2The aim of this work is to study the existence of a periodic solutions of neutral integro-differential partial functional differential equations with two variable: ∂ ∂x ∂ ∂y (u(x, y)−Bu(x− r, y − r)) + Au(x, y) = x −∞ y −∞ a(x − s, y − ξ)u(s, ξ)dξds + f (x, y), x, y ∈ [0, 2π]. Our approach is based on the L p-multipliers of linear operators

Bahaj, Mohamed

Bahloul, Rachid

Cherti, Ilias

Sidki, Omar

English

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Existence of Periodic solutions of integro-differential delay equations with two variable in vector-valued function space

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Existence of Periodic solutions of integro-di fferential
delay equations with two variable in vector-valued
function space
Rachid Bahloul, Mohamed Bahaj, Ilias Cherti, Omar Sidki
To cite this version:
Rachid Bahloul, Mohamed Bahaj, Ilias Cherti, Omar Sidki. Existence of Periodic solutions of integro-
di fferential delay equations with two variable in vector-valued function space. 2017. ￿hal-01522143￿
Existence of Periodic solutions of
integro-differential delay equations with two
variable in vector-valued function space
Bahloul Rachid
Faculty of Sciences and Technology, Fez, Morocco
bahloulr33@hotmail.com
Bahaj Mohamed
Faculty of Sciences and Technology, Settat, Morocco
mohamedbahai@gmail.com
Cherti Ilias
Faculty of Sciences and Technology, Settat, Morocco
iliascherti10@gmail.com
Sidki Omar
Faculty of Sciences and Technology, Fez, Morocco
osidki@hotmail.com
ABSTRACT
The aim of this work is to study the existence of a periodic solutions of neutral integro-
differential partial functional differential equations with two variable: ∂∂x
∂
∂y (u(x,y)−Bu(x−
r,y − r)) + Au(x,y) =
∫
x
−∞
∫
y
−∞
a(x − s,y − ξ)u(s, ξ)dξds + f(x,y),x,y ∈ [0,2π]. Our
approach is based on the Lp-multipliers of linear operators.
Keywords: neutral integro-differential equations, Lp-multipliers, vector-valued function
space.
1 Introduction
Motivated by the fact that neutral functional differential equations (abbreviated, NFDE)
with finite delay arise in many areas of applied mathematics, this type of equations has
received much attention in recent years. In particular, the problem of existence of periodic
solutions, has been considered by several authors. We refer the readers to papers [[1], [2],
[3], [5], [6], [7], [9], [13], [15], [19]] and the references listed therein for information on this
subject.
In this work, we study the existence of periodic solutions for the following neutral partial
functional differential equations of the following form
∂
∂x
∂
∂y
(u(x,y) −Bu(x− r,y − r)) +Au(x,y) =
∫
x
−∞
∫
y
−∞
a(x− s,y − ξ)u(s, ξ)dξds + f(x,y).
(1.1)
Where (A,D(A)) and (B,D(B)) are a closed linear operators on a Banach spase X such
that D(A) ⊂ D(B)), a ∈ L1(R+ × R+) and f : R × R → X is a locally p-integrable and
2π-periodic function for 1 ≤ p < ∞.
In [4], Bahloul established results on the existence of periodic solutions of Eq. ((1.1)) when
a = 0, namely, for the following partial functional differential equation
∂
∂x
∂
∂y
u(x, y) +Au(x, y) = f(x, y)
where (A,D(A)) is a linear operator.
In [3], Bahloul et al established results on the existence for some degenerate differential
equation. Resently in [5], Bahloul et al established the existence of periodic solution for the
some functional differential equations in UMD space.
d
dt
(x(t) − L(xt)) = A(x(t) − L(xt)) +G(xt) + f(t) for t ∈ R,
In [6], Sylvain Koumla, Khalil Ezzinbi and Rachid Bahloul, studied the existence of mild
solutions for some partial functional integrodifferential equations with finite delay in Frechet
spaces described in the following form:
x′(t) = Ax(t) +
∫ t
0
B(t− s)x(s)ds+ f(t, xt) + h(t, xt) for t ≥ 0
where A : D(A) → X is the infinitesimal generator of a C0-semigroup (T (t))t≥0 on a Banach
space X, B(t) is a closed linear operator with domain D(B) such that D(A) ⊂ D(B).
