CUBO A Mathematical Journal

Vol.10, N
o
¯ 02, (61–74). July 2008

C(n)-Almost Automorphic Solutions of Some

Nonautonomous Differential Equations

Khalil Ezzinbi

Université Cadi Ayyad, Faculté des Sciences Semlalia,

Département de Mathématiques, BP. 2390, Marrakech, Morocco

email: ezzinbi@ucam.ac.ma

Valerie Nelson

Department of Mathematics, Morgan State University,

1700 E. Cold Spring Lane, Baltimore, MD 21251, USA

email: valerie.nelson@morgan.edu

and

Gaston N’Guérékata

Department of Mathematics, Morgan State University,

1700 E. Cold Spring Lane, Baltimore, MD 21251, USA

email: gaston.n’guerekata@morgan.edu

ABSTRACT

This paper is concerned with the study of properties of C(n)-almost automorphic func-

tions and their uniform spectra. We apply the obtained results to prove Massera type

theorems for the nonautonomous differential equation in C
k
: x′(t) = A(t)x(t)+f (t), t ∈

R and A(t) is τ periodic and the equation x′(t) = Ax(t) + f (t), t ∈ R where the oper-

ator A generates a quasi-compact semigroup in a Banach space, and in both cases f is

C(n)-almost automorphic.



62 Khalil Ezzinbi, Valerie Nelson and Gaston N’Guérékata CUBO
10, 2 (2008)

RESUMEN

En este art́ıculo estudiamos las propriedades de funciones C(n)-casi automoficas. Apli-

camos los resultados obtenidos para provar teoremas de tipo Massera para la ecuación

diferencial no autonoma en C
k
: x′(t) = A(t)x(t) + f (t), t ∈ R, A(t) es τ -periódica y

para la ecuación x′(t) = Ax(t) + f (t), t ∈ R donde el operador A genera un semigrupo

casi compacto en un espacio de Banach, en ambos casos f es una función C(n)-casi

automorfica.

Key words and phrases: Evolution equation, mild solution, almost automorphy, uniform spec-

trum.

Math. Subj. Class.: 47D06, 34G10, 45M05

1 Introduction

Let us consider in C
k

equations of the form

dx

dt
= A(t)x + f (t), (1.1)

where A(t) is a (generally unbounded) linear operator which is τ -periodic, and f is a C(n)-almost

automorphic) function on R. We will prove a Massera type result for the above differential equation

and present conditions under which every bounded solution of this equation is C(n+1)-almost

automorphic.

The concept of C(n)-almost automorphic functions was introduced by Ezzinbi, Fatajou and

N’Guérékata in [9] as a generalization of C(n)-almost periodicity (see for instance [1, 2, 3, 5, 13]).

In their work [9] , the authors study the existence of C(n)-almost automorphic solutions,

(n ≥ 1), for the following partial neutral functional differential equation

d

dt
Dut = ADut + L(ut) + f (t) for t ∈ R (1.2)

where A is a linear operator on a Banach space X satisfying the following well-known Hille-Yosida

condition

(H
0
) there exist M̄ ≥ 1 and ω ∈ R such that (ω, +∞) ⊂ ρ(A) and

|R(λ, A)n| ≤
M̄

(λ − ω)n
for n ∈ N and λ > ω,

where ρ(A) is the resolvent set of A and R(λ, A) = (λI − A)−1 for λ ∈ ρ(A). D : C → X is a

bounded linear operator, where C = C([−r, 0] ; X) is the space of continuous functions from [−r, 0]



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C(n)-Almost Automorphic Solutions ... 63

to X endowed with the uniform norm topology. For the well posedness of equation (1.2), we

assume that D has the following form

Dϕ = ϕ(0) −

∫ 0

−r

[dη(θ)] ϕ(θ) for ϕ ∈ C,

for a mapping η : [−r, 0] → L(X) of bounded variation and non atomic at zero, which means that

there exists a continuous nondecreasing function δ : [0, r] → [0, +∞) such that δ(0) = 0 and

∣∣∣∣
∫ 0

−s

[dη(θ)] ϕ(θ)

∣∣∣∣ ≤ δ(s) sup
−r≤θ≤0

|ϕ(θ)| for ϕ ∈ C and s ∈ [0, r] ,

where L(X) denotes the space of bounded linear operators from X to X. For every t ≥ σ, the

history function ut ∈ C is defined by

ut(θ) = u(t + θ) for θ ∈ [−r, 0] .

