finite set of the eigenvalues of AU which is not in the essential spectrum More precisely, λ ∈ σ + (AU ) if and only if there exists x ∈ D(A)\{0} which solves the following characteristic equation (λ)x = λx − Ax − L(eλ· x) = We have the following spectral decomposition result Theorem 3.7 ([2]) Assume that (H0 ), (H1 ) hold and B is a uniform fading memory space Then the phase space B A is decomposed as B A = S ⊕ V, where S, V are two closed subspaces of B A which are invariant under the semigroup (U (t))t≥0 Let U S (t) be the restriction of U (t) on S Then there exist positive constants N and µ such that U S (t)φ ≤ N e−µt φ for φ ∈ S On the other hand, V is a finite dimensional space Then the restriction U V (t) of U (t) on V becomes a group Let Π S and Π V denote the projections on S and V respectively and d = dim V Take a basis {φ1 , , φd } in V Then there exist d-elements {ψ1 , , ψd } in the dual space B ∗A of B A such that ψi , φ j = δi j , where δi j = if i = j if i = j, and ψi , φ = 0, for φ ∈ S and i = 1, , d, with ·, · being the canonical pairing between the dual space and the original space Denote by Φ = (φ1 , , φd ) and by Ψ the transpose of (ψ1 , , ψd ) One has Ψ , Φ = I Rd , where IRd is the identity d × d-matrix For each φ ∈ B A , Π V φ is computed as Π Vφ = Φ Ψ, φ , d ψi , φ φi = i=1 K Ezzinbi et al / Nonlinear Analysis 69 (2008) 1413–1424 1419 Let u be the solution of Eq (2.1) and ζ (t) = (ζ1 (t), , ζd (t)) be the components of Π V u t in the basis Φ Then Π V u t = Φζ (t) Since U V (t) t≥0 ζ (t) = Ψ , u t and is a group on the finite dimensional space V, then there exists a d × d-matrix G such that U V (t)φ = ΦeGt Ψ , φ for t ∈ R and φ ∈ V This means that U V (t)Φ = ΦeGt for t ∈ R Let n ∈ N such that n > ω We define, for n ∈ N such that n ≥ n and i ∈ {1, , d}, the functional xn∗i by xn∗i , x = ψi , Bn (X x) for all x ∈ X By Theorem 2.6, we have Bn (X x) = nen −1 (n)x, for n ≥ n Then, we choose n large enough such that for all x ∈ X and n ≥ n Bn (X x) ≤ M0 |x| This implies that xn∗i is a bounded linear operator on X with xn∗i ≤ M0 |ψi | Define the d-columns vector xn∗ as an element of L(X, Rd ) given by the transpose of xn∗1 , , xn∗d Then, for all n ≥ n and x ∈ X , we have xn∗ , x = Ψ , Bn (X x) and sup xn∗ ≤ M0 sup |ψi | < ∞ n≥n i=1, ,d We have the following important result Theorem 3.8 ([2]) Assume that (H0 ), (H1 ) hold and B is a uniform fading memory space Then the sequence xn∗ n≥n converges weakly in L(X, Rd ), in the sense that there exists x ∗ ∈ L(X, Rd ) such that xn∗ , x −→ x ∗ , x for x ∈ X n→∞ Corollary 3.9 ([2]) Assume that (H0 ), (H1 ) hold and B is a uniform fading memory space Then for any continuous function h : [σ, T ] → X , we have for all t ∈ [σ, T ] t lim n→∞ σ U V (t − s)Π V Bn (X h(s)) ds = Φ t σ e(t−s)G x ∗ , h(s) ds In the next theorem, we state a finite dimensional reduction principle of Eq (1.1) Theorem 3.10 ([2]) Assume that (H0 ), (H1 ) hold and B is a uniform fading memory space Let u be a solution of Eq (1.1) on R Then ζ (t) = Ψ , u t for t ∈ R, is a solution of the following ordinary differential equation ζ˙ (t) = Gζ (t) + x ∗ , f (t) for t ∈ R (3.1) Conversely, if f is bounded and ζ is a solution of (3.1), then the function Φζ (t) + lim t n→+∞ −∞ U S (t − s)Π S Bn (X f (s)) ds (0) (3.2) is a solution of Eq (1.1) on R C (n) -almost-periodic functions We recall some properties about C (n) -almost-periodic functions Let BC(R, X ) be the space of all bounded and continuous functions from R to X , equipped with the uniform norm topology Let h ∈ BC(R, X ) and τ ∈ R; we define the function h τ by h τ (s) = h(τ + s) for s ∈ R 1420 K Ezzinbi et al / Nonlinear Analysis 69 (2008) 