()


CUBO A Mathematical Journal
Vol.16, No¯ 03, (55–65). October 2014

S-asymptotically ω-periodic solution for a nonlinear
differential equation with piecewise constant argument in a

Banach space

William Dimbour & Jean-Claude Mado

Laboratoire C.E.R.E.G.M.I.A.

Université des Antilles et de la Guyane,

Campus Fouillole 97159 Pointe-à-Pitre Guadeloupe (FWI)

William.Dimbour@univ-ag.fr, Jean-Claude.Mado@univ-ag.fr

ABSTRACT

In this paper, we give some sufficient conditions for the existence and uniqueness of

S-asymptotically ω-periodic (mild) solutions for a differential equation with piecewise

constant argument, when ω is an integer. An example is also given in order to illustrate

the result.

RESUMEN

En este art́ıculo entregamos algunas condiciones suficientes para la existencia y uni-

cidad de las soluciones mild ω-periódicas S-asintóticas para una ecuación diferencial

semilineal con argumento constante por tramos en un espacio de Banach cuando ω es

un entero. Luego, entregamos ejemplos para ilustrar nuestros resultados.

Keywords and Phrases: S-asymptotically ω-periodic function, differential equations with piece-

wise constant argument, semigroup

2010 AMS Mathematics Subject Classification: 34K05; 34A12; 34A40.



56 William Dimbour & Jean-Claude Mado CUBO
16, 3 (2014)

1 Introduction

Let (X, ||.||) be a Banach space. This work is concerned with the existence of S-asymptotically

ω-periodic solutions to the differential equations with piecewise constant argument of the form

(2)

{
x′(t) = Ax(t) + A0x([t]) + g(t,x(t))

x(0) = c0

where A is the infitesimal generator of an exponentially stable C0-semigroup acting on X, [·]

is the largest interger function and g : R+ × X → X is an appropriate function that will be defined
later.

They are a some papers dealing wiht S-asymptotically ω-periodic functions. Qualitative properties

of such functions are discussed for instance in [1] and [5]. In [5], a new composition theorem for such

functions is also presented. In [7], Lizama and N’Guérékata created a chart establishing a general

relationship between S-asymptotically ω-periodic functions and various subspaces of BC(R,X).

[1], [3], [4],[5], [6],[7] study the existence of S-asymptotically ω-periodic solutions of diffenretial

equations in finite as well infinite dimensional spaces.

They are also some papers dealing with the existence of almost automorphic solutions for

differential equation with piecewise constant argument. In [8], Nguyen Van Minh and Tran Tat Dat

give sufficient spectral conditions for the almost automorphy of bounded solutions to differential

equations wtih piecewise constant argument of the form

x′(t) = Ax(t) + f(t),t ∈ R,

where A is a bounded linear operator in X and f in an X-valued almost automorphic function.

In [2], Dimbour generalize the work of Nguyen Van Minh and Tran Tat Dat, giving also sufficient

spectral conditions for the almost automorphy of bounded solutions to differential equations wtih

piecewise constant argument of the form

x′(t) = A(t)x(t) + f(t),t ∈ R,

where A(t) is an almost automorphy operator and f in an X-valued almost automorphic function.

Following this work, we study in this paper S-asymptotically ω-periodic solutions of (2). We

first study the linear system associated to (2). Then using the Banach’s theorem, we showed the

existence of S-asymptotically ω-periodic solutions for the following equation

(1)

{
x′(t) = Ax(t) + A0x([t]) + f(t)

x(0) = c0

The rest of the paper is organise as follows. In section 2, we recall some results on S-asymptotically

ω-periodic functions. In section 3, first of all considering the c0 semigroup theory ([9]), we define

a mild solution of (1). We give some sufficient conditions for the existence and uniqueness of

S-asymptotically ω-periodic solutions of (1) and (2). These results are obtained by mean of the

Banach fixed point principle, when ω is an integer. In the section 4, we give an example.



