Existence and uniqueness of pseudo almost periodic solutions to some abstract partial neutral functional–differential equations and applications

J. Math. Anal. Appl. 327 (2007) 776–791 www.elsevier.com/locate/jmaa

Existence and uniqueness of pseudo almost periodic solutions to some abstract partial neutral functional–differential equations and applications Toka Diagana a,∗ , Eduardo M. Hernández b a Department of Mathematics, Howard University, 2441 6th Street N.W., Washington, DC 20059, USA b Departamento de Matemática, I.C.M.C. Universidade de São Paulo, Caixa Postal 668,

13560-970 São Carlos SP, Brazil Received 8 February 2006 Available online 30 May 2006 Submitted by G. Chen

Abstract The paper considers the existence and uniqueness of pseudo almost periodic (mild) solutions to some classes of first-order partial neutral functional–differential equations. Upon making some suitable assumptions, existence and uniqueness results are obtained. Applications include both a partial integro-differential equation arising in control systems and the scalar reaction–diffusion equation with delay. © 2006 Elsevier Inc. All rights reserved. Keywords: Partial functional–differential equations; Pseudo almost periodic function

1. Introduction Let (X,  · ) be a Banach space. In this paper we study the original problem, which consists of the existence and uniqueness of pseudo almost periodic (mild) solutions to the following abstract partial neutral functional–differential equations  d u(t) + f (t, ut ) = Au(t) + g(t, ut ) dt

and

* Corresponding author.

E-mail addresses: [email protected] (T. Diagana), [email protected] (E.M. Hernández). 0022-247X/$ – see front matter © 2006 Elsevier Inc. All rights reserved. doi:10.1016/j.jmaa.2006.04.043

(1.1)

T. Diagana, E.M. Hernández / J. Math. Anal. Appl. 327 (2007) 776–791

      d u(t) + f t, u t − γ1 (t) = Au(t) + g t, u t − γ2 (t) , dt

777

(1.2)

where A is the infinitesimal generator of an uniformly exponentially stable semigroup of linear operators on X, the history ut ∈ C([−p, 0]; X) with p > 0 (ut being defined by ut (θ ) = u(t + θ ) for each θ ∈ [−p, 0]) and f, g, γi (i = 1, 2) are some suitable functions to be defined later on in the text. The existence of almost periodic, asymptotically almost periodic, almost automorphic, asymptotically almost automorphic, and pseudo almost periodic solutions is one of the most attracting topics in the qualitative theory of differential equations due to their significance and applications in physics, mathematical biology, control theory, and others. The concept of the pseudo almost periodicity, which is the central question in this paper was introduced in the literature in the early nineties by Zhang [34–36] as a natural generalization of the well-known (Bochner) almost periodicity. Thus this new concept is welcome to implement another existing generalization of the (Bochner) almost periodicity, the notion of asymptotically almost periodicity due to Fréchet [5,9,22,32]. For more on these concepts and related topics, see, e.g., [1–4,6,7,19,23,25,28,34] and references therein. Some recent contributions on almost periodic, asymptotically almost periodic, and pseudo almost periodic solutions to abstract differential and partial differential equations have been made in [4,6–8,16–19]. Existence results concerning almost periodic and asymptotically almost periodic solutions to ordinary neutral differential equations and abstract partial neutral differential equations have recently been established in [15,21,27]. However, the existence of pseudo almost periodic to functional–differential equations with delay, especially, abstract partial neutral differential equations, is an untreated topic and this is the main motivation of the present paper. Neutral differential equations arise in many areas of applied mathematics. For this reason, those equations have been of a great interest during the last few decades. The literature relative to ordinary neutral differential equations is quite extensive; for more on this topic and related applications we refer the reader to Halle [10], which contains a comprehensive presentation on those equations. Similarly, for more on partial neutral functional differential equations we refer to Hale [11], Wu [29–31], Adimy [1] for finite delay equations, and Hernández and Henríquez [12,13] and Hernández [14] for unbounded delays. 2. Preliminaries In what follows we recall some definitions, notations, and new pseudo almost periodic spaces that we need in the sequel. From now on, (X,  · ) stands for a Banach space and A : D(A) ⊂ X → X denotes the infinitesimal generator of an uniformly asymptotically stable semigroup of linear operators (T (t))t0 such that there exist some positive constants M, w with   T (t)  Me−wt , t  0. To deal with pseudo almost periodic solutions we will need to introduce some classical and new concepts. In what follows, – both (Z,  · Z ) and (W,  · W ) are Banach spaces; – C(R, Z) denotes the collection of all continuous functions from R into Z;