1413–1424 Let C (n) (R, X ) be the space of all continuous functions which have a continuous nth derivative on R and Cbn (R, X ) be the subspace of C (n) (R, X ) of functions satisfying i=n sup f (i) (t) < ∞, t∈R i=0 f (i) denotes the ith derivative of f Then Cbn (R, X ) is a Banach space equipped with the following norm i=n | f |n = sup f (i) (t) t∈R i=0 Definition 4.1 ([6]) A bounded continuous function h : R → X is said to be almost periodic if {h τ : τ ∈ R} is relatively compact in BC(R, X ) Definition 4.2 ([3]) Let ε > and f ∈ Cbn (R, X ) A number τ ∈ R is said to be a |.|n − ε almost periodic of the function f if | f τ − f |n < ε The set of all |.|n − ε almost periodic of the function f is denoted by E (n) (ε, f ) Definition 4.3 ([3]) A function f ∈ Cbn (R, X ) is said to be a C (n) -almost-periodic function if for every ε > 0, the set E (n) (ε, f ) is relatively dense in R Definition 4.4 A P (n) (R, X ) denotes the space of the all C (n) -almost-periodic functions R → X Since it is well known that for any almost-periodic functions f and g and ε > 0, there exists a relatively dense set of their common ε almost periods, we get the following result Proposition 4.5 f ∈ A P (n) (R, X ) if and only if f (i) ∈ A P(R, X ) for i = 1, , n Proposition 4.6 ([4]) A P (n) (R, X ) provided with the norm |.|n is a Banach space The following example of a C (n) -almost-periodic function has been given in [4] Example Let g(t) = sin(αt) + sin(βt), where βα ∈ Q Then the function f (t) = eg(t) is a C (n) -almost-periodic function for any n ≥ In [4], one can find example of a function which is C (n) -almost periodic but not C (n+1) -almost periodic The following theorem provides the sufficient and necessary condition for the existence of C (n) -almost-periodic solutions of Eq (1.3) Theorem 4.7 Assume that e is a C (n) -almost-periodic function If Eq (1.3) has a bounded solution on R+ , then it has an C (n+1) -almost-periodic solution Moreover, every bounded solution of Eq (1.3) on R is C (n+1) -almost periodic Proof We assume that Eq (1.3) has a bounded solution χ in R+ Since e is an almost-periodic solution, then there exists a sequence (tn )n , tn → ∞ such that e(t + tn ) → e(t), uniformly in t ∈ R Using the diagonal extraction process, we show that the sequence χ (t + tn ) has a subsequence which converges compactly to the solution of Eq (1.3) which is bounded and defined in R For m = 0, it has been proved in [6], that if e is an almost-periodic function then every bounded solution of equation in R is almost periodic Let x be a bounded solution of Eq (1.3), then x is almost periodic Gx is also almost periodic and x (t) = Gx(t) + e(t) for t ∈ R, K Ezzinbi et al / Nonlinear Analysis 69 (2008) 1413–1424 1421 it follows that x is almost periodic Since the function f is C (n) -almost periodic and for i = 1, , n, we have the formula, x (i) (t) = Gx (i−1) (t) + e(i−1) (t) for t ∈ R Consequently, we deduce that x (i) is almost periodic for i = 1, , n and by Proposition 4.5, we deduce that x is C (n) -almost periodic Moreover x (n+1) (t) = Gx (n) (t) + e(n) (t) which implies that x (n+1) for t ∈ R, is almost periodic and x is C (n+1) -almost periodic C (n) -almost-periodic solutions In the following, we assume that (H2 ) f is a C (n) -almost-periodic function Theorem 5.1 Assume that (H0 ), (H1 ), (H2 ) hold and B is a uniform fading memory space If there is at least φ ∈ B such that Eq (2.1) has a bounded solution on R+ , then Eq (1.1) has a C (n) -almost-periodic solution Moreover every bounded solution on the whole line is a C (n) -almost-periodic solution Proof Let u be a bounded solution of Eq (1.1) on R+ By Theorem 3.10, the function