CUBO, A Mathematical Journal
Vol. 23, no. 03, pp. 469–487, December 2021

DOI: 10.4067/S0719-06462021000300469

On the periodic solutions for some retarded partial
differential equations by the use of semi-Fredholm
operators

Abdelhai Elazzouzi1

Khalil Ezzinbi1,2

Mohammed Kriche1

1 Département de Mathématiques,

Laboratoire des Sciences de

l’Ingénieur (LSI), Faculté

Polydisciplinaire de Taza,

Université Sidi Mohamed Ben

Abdellah (USMBA) - Fes, BP.

1223, Taza, Morocco.

abdelhai.elazzouzi@usmba.ac.ma

mohammed.kriche@usmba.ac.ma

2Département de Mathématiques,

Faculté des Sciences Semlalia,

Université Cadi Ayyad, B.P.

2390, Marrakesh, Morocco.

ezzinbi@uca.ac.ma

ABSTRACT

The main goal of this work is to examine the periodic dynamic
behavior of some retarded periodic partial differential equations
(PDE). Taking into consideration that the linear part realizes the
Hille-Yosida condition, we discuss the Massera’s problem to this
class of equations. Especially, we use the perturbation theory of
semi-Fredholm operators and the Chow and Hale’s fixed point the-
orem to study the relation between the boundedness and the peri-
odicity of solutions for some inhomogeneous linear retarded PDE.
An example is also given at the end of this work to show the appli-
cability of our theoretical results.

RESUMEN

El principal objetivo de este trabajo es examinar el comportamiento
dinámico periódico de algunas ecuaciones diferenciales parciales
(EDP) periódicas con retardo. Tomando en consideración que la
parte lineal cumple la condición de Hille-Yosida, discutimos el prob-
lema de Massera para esta clase de ecuaciones. Especialmente us-
amos la teoría de perturbaciones de operadores semi-Fredholm y
el teorema de punto fijo de Chow y Hale para estudiar la relación
entre el acotamiento y la periodicidad de soluciones para algunas
EDP no homogéneas lineales con retardo. Se entrega un ejemplo al
final de este trabajo para mostrar la aplicabilidad de los resultados
teóricos.

Keywords and Phrases: Hille-Yosida condition, Integral solutions, Semigroup, Semi-Fredholm operators, Periodic

solution, Poincaré map.

2020 AMS Mathematics Subject Classification: 34K14, 34K30, 35B10, 35B40.

Accepted: 07 October, 2021

Received: 25 March, 2021

©2021 A. Elazzouzi 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/S0719-06462021000300469
https://orcid.org/0000-0002-6952-3112
https://orcid.org/0000-0001-5334-2264


470 A. Elazzouzi, K. Ezzinbi & M. Kriche CUBO
23, 3 (2021)

1 Introduction

Along this work, we investigate the periodicity of solutions of the following inhomogeneous linear

retarded PDE 


d

dt
y(t) = Ay(t) + L(yt) + h(t) for t ≥ 0

y(t) = ψ(t) for − r ≤ t ≤ 0.
(1.1)

We assume that the generator A is not necessarily dense on a Banach space E and realizes the
following Hille-Yosida condition:

(i) there exist M ≥ 1, ω̂ ∈ R such that (ω̂,∞) ⊂ ρ(A) ,

(ii) the operator A satisfies for n ∈ N, λ > ω̂, the following inequality

| (λI − A)−n | ≤
M

(λ − ω̂)n
,

where ρ(A) is the resolvent set of A. The history function yt : [−r,0] → E defined for each
θ ∈ [−r,0] by yt(θ) = y(t + θ), belongs to C([−r,0],E) the space of continuous functions equipped
with the supremum norm. L : C → E is a linear bounded operator and h is a continuous function
from R to E.

Almost periodic and periodic solutions remain the most interesting subject in the qualitative

analysis of PDE in view of their important applications in many real phenomena and fields. Recall

that the concept of almost periodic is more general than the one of periodicity. It was introduced

by Bochner and studied by many authors. For more details on almost periodic function we refer

to [9, 16, 17, 18]. For the periodicity, there is an extensive literature related to this topic, see

for example [10, 11, 25] for more details. Moreover, the choice of a suitable fixed point theorem

is a fundamental tool to establish the periodicity of solutions for different classes of differential

equations, in fact, to find a fixed point of the well known Poincaré map is equivalent to find

the initial data of the periodic solution of the equation. After a long period of research and

development, Massera’s theorem [24] is the first result explaining the relation between the existence

of bounded and periodic solutions for periodic differential equations. In finite dimensional spaces,