L is a bounded linear operator from C to X and f is a continuous function from R to X.

Another important problem studied in [9] is the following Massera type result.

Consider the differential equations

dx

dt
= Dx(t) + e(t), (1.3)

where D is a constant d × d matrix and e :→ Rd is C(n)-almost automorphic function. Then if

Equ. (1.3) has a bounded solution on R
+
, it has a C(n+1)-almost automorphic solution. Moreover

every bounded solution on R is C(n+1)-almost automorphic.

In the present paper we continue the study of elementary properties of C(n)-almost automor-

phic functions and apply them to investigate the C(n)-almost automorphic functions solutions to

the non autonomous periodic equation (1.1).

The work is organized as follows. In Section 2, we review the concept of C(n)-almost periodic

functions and present further properties of C(n)-almost automorphic functions with values in a

Hilbert space. In Section 3, we discuss some results related to the uniform spectrum of C(n)-

almost automorphic functions. Our main results (Theorem 4.2 and 4.11)are presented in Section

4.

2 C(n)-almost periodic and C(n)-almost automorphic func-

tions

We recall some properties about C(n)-almost periodic and C(n)-almost automorphic 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 hτ by

hτ (s) = h(τ + s) for s ∈ R.



64 Khalil Ezzinbi, Valerie Nelson and Gaston N’Guérékata CUBO
10, 2 (2008)

Let Cn(R, X) be the space of all continuous function which have a continuous n-th derivative on

R and Cn
b
(R, X) be the subspace of Cn(R, X) of functions satisfying

sup
t∈R

n∑

i=0

‖h(i)(t)‖ < ∞,

h(i) denotes the i-the derivative of h. Then Cn
b
(R, X) is a Banach space provided with the following

norm

‖h‖n = sup
t∈R

n∑

i=0

‖h(i)(t)‖.

Definition 2.1. A bounded continuous function h : R → X is said to be almost periodic if

{hτ : τ ∈ R} is relatively compact in BC(R, X).

Definition 2.2. A continuous function θ : R×X → X is said to be almost periodic in t uniformly

in x if for any compact K in X and for every sequence of real numbers (s′
n
)n there exists a

subsequence (sn)n such that

lim
n→∞

θ(t + sn, x) exists uniformly in (t, x) ∈ R×K.

Definition 2.3. [3] Let ε > 0 and h ∈ Cn
b
(R, X). A number τ ∈ R is said to be a ‖ · ‖n − ε almost

period of the function f if

‖hτ − h‖n < ε.

The set of all ‖ · ‖n − ε almost period of the function h is denoted by E
(n)

(ε, f ).

Definition 2.4. [3] A function h ∈ Cn
b
(R, X) is said to be a almost periodic function if for every

ε > 0, the set E(n)(ε, h) is relatively dense in R.

Definition 2.5. AP (n)(X) is the space of the Cn-almost periodic functions.

Since it is well known that for any almost periodic functions h1 and h2 and ε > 0, there exists a

relatively dense set of their common ε almost period. Consequently, we get the following result.

Proposition 2.6. h ∈ AP (n)(X) if and only if h(i) ∈ AP (X) for i = 0, 1, 2, ..., n.

Since AP (X) equipped with uniform norm topology is a Banach space, then we get the following

result.

Proposition 2.7. AP (n)(X) provided with the norm ‖ · ‖n is a Banach space.

Example. The following example of a Cn-almost periodic function has been given in [5]. Let

g(t) = sin(αt) + sin(βt),

where
α

β
/∈ Q. Then the function h(t) = eg(t) is Cn-almost periodic for any n ≥ 1. In [5], one can

find example of function which is Cn-almost periodic but not Cn+1-almost periodic.



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C(n)-Almost Automorphic Solutions ... 65

Definition 2.8. [18] A continuous function h : R → X is said to be almost automorphic if for

every sequence of real numbers (s′
n
)n there exists a subsequence (sn)n such that

k(t) = lim
n→∞

h(t + sn) exists for all t in R

and

lim
n→∞

k(t − sn) = h(t) for all t in R.