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– BC(R, Z) denotes the Banach space of all bounded continuous functions from R into Z equipped with the sup norm defined by u∞ := supt∈R u(t); – Br (x, Z) stands for the open ball centered at x with radius r > 0 in Z. Similar definitions as above apply for both C(R × Z, W) and BC(R × Z, W). Definition 2.1. A function f ∈ C(R, Z) is almost periodic if for each ε > 0 there exists a relatively dense subset of R denoted by H(ε, f, Z) (i.e., there exists δ > 0 such that [a, a + δ] ∩ H(ε, f, Z) = {∅} for each a ∈ R) such that   f (t + τ ) − f (t) < ε Z

for each t ∈ R and each τ ∈ H(ε, f, Z). The collection of such functions will be denoted by AP (Z). The next lemma is also a characterization of almost periodic functions. Lemma 2.2. [33, p. 25] A function f ∈ C(R, Z) is almost periodic if and only if the set of functions {στ f : τ ∈ R}, where (στ f )(t) = f (t + τ ), is relatively compact in C(R, Z). Similarly, Definition 2.3. A function F ∈ C(R × Z, W) is almost periodic in t ∈ R uniformly in z ∈ Z if for each ε > 0 and for all compact K ⊂ Z there exists a relatively dense subset of R denoted by H(ε, F, K) such that   F (t + τ, z) − F (t, z) < ε W

for all t ∈ R, z ∈ K, and τ ∈ H(ε, F, K). The collection of such functions will be denoted by AP(Z, W). The notation PAP0 (Z) stands for the space of functions  r  1  u(t) dt = 0 . PAP0 (Z) = u ∈ BC(R, Z): lim Z r→∞ 2r −r

To study issues related to delay we need to introduce the new space of functions defined for each p > 0 by  r    1

PAP0 (Z, p) := u ∈ BC(R, Z): lim sup u(θ )Z dt = 0 . r→∞ 2r θ∈[t−p,t] −r

In addition to the above-mentioned spaces, the present setting requires the introduction of the following function spaces  r  1    PAP0 (Z, W) = u ∈ BC(R × Z, W): lim u(t, z) W dt = 0 and r→∞ 2r 

−r

1 PAP0 (Z, W, p) := u ∈ BC(R × Z, W): lim r→∞ 2r

r

−r

   sup u(θ, z)W dt = 0 ,

θ∈[t−p,t]

