CUBO A Mathematical Journal
Vol.16, No¯ 02, (01–31). June 2014

Pseudo-Almost Periodic and Pseudo-Almost Automorphic
Solutions to Some Evolution Equations Involving

Theoretical Measure Theory

Toka Diagana

Howard University,

2441 6th Street N.W.,

Washington, D.C. 20059,

USA.

tdiagana@howard.edu

Khalil Ezzinbi

Université Cadi Ayyad,

Faculté des Sciences Semlalia,

Département de

Mathématiques,

BP 2390, Marrakesh, Maroc

Mohsen Miraoui

Institut supérieur des Etudes

technologiques de Kairouan,

Rakkada-3191 Kairouan,

Tunisie.

ABSTRACT

Motivated by the recent works by the first and the second named authors, in this paper

we introduce the notion of doubly-weighted pseudo-almost periodicity (respectively,

doubly-weighted pseudo-almost automorphy) using theoretical measure theory. Basic

properties of these new spaces are studied. To illustrate our work, we study, under

Acquistapace–Terreni conditions and exponential dichotomy, the existence of (µ,ν)-

pseudo almost periodic (respectively, (µ,ν)-pseudo almost automorphic) solutions to

some nonautonomous partial evolution equations in Banach spaces. A few illustrative

examples will be discussed at the end of the paper.

RESUMEN

Motivado por los trabajos recientes del primer y segundo autor, en este art́ıculo intro-

ducimos la noción de seudo-casi periodicidad con doble peso (seudo-casi automorf́ıa con

doble peso respectivamente) usando Teoŕıa de la Medida. Se estudian las propiedades

básicas de estos espacios nuevos. Para ilustrar nuestro trabajo, bajo las condiciones de

Acquistapace-Terreni y dicotomı́a exponencial estudiamos la existencia de soluciones

(respectivamente, (µ,ν) seudo-casi periódicas (µ,ν) seudo-casi automórficas) para al-

gunas ecuaciones parciales de evolución autónomas en espacios de Banach. Algunos

ejemplos ilustrativos se discutirán al final del art́ıculo.

Keywords and Phrases: Evolution family; exponential dichotomy; Acquistapace–Terreni con-

ditions; pseudo-almost periodic; pseudo-almost automorphic; evolution equation; nonautonomous

equation; doubly-weighted pseudo-almost periodic; doubly-weighted pseudo-almost automorphy;

(µ,ν)-pseudo-almost periodicity; (µ,ν)-pseudo-almost automorphy; neutral systems; positive mea-

sure.
2010 AMS Mathematics Subject Classification: 34C27; 34K14; 34K30; 35B15; 43A60;

47D06; 28Axx; 58D25; 65J08.



2 Toka Diagana, Khalil Ezzinbi & Mohsen Miraoui CUBO
16, 2 (2014)

1 Introduction

Motivated by the recent works by Ezzinbi et al. [12, 13] and Diagana [30], in this paper we make

extensive use of theoretical measure theory to introduce and study the concept of doubly-weighted

pseudo almost periodicity (respectively, doubly-weighted pseudo almost automorphy). Obviously,

these new notions generalize all the different notions of weighted pseudo-almost periodicity (respec-

tively, weighted pseudo-almost automorphy) recently introduced in the literature. In contrast with

[12, 13], here the idea consists of using two positive measures instead of one. Doing so will provide

us a larger and richer class of weighted ergodic spaces. Basic properties of these new functions will

be studied including their translation invariance and compositions etc.

To illustrate our study, we study the existence of (µ,ν)-pseudo-almost periodic (respectively,

(µ,ν)-pseudo-almost automorphic) solutions to the following nonautonomous differential equations,

d

dt
u(t) = A(t)u(t) + F(t, u(t)), t ∈ R, (1.1)

and

d

dt

!

u(t) − G(t, u(t))
"

= A(t)
!

u(t) − G(t, u(t))
"

+ F(t, u(t)), t ∈ R, (1.2)

where A(t) : D(A(t)) ⊂ X #→ X for t ∈ R is a family of closed linear operators on a Banach
space X, satisfying the well-known Acquistapace–Terreni conditions, and F, G : R × X #→ X are
jointly continuous functions satisfying some additional conditions. One should indicate that the

autonomous case, i.e., A(t) = A for all t ∈ R, and the periodic case, that is, A(t + θ) = A(t) for
some θ > 0, have been extensively studied, see [8, 10, 40, 41, 53, 56] for the almost periodic case

and [18, 22, 39, 42, 50, 51] for the almost automorphic case. Recently, Diagana [24, 25, 26, 32]

studied the existence and uniqueness of weighted pseudo-almost periodic and weighted pseudo-

almost automorphic solutions to some classes of nonautonomous partial evolution equations of

type Eq. (1.1). Similarly, in Diagana [33], the existence of pseudo-almost periodic solutions to Eq.

(1.2) has been studied in the particular case when G = 0. In this paper it goes back to studying the

existence of doubly-weighted pseudo-almost periodic (respectively, doubly-weighted pseudo-almost

automorphic) solutions in the general case as outlined above using theoretical measure theory.

The existence and uniqueness of almost periodic, almost automorphic, pseudo-almost peri-

odic and pseudo-almost automorphic solutions is one of the most attractive topics in the quali-

tative theory of ordinary or functional differential equations due to applications in the physical

sciences, mathematical biology, and control theory. The concept of almost automorphy, which

was introduced by Bochner [15], is an important generalization of the classical almost period-

icity in the sense of Bohr. For basic results on almost periodic and almost automorphic func-

tions we refer the reader to [7, 59, 61], where the authors give an important overview about

their applications to differential equations. In recent years, the existence of almost periodic,

pseudo-almost periodic, almost automorphic, and pseudo-almost automorphic solutions to dif-

ferent kinds of differential equations have been extensively investigated by many people, see, e.g.,



CUBO
16, 2 (2014)

Pseudo-Almost Periodic and Pseudo-Almost Automorphic . . . 3

[3, 4, 5, 16, 17, 19, 20, 30, 23, 24, 32, 33, 34, 35, 36, 37, 39, 43, 44, 45, 46, 48, 57, 58, 60] and the

references therein.

The concept of weighted pseudo-almost periodicity, which was introduced by Diagana [25,

26, 27, 29] is a natural generalization of the classical pseudo-almost periodicity due to Zhang

[59, 60, 61]. A few years later, Blot et al. [11], introduced the concept of weighted pseudo-almost

automorphy as a generalization of weighted pseudo-almost periodicity. More recently, Ezzinbi

et al. [12, 13] presented a new approach to study weighted pseudo-almost periodic and weighted

pseudo-almost automorphic functions using theoretical measure theory, which turns out to be more

general than Diagana’s approach.

Let us explain the meaning of this notion as introduced by Ezzinbi et al.et al. [12, 13]. Let

µ be a positive measure on R. We say that a continuous function f : R #→ X is µ-pseudo-almost
periodic (respectively, µ-pseudo almost automorphic) if f = g + ϕ, where g is almost periodic

(respectively, almost automorphic) and ϕ is ergodic with respect to the measure µ in the sense

that

lim
r→∞

1

µ(Qr)

∫

Qr

∥ϕ(s)∥dµ(s) = 0,

where Qr := [−r, r] and µ(Qr) :=

∫

Qr

dµ(t).

One can observe that a ρ-weighted pseudo almost automorphic function is µ-pseudo almost

automorphic, where the measure µ is absolutely continuous with respect to the Lebesgue measure

and its RadonNikodym derivative is ρ,

dµ(t)

dt
= ρ(t).

Here we generalize the above-mentioned notion of µ-pseudo-almost periodicity. Fix two pos-

itive measures µ,ν in R. We say that a function f : R #→ X is (µ,ν)-pseudo-almost periodic
(respectively, (µ,ν)-pseudo-almost automorphic) if

f = g + ϕ,

where g is almost periodic (respectively, almost automorphic) and ϕ is (µ,ν)-ergodic in the sense

that

lim
r→∞

1

ν(Qr)

∫

Qr

∥ϕ(s)∥dµ(s) = 0,

Clearly, the (µ, µ)-pseudo-almost periodicity coincides with the µ-pseudo-almost periodicity. More

generally, the (µ,ν)-pseudo-almost periodicity coincides with the µ-pseudo-almost periodicity when

the measures µ and ν are equivalent.