Remark. By the pointwise convergence, the function k is just measurable and not necessarily

continuous. If the convergence in both limits is uniform, then h is almost periodic. The concept of

almost automorphy is then larger than the one of the almost periodicity. If h is almost automorphic,

then its range is relatively compact, thus bounded in norm. Let p(t) = 2 + cost + cos
√

2t and

h : R → R such that h = sin 1
p
. Then h is almost automorphic, but h is not uniformly continuous

on R, it follows that h is not almost periodic.

Definition 2.9. [18] A continuous function h : R → X is said to be compact almost automorphic

if for every sequence of real numbers (s′
n
)n, there exists a subsequence (sn)n such that

lim
m→∞

lim
n→∞

h(t + sn − sm) = h(t) uniformly on any compact set in R.

Theorem 2.10. [18] If we equip AAc(X), the space of compact almost automorphic X-valued

functions, with the sup norm, then AAc(X) is a Banach space.

Theorem 2.11. [18] If we equip AA(X), the space of almost automorphic X-valued functions,

with the sup norm, then AA(X) turns out to be a Banach space.

Definition 2.12. A continuous function θ : R×X → X is said to be almost automorphic in t with

respect to x if for every sequence of real numbers (s′
n
)n, there exists a subsequence (sn)n such that

lim
m→∞

lim
n→∞

θ(t + sn − sm, x) = θ(t, x) for t ∈ R and x ∈ X.

Now we recall the concept of Cn-almost automorphic functions recently introduced in [9] as a

generalization of the one of Cn-almost periodic functions.

Definition 2.13. A continuous function h : R → X is said to be Cn- almost automorphic for

n ≥ 1 if for i = 0, 1, ..., n, the i-th derivative h(i) of h is almost automorphic.

We will denote by AA(n)(X) the space of all Cn-almost automorphic X-valued functions.

Definition 2.14. ([9]) A continuous function h : R → X is said to be Cn-compact almost auto-

morphic if for i = 0, 1, ..., n, the i-th derivative h(i) of h is compact almost automorphic.

We denote by AA
(n)
c (X) the space of all C

n
-compact almost automorphic X-valued functions.

Since AA(X) and AAc(X) are Banach spaces, then we get also the following result.



66 Khalil Ezzinbi, Valerie Nelson and Gaston N’Guérékata CUBO
10, 2 (2008)

Proposition 2.15. ([9]) AA(n)(X) and AA
(n)
c (X) provided with the norm |.|n are Banach spaces.

The following superposition result is easy to prove.

Proposition 2.16. Let f ∈ AA(n)(X) and A ∈ B(X). Then Af ∈ AA(n)(X).

Proposition 2.17. Let λ ∈ AA(n)(R, K) and f ∈ A(n)(X) where X is a Banach space over the

field K. Then (λf )(t) := λ(t)f (t) is in AA(n)(X).

We also have the following results

Theorem 2.18. Let X be a Hilbert space and f ∈ AA(n)(X). Then the function F (t) =
∫

t

0
f (s)ds ∈

AA(n+1)(X) iff RF is bounded in X.

Proof. We have just to prove the only if part. It comes by induction. The case n = 0 is known

([18] Theorem 2.4.6). Assume now that f is in AA(n)(X), and that the theorem is true for n − 1;

then F ∈ AA(n)(X). But we have F ′ = f and so F ′ ∈ AA(n)(X), from which we conclude that

F ∈ AA(n+1)(X).

Theorem 2.19. Let ν ∈ AA(n)(R, Ls(X, Y )) and f ∈ AA
(n)

(R, X). Then νf ∈ AA(n)(R, Y ) for

two Banach spaces X and Y .

Proof. It suffices to observe that ν(i)f (n−i) : R → Y is almost automorphic, for each i = 0, 1, ...n.

3 Uniform spectrum of a function in BC(R, X)

Let us consider the following simple ordinary differential equation in a complex Banach space X

x
′

(t) − λx = f (t), (3.1)

where f ∈ BC(X). If Reλ 6= 0, the homogeneous equation associated with this has an exponential

dichotomy; so, (3.1) has a unique bounded solution which we denote by xf,λ(·). Moreover, from

the theory of ordinary differential equations, it follows that for every fixed ξ ∈ R,

xf,λ(ξ) :=

{ ∫
ξ

−∞
eλ(ξ−t)f (t)dt (if Reλ < 0)

−
∫ +∞

ξ
eλ(ξ−t)f (t)dt (if Reλ > 0).