In [14], Lizama gave necessary and sufficient conditions for the existence of periodic solutions
of the following equation.
d
dt
x(t) = Ax(t) + L(xt) + f(t) for t ∈ R,
where A is a closed linear operator on an UMD-space Y . We note that, starting with the
work [14], the problem of characterization of maximal regularity for avolution equations
with periodic initial conditions have been studied intensively in the last years. See e.g [8],
[12] and [16], see alsoo [10], [11], [17] and [18] and references therien.
the main novelty in this paper relies in the presence of two variable. This work can be
considered as a progress on the treatment of such kind of problems.
This work is organized as follows : After a creterion in the second section, we are able
to characterize in section 3 the existence and uniqueness of strong Lp-solution for the
problem ((1.1)) solely in terms of a property of Lp-multiplier for the sequence of operators
−kz(−kzI + kzBkz +A−Lk,z)
−1. We optain that the following assertion are equivalent in
general Banach spaces:
(1) (−kzI+kzBkz+A−˜̃a(k, z)) is invertible and {−kz(−kzI+kzBkz+A−˜̃a(k, z))
−1, (k, z) ∈
Z× Z} is an Lp-muliplier.
2
(2) For every f ∈ Lp(T×T;X) there exist a unique function u ∈ H1,p(T×T;X) such that
u ∈ D(A) and equation ((1.1)) holds for a.e (x, y) ∈ [0, 2π]× [0, 2π].
In section 4, we give the main abstract result ( theorem ((4.1))) of this work.
2 A creterion for periodic solutions
Let X be a Banach Space. Firstly, we denote By T the group defined as the quotient
R/2πZ. There is an identification between functions on T and 2π-periodic functions on
R. We consider the interval [0; 2π) as a model for T. Given 1 ≤ p < ∞, we denote by
Lp(T×T;X) the space of 2π-periodic locally p-integrable functions from R×R into X , with
the norm:
‖f‖p := (
∫ 2π
0
∫ 2π
0 ‖f(x, y)‖
p
dxdy)1/p
For f ∈ Lp(T× T;X), we denote by f̂(k, z), (k, z) ∈ Z the (k, z)-th Fourier coefficient of f
that is defined by:
̂̂
f(k, z) = ( 12π )
2
∫ 2π
0
∫ 2π
0
e−i(kx+zt)f(x, t)dxdt.
for (k, z) ∈ Z× Z and (x, y) ∈ R× R.
Remark 2.1. [4]
For all (k, z) ∈ Z× Z and (x, y) ∈ T× T, we have
(1)
̂̂
f(0, 0) = ( 12π )
2
∫ 2π
0
∫ 2π
0
f(x, y)dxdy.
(2)
̂̂
f(k, 0) = ( 12π )
2
∫ 2π
0
∫ 2π
0 e
−ikxf(x, y)dxdy.
(3)
̂̂
f(0, z) = ( 12π )
2
∫ 2π
0
∫ 2π
0 e
−iztf(x, y)dxdy.
(4) f̂(k, y) = 12π
∫ 2π
0 e
−ikxf(x, y)dx.
(5) f̂(x, z) = 12π
∫ 2π
0
e−izyf(x, y)dy.
Notation 2.1. : Let (k, z) ∈ Z× Z and 1 ≤ p < ∞. Denote by
H1,p(T× T;X) =
{
u ∈ Lp(T× T;X) : ∃v ∈ Lp(T× T;X), ̂̂v(k, z) = −kẑ̂u(k, z)
}
. and
Bkz = e
−(ikx+izy)B
Remark 2.2. [9]
Let f ∈ L1(T;X). If g(t) =
∫ t
0
f(s)ds and k ∈ Z, k 6= 0, then
ĝ(k) = ik f̂(0)−
i
k f̂(k)
Lemma 2.2. [4]
Let f ∈ L1(T× T;X).