CUBO
16, 3 (2014)

S-asymptotically ω-periodic solution for a nonlinear differential . . . 57

2 PRELIMINARIES

Let X be a Banach space. BC(R+,X) denotes the space of the continuous bounded functions from

R
+ into X; endowed with the norm ||f||∞ := supt≥0 ||f(t)||, it is a Banach space. C0(R

+,X) denotes

the space of the continuous functions from R into X such that lim
t→∞

f(t) = 0; it is a Banach sub-

space of BC(R+,X). When we fix a positive number ω, Pω(X) denotes the space of all continuous

ω-periodic functions from R+ into X; it is a Banach subspace of BC(R+,X) under the sup norm.

Definition 1. Let f ∈ BC(R+,X) and ω > 0. We say that f is asymptotically ω-periodic if

f = g + h where g ∈ Pω(X) and h ∈ C0(R
+,X).

We denote by AP(X) the set of all asymptotically ω-periodic functions from R+ to X. It is

a Banach space under the sup norm.

Definition 2. A function f ∈ BC(R+,X) is called S-asymptotically ω-periodic if there exists ω

such that lim
t→∞

(f(t + ω) − f(t)) = 0. In this case we say that ω is an asymptotic period of f and

that f is S-asymptotically ω-periodic.

We will denote by SAPω(X), the set of all S-asymptotically ω-periodic functions from R
+ to

X. Then we have

APω(X) ⊂ SAPω(X).

The inclusion is strict. Indeed consider the function f : R+ → c0 where c0 = {x = (xn)n∈N :
lim

n→∞
xn = 0} equipped with the norm ||x|| = supn∈N |x(n)|, and (f(t) =

2nt
2

t2+n2
)n∈N. Then

f ∈ SAPω(X) but f /∈ APω(X)(see [5] Example 3.1).

The following result is due to Henriquez-Pierri-Tàboas; Proposition 3.5 in [5].

Theorem 1. Endowed with the norm || · ||∞, SAPω(X) is a Banach space.

Corollary 1. (see [1], Corollary 3.10 p.5) Let X and Y be two Banach spaces, and let A ∈

L(X,Y). Then when f ∈ SAPω(X), we have Af := [t → Af(t)] ∈ SAPω(Y).

For the sequel we consider asymptotically ω-periodic functions with parameters.

Definition 3. (see [5]) A continuous function g : [0,∞[×X → X is said to be uniformly S-
asymptotically ω-periodic on bounded sets if for every bounded set K ⊂ X, the set {f(t,x) : t ≥

0,x ∈ K} is bounded and limt→∞(f(t,x) − f(t + ω,x)) = 0 uniformly on x ∈ K.

Definition 4. (see [5]) A continuous function g : [0,∞[×X → X is said to be asymptotically
uniformly continuous on bounded sets if for every ǫ > 0 and every bounded set K ⊂ X, there exist

Lǫ,K > 0 and δǫ,K > 0 such that ||f(t,x) − f(t,y)|| < ǫ for all t ≥ Lǫ,K and all x,y ∈ K with

||x − y|| < δǫ,K.



58 William Dimbour & Jean-Claude Mado CUBO
16, 3 (2014)

Theorem 2. (see [5]) Let g : [0,∞[×X → X be a function which uniformly S-asymptotically
ω-periodic on bounded sets and asymptotically uniformly continuous on bounded sets. Let u :

[0,∞[→ X be S-asymptotically ω-periodic function. Then the Nemytskii function φ(.) := f(.,u(.))
is S-asymptotically ω-periodic function.

3 Main result

3.1 The linear case

Definition 5. A solution of Eq.(1) on R+ is a function x(t) that satisfies the conditions:

1-x(t) is continuous on R+.

2-The derivative x′(t) exists at each point t ∈ R+, with possible exception of the points [t] ∈ R+

where one-sided derivatives exists.

