CUBO, A Mathematical Journal

Vol. 23, no. 01, pp. 63–85, April 2021

DOI: 10.4067/S0719-06462021000100063

Convolutions in (µ,ν)-pseudo-almost periodic and
(µ,ν)-pseudo-almost automorphic function spaces
and applications to solve integral equations

David Békollè1

Khalil Ezzinbi2

Samir Fatajou3

Duplex Elvis Houpa Danga4

Fritz Mbounja Béssémè5

1,4 Department of Mathematics, Faculty

of Science, University of Ngaoundéré

P.O. Box 454, Ngaoundéré, Cameroon.

dbekolle@univ-ndere.cm;

e houpa@yahoo.com

2,3 Department of Mathematics, Faculty

of Science Semlalia, Cadi Ayyad

University, B.P. 2390 Marrakesh,

Morocco.

ezzinbi@uca.ac.ma;

fatajou@yahoo.fr

5 Department of Mines and Geology,

School of Geology and Mining

Engineering, University of Ngaoundéré

P.O. Box 454, Ngaoundéré, Cameroon.

mbounjafritz@gmail.com

ABSTRACT

In this paper we give sufficient conditions on k ∈ L1(R)
and the positive measures µ, ν such that the doubly-measure

pseudo-almost periodic (respectively, doubly-measure pseudo-

almost automorphic) function spaces are invariant by the con-

volution product ζf = k ∗ f. We provide an appropriate
example to illustrate our convolution results. As a conse-

quence, we study under Acquistapace-Terreni conditions and

exponential dichotomy, the existence and uniqueness of (µ,ν)-

pseudo-almost periodic (respectively, (µ,ν)- pseudo-almost

automorphic) solutions to some nonautonomous partial evo-

lution equations in Banach spaces like neutral systems.

RESUMEN

En este art́ıculo damos condiciones suficientes sobre k ∈ L1(R)
y las medidas positivas µ, ν tales que los espacios de fun-

ciones pseudo-casi periódicas que duplican la medida (respec-

tivamente, pseudo-casi automorfas que duplican la medida)

son invariantes por el producto de convolución ζf = k ∗ f.
Entregamos un ejemplo apropiado para ilustrar nuestros re-

sultados de convolución. Como consecuencia, estudiamos bajo

condiciones de Acquistapace-Terreni y dicotomı́a exponen-

cial, la existencia y unicidad de soluciones (µ,ν)- pseudo-casi

periódicas (respectivamente, (µ,ν)- pseudo-casi automorfas)

de algunas ecuaciones de evolución parciales no autónomas en

espacios de Banach como sistemas neutrales.

Keywords and Phrases: Measure theory, (µ,ν)-ergodic, (µ,ν)-pseudo almost periodic and automorphic functions,

evolution families, nonautonomous equations, neutral systems.

2020 AMS Mathematics Subject Classification: 34C27, 34K14, 35B15, 35K57, 37A30, 43A60.

Accepted: 09 January, 2021

Received: 22 August, 2020

©2021 D. Békollè et al. This open access article is licensed under a Creative Commons

Attribution-NonCommercial 4.0 International License.

http://cubo.ufro.cl/
http://dx.doi.org/10.4067/S0719-06462021000100063
https://orcid.org/0000-0001-5959-098X
https://orcid.org/0000-0003-4779-5800
https://orcid.org/0000-0002-6052-6755


64 D. Békollè, K. Ezzinbi, S. Fatajou, D.E. Houpa Danga & F. Mbounja B. CUBO
23, 1 (2021)

1 Introduction

The existence and uniqueness of pseudo almost periodic and pseudo almost automorphic solu-

tions is one of the most powerful tools in the qualitative theory of differential equations due to

applications in mathematical biology, control theory and physical sciences. Recently, Diagana,

Ezzinbi and Miraoui [11] applied the abstract measure theory to define the notion of double-weight

pseudo almost periodicity (respectively double-weight pseudo almost automorphy) functions, and

thus the classical theory of µ-pseudo almost periodic (respectively µ-pseudo almost automorphic)

introduced by [4, 5], and double-weight pseudo almost periodicity [8] become particular cases of

this approach. See the section 2.1 for technical details about this concept of double-weight pseudo

almost periodicity (respectively double-weight pseudo almost automorphy) functions. We note

that for f ∈ PAP(R ×X,X,µ,ν) or f ∈ PAA(R ×X,X,µ,ν), k ∈ L1(R), k ∗f = k ∗g + k ∗φ.
We have that k ∗ g is almost periodic or almost automorphic function, but k ∗φ is not necessar-
ily in E(R,X,µ,ν). Then, the convolution invariance of the spaces PAP(R × X,X,µ,ν) (resp.
PAA(R×X,X,µ,ν)) is equivalent to the convolution invariance of E(R,X,µ,ν).
During the last decade, many research results about pseudo almost periodic and pseudo almost

atomorphic was produce see [4, 5, 7, 9, 10]. Inspired by the work of Ezzinbi et al. [11] who studied

the translation invariance of PAA(R×X,X,µ,ν) (resp. PAP(R×X,X,µ,ν)) functions and the
recent work of Mbounja et al. [15] who gave some several hypotheses for convolution invariance of

PAP(R×X,X,µ) and PAA(R×X,X,µ), in this work we established new sufficient conditions on
µ,ν ∈M and k ∈ L1(R) ensuring that, the space PAP(R,X,µ,ν) of (µ,ν)-pseudo almost periodic
functions and the space PAA(R,X,µ,ν) of (µ,ν)-pseudo almost automorphic functions are invari-

ant by the convolution product ζf = k∗f. Our obtained conditions are more general than [15] and
helped to show that the integral solution of some differential equations is a (µ,ν)-pseudo almost

periodic (respectively (µ,ν)-pseudo almost automorphic) solutions. To illustrate our investigation,

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

almost automorphic) solutions of the following nonautomous 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 7−→ X for t ∈ R is a family of closed linear operators on a Ba-
nach space X, satisfying the well-known Acquitaspace-Terreni conditions developed in [1, 2], and

F,G : R × X 7−→ X are jointly continuous functions satisfying some additional conditions. The
study of equation (1.1) in an non-autonomous case is new even in the case of one measure, µ = ν.