If w(x, y) =
∫ x
0
∫ y
0 f(s, ξ)dξds and (k, z) ∈ Z× Z− {(0, 0)} then
3
̂̂w(k, z) = − 1kz
̂̂
f(0, 0) + 1kz
̂̂
f(0, z) + 1kz
̂̂
f(k, 0)− 1kz
̂̂
f(k, z)
Lemma 2.3. [4] ((2.2))
Let 1 ≤ p < ∞ and let f, g ∈ L1(T× T;X). The following are equivalent:
(1) ̂̂v(0, 0) = ̂̂v(k, 0) = ̂̂v(0, z) = 0 and there exists a ∈ X such that
u(x, y) = a+
∫ x
0
∫ y
0 v(s, ξ)dξds, (x, y) ∈ T× T
(2) ̂̂v(k, z) = −kẑ̂u(k, z), ((k, z) ∈ Z× Z)
Remark 2.3. By Lemma ((2.3)) we have u ∈ H1,p(T×T;X) ⇔ there exists v ∈ Lp(T×T;X)
such that ̂̂v(0, 0) = ̂̂v(k, 0) = ̂̂v(0, z) = 0 and there exists a ∈ X such that
u(x, y) = a+
∫ x
0
∫ y
0 v(s, ξ)dξds, (x, y) ∈ T× T
Now we can formulate the following multiplier definition.
Definition 2.1. For 1 ≤ p < ∞, a sequence {Mk,z}(k,z)∈Z×Z is said to be an L
p-multiplier
(or (Lp, Lp)-multiplier) if for each f ∈ Lp(T×T, X), there exists u ∈ Lp(T×T, Y ) such that
̂̂g(k, z) = Mk,z ̂̂f(k, z) for all (k, z) ∈ Z× Z.
Definition 2.2. : For 1 ≤ p < ∞ , a sequence {Mk,z}(k,z)∈Z×Z is said to be an (L
p, H1,p)-
multiplier if for each f ∈ Lp(T× T, X), there exists u ∈ H1,p(T× T, Y ) such that ̂̂g(k, z) =
Mk,z
̂̂
f(k, z) for all (k, z) ∈ Z× Z.
We recall the following results
Lemma 2.4. [4] Let f, g ∈ Lp([0, 2π] × [0, 2π];X), where 1 ≤ p < ∞. Then the following
are equivalent.
(i) f(x, y) ∈ D(A) and Af(x, y) = g(x, y),
(ii)
̂̂
f(k, z) ∈ D(A) and A
̂̂
f(k, z) = ̂̂g(k, z) for all (k, z) ∈ Z× Z
Next we give some preliminaries. Given a ∈ L1(R+ × R+) and u : [0, 2π]× [0, 2π] → X
(extended by periodicity to R), we define
F (x, y) =
∫ x
−∞
∫ y
−∞
a(x− s, y − ξ)u(s, ξ)dsdξ (2.1)
Let ˜̃a(λ, µ) =
∫ +∞
0
∫ +∞
0 e
−(λx+µy)a(x, y)dxdy, λ, µ ∈ C, be the Laplace transform of a. An
easy computation shows that:
̂̂
F (k, z) = ˜̃a(ik, iz)̂̂u(k, z), four k, z ∈ Z (2.2)
Remark 2.4. For all (λ, µ) ∈ C× C and (x, t) ∈ R× R, we have
(1)
˜̃
f(0, 0) =
∫ +∞
0
∫ +∞
0 f(x, y)dxdy.
(2)
˜̃
f(λ, 0) =
∫ +∞
0
∫ +∞
0
e−iλxf(x, y)dxdy.
(3) ˜̃f(0, µ) =
∫ +∞
0
∫ +∞
0
e−iµyf(x, y)dxdy.
(4) f̃(λ, y) =
∫ +∞
0 e
−iλxf(x, y)dx.
(5) f̃(x, µ) =
∫ +∞
0 e
−iµyf(x, y)dy.
4
3 A caracterization of stong Lp-solution
Definition 3.1. : Let f ∈ Lp(T×T;X). A function u ∈ H1,p(T×T;X) is said to be a 2π-
periodic strong Lp-solution of Eq. ((1.1)) if u ∈ D(A), u(0, y) = u(2π, y), u(x, 0) = u(x, 2π)
for all (x, y) ∈ T× T and Eq.((1.1)) holds almost every where.
Proposition 3.1. Let X be a Banach space. Suppose that for every f ∈ Lp(T×T;X) there
exists a unique strong solution of Eq. ((1.1)) for 1 ≤ p < ∞. Then
(1) for every (k, z) ∈ Z× Z the operator (−kzI + kzBkz + A− ˜̃a(ik, iz)) has invertible.