3-Eq.(1) is satisfied on each interval [n,n + 1[ with n ∈ N.

Let T(t) be the C0 semigroup generated by A and x a solution of (1). We assume that

f ∈ L1(R+,X). Then the function g defined by g(s) = T(t − s)x(s) is differentiable for s < t.

dg(s)

ds
= −AT(t − s)x(s) + T(t − s)x′(s)

= −AT(t−s)x(s)+T(t−s)Ax(s)+ T(t−s)A0x([s])+T(t−s)f(s)

(3) = T(t − s)A0x([s]) + T(t − s)f(s)

Since f ∈ L1(R,X), T(t − s)f(s) is integrable on [0,t] with t ∈ R+. The function x([s]) is

a step function. Therefore x([s]) is integrable on [0,t] with t ∈ R+. Integrating (3) on [0,t], we

obtain that

x(t) − T(t)x(0) =

∫t

0

T(t − s)A0x([s])ds +

∫t

0

T(t − s)f(s)ds.

Therefore, we define

Definition 6. Let T(t) be the C0 semigroup generated by A and f ∈ L
1(R+,X). The function

x ∈ C(R+,X) given by

x(t) = T(t)c0 +

∫t

0

T(t − s)A0x([s])ds +

∫t

0

T(t − s)f(s)ds

is the mild solution of the equation (1).

Now we make the following hypothesis.



CUBO
16, 3 (2014)

S-asymptotically ω-periodic solution for a nonlinear differential . . . 59

(H.1) The operator A is the infinitesimal generator of an exponentially stable semigroup

(T(t))t≥0 such that there exist constants M > 0 and δ > 0 with

||T(t)||B(X) ≤ Me
−δt, ∀t ≥ 0.

Lemma 1. We assume that the hypothesis (H.1) is satisfied. Then the function L defined by

L(t) = T(t)x(0)

belongs to SAPω(X).

Proof.

||L(t + ω) − L(t)|| = ||T(t + ω)x(0) − T(t)x(0)||

≤ ||T(t + ω)x(0)|| + ||T(t)x(0)||

≤ Me−δ(t+ω) + Me−δt

Since δ > 0, we deduce that

lim
t→∞

||L(t + ω) − L(t)|| = 0.

Then L ∈ SAPω(X).✷

Lemma 2. We assume that the hypothesis (H.1) is satisfied. We assume also that A0 is a linear

bounded operator and ω ∈ N. We define the nonlinear operator ∧1 by: for each φ ∈ SAPω(X)

(∧1φ)(t) =

∫t

0

T(t − s)A0φ([s])ds.

Then the operator ∧1 maps SAPω(X) into itself.

Proof. We put v(t) =
∫t
0
T(t − s)A0φ([s])ds. For t ≥ 0, we have

v(t + ω) − v(t) =

∫t+ω

0

T(t + ω − s)A0φ([s])ds −

∫t

0

T(t − s)A0φ([s])ds

=

∫ω

0

T(t + ω − s)A0φ([s])ds +

∫t+ω

ω

T(t + ω − s)A0φ([s])ds

−

∫t

0

T(t − s)A0φ(s)ds.

Then we have

||v(t + ω) − v(t)|| ≤ ||I1(t)|| + ||I2(t)||

where

I1(t) =

∫ω

0

T(t + ω − s)A0φ([s])ds



60 William Dimbour & Jean-Claude Mado CUBO
16, 3 (2014)

and

I2(t) =

∫t+ω

ω

T(t + ω − s)A0φ([s])ds −

∫t

0

T(t − s)A0φ([s])ds.

Observing that

I1(t) = T(t)

∫ω

0

T(ω − s)A0φ([s])ds

and using the fact that
(

T(t)
)

t≥0
is exponentially stable, we deduce that

||I1(t)|| ≤ Me
−δt||v(ω)||.

Therefore lim
t→∞

I1(t) = 0.