Also, equation (1.2) is treated here.

The rest of this work is organized as follows. In section 2, we recall some basic results which will

be used throughout this work. In section 3, we state and prove main results about the convolution



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Convolutions in (µ,ν)-PAP and (µ,ν)-PAA Functions 65

invariance. In section 4 we study the existence and uniqueness of (µ,ν)-pseudo almost periodic (re-

spectively (µ,ν)-pseudo almost automorphic) solutions to both equation (1.1) and equation (1.2)

which illustrate our new results.

2 Preliminaries

2.1 Notation and terminology

Let (X,‖ ·‖) a Banach space and let BC(R,X) be the space of bounded continuous functions
f : R −→ X. The space BC(R,X), equipped with the supremum norm ‖f‖∞ = sup

t∈R
‖f(t)‖, is a

Banach space.

We denote by B the Lebesgue σ-field of R and by M the space of all positive measures ϑ on B
satisfying ϑ(R) = +∞ and ϑ([a,b]) < ∞, for all a,b ∈ R (a ≤ b).

Definition 2.1 ([6]). A continuous function f : R → X is said to be almost periodic if for every
ε > 0 there exists a positive number lε such that every interval of length lε contains a number τ

such that:

‖f(t + τ) −f(t)‖ < ε, ∀t ∈ R.

Let AP (R,X) denote the collection of almost periodic functions from R to X. We recall that

(AP (R,X) ,‖·‖∞) is a Banach space.

Definition 2.2 ([11]). Let µ,ν ∈ M. A bounded continuous function f : R → X is said to be
(µ,ν)-ergodic if

lim
r→+∞

1

ν([−r,r])

∫ r
−r
‖f(t)‖dµ(t) = 0.

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

The space (E(R,X,µ,ν),‖.‖∞) is a Banach space for the supremum norm.

Definition 2.3 ([11]). Let µ,ν ∈M. A continuous function f : R → X is said to be (µ,ν)-pseudo
almost periodic if f admits the following decomposition:

f = g + φ,

where g ∈ AP(R,X) and φ ∈E(R,X,µ,ν).
We denote the space of all such functions by PAP(R,X,µ,ν).



66 D. Békollè, K. Ezzinbi, S. Fatajou, D.E. Houpa Danga & F. Mbounja B. CUBO
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We have AP(R,X) ⊂ PAP(R,X,µ,ν) ⊂ BC(R,X).

Let (Y,‖·‖) a Banach space and let BC(R × Y,X) be the space of jointly bounded continu-
ous functions f : R × Y −→ X. The space BC(R × Y,X) equipped with the supremum norm
‖f‖∞ = sup

t∈R,x∈Y
‖f(t,x)‖ is a Banach space.

Definition 2.4 ([12]). A jointly continuous function f : R×Y → X is said to be almost periodic
in t uniformly with respect to x ∈ Y , if for every ε > 0, and any compact subset K of Y , there
exists a positive number lK(ε) such that every interval of length lK(ε) contains a number τ such

that:

‖f(t + τ,x) −f(t,x)‖ < ε, ∀(t,x) ∈ R×K.

We denote the space of such functions by APU(R×Y,X).

Definition 2.5 ([11]). Let µ,ν ∈ M. A continuous function f : R × Y → X is said to be
(µ,ν)-ergodic in t uniformly with respect to x ∈ Y , if the following two conditions are true:

(i) f is uniformly continuous on each compact set K in Y with respect to the second variable x.

(ii) ∀x ∈ Y , f(.,x) ∈E(R,X,µ,ν).

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

Definition 2.6 ([11]). Let µ,ν ∈ M. A continuous function f : R × Y → X is said to be
(µ,ν)-pseudo almost periodic in t uniformly for x ∈ Y , if f admits the following decomposition:

f = g + φ, (2.1)

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

We have APU(R×Y,X) ⊂ PAPU(R×Y,X,µ,ν) ⊂ BC(R×Y,X,µ,ν).

Definition 2.7 ([16]). A continuous function f : R → X is said to be almost automorphic if for
every sequence of real numbers (s′n)n∈N, there exists a subsequence (sn)n∈N ⊂ (s′n)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) exists ∀t ∈ R and f(t) = lim
n→∞

g(t−sn) ∀t ∈ R.

We denote the space of such functions by AA(R,X).



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Convolutions in (µ,ν)-PAP and (µ,ν)-PAA Functions 67

We recall that (AA (R,X) ,‖·‖∞) is a Banach space.

Definition 2.8 ([11]). Let µ,ν ∈M. A continuous function f : R → X is said to be (µ,ν)-pseudo
almost automorphic if f admits the following decomposition:

f = g + φ,

where g ∈ AA(R,X) and φ ∈E(R,X,µ,ν).
We denote the space of all such functions by PAA(R,X,µ,ν).

We have AA(R,X) ⊂ PAA(R,X,µ,ν) ⊂ BC(R,X).

Definition 2.9 ([16]). A continuous function f : R×Y → X is said to be almost automorphic in
t uniformly for x ∈ Y , if the following conditions hold:

(i) f is uniformly continuous on each compact set K in Y with respect to the second variable x,

namely, for each compact set K in Y , 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.

(ii) for all x ∈ Y , f(.,x) ∈ AA(R,X).

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

Definition 2.10 ([11]). Let µ,ν ∈ M. A continuous function f : R × Y → X is said to be
(µ,ν)-pseudo almost periodic in t uniformly for x ∈ Y , if f admits the following decomposition:

f = g + φ, (2.2)

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

We have AAU(R×Y,X) ⊂ PAAU(R×Y,X,µ,ν) ⊂ BC(R×Y,X,µ,ν)

2.2 Some useful results on the space functions

For µ ∈M and τ ∈ R, we denote by µτ the positive measure on (R,B) defined by:

µτ (A) = µ ({a + τ : a ∈ A}) , ∀A ∈B.