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where in both cases the limit (as r → ∞) is uniform in compact subset of Z. In view of the previous definitions it is clear that PAP0 (Z, p) and PAP0 (Z, W, p) are continuously embedded in PAP0 (Z) and PAP0 (Z, W), respectively. Furthermore, it is not hard to see that PAP0 (Z, p) and PAP0 (W, Z, p) are closed in PAP0 (Z) and PAP0 (W, Z), respectively. Consequently, using [19, Lemma 1.2], one obtains the following: Lemma 2.4. The spaces PAP0 (Z, p) and PAP0 (W, Z, p) endowed with the uniform convergence topology are Banach spaces. Definition 2.5. A function f ∈ BC(R, Z) is called pseudo almost periodic if f = g + ϕ, where g ∈ AP(Z) and ϕ ∈ PAP0 (Z). The class of such functions will be denoted by PAP(Z). Definition 2.6. A function F ∈ BC(R × Z, W) is called pseudo almost periodic if F = G + Φ, where G ∈ AP(Z, W) and Φ ∈ PAP0 (Z, W). The class of such functions will be denoted by PAP(Z, W). We need to introduce two new notions of pseudo almost periodicity that we will use in the sequel. Definition 2.7. A function F ∈ BC(R, Z) is called pseudo almost periodic of class p if F = G + ϕ, where G ∈ AP(Z) and ϕ ∈ PAP0 (Z, p). The class of such functions will be denoted by PAP(Z, p). Definition 2.8. A function F ∈ BC(R × Z, W) is called pseudo almost periodic of class p if F = G + ϕ, where G ∈ AP(R × Z, W) and ϕ ∈ PAP0 (Z, W, p). The class of such functions will be denoted by PAP(Z, W, p). This paper is organized as follows: Section 3 is devoted to the proofs of some preliminaries results related to the composition of pseudo almost periodic functions of class p. The existence of pseudo almost periodic solutions for the neutral systems (1.1) and (1.2) will be investigated in Sections 4 and 5. Finally, Section 6 considers some applications. 3. Preliminary results Our main results on the existence of pseudo almost periodic solutions require some preliminaries results related to the composition of pseudo almost periodic functions of class p. Basically, those results are inspired from ideas and estimates given in [19]. Thus, for the sake of clarity, proofs of those results will be given. Throughout this section we consider Banach spaces (Z,  · Z ), (W,  · W ), previously introduced. Theorem 3.1. Let F ∈ PAP(Z, W, p) and let h ∈ PAP(W, p). Assume that there exists a function LF : R → [0, ∞) satisfying   F (t, z1 ) − F (t, z2 )  LF (t)z1 − z2 Z , ∀t ∈ R, ∀z1 , z2 ∈ Z. (3.1) W

If

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1 lim sup r→∞ 2r 1 r→∞ 2r

r

 LF (θ ) dt < ∞ and

sup −r

r

lim

(3.2)

θ∈[t−p,t]

sup

 LF (θ ) ξ(t) dt = 0

(3.3)

θ∈[t−p,t]

−r

for each ξ ∈ PAP0 (R), then the function t → F (t, h(t)) belongs to PAP(W, p). Proof. Assume that F = F1 + ϕ, h = h1 + h2 , where F1 ∈ AP(Z, W), ϕ ∈ PAP0 (Z, W, p), h1 ∈ AP(Z) and h2 ∈ PAP0 (Z, p). Consider the decomposition          F t, h(t) = F1 (t, h1 (t)) + F t, h(t) − F t, h1 (t) + ϕ t, h1 (t) . Since F1 (·, h1 (·)) ∈ AP(W), it remains to prove that both [F (·, h(·)) − F (·, h1 (·))] and ϕ(·, h1 (·)) belong to PAP0 (W, p). Indeed, using (3.1) above it follows that 1 2r

r

     sup F θ, h(θ ) − F θ, h1 (θ )  dt θ∈[t−p,t]

−r

1  2r 

1 2r

r

sup −r r

θ∈[t−p,t]

sup

−r

  LF (θ )h2 (θ ) dt

θ∈[t−p,t]



LF (θ ) ·

  sup h2 (θ ) dt, θ∈[t−p,t]

which implies that [F (·, h(·)) − F (·, h1 (·))] ∈ PAP0 (W, p), by (3.3). Since h1 (R) is relatively compact in Z and F1 is uniformly continuous on sets of the form R × K where K ⊂ Z is compact subset, for ε > 0 there exists δ ∈ (0, ε) such that   F1 (t, z) − F1 (t, z¯ )  ε, z, z¯ ∈ h1 (R), with z − z¯  < δ. Now, fix z1 , . . . , zn ∈ h1 (R) such that h1 (R) ⊂ ni=1 Bδ (zi , Z). Obviously, the sets Ei = h−1 an open covering of R, and therefore using the sets B1 = E1 , B2 = E2 \ E1 1 (Bδ (zi )) form i−1 and Bi = Ei \ j =1 Ej one obtains a covering of R by disjoint open sets. For t ∈ Bi , h1 (t) ∈ Bδ (zi ),              ϕ t, h1 (t)   F t, h1 (t) − F (t, zi ) + −F1 t, h1 (t) + F1 (t, zi ) + ϕ(t, zi )      LF (t)h1 (t) − zi  + ε + ϕ(t, zi )    LF (t)ε + ε + ϕ(t, zi ). Now using the previous inequality it follows that 1 2r

r

−r

   sup ϕ t, h1 (t)  dt θ∈[t−p,t]