Now, show that lim
t→∞

||φ([t + ω]) − φ([t])|| = 0.

We have that lim
t→∞

||φ(t + ω) − φ(t)|| = 0.

Therefore:

∀ǫ > 0, ∃ T0ǫ ∈ R
+,∀ t > T0ǫ ⇒ ||φ(t + ω) − φ(t)|| < ǫ.

We put Tǫ = [T
0
ǫ] + 1. Let ǫ > 0. For t > Tǫ, we observe that [t] ≥ Tǫ because Tǫ is an integer.

We deduce so that

∀ǫ > 0, ∃Tǫ ∈ R
+,∀ t > Tǫ ⇒ ||φ([t] + ω) − φ([t])|| < ǫ.

Since ω is an integer, we observe that

∀ǫ > 0, ∃Tǫ ∈ R
+,∀ t > Tǫ ⇒ ||φ([t + ω]) − φ([t])|| < ǫ.

Let ǫ > 0, we can find Tǫ sufficiently large such that

||φ([t + ω]) − φ([t])|| <
δ

M ||A0 ||
ǫ, for t > Tǫ.

Let’s write

I2(t) =

∫t

0

T(t − s))A0(φ([s + ω]) − φ([s]))ds.

then we obtain

||I2(t)|| ≤ ||

∫Tǫ

0

T(t − s))A0(φ([s + ω]) − x([s]))ds|| + ||

∫t

Tǫ

T(t − s))A0(x([s + ω]) − x([s]))ds||.

Observing that

||

∫Tǫ

0

T(t − s)A0(φ([s + ω]) − φ([s]))ds|| ≤

∫Tǫ

0

||T(t − s)|| ||A0|| ||φ([s + ω]) − φ([s])||

≤

∫Tǫ

0

Me−δ(t−s) ||A0||2||φ||∞ds



CUBO
16, 3 (2014)

S-asymptotically ω-periodic solution for a nonlinear differential . . . 61

≤
M ||A0||2||φ||∞

δ
(e−δ(t−Tǫ) − e−δt)

we deduce that

lim
t→∞

∫Tǫ

0

T(t − s)(Φ([s + ω]) − Φ([s]))ds = 0.

We have also that

||

∫t

Tǫ

T(t − s)A0(Φ([s + ω]) − Φ([s]))ds|| ≤

∫t

Tǫ

Me−δ(t−s) ||A0||
δ

M ||A0||
ǫds

≤ ǫ

∫t

Tǫ

δe−δ(t−s)ds

≤ ǫ(1 − e−δ(t−Tǫ))

≤ ǫ

Therefore lim
t→∞

∫t

Tǫ

T(t − s)A0(Φ([s + ω]) − Φ([s]))ds = 0.

We deduce so that lim
t→∞

I2(t) = 0, this proves that ∧1 ∈ SAPω(X).✷

Theorem 3. We assume that the hypothesis (H.1) is satisfied. Let ω ∈ N. We assume also that

f is a S asymptotically ω-periodic function. Then the equation (1) has a unique S asymptotically

ω-periodic solution if

Θ :=
M

δ
||A0|| < 1.

Proof. Define the nonlinear operator Γ : SAPω(X) 7→ SAPω(X)

(Γu)(t) := L(t) + (∧1u)(t) + ∧2(t)

for every u ∈ SAPω(X), where

(∧1u)(t) =

∫t

0

T(t − s)A0φ([s])ds

and

∧2(t) =

∫t

0

T(t − s)f(s)ds.

We satisfy that the nonlinear operator Γ is well defined.

The lemma 1 show that L(t) is is a S asymptotically ω-periodic. The lemma 2 show that the

operator ∧1 maps SAPω(X) into itself. Then the nonlinear operator Γ maps SAPω(X) into itself.

Since ||L(t)|| ≤ Me−δt, ∀t ≥ 0, we observe that L(t) ∈ C0(R
+,X).