(2)
{
−kz(−kzI + kzBkz +A− ˜̃a(ik, iz))
−1
}
(k,z)∈Z×Z
is an Lp-multiplier.
Proof. 1. Let k, z ∈ Z and w ∈ X . Then for f(x, y) = ei(kx+zy)w , there exists x ∈
H1,p(T× T;X) such that:
∂
∂x
∂
∂y (u(x, y)−Bu(x− r, y − r)) +Au(x, y) =
∫ x
−∞
∫ y
−∞
a(x− s, y − ξ)u(s, ξ)dξds+ f(x, y)
Taking Fourier transform. Then we obtain
(−kzI + kzBkz + A − ˜̃a(ik, iz))̂̂u(k, z) = ̂̂f(k, z) = w ⇒ (−kzI + kzBkz + A − ˜̃a(ik, iz)) is
surjective.
if (−kzI + kzBkz + A− ˜̃a(ik, iz))(v) = 0, then, u(x, y) = e
i(kx+zy)v is a 2π-periodic strong
Lp-solution of Eq. ((1.1)) corresponding to the function f = 0 Hence u(x, y) = 0 and v = 0
then (−kzI + kzBkz +A− ˜̃a(ik, iz)) is injective.
2) Let f ∈ Lp(T × T;X). By hypothesis, there exists a unique u ∈ H1,p(T × T, X) such
that the Eq.((1.1)) is valid. Taking Fourier transforms, we deduce that ̂̂u(k, z) = (−kzI +
kzBkz+A−˜̃a(ik, iz))
−1̂̂f(k, z) for all k, z ∈ Z, then
{
(−kzI + kzBkz + A− ˜̃a(ik, iz))
−1
}
(k,z)∈Z×Z
is an Lp-multiplier. Hence −kẑ̂u(k, z) = −kz(−kzI + kzBkz + A − ˜̃a(ik, iz))−1̂̂f(k, z) for
all k, z ∈ Z. On the other hand, since u ∈ H1,p(T × T;X) , there exists w ∈ Lp(T ×
T;X) such that ̂̂w(k, z) = −kẑ̂u(k, z) = −kz(−kzI + kzBkz + A − ˜̃a(ik, iz))−1̂̂f(k, z) i.e{
−kz(−kzI + kzBkz +A− ˜̃a(ik, iz))
−1
}
(k,z)∈Z×Z
is an Lp-multiplier.
4 Main result
The next Theorem is the main result of this section.
Theorem 4.1. Let A be a closed linear operator and 1 ≤ p < ∞. If
(1) for every (k, z) ∈ Z × Z the operator Mk,z = (−kzI + kzBkz + A − ˜̃a(ik, iz)) has
invertible.
(2)
{
−kz(−kzI + kzBkz +A− ˜̃a(ik, iz))
−1
}
(k,z)∈Z×Z
is an Lp-multiplier.
Then for every f ∈ Lp(T×T;X) there exists a unique 2π-periodic strong Lp-solution of Eq.
((1.1)).
Lemma 4.2. [4] Let 1 ≤ p < ∞ and X be a Banach space. The following assertions are
equivalent
5
(1) {Mk,z}(k,z)∈Z×Z is an (L
p, Hp)-multiplier.
(2) {−kzMk,z}(k,z)∈Z×Z is an L
p-multiplier.
Proof. 1 ⇒ 2. Let f ∈ Lp(T × T;X), then there exists g ∈ H1,p(T × T;X) ( in particular
g ∈ Lp(T × T;X)) such that ̂̂g(k, z) = −kzMk,ẑ̂f(k, z), then {−kzMk,z}(k,z)∈Z×Z is an Lp-
multiplier.
2 ⇒ 1. Let f ∈ Lp(T × T;X), by hypothesis ∃v ∈ Lp(T × T;X) such that ̂̂v(k, z) =
−kzMk,z
̂̂
f(k, z). In particular ̂̂v(0, 0) = ̂̂v(0, z) = ̂̂v(k, 0) = 0.