T. Diagana, E.M. Hernández / J. Math. Anal. Appl. 327 (2007) 776–791



n 1  2r

   sup ϕ θ, h1 (θ )  dt



θ∈[t−p,t]

i=1B ∩[−r,r] i



n 1  2r





i=1B ∩[−r,r] i





−r

    F1 θ, h1 (θ ) − F1 (θ, zj ) dt

sup

  ϕ(θ, zj ) dt

j =1,...,n θ∈[t−p,t]∩Bj



 sup

i=1B ∩[−r,r] i

r

sup

 sup

i=1B ∩[−r,r] i

1  2r

    F θ, h1 (θ ) − F (θ, zj ) dt

j =1,...,n θ∈[t−p,t]∩Bj



n 1 + 2r

sup

 sup

n 1 + 2r

   ϕ θ, h1 (θ )  dt

j =1,...,n θ∈[t−p,t]∩Bj

i=1B ∩[−r,r] i



sup

 sup

n 1  2r

781

j =1,...,n θ∈[t−p,t]∩Bj

r n

  1

sup LF (θ )ε + ε dt + sup ϕ(θ, zj ) dt. 2r θ∈[t−p,t] θ∈[t−p,t] 

i=1

−r

In view of the above it is clear that ϕ(·, h1 (·)) belongs to PAP0 (W, p).

2

Remark 3.2. Note that assumptions (3.2) and (3.3) are verified by many functions. Examples include constants functions, functions in PAP(R, p), and functions of L1 (R) that are decreasing on [0, ∞) and increasing on [−∞, 0), among others. To study the system (1.1) we need the following result: Theorem 3.3. If u ∈ PAP(Z, p), then t → ut belongs to PAP(C([−p, 0], Z), p). Proof. Assume that u = h + g where h ∈ AP(Z) and g ∈ PAP0 (Z, p). Clearly, ut = ht + gt and from Lemma 2.2 we infer that ht is almost periodic. On the other hand, for r > 0 we see that 1 2r

r 

  sup g(θ + ξ ) dt

sup

−r

θ∈[t−p,t] ξ ∈[−p,0]

1  2r 1  2r 1  2r

r

sup −r

  g(θ ) dt

θ∈[t−2p,t]

r−p 

−r−p r−p 

−r−p

  sup g(θ ) + θ∈[t−p,t]

  sup g(θ ) dt θ∈[t,t+p]

  1 sup g(θ ) dt + 2r θ∈[t−p,t]

r−p 

−r−p

  sup g(θ ) dt θ∈[t,t+p]

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1 r +p  r 2(r + p)

r+p 

−r−p

  1 sup g(θ ) dt + 2r θ∈[t−p,t]

r

−r

  sup g(θ ) dt, θ∈[t−p,t]

2

which enables to complete the proof.

Our existence results require the next theorem. Theorem 3.4. Let u ∈ PAP0 (Z, p). If v is the function defined by t T (t − s)u(s) ds,

v(t) :=

∀t ∈ R,

−∞

then v ∈ PAP0 (Z, p). Proof. For r > 0 we get 1 2r

r 

θ sup θ∈[t−p,t] −∞

−r

1  2r 1  2r 1  2r 

r 

θ sup

−r

e

θ sup

e

θ∈[t−p,t]

−r r 

sup −r

ewp 2r

  u(s) ds dt



−w(θ−s) 

θ∈[t−p,t] −∞

r 

ewp  2r 

    T (θ − s)u(s) ds dt

r  t

e

e

−w(t−s) 

  u(s) ds dt



−∞

  e−w(t−s) u(s) ds dt

−r −∞ −r r

e

 ewp u(s) dt ds + 2r



−w(t−s) 

−∞ −r

ewp u∞ 2rw

wp

   u(s) ds dt



−w(t−s) 

−∞ t

e

θ∈[t−p,t]

wp

−r ew(s+r) ds + −∞

ewp 2rw

r

r r

  e−w(t−s) u(s) dt ds

−r s

  u(s) dt.

−r

Consequently, 1 2r

r

−r

r   ewp u∞ ewp  u(s) dt, sup v(θ ) dt  + 2 2r 2rw θ∈[t−p,t]

which converges to zero as r → ∞.