several works have been developed on this subject. The authors in [4, 12] proved the periodicity of

solutions when the solutions of periodic system are just bounded and ultimately bounded by the

use of the Horn’s fixed point theorem. Especially, in infinite dimensional spaces, the authors in [8],

used the Poincaré map approach to get the periodicity of solutions for a class of retarded differential

equation, they supposed that the infinitesimal generator satisfies the Hille-Yosida condition and

generates a compact semigroup (T (t))t≥0 by applying an appropriate fixed point theorem. In
[22], the authors proved the periodicity for a nonhomogeneous linear differential equation when

the linear part generates a C0-semigroup on E and they obtained the existence and uniqueness of

periodic solutions for this class of equations.



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On the periodic solutions for some retarded partial differential... 471

The present work would be a continuation and extension of the work [8] for inhomogeneous linear

retarded PDE, we establish the periodicity of solution for Equation (1.1) by using the perturbation

theory of semi-Fredholm operators and without considering the compactness condition of (T (t))t≥0.
To achieve this goal, we suppose that Equation (1.1) admits a bounded solution on the positive

real line and under suitable estimations on the norm of the operator L, we derive periodic solution

of Equation (1.1) from bounded ones on the positive real line by using the perturbation theory of

semi-Fredholm operators and the Chow and Hale’s fixed point theorem.

This work is treated as follows, in Section 2, we give some definitions and results about integral

solutions of Equation (1.1). Moreover, we give some definitions and properties concerning the semi-

Fredholm operators. Section 3 is devoted to prove and introduce some useful estimations on the

integral solutions of Equation (1.1). In Section 4, we discuss the problem of existence of periodic

solutions of Equation (1.1). Finally, we apply our theoretical results to an equation appearing in

physical systems.

2 Preliminary results

In this article, we assume that:

(H0) A satisfies the Hille-Yosida condition.

We consider the following definition and results.

Definition 2.1 ([1]). A continuous function y : [0,+∞) → E is said to be an integral solution of
Equation (1.1) if the following conditions hold:

(i) y : [0,+∞) → E is continuous, such that y0 = ψ,

(ii)
∫ t
0

y(s)ds ∈ D(A) for t ≥ 0,

(iii) y(t) = ψ(0) + A
∫ t
0

y(s)ds +

∫ t
0

(L(ys) + h(s)) ds for t ≥ 0.

We can deduce from the continuity of the integral solution y that y(t) ∈ D(A), for all t ≥ 0.
Moreover ψ(0) ∈ D(A). In the next we define the part A0 of the operator A in D(A) as follows

D(A0) = {y ∈ D(A) : Ay ∈ D(A)},

and

A0y = Ay for y ∈ D(A0).

Lemma 2.2 ([2]). The operator A0 is the infinitesimal generator of a strongly continuous semi-
group denoted by (T0(t))t≥0 on D(A).



472 A. Elazzouzi, K. Ezzinbi & M. Kriche CUBO
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Theorem 2.3 ([1]). Under the assumption (H0). For all ψ ∈ C such that ψ(0) ∈ D(A), Equation
(1.1) has a unique integral solution y(.) on [−r,+∞). Furthermore, y(.) is given by

y(t) = T0(t)ψ(0) + lim
λ→+∞

∫ t
0

T0(t − s)λR(λ,A) (L(ys) + h(s)) ds for t ≥ 0.

Through this work, the integral solutions of Equation (1.1) are called plainly solutions. Let

y(.,ψ,L,h) be the solution of Equation (1.1).

We define C0 the phase space of Equation (1.1) as C0 = {ψ ∈ C : ψ(0) ∈ D(A)}.

Let X(t) be the linear operator defined on C0 for each t ≥ 0, by

X(t)ψ = yt(.,ψ,0,0),

where yt(.,ψ,0,0) is the solution of the following equation


d

dt
y(t) = Ay(t) for t ≥ 0,

y0 = ψ.

Theorem 2.4 ([1]). (X(t))t≥0 is a linear strongly continuous semigroup on C0:

(i) for all t ≥ 0, X(t) is a bounded linear operator on C0 such that X(0) = I and X(t + s) =
X(t)X(s) for all t,s ≥ 0,

(ii) for t ≥ 0 and θ ∈ [−r,0], (X(t))t≥0 satisfies:

[X(t)ϕ](θ) =


 [X(t + θ)ϕ](0), if t + θ ≥ 0,ϕ(t + θ), if t + θ ≤ 0.

(iii) for all ψ ∈ C0 and t ≥ 0, X(t)ψ is a continuous function with values in C0.

