CUBO A Mathematical Journal
Vol.15, No¯ 01, (49–75). March 2013

Existence and stability in the α-norm for nonlinear neutral
partial differential equations with finite delay

Taoufik Chitioui,

Université de Sfax,

Faculté des Sciences de Sfax,

Sfax, Tunisie.

chtioui.taoufik@yahoo.fr

Khalil Ezzinbi,

Caddy Ayyad University,

Université Cadi Ayyad,

Faculté des Sciences Semlalia,

BP 2390, Marrakech, Morocco,

ezzinbi@gmail.com

Amor Rebey

Université de Kairouan

Institut Supérieur des

Mathématiques Appliquées et

de l’Informatique,

Tunisie.

ezzinbi@gmail.com

ABSTRACT

In this work, we study the existence, regularity and stability of solutions for some

nonlinear class of partial neutral functional differential equations. We assume that

the linear part generates a compact analytic semigroup on a Banach space X, the

delayed part is assumed to be continuous with respect to the fractional power of the

generator. For illustration, some application is provided for some model with diffusion

and nonlinearity in the gradient.

RESUMEN

En este trabajo estudiamos la existencia, regularidad y estabilidad de soluciones para

una clase de ecuaciones diferenciales parciales funcionales neutrales. Asumimos que la

parte lineal genera un semigrupo compacto anaĺıtico en un espacio de Banach X, la

parte retardada se asume continua respecto de la potencia fraccional del generador.

Como ejemplo, se muestra una aplicación para un modelo con difusión y no linealidad

en el gradiente.

Keywords and Phrases: Neutal equation; Analytic semigroup; Fractional power; Mild solution;

Strict solution.

2010 AMS Mathematics Subject Classification:



50 Taoufik Chitioui, Khalil Ezzinbi and Amor Rebey CUBO
15, 1 (2013)

1 Introduction

In this paper, we study the existence, regularity and stability of solutions in the α-norm for partial

differential equations with finite delay. The following model provides an example of such a situation





∂

∂t

[
v(t,x) − qv(t − r,x) + g(

∂

∂x
v(t − r,x))

]
=
∂2

∂x2

[
v(t,x) − qv(t − r,x)

+g(
∂

∂x
v(t − r,x))

]
+ f

(
v(t − r,x),

∂

∂x
[v(t,x) − qv(t − r,x)]

)

for t ≥ 0 and x ∈ [0,π],

v(t,0) − qv(t − r,0) = v(t,π) − qv(t − r,π) = 0 for t ≥ 0,

v(θ,x) = v0(θ,x) for − r ≤ θ ≤ 0 and x ∈ [0,π],

(1)

where q, r are positive constants, the initial data v0 is given function and f, g are continuous

functions. The previous system can be written as a neutral partial differential equation of the

following form





d

dt

[
x(t) − G(t,xt)

]
= −A

[
x(t) − G(t,xt)

]
+ F(t,xt) for t ≥ 0,

x0 = ϕ ∈ Cα,
(2)

where −A generates an analytic semigroup (T(t))t>0 on a Banach space X, Cα := C([−r,0],D(A
α)), r >

0, and 0 < α < 1, denotes the space of continuous functions from [−r,0] into D(Aα), and the op-

erator Aα is the fractional α-power of A. This operator (Aα,D(Aα)) will be describe later. For

x ∈ C([−r,b],D(Aα)),b > 0, and t ∈ [0,b], xt denotes, as usual, the element of Cα defined by

xt(θ) = x(t + θ) for θ ∈ [−r,0].G and F are continuous functions from R+ × Cα with values

respectively in Xα and X.

This work was motivated by [4, 18]. In [4] the authors have developed a basic theory of partial

neutral functional differential equations in fractional power spaces, they proved the existence and

regularity of the solution of Eq. (2) , but only in the case where G : Cα → D(A
α) is a bounded

linear operator. They considered the following neutral partial differential equation






d

dt
D(xt) = −AD(xt) + F(xt) for t ≥ 0,

x0 = ϕ ∈ Cα,
(3)

where D is a bounded linear operator from Cα into Xα defined by Dϕ = ϕ(0) −D0ϕ, for ϕ ∈ Cα,

where D0 is a bounded linear operator given by:

D0ϕ =

∫0

−r

dη(θ)ϕ(θ) for ϕ ∈ Cα,

and η : [−r,0] → L(Xα) is of bounded variation and non-atomic at zero. That is

var[−ε,0](η) → 0 as ε → 0.



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Existence and stability in the α-norm for nonlinear neutral ... 51

Which F is a globally Lipschitz continuous mapping from Cα into D(A
α), and if x ∈ D(Aα)

and θ ∈ [−r,0] then η(θ)x ∈ D(Aα) and Aαη(θ)x = η(θ)Aαx.

It is well known, that if the phase space Cα is the space of all continuous functions from [−r,0]

into X (i.e. α = 0), Equation (3) has been studied by several authors. For more details, we

refer to the book of Wu [29]. For example, Wu and Xia considered in [30] a system of partial

neutral functional differential-difference equations defined on the unit circle S1, which is a model

for a continuous circular array of resistively coupled transmission lines with mixed initial boundary

conditions. They obtained equations of the form

∂

∂t
[x(.,t) − qx(.,t − r)] = K

∂2

∂ξ2
[x(.,t) − qx(.,t − r)] + f(xt) for t ≥ 0,

where ξ ∈ S1, K a positive constant and 0 ≤ q < 1. The space of initial data was chosen to be

C([−r,0];H1(S1)). Motivated by this work, Hale presented, in [19, 20], the basic theory of existence

and uniqueness, and properties of the solution operator, as well as Hopf bifurcation and conditions

for the stability and instability of periodic orbits for a more general class of PNFDE on the unit

circle S1. For the sake of comparison, let us briefly restate the equations considered by Hale in

[19, 20]. If ϕ ∈ C([−r,0];H1(S1)), we write it as ϕ(ξ,θ) for ξ ∈ S1 and θ ∈ [−r,0]. For any

function f̃ ∈ Ck+1(C([−r,0]; R); R), k ≥ 1, we let f ∈ Ck+1(C([−r,0];H1(S1));L2(S1)) be defined

by f(ϕ)(ξ) = f̃(ϕ(ξ,.)), ξ ∈ S1. Let D̃ ∈ L(C([−r,0]; R); R) be defined by






D̃ψ = ψ(0) − g̃(ψ),

g̃(ψ) =

∫0

−r

dη(θ)ψ(θ),

where η is of bounded variation and non-atomic at 0.

We define D ∈ L(C([−r,0];H1(S1));H1(S1)) as

D(ϕ)(ξ) = D̃(ϕ(ξ,.)) for ξ ∈ S1.