−r

2

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4. Existence results for a neutral system with finite delay This section is devoted to the existence and uniqueness of pseudo almost periodic solutions to the neutral system  d u(t) + f (t, ut ) = Au(t) + g(t, ut ), dt   uσ = φ ∈ B = C [−p, 0]; X .

t ∈ [σ, σ + a),

(4.1) (4.2)

In what follows, we adopt the notion of mild solution to (4.1)–(4.2) from the one given in Hernández and Henríquez [12]. Definition 4.1. A continuous function, u : [σ, σ + a) → X, a > 0, is a mild solution of the neutral system (4.1)–(4.2) on [σ, σ + a), if the function, s → AT (t − s)f (s, us ) is integrable on [0, t) for every σ < t < σ + a, and   u(t) = T (t − σ ) ϕ(σ ) + f (σ, ϕ) − f (t, ut ) −

t AT (t − s)f (s, us ) ds σ

t +

T (t − s)g(s, us ) ds,

t ∈ [σ, σ + a).

σ

To discuss the existence of pseudo almost periodic solutions to (4.1)–(4.2) we need to set some assumptions on f and g. In what follows, [D(A)] denotes the domain of the linear operator A endowed with the graph norm defined by: u[D(A)] = u + Au for each u ∈ D(A). (H1) The functions f, g : R × B → X are continuous, f is D(A)-valued and there exist a positive constant Lf and a continuous functions Lg : R → [0, ∞) such that   f (t, ψ1 ) − f (t, ψ2 )  Lf ψ1 − ψ2 B , [D(A)]   g(t, ψ1 ) − g(t, ψ2 )  Lg (t)ψ1 − ψ2 B , for all t ∈ R, ψi ∈ B. Remark 4.2. The assumption on f is linked to the integrability of the function s → AT (t − s) × f (s, us ) over [0, t). In general, except trivial cases, the operator function t → AT (t) is not integrable over [0, a]. If f satisfies (H1), then from the Bochner’s criterion for integrable functions and the estimate     AT (t − s)f (s, us ) = T (t − s)Af (s, us )    Me−w(t−s) Af (s, us )    Me−w(t−s) f (s, us )[D(A)] , it follows that the function s → AT (t − s)f (s, us ) is integrable over (−∞, t) for each t > 0. For additional remarks related this type of conditions in partial neutral differential equations, see, e.g., [1,12,13], in particular, [14].

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Definition 4.3. A function u ∈ BC(R, X) is a mild pseudo almost periodic solution to the neutral system (4.1)–(4.2) provided that the function s → AT (t − s)f (s, us ) is integrable on (−∞, t) for each t ∈ R, and t u(t) = −f (t, ut ) −

t AT (t − s)f (s, us ) ds +

−∞

T (t − s)g(s, us ) ds,

t ∈ R.

−∞

Theorem 4.4. Under assumption (H1), there exist a unique pseudo almost periodic solution to (4.1)–(4.2) whenever     t M −w(t−s) + M sup Θ := Lf 1 + e Lg (s) ds < 1. (4.3) w t∈R −∞

Proof. In PAP(X, p) define the operator Γ : PAP(X, p) → C(R, X) by setting t

t

Γ u(t) := −f (t, ut ) −

AT (t − s)f (s, us ) ds +

−∞

T (t − s)g(s, us ) ds,

t ∈ R.

−∞

From previous assumptions one can easily see that Γ u is well-defined and continuous. Moreover, from Theorems 3.1, 3.3 and 3.4 we infer that Γ u ∈ PAP(X, p), that is, Γ : PAP(X, p) → PAP(X, p). It remains to prove that Γ is a strict contraction on PAP(X, p). For u, v ∈ PAP(X, p) we get   Γ u(t) − Γ v(t)  Lf ut − vt B + M

t

Lf e−w(t−s) us − vs B ds

−∞

t +M   Lf

e−w(t−s) Lg (s)us − vs B ds

−∞



  t M −w(t−s) e Lg (s) ds u − v∞ 1+ + M sup w t∈R −∞

 Θu − v∞ . Clearly, from (4.3) and using the classical Banach fixed-point principle it follows that the system (4.1)–(4.2) has a unique fixed-point, which obviously is in PAP(X, p). 2 5. A functional neutral differential system In this section we discuss briefly the existence and uniqueness of a pseudo almost periodic solution to the abstract neutral differential equations of the form       d u(t) + f t, u γ1 (t) = Au(t) + g t, u γ2 (t) , dt u(0) = u0 ∈ X.