In this paper, we introduce and study properties of (µ,ν)-pseudo-almost periodic functions

and make use of these new functions to study the existence and uniqueness of (µ,ν)-pseudo-almost

periodic (respectively, (µ,ν)- pseudo-almost automorphic) solutions of the nonautonomous partial

evolution equations Eq. (1.1) and Eq. (1.2) in a Banach space.



4 Toka Diagana, Khalil Ezzinbi & Mohsen Miraoui CUBO
16, 2 (2014)

The organization of this paper is as follows. In Section 2, we recall some definitions and

lemmas of (µ,ν)-pseudo almost periodic functions, (µ,ν)-pseudo-almost automorphic functions,

and the basic notations of evolution family and exponential dichotomy. In Section 3, we study

the existence and uniqueness of (µ,ν)-pseudo almost periodic (respectively, (µ,ν)-pseudo almost

automorphic) solutions to both Eq. (1.1) and Eq. (1.2). In Section 4, we give some examples to

illustrate our abstract results.

2 Preliminaries

2.1 (µ,ν)-Pseudo-Almost Periodic and (µ,ν)-Pseudo-Almost Automor-
phic Functions

Let (X, ∥ · ∥), (Y, ∥ · ∥) be two Banach spaces and let BC(R, X) (respectively, BC(R × Y, X)) be
the space of bounded continuous functions f : R −→ X (respectively, jointly bounded continuous
functions f : R × Y −→ X). Obviously, the space BC(R, X) equipped with the super norm

∥f∥∞ := sup
t∈R

∥f(t)∥

is a Banach space. Let B(X, Y) denote the Banach spaces of all bounded linear operator from X

into Y equipped with natural topology with B(X, X) = B(X).

Definition 2.1. [21] A continuous function f : R #→ X is said to be almost periodic if for every
ε > 0 there exists a positive number l(ε) such that every interval of length l(ε) contains a number

τ such that

∥f(t + τ) − f(t)∥ < ε for t ∈ R.

Let AP(R, X) denote the collection of almost periodic functions from R to X. It can be easily

shown that (AP(R, X), ∥ · ∥∞) is a Banach space.

Definition 2.2. [38] A jointly continuous function f : R × Y #→ X is said to be almost periodic in
t uniformly for y ∈ Y, if for every ε > 0, and any compact subset K of Y, there exists a positive
number l(ε) such that every interval of length l(ε) contains a number τ such that

∥f(t + τ, y) − f(t, y)∥ < ε for (t, y) ∈ R × K.

We denote the set of such functions as APU(R × Y, X).

Let µ,ν ∈ M. If f : R #→ X is a bounded continuous function, we define its doubly-weighted
mean, if the limit exists, by

M(f, µ,ν) := lim
r→∞

1

ν(Qr)

∫

Qr

f(t)dµ(t).



CUBO
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Pseudo-Almost Periodic and Pseudo-Almost Automorphic . . . 5

It is well-known that if f ∈ AP(R, X), then its mean defined by

M(f) := lim
r→∞

1

2r

∫

Qr

f(t)dt

exists [15]. Consequently, for every λ ∈ R, the following limit

a(f,λ) := lim
r→∞

1

2r

∫

Qr

f(t)e−iλtdt

exists and is called the Bohr transform of f.

It is well-known that a(f,λ) is nonzero at most at countably many points [15]. The set defined

by

σb(f) :=
{
λ ∈ R : a(f,λ) ̸= 0

}

is called the Bohr spectrum of f [47].

Theorem 2.3. [47] Let f ∈ AP(R, X). Then for every ε > 0 there exists a trigonometric polynomial

Pε(t) =
n∑

k=1

ake
iλkt

where ak ∈ X and λk ∈ σb(f) such that ∥f(t) − Pε(t)∥ < ε for all t ∈ R.

Theorem 2.4. Let µ,ν ∈ M and suppose that lim
r→∞

µ(Qr)

ν(Qr)
= θµν. If f : R #→ X is an almost

periodic function such that

lim
r→∞

#

#

#

#

#

1

ν(Qr)

∫

Qr

eiλtdµ(t)

#

#

#

#

#

= 0 (2.1)

for all 0 ̸= λ ∈ σb(f), then the doubly-weighted mean of f,

M(f, µ,ν) = lim
T→∞

1

ν(QT )

∫

QT

f(t)dµ(t)

exists. Furthermore, M(f, µ,ν) = θµνM(f).

Proof. The proof of this theorem was given in [30] in the case of measures of the form ρ(t)dt.

For the sake of completeness we reproduce it here for positive measures. If f is a trigonometric

polynomial, say, f(t) =
∑n

k=0 ake
iλkt where ak ∈ X − {0} and λk ∈ R for k = 1, 2, ..., n, then

σb(f) = {λk : k = 1, 2, ..., n}. Moreover,

1

ν(Qr)

∫

Qr

f(t)dµ(t) = a0
µ(Qr)

ν(Qr)
+

1

ν(Qr)

∫

Qr

$

n∑

k=1

ake
iλkt

%

dµ(t)

= a0
µ(Qr)

ν(Qr)
+

n∑

k=1

ak
$ 1

ν(Qr)

∫

Qr

eiλktdµ(t)
%



6 Toka Diagana, Khalil Ezzinbi & Mohsen Miraoui CUBO
16, 2 (2014)

and hence
&

&

&

&

1

ν(Qr)

∫

Qr

f(t)dµ(t) − a0
µ(Qr)

ν(Qr)

&

&

&

&

≤
n∑

k=1

∥ak∥
#

#

#

1

ν(Qr)

∫

Qr

eiλktdµ(t)
#

#

#

which by Eq. (2.1) yields
&

&

&

&

1

ν(Qr)

∫

Qr

f(t)dµ(t) − a0θµν

&

&

&

&

→ 0 as r → ∞

and therefore M(f, µ,ν) = a0θµν = θµνM(f).

If in the finite sequence of λk there exist λnk = 0 for k = 1, 2, ...l with am ∈ X − {0} for all
m ̸= nk (k = 1, 2, ..., l), it can be easily shown that

M(f, µ,ν) = θµν
l∑

k=1

ank = θµνM(f).

Now if f : R #→ X is an arbitrary almost periodic function, then for every ε > 0 there exists a
trigonometric polynomial (Theorem 2.3) Pε defined by

Pε(t) =
n∑

k=1

ake
iλkt

where ak ∈ X and λk ∈ σb(f) such that

∥f(t) − Pε(t)∥ < ε (2.2)

for all t ∈ R.

Proceeding as in Bohr [15] it follows that there exists r0 such that for all r1, r2 > r0,
&

&

&

1

ν(Qr1)

∫

Qr1

Pε(t)dµ(t) −
1

ν(Qr2)

∫

Qr2

Pε(t)dµ(t)
&

&

&
= θµν

&

&

&
M(Pε) − M(Pε)

&

&

&
= 0 < ε.

In view of the above it follows that for all r1, r2 > r0,

&

&

&

1

ν(Qr1)

∫

Qr1

f(t)dµ(t) −
1

ν(Qr2)

∫

Qr2

Pε(t)dµ(t)
&

&

&
≤

1

ν(Qr1)

∫

Qr1

∥f(t) − Pε(t)∥dµ(t)

+
&

&

&

1

ν(Qr1)

∫

Qr1

Pε(t)dµ(t) −
1

ν(Qr2)

∫

Qr2

Pε(t)dµ(t)
&

&

&
< ε.

Now for all r > r0,
&

&

&

1

ν(Qr)

∫

Qr

f(t)dµ(t) −
1

ν(Qr)

∫

Qr

Pε(t)dµ(t)
&

&

&
< ε

and hence M(f, µ,ν) = M(Pε, µ,ν) = θµνM(Pε) = θµνM(f). The proof is complete.



CUBO
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Pseudo-Almost Periodic and Pseudo-Almost Automorphic . . . 7

Definition 2.5. [51] A continuous function f : R → X is called almost automorphic if for every

sequence (σn)n∈N there exists a subsequence (sn)n∈N ⊂ (σn)n∈N such that

lim
n,m→∞

f(t + sn − sm) = f(t) for each t ∈ R.

Equivalently,

g(t) := lim
n→∞

f(t + sn) and f(t) = lim
n→∞

g(t − sn)

are well defined for each t ∈ R.

Let AA(R, X) denote the collection of all almost automorphic functions from R to X. It can

be easily shown that (AA(R, X), ∥.∥∞) is a Banach space.