68 D. Békollè, K. Ezzinbi, S. Fatajou, D.E. Houpa Danga & F. Mbounja B. CUBO
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Now we introduce the following hypotheses on µ,ν ∈M.
(H0): For all τ ∈ R, there exists δ > 0 and a bounded interval I such that

µτ (A) ≤ δµ(A), ντ (A) ≤ δν(A), ∀A ∈B satisfied A∩ I = ∅.

(H1):

lim sup
r→∞

µ([−r,r])
ν([−r,r])

< ∞.

Remark 2.11.

i) Without assumptions on µ and ν, like (H0), the decomposition (2.1) (resp. (2.2)) of the (µ,ν)-

pseudo almost periodic and automorphic functions is not unique, (see [11]).

ii) The spaces E(R,X,µ,ν), E(R×Y,X,µ,ν), PAP(R,X,µ,ν), PAP(R×Y,X,µ,ν), PAA(R,X,µ,ν),
and PAA(R ×Y,X,µ,ν) coincides when µ = ν, with the spaces E(R,X,µ), E(R ×Y,X,µ),
PAP(R,X,µ), PAP(R×Y,X,µ), PAA(R,X,µ), and PAA(R×Y,X,µ).

We recall the following six theorems proved in [11].

Theorem 2.12 ([11]). Consider that µ,ν ∈M and k ∈ L1(R) and f ∈ PAP(R,X,µ,ν) (respec-
tively f ∈ PAA(R,X,µ,ν). If (H0) is valid then PAP(R,X,µ,ν) (respectively PAA(R,X,µ,ν))
is translation invariant. Moreover,

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

Theorem 2.13 ([11]). If (H0) is valid, then the decomposition (2.1)(resp. (2.2)) of PAP(R,X,µ,ν)

and PAA(R,X,µ) is unique.

Theorem 2.14 ([11]). If (H1) holds, then (E(R,X,µ,ν),‖ ·‖∞) is a Banach space with respect
to the sup norm.

Theorem 2.15 ([11]). Let µ,ν ∈ M satisfy (H1). If (H0) holds, then PAP(R,X,µ,ν) and
PAA(R,X,µ,ν) are Banach spaces with respect to the sup norm.

Theorem 2.16 ([11]). Let µ,ν ∈ M, F ∈ PAPU(R × Y,X,µ,ν) and h ∈ PAP(R,X,µ,ν).
Assume that (H1) and the following hypothesis holds:

For all bounded subsets B of X, F is bounded on R×B.
Then t 7−→ F(t,h(t)) ∈ PAP(R,X,µ,ν).

Theorem 2.17 ([11]). Let µ,ν ∈ M, F ∈ PAAU(R × Y,X,µ,ν) and h ∈ PAA(R,X,µ,ν).
Assume that for all bounded subsets B of X, F is bounded on R × B. Then t 7−→ F(t,h(t)) ∈
PAA(R,X,µ,ν).



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Convolutions in (µ,ν)-PAP and (µ,ν)-PAA Functions 69

2.3 Measure theory results

Let µ,ν ∈ M; if f : R −→ X is a bounded continuous function, we define the following doubly-
weight mean, if the limit exists, by:

M(f,µ,ν) := lim
r→+∞

1

ν([−r,r])

∫ r
−r
‖f(t)‖Xdµ(t).

Definition 2.18 ([17]). Let (E,B) be a Borel space. If µ and ν are two measures defined on
(E,B), we say that:

(i) µ and ν are mutually singular, if there are disjoint sets A and B in B such that E = A∪B
and

ν(A) = µ(B) = 0.

(ii) ν is absolutely continuous with respect to µ, if for each A ∈B,

(µ(A) = 0) =⇒ (ν(A) = 0).

We recall the following theorems of measure theory.

Theorem 2.19 (Radon-Nikodym [17]). Let (E,B,µ) be a σ-finite measure space, and let ν be
a measure defined on B which is absolutely continuous with respect to µ. Then there is a unique
nonnegative measurable function f such that for each set B in B we have:

ν(B) =

∫
B

fdµ.

The function f is called the Radon-Nikodym derivative of ν with respect of µ.

Example 2.20.

Let ρ be a nonnegative B-measurable function. Denote by µ the positive measure defined by:

µ(A) =

∫
A

ρ(t)dt, for A ∈B

where dt is the Lebesgue measure on R. The function ρ is the Radon-Nikodym derivative of µ

with respect to the Lebesgue measure dt on R, i.e. dµ(t) = ρ(t)dt. In this case, µ ∈M if and only
if its Radon-Nikodym derivative ρ is locally Lebesgue integrable on R and it satisfies∫ +∞

−∞
ρ(t)dt = +∞.

Theorem 2.21 (Lebesgue-Radon-Nikodym [17]). Let (X,B,ϑ) be a σ-finite measure space, and
µ a σ-finite measure defined on B. Then, we can find a measure µ0, singular with respect to ϑ,
and a measure µ1, absolutely continuous with respect to ϑ, such that µ = µ0 + µ1. The measures

µ0 and µ1 are unique.



70 D. Békollè, K. Ezzinbi, S. Fatajou, D.E. Houpa Danga & F. Mbounja B. CUBO
23, 1 (2021)

In this section, by using the previous theorem, we consider that for a given µ ∈M, µ = µ0 +µ1
where µ0 is the µ-measure component which is absolutely continuous with respect to the Lebesgue

measure and its Radon-Nikodym derivative is ρ, that is dµ0(t) = ρ(t)dt and µ1 is the µ-measure

component such that µ1 is singular to Lebesgue measure.

We give new general hypotheses on µ,ν ∈M and k ∈ L1(R) such that:

(ζf)(t) =

∫ +∞
−∞

k(t−s)f(s)ds, ∀k ∈ L1(R) (2.3)

maps E(R,X,µ,ν) into itself.
In particular, our hypotheses on µ,ν ∈M and k ∈ L1(R) will imply that for every f ∈E(R,X,µ,ν),
the (µ,ν)-mean,

M(ζf,µ,ν) := lim
r→+∞

1

ν([−r,r])

∫ r
−r

∣∣∣∣
∣∣∣∣
∫ +∞
−∞

k(t−s)f(s)ds
∣∣∣∣
∣∣∣∣
X

dµ(t)

exists.