z(t) = Ψ , u t , for t ≥ 0, is a solution of the ordinary differential equation (3.1) and z is bounded on R+ Moreover, the function (t) = x ∗ , f (t) for t ∈ R, C (n) -almost is periodic from R to Rd By Theorem 4.7, we get that the reduced system (3.1) has a C (n) -almostperiodic solution z and Φz(.) is a C (n) -almost-periodic function on R From Theorem 3.10, we know that the function u(t) = v(t)(0), where v(t) = Φz(t) + lim t n→+∞ −∞ U S (t − s)Π S Bn X f (s) ds for t ∈ R, is a solution of Eq (1.1) on R We claim that v is C (n) -almost periodic In fact, let y be defined by t y(t) = lim n→+∞ −∞ U S (t − s)Π S Bn X f (s) ds for t ∈ R Then y ∈ Cbn (R, X ) Since f is C (n) -almost periodic Let ε > and τ be a |.|n − ε almost periodic of the function f that is | f τ − f |n < ε It follows that ∞ y(t + τ ) − y(t) = lim n→+∞ U S (s)Π S Bn X ( f (t + τ − s) − f (t − s)) ds and we get for some positive constant γ that |yτ − y|n ≤ γ ε | f τ − f |n , which implies that y is C (n) -almost periodic Let v be a bounded solution on the whole line then v is given by the v(t) = Φz(t) + lim t n→+∞ −∞ where z(t) = Ψ , u t for t ∈ R U S (t − s)Π S Bn X f (s) ds for t ∈ R, 1422 K Ezzinbi et al / Nonlinear Analysis 69 (2008) 1413–1424 is a solution of the reduced system (3.1), which is C (n) -almost periodic by Theorem 4.7, and arguing as above, one can prove that the function t t → lim n→+∞ −∞ U S (t − s)Π S Bn X f (s) ds for t ∈ R, is also C (n) -almost periodic Exponential dichotomy Definition 6.1 The semigroup (U (t))t≥0 is said to have an exponential dichotomy if σ (AU ) ∩ iR = ∅ Since ωess (U ) < 0, then we get the following result on the spectral decomposition of the phase space B A Theorem 6.2 ([2]) Assume that (H0 ), (H1 ) hold and B is a uniform fading memory space If the semigroup (U (t))t≥0 has an exponential dichotomy, then the space B A is decomposed as a direct sum B A = S ⊕ U of two U (t) invariant closed subspaces S and U such that the restricted semigroup on U is a group and there exist positive constants N0 and ε0 such that |U (t)ϕ| ≤ N0 e−ε0 t |ϕ| for t ≥ and ϕ ∈ S |U (t)ϕ| ≤ N0 eε0 t |ϕ| for t ≤ and ϕ ∈ U As a consequence of the exponential dichotomy, we get the following result on the uniqueness of the bounded solution of Eq (1.1) Theorem 6.3 Assume that (H0 ), (H1 ) hold and B is a uniform fading memory space If the semigroup (U (t))t≥0 has an exponential dichotomy, then for any bounded continuous function f on R, Eq (1.1) has a unique bounded solution on R Moreover, this solution is C (n) -almost periodic if f is C n -almost periodic Proof Since the semigroup (U (t))t≥0 has an exponential dichotomy, then Eq (1.1) has one and only one bounded solution on R which is given for t ∈ R by the following formula t lim n→+∞ −∞ U S (t − s)Π S Bn X f (s) ds + lim t n→+∞ +∞ U U (t − s)Π U Bn X f (s) ds (0) By Theorem 3.10, we conclude that this solution is C (n) -almost periodic when f is C (n) -almost periodic Application To illustrate the previous results, we consider the following Lotka–Volterra model with diffusion  ∂2 ∂ v(t, ξ ) = v(t, ξ ) + η(θ )v(t + θ, ξ )dθ + ρ(t)F(ξ ) for t ∈ R and ≤ ξ ≤ π, ∂ξ −∞  ∂t v(t, 0) = v(t, π ) = for t ∈ R (7.1) where η is a positive function on (−∞, 0] , ρ : R → R is C n -almost periodic, for example, we consider g(t) = sin(αt) + sin(βt) for t ∈ R where βα ∈ Q Then the function ρ(t) = eg(t) F : [0, π] → R is a continuous function Let X = C([0, π] ; R) be the space of all continuous functions from [0, π] to R endowed with the uniform norm topology Consider the operator A : D(A) ⊂ X → X defined by D(A) = {z ∈ C ([0, π] ; R) : z(0) = z (π ) = 0}, Az = z 1423 K Ezzinbi et al / Nonlinear