Theorem 2.5 ([1]). Under the assumption (H0). The solution Y(t)ψ = yt(.,ψ,L,0) of Equation
(1.1) with h = 0 can be decomposed as follows:

Y(t)ψ = X(t)ψ + Z(t)ψ for t ≥ 0,

where Z(t), is a bounded linear operator defined on C0, by

[Z(t)ψ](θ) =




lim
λ→+∞

∫ t+θ
0

T0(t + θ − s)λR(λ,A)L(ys(ψ))ds t + θ ≥ 0,

0 t + θ ≤ 0.
(2.1)

for each t ≥ 0.

To discuss the existence of periodic solutions, we use the theory of semi-Fredholm operators. We

consider some definitions and propositions which are taken from [21].



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On the periodic solutions for some retarded partial differential... 473

Definition 2.6. Let E, F be two Banach spaces. A bounded linear operator F from E to F, denoted

by F ∈ L(E,F), is said to be semi-Fredholm from E to F if

(i) dim ker(F) < ∞, where ker(F) is the null space.

(ii) Im(F) the range of F is closed in F.

We designate by Φ+(E,F) the set of all semi-Fredholm operators and Φ+(E) = Φ+(E,E). Now

we recall some well known theorems for the closed range operators. Let F ∈ B(E,F). Then the
quotient space E/ker(F) is a Banach space equipped with the norm

|[x]| = inf{|x + m| : m ∈ ker(F)},

where

[x] = x + ker(F) := {x + m : m ∈ ker(F)}.

Furthermore, if dim ker(F) < ∞, then there exists a closed subspace M of E such that

E = ker(F) ⊕ M.

Theorem 2.7 ([21]). Let F be a bounded linear operator in E. Then, Im(F) is closed if and only

if there exists a constant c̃ such that

|[x]| ≤ c̃∥Fx∥ for all x ∈ E.

Theorem 2.8 ([21]). Let F be a bounded linear operator in E such that dim ker(F) < ∞. Then,
the following assertions are equivalent.

(i) F ∈ Φ+(E).

(ii) There exists a constant c̃ such that

|[x]| ≤ c̃∥Fx∥ for all x ∈ E.

(iii) There exists a constant b such that

∥(I − P)x∥ ≤ b∥Fx∥ for all x ∈ E,

where P is the projection operator onto ker(F) along M.

We present now a result for bounded perturbation of Semi-Fredholm operators.

Theorem 2.9 ([21]). Let F be an operator in Φ+(E,F). If S ∈ L(E) satisfying

∥S∥ <
1

2b
,

where b is the constant given in Theorem 2.8. Then,

F + S ∈ Φ+(E,F) with dim ker(F + S) ≤ dim ker(F).



474 A. Elazzouzi, K. Ezzinbi & M. Kriche CUBO
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Now, we need to introduce some well known definitions and results about the spectral theory. Let

J be a linear bounded operator on F, we define the measure of Kuratowski of noncompactness of
the operator J as follows

|J |α = inf{ϵ > 0 : α(J (B)) ≤ ϵα(B), for every bounded set B ⊂ F},

where α(.) is the measure of Kuratowski of noncompactness of bounded sets B ⊂ F defined by

α(J) = inf{ϵ > 0 : B has a finite cover of ball of diameter < ϵ}.

The essential radius ress(J ) is characterized by the following Nussbaum Formula introduced in
[19]:

ress(J ) = lim
n→+∞

[|J n|α]1/n.

Moreover, if J is bounded and ress(J ) < 1, then I − J ∈ Φ+(F).

Let us define the essential growth bound of a strongly continuous semigroup S := (S(t))t≥0 on a
Banach space F as

ωess(S) := lim
t→+∞

1

t
log |S(t)|α.

It is well know that

ress(S(t)) = exp (tωess(S)) , t > 0.

3 Several estimates

Before discussing the periodicity of solution of Equation (1.1), we need some preparatory estimates.

Proposition 3.1. Suppose that |T0(t)| ≤ M0 eω0t for t ≥ 0. Then

∥Z(t)∥C ≤ M0 eω
+
0 t(eM0|L|M t − 1) for t ≥ 0,

where ω+0 = max{ω0,0}.

To prove the above Proposition, we need to introduce the following Lemma.

Lemma 3.2. Let |T0(t)| ≤ M0eω0t for t ≥ 0. Then, the solution of Equation (1.1) in the case
where h = 0 is estimated as

|Y(t)| ≤ M0 e(ω
+
0 +M0|L|M)t.