(3.2)

=

{ ∫ 0
−∞

e−ληf (ξ + η)dη (if Reλ < 0)

−
∫ +∞
0

e−ληf (ξ + η)dη (if Reλ > 0).
(3.3)

As is well known, the differentiation operator D is a closed operator on BC(R, X). The above

argument shows that ρ(D) ⊃ C\iR and xf,λ = (D − λ)
−1f for every λ ∈ C\iR and f ∈ BC(R, X).



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C(n)-Almost Automorphic Solutions ... 67

Hence, for every λ ∈ C with Reλ 6= 0 and f ∈ BC(R, X) the function [(λ − D)−1f ](t) =

Ŝ(t)f (λ) ∈ BC(R, X). Moreover, (λ − D)−1f is analytic on C\iR.

Definition 3.1. Let f be in BC(R, X). Then,

i) α ∈ R is said to be uniformly regular with respect to f if there exists a neighborhood U of

iα in C such that the function (λ − D)−1f , as a complex function of λ with Reλ 6= 0, has

an analytic continuation into U.

ii) The set of ξ ∈ R such that ξ is not uniformly regular with respect to f ∈ BC(R, X) is called

uniform spectrum of f and is denoted by spu(f ).

Observe that, if f ∈ BU C(R, X), then α ∈ R is uniformly regular if and only if it is regular

with respect to f (cf. [15]).

We now list some properties of uniform spectra of functions in BC(R, X).

Proposition 3.2. Let g, f, fn ∈ BC(R, X) such that fn → f as n → +∞ and let Λ ⊂ R be a

closed subset satisfying spu(fn) ⊂ Λ for all n ∈ N. Then the following assertions hold:

i) spu(f ) = spu(f (h + ·));

ii) spu(αf (·)) ⊂ spu(f ), α ∈ C;

iii) sp(f ) ⊂ spu(f );

iv) spu(Bf (·)) ⊂ spu(f ), B ∈ L(X);

v) spu(f + g) ⊂ spu(f ) ∪ spu(g);

vi) spu(f ) ⊂ Λ.

We also recall the following important result (see [15] for the proof).

Proposition 3.3. Let f ∈ BC(R, X). Then

spu(f ) = spc(f ),

where spc(f ) denotes the Carleman spectrum of f .

From the above properties, the following is obtained:

Proposition 3.4. ([3]) Let f ∈ C
(n)
b

(X). Then

spu(f
(i)

) ⊂ spu(f
(i−1)

), f or every i = 1, 2, ..., n.

Now we can state and prove.



68 Khalil Ezzinbi, Valerie Nelson and Gaston N’Guérékata CUBO
10, 2 (2008)

Lemma 3.5. Let f ∈ AA(n)(X) and φ ∈ L1(R) whose Fourier transform has compact support

supp(φ) . Then the function g := φ ∗ f ∈ AA(n)(X); moreover spu(g) ⊂ spu(f ) ∩ supp(φ).

Proof. Let’s assume n = 0. And let (s′
n
) be an arbitrary sequence of real numbers. Since f ∈

AA(X), there exists a subsequence (sn) such that

h(t − s) := lim
n→∞

f (t − s + sn)

is well-defined for each t, s ∈ R, and

lim
n→∞

h(t − s − sn) = f (t − s)

each t, s ∈ R.

Note that ‖f (t − s + sn)φ(s)‖ ≤ ‖f‖∞‖φ(s)‖. And since φ ∈ L
1
(R), we may deduce by the

Lebesgue’ dominated convergence theorem that

lim
n→∞

g(t + sn) =

∫

R

lim
n→∞

f (t − s + sn)φ(s)ds =

∫

R

h(t − s)φ(s)ds = (h ∗ φ)(t)

for each t ∈ R.

Similarly we can prove that

lim
n→∞

(h ∗ φ)(t − sn) = (φ ∗ f )(t)

for each t ∈ R.

Thus φ ⋆ f ∈ AA(X). Now we know that g is Cn with derivatives: g(k) = φ ∗ f (k) (if k ≤ n).

So, for each k ≤ n, g(k) ∈ AA(X), and the lemma follows.

4 Applications to Differential Equations

Consider in a (complex) Banach space X the linear equation

x
′

(t) = Ax(t) + f (t), t ∈ R, (4.1)

where A : D(A) ⊂ X → X is a linear operator, and f ∈ C(R, X).