Definition 2.6. [13] A function f : R × X → Y is said to be almost automorphic in t uniformly
with respect to x in X if the following two conditions hold:

(i) for all x ∈ X, f(., x) ∈ AA(R, Y),
(ii) f is uniformly continuous on each compact set K in X with respect to the second variable x,

namely, for each compact set K in X, for all ε > 0, there exists δ > 0 such that for all x1, x2 ∈ K,
one has

∥x1 − x2∥ ≤ δ ⇒ sup
t∈R

∥f(t, x1) − f(t, x2)∥ ≤ ε.

Denote by AAU(R × X, Y) the set of all such functions.

Remark 2.7. [13] Note that in the above limit the function g is just measurable. If the convergence

in both limits is uniform in t ∈ R, then f is almost periodic. The concept of almost automorphy is
then larger than almost periodicity. If f is almost automorphic, then its range is relatively compact,

thus bounded in norm.

Example 2.8. [49] Let k : R → R be such that

k(t) = sin
! 1

2 + cos(t) + cos(
√
2t)

"

, t ∈ R.

Then k is almost automorphic, but it is not uniformly continuous on R. Then, it is not almost

periodic.

In what follows, we introduce a new concept of ergodicity, which will generalize those given

in [12] and [29, 31].

Let B denote the Lebesque σ-field of R and let M be the set of all positive measures µ on B
satisfying µ(R) = +∞ and µ([a, b]) < ∞, for all a, b ∈ R (a ≤ b).

Definition 2.9. [12] Let µ,ν ∈ M. The measures µ and ν are said to be equivalent there exist
constants c0, c1 > 0 and a bounded interval Ω ⊂ R (eventually ∅) such that

c0ν(A) ≤ µ(A) ≤ c1ν(A)

for all A ∈ B satisfying A ∩ Ω = ∅.



8 Toka Diagana, Khalil Ezzinbi & Mohsen Miraoui CUBO
16, 2 (2014)

We introduce the following new space.

Definition 2.10. Let µ,ν ∈ M. A bounded continuous function f : R → X is said to be (µ,ν)-
ergodic if

lim
r→∞

1

ν(Qr)

∫

Qr

∥f(s)∥dµ(s) = 0.

We then denote the collection of all such functions by E(R, X, µ,ν).

We are now ready to introduce the notion of (µ,ν)-pseudo-almost periodicity (respectively,

(µ,ν)-pseudo-almost automorphy) for two positive measures µ,ν ∈ M.

Definition 2.11. Let µ,ν ∈ M. A continuous function f : R → X is said to be (µ,ν)-pseudo
almost periodic if it can be written in the form

f = g + h,

where g ∈ AP(R, X) and h ∈ E(R, X, µ,ν). The collection of such functions is denoted by
PAP(R, X, µ,ν).

Definition 2.12. Let µ,ν ∈ M. A continuous function f : R → X is said to be (µ,ν)-pseudo
almost automorphic if it can be written in the form

f = g + h,

where g ∈ AA(R, X) and h ∈ E(R, X, µ,ν). The collection of such functions will be denoted by
PAA(R, X, µ,ν).

We formulate the following hypotheses.

(M.1) Let µ,ν ∈ M such that

lim sup
r→∞

µ(Qr)

ν(Qr)
< ∞. (2.3)

(M.2) For all τ ∈ R, there exist β > 0 and a bounded interval I such that

µ({a + τ : a ∈ A}) ≤ βµ(A) when A ∈ B satisfies A ∩ I = ∅.

Theorem 2.13. Let µ,ν ∈ M satisfy (M.2). Then the spaces PAP(R, X, µ,ν) and PAA(R, X, µ,ν)
are translation invariants.

Proof. We show that E(R, X, µ,ν) is translation invariant. Let f ∈ E(R, X, µ,ν), we will show that
t #→ f(t + s) belongs to E(R, X, µ,ν) for each s ∈ R.



CUBO
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Pseudo-Almost Periodic and Pseudo-Almost Automorphic . . . 9

Indeed, letting µs = µ({t + s : t ∈ A}) for A ∈ B it follows from (M.2) that µ and µs are
equivalent (see [12]).

Now

1

ν(Qr)

∫

Qr

∥f(t + s)∥dµ(t) =
ν(Qr+|s|)

ν(Qr)
.

1

ν(Qr+|s|)

∫

Qr

∥f(t + s)∥dµ(t)

=
ν(Qr+|s|)

ν(Qr)
.

1

ν(Qr+|s|)

∫

Qr+|s|

∥f(t)∥dµ−s(t)

≤
ν(Qr+|s|)

ν(Qr)
.

cst.

ν(Qr+|s|)

∫

Qr+|s|

∥f(t)∥dµ(t).

Since ν satisfies (M.2) and f ∈ E(R, X, µ,ν), we have

lim
r→∞

1

ν(Qr)

∫

Qr

∥f(t + s)∥dµ(t) = 0.

Therefore, E(R, X, µ,ν) is translation invariant. Since AP(R, X) and AA(R, X) are translation
invariants, then PAP(R, X, µ,ν) and PAA(R, X, µ,ν) are translation invariants.

Theorem 2.14. Let µ,ν ∈ M satisfy (M.1), then (E(R, X, µ,ν), ∥.∥∞) is a Banach space.

Proof. It is clear that (E(R, X, µ,ν) is a vector subspace of BC(R, X). To complete the proof, it is
enough to prove that (E(R, X, µ,ν) is closed in BC(R, X). If (fn)n be a sequence in (E(R, X, µ,ν)
such that

lim
n→∞

fn = f

uniformly in R.

From ν(R) = ∞, it follows ν(Qr) > 0 for r sufficiently large. Using the inequality
∫

Qr

∥f(t)∥dµ(t) ≤
∫

Qr

∥f(t) − fn(t)∥dµ(t) +
∫

Qr

∥fn(t)∥dµ(t)

we deduce that

1

ν(Qr)

∫

Qr

∥f(t)∥dµ(t) ≤
µ(Qr)

ν(Qr)
∥f − fn∥∞ +

1

ν(Qr)

∫

Qr

∥fn(t)∥dµ(t),

then from (M.1) we have

lim sup
r→∞

1

ν(Qr)

∫

Qr

∥f(t)∥dµ(t) ≤ cst.∥f − fn∥∞

for all n ∈ N. Since lim
n→∞

∥f − fn∥∞ = 0, we deduce that

lim
r→∞

1

ν(Qr)

∫

Qr

∥f(t)∥dµ(t) = 0.



10 Toka Diagana, Khalil Ezzinbi & Mohsen Miraoui CUBO
16, 2 (2014)

Lemma 2.15. [13] Let g ∈ AA(R, X) and ε > 0 be given. Then there exist s1, ..., sm ∈ R such
that

R =

i=1
'

m

(si + Cε), where Cε := {t ∈ R : ∥g(t) − g(0)∥ < ε}.

Theorem 2.16. Let µ,ν ∈ M and f ∈ PAA(R, X, µ,ν) be such that

f = g + φ,

where g ∈ AA(R, X) and φ ∈ E(R, X, µ,ν). If PAA(R, X, µ,ν) is translation invariant, then

{g(t); t ∈ R} ⊂ {f(t); t ∈ R}, (the closure of the range of f). (2.4)

Proof. The proof is similar to the one given in [13]. Indeed, if we assume that (2.4) does not hold,

then there exists t0 ∈ R such that g(t0) is not in {f(t); t ∈ R}. Since the spaces AA(R, X) and
E(R, X, µ,ν) are translation invariants, we can assume that t0 = 0, then there exists ε > 0 such
that ∥f(t) − g(0)∥ > 2ε for all t ∈ R. Then we have

∥φ(t)∥ = ∥f(t) − g(t)∥ ≥ ∥f(t) − g(0)∥ − ∥g(t) − g(0)∥ ≥ ε

for all t ∈ Cε. Therefore,

∥φ(t − si)∥ ≥ ε, for all i ∈ {1, ..., m}, and t ∈ si + Cε.

Let φ be the function defined by

φ(t) :=
i=m∑

i=1

∥φ(t − si)∥.

From Lemma 2.15, we deduce that

∥φ(t)∥ ≥ ε for all t ∈ R. (2.5)

Since E(R, X, µ,ν) is translation invariant, then [t → φ(t − si)] ∈ E(R, X, µ,ν) for all i ∈ {1, ..., m},
then φ ∈ E(R, X, µ,ν) which is a contradiction. Consequently (2.4) holds.

Theorem 2.17. Let µ,ν ∈ M satisfy (M.2), then the decomposition of a (µ,ν)-pseudo almost
automorphic function in the form f = g + h, where g ∈ AA(R, X) and h ∈ E(R, X, µ,ν), is unique.