Proof. For t ≥ 0, θ ∈ [−r,0], one has

∥Y(t)ψ∥C = sup
θ∈[−r,0]

|y(t + θ,ψ)| = sup
ξ∈[t−r,t]

|y(ξ,ψ)|.



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On the periodic solutions for some retarded partial differential... 475

Then for t ≥ r,

sup
ξ∈[t−r,t]

|y(ξ,ψ)| ≤ sup
ξ∈[t−r,t]

|T0(ξ)ψ(0)| + sup
ξ∈[t−r,t]

∣∣∣∣∣ limλ→+∞ λ
∫ ξ
0

T0(ξ − s)R(λ,A)L(Y(s)ψ)ds

∣∣∣∣∣
≤ M0eω

+
0 t∥ψ∥C + M0|L|M

(∫ t
0

eω
+
0 (t−s) ∥Y(s)ψ∥C ds

)
.

For t < r,

sup
ξ∈[t−r,t]

|y(ξ,ψ)| = max

{
sup

ξ∈[t−r,0]
|y(ξ,ψ)|; sup

ξ∈[0,t]
|y(ξ,ψ)|

}

≤ max

{
∥ψ∥C; sup

ξ∈[0,t]
|y(ξ,ψ)|

}
,

and

sup
ξ∈[0,t]

|y(ξ,ψ)| ≤ M0eω
+
0 t∥ψ∥C + M0|L|M

(∫ t
0

eω
+
0 (t−s) ∥Y(s)ψ∥Cds

)
.

Finally, we obtain that

∥Y(t)ψ∥C ≤ M0eω
+
0 t |ψ|C + M0 |L|M

∫ t
0

eω
+
0 (t−s) ∥Y(s)ψ∥C ds.

Hence,

∥e−ω
+
0 t Y(t)ψ∥C ≤ M0 ∥ψ∥C + M0 |L|M

∫ t
0

e−ω
+
0 s ∥Y(s)ψ∥C ds.

Gronwall’s inequality implies that

∥e−ω
+
0 t Y(t)ψ∥C ≤ M0 eM0 |L| M t ∥ψ∥C,

and then

∥Y(t)ψ∥C ≤ M0 e(ω
+
0 +M0 |L| M) t ∥ψ∥C

Proof of Proposition 3.1. Let t ≥ 0, t + θ ≥ 0. Then

∥Z(t)ψ∥C = sup
θ∈[−r,0]

|(Z(t)ψ)(θ)|

≤ M0|L|M
(∫ t

0

eω
+
0 (t−s)∥Y(s)ψ∥C ds

)
.

From Lemma 3.2, we have

∥Z(t)ψ∥C ≤ M20 |L|M e
ω

+
0 t

(∫ t
0

eM0|L| M s ds

)
∥ψ∥C

≤ M0 eω
+
0 t(eM0|L| M t − 1) ∥ψ∥C.

This implies our inequality.

Proposition 3.3. A function ϕ ∈ ker(I − X(ω)) if and only if ϕ(0) ∈ ker(I − T0(ω)), furthermore

dim ker(I − X(ω)) = dim ker(I − T0(ω)).



476 A. Elazzouzi, K. Ezzinbi & M. Kriche CUBO
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Proof. Let ϕ ∈ ker(I − X(ω)). Then,

X(ω)ϕ = ϕ and (X(ω)ϕ)(θ) = ϕ(θ) for θ ∈ [−r,0].

Since

(X(ω)ϕ)(0) = T0(ω)ϕ(0),

then

ϕ(0) = T0(ω)ϕ(0),

and hence

ϕ(0) ∈ ker(I − T0(ω)).

Conversely, let x ∈ ker(I − T0(ω)) and ϕn(θ) = T0(nω + θ)x for n ≥ [ rω ] + 1, where [.] denotes
the integer part. Then,

T0(t + ω)x = T0(t)T0(ω)x = T0(t)x for t ≥ 0,

and ϕn(θ) is independent of the integer n and then

ϕn(θ) = T0(ω + θ)x = ϕ(θ) and ϕ(0) = x.

If ω + θ ≥ 0, then
(X(ω)ϕ)(θ) = T0(ω + θ)ϕ(0) = T0(ω + θ)x = ϕ(θ).

If ω + θ ≤ 0, then

(X(ω)ϕ)(θ) = ϕ(ω + θ) = ϕn(ω + θ)

= T0(θ + ω + nω)x = T0(nω + θ)x

= ϕn(θ) = ϕ(θ),

hence,

X(ω)ϕ = ϕ,

which implies that ϕ ∈ ker(I − X(ω)). Moreover, ker(I − T0(ω)) is mapped bijectively onto the
space ker(I − X(ω)). Therefore,

dim ker(I − X(ω)) = dim ker(I − T0(ω)).