3 Main results of convolution and translation invariance

3.1 Convolution invariance on E(R,X,µ,ν)

Theorem 3.1. Let k ∈ L1(R) and ν ∈M. Consider that µ ∈M, with Radon-Nikodym derivative
ρ with respect to dt and ζ is defined in (2.3). Assume that ρ,µ,ν and k satisfy the following

requirements: 


sup
|s|≤r,r∈R+

1

ρ(s)

∫ r
s
|k(t−s)|dµ(t) < ∞, (3.1.1)

sup
|s|≤r,r∈R+

1

ρ(s)

∫ s
−r|k(t−s)|dµ(t) < ∞, (3.1.2)

(3.1)




lim
r→+∞

1

ν([−r,r])
∫−r
−∞

(∫ r
−r|k(t−s)|dµ(t)

)
ds = 0, (3.2.1)

lim
r→+∞

1

ν([−r,r])
∫ +∞
r

(∫ r
−r|k(t−s)|dµ(t)

)
ds = 0. (3.2.2)

(3.2)

If f ∈E(R,X,µ,ν), then ζf ∈E(R,X,µ,ν).

Proof. We adapt the proof in [15], Theorem 3.5. By the properties of convolution we have

that f ∈ BC(R,X) implies that k ∗ f ∈ BC(R,X), ∀k ∈ L1(R). Then, in order to get that
k ∗f ∈E(R,X,µ,ν) we must prove that M(ζf,µ,ν) = 0.
We consider µ ∈ M and ρ its Radon-Nikodym derivative, ν ∈ M. In the first stage, we assume
that k(t) = 0 on R∗−. From ν(R) = +∞, we deduce the existence of r0 ≥ 0 such that ν([−r,r]) >
0, ∀r ≥ r0. Then by applying the Fubini’s Theorem, we deduce that for f ∈ BC(R,X), ∀r ≥ r0.



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Convolutions in (µ,ν)-PAP and (µ,ν)-PAA Functions 71

We notice that

M(ζf,µ,ν) = lim
r→+∞

1

ν([−r,r])

∫ r
−r
‖k ∗f‖Xdµ(t)

≤ lim
r→+∞

1

ν([−r,r])

∫ r
−r

∫ t
−∞
‖ f(s) ‖X| k(t−s) | dsdµ(t)

= lim
r→+∞

1

ν([−r,r])

∫ r
−r

(∫ −r
−∞
‖ f(s) ‖X| k(t−s) | ds

)
dµ(t)

+ lim
r→+∞

1

ν([−r,r])

∫ r
−r

(∫ t
−r
‖ f(s) ‖X| k(t−s) | ds

)
dµ(t)

≤ ‖ f ‖∞ lim
r→+∞

1

ν([−r,r])

∫ r
−r

(∫ −r
−∞
| k(t−s) | ds

)
dµ(t)

+ lim
r→+∞

1

ν([−r,r])

∫ r
−r
‖ f(s) ‖X

(∫ r
s

| k(t−s) | dµ(t)
)
ds

≤ ‖ f ‖∞ lim
r→+∞

1

ν([−r,r])

∫ −r
−∞

(∫ r
−r
| k(t−s) | dµ(t)

)
ds

+ lim
r→+∞

1

ν([−r,r])

∫ r
−r
‖ f(s) ‖X

[
1

ρ(s)

∫ r
s

| k(t−s) | dµ(t)
]
ρ(s)ds

≤ sup
|s|≤r,r∈R+

1

ρ(s)

∫ r
s

|k(t−s)|dµ(t) lim
r→+∞

1

ν([−r,r])

∫ r
−r
‖f(s)‖Xρ(s)ds

+ ‖ f ‖∞ lim
r→+∞

1

ν([−r,r])

∫ −r
−∞

(∫ r
−r
| k(t−s) | dµ(t)

)
ds

≤ sup
|s|≤r,r∈R+

1

ρ(s)

∫ r
s

|k(t−s)|dµ(t) lim
r→+∞

1

ν([−r,r])

∫ r
−r
‖f(s)‖Xdµ(s)

+ ‖ f ‖∞ lim
r→+∞

1

ν([−r,r])

∫ −r
−∞

(∫ r
−r
| k(t−s) | dµ(t)

)
ds.

Using assumptions (3.1.1), (3.2.1) and the fact that f ∈E(R,X,µ,ν), we have proved that
M(ζf,µ,ν) = 0 + 0 = 0. This settles the first stage for every k ∈ L1(R) such that k(t) = 0 on R∗−.
Now, in the second stage, proceeding similarly like in the first stage, we assume that k(t) = 0 on

R∗+ we obtain:

M(ζf,µ,ν) = lim
r→+∞

1

ν([−r,r])

∫ r
−r
‖k ∗f‖Xdµ(t)

≤ ‖ f ‖∞ lim
r→+∞

1

ν([−r,r])

∫ +∞
r

(∫ r
−r
| k(t−s) | dµ(t)

)
ds

+ sup
|s|≤r,r∈R+

1

ρ(s)

∫ s
−r
|k(t−s)|dµ(t) lim

r→+∞

1

ν([−r,r])

∫ r
−r
‖f(s)‖Xdµ(s).

Then, using the fact that f ∈ E(R,X,µ,ν) and hypotheses (3.1.2), (3.2.2), we have that
M(ζf,µ,ν) = 0.

In the general case of k, we deduce the result using the fact that k(t) = kχt≥0(t) + kχt<0(t).

Theorem 3.2. Assume that µ,ν ∈M and (H1) holds. Then the condition (3.2.1) (resp. (3.2.2))



72 D. Békollè, K. Ezzinbi, S. Fatajou, D.E. Houpa Danga & F. Mbounja B. CUBO
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is valid for every k ∈ L1(R) if and only if the following condition (3.3.1) (resp. (3.3.2)) is true:


lim
r→+∞

µ([−r,σ −r])
ν([−r,r])

= 0, ∀σ > 0 (3.3.1)

lim
r→+∞

µ([σ + r,r])

ν([−r,r])
= 0, ∀σ < 0 (3.3.2).