Proof. Suppose that f = g1 + φ1 = g2 + φ2, where g1, g2 ∈ AA(R, X) and φ1,φ2 ∈ E(R, X, µ,ν).
Then 0 = (g1 − g2) + (φ1 − φ2) ∈ PAA(R, X, µ,ν) where g1 − g2 ∈ AA(R, X) and φ1 − φ2 ∈
E(R, X, µ,ν). From Theorem 2.16, we obtain (g1 − g2)(R) ⊂ {0}, therefore we have g1 = g2 and
φ1 = φ2.

From Theorem 2.17, we deduce



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Pseudo-Almost Periodic and Pseudo-Almost Automorphic . . . 11

Theorem 2.18. Let µ,ν ∈ M satisfy (M.2), then the decomposition of a (µ,ν)-pseudo almost
periodic function in the form f = g + h, where g ∈ AP(R, X) and h ∈ E(R, X, µ,ν), is unique.

Theorem 2.19. Let µ,ν ∈ M satisfy (M.1) and (M.2). Then, the spaces (PAP(R, X, µ,ν), ∥.∥∞)
and (PAA(R, X, µ,ν), ∥.∥∞) are Banach spaces.

Proof. The proof is similar to the one given in [13], in fact we assume that (fn)n is a Cauchy

sequence in PAA(R, X, µ,ν). We have fn = gn +φn where gn ∈ AA(R, X) and φn ∈ E(R, X, µ,ν).
From Theorem 2.16 we see that

∥gn − gm∥∞ ≤ ∥fn − fm∥∞,

therefore (gn)n is a Cauchy sequence in the Banach space (AA(R, X), ∥.∥∞). So, φn = fn − gn
is also a Cauchy sequence in the Banach space E((R, X, µ,ν), ∥.∥∞). Then we have limn→∞ gn =
g ∈ AA(R, X) and limn→∞ φn = φ ∈ E(R, X, µ,ν). Finally we have

lim
n→∞

fn = g + φ ∈ PAA(R, X, µ,ν).

The proof for PAP(R, X, µ,ν) is similar to that of PAA(R, X, µ,ν).

Definition 2.20. Let µ,ν ∈ M. A continuous function f : R × Y → X is said to be (µ,ν)-ergodic
in t uniformly with respect to y ∈ Y if the following conditions are true
(i) For all y ∈ Y, f(., y) ∈ E(R, X, µ,ν).
(ii) f is uniformly continuous on each compact set K in Y with respect to the second variable y.

The collection of such function is denoted by EU(R × Y, X, µ,ν).

Definition 2.21. Let µ,ν ∈ M. A continuous function f : R × Y → X is said to be (µ,ν)-pseudo
almost periodic if is written in the form

f = g + h,

where g ∈ APU(R × Y, X) and h ∈ EU(R × Y, X, µ,ν). The collection of such functions is denoted
by PAPU(R × Y, X, µ,ν).

Theorem 2.22. Let µ, ν ∈ M and I be a bounded interval (eventually I = Ø). Assume that (M1)
and f ∈ BC(R, X). Then the following assertions are equivalent:
(i) f ∈ E(R, X, µ,ν).
(ii) lim

r→∞

1

ν(Qr \ I)

∫

Qr\I

∥f(t)∥dµ(t) = 0.

(iii) For any ε > 0, lim
r→∞

µ({t ∈ Qr \ I : ∥f(t)∥ > ε})
ν({Qr \ I)

= 0.

Proof. The proof is similar to the one given in [13], in fact we have

(i) ⇔ (ii) : Denote by A = ν(I), B =
∫
I
∥f(t)∥dµ(t) and C = µ(I). Since the interval I is bounded



12 Toka Diagana, Khalil Ezzinbi & Mohsen Miraoui CUBO
16, 2 (2014)

and the function f is bounded and continuous, then A, B and C are finite. For r > 0 such that

I ⊂ Qr and ν(Qr \ I) > 0, we have

1

ν(Qr \ I)

∫

Qr\I

∥f(t)∥dµ(t) =
1

ν(Qr) − A

!

∫

Qr

∥f(t)∥dµ(t) − B
"

=
ν(Qr)

ν(Qr) − A

! 1

ν(Qr)

∫

Qr

∥f(t)∥dµ(t) −
B

ν(Qr)

"

.

Since ν(R) = ∞, we deduce that (ii) is equivalent to (i).

(iii) ⇒ (ii) Denote by Aεr and B
ε
r the following sets

Aεr = {t ∈ Qr \ I : ∥f(t)∥ > ε} and B
ε
r = {t ∈ Qr \ I : ∥f(t)∥ ≤ ε}.

Assume that (iii) holds, that is

lim
r→∞

µ(Aεr)

ν(Qr \ I)
= 0.

From the following equality

∫

Qr\I

∥f(t)∥dµ(t) =
∫

Aεr

∥f(t)∥dµ(t) +
∫

Bεr

∥f(t)∥dµ(t),

and (M.1), we deduce for r large enough that

,

1

ν(Qr \ I)

∫

Qr\I

∥f(t)∥dµ(t) ≤ ∥f∥∞
µ(Aεr)

ν(Qr \ I)
+

µ(Bεr)

ν(Qr \ I)
ε

≤ ∥f∥∞
µ(Aεr)

ν(Qr \ I)
+

µ(Qr \ I)

ν(Qr \ I)
ε

= ∥f∥∞
µ(Aεr)

ν(Qr \ I)
+

µ(Qr) − C

ν(Qr) − A
ε

= ∥f∥∞
µ(Aεr)

ν(Qr \ I)
+

µ(Qr)

ν(Qr)

1 − C
µ(Qr)

1 − A
ν(Qr)

ε

≤ ∥f∥∞
µ(Aεr)

ν(Qr \ I)
+ cst.

1 − C
µ(Qr)

1 − A
ν(Qr)

ε.

Since µ(R) = ν(R) = ∞, then for all ε > 0 we have

lim sup
r→∞

1

ν(Qr \ I)

∫

Qr\I

∥f(t)∥dµ(t) ≤ cst.ε,



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Pseudo-Almost Periodic and Pseudo-Almost Automorphic . . . 13

consequently (ii) holds.

(ii) ⇒ (iii) Assume that (ii) holds. From the following inequality:

1

ν(Qr \ I)

∫

Qr\I

∥f(t)∥dµ(t) ≥
1

ν(Qr \ I)

∫

Aεr

∥f(t)∥dµ(t)

≥ ε
µ(Aεr)

ν(Qr \ I)
,

for r sufficiently large, we obtain (iii).

Theorem 2.23. [12] Let F ∈ APU(R×X, Y) and h ∈ AP(R, X). Then [t #−→ F(t, h(t))] ∈ AP(R, Y).

Proposition 2.24. [12] Let f : R × X → Y be a continuous function. Then f ∈ APU(R × X, Y) if
and only if the two following conditions hold:

(i) for all x ∈ X, f(., x) ∈ AP(R, Y),
(ii) f is uniformly continuous on each compact set K in X with respect to the second variable x,

namely, for each compact set K in X, for all ε > 0, there exists δ > 0 such that for all x1, x2 ∈ K,
one has

∥x1 − x2∥ ≤ δ ⇒ sup
t∈R

∥f(t, x1) − f(t, x2)∥ ≤ ε.

The proof of our result of composition of (µ,ν)-pseudo-almost periodic functions is based on

the following lemma due to Schwartz [54].

Lemma 2.25. If Ψ ∈ C(X, Y), then for each compact set K in X and all ε > 0, there exists δ > 0,
such that for any x1, x2 ∈ X, one has

x1 ∈ K and ∥x1 − x2∥ ≤ δ ⇒ ∥Ψ(x1) − Ψ(x2)∥ ≤ ε.

Theorem 2.26. Let µ, ν ∈ M, F ∈ PAPU(R × X, Y, µ,ν) and h ∈ PAP(R, Y, µ,ν). Assume that
(M.1) and the following hypothesis holds:

(C) For all bounded subset B of Y, F is bounded on R × B.

Then t #−→ F(t, h(t)) ∈ PAP(R, Y, µ,ν).