Let w(x, y) =
∫ x
0
∫ y
0
v(s, ξ)dξds, then we have
ŵ(x, z) =
∫ x
0
( iz v̂(x, 0)−
i
z v̂(x, z))ds, for all z 6= 0
̂̂w(k, z) = ik [ iz ̂̂v(0, 0)− iz ̂̂v(0, z))]− ik [ iz ̂̂v(k, 0)− iz ̂̂v(K, z))], for all k 6= 0
̂̂w(k, z) = ik iz ̂̂v(k, z) = −1kz ̂̂v(k, z) = Mk,z
̂̂
f(k, z), for all k 6= 0 and z 6= 0
Let u(x, y) = w(x, y) +M0,0
̂̂
f(0, 0)− ̂̂w(0, 0) i.e
u(x, y) = a+
∫ x
0
∫ y
0 v(s, ξ)dξds, a = M0,0
̂̂
f(0, 0)− ̂̂w(0, 0). Then by Remark ((2.2)) we have
u ∈ H1,p(T× T;X).
̂̂u(k, z) = ̂̂w(k, z) + 0 = ̂̂w(k, z) = Mk,z ̂̂f(k, z), for all (k, z) 6= (0, 0),
Proof of theorem ((4.1)). Let f ∈ Lp(T × T;X). Define Mk,z = (−kzI + kzBkz + A −
˜̃a(ik, iz))−1 by theorem ((4.1)), the family
{
−kz(−kzI + kzBkz +A− ˜̃a(ik, iz))
−1
}
(k,z)∈Z×Z
is an Lp-multiplier and by Lemma ((4.2)), it is equivalent to the fact that the family
{Mk,z}(k,z)∈Z×Z is an (L
p, Hp)-multiplier, i.e, there exists u ∈ H1,p(T× T;X) such that
̂̂u(k, z) = Mk,z ̂̂f(k, z) = (−kzI + kzBkz +A− ˜̃a(ik, iz))−1̂̂f(k, z) (4.1)
In particuler, u ∈ Lp(T × T;X) and there exists v ∈ Lp(T× T;X) such that
̂̂v(k, z) = −kẑ̂u(k, z) (4.2)
Moreover, ∂̂∂x
∂
∂tu(k, z) = −kz
̂̂v(k, z).
By Fejer’s theorem one has in Lp(T× T;X)
̂̂u(k, z) = limn→∞ limm→∞ σn,m(u)(x, y)
such that σn,m(u)(x, y) =
1
n+1
1
m+1
∑n
p=0
∑p
k=−p
∑m
l=0
∑l
z=−l e
i(kx+zt)̂̂u(k, z).
By ((4.1)) and ((4.2)) we have
−kẑ̂u(k, z) + kzBkẑ̂u(k, z) +Â̂u(k, z) = ˜̃a(ik, iz)̂̂f(k, z) + ̂̂f(k, z)
i.e
−kẑ̂u(k, z)− (−kz)Bkz ̂̂u(k, z) + Â̂u(k, z) = ̂̂F (k, z) + ̂̂f(k, z)
For all (k, z) ∈ Z×Z. Then using that A and B are closed, we conclude that u ∈ D(A) and
∂
∂x
∂
∂y
(u(x, y)−Bu(x− r, y− r)) +Au(x, y) =
∫ x
−∞
∫ y
−∞
a(x− s, y− ξ)u(s, ξ)dξds+ f(x, y)
6
Suppose (1) has two strong Lp-solution u1 and u2, then u = u1 − u2 is a solution of Eq.
((1.1)) corresponding to the fonction f = 0. Taking Fourier transform on both sides of this
equality we get (−kzI + kzBkz +A− ˜̃a(ik, iz))̂̂u(k, z), which implies that ̂̂u(k, z) = 0 for all
(k, z) ∈ Z× Z, consequently, u = 0. Then u1 = u2.
5 Conclusion
For every function f ∈ Lp(T×T;X), this paper deals with the existence of periodic solutions
of neutral integro-differential equation with two variable ∂∂x
∂
∂y (u(x,y)−Bu(x−r,y−r))+
Au(x,y) =
∫
x
−∞
∫
y
−∞
a(x − s,y − ξ)u(s, ξ)dξds + f(x,y). Using the method of Fourier
transform of equation ((1.1)) and Lp-multipliers of linear operators. the results obtained
in this paper generalizes the result of Bahloul [4].
Competing Interests
The author declare that no competing interests exist.
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Existence of Periodic solutions of integro-differential delay equations with two variable in vector-valued function space

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