(3.3)

Proof. We first prove that (3.3.1) =⇒ (3.2.1), for every k ∈ L1(R).
In the first stage, we assume that k(t) = 0 on R∗−. Let σ = t−s > 0 fixed. From ν(R) = +∞, we
deduce the existence of r0 ≥ 0 such that ν([−r,r]) > 0, ∀r ≥ r0. In the sequel, for all r ≥ r0, we
shall assume that

B :=
1

ν([−r,r])

∫ r
−r

(∫ −r
−∞
| k(t−s) | ds

)
dµ(t).

Then, by applying the Fubini’s Theorem we deduce that:

B =
1

ν([−r,r])

∫ r
−r

(∫ +∞
t+r

| k(σ) | dσ
)
dµ(t)

=

∫ +∞
0

(∫ min(σ−r,r)
−r dµ(t)

ν([−r,r])

)
| k(σ) | dσ

=

∫ +∞
0

(
µ([−r, min(σ −r,r)])

ν([−r,r])

)
| k(σ) | dσ.

By using assumption (3.3.1), we have:

lim
r→+∞

µ([−r, min(σ −r,r)])
ν([−r,r])

= 0, ∀σ > 0.

Since µ([−r, min(σ−r,r)]) ≤ µ([−r,r]) and the fact that (H1) holds, there exists β > 0 such that:

0 ≤
(
µ([−r, min(σ −r,r)])

ν([−r,r])

)
| k(σ) |≤ β | k(σ) |,

where k ∈ L1(R), ∀σ > 0. Then, by the Lebesgue dominated convergence Theorem, we obtain:

lim
r→+∞

∫ +∞
0

(
µ([−r, min(σ −r,r)])

ν([−r,r])

)
| k(σ) | dσ = 0

This concludes this stage of (3.2.1).

Now, in the second stage, proceeding similarly like the first stage, we assume that k(t) = 0

on R∗+. Let σ = t− s < 0 fixed. From ν(R) = +∞, we deduce the existence of r0 ≥ 0 such that
ν([−r,r]) > 0, ν([−r,r]) > 0, ∀r ≥ r0. We set:

A :=
1

ν([−r,r])

∫ r
−r

(∫ +∞
r

| k(t−s) | ds
)
dµ(t)



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Convolutions in (µ,ν)-PAP and (µ,ν)-PAA Functions 73

Then, by applying the Fubini’s Theorem, we deduce that:

A =
1

ν([−r,r])

∫ r
−r

(∫ t−r
−∞

| k(σ) | dσ
)
dµ(t)

=

∫ 0
−∞

(∫ r
max(σ+r,r)

dµ(t)

ν([−r,r])

)
| k(σ) | dσ

=

∫ 0
−∞

(
µ([max(σ + r,r),r])

ν([−r,r])

)
| k(σ) | dσ.

Like in the first part, we use assumptions (3.3.2), and the Lebesgue dominated convergence The-

orem. This concludes this second stage of (3.2.2).

Let us prove (3.2.1) =⇒ (3.3.1).
Let σ = t−s, by (3.2.1) and Fubini’s Theorem we have that:

0 = lim
r→+∞

1

ν([−r,r])

∫ r
−r

(∫ −r
−∞
| k(t−s) | ds

)
dµ(t)

= lim
r→+∞

∫ +∞
0

(
µ([−r, min(σ −r,r)])

ν([−r,r])

)
| k(σ) | dσ.

Let τ > 0 such that σ ∈ [τ,τ + 1] and r >
τ

2
. We have also [−r,τ − r] ⊆ [−r,σ − r] and

[−r,τ −r] ⊆ [−r,r], that implies µ([−r,τ −r]) ≤ min{µ([−r,σ −r]),µ([−r,r])}, i.e.

µ([−r,τ −r])
ν([−r,r])

≤
µ([−r, min(σ −r,r)])

ν([−r,r])
.

Let k(σ) = χ[τ,τ+1](σ). We have that:

0 ≤
µ([−r,τ −r])
ν([−r,r])

∫ τ+1
τ

dσ

≤
∫ τ+1
τ

µ([−r, min(σ −r,r)])
ν([−r,r])

dσ

= lim
r→+∞

1

ν([−r,r])

∫ r
−r

(∫ −r
−∞
| k(t−s) | ds

)
dµ(t).

Then by (3.2.1):

lim sup
r→+∞

µ([−r,τ −r])
ν([−r,r])

∫ τ+1
τ

dσ ≤ lim
r→+∞

∫ τ+1
τ

µ([−r, min(σ −r,r)])
ν([−r,r])

dσ = 0

then:

lim
r→+∞

µ([−r,τ −r])
ν([−r,r])

= 0.

So (3.3.1) is verified. In the second stage (3.3.2), we do the same proof as above.



74 D. Békollè, K. Ezzinbi, S. Fatajou, D.E. Houpa Danga & F. Mbounja B. CUBO
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Remark 3.3. Hypothesis (H1) was used only in the proof of the implication

(3.3.1) =⇒ (3.2.1).

Corollary 3.4. Let µ,ν ∈M be such that the nonnegative B-measurable function ρ be the Radon-
Nikodym derivative of µ. Assume that for all k ∈ L1(R) the requirements (3.1) and (3.2) are
satisfied. Then E(R,X,µ,ν) is convolution invariant.

Corollary 3.5. Consider that µ,ν ∈ M, such that the nonnegative B-measurable function ρ be
the Radon-Nikodym derivative of µ. Assume that (H1) holds and the requirements (3.1) and (3.3)

are satisfied. Then E(R,X,µ,ν) is convolution invariant.

Example 3.6.

We check that Theorem 3.1 and Corollary 3.5 hold.

Let

k(t) =




1

10
e−2t, for t ∈ [0, +∞[

0, for t ∈] −∞, 0[.