t ∈ R,

(5.1) (5.2)

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where f, g : R × X → X and γi : R → [0, ∞) for i = 1, 2, are some suitable continuous functions. The purpose here is to look for some weaker assumptions on the function f . Namely, one assumes the existence of a Banach space Y → X, not necessarily [D(A)], such that f is a Y-valued continuous function. From now on, (Y,  · Y ) denotes an arbitrary Banach space continuously embedded into X. In this event, L(Y, X) and L(Y) stand respectively for the class of all bounded linear operators, which go from Y into X, and the class of all bounded linear operators from Y into itself. We also require the following assumptions: (H2) The function s → T (s)y ∈ C([0, ∞), Y) for each y ∈ Y and there are positive constants  α˜ such that T (s)L(Y)  Me  −αs for each s  0. Moreover, the function s → AT (s) M, defined from (0, ∞) into L(Y, X) is strongly measurable and there exist a nondecreasing function H : [0, ∞) → [0, ∞) and δ > 0 with e−δs H (s) ∈ L1 ([0, ∞)) and such that   −δs AT(s) H (s) L(Y,X)  e for every s > 0. (H3) The functions f, g : R × X → X are continuous, f (·) is Y-valued, f : R × X → Y is continuous and there are a constant Lf ∈ (0, 1) and a continuous function Lg : R → (0, ∞) such that   f (t, y1 ) − f (t, y2 )  Lf y1 − y2 , t ∈ R, yi ∈ X, i = 1, 2, Y  g(t, y1 ) − g(t, y2 )  Lg (t)y1 − y2 , t ∈ R, yi ∈ X, i = 1, 2. (H4) The functions γi : R → [0, ∞) for i = 1, 2, are nondecreasing, continuously differentiable with uniformly bounded derivatives on R, inft∈R γi (t) > 0, and for each u ∈ AP(X) the function u(γi (·)) ∈ AP(X) with   |γi (−r)| + |γi (r)| < ∞. lim sup r r→∞ Remark 5.1. Note that the assumption (H2) is achieved in many cases, see, for instance, Lunardi [20], and the section devoted to applications in Rankin [26]. Remark 5.2. Note also that if γi (t) = t − p, where p ∈ R, then assumption (H4) is achieved. Lemma 5.3. Assume that both γ1 and γ2 satisfy (H4). If u ∈ PAP(X), then u(γi (·)) ∈ PAP(X) for i = 1, 2. Proof. Consider the decomposition u = v + w, where v ∈ AP(X) and w ∈ PAP0 (X). Obviously, v(γi (·)) ∈ AP(X). On the other hand, for r > 0 one can easily see that 1 2r

r

   w γi (t)  dt

−r



1 2r inft∈R γi (t)

r

   w γi (t) γ (t) dt i

−r

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2r inft∈R γi (t)

|γi (−r)|+|γ  i (r)|

2r inft∈R γi (t)

= 1 2r

  w(t) dt

−|γi (−r)|

1



and hence

|γi (r)|

1

  w(t) dt

−|γi (−r)|−|γi (r)|

 (inft∈R γi (t))−1 |γi (−r)| + |γi (r)| r 2(|γi (−r)| + |γi (r)|) r

−r

|γi (r)|+|γ  i (r)| −|γi (−r)|−|γi (r)|

w(γi (t)) dt converges to zero as r → ∞. This completes the proof.