Let us define the constant mω by

mω = sup
0≤t≤ω

|T0(t)|.

As it is shown in [22], the proof is omitted here, if I − T0(ω) is semi-Fredholm on D(A), then, the
operator defined by

SM := I − T0(ω) : M → Im(I − T0(ω)),



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On the periodic solutions for some retarded partial differential... 477

is bijective, such that M is a subset of E and D(A) is decomposed as

D(A) = ker(I − T0(ω)) ⊕ M.

Let S−1M be the inverse operator of SM and let k ∈ N
∗ such that

(k − 1)ω < r ≤ kω.

We set

Ik = [−r,−(k − 1)ω) and Ip = [−pω,−(p − 1)ω) for p = 1,2, . . . ,k − 1 with k ≥ 2.

Let G be a linear operator defined from D(G) to C0 by

(Gϕ)(θ) =
p−1∑
j=0

ϕ(θ + jω) + T0(θ + pω)S−1M ϕ(0) for θ ∈ Ip,

with

D(G) = {ϕ ∈ C0 : ϕ(0) ∈ Im(I − T0(ω))}.

Clearly, for θ ∈ Ip,p = 1,2, . . . ,k,

∥Gϕ∥C = sup
θ∈[−r,0]

|(Gϕ)(θ)| ≤
p−1∑
j=0

∥ϕ∥C + sup
s∈[0,ω]

|T0(s)||S−1M ||ϕ(0)|.

Then

∥Gϕ∥C ≤
(
k + mω|S−1M |

)
∥ϕ∥C. (3.1)

Consequently, we have the following Theorem.

Theorem 3.4. I − T0(ω) is semi-Fredholm on D(A) if and only if I − X(ω) is semi-Fredholm on
C0 .

To prove Theorem 3.4, we need the following Lemma

Lemma 3.5 ([14]).

Im(I − X(ω)) = D(G).

Proof of Theorem 3.4. Suppose that Im(I − T0(ω)) is closed, let ϕn ∈ D(G) such that ϕn → ϕ as
n → ∞. Then

ϕn(0) ∈ Im(I − T0(ω)) and ϕn(0) → ϕ(0) ∈ Im(I − T0(ω)),

which implies that ϕ ∈ D(G) and D(G) is closed. Lemma 3.5 implies that Im(I − X(ω)) is closed.
Now, we suppose that Im(I − X(ω)) is closed and xn ∈ Im(I − T0(ω)) such that xn → x as
n → ∞. Let ϕn,ϕ ∈ C0 be such that ϕn(θ) = xn and ϕ(θ) = x. It is clear that ϕn → ϕ as
n → ∞ and by Lemma 3.5 we have that (ϕn) ∈ Im(I − X(ω)). Then ϕ ∈ Im(I − X(ω)) and
ϕ(0) = x ∈ Im(I − T0(ω)). Consequently, Im(I − T0(ω)) is closed. Therefore, by the use of
Proposition 3.3 we obtain the desired result.



478 A. Elazzouzi, K. Ezzinbi & M. Kriche CUBO
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In the nondensely defined case, we can prove the following result as in [22], the proof is omitted

here.

Theorem 3.6. Suppose that I−T0(ω) is semi-Fredholm on D(A) such that dim ker(I−T0(ω)) = n.
If

|Z(ω)| <
1

2c̃(1 +
√
n)
.

Then,

I − Y(ω) ∈ Φ+(C0) and dim ker(I − Y(ω)) ≤ n.

Proposition 3.7. Suppose that I − T0(ω) is semi-Fredholm on D(A). If |L| satisfies

|L| <
log

(
e−ω

+
0 ω

2M0c̃(1 +
√
n)

+ 1

)
M0Mω

. (3.2)

Then,

I − Y(ω) ∈ Φ+(C0) and dim ker(I − Y(ω)) ≤ n.

Proof. By the inequality (3.2), it follows that

M0 e
ω

+
0 ω(eM0M|L| ω − 1) <

1

2c̃(1 +
√
n)
,

and by Proposition 3.1, one has

|Z(ω)| <
1

2c̃(1 +
√
n)
.

Theorem 3.6 gives that

I − Y(ω) ∈ Φ+(C0) and dim ker(I − Y(ω)) ≤ n.

4 Periodic solutions for Equation (1.1)

To discuss the existence of periodic solutions of Equation (1.1), we introduce the following fixed

point Theorem for a linear affine map T defined from E to E by

Tu = Tu + v for u ∈ E,

where T ∈ B(E) and v ∈ E is fixed. Let FT be the set of all fixed points of the map T .