Hale considered, in [19, 20], PNFDE of the form

∂

∂t
Dxt = K

∂2

∂ξ2
Dxt + f(xt) for t ≥ 0, (4)

with C([−r,0];H1(S1)) as the space of initial data. He considered the Laplace operator

A0 = K
∂
2

∂ξ2
with domain H2(S1), which yields an operator generating an analytic semigroup.

In [1, 2, 3], authors considered a natural generalization of the work of Hale [19, 20]. We extended

the study to the case when the linear part of PNFDE is non-densely defined Hille-Yosida operator.

In[27], Travis and Webb investigated the local existence of mild solutions and strong solutions of

Eq. (2) with respect to the α-norm, but in the particular case when G(., .) = 0. The existence of

strong solutions is considered when F is locally Hölder continuous in both of its variables, also in

[26], they studied the existence and regularity of mild solution when F is Lipschitz continuous with



52 Taoufik Chitioui, Khalil Ezzinbi and Amor Rebey CUBO
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the X-norm.

Here, we assume that G is a nonlinear function and is defined in a smaller space than CX,

that is Cα for some 0 < α < 1, the space of continuous function from [−r,0] into Xα, which will

be specified later. We prove the existence of the mild and strict solution.

This paper is organized as follows. In Section 2, we recall some preliminary results about analytic

semigroups and fractional power associated to its generator and the definition of the measure of

noncompactness. After that, we start to prove the existence and uniqueness of mild solutions

in the α-norm for Eq. (2). In Section 3, we study the regularity of solution, we give sufficient

conditions to get the existence of the strict solutions. In Section 4, we state some properties of the

solution operator associated to the autonomous case of Eq. (2). Also, we investigate the stability

near an equilibrium. Mainly, we prove that the equilibrium of the solution semigroup associated to

the autonomous case is locally exponentially stable when its linearized solution semigroup around

this equilibrium is exponentially stable. Finally, to illustrate our results, we give in Section 5 an

application to a reaction diffusion equation.

2 Existence of mild solutions

Let (X,‖.‖) be a Banach space, and α be a constant such that 0 < α < 1 and −A be the infinitesimal

generator of a bounded analytic semigroup of linear operator (T(t))t≥0 on X. We assume without

loss of generality that 0 ∈ ρ(A). Note that if the assumption 0 ∈ ρ(A) is not satisfied, one can

substitute the operator A by the operator (A − σI) with σ large enough such that 0 ∈ ρ(A − σ).

This allows us to define the fractional power Aα for 0 < α < 1, as a closed linear invertible

operator with domain D(Aα) dense in X. The closeness of Aα implies that D(Aα), endowed with

the graph norm of Aα, |x| = ‖x‖ + ‖Aαx‖, is a Banach space. Since Aα is invertible, its graph

norm |.| is equivalent to the norm |x|α = ‖A
αx‖. Thus, D(Aα) equipped with the norm |.|α, is a

Banach space, which we denote by Xα. The space Cα := C([−r,0],Xα), r > 0 denotes the space

of continuous functions from [−r,0] into Xα endowed with the uniform norm topology:

‖ϕ‖α := sup
θ∈[−r,0]

|ϕ(θ)|α for ϕ ∈ Cα.

Also, the following properties are well known.

Theorem 2.1. [24] Let 0 < α < 1. Assume that the operator −A is the infinitesimal generator of

an analytic semigroup (T(t))t≥0 on the Banach space X satisfying 0 ∈ ρ(A). Then we have

i) T(t) : X −→ D(Aα) for every t > 0,

ii) T(t)Aαx = AαT(t)x for every x ∈ D(Aα) and t ≥ 0,

iii) for every t > 0, AαT(t) is bounded on X and there exist Mα > 0 and δ > 0 such that

‖AαT(t)‖ ≤ Mαe
−δtt−α ≤ Mαt

−α for t > 0,



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Existence and stability in the α-norm for nonlinear neutral ... 53

iv) If 0 < α ≤ β < 1, then D(Aβ) →֒ D(Aα).

v) There exists Nα > 0 such that

‖(T(t) − I)A−α‖ ≤ Nαt
α for t > 0.

vi) If T(t) is compact for each t > 0, then A−α is compact.

Now, we propose to find the existence of a mild solution for problem (2) using the sadovskii’s

fixed point theorem. Then, we obtain the uniqueness result of the solution by adding a hypothesis

of Lipschitz continuous on F.

Let E be a Banach space. We introduce the Kuratowskii measure of noncompactness χ(Ω) of

a set Ω ⊂ E by

χ(Ω) = inf{ε > 0 : Ω has a finite cover of diameter < ε}.

Lemma 2.1. [10] Let E be a Banach Space and B, C ⊆ E be bounded set. Then, the following

properties are true :

(1) B is relatively compact if and only if χ(B) = 0,

(2) χ(B + C) ≤ χ(B) + χ(C), where B + C = {x + y : x ∈ B, y ∈ C},

(3) Every Lipschitz continuous function K from C to F satisfies:

χ[K(Ω)] ≤ lipKχ(Ω),

where lip K decides the smallest Lipschitz constant of K.

Definition 2.2. [25] A mapping K from a set C in a Banach space E is called a condensing

operator if it is continuous and for every bounded noncompact set Ω ⊆ C the inequality holds

χ[K(Ω)] < χ(Ω).

Theorem 2.2. [25](Sadovskii’s fixed point theorem). If a condensing mapping K maps a bounded

convex closed set C in a Banach space E into itself, then K has at least one fixed point in T.

First of all, we study the existence of mild solutions, in order to do that, we assume the

following assumptions.

(H0) The operator −A is the infinitesimal generator of an analytic semigroup (T(t))t≥0 on the

Banach space X , moreover, we assume that 0 ∈ ρ(A).

(H1) The semigroup (T(t))t≥0 is compact on X for t > 0. It means that T(t) is compact on X for

t > 0.



54 Taoufik Chitioui, Khalil Ezzinbi and Amor Rebey CUBO
15, 1 (2013)

(H2) G : [0,a] × Cα → Xα is continuous and for each a > 0 there exists 0 < Lg < 1 such that

|G(t,ϕ) − G(t,ψ)|α ≤ Lg‖ϕ − ψ‖α for every t ∈ [0,a] and ϕ, ψ ∈ Cα.

(H3) The function F : [0,a] × Cα → X satisfies the following conditions

i) F : [0,a] × Cα → X is continuous.

ii) There exists a continuous nondecreasing function β : [0,a] → R+ such that ‖F(t,ϕ)‖ ≤

β(t)‖ϕ‖α for (t,ϕ) ∈ [0,a] × Cα.