Proof. The function [t #→ F(t, h(t))] is continuous and by Hypothesis (C), it is bounded. Since
h ∈ PAP(R, X, µ,ν), we can write

h = h1 + h2,

where h1 ∈ AP(R, X) and h2 is (µ,ν)-ergodic. Since F ∈ PAPU(R × X, Y, µ,ν), we have

F = F1 + F2



14 Toka Diagana, Khalil Ezzinbi & Mohsen Miraoui CUBO
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where F1 ∈ APU(R × X, Y) and F2 ∈ E(R × X, Y, µ,ν). The function F can be written in the form

F(t, h(t)) = F1(t, h1(t)) + [F(t, h(t)) − F(t, h1(t))] + [F(t, h1(t)) − F1(t, h1(t))]

= F1(t, h1(t)) + [F(t, h(t)) − F(t, h1(t))] + F2(t, h1(t)).

From Theorem 2.23, we have [t #−→ F1(t, h1(t))] ∈ AP(R, Y). Denote by K the closure of the
range of h1: K = ¯{h1(t) : t ∈ R} . Since h1 is almost periodic, K is a compact subset of X. Denote
by Φ the function defined by

Φ : X → PAP(R, Y, µ,ν)

x #→ F(., x)

Since F ∈ PAPU(R × X, Y, µ,ν) ), by using Proposition 2.24, we deduce that the restriction of Φ
on all compact K of X, is uniformly continuous, which is equivalent to saying that the function Φ

is continuous on X. From Lemma 2.25 applied to Φ, we deduce that for given ε > 0, there exists

δ > 0 such that, for all t ∈ R, ξ1 and ξ2 ∈ X, one has

ξ1 ∈ K and ∥ξ1 − ξ2∥ ≤ δ ⇒ ∥F(t,ξ1) − F(t,ξ2)∥ ≤ ε.

Since h(t) = h1(t) + h2(t) and h1(t) ∈ K, we have

t ∈ R and ∥h2(t)∥ ≤ δ ⇒ ∥F(t, h(t)) − F(t, h2(t))∥ ≤ ε,

therefore, we have

µ{t ∈ Qr : ∥F(t, h(t)) − F(t, h1(t))∥ > ε}
ν(Qr)

≤
µ{t ∈ Qr : ∥h2(t)∥ > δ}

ν(Qr)
.

Since h2 is (µ,ν)-ergodic, Theorem 2.22 yields that for the above-mentioned δ we have

lim
r→∞

µ{t ∈ Qr : ∥h2(t)∥ > δ}
ν(Qr)

= 0,

then we obtain

lim
r→∞

µ{t ∈ Qr : ∥F(t, h(t)) − F(t, h1(t))∥ > ε}
ν(Qr)

= 0.

By Theorem 2.22 we have t #→ F(t, h(t))−F(t, h1(t)) is (µ,ν)-ergodic. Now to complete the proof,
it is enough to prove that t #→ F2(t, h1(t)) is (µ,ν)-ergodic. Since F2 is uniformly continuous on
the compact set K = ¯{h1(t) : t ∈ R} with respect to the second variable x, we deduce that for given
ε > 0, there exists δ > 0 such that, for all t ∈ R, ξ1 and ξ2 ∈ K, one has

∥ξ1 − ξ2∥ ≤ δ ⇒ ∥F2(t,ξ1) − F2(t,ξ2)∥ ≤ ε.

then, there exist n(ε) and {xi}
n(ε)
i=1 ⊂ K, such that

K ⊂
n(ε)
'

i=1

B(xi,δ),



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Pseudo-Almost Periodic and Pseudo-Almost Automorphic . . . 15

and then

∥F2(t, h1(t))∥ ≤ ε +
n(ε)∑

i=1

∥F2(t, xi)∥.

Since

∀i ∈ {1, ..., n(ε)}, lim
r→+∞

1

ν(Qr)

∫

Qr

∥F2(t, xi)∥dµ(t) = 0,

we deduce that

∀ε > 0, lim sup
r→∞

1

ν(Qr)

∫

Qr

∥F2(t, h1(t))∥dµ(t) ≤ ε,

that implies

lim
r→∞

1

ν(Qr)

∫

Qr

∥F2(t, h1(t))∥dµ(t) = 0,

then t #→ F2(t, h1(t)) is (µ,ν)-ergodic and the theorem is proved.

Definition 2.27. Let µ,ν ∈ M. A continuous function f : R × Y → X is said to be (µ,ν)-pseudo
almost automorphic if is written in the form

f = g + h,

where g ∈ AAU(R × Y, X) and h ∈ EU(R × Y, X, µ,ν). The collection of such functions is denoted
by PAAU(R × Y, X, µ,ν).

Theorem 2.28. Let µ,ν ∈ M, F ∈ PAAU(R × X, Y, µ,ν) and h ∈ PAA(R, Y, µ,ν). Assume that,
for all bounded subset B of Y, F is bounded on R × B. Then t #−→ F(t, h(t)) ∈ PAA(R, X, µ,ν).

Proof. The proof for PAA(R, Y, µ,ν) is similar to that of PAP(R, Y, µ,ν).

2.2 Evolution Families and Exponential Dichotomy

(H0) A family of closed linear operators A(t) for t ∈ R on X with domain D(A(t)) (possibly not
densely defined) is said to satisfy the so-called Acquistapace–Terreni conditions, if there exist

constants ω ∈ R, θ ∈ (π
2
,π), K, L ≥ 0 and µ0,ν0 ∈ (0, 1] with µ0 + ν0 > 1 such that

Σθ ∪ {0} ⊂ ρ(A(t) − ω) ∋ λ, ∥R(λ, A(t) − ω)∥ ≤
K

1 + |λ|
(2.6)

and

∥(A(t) − ω)R(λ, A(t) − ω) [R(ω, A(t)) − R(ω, A(s))]∥ ≤ L
|t − s|µ0

|λ|ν0
(2.7)

for t, s ∈ R, λ ∈ Σθ := {λ ∈ C \ {0} : | argλ| ≤ θ}.



16 Toka Diagana, Khalil Ezzinbi & Mohsen Miraoui CUBO
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Note that in the particular case when A(t) has a constant domain

D = D(A(t)), it is well-known on [6] that condition (2.7) can be replaced with the following one:

There exist constants L and 0 < γ ≤ 1 such that

∥(A(t) − A(s))R(ω, A(r))∥ ≤ L|t − s|γ, for all, s, t, r ∈ R. (2.8)

For a given family of linear operators A(t), the existence of an evolution family associated

with it is not always guaranteed. However, if A(t) satisfies Acquistapace–Terreni, then there exists

a unique evolution family(see[1, 2, 52])

U = {U(t, s) : t, s ∈ R, t ≥ s}

on X associated with A(t) such that U(t, s)X ⊆ D(A(t)) for all t, s ∈ R with t ≥ s, and,

(1) U(t, r)U(r, s) = U(t, s) and U(s, s) = I for all t ≥ r ≥ s and t, r, s ∈ R;

(2) the map (t, s) → U(t, s)x is continuous for all x ∈ X, t ≥ s and t, s ∈ R;

(3) U(·, s) ∈ C1((s, ∞), B(X)),
∂U

∂t
(t, s) = A(t)U(t, s) and

&

&

&
A(t)kU(t, s)

&

&

&
≤ K (t − s)−k

for 0 < t − s ≤ 1, k = 0, 1.

Definition 2.29. An evolution family (U(t, s))t≥s on a Banach space X is X is called hyperbolic (or

has exponential dichotomy) if there exist projections P(t), t ∈ R, uniformly bounded and strongly
continuous in t, and constant N ≥ 1, δ > 0 such that

(1) U(t, s)P(s) = P(t)U(t, s) for t ≥ s;

(2) the restriction UQ(t, s) : Q(s)X → Q(t)X of U(t, s) is invertible for t, s ∈ R and we set
UQ(t, s) = U(s, t)

−1;

(3)

∥U(t, s)P(s)∥ ≤ Ne−δ(t−s) (2.9)

and

∥UQ(s, t)Q(t)∥ ≤ Ne−δ(t−s) (2.10)

for t ≥ s and t, s ∈ R, were Q(t) := I − P(t).

3 Existence Results

To study the existence and uniqueness of (µ,ν)-pseudo almost periodic (respectively, (µ,ν)-pseudo

almost automorphic) solutions to equation (1.1), in addition to above, we also assume that the

next assumption holds:



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Pseudo-Almost Periodic and Pseudo-Almost Automorphic . . . 17

(H1) The evolution family U generated by A(.) has an exponential dichotomy with constants

N ≥ 1, δ > 0 and dichotomy projections P(t).

We recall from [48, 55], the following sufficient conditions to fulfill the assumption (H1).