We take dµk,η(t) = e
σtdt + η

∞∑
n=−∞

eσnδn, where 0 ≤ σ < 2, η > 0 and δn denotes the Dirac

measure at the integer n (
∑∞
n=−∞e

σnδn is a ’generalized Dirac comb’, it is called a Dirac comb

when σ = 0). Then µσ,η ∈M and its Radon-Nikodym derivative is ρσ,η(t) = eσt. Let νσ,η = γµσ,η,
where γ > 0. Then νσ,η ∈M.
First, if for |s| ≤ r, r > 0, we write:

Jσ,η(r,s) :=
1

ρσ,η(s)

∫ r
s

|k(t−s)|dµσ(t),

we must prove that:

sup
|s|≤r, r>0

Jσ,η(r,s) < ∞.

In fact,

Jσ,η(r,s) =
1

10eσs


∫ r

s

e−2(t−s)eσtdt + η
∑

s≤n≤r

e−2(n−s)eσn




=
1

10


∫ r

s

e−(2−σ)(t−s)dt + η
∑

s≤n≤r

e−(2−σ)(n−s)




≤
1

10


∫ r−s

0

e−(2−σ)udu + η
∑

[s]≤n≤[r]

e−(2−σ)(n−[s]−1)


 ,

where we applied the change of integral u = t− s in the integral and we denoted [x] the integral
part of the real number x. We next apply the change of index m = n− [s] in the latter sum; this
implies:



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Convolutions in (µ,ν)-PAP and (µ,ν)-PAA Functions 75

Jσ,η(r,s) ≤
1

10


∫ r−s

0

e−(2−σ)(u)du + ηe2−σ
[r]−[s]∑
m=0

e−(2−σ)m


 .

Then:

sup
|s|≤r, r>0

Jσ,η(r,s) ≤
1

10

(∫ ∞
0

e−(2−σ)(u)du + ηe2−σ
∞∑
m=0

e−(2−σ)m

)

=
1

10(2 −σ)
+ ηe2−σ

1

10(1 −e−(2−σ))
< ∞.

This proves the estimate (3.1.1).

Secondly, we shall show that for all α > 0, we have:

lim
r→∞

µσ,η([−r,α−r])
νσ,η([−r,r])

= 0.

It actually suffices to prove this estimate when α is a positive integer. In fact,

µσ,η([−r,α−r])
νσ,η([−r,r])

=

∫ α−r
−r

eσtdt + η
∑

−r≤n≤α−r

eσn

γ


∫ r
−r
eσtdt + η

∑
−r≤n≤r

eσn




≤

1

σ

(
eσ(α−r) −e−σr

)
+ η

∑
−[r]−1≤n≤α−[r]

eσn

γ


 1
σ

(eσr −e−σr) + η
∑

−[r]≤n≤[r]

eσn




=

1

σ
e−σr (eσα − 1) + ηe−σ([r]+1)

α+1∑
m=0

eσm

γ


 1
σ

(eσr −e−σr) + ηe−σ[r]
2[r]∑
m=0

eσm



,

where we applied the change of index m = n + [r] + 1 on the numerator and the change of index

m = n + [r] on the denominator. So

µσ,η([−r,α−r])
µσ,η([−r,r])

≤

1

σ
e−σr (eσα − 1) + η

eσ(α+2) − 1
eσ − 1

e−σ([r]+1)

γ

σ
eσr (1 −e−2σr) + γηe−σ[r]

eσ(2[r]+1) − 1
eσ − 1

.

The estimation (3.3.1) easily follows.

Thirdly, we show that (H1) holds.

lim sup
r→∞

µσ,η([−r,r])
νσ,η([−r,r])

=
1

γ
< ∞.

Then, Theorem 3.1 and Corollary 3.5 hold.



76 D. Békollè, K. Ezzinbi, S. Fatajou, D.E. Houpa Danga & F. Mbounja B. CUBO
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3.2 Translation invariance and convolution invariance of PAP(R,X,µ,ν)
and PAA(R,X,µ,ν)

Theorem 3.7. Assume that µ,ν ∈M and (H1) holds. If the space E(R,X,µ,ν) is translation
invariant, then E(R,X,µ,ν) is convolution invariant.

Proof. Let f ∈ E(R,X,µ,ν). Let us prove that if f(t − τ) ∈ E(R,X,µ,ν), for τ ∈ R,
then ζf ∈ E(R,X,µ,ν), i.e. M(ζf,µ,ν) = 0. By the properties of convolution we have that
f ∈ BC(R,X) implies that k ∗f ∈ BC(R,X), ∀k ∈ L1(R). By the Fubini’s Theorem we have,

M(ζf,µ,ν) = lim
r→+∞

1

ν([−r,r])

∫ r
−r
‖k ∗f‖Xdµ(t)

≤ lim
r→+∞

∫ ∞
−∞

|k(s)|
ν([−r,r])

(∫ r
−r
‖f(t−s)‖Xdµ(t)

)
ds

Since f is invariant by translation we have for all s ∈ R:

lim
r→+∞

1

ν([−r,r])

∫ r
−r
‖f(t−s)‖Xdµ(t) = 0.

Since (H1) holds for all s ∈ R, we have that:

0 ≤
|k(s)|

ν([−r,r])

∫ r
−r
‖f(t−s)‖Xdµ(t)ds ≤ β|k(s)|‖f‖∞,

where k ∈ L1(R). Then by the Lebesgue dominated convergence Theorem, we obtain that
M(ζf,µ,ν) = 0.

Theorem 3.8. Let (H1) holds. If the space PAP(R,X,µ,ν) (resp. PAA(R,X,µ,ν)) is transla-

tion invariant, then E(R,X,µ,ν) is convolution invariant.

Proof. For f ∈ AP(R,X) or f ∈ AA(R,X), then f is invariant by ζ i.e. ζf ∈ AP(R,X) or
ζf ∈ AA(R,X). We use the previous theorem to conclude.

Corollary 3.9. Let (H0) and (H1) hold. Then E(R,X,µ,ν), PAP(R,X,µ,ν) and PAA(R,X,µ,ν)
are convolution invariant.

Proof. Combine Theorem 2.12 and Theorem 3.8.



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Convolutions in (µ,ν)-PAP and (µ,ν)-PAA Functions 77

4 Existence, Uniqueness results and Applications

This section is similar to section 3 in [11], but here we applied our new results obtained in the

above section.