Definition 2.3. A continuous function x : [−r,a] −→ Xα, for a > 0 is said to be a mild solution

of Eq. (2), if

i) x(t) = T(t)
[
ϕ(0) − G(0,ϕ)

]
+ G(t,xt) +

∫t

0

T(t − s)F(s,xs)ds for t ∈ [0,a],

ii) x0 = ϕ.

Definition 2.4. A continuous function x : [−r,a] −→ Xα is said to be a strict solution of Eq. (2),

if

i) x(.) − G(.,x(.)) ∈ C
1([0,a],Xα),

ii)
d

dt
(x(t) − G(t,xt)) = −A(x(t) − G(t,xt)) + F(t, xt) for t ∈ [0,a],

iii) x0 = ϕ.

Now, we state our first result.

Theorem 2.3. Assume that the hypothesis (H0)-(H3) hold. Let ϕ ∈ Cα. Assume that the

following condition holds

Lg + Mα

∫a

0

β(s)

(a − s)α
ds < 1. (5)

Then Eq. (2) has at least one mild solution on [0,a].

Proof. Let k > ‖ϕ‖α. We define the following set

Bk = {x ∈ C([0,a],Xα) : x(0) = ϕ(0) and |x|∞ ≤ k},

where |x|∞ = sup
t∈[0,a]

|x(t)|α. For x ∈ Bk, define the mapping x̃ : [−r,a] → Xα by

x̃(t) =

{
x(t) for t ∈ [0,a]

ϕ(t) for t ∈ [−r,0].



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Existence and stability in the α-norm for nonlinear neutral ... 55

The function t 7→ x̃t is continuous from [0,a] to Cα.

Now, define the operator K on Bk by

K(x)(t) = T(t)(ϕ(0) − G(0,ϕ)) + G(t, x̃t) +

∫t

0

T(t − s)F(s, x̃s)ds for t ∈ [0,a].

It is sufficient to show that K has a fixed point in Bk. We first show that there is a positive

number k > ‖ϕ‖α such that K(Bk) ⊆ Bk. If not, then for each k > ‖ϕ‖α, there exist xk ∈ Bk and

tk ∈ [0,a] such that |(Kxk)(tk)|α > k. It follows that

k < |(Kxk)(tk)|α

≤ |T(tk)(ϕ(0) − G(0,ϕ))|α + |G(tk, x̃tk)|α +

∫tk

0

|T(tk − s)F(s, x̃s)|αds.

Let M = sup{‖T(t)‖ : t ∈ [0,a]}. Then

k < M|ϕ(0) − G(0,ϕ)|α + |G(tk, x̃tk) − G(tk,0)|α + |G(tk,0)|α

+

∫tk

0

Mα

(tk − s)
α
β(s)‖x̃s‖αds.

Moreover ‖x̃s‖α ≤ k for all s ∈ [0,a] and x ∈ Bk. Then, we obtain

k < M|ϕ(0) − G(0,ϕ)|α + |G(tk, x̃tk) − G(tk,0)|α + |G(tk,0)|α +

∫tk

0

kMα

(tk − s)
α
β(s)ds.

We shall show that the function g : t 7→

∫t

0

β(s)

(t − s)α
ds is nondecreasing on [0,a]. Let t,t′ ∈ [0,a]

be such that t < t′. Then we have

g(t) =

∫t

0

β(t − s)

sα
ds ≤

∫t

0

β(t′ − s)

sα
ds ≤

∫t′

0

β(t′ − s)

sα
ds = g(t′).

Therefore

k ≤ M|ϕ(0) − G(0,ϕ)|α + Lg‖x̃tk‖α + sup
0≤s≤a

|G(s,0)|α +

∫a

0

kMα

(a − s)α
β(s)ds.

Dividing both sides by k and taking the lower limit as k → +∞, then we get that

Lg + Mα

∫a

0

β(s)

(a − s)α
ds ≥ 1,

which contradicts (5). Consequently, there exists k ≥ 0 such K(Bk) ⊆ Bk.

To prove that K has at least a fixed point on Bk, we decompose K as follows K := K1 + K2, where

K1(x)(t) = G(t, x̃t) for t ∈ [0,a].

and

K2(x)(t) = T(t)(ϕ(0) − G(0,ϕ)) +

∫t

0

T(t − s)F(s, x̃s)ds for t ∈ [0,a].



56 Taoufik Chitioui, Khalil Ezzinbi and Amor Rebey CUBO
15, 1 (2013)

We claim that K1 is a strict contraction and K2 is compact.

To see this, observe that for t ∈ [0,a] and x, y ∈ Bk, we have by assumption (H2).

|K1x(t) − K1y(t)|α = |G(t, x̃t) − G(t, ỹt)|α

≤ Lg‖x̃t − ỹt‖α

≤ Lg|x − y|∞

Then K1 is a strict contraction. We will prove now the continuity of K2. Let (x
n)n ⊂ Bk with

xn → x in Bk. Then, the set Λ = {(s, x̃
n
s ),(s, x̃s) : s ∈ [0,a],n ≥ 1} is compact in [0,a] × Cα. By

Hëıne’s theorem implies that F is uniformly continuous in Λ and

|K2(x
n) − K2(x)|∞ = sup

t∈[0,a]

∫t

0

AαT(t − s)
(
F(s, x̃ns ) − F(s, x̃s)

)
ds

≤ Mα

∫a

0

ds

sα
sup

s∈[0,a]

‖F(s, x̃ns ) − F(s, x̃s)‖ → 0 as n → +∞.

and this yield the continuity of K2, then the continuity of K on Bk.

We next show that the operator K2 is compact.

In order to apply Ascoli theorem we have to show that the set {K2(x)(t) : x ∈ Bk} is relatively

compact for each t ∈]0,a].

Let t ∈]0,a] be fixed, and γ > 0 be such that α < γ < 1. Then

‖ (AγK2(x))(t) ‖ ≤ ‖ A
γT(t)(ϕ(0) − G(0,ϕ)) ‖ + ‖

∫t

0

AγT(t − s)F(s, x̃s)ds ‖

≤ Mγt
−γ ‖ ϕ(0) − G(0,ϕ) ‖ +kMγ

∫t

0

(t − s)−γβ(s)ds < +∞.

Then for fixed t ∈]0,a], {(AγK2x)(t)} is bounded in X. Appealing (H1) and (vi) of Theorem 2.1,

we deduce that A−γ : X → Xα is compact, it follows that {K2(x)(t) : x ∈ Bk} is relatively compact

set in Xα.