For every φ,ψ ∈ SAPω(X)

||Γ(φ)(t) − Γ(ψ)(t)||



62 William Dimbour & Jean-Claude Mado CUBO
16, 3 (2014)

= ||T(t)c0 +

∫t

0

T(t − s)A0φ([s])ds +

∫t

0

T(t − s)f(s)ds

−T(t)c0 −

∫t

0

T(t − s)A0ψ([s])ds −

∫t

0

T(t − s)f(s)ds||

≤ ||

∫t

0

T(t−s)A0
(

φ([s])−ψ([s])
)

ds||

≤

∫t

0

||T(t−s)|| ||A0|| ||φ([s])−ψ([s])||ds

≤

∫t

0

||T(t−s)|| ||A0|| ||φ−ψ||∞ ds

≤

∫t

0

Me−δ(t−s)ds||A0|| ||φ − ψ||∞

≤
M

δ
||A0|| ||φ − ψ||∞.

Therefore, if Θ < 1, then the equation (1) has a unique S asymptotically ω-periodic solution.

3.2 The nonlinear case

Definition 7. A solution of Eq.(2) on R+ is a function x(t) that satisfies the conditions:

1-x(t) is continuous on R+.

2-The derivative x′(t) exists at each point t ∈ R+, with possible exception of the points [t] ∈ R+

where one-sided derivatives exists.

3-Eq.(2) is satisfied on each interval [n,n + 1[ with n ∈ N.

We now make the following assumption.

(H.2) The function g : R+×X → X,(t,u) → g(t,u) is uniformly S-asymptotically ω-periodic
on bounded sets and asymptotically uniformly continuous on bounded sets. There exist constant

Kg ≥ 0 such that

||g(t,u) − g(t,v)|| ≤ Kg||u − v||

for all t ∈ R+, and ∀u,v ∈ X.

Definition 8. Let T(t) be the C0 semigroup generated by A. The function x ∈ C(R
+,X) given by

x(t) = T(t)c0 +

∫t

0

T(t − s)A0x([s])ds +

∫t

0

T(t − s)g(s,x(s))ds

is the mild solution of the equation (2).



CUBO
16, 3 (2014)

S-asymptotically ω-periodic solution for a nonlinear differential . . . 63

Theorem 4. We assume that the hypothesis (H.1) and (H.2) are satisfied. Let ω ∈ N. Then

the equation (2) has a unique S asymptotically ω-periodic solution if

Θ :=
M

δ

(

||A0|| + Kg
)

< 1.

Proof. Define the nonlinear operator Γ : SAPω(X) 7→ SAPω(X)

(Γu)(t) := L(t) + (∧1u)(t) + ∧2(t)

for every u ∈ SAPω(X), where

(∧1u)(t) =

∫t

0

T(t − s)A0φ([s])ds

and

∧2(t) =

∫t

0

T(t − s)g(s,x(s))ds.

Since the hypothesis (H.2) is satisfied, the nonlinear operator Γ is well defined.

For every φ,ψ ∈ SAPω(X)

||Γ(φ)(t) − Γ(ψ)(t)||

= ||T(t)c0+

∫t

0

T(t−s)A0φ([s])ds+

∫t

0

T(t−s)g(s,φ(s))ds

−T(t)c0 −

∫t

0

T(t−s)A0ψ([s])ds−

∫t

0

T(t−s)g(s,ψ(s))ds||

≤ ||

∫t

0

T(t−s)A0
(

φ([s])−ψ([s])
)

ds||+||

∫t

0

T(t−s)
(

g(s,φ(s))−g(s,ψ(s))
)

ds

≤

∫t

0

||T(t−s)|| ||A0|| ||φ([s])−ψ([s])||ds+

∫t

0

||T(t−s)|| ||g(s,φ(s))−g(s,ψ(s))||ds

≤

∫t

0

||T(t−s)|| ||A0|| ||φ−ψ||∞ ds+

∫t

0

||T(t−s)||Kg ||φ−ψ||∞ ds

≤

∫t

0

Me−δ(t−s)ds||A0|| ||φ−ψ||∞+

∫t

0

MKge
−δ(t−s)ds ||φ−ψ||∞

≤
M

δ
||A0|| ||φ−ψ||∞ +

M

δ
Kg ||φ−ψ||∞

≤
M

δ

(

||A0||+Kg
)

||φ−ψ||∞

Therefore, if Θ < 1, then the equation (2) has a unique S asymptotically ω-periodic solution.