4.1 Evolution Families and Exponential Dichotomy

(H2): 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 1 < µ0 + ν0 such that

Σθ ∪{0}⊂ ρ(A(t) −ω) 3 λ,‖R(λ,A(t) −ω)‖≤
K

1 + |λ|
, (4.1)

and

‖(A(t) −ω)R(λ,A(t) −ω)[R(ω,A(t)) −R(ω,A(s))]‖≤ L
|t−s|µ0
|λ|ν0

, (4.2)

for t,s ∈ R,λ ∈ Σθ := {λ ∈ C/{0} : |argλ| ≤ θ}.
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) satisfied Acquistapace-Terreni conditions, then there

exists a unique evolution family

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,

i) U(t,r)U(r,s) = U(t,s) and U(s,s) = I ∀t ≥ r ≥ s and t,r,s ∈ R;

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

iii) 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 4.1 ([3]). An evolution family (U(t,s))t≥s on a Banach space X is called hyperbolic (or
has exponential dichotomy) if there exist projections P(t), t ∈ R, uniformly bounded and strongly
continuous in t, and constants N ≥ 1, δ > 0 such that

i) U(t,s)P(s) = P(t)U(t,s) for t ≥ s;

ii) the restriction UQ(t,s) : Q(s)X −→ Q(t)X for U(t,s) is inversible for t,s ∈ R and we set
UQ(t,s) = U(s,t)−1;



78 D. Békollè, K. Ezzinbi, S. Fatajou, D.E. Houpa Danga & F. Mbounja B. CUBO
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iii)

‖U(t,s)P(s)‖≤ Ne−δ(t−s) (4.3)

and

‖UQ(s,t)Q(t)‖≤ Ne−δ(t−s) (4.4)

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

4.2 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), we also assume that the next hypoth-

esis holds:

(H3) The evolution family U generated by A(.) has an exponential dichotomy with constants

N ≥ 1, δ > 0 and dichotomy projections P(t).
We recall the following sufficient conditions to fulfill the assumption (H3).

(H3.1) Let (A(t),D(A(t)))t∈R be generators of analytic semigroups on X of the same type. Sup-

pose that D(A(t)) = D(A(0)), A(t) is inversible, 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.
(H3.2) The semigroup (e

τA(t))τ≥0, t ∈ R, are hyperbolic with projection Pt and constants N,δ >
0. Moreover, let

‖A(t)(eτA(t)Pt)‖≤ Ψ(τ), ‖A(t)(eτA(t)Qt)‖≤ Ψ(−τ)

for τ > 0 and a function Ψ such that R 3 s −→ ϕ(s) := |s|µΨ(s) is integrable with
L0‖ϕ‖L1(R) < 1.
We introduce here the defnition of the mild solution of equation (1.1).

Definition 4.2 ([3]). A continuous function u : R 7−→ 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 ∈ R,with t ≥ s. (4.5)

Theorem 4.3 ([11]). Assume that (H2) and (H3) 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 7−→ X
given by

u(t) =

∫
R

Γ(t,s)F(s,u(s))ds, t ∈ R,



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Convolutions in (µ,ν)-PAP and (µ,ν)-PAA Functions 79

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 ∈ R, with t ≥ s,

Γ(t,s) = −UQ(t,s)Q(s), ∀t,s ∈ R, with t < s.

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 this work, we fix µ,ν ∈M to satisfy (H1).

4.3 Existence of (µ,ν)-pseudo-almost periodic solutions

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

(H4): R(ω,A(.)) ∈ AP(R,L(X)).
(H5): We propose F : R×X 7−→ 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 the main results of this study.

Lemma 4.4 ([13]). Assume that (H2)-(H4) hold. Then r −→ Γ(t+r,s+r) belongs to AP(R,L(X))
for all 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 4.5. Assume that (H2)-(H5) hold. If (3.1) and (3.2), or (3.1) and (3.3) hold, 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 the assumption (H5) and
Theorem 2.16 it follows that h ∈ PAP(R,X,µ,ν). 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

ΞΨi(t) :=

∫ t
−∞

U(t,s)P(s)Ψi(s)ds,for i ∈{1, 2}.



80 D. Békollè, K. Ezzinbi, S. Fatajou, D.E. Houpa Danga & F. Mbounja B. CUBO
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From Lemma 4.4, 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. From equation (4.3), we have:

1

ν([r,−r])

∫ r
−r
‖(Ξ(Ψ2)(t)‖dµ(t) ≤

1

ν([r,−r])

∫ r
−r

∫ t
−∞

U(t,s)P(s)Ψ2(s)dsdµ(t)

≤
N

ν([r,−r])

∫ r
−r

∫ t
−∞

e−δ(t−s)‖Ψ2(s)‖dsdµ(t)

Since µ and ν satisfy (3.1.1) and (3.2.1), (3.1.1) and (3.3.1), with k(t) = e−δt, then by Theorem

3.1 or Corollary 3.5, we conclude that:

lim
r→+∞

1

ν([r,−r])

∫ r
−r
‖(Ξ(Ψ2)(t)‖dµ(t) = 0.

The proof for Γ2u(.) is similar to that of Γ1u(.) except that one makes use of equation (4.4) instead

of (4.3), (3.1.2) and (3.2.2), or (3.1.2) and (3.3.2).

Theorem 4.6. Assume that (H2)-(H5) hold. If (3.1) and (3.2), or (3.1) and (3.3) hold, then

equation (1.1) has a unique (µ,ν)-pseudo almost periodic mild solution whenever KF is small

enough.

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

Ku(t) =
∫ t
−∞

U(t,s)P(s)F(s,u(s))ds−
∫ +∞
t

UQ(t,s)Q(s)F(s,u(s))ds, ∀t ∈ R.

By Lemma 4.5, it follows that K maps PAP(R,X,µ,ν) into itself. To complete the proof one has

to show that K is a contraction map on PAP(R,X,µ,ν).