Next, we will show that {K2x : x ∈ Bk} is an equicontinuous family of functions. For 0 ≤ t1 <

t2 ≤ a,

K2x(t2) − K2x(t1) = (T(t2) − T(t1))(ϕ(0) − G(0,ϕ)) +

∫t2

t1

T(t2 − s)F(s, x̃s)ds

+

∫t1

0

(T(t2 − s) − T(t1 − s))F(s, x̃s)ds

= (T(t2) − T(t1))(ϕ(0) − G(0,ϕ)) +

∫t2

t1

T(t2 − s)F(s, x̃s)ds

+(T(t2 − t1) − I)

∫t1

0

T(t1 − s)F(s, x̃s)ds.



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Existence and stability in the α-norm for nonlinear neutral ... 57

We obtain that

|K2x(t2) − K2x(t1)|α ≤ ‖(T(t2) − T(t1))A
α(ϕ(0) − G(0,ϕ))‖ + kMα‖β‖∞

∫t2

t1

(t2 − s)
−αds

+‖(T(t2 − t1) − I)

∫t1

0

AαT(t1 − s)F(s, x̃s)ds‖

It’s clair to prove the first part tend to zero as |t2 − t1| → 0. Since for t1 > 0 the set

{ ∫t1

0

AαT(t1 − s)F(s, x̃s)ds : x ∈ Bk

}

is relatively compact in X, there is a compact set K̃ in X such that

∫t1

0

AαT(t1 − s)F(s, x̃s)ds ∈ K̃ for x ∈ Bk.

By Banach-Steinhaus’s theorem, we have

∥∥∥(T(t2 − t1) − I)
∫t1

0

AαT(t1 − s)F(s, x̃s)ds
∥∥∥ → 0 as t2 → t1,

uniformly in x ∈ Bk. Using similar argument for 0 ≤ t2 < t1 ≤ a, we can conclude that

{K2x(t), x ∈ Bk} is an equicontinuous . Using Ascoli-Arzla theorem, we deduce that K2 : Bk → Bk

is compact, and K = K1 + K2 is a condensing operator. By the Sadovskii’s fixed-point theorem

2.2, we conclude that K has at least one fixed point in Bk, which is a mild solutions of Eq. (2) on

[0,a].

To prove result on uniqueness, we to assume that

(H4) F : [0,a] × Cα → X is continuous and Lipschitzian with respect to the second variable. Let

Lf > 0 be such that

‖F(t,ψ1) − F(t,ψ2)‖ 6 Lf‖ψ1 − ψ2‖α (6)

for every ψ1,ψ2 ∈ Cα and t ∈ [0,a].

Theorem 2.4. Let ϕ ∈ Cα. If the assumptions (H0), (H2) and (H4) are satisfied, then Eq.

(2) has a unique mild solution provided that

Lg + MαLf
a1−α

1 − α
< 1. (7)

Proof. Consider the nonempty closed subset of C([0,a],Xα) defined by

Ω(ϕ) := {x ∈ C([0,a],Xα) : x(0) = ϕ(0)}.

For x ∈ Ω(ϕ), define the mapping x̃ : [−r,a] → Xα by

x̃(t) =

{
x(t) for t ∈ [0,a]

ϕ(t) for t ∈ [−r,0].



58 Taoufik Chitioui, Khalil Ezzinbi and Amor Rebey CUBO
15, 1 (2013)

Define the operator K : Ω(ϕ) → Ω(ϕ) by

K(x)(t) = T(t)(ϕ(0) − G(0,ϕ)) + G(t, x̃t) +

∫t

0

T(t − s)F(s, x̃s)ds for t ∈ [0,a].

We shall show that it is a strict contraction. Let x,y ∈ Ω(ϕ) and t ∈ [0,a]. Then

|Kx(t) − Ky(t)|α 6 |G(t, x̃t) − G(t, ỹt)|α +

∫t

0

|T(t − s){F(s, x̃s) − F(s, ỹs)}|αds

6 Lg‖x̃t − ỹt‖α + Mα

∫t

0

‖F(s, x̃s) − F(s, ỹs)‖(t − s)
−αds

6

(
Lg + MαLf

a1−α

1 − α

)
|x − y|∞

Then

|Kx − Ky|∞ 6
(
Lg + MαLf

a1−α

1 − α

)
|x − y|∞.

It follows that K is a strict contraction since Lg + MαLf
a1−α

1 − α
< 1. By the contraction principle,

we conclude that there exists a unique fixed point x for K in Ω(ϕ), therefore Eq. (2) has a unique

mild solution on [−r,a]. The proof is completed.

3 Existence of strict solutions

For the regularity of the integral solutions, we suppose moreover the following assumptions:

(H5) G and F are continuously differentiable and their partial derivatives are locally Lipschitzian

with respect to the second argument in the sense that; for any compact set K ⊂ [0,a] × Cα,

there exist positive constants L1,  L2, L3 and L4 such that

|D1G(t,ψ1) − D1G(t,ψ2)|α 6 L1‖ψ1 − ψ2‖α,

‖D2G(t,ψ1) − D2G(t,ψ2)‖L(Cα,Xα) 6 L2‖ψ1 − ψ2‖α,

‖D1F(t,ψ1) − D1F(t,ψ2)‖ 6 L3‖ψ1 − ψ2‖α,

‖D2F(t,ψ1) − D2F(t,ψ2)‖L(Cα,X) 6 L4‖ψ1 − ψ2‖α,

for (t,ψ1), (t,ψ2) ∈ K and t ∈ [0,a]. Where D1 and D2 are the partial derivatives with

respect to the first and second argument.

Theorem 3.1. Assume that (H0),(H2), (H4), (H5) hold and condition (7) is true. Let ϕ ∈

C1([−r,0],Xα) be such that ϕ(0) − G(0,ϕ) ∈ D(A) and

ϕ′(0) − D1G(0,ϕ) − D2G(0,ϕ)ϕ
′ = −A[ϕ(0) − G(0,ϕ)] + F(0,ϕ)

Then Eq. (2) has a unique strict solution on [0,a].



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Existence and stability in the α-norm for nonlinear neutral ... 59

Proof. Let x be the mild solution of Eq. (2). Consider the equation






y(t) = T(t)[−A(ϕ(0) − G(0,ϕ)) + F(0,ϕ)] + D1G(t,xt) + D2G(t,xt)yt

+

∫t

0

T(t − s)[D1F(s,xs) + D2F(s,xs)ys]ds for t ∈ [0,a],

y0 = ϕ
′ ∈ Cα.