64 William Dimbour & Jean-Claude Mado CUBO
16, 3 (2014)

4 Application

As an application, we consider

(4)






∂u
∂t

(t,x) = ∂
2
u

∂x2
(t,x) + αu([t],x) + g(t,u(t,x)) t ∈ R+,x ∈ [0,π], α ∈ R

u(t,0) = u(t,π) = 0 t ∈ R+

u(0) = c0 ∈ X

We assume that (X, || · ||) = (L2(0,π), || · ||2) and define

D(A) = {u ∈ L2[0,π], u(0) = u(π) = 0}

Au(·) = △u = u′′(·), ∀u(·) ∈ D(A).

A is the infinetesimal generator of a semigroup T(t) on L2[0,π] with ||T(t)|| ≤ e−t for t ≥ 0.

Put u(t) = u(t, ·) that is u(t)x = u(t,x), (t,x) ∈ R+ × (0,π). Considering A0 : L
2[0,π] 7→

L2[0,π], y → αy, we observe that A0 is a linear bounded operator such that ||A0|| = |α| and
A0u([t]) = αu([t], ·).

Theorem 5. We assume that ω ∈ N. Then the system (4) has a unique mild solution S asymp-

totically ω-periodic if |α| < 1.

Proof. We have M = 1, δ = 1, ||A0|| = |α|. Then we apply the theorem (4) for the system (4).

Received: April 2014. Accepted: May 2014.

References

[1] J.Blot, P.Cieutat, G.N’Guérékata S-asymptotically ω-periodic functions and applications to

evolution equations Afr. diaspora. J.Math. 12 (2011), 113-121.

[2] W. Dimbour, Almost automorphic solutions for a differential equations with piecewise constant

argument in a Banach space, Nonlinear Analysis, Vol.74 (2011) 2351-2357.

[3] W. Dimbour, G. N’Guérékata, S-asymptotically ω-periodic solutions to some classes of partial

evolution equation, Applied Mathematics and Computation, Vol.218 (2012), 7622-7628.

[4] W. Dimbour, G. Mophou, G. N’Guérékata, S-asymptotically ω-periodic solutions for partial

differential equations with finite delay, Electronic Journal of Differential Equation, Vol.2011

(2011), 1-12.



CUBO
16, 3 (2014)

S-asymptotically ω-periodic solution for a nonlinear differential . . . 65

[5] H.R.Henŕıquez, M.Pierre, P.Táboas On S-asymptotically ω-periodic function on Banach

spaces and applications, J.Math.Anal.Appl 343(2008), 1119-1130.

[6] H.R.Henŕıquez, M.Pierre, P.Táboas Existence of S-asymptotically ω-periodic solutions for

abstract neutral equations, Bull.Austr.Math.Soc 78(2008), 365-382.

[7] C.Lizama, G.N’Guérékata Bounded mild solutions for semilinear integrodifferential equations

in Banach space, Integr.Eq.Oper. Theory 68(2010) 207-227.

[8] N.Van Minh, T. Tat Dat, On the Almost Automorphy of bounded solutions of differential

equations with piecewise constant argument, Journal of Mathematical analysis and appliction

326(2007), 165-178.

[9] K.Yosida, Functional Analysis, Springer-Verlag (1968).


	Introduction
	PRELIMINARIES
	Main result
	The linear case
	The nonlinear case

	Application