Theorem 4.1 ([5]). Let T be a linear affine map on a Banach space E such that the range Im(I−T)
is closed. If there is an u0 ∈ E such that {Tku0,k ∈ N} is bounded in E, then FT ̸= ∅.

If there exists some v ∈ FT , then
FT = v + ker(I − T).



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On the periodic solutions for some retarded partial differential... 479

dim FT is defined as
dim FT = dim ker(I − T).

If I − T ∈ Φ+(X). Then, Theorem 4.1 is refined as follows

Theorem 4.2 ([22]). Let T be a linear affine map on a Banach space E. If I − T ∈ Φ+(E) and if
there exists an u0 ∈ E such that {Tku0,k ∈ N} is bounded, then FT ̸= ∅ and dim FT is finite.

Through the rest of this work, we suppose that

(H1) h is an ω-periodic function.

Furthermore, by property (R) we mean the following equivalence:

Equation (1.1) has an ω−periodic solution if and only if it has a bounded one on the positive real
line. Then, we have the following result.

Theorem 4.3. Under assumptions (H0) and (H1). If I − T0(ω) is semi-Fredholm on D(A) and
if the operator L satisfies the following estimate

|L| <
log

(
e−ω

+
0 ω

2M0c̃(1 +
√
n)

+ 1

)
M0Mω

,

where c̃ and n are the constants given in Theorem 3.6. Then, Equation (1.1) satisfies the property

(R).

Proof. Let y(.,ψ,h) be the solution of Equation (1.1). We introduce the Poincaré map P defined
from C0 to C0 as follows

Pω(ψ) = yω(.,ψ,h),

Then,

Pωψ = yω(.,ψ,0) + yω(.,0,h),

and hence Pω is an affine map such that

Pωψ = Pψ + φ,

with

Pψ = yω(.,ψ,0) and φ = yω(.,0,h).

According to the second section, P is decomposed as

P = X(ω) + Z(ω).

Moreover, Proposition 3.7 gives that

I − P ∈ Φ+(C0).



480 A. Elazzouzi, K. Ezzinbi & M. Kriche CUBO
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Now, let y(.,ψ,h) denote the bounded solution of Equation (1.1) on R+. Thus, for each n ∈ N, we
have

Pnωψ = ynω(.,ψ,h),

and then (Pnωψ)n≥0 is a bounded sequence in C0. All conditions of Theorem 4.2 are satisfied and
then FPω ̸= ∅, which yields an ω-periodic solution of Equation (1.1).

Corollary 4.4. Under assumptions (H0) and (H1). If I − T0(ω) is semi-Fredholm in D(A) and
if |L| satisfies the following inequality

|L| <
log

(
e−ω

+
0 ω

2M0
(
k + mω|S−1M |

)
(1 +

√
n)

+ 1

)
M0Mω

.

Then, Equation (1.1) satisfies the property (R).

To establish the proof, we need the following Lemma.

Lemma 4.5 ([14]). Suppose that I − T0(ω) is semi-Fredholm on D(A). If there exists a constant
c̃ > 0 such that

∥Gψ∥C ≤ c̃∥ψ∥C for all ψ ∈ D(G).

Then,

|[φ]| ≤ c̃∥(I − X(ω))φ∥C for all φ ∈ C0.

Proof of Corollary 4.4: Since,

|L| <
log

(
e−ω

+
0 ω

2M0
(
k + mω|S−1M |

)
(1 +

√
n)

+ 1

)
M0Mω

.

it follows that,

(k + mω|S−1M |)(e
M0Mω|L| − 1) <

e−ω
+
0 ω

2M0 (1 +
√
n)
.

Lemma 4.5 and estimation (3.1) implies that

c̃ ≤ k + mω|S−1M |,

and then

c̃(eM0Mω|L| − 1) <
e−ω

+
0 ω

2M0 (1 +
√
n)
.

Finally

|L| <
log

(
e−ω

+
0 ω

2M0c̃(1 +
√
n)

+ 1

)
M0Mω

.

Now, Theorem 4.3 shows that Equation (1.1) satisfies the property (R).



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On the periodic solutions for some retarded partial differential... 481

In the particular case where the semigroup (T0(t))t≥0 is exponentially stable, we have the following
Theorem.

Theorem 4.6. Under assumptions (H0) and (H1). If the semigroup (T0(t))t≥0 is exponentially
stable and if the operator L satisfies the following inequality

|L| <
log

(
1

2M0
(
k + mω|S−1M |

) + 1
)

M0Mω
.

Then, Equation (1.1) satisfies the property (R).