(H1.1) Let (A(t), D(A(t)))t∈R be generators of analytic semigroups on X of the same type. Suppose

that D(A(t)) = D(A(0)), A(t) is invertible, supt,s∈R ∥A(t)A(s)−1∥ is finite, and

∥A(t)A(s)−1 − I∥ ≤ L0|t − s|µ1

for t, s ∈ R and constants L0 ≥ 0 and 0 < µ1 ≤ 1.

(H1.2) The semigroups (eτA(t))τ≥0, t ∈ R, are hyperbolic with projection Pt and constants N,δ > 0.
Moreover, let

∥A(t)eτA(t)Pt∥ ≤ Ψ(τ) and ∥A(t)eτAQ(t)Qt∥ ≤ Ψ(−τ)

for τ > 0 and a function Ψ such that R ∋ s → ϕ(s) := |s|µΨ(s) is integrable with
L0∥ϕ∥L1(R) < 1.

Now, we first introduce the definition of the mild solution to Eq. (1.1).

Definition 3.1. A continuous function u : R #→ X is called a bounded mild solution of equation
(1.1) if:

u(t) = U(t, s)u(s) +

∫t

s

U(t,τ)F(τ, u(τ))dτ, t ≥ s, t, s ∈ R. (3.1)

Theorem 3.2. Assume that (H0) and (H1) hold. If there exists 0 < KF <
δ
2N

such that

∥F(t, u) − F(t, v)∥ ≤ KF∥u − v∥,

for all u, v ∈ X and t ∈ R, then the equation (1.1) has a unique bounded mild solution u : R #→ X
given by

u(t) =

∫

R

Γ(t, s)F(s, u(s))ds, t ∈ R,

where the operator family Γ(t, s), called Green’s function corresponding to U and P(.), is given by

{
Γ(t, s) = U(t, s)P(s), t ≥ s, t, s ∈ R
Γ(t, s) = −UQ(t, s)Q(s), t < s, t, s ∈ R.

Proof. If one supposes

u(t) =

∫

R

Γ(t,τ)F(τ, u(τ))dτ, t ∈ R,



18 Toka Diagana, Khalil Ezzinbi & Mohsen Miraoui CUBO
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Thus we have

u(t) =

∫t

−∞

U(t,τ)P(τ)F(τ, u(τ))dτ −

∫+∞

t

UQ(t,τ)Q(τ)F(τ, u(τ))dτ, for all t ∈ R.

For t = s, one obtains

u(s) =

∫s

−∞

U(s,τ)P(τ)F(τ, u(τ))dτ −

∫+∞

s

UQ(s,τ)Q(τ)F(τ, u(τ))dτ,

and

U(t, s)u(s) =

∫s

−∞

U(t,τ)P(τ)F(τ, u(τ))dτ −

∫+∞

s

UQ(t,τ)Q(τ)F(τ, u(τ))dτ.

Now, we have

u(t) − U(t, s)u(s) =

∫t

s

U(t,τ)f(τ, u(τ))dτ.

Then, u(t) =
∫
R
Γ(t,τ)F(τ, u(τ))dτ checks equation (3.1).

Consider the nonlinear operator K defined on X by

Ku(t) =

∫

R

Γ(t,τ)F(τ, u(τ))dτ, t ∈ R.

To complete the proof, one has to show that K is a contraction map on X.

From assumption (H1), there exist two constant N ≥ 1 and δ > 0 such that

∥Γ(t, s)∥ ≤ Ne−δ|t−s| for all t, s ∈ R

If u, v ∈ X, then one has

∥Kv − Ku∥∞ <
2NKF
δ

∥v − u∥∞,

and K is a contraction map on X. Therefore, K has unique fixed point in X, that is, there

exists unique u ∈ X such that Ku = u. Therefore, Eq.(1.1) has unique mild solution.

Denote by Γ1 and Γ2 the nonlinear integral operators defined by,

(Γ1u)(t) :=

∫t

−∞

U(t, s)P(s)F(s, u(s))ds,

and

(Γ2u)(t) :=

∫+∞

t

UQ(t, s)Q(s)F(s, u(s))ds.

In the rest of the paper, we fix µ,ν ∈ M satisfy (M1) and (M2).



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Pseudo-Almost Periodic and Pseudo-Almost Automorphic . . . 19

3.1 Existence of (µ,ν)-Pseudo-Almost Periodic Solutions

In addition to the previous assumptions, we require the following additional ones:

(H2) R(ω, A(·)) ∈ AP(R, L(X)).

(H3) We suppose F : R × X #→ X belongs to PAP(R × X, X, µ,ν) and there exists KF > 0 such that

∥F(t, u) − F(t, v)∥ ≤ KF∥u − v∥,

for all u, v ∈ X and t ∈ R.

The following Lemma plays an important role to prove main results of this work.

Lemma 3.3. [48] Assume that (H0), (H1)and (H2) hold. Then r → Γ(t + r, s + r) belongs to

AP(R, L(X)) for t, s ∈ R, where we may take the same pseudo periods for t, s with |t − s| ≥ h > 0.
If f ∈ AP(R, L(X)), then the unique bounded mild solution u(t) =

∫
R
Γ(t, s)f(s)ds of the following

equation

u′(t) = A(t)u(t) + f(t), t ∈ R,

is almost periodic.

Lemma 3.4. Under assumptions (H0)–(H3), then the integral operators Γ1 and Γ2 defined above

map PAP(R, X, µ,ν) into itself.

Proof. Let u ∈ PAP(R, X, µ,ν). Setting h(t) = F(t, u(t)), using assumption (H3) and Theorem
2.26 it follows that h ∈ PAP(R, X, µ). Now write h = ψ1 + ψ2 where ψ1 ∈ AP(R, X) and
ψ2 ∈ E(R, X, µ,ν). That is, Γ1h = Ξ(ψ1) + Ξ(ψ2) where

Ξψ1(t) :=

∫t

−∞

U(t, s)P(s)ψ1(s)ds, and Ξψ2(t) :=

∫t

−∞

U(t, s)P(s)ψ2(s)ds.

From Lemma 3.3, we have Ξ(ψ1) ∈ AP(R, X). To complete the proof, we will prove that
Ξ(ψ2) ∈ E(R, X, µ,ν). Now, let r > 0. Again from Eq. (2.9), we have

1

ν(Qr)

∫

Qr

∥(Ξψ2)(t)∥dµ(t) ≤
1

ν(Qr)

∫

Qr

∫+∞

0

∥U(t, t − s)P(t − s)ψ2(t − s)∥dsdµ(t)

≤
N

ν(Qr)

∫

Qr

∫+∞

0

e−δs∥ψ2(t − s)∥dsdµ(t)

≤ N
∫+∞

0

e−δs
(

1

ν(Qr)

∫

Qr

∥ψ2(t − s)∥dµ(t)
)

ds.



20 Toka Diagana, Khalil Ezzinbi & Mohsen Miraoui CUBO
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Since µ and ν satisfy (M2), then from Theorem 2.13, we have t #→ ψ2(t − s) ∈ E(R, X, µ,ν) for
every s ∈ R. By the Lebesgue’s Dominated Convergence Theorem, we have

lim
r→∞

1

ν(Qr)

∫

Qr

∥(Ξψ2)(t)∥dµ(t) = 0.

The proof for Γ2u(·) is similar to that of Γ1u(·) except that one makes use of equation (2.10) instead
of equation (2.9).

Theorem 3.5. Under assumptions (H0)—(H3), then Eq. (1.1) has a unique (µ,ν)-pseudo almost

periodic mild solution whenever KF is small enough.

Proof. Consider the nonlinear operator M defined on PAP(R, X, µ,ν) by

Mu(t) =

∫t

−∞

U(t, s)P(s)F(s, u(s))ds −

∫+∞

t

UQ(t, s)Q(s)F(s, u(s))ds, for all t ∈ R.

In view of Lemma 3.4, it follows that M maps PAP(R, X, µ,ν) into itself. To complete the proof one

has to show that M is a contraction map on PAP(R, X, µ,ν). Let u, v ∈ PAP(R, X, µ,ν). Firstly,
we have

∥Γ1(v)(t) − Γ1(u)(t)∥ ≤
∫t

−∞

∥U(t, s)P(s) [F(s, v(s)) − F(s, u(s))] ∥ds

≤ NKF
∫t

−∞

e−δ(t−s)∥v(s) − u(s)∥ds

≤ NKFδ−1∥v − u∥∞.