We first generalize [9] Theorem 3.20 as follows.

Lemma 4.1. Suppose f ∈ AA(n)(X) and A ∈ L(X). Then every bounded solution of Eq.(4.2) is

in AA(n+1)(X).

Proof. It suffices to observe that since A is bounded, then

x(n+1)(t) = Ax(n)(t) + f (n)(t).



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C(n)-Almost Automorphic Solutions ... 69

We have the following Massera type result.

Theorem 4.2. Let f ∈ AA(n)(Ck). If Eq. (4.1) has a bounded solution on R+, then it has a

AA(n+1)(Ck) solution. Moreover every bounded solution of the differential equation

x′(t) = A(t)x(t) + f (t), t ∈ R, (4.2)

where A(t) : R → Mk(C) is τ -periodic, is in AA
(n+1)

(C
k
).

Proof. The proof is similar to Theorem 3.1 [14]. First let us note that by Floquet’s theory and

without loss of generality we may assume that A(t) = A is independent of t. Next we will show

that the problem can be reduced to the one-dimensional case. In fact, if A is independent of t, by

a change of variable if necessary, we may assume that A is of Jordan normal form. In this direction

we can go further with assumption that A has only one Jordan box. That is, we have to prove the

theorem for equations of the form




x′1(t)

x′2(t)
.
.
.

x′
k
(t)




=




λ 1 0 . . . 0

0 λ 1 . . . 0

. . . . . . . . . . . .

0 0 0 . . . λ







x1(t)

x2(t)
.
.
.

xk(t)




+




f1(t)

f2(t)
.
.
.

fk(t)




, t ∈ R.

Now if x is a bounded solution of the above system on R+, then by Theorem 3.14 [9], it has

an almost almost automorphic solution on R. Since f ∈ AA(n)(C), then by Lemma 4.1 above, we

deduce that x ∈ AA(n+1)(C).

The following is easy to establish.

Corollary 4.3. Consider the Differential Equation

x′(t) = Ax(t) + f (t), t ∈ R (4.3)

where f ∈ AA(n)(Rk), and A ∈ B(Rk) such that the real part of each of its eigenvalues is negative.

Then Eq.(4.3) has a unique solution in x ∈ AA(n+1)(Rk).

We also have the following result.

Theorem 4.4. Let A ∈ B(Rk) and suppose that Eq.(4.3) has a unique AA(1)(Rk) solution for

each f ∈ AAk. Then the map T : AA(Rk) → AA(1)(Rk), f → x is linear and continuous, that is

there exists c > 0 such that

‖x‖1 ≤ c‖f‖0

where ‖ · ‖0 denotes the usual sup norm in AA(R
k
)

Proof. Linearity of T is obvious. Let us prove its continuity.



70 Khalil Ezzinbi, Valerie Nelson and Gaston N’Guérékata CUBO
10, 2 (2008)

First, let us consider the map S : AA(1)(Rk) → AA(Rk) given by

Sx(t) = f (t).

That is, x is the unique AA(1)(Rk) solution to ACP. S is defined as

(Sx)(t) = x′(t) − Ax(t) = f (t),

thus Sx = f so ST f = f . Also T Sx = T f = x. We deduce that S = T −1.

On another hand we have

‖Sx‖0 ≤ ‖x
′

‖0 + K‖x‖0 ≤ K1(‖x
′

‖0 + ‖x‖0)

where K1=max (1, K). Thus we have

‖Sx‖0 ≤ K1‖x‖1.

That means S is continuous. And since S is injective, then S−1 = T is continuous ([16] 1.6.6

Corollary page 44) This ends our proof.

Now we investigate the existence of C(n) almost automorphic solutions for the following equa-

tion

x′(t) = Ax(t) + f (t) for t ∈ R (4.4)

where A is the infinitesimal generator of a strongly continuous semigroup (T (t))
t≥0 in a Banach

space X.

Definition 4.5. We say that a functon is a mild solution of equation (4.4) if for any σ and t ≥ σ,

we have

x(t) = T (t − σ)x(σ) +

∫
t

σ

T (t − s)f (s)ds.

For simplicity, mild solution will be called solution in the sequel.