Lemma 5.4. Let u ∈ PAP0 (Y). If v is the function defined by t AT (t − s)u(s) ds, v(t) = −∞

then v ∈ PAP0 (X). Proof. For r > 0 we get  r   t  1    AT (t − s)u(s) ds  dt    2r −r −∞ r

1  2r





1 2r 1 2r +

t

  u(s) ds dt Y

  e−δ(t−s) H (t − s)u(s)Y dt ds

−∞ −r r r

  e−δ(t−s) H (t − s)u(s)Y dt ds

−r s

−r

r−s H (−r − s)

−∞

ηuY,∞ η + 2rδ 2r ∞

L(Y,X)

  e−δ(t−s) H (t − s)u(s)Y ds dt

−r −∞ −r r

uY,∞  2r 

  AT (t − s)

−r −∞ r  t

1 2r

  w(t) dt,

r

−r−s

e

−δξ

η dξ ds + 2r

r

  u(s) dt

−r

  u(s) dt,

−r

e−δs H (s) ds

where η = 0 and uY,∞ = sups∈R (u(s)Y ). The previous inequality proves our claim. 2

2

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Definition 5.5. A function u ∈ BC(R, X) is a mild pseudo almost periodic solution of neutral system (5.1), if the function s → AT (t − s)f (s, u(γ1 (s))) is integrable on (−∞, t) for each t ∈ R and t

  u(t) = −f t, u γ1 (t) − 

   AT (t − s)f s, u γ1 (s) ds

−∞

t

   T (t − s)g s, u γ2 (t) ds,

+

t ∈ R.

−∞

We now prove the main results of this section. Theorem 5.6. Under assumptions (H2)–(H4), the system (5.1)–(5.2) has a unique pseudo almost periodic solution whenever     t t e−δ(t−s) H (t − s) ds + M sup

Θ = Lf 1 + sup t∈R −∞

e−w(t−s) Lg (s) ds < 1.

t∈R −∞

Proof. In PAP(X) define the operator Γ : PAP(X) → C(R, X) by setting    Γ u(t) = −f t, u γ1 (t) −

t

   AT (t − s)f s, u γ1 (s) ds

−∞

t +

   T (t − s)g s, u γ2 (s) ds,

t ∈ R.

−∞

From the estimate           AT (t − s)f s, u γ1 (s)   AT (t − s)   L(Y,X) f s, u γ1 (s) Y     −δ(t−s)   e H (t − s) f s, u γ1 (s) Y , we infer the Γ u is well defined, continuous and that Γ u ∈ BC(R, X). Moreover, from Lemmas 5.3 and 5.4 it follows that Γ is PAP(X)-valued. To complete the proof, we must show that Γ is a strict contraction. For u, v ∈ PAP(X),   Γ u(t) − Γ v(t)  Lf u − v∞ +

t

Lf e−δ(t−s) H (t − s)u − v∞ ds

−∞

t +M 

e−w(t−s) Lg (s)u − v∞ ds

−∞



t

 Lf 1 + sup t∈R −∞

e

−δ(t−s)

H (t − s) ds u − v∞

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t + M sup t∈R −∞

e−w(t−s) Lg (s) dsu − v∞

 Θu − v∞ . From Θ < 1 it follows that Γ is a strict contraction, and hence by the Banach fixed-point principle, there exists a unique mild solution to (4.2)–(5.2) which obviously is pseudo almost periodic. 2 6. Examples In this section we provide with two examples to illustrate our previous abstract results. For that, we first introduce the required background needed in the sequel. Throughout the rest of this section, we take X = L2 ([0, π]) and let A be the operator given by Af = f

with domain       D(A) := f ∈ L2 [0, π] : f

∈ L2 [0, π] , f (0) = f (π) = 0 . It is well known that A is the infinitesimal generator of an analytic semigroup (T (t))t0 on X. Furthermore, A has a discrete spectrum with eigenvalues of the form −n2 , n ∈ N, and corresponding normalized eigenfunctions given by  2 sin(nξ ). zn (ξ ) := π In addition to the above, the following properties hold: (a) {zn : n ∈ N} is an orthonormal basis for X;  ∞ 2 −n2 t f, z z (b) for f ∈ X, T (t)f = ∞ n n and Af = − n=1 e n=1 n f, zn zn , for every f ∈ D(A). Moreover, it is possible to define fractional powers of A, see, e.g., [20] and [24, Chapter 2]. In particular,  1 (c) for f ∈ X and α ∈ (0, 1), (−A)−α f = ∞ n=1 n2α f, zn zn ; α α (d) the operator (−A) : D((−A) ) ⊆ X → X is given by (−A)α f =

∞

n2α f, zn zn ,

  ∀f ∈ D (−A)α ,

n=1

where D((−A)α ) = {f (·) ∈ X:

∞

n=1 n

2α f, z z n n

6.1. Case of a first-order boundary-value problem Consider the first-order boundary value problem

∈ X}.