Next, we have

∥Γ2(v)(t) − Γ2(u)(t)∥ ≤
∫+∞

t

∥UQ(t, s)Q(s) [F(s, v(s)) − F(s, u(s))] ∥ds

≤ N
∫+∞

t

eδ(t−s)∥F(s, v(s)) − F(s, u(s))∥ds

≤ NKF
∫+∞

t

eδ(t−s)∥v(s) − u(s)∥ds

≤ NKF∥v − u∥∞
∫+∞

t

eδ(t−s)ds

= NKFδ
−1∥v − u∥∞.

Finally, combining previous approximations it follows that

∥Mv − Mu∥∞ < 2NKFδ−1∥v − u∥∞.

Thus if KF is small enough, that is, KF < δ(2N)
−1, then M is a contraction map on PAP(R, X, µ,ν).

Therefore, M has unique fixed point in PAP(R, X, µ,ν), that is, there exists unique function u

satisfying Mu = u, which is the unique (µ,ν)-pseudo almost periodic mild solution to Eq. (1.1).



CUBO
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Pseudo-Almost Periodic and Pseudo-Almost Automorphic . . . 21

3.2 Existence of (µ,ν)-Pseudo-Almost Automorphic Solutions

In this section we consider the following conditions:

(H’2) R(ω, A(·)) ∈ AA(R, L(X)).

(H’3) We suppose F : R × X #→ X belongs to PAA(R × X, X, µ,ν) and there exists KF > 0 such that

∥F(t, u) − F(t, v)∥ ≤ KF∥u − v∥∞,

for all u, v ∈ X and t ∈ R.

Lemma 3.6. [9] Assume that (H0), (H1)and (H’2) hold. Let a sequence (s′l)l∈N ∈ R there is a
subsequence (sl)l∈N such that for every h > 0

∥Γ(t + sl − sk, s + sl − sk) − Γ(t, s)∥ → 0, k, l → ∞

for |t − s| ≥ h.

Lemma 3.7. Under assumptions (H0), (H1) , (H’2) and (H’3), then the integral operators Γ1
and Γ2 defined above map PAA(R, X, µ,ν) into itself.

Proof. Let u ∈ PAA(R, X, µ,ν). Setting g(t) = F(t, u(t)), using assumption (H’3) and Theorem
2.28 it follows that g ∈ PAA(R, X, µ,ν). Now write g = u1 + u2 where u1 ∈ AA(R, X) and
u2 ∈ E(R, X, µ,ν). That is, Γ1g = Su1 + Su2, where

Su1(t) :=

∫t

−∞

U(t, s)P(s)u1(s)ds, and Su2(t) :=

∫t

−∞

U(t, s)P(s)u2(s)ds.

From Eq. (2.9), we obtain

∥Su1(t)∥ ≤ Nδ−1∥u1∥∞ and ∥Su2(t)∥ ≤ Nδ−1∥u2∥∞ for all t ∈ R.

Then Su1, Su2 ∈ BC(R, X). Now, we prove that Su1 ∈ AA(R, X). Since u1 ∈ AA(R, X), then
for every sequence (τ′n)n∈N ∈ R there exists a subsequence (τn)n∈N such that

v1(t) := lim
n→∞

u1(t + τn), (3.2)

is well defined for each t ∈ R, and

lim
n→∞

v1(t − τn) = u1(t), (3.3)

for each t ∈ R. Set

M(t) =

∫t

−∞

U(t, s)P(s)u1(s)ds and N(t) =

∫t

−∞

U(t, s)P(s)v1(s)ds, t ∈ R.



22 Toka Diagana, Khalil Ezzinbi & Mohsen Miraoui CUBO
16, 2 (2014)

Now, we have

M(t + τn) − N(t) =

∫t+τn

−∞

U(t + τn, s)P(s)u1(s)ds −

∫t

−∞

U(t, s)P(s)v1(s)ds

=

∫t

−∞

U(t + τn, s + τn)P(s + τn)u1(s + τn)ds −

∫t

−∞

U(t, s)P(s)v1(s)ds

=

∫t

−∞

U(t + τn, s + τn)P(s + τn)
$

u1(s + τn) − v1(s)
%

ds

+

∫t

−∞

$

U(t + τn, s + τn)P(s + τn) − U(t, s)P(s)
%

v1(s)ds.

Using Eq. (2.9), Eq. (3.2) and the Lebesgue’s Dominated Convergence Theorem, it follows that

∥
∫t

−∞

U(t + τn, s + τn)P(s + τn)
$

u1(s + τn) − v1(s)
%

ds∥ → 0 as n → ∞, t ∈ R. (3.4)

Similarly, using Lemma 3.6 it follows that

∥
∫t

−∞

$

U(t + τn, s + τn)P(s + τn) − U(t, s)P(s)
%

v1(s)ds∥ → 0 as n → ∞, t ∈ R. (3.5)

Therefore, we have

N(t) := lim
n→∞

M(t + τn), t ∈ R. (3.6)

Using similar ideas as the previous ones, then

M(t) := lim
n→∞

N(t − τn), t ∈ R. (3.7)

Therefore, Su1 ∈ AA(R, X). Finally, by using the same stages that the proof of Lemma 3.4 one
obtains Su2 ∈ E(R, X, µ,ν). The proof for Γ2u(·) is similar to that of Γ1u(·) except that one makes
use of equation (2.10) instead of equation 2.9.

Theorem 3.8. Under assumptions (H0), (H1) , (H’2) and (H’3), then Eq. (1.1) has a unique

(µ,ν)-pseudo almost automorphic mild solution whenever KF is small enough.

Proof. The proof for Theorem 3.8 is similar to that of Theorem 3.5 except that one makes use of

Lemma 3.7 instead of Lemma 3.4.



CUBO
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Pseudo-Almost Periodic and Pseudo-Almost Automorphic . . . 23

3.3 Neutral Systems

In this subsection, we establish the existence and uniqueness of (µ,ν)-pseudo-almost periodic

(respectively, (µ,ν)-pseudo-almost automorphic) solutions for the nonautonomous neutral partial

evolution equation (1.2). For that, we need the following assumptions:

(H4) We suppose G : R × X #→ X belongs to PAP(R × X, X, µ,ν) and there exists KG > 0 such that

∥G(t, u) − G(t, v)∥ ≤ KG∥u − v∥,

for all u, v ∈ X and t ∈ R.

(H’4) G : R × X #→ X belongs to PAA(R × X, X, µ,ν) and there exists KG > 0 such that

∥G(t, u) − G(t, v)∥ ≤ KG∥u − v∥,

for all u, v ∈ X and t ∈ R.

Definition 3.9. A function v : R #→ X is said to be a bounded mild solution to equation (1.2) and
we have:

v(t) = G(t, v(t)) +

∫t

−∞

U(t, s)P(s)F(s, v(s))ds −

∫+∞

t

UQ(t, s)Q(s)F(s, v(s))ds

for all t ∈ R.

Theorem 3.10. If assumptions (H0), (H1), (H2), (H3) and (H4) hold and
!

KG+2NKFδ
−1

"

<

1, then Eq. (1.2) has a unique (µ,ν)-pseudo almost periodic mild solution.

Proof. We consider the nonlinear operator W defined on PAP(R, X, µ) by

Wv(t) = G(t, v(t)) +

∫t

−∞

U(t, s)P(s)F(s, v(s))ds −

∫+∞

t

UQ(t, s)Q(s)F(s, v(s))ds

for all t ∈ R.

From (H4), Theorem (2.26), and Lemma 3.4 it follows that W maps PAP(R, X, µ,ν) into

itself. To complete the proof we need to show that W is a contraction map on PAP(R, X, µ,ν).

For that, letting u, v ∈ PAP(R, X, µ,ν), we obtain

∥Wv − Wu∥∞ ≤
!

KG + 2NKFδ
−1

"

∥v − u∥∞,

which yields W is a contraction map on PAP(R, X, µ,ν). Therefore, W has unique fixed point in

PAP(R, X, µ,ν). Therefore, Eq.(1.2), has unique (µ,ν)-pseudo-almost periodic mild solution.

Theorem 3.11. If assumptions (H0), (H1), (H’2), (H’3) and (H’4) hold and
!

KG+2NKFδ
−1

"

<

1, then Eq. (1.2) has a unique (µ,ν)-pseudo almost automorphic mild solution.



24 Toka Diagana, Khalil Ezzinbi & Mohsen Miraoui CUBO
16, 2 (2014)

Proof. Similarly the proof of Theorem 3.10, we can show, by using the assumption (H’4), Theorem

2.28 and Lemma 3.7, that the Eq. (1.2) has a unique (µ,ν)-pseudo almost automorphic mild

solution.