We need to recall some preliminary results on quasi compact semigroups.We first introduce

the Kuratowski measure of noncompactness α(.) of bounded sets K in a Banach space X by

α(K) = inf {ε > 0 : K has a finite cover of balls of diameter < ε} .

For a bounded linear operator B on X, |B|
α

is defined by

|B|
α

= inf {ε > 0 : α(B(K)) ≤ εα(K) for any bounded set K of X} .

The essential growth bound ωess (T ) of the semigroup (T (t))t≥0 is defined by

ωess (T ) = lim
t→+∞

1

t
log |T (t)|

α
,

= inf
t>0

1

t
log |T (t)|

α
.



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C(n)-Almost Automorphic Solutions ... 71

Definition 4.6. The essential spectrum σess(A) of A is the set of λ ∈ σ(A) : the spectrum of A,

such that one of the following conditions holds:

(i) Im(λI − A) is not closed,

(ii) the generalized eigenspace Mλ(A) =
⋃

k≥1

Ker(λI − A)k is of infinite dimension,

(iii) λ is a limit point of σ(A) \ {λ}.

The essential radius of any bounded operator T in Y is defined by

ress(T ) = sup{|λ| : λ ∈ σess(T )}.

Definition 4.7. We say that the semigroup (T (t))t≥0 is quasi compact if

ωess (T ) < 0.

Theorem 4.8. The semigroup (T (t))t≥0 is quasi compact if for some t0 > 0, we have

ress(T (t0)) < 1.

Lemma 4.9. If the semigroup (T (t))t≥0 is quasi compact. Then,

σ+(A) = {λ ∈ σ(A) : Re(λ) ≥ 0}

is a finite set of the eigenvalues of A which are not in the essential spectrum.

Theorem 4.10. [9] Assume that the semigroup (T (t))t≥0 is quasi compact. Then X is decomposed

as follows

X = S ⊕ V,

where X is T −invariant and there are positive constants α and N such that

|T (t) x| ≤ N e−αt |x| for t ≥ 0 and x ∈ S. (4.5)

Moreover V is a finite dimensional space and the restriction of T to V becomes a group.

Let P − and P + denote respectively the projection operators respectively of X into S and V .

Theorem 4.11. Assume that the semigroup (T (t))t≥0 is quasi compact and the input function

f is C(n)-almost automorphic. If equation (4.4) has a bounded solution on R+, then it has a

C(n)-almost automorphic solution. Moreover every bounded solution of equation (4.4) on R is a

C(n)-almost automorphic solution.

Proof of Theorem. Let B be a matrix be such that

T (t) = etB in V .



72 Khalil Ezzinbi, Valerie Nelson and Gaston N’Guérékata CUBO
10, 2 (2008)

Let u be a bounded solution of equation (4.4) on R+. The function z(t) = P +u(t) is a bounded

solution on R
+

of the following ordinary differential equation

z′(t) = Bz(t) + P +f (t) for t ≥ 0. (4.6)

Moreover, the function t → P +f (t) is C(n)-almost automorphic from R to Rd. By Theorem 4.2

we get that the reduced system (4.6) has a C(n)-almost automorphic solution z̃ and the function

v defined by

v(t) = z̃ (t) +

∫
t

−∞

T (t − s) P −f (s) ds for t ∈ R,

is a bounded solution of equation (4.4) on R. We claim that v is C(n)-almost automorphic. In

fact, let y be defined by

y(t) =

∫
t

−∞

T (t − s) P −f (s) ds for t ∈ R.

Then y ∈ C
(n)
b

(R, X). Clearly y is a. a. by [19]. Also we have y′(t) = P −f (t) + y(t). So y′ is

a. a. In general y(i) = P −f (i−1)(t) + y(i−1)(t), i = 1, 2, ..., n, which implies that y is C(n) almost

automorphic. Let u be a bounded solution on R, then u is given by the following formula

u (t) = z (t) +

∫
t

−∞

T (t − s) P −f (s) ds for t ∈ R,

where

z(t) = P +u(t) for t ∈ R

is a solution of the reduced system (4.6), which is C(n)-almost automorphic by Theorem 4.2 and

arguing as above, one can prove that the function

t →

∫
t

−∞

T (t − s) P
−

f (s) ds for t ∈ R,

is also C(n)-almost automorphic.

Received: December 2007. Revised: February 2008.

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