(8)

We claim that Eq. (8) has a unique solution on [0,a]. In fact, consider the operator P defined

on Λ := {x ∈ C([−r,a];Xα) : x(t) = ϕ
′(t) for t ∈ [−r,0]} by

Py(t) =






T(t)[−A(ϕ(0) − G(0,ϕ)) + F(0,ϕ)] + D1G(t,xt)

+D2G(t,xt)yt +

∫t

0

T(t − s)[D1F(s,xs) + D2F(s,xs)ys]ds for t ∈ [0,a],

ϕ′(t) for t ∈ [−r,0].

Let u,v ∈ Λ. Then for each t ∈ [0,a], we have

|Pu(t) − Pv(t)|α 6 ‖D2G(t,xt)‖L(Cα,Xα) ‖ut − vt‖α

+Mα

∫t

0

‖D2F(s,xs)‖L(Cα,X)‖us − vs‖α
ds

(t − s)α

6

(
Lg + MαLf

a1−α

1 − α

)
|u − v|∞.

Then P is a strict contraction. Consequently, it has a unique mild solution y.

Define z : [−r,a] → Xα by

z(t) =






ϕ(0) +

∫t

0

y(s)ds for t ∈ [0,a]

ϕ(t) for t ∈ [−r,0],

we will show that z(t) = x(t) on [0,a].

For t ∈ [0,a], we have

z(t) = ϕ(0) +

∫t

0

T(s)(−A)(ϕ(0) − G(0,ϕ))ds +

∫t

0

T(s)F(0,ϕ)ds

+

∫t

0

D1G(s,xs) + D2G(s,xs)ysds

+

∫t

0

∫s

0

T(s − τ)[D1F(τ,xτ) + D2F(τ,xτ)yτ]dτds.

Moreover, we can see that

zt = ϕ +

∫t

0

ysds for t ∈ [0,a]. (9)



60 Taoufik Chitioui, Khalil Ezzinbi and Amor Rebey CUBO
15, 1 (2013)

Then t 7→ zt and t 7→
∫t
0
T(t − s)F(s,zs)ds are continuously differentiable on [0,a] and satisfy

d

dt

∫t

0

T(t − s)F(s,zs)ds = T(t)F(0,ϕ) +

∫t

0

T(t − s)[D1F(s,zs) + D2F(s,zs)ys]ds, (10)

then (10) yields
∫t

0

T(s)F(0,ϕ)ds =

∫t

0

T(t − s)F(s,zs)ds

−

∫t

0

∫s

0

T(s − τ)[D1F(τ,zτ) + D2F(τ,zτ)yτ]dτds.

On the other hand

G(t,zt) = G(0,ϕ) +

∫t

0

d

ds
G(s,zs)ds

= G(0,ϕ) +

∫t

0

D1G(s,zs) + D2G(s,zs)ysds.

Then

z(t) = T(t)(ϕ(0) − G(0,ϕ)) + G(t,zt) −

∫t

0

D1G(s,zs) + D2G(s,zs)ysds

+

∫t

0

T(t − s)F(s,zs)ds −

∫t

0

∫s

0

T(s − τ)[D1F(τ,zτ) + D2F(τ,zτ)yτ]dτds

+

∫t

0

(D1G(s,xs) + D2G(s,xs)ys)ds +

∫t

0

∫s

0

T(s − τ)[D1F(τ,xτ) + D2F(τ,xτ)yτ]dτds.

Therefore

|z(t) − x(t)|α ≤ |G(t,zt) − G(t,xt)|α +

∫t

0

|D1G(s,zs) − D1G(s,xs)|αds

+

∫t

0

|D2G(s,zs)ys − D2G(s,xs)ys|αds

+

∫t

0

|T(t − s)(F(s,zs) − F(s,xs))|αds

+

∫t

0

∫s

0

|T(s − τ)[D1F(τ,zτ) − D1F(τ,xτ)]|αdτds

+

∫t

0

∫s

0

|T(s − τ)[D2F(τ,zτ)yτ − D2F(τ,xτ)yτ]|αdτds.

Note that the sets {(s,zs) : s ∈ [0,a]} and {(s,xs) : s ∈ [0,a]} are compacts in [0,a] ×Cα, since

the mapping t → zt and t → xt are continuous on [0,a]. Then, we deduce that

‖D1G(s,zs) − D1G(s,xs)‖α 6 L1‖zs − xs‖α,

‖D2G(s,zs) − D2G(s,xs)‖L(Cα,Xα) 6 L2‖zs − xs‖α,

‖D1F(s,zs) − D1F(s,xs)‖ 6 L3‖zs − xs‖α,

‖D2F(s,zs) − D2F(s,xs)‖L(Cα,X) 6 L4‖zs − xs‖α,



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Existence and stability in the α-norm for nonlinear neutral ... 61

for all s ∈ [0,a], x ∈ Λ and z given in (9). Let L = max{Lf, L1,  L2, L3,L4}. Then

|z(t) − x(t)|α ≤ Lg + L
(
t + ‖y‖∞t +

Mα

1 − α
t1−α +

Mα

(1 − α)(2 − α)
t2−α

+
Mα‖y‖∞

(1 − α)(2 − α)
t2−α

)
sup

0≤s≤t

|z(s) − x(s)|α.

We can choose t0 ∈ [0,a] such that

Lg + L
(
t0 + ‖y‖∞t0 +

Mα

1 − α
t1−α0 +

Mα

(1 − α)(2 − α)
t2−α0 +

Mα‖y‖∞
(1 − α)(2 − α)

t2−α0

)
< 1.

we deduce that x = z on [0,t0]. We claim that x(t) = z(t) for t ∈ [0,a]. We proceed by

contradiction and assume that there exists t1 ∈ [0,a] such that x(t1) 6= z(t1). Let t
∗ be the

smallest number such that x(t) 6= z(t). Then

t∗ = inf{t ∈ [0,a] : |z(t) − x(t)|α > 0}.

By continuity, one has x(t) = z(t) for t ∈ [0,t∗] and there exists ε > 0 such that

|z(t) − x(t)|α > 0 for t ∈]t
∗,t∗ + ε[.

It follows for t ∈ [t∗,t∗ + ε] that

|z(t) − x(t)|α ≤ |G(t,zt) − G(t,xt)|α +

∫t

t∗
|D1G(s,zs) − D1G(s,xs)|αds

+

∫t

t∗
|D2G(s,zs)ys − D2G(s,xs)ys|αds

+

∫t

t∗
|T(t − s)(F(s,zs) − F(s,xs))|αds

+

∫t

t∗

∫s

t∗
|T(s − τ)[D1F(τ,zτ) − D1F(τ,xτ)]|αdτds

+

∫t

t∗

∫s

t∗
|T(s − τ)[D2F(τ,zτ)yτ − D2F(τ,xτ)yτ]|αdτds.