Proof. From the exponential stability of (T0(t))t≥0, we have

ωess(T0) = lim
t→+∞

1

t
log |T0(t)|α ≤ lim

t→+∞

1

t
log |T0(t)| = −ω0 < 0.

Consequently,

ress(T0(ω)) = exp (ωωess(T0)) < 1.

Which implies that Im(I − T0(ω)) is closed. On the other hand, one has

|T0(ω)n| = |T0(nω)| ≤ M0e−ω0nω

and

|T0(nω)|
1
n ≤ M

1
n
0 e

−ω0ω,

which implies that the spectral radius is estimated as

r(T0(ω)) = lim
n→+∞

|T0(ω)n|
1
n ≤ e−ω0ω lim

n→+∞
M

1
n
0 < lim

n→+∞
M

1
n
0 < 1.

Consequently

1 /∈ σ(T0(ω)) and n = dim ker(I − T0(ω)) = 0.

All conditions of Corollary 4.4 are satisfied with n = 0. Then, Equation (1.1) satisfies the property

(R).

5 Application

In order to apply our theoretical results, we consider the following delayed partial differential

equation:


∂

∂t
y(t,ζ) =

∂2

∂ζ2
y(t,ζ) − ay(t,ζ) + by(t − r,ζ) + g(t,ζ) for t ∈ R+ and ζ ∈ R,

y(θ,ζ) = ψ0(θ,ζ) for θ ∈ [−r,0] and ζ ∈ R,
(5.1)

where a and b are positive constants, g : R × R → R and ϕ : [−r,0] × R → R are continuous
functions where ϕ(θ,ζ) has a finite limit at ±∞.



482 A. Elazzouzi, K. Ezzinbi & M. Kriche CUBO
23, 3 (2021)

Note that Equation (5.1) can be written in the form of Equation (1.1). In fact: we set R :=

[−∞,+∞] and we say that z ∈ Ck(R) if z ∈ Ck(R) and all derivatives of z up to the order k have
finite limits at ±∞. Then, the space of continuous functions on R, denoted by E = C(R), endowed
with the norm

∥z∥∞ = sup
−∞<ζ<+∞

|z(ζ)|

becomes a Banach space. we consider the linear operator ∆ defined from D(∆) ⊂ E to E by


D(∆) =

{
z ∈ C2

(
R
)
: lim
ζ→±∞

z(ζ) = 0

}
,

∆z = z′′.

Then, we have

Lemma 5.1 ([7]).

(0,+∞) ⊂ ρ(∆)

and for each λ > 0 ∣∣∣(λI − ∆)−1∣∣∣ ≤ 1
λ
.

Clearly

D(∆) =

{
z ∈ C

(
R
)
: lim
ζ→±∞

z(ζ) = 0

}
.

We write the part ∆0 of ∆ in D(∆) as


D(∆0) =

{
z ∈ C2

(
R
)
: lim
ζ→±∞

z(ζ) = lim
ζ→±∞

z′′(ζ) = 0

}
,

∆0z = z
′′.

Lemma 5.2 ([7]). ∆0 is the infinitesimal generator of a strongly continuous semigroup (T∆0(t))t≥0
on D(∆). Furthermore,

|T∆0(t)| ≤ 1 for t ≥ 0.

Let A : D(A) ⊂ E → E defined by:


D(A) =
{
z ∈ C2

(
R
)
: lim
ζ→±∞

z(ζ) = 0

}
,

Az = z′′ − az.

By Lemma 5.1, it is clear that

Lemma 5.3.

(−a,+∞) ⊂ ρ(A)

and for each λ > −a ∣∣∣(λI − A)−1∣∣∣ ≤ 1
λ + a

.



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On the periodic solutions for some retarded partial differential... 483

Lemma 5.3 guarantees that the assumption (H0) is satisfied with ω̂ = −a and M = 1. Moreover,

D(A) =
{
z ∈ C

(
R
)
: lim
ζ→±∞

z(ζ) = 0

}
⊊ E.

Moreover, we write the part A0 of the linear operator A in D(A) as:


D(A0) =
{
z ∈ C2

(
R
)
: lim
ζ→±∞

z(ζ) = 0 = lim
ζ→±∞

z′′(ζ) = 0

}
,

A0z = z′′ − az.

Lemma 5.4. A0 is the infinitesimal generator of an exponentially stable continuous semigroup
(T0(t))t≥0 on D(A). Moreover, for t ≥ 0, we have

|T0(t)| ≤ e−at.