Let u,v ∈ PAP(R,X,µ,ν). Firstly, we have that:

‖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 that:

‖Γ2(v)(t) − Γ2(u)(t)‖ ≤
∫ +∞
t

‖UQ(t,s)Q(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‖∞
∫ +∞
t

e−δ(t−s)ds

= NKFδ
−1‖v −u‖∞.

Finally, combining previous approximations it follows that:

‖Kv −Ku‖∞ < 2NKFδ−1‖v −u‖∞.



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Convolutions in (µ,ν)-PAP and (µ,ν)-PAA Functions 81

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

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

u satisfying Ku = u, which is the unique (µ,ν)-pseudo almost periodic mild solution to equation

(1.1).

Theorem 4.7 ([11]). Assume that (H2)-(H5) hold. If (H0) holds, then equation (1.1) has a unique

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

4.4 Existence of (µ,ν)-pseudo-almost automorphic solutions

In this section we consider the following assumptions:

(H6): R(ω,A(.)) ∈ AA(R,L(X)).
(H7): We propose F : R×X 7−→ 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 4.8 ([14]). Assume that (H2), (H3) and (H6) hold. Let a sequence (s
′
l)l∈N ⊂ R there is

a sub-sequence (sl)l∈N such that for every h > 0

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

Lemma 4.9. Assume that (H2), (H3), (H6) and (H7) hold. If (3.1) and (3.2) or (3.1) and (3.3)

or (H0) hold, then the integral operators Γ1 and Γ2 defined above map PAA(R ×X,X,µ,ν) into
itself.

Proof. Let u ∈ PAA(R,X,µ,ν). Setting g(t) = F(t,u(t)), by assumption (H7) and Theo-
rem 2.17 we obtain 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, Su2(t) :=
∫ t
−∞

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

From equation (4.3), we obtain:

‖Su1(t)‖≤ Nδ−1‖u1‖∞, ‖Su2(t)‖≤ Nδ−1‖u2‖∞, ∀t ∈ R.

Then Su1(t),Su2(t) ∈ BC(R,X). Now, we prove that Su1(t) ∈ 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), (4.6)

is well defined for each t ∈ R, and

lim
n→∞

v1(t− τn) = u1(t), ∀t ∈ R. (4.7)



82 D. Békollè, K. Ezzinbi, S. Fatajou, D.E. Houpa Danga & F. Mbounja B. CUBO
23, 1 (2021)

Set for t ∈ R,

M(t) :=

∫ t
−∞

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

∫ t
−∞

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

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 equation (4.3), equation (4.6) and the Lebesgue’s Dominated Convergence Theorem, it

follows that:

lim
n→+∞

∣∣∣∣
∣∣∣∣
∫ t
−∞

U(t + τn,s + τn)P(s + τn)[u1(s + τn) −v1(s)]ds
∣∣∣∣
∣∣∣∣ = 0, for t ∈ R. (4.8)

Similary, using Lemma 4.8 it follows that:

lim
n→+∞

∣∣∣∣
∣∣∣∣
∫ t
−∞

[U(t + τn,s + τn)P(s + τn) −U(t,s)P(s)]v1(s)ds
∣∣∣∣
∣∣∣∣ = 0, for t ∈ R. (4.9)

Therefore, we have that:

N(t) := lim
n→∞

M(t + τn),∀t ∈ R. (4.10)

Using similar ideas as the previous ones, then:

M(t) := lim
n→∞

N(t− τn),∀t ∈ R. (4.11)

Therefore, Su1(t) ∈ AA(R,X). Arguing as in Lemma 4.5, we get that Su2(t) ∈E(R,X,µ,ν). The
proof for Γ2u(.) is similar to that of Γ1u(.) except that one makes use of equation (4.4) instead of

equation (4.3) and, (3.1.2) and (3.2.2), (3.1.2) and (3.3.2).

Theorem 4.10. Under assumptions (H2), (H3), (H6) and (H7), if (3.1) and (3.2) or (3.1) and

(3.3) or (H0) then equation (1.1) has a unique (µ,ν)-pseudo almost automorphic mild solution

whenever KF is small enough.

Proof The proof of Theorem 4.10 is similar to that Theorem 4.6 except that one makes use

of Lemma 4.9 instead of Lemma 4.5.



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Convolutions in (µ,ν)-PAP and (µ,ν)-PAA Functions 83

4.5 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:

(H8): 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.
(H9) We suppose 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 4.11. A function v : R 7−→ X is said a mild solution of (1.2) on R if :

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 4.12. Assume that assumptions (H2)-(H5) and (H8) hold. If (3.1) and (3.2) or (3.1)

and (3.3) or (H0) hold, and (KG + 2NKFδ
−1) < 1, then equation (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,ν) 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 (H9), Theorem 2.16, and Lemma 4.5 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, equation (1.2) has unique (µ,ν)-pseudo almost periodic mild solution.

Theorem 4.13. Assume that (H2), (H3),(H6), (H7) and (H9) hold and (KG + 2NKFδ
−1) < 1.

If (3.1) and (3.2)or (3.1) and (3.3) or (H0) hold, then equation (1.2) has a unique (µ,ν)-pseudo

almost automorphic mild solution.

Proof. Similarly, we can show, by using the assumption (H9), Theorem 2.17 and Lemma 4.9,

that the equation (1.2) has a unique (µ,ν)-pseudo almost automorphic mild solution.



84 D. Békollè, K. Ezzinbi, S. Fatajou, D.E. Houpa Danga & F. Mbounja B. CUBO
23, 1 (2021)

Acknowledgments

The authors are grateful to the anonymous referee for the careful reading of this paper, for very

helpful suggestions and comments which improved of quality of this article.

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	Introduction
	Preliminaries
	Notation and terminology
	Some useful results on the space functions
	Measure theory results

	Main results of convolution and translation invariance
	Convolution invariance on E(R,X,, )
	Translation invariance and convolution invariance of PAP(R,X,, ) and PAA(R,X,, ) 

	Existence, Uniqueness results and Applications
	Evolution Families and Exponential Dichotomy
	Existence Results
	Existence of (,)-pseudo-almost periodic solutions
	Existence of (,)-pseudo-almost automorphic solutions
	Neutral Systems