Consequently,

|z(t) − x(t)|α ≤ Lg + L
(
ε + ‖y‖∞ε +

Mα

1 − α
ε1−α +

Mα

(1 − α)(2 − α)
ε2−α

+
Mα‖y‖∞

(1 − α)(2 − α)
ε2−α

)
sup

t∗≤s≤t∗+ε

|z(s) − x(s)|α.

If we choose ε such that

Lg + L
(
ε + ‖y‖∞ε +

Mα

1 − α
ε1−α +

Mα

(1 − α)(2 − α)
ε2−α +

Mα‖y‖∞
(1 − α)(2 − α)

ε2−α
)
< 1

then x(t) = z(t) for t ∈ [t∗,t∗ + ε] which gives a contradiction. Consequently x(t) = z(t) for

t ∈ [0,a] and t 7→ xt is continuously differentiable in [0,a] and t 7→ F(t,xt) ∈ C
1([0,a],X). To end

the proof, we use the following lemma.



62 Taoufik Chitioui, Khalil Ezzinbi and Amor Rebey CUBO
15, 1 (2013)

Lemma 3.1. [24] Let h : [0,a] → X be continuously differentiable and u satisfy

u(t) = T(t)u0 +

∫t

0

T(t − s)h(s)ds for t ∈ [0,a].

If u0 ∈ D(A), then u is continuously differentiable on [0,a] and

u′(t) = −Au(t) + h(t) for t ∈ [0,a].

In our case, we have ϕ(0) − G(0,ϕ) ∈ D(A), t 7→ F(t,xt) is continuously differentiable on

[0,a] and

x(t) − G(t,xt) = T(t)
[
ϕ(0) − G(0,ϕ)

]
+

∫t

0

T(t − s)F(s,xs)ds for t ∈ [0,a].

By Lemma 3.1, the mapping t 7→ x(t) − G(t,xt) is continuously differentiable on [0,a] and for

t ∈ [0,a],
d

dt

[
x(t) − G(t,xt)

]
= −A

[
x(t) − G(t,xt)

]
+ F(t,xt) for t ∈ [0,a].

These implies that x is a strict solution of Eq. (2) on [0,a].

4 The solution semigroup in the autonomous case and the

linearized stability principle

In this section, we suppose that F and G are autonomous. Then Eq. (2) becomes





d

dt

[
x(t) − G(xt)

]
= −A

[
x(t) − G(xt)

]
+ F(xt) for t ≥ 0,

x0 = ϕ ∈ Cα.
(11)

We can see that the mild solutions of Eq. (11) satisfy the properties of a nonlinear strongly

continuous semigroup on Cα and we prove that this semigroup satisfies the translation property

and a Lipschitz property.

For each t ≥ 0, define the nonlinear operator U(t) on Cα by

U(t)(ϕ) = xt(.,ϕ)

where x(.,ϕ) is the unique mild solution of Eq. (11) for the initial condition ϕ ∈ Cα. One can

prove the proposition.

Proposition 4.1. Under the assumption as in the Theorem (2.4), the family (U(t))t>0 is a non-

linear strongly continuous semigroup on Cα. Moreover

(i) (U(t))t>0 satisfies the following translation property, for t > 0 and θ ∈ [−r,0],

(U(t)(ϕ))(θ) =

{
(U(t + θ)(ϕ))(0), if t + θ > 0

ϕ(t + θ), if t + θ 6 0



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Existence and stability in the α-norm for nonlinear neutral ... 63

(ii) for all T > 0, there are two functions p, q ∈ L∞([0,T],R+) such that, for all ϕ1, ϕ2 ∈ Cα,

‖U(t)(ϕ1) − U(t)(ϕ2)‖α ≤ p(t)e
q(t)‖ϕ1 − ϕ2‖α, t ∈ [0,T]. (12)

Proof. Proof of (ii). Let x1 := x(.,ϕ1), x
2 := x(.,ϕ2), T > 0 and M > 1 such that sup{‖T(t)‖, t ∈

[0,T]} ≤ M. For t ∈ [0,T], we have

‖U(t)(ϕ1) − U(t)(ϕ2)‖α = ‖x
1
t − x

2
t‖α

= sup
−r≤θ≤0

|x1(t + θ) − x2(t + θ)|α

≤ (M + MLg)‖ϕ1 − ϕ2‖α + Lg sup
−r≤θ≤0

‖x1t+θ − x
2
t+θ‖α

+ MLf

∫t

0

‖x1s − x
2
s‖αds.

Letting t ∈ [0,r]. Then, for θ ∈ [−r,0] such that t + θ ≥ 0, we have

‖x1t+θ − x
2
t+θ‖α = sup

−r≤τ≤0

|x1(t + θ + τ) − x2(t + θ + τ)|α

= sup
−r+t+θ≤τ≤t+θ

|x1(τ) − x2(τ)|α

= max{‖ϕ1 − ϕ2‖α, sup
0≤τ≤t+θ

|x1(τ) − x2(τ)|α}

≤ ‖ϕ1 − ϕ2‖α + ‖x
1
t − x

2
t‖α.

Then,

‖x1t − x
2
t‖α ≤ (

M + MLg + 1

1 − Lg
)‖ϕ1 − ϕ2‖α +

MLf

1 − Lg

∫t

0

‖x1s − x
2
s‖αds.

Using Gronwall’s lemma, we obtain

‖x1t − x
2
t‖α ≤ (

M + MLg + 1

1 − Lg
)e

MLf
1−Lg

t
‖ϕ1 − ϕ2‖α.

We can repeat the previous argument for t ∈ [r,2r], to see that for every t ∈ [r,2r],

‖U(t)(ϕ1) − U(t)(ϕ2)‖α ≤ ‖U(r)‖‖U(t − r)(ϕ1) − U(t − r)(ϕ2)‖α

≤ (
M + MLg + 1

1 − Lg
)2e

MLf
1−Lg

t
‖ϕ1 − ϕ2‖α.

For t ∈ [2r,3r]

‖U(t)(ϕ1) − U(t)(ϕ2)‖α ≤ ‖U(2r)‖‖U(t − 2r)(ϕ1) − U(t − 2r)(ϕ2)‖α

≤ (
M + MLg + 1

1 − Lg
)3e

MLf
1−Lg

t
‖ϕ1 − ϕ2‖α.



64 Taoufik Chitioui, Khalil Ezzinbi and Amor Rebey CUBO
15, 1 (2013)

Inductively, for t ∈ [nr,(n + 1)r] with n ≥ 2, we obtain

‖U(t)(ϕ1) − U(t)(ϕ2)‖α ≤ ‖U(nr)‖‖U(t − nr)(ϕ1) − U(t − nr)(ϕ2)‖α

≤ (
M + MLg + 1

1 − Lg
)n+1e

MLf
1−Lg

t
‖ϕ1 − ϕ2‖α.