4 Examples

Example 4.1. Let X = L2([0, 1]) be equipped with its natural topology. In order to illustrate

Theorem 3.5, we consider the following one-dimensional heat equation with Dirichlet conditions,

∂

∂t

$

v(t, x)
%

=
$ ∂2

∂x2
+ ξ

!

sin(at) + sin(bt)
"%

v(t, x) + f(t, v(t, x)), on R × (0, 1)

v(t, 0) = v(t, 1) = 0, t ∈ R,
(4.1)

where the coefficient ξ ∈]0, 1
2
[, the constants a, b ∈ R with a

b
/∈ Q, and the forcing term f : R×X #→

X is continuous function.

In order to write Eq.(4.1) in the abstract form Eq.(1.1), we consider the linear operator

A : D(A) ⊂ X −→ X, given by

D(A) = H2(0, 1) ∩ H10(0, 1) and Au = u
′′ for u ∈ D(A).

It is well known that A is the infinitesimal generator of an exponentially stable C0-semigroup
*

T(t)
+

t≥0
such that ∥T(t)∥ ≤ e−π

2t for t ≥ 0.

Define a family of linear operator A(t) as follows:
⎧
⎪⎪⎨

⎪⎪⎩

D(A(t)) = D(A) = H2[0, 1] ∩ H10[0, 1]

A(t)v =
$

A + ξ
!

sin(at) + sin(bt)
"%

v, v ∈ D(A).

Obviously, D(A(t)) = D(A). Furthermore,

∥A(t) − A(s)∥ = ∥ξ
!

sin(at) − sin(as) + sin(bt) − sin(bs)
"

∥ ≤ ξ
!

|a| + |b|
"

|t − s|,

for all s, t ∈ R and hence (H0) holds. Consequently, A(t) generates an evolution family, which we
denote by U(t, s)t≥s and which satisfies

U(t, s)v = T(t − s) exp
$

∫t

s

ξ
!

sin(aτ) + sin(bτ)
"

dτ
%

v.

Since ∥U(t, s)∥ ≤ e−(π
2−1)(t−s) for t ≥ s and t, s ∈ R, it follows that (H1) holds with N = 1,

δ = π2 − 1 > 0. And since t #→ sin(at) + sin(bt) is almost periodic, then R(ω, A(·)) ∈ AP(R, L(X))
and so (H2) holds.



CUBO
16, 2 (2014)

Pseudo-Almost Periodic and Pseudo-Almost Automorphic . . . 25

Let F : R × X #→ X be the mapping defined by F(t,ϕ)(x) = f(t,ϕ(x)) for x ∈ [0, 1], and let
y : R → X be the function defined by y(t) = v(t, .), for t ∈ R. Then Eq. (4.1) takes the abstract
form,

d

dt

,

y(t)
-

= A(t)y(t) + F(t, y(t)), t ∈ R. (4.2)

Let µ = ν and suppose that its Radon-Nikodym derivative is given by

ρ(t) =

{
et if t ≤ 0,
1 if t > 0.

Then from [12], µ ∈ M satisfies (M1) and (M2).

If we assume that f is µ-pseudo almost periodic in t ∈ R uniformly in u ∈ X and is globally
Lipschitz with respect to the second argument in the following sense: there exists Kf > 0 such that

&

&

&
f(t, u) − f(t, v)

&

&

&
≤ Kf

&

&

&
u − v

&

&

&
for all t ∈ R and u, v ∈ X,

then F satisfies (H3).

We deduce all assumptions (H0),(H1),(H2),(H3), (M.1) and (M.2) of Theorem 3.5 are

satisfied and thus equation (4.1) has a unique (µ, µ)-pseudo almost periodic solution whenever Kf
is small enough (Kf <

π2−1
2

).

To illustrate the result in Theorem 3.11, we consider the following equation

∂

∂t

$

v(t, x) − g1(t, v(t, x))
%

=
! ∂2

∂x2
+ ξ(sin(at) + sin(bt))

"$

v(t, x) − g1(t, v(t, x))
%

+f1(t, v(t, x)), on R × (0, 1)
v(t, 0) = v(t, 1) = 0, t ∈ R,

(4.3)

where the coefficient ξ ∈ (0, 1
2
) , a, b ∈ R and a

b
/∈ Q, f1, g1 : R × X #→ X are given by

f1(t, x) = x sin
1

2 + cos t + cos
√
2t

+ max
k∈Z

{e−(t±k
2)2} cos x, t ∈ R , x ∈ X,

g1(t, x) =
x

4
sin

1

2 + sin t + sin
√
2t

+
1

4
max
k∈Z

{e−(t±k
2)2} cosx, t ∈ R , x ∈ X.

Clearly, f1, g1 ∈ PAA(R × X, X, µ, µ) and satisfies the Lipschitz condition in Theorem 3.11
with N = 1, δ = π2 − 1, Kf1 = 2 and Kg1 =

1
2
. By Theorem 3.11, the evolution equation (4.3)

has a unique (µ, µ)-pseudo almost automorphic solution, with µ being the measure defined in the

example above.

Example 4.2. Let Ω ⊂ RN (N ≥ 1) be an open bounded subset with regular boundary Γ = ∂Ω
and let X = L2(Ω) equipped with its natural topology ∥ · ∥2.



26 Toka Diagana, Khalil Ezzinbi & Mohsen Miraoui CUBO
16, 2 (2014)

We study the existence of (µ,ν)-pseudo-almost automorphic solutions to the N-dimensional

heat equation

⎧
⎪⎪⎨

⎪⎪⎩

∂ϕ

∂t
= a(t, x)∆ϕ + g(t,ϕ), in R × Ω

ϕ = 0, on R × Γ

(4.4)

where a : R × Ω #→ R is almost automorphic, and g : R × L2(Ω) #→ L2(Ω) is (µ,ν)-pseudo almost
automorphic function.

Define the linear operator appearing in Eq. (4.4) as follows:

A(t)u = a(t, x)∆u for all u ∈ D(A(t)) = H10(Ω) ∩ H
2(Ω),

where a : R×Ω #→ R, in addition of being almost automorphic satisfies the following assumptions:

(h.1) inf
t∈R,x∈Ω

a(t, x) = m0 > 0, and

(h.2) there exists L > 0 and 0 < µ ≤ 1 such that

|a(t, x) − a(s, x)| ≤ L|s − t|µ

for all t, s ∈ R uniformly in x ∈ Ω.

(h.3) sup
t∈R,x∈Ω

a(t, x) < ∞.

(h.4) g is µ-pseudo-almost periodic in t ∈ R uniformly in u ∈ L2(Ω) and satisfying globally
Lipschitz with respect to the second argument in the following sense: there exists Kg > 0

such that

∥g(t, u) − g(t, v)∥2 ≤ Kg∥u − v∥2 for all t ∈ R and u, v ∈ L2(Ω),

A classical example of a function a satisfying the above-mentioned assumptions is for instance

a(t, x) = 3 + sin(t) + sin(
√
2t) + l(x), for x ∈ Ω and t ∈ R,

where l : Ω #−→ R+, continuous and bounded on Ω.

Under previous assumptions, it is clear that the operators A(t) defined above are invertible

and satisfy Acquistapace–Terreni conditions. Moreover, it can be easily shown that

R
!

ω, a(·, x)∆
"

ϕ =
1

a(·, x)
R
! ω

a(·, x)
,∆

"

ϕ ∈ AA(R, L2(Ω))

for each ϕ ∈ L2(Ω) with
&

&

&
R
!

ω, a∆
"
&

&

&

B(L2(Ω))
≤

const.

|ω|
.



CUBO
16, 2 (2014)

Pseudo-Almost Periodic and Pseudo-Almost Automorphic . . . 27

Let F : R × L2(Ω) #→ L2(Ω) be the mapping defined by

F(t,φ)(x) = g(t,φ(x)) for x ∈ Ω,

and z : R → L2(Ω) be the function defined by z(t) = ϕ(t, .), for t ∈ R. Then the Eq.(4.4) takes
the abstract form

d

dt

,

z(t)
-

= A(t)z(t) + F(t, z(t)), t ∈ R, (4.5)

Furthermore, all assumptions (H0), (H1), (H’2), (H’3), (M1) and (M2) of Theorem 3.8 are

fulfilled. Then, the evolution equation (4.4) has a unique (µ,ν)-pseudo almost automorphic solution

whenever Kg is small enough, with µ,ν being arbitrary positive measures of M satisfying (M.1)
and (M.2).

Received: January 2014. Revised: April 2014.

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