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789

  0 π ∂ b(s, η, ξ )u(t + s, η) dη ds u(t, ξ ) + ∂t −p 0

∂2 = 2 u(t, ξ ) + a0 (ξ )u(t, ξ ) + ∂ξ

0 a(s)u(t + s, ξ ) ds,

(6.1)

−p

u(t, 0) = u(t, π) = 0,

(6.2)

for t ∈ R and ξ ∈ I = [0, π]. Note that equations of type (6.1)–(6.2) arise for instance in control systems described by abstract retarded functional–differential equations with feedback control governed by proportional integro-differential law, see [12, Examples 4.2] for details. To study (6.1)–(6.2) we put X = L2 ([0, π]) and B = C([−p, 0]; X). In addition to that we suppose that the functions a, a0 , a1 are continuous and that the following holds: i

∂ (i) The functions b(·), ∂ζ i b(τ, η, ζ ), i = 1, 2, are (Lebesgue) measurable, b(τ, η, π) = 0, b(τ, η, 0) = 0 for every (τ, η) and π 0 π  2 ∂i N1 := max b(τ, η, ζ ) dη dτ dζ : i = 0, 1, 2 < ∞. ∂ζ i 0 −p 0

Define f, g: C([−p, 0]; X) by setting 0 π f (t, ψ)(ξ ) :=

b(s, η, ξ )ψ(s, η) dη ds,

−p 0

0 g(t, ψ)(ξ ) := a0 (ξ )u(t, ξ ) +

a(s)ψ(s, ξ ) ds.

−p

In view of the above, it is clear that the system (6.1)–(6.2) can be rewritten as an abstract system of the form (1.1). By a straightforward estimation that uses (i) one can show that f has values in D(A) and that f (t, ·) : C([−p, 0]; X) → [D(A)] is a bounded linear operator with Af (t, ·)  (N1 p)1/2 for each t ∈ R. Furthermore, g is a bounded linear operator on X with    0     2 g(t, ·)  a0 ∞ + p a (s) ds −p

for every t ∈ R. The next result is a consequence of Theorem 4.4. Theorem 6.1. Under the previous assumptions, the system (6.1)–(6.2) has a unique pseudo almost periodic solution whenever    0    !  2 N1 p + a0 ∞ + p a 2 (s) ds < 1. −p

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6.2. Reaction–diffusion equations with delay Most of different differential equations, reaction–diffusion equations with delay, wave equations, age-dependent population equations, can be described through abstract semilinear functional–differential equations, see Wu [31]. Here, we make use of Theorem 4.4 to study the existence and uniqueness of pseudo almost periodic solutions to the scalar reaction–diffusion equation with delay given by   ∂ ∂2 (t, ξ ) = 2 u(t, ξ ) + g t, u(t − p, ξ ) , ∂t ∂ξ u(t, 0) = u(t, π) = 0, u(τ, ξ ) = ϕ(τ, ξ ),

(6.3) (6.4)

τ ∈ [−p, 0], ξ ∈ [0, π].

(6.5)

We have Theorem 6.2. Assume that g : R × C([−p, 0]; X) → X is continuous and the existence of a positive and integrable function Lg : R → R such that   g(t, ψ1 ) − g(t, ψ2 )  Lg (t)ψ1 − ψ2 ∞ for every t ∈ R and all ψ1 , ψ2 ∈ C([−p, 0]; X). If   t e−(t−s) Lg (s) ds < 1,

Θ := sup t∈R

−∞

then there exists a unique pseudo almost periodic mild solution to the problem (6.3)–(6.5). Proof. This is a straightforward consequence of Theorem 4.4.

2

Acknowledgment The authors thank to the referee for his/her valuable comments and suggestions on the paper.

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