Consequently, the estimate (12) is true. This ends the proof.

In what follows, we study the stability of an equilibrium of the following autonomous equation:





d

dt

[
D(xt) − G(xt)

]
= −A

[
D(xt) − G(xt)

]
+ F(xt) for t ≥ 0,

x0 = ϕ ∈ Cα,
(13)

where F and G are Lipschitz continuous on Cα with constants respectively LF and LG and D :

Cα −→ Xα is an operator defined by Dϕ = ϕ(0) − D0ϕ with D0 a bounded linear operator from

Cα into Xα such that LG + ‖D0‖ < 1.

We are now interested by the stability of the equilibriums of Equation (13). By equilibrium, we

mean a constant mild solution x∗ of (13). Without loss of generality, we can assume that x∗ = 0

and G(0) = F(0) = 0:

We need the following assumption.

(H6) F and G are Fréchet-differentiable at 0 and G′(0) = 0.

Let L = F′(0). Then, the linearized equation of Eq. (13) around the equilibrium 0 is the following:





d

dt
Dyt = −ADyt + L(yt) for t ≥ 0,

y0 = ϕ ∈ Cα.
(14)

Let (U(t))t>0 the nonlinear semigroup associated to Eq. (13) and the linear semigroup (V(t))t>0
associated to the linear equation (14) in the same space Cα. Then, we have the following result.

Theorem 4.1. Assume that the conditions (H0), (H2), (H4), (H5) and (H6) hold. Then, for

every t > 0 the derivative at zero of U(t) is V(t).

The proof of this theorem is based on the following fundamental lemma

Lemma 4.2. Let H : Cα −→ Xα be a continuous function such that there exists 0 < µ0 < 1

satisfying

|H(ϕ1) − H(ϕ2)|α 6 µ0‖ϕ1 − ϕ2‖α

Let ϕ ∈ Cα and h : [0,+∞[−→ Xα be a continuous function. Suppose that there exist continuous

functions x,y : [−r,+∞[−→ Xα such that

{
x(t) − y(t) = H(xt) − H(yt) + h(t), t ≥ 0,

x0 = y0 = ϕ.



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Existence and stability in the α-norm for nonlinear neutral ... 65

Then, for each 0 < T ≤ r we have

‖xt − yt‖α 6
1

1 − µ0
sup

06s6t

|h(s)|α, t ∈ [0,T].

Proof. For t ≥ 0, we have

‖xt − yt‖α = sup
−r≤θ≤0

|x(t + θ) − y(t + θ)|α

= sup
t−r≤s≤t

|x(s) − y(s)|α

= sup
0≤s≤t

|x(s) − y(s)|α

≤ sup
0≤s≤t

|H(xs) − H(ys)|α + sup
0≤s≤t

|h(s)|α

≤ µ0 sup
0≤s≤t

‖xs − ys‖α + sup
0≤s≤t

|h(s)|α

= µ0‖xt − yt‖α + sup
0≤s≤t

|h(s)|α

Then ‖xt − yt‖α 6
1

1 − µ0
sup

06s6t

|h(s)|α

Proof. (of Theorem 4.1) It suffices to show that for each ϕ ∈ Cα, t ≥ 0 and ε > 0, there exists

δ > 0 such that

‖U(t)ϕ − V(t)ϕ‖α 6 ε‖ϕ‖α, for ‖ϕ‖α 6 δ

Let t ≥ 0 be fixed and ϕ ∈ Cα. We have

(D − G)(U(t)ϕ) − D(V(t)ϕ)

=

∫t

0

T(t − s)[F(U(s)ϕ) − F(V(s)ϕ)]ds − T(t)G(ϕ)

+

∫t

0

T(t − s)[F(V(s)ϕ) − L(V(s)ϕ)]ds

Then,

(D − G)(U(t)ϕ) − (D − G)(V(t)ϕ)

= G(V(t)ϕ) − T(t)G(ϕ) +

∫t

0

T(t − s)[F(U(s)ϕ) − F(V(s)ϕ)]ds

+

∫t

0

T(t − s)[F(V(s)ϕ) − L(V(s)ϕ)]ds

Let x,y : [−r,+∞[−→ Xα and h : [0,+∞[−→ Xα be defined by

x(t) =

{
(U(t)ϕ)(0) if t ∈ [0,+∞[

ϕ(t) if t ∈ [−r,0]
y(t) =

{
(V(t)ϕ)(0) if t ∈ [0,+∞[

ϕ(t) if t ∈ [−r,0]



66 Taoufik Chitioui, Khalil Ezzinbi and Amor Rebey CUBO
15, 1 (2013)

and

h(t) = G(V(t)ϕ) − T(t)G(ϕ) +

∫t

0

T(t − s)[F(U(s)ϕ) − F(V(s)ϕ)]ds

+

∫t

0

T(t − s)[F(V(s)ϕ) − L(V(s)ϕ)]ds

Then, {
(D − G)(xt) − (D − G)(yt) = h(t), t ≥ 0,

x0 = y0 = ϕ.

which is equivalent to

{
x(t) − y(t) = (D0 + G)(xt) − (D0 + G)(yt) + h(t), t ≥ 0,

x0 = y0 = ϕ.

Using Lemma (4.2), we obtain

‖xt − yt‖α 6
1

1 − (LG + ‖D0‖)
sup

06s6t

|h(s)|α, t ≥ 0.

By virtue of the continuous differentiability of G and F at 0, we deduce that for ε > 0, there exists

δ > 0 such that

|G(V(t)ϕ) − T(t)G(ϕ)|α 6 ε‖ϕ‖α for ‖ϕ‖α 6 δ,

and

Mα

∫t

0

|F(V(s)ϕ) − L(V(s)ϕ)|
ds

(t − s)α
6 ε‖ϕ‖α for ‖ϕ‖α 6 δ.

Then, for ‖ϕ‖α 6 δ,

|h(t)|α ≤ 2ε‖ϕ‖α + MαLF

∫t

0

‖U(s)ϕ − V(s)ϕ‖α
ds

(t − s)α
,

Since for s ∈ [0,t] and t ∈ [0,r],

‖U(s)ϕ − V(s)ϕ‖α = sup
−r≤θ≤0

|xs(θ) − ys(θ)|α

= sup
−r+s≤τ≤s

|x(τ) − y(τ)|α

= sup
0≤τ≤s

|x(τ) − y(τ)|α

≤ sup
0≤τ≤t

|x(τ) − y(τ)|α

= ‖U(t)ϕ − V(t)ϕ‖α.