Analysis 69 (2008) 1413–1424 Lemma 7.1 ([2]) A satisfies the Hille–Yosida condition on X On the other hand, one can see that D(A) = {ψ ∈ C([0, π]; R) : ψ(0) = ψ(π ) = 0} We introduce the following space B = Cγ , γ > 0, where Cγ = φ ∈ C((−∞, 0]; X ) : lim eγ θ φ(θ ) exists in X θ→−∞ provided with the following norm φ γ eγ θ |φ(θ )| sup = −∞ 0, is a uniform fading memory space satisfying (D1 ) and (D2 ) We make the assumption (E1 ) η(·)e−γ · is integrable on (−∞, 0] Define   η(θ )φ(θ )(ξ )dθ for ξ ∈ [0, π] and φ ∈ B, (L(φ))(ξ ) = −∞  f (t)(ξ ) = ρ(t)F(ξ ) for t ∈ R and ξ ∈ [0, π] By assumption (E1 ), one can see that L is a bounded linear operator from B to X and f : R → X is C n -almost periodic We put x(t)(ξ ) = v(t, ξ ) for t ∈ R and ξ ∈ [0, π] Then Eq (7.1) takes the abstract form d x(t) = Ax(t) + L(xt ) + f (t) dt for t ∈ R (7.2) The part A0 of A in D(A) is given by D(A0 ) = {z ∈ C ([0, π] ; R) : z(0) = z(π ) = z (0) = z (π ) = 0}, A0 z = z Then it is well known that A0 generates a compact C0 -semigroup on D(A) In order to prove the existence and uniqueness of a C (n) -almost- periodic solution of Eq (7.2), we make the assumption (E2 ) −∞ η(θ )dθ < Theorem 7.3 If (E1 ) and (E2 ) hold Then the semigroup solution associated to (7.2) with f = has an exponential dichotomy Proof By Theorem 3.6, it suffices to show that σ + (AU ) = ∅ In fact, we proceed by contradiction and assume that there exists λ ∈ σ + (AU ) Then there exists ϑ ∈ D(A)\{0} such that (λ)ϑ = 0, which is equivalent to λ− A− η(θ )eλθ dθ ϑ = (7.3) −∞ The spectrum σ (A) is reduced to the point spectrum σ p (A) and σ p (A) = {−n : n ∈ N∗ } Then λ is a solution of the characteristic equation (7.3) with Re(λ) ≥ if and only if λ satisfies λ− −∞ η(θ )eλθ dθ = −n for some n ∈ N∗ (7.4) 1424 K Ezzinbi et al / Nonlinear Analysis 69 (2008) 1413–1424 It follows that Re(λ) = −∞ ≤ η(θ )eRe(λ)θ cos(Im(λ)θ )dθ − n η(θ )dθ − n −∞ Since −∞ η(θ )dθ < 1, then a contradiction is obtained with the fact that Re(λ) ≥ Consequently, the semigroup solution associated to (7.2) with f = has an exponential dichotomy By Theorem 6.2, we obtain the following existence and uniqueness result on C (n) -almost-periodic solution of Eq (7.2) Corollary 7.4 Assume that (E1 ) and (E2 ) hold Then Eq (7.2) has a unique C (n) -almost-periodic solution References [1] M Adamczak, C (n) -almost periodic functions, Commentationes Mathematicae Prace Matematyczne 37 (1997) 1–12 [2] M Adimy, K Ezzinbi, A Ouhinou, Variation of constants formula and almost periodic solutions for some partial functional differential equations with infinite delay, Journal of Mathematical Analysis and Applications 317 (2006) 668–689 [3] J.B Baillon, J Blot, G.M N’Gu´er´ekata, D Pennequin, On C (n) -almost periodic solutions to some nonautonomous differential equations in Banach spaces, Annales Societatis 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Analysis, Theory, Methods & Applications 66 (2007) 1899–1910 [10] J.L Massera, The existence of periodic solutions of systems of differential equations, Duke Mathematical Journal 17 (1950) 457–475  ... n-matrix and e : R → Rn is C (n) -almost periodic We then prove the Massera-type theorem for the existence of C (n) -almost-periodic solution, more precisely, we prove that the existence of a... following theorem provides the sufficient and necessary condition for the existence of C (n) -almost-periodic solutions of Eq (1.3) Theorem 4.7 Assume that e is a C (n) -almost-periodic function... Integral solutions and variation of constants formula We use the (classical) axiomatic approach of Hale and Kato [7] for the phase space B We assume that (B, · ) is a normed space of functions