Consider the following notations:
 y(t)(ζ) = y(t,ζ) for t ∈ R

+, ζ ∈ R,
ψ(θ)(ζ) = ψ0(θ,ζ) for θ ∈ [−r,0], ζ ∈ R,

and define the function L : C → E as follows

L(ϕ)(ζ) = bϕ(−r)(ζ) for ζ ∈ R and ϕ ∈ C.

h : R −→ E is defined by

h(t)(ζ) = g(t,ζ) for t ∈ R and ζ ∈ R.

Clearly, L is a linear bounded operator from C to E. Then, Equation (5.1) can be written in E as
follows 


d

dt
y(t) = Ay(t) + L(yt) + h(t) for t ≥ 0,

y0 = ψ ∈ C.
(5.2)

We suppose that lim
ζ→±∞

ψ0(0,ζ) = 0, then Equation (5.2) has a unique integral solution y on

[−r,+∞).

To get the periodicity of solutions of Equation (5.2), we suppose that

(H2) b < a.

Let ρ = 1 +
|h|∞
a − b

where |h|∞ = sup
0≤t≤ω

|h(t)|. Then, we have

Lemma 5.5. Under assumption (H2). For every ψ ∈ C such that ∥ψ∥C < ρ, the solution of
Equation (5.2) is bounded by ρ on R+.



484 A. Elazzouzi, K. Ezzinbi & M. Kriche CUBO
23, 3 (2021)

Proof. We proceed by contradiction. Let

t∗ = inf{t > 0 : |y(t,ψ)| > ρ}.

From the continuity of y, one has

|y(t∗,ψ)| = ρ,

and there is α > 0, with

|y(t,ψ)| > ρ for each t ∈ (t∗, t∗ + α).

Applying the variation-of-constants formula for Equation (5.2) with the initial value ψ,

y(t) = T0(t)ψ(0) + lim
λ→+∞

λ

∫ t
0

T0(t − s)R(λ,A) (L(ys) + h(s)) ds.

Then, for t ≥ 0

|y(t∗,ψ)| ≤ |T0(t∗)| |ψ(0)| +
∫ t∗
0

|T0(t∗ − s)| (|L(ys)| + |h(s)|) ds.

Since for 0 < s < t∗, it follows that −r ≤ s − r ≤ t∗ − r < t∗ and then

|L(ys)| = b|y(s − r)| ≤ bρ,

hence

|y(t∗,ψ)| ≤ ρe−at∗ + (bρ + |h|∞)
∫ t∗
0

e−a(t∗−s) ds

≤ ρe−at∗ +
(1 − e−at∗)

a
(bρ + |h|∞) .

Consequently,

|y(t∗,ψ)| ≤ ρe−at∗ + (bρ + (a − b)(ρ − 1))
(1 − e−at∗)

a

≤ ρe−at∗ +
(
ρ − 1 +

b

a

)
(1 − e−at∗)

≤ ρe−at∗ + ρ(1 − e−at∗)

≤ ρ,

which contradicts the definition of t∗, and we deduce that

|y(t,ψ)| ≤ ρ for t ≥ 0.

To discuss the periodicity of solutions of Equation (5.2), we assume that:

(H3) h is an ω-periodic function in t.

Theorem 5.6. Suppose that (H2) and (H3) hold true. If

|L| < ω−1 log
(
(1 − e−aω)(1 + 2k) + 2

2 + 2k(1 − e−aω)

)
,

then, Equation (5.2) has an ω-periodic solution.



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On the periodic solutions for some retarded partial differential... 485

Proof. Let mω be the constant defined by

mω = sup
0≤t≤ω

|T0(t)|.

Then

mω ≤ sup
0≤t≤ω

e−at = 1.

Moreover, since |T0(ω)| < 1, one has

|S−1M | = |(I − T0(ω))
−1| ≤

1

I − |T0(ω)|

≤
1

1 − e−aω
.

Thus,

k + mω|S−1M | ≤ k +
1

1 − e−aω
,

and

|L| < ω−1 log
(
(1 − e−aω)(1 + 2k) + 2

2 + 2k(1 − e−aω)

)
< ω−1 log

(
1

2
(
k + mω|S−1M |

) + 1
)
.

All condition of Theorem 4.6 are satisfied. Then, Lemma 5.5 implies that Equation (5.2) has an

ω−periodic solution.

Acknowledgement

The authors would like to thank the anonymous referees for their constructive comments and

valuable suggestions, which are helpful to improve the quality of this paper.



486 A. Elazzouzi, K. Ezzinbi & M. Kriche CUBO
23, 3 (2021)

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	Introduction
	Preliminary results
	Several estimates
	Periodic solutions for Equation (1.1)
	Application