Then for t ∈ [0,r] fixed

‖U(t)ϕ − V(t)ϕ‖α ≤
2ε‖ϕ‖α

1 − (LG + ‖D0‖)
+

MαLF

1 − (LG + ‖D0‖)

∫t

0

‖U(s)ϕ − V(s)ϕ‖α
(t − s)α

ds



CUBO
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Existence and stability in the α-norm for nonlinear neutral ... 67

Using Gronwall’s lemma, we obtain

‖U(t)ϕ − V(t)ϕ‖α ≤
2ε‖ϕ‖α

1 − (LG + ‖D0‖)
exp(

MαLFt
1−α

(1 − (LG + ‖D0‖))(1 − α)
)

for ‖ϕ‖α 6 δ. We conclude that U(t) is differentiable at 0, for each t ∈ [0,T] and DϕU(t)(0) =

V(t).

Now, suppose that t ∈ [T,2T] fixed. It follows that, for max {‖ϕ‖α,‖U(t − T)(ϕ)‖α} ≤ δ0, where

δ0 > 0 is small enough

‖U(t)ϕ − V(t)ϕ‖α ≤ ‖U(T)U(t − T)(ϕ) − V(T)U(t − T)(ϕ)‖α

+‖V(T)‖‖U(t − T)(ϕ) − V(t − T)(ϕ)‖α

≤ ε‖ϕ‖α.

By steps, we conclude that U(t) is differentiable at 0, for each t ≥ 0 and DϕU(t)(0) = V(t).

Theorem 4.2. Under the assumption as in the Theorem (4.1), if the zero equi- librium of (V(t))t≥0
is exponentially stable, then the zero equilibrium of (U(t))t≥0 is locally exponentially stable, in the

sense that there exist δ > 0, µ > 0 and k ≥ 1 such that

‖U(t)(ϕ)‖α ≤ ke
−µt‖ϕ‖α for t ≥ 0 and ϕ ∈ Cα with ‖ϕ‖α ≤ δ.

Moreover, if Cα can be decomposed as Cα = H1
⊕

H2 where Hi are V-invariant subspaces of

Cα, H1 is fnite-dimensional and with

ω = lim
h→∞

1

h
log‖V(h)/H2‖α,

we have

inf{|λ| : λ ∈ σ(V(t)/H1)} > e
ωt,

then, the zero equilibrium of (U(t))t≥0 is not stable, in the sense that there exist ε > 0 and a se-

quence (ϕn)n converging to 0 and a sequence (tn)n of positive reel numbers such that ‖U(tn)ϕn‖α >

ε.

The proof of this theorem is based on Proposition 4.1 ,Theorem 4.1 and the following result.

Theorem 4.3. (Desch and Schappacher [13]). Let (V(t))t≥0 be a nonlinear strongly continuous

semigroup on a subset Ω of a Banach space Z. Assume that x0 ∈ Ω is an equilibrium of (V(t))t≥0

such that V(t) is Fréchet-differentiable at x0 for each t ≥ 0, with W(t) the derivative at x0 of

V(t), t ≥ 0. Then, (W(t))t≥0 is a strongly continuous semigroup of bounded linear operators

on Z and, if the zero equilibrium of (W(t))t≥0 is exponentially stable, then the equilibrium x0 of

(V(t))t≥0 is locally exponentially stable. Moreover, if Z can be decomposed as Z = Z1
⊕
Z2 where

Zi are W-invariant subspaces of Z and Z1 is fnite-dimensional and with

ω = lim
h→∞

1

h
log‖W(h)/Z2‖,



68 Taoufik Chitioui, Khalil Ezzinbi and Amor Rebey CUBO
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we have

inf{|λ| : λ ∈ σ(W(t)/Z1)} > e
ωt,

then, the zero equilibrium x0 of (V(t))t≥0 is not stable, in the sense that there exist ε > 0 and a

sequence (xn)n converging to x0 and a sequence (tn)n of positive reel numbers such that ‖V(tn)xn−

x0‖ > ε.

In the following, we will concentrate our study on the linear equation (14). Let (AV,D(AV))

be the generator of the semigroup (V(t))t≥0 on Cα. We have the result

Theorem 4.4. [4] Assume that the conditions (H0), (H2), (H4), (H5) and (H6) hold. Then,

the operator (AV,D(AV)) is given by

{
D(AV) = {ϕ ∈ Cα, ϕ

′ ∈ Cα, D(ϕ) ∈ D(A) and D(ϕ
′) = −AD(ϕ) + L(ϕ)},

AVϕ = ϕ
′, ϕ ∈ D(AV).

Let C be the space of continuous functions from [−r,0] into X provided with the uniform norm

topology and let

CD = {ϕ ∈ C : D(ϕ) = 0}.

Definition 4.3. [22] D is said to be stable if the zero solution of the difference equation

{
D(yt) = 0, t ≥ 0,

y0 = ϕ ∈ CD,

is exponentially stable.

Lemma 4.4. [4] If D is stable, then there exist positive constants a, b, c and d such that for any

ε ∈]0,r] sufficiently small and any continuous function h from [0,+∞[ into X, the solution v of

the equation

D(vt) = h(t), t ≥ 0,

satisfies the inequality

‖vt‖ ≤ e
−a(t−ε)

[
b‖v0‖ + c sup

0≤s≤ε

|h(s)|
]
+ d sup

max(ε,t−r)≤s≤t

|h(s)|, t ≥ ε. (15)

The estimate (15) is very interesting because, if |h(s)| is bounded on [0,+∞[, then the ultimate

bound on vt as t → +∞ is determined by the bound on |h(s)| for s in the delay interval [t − r,t]

as t → +∞.

Proposition 4.5. [20] Let D(ϕ) =

p∑

k=0

akϕ(−rk). Then, D is stable iff

p∑

k=0

|ak| < 1.

In the sequel, we assume that



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Existence and stability in the α-norm for nonlinear neutral ... 69

(H7) The operator D is stable.

Theorem 4.5. [4] Assume that (H0), (H1), (H2), (H4), (H5) and (H7) hold. Then the

semigroup (U(t))t≥0 can be decomposed as follows

U(t) = U1(t) + U2(t) for t ≥ 0,

where U1(t) is an exponentially stable semigroup on Cα and U2(t) is compact on Cα for every

t > 0.

Let (Y,‖.‖) be a Banach space. For a bounded linear operator B in Y, we define

‖B‖ess := inf{c > 0 : χ(B(H)) ≤ cχ(H), for every bounded set H of Y},

where χ(.) denotes the measure of noncompactness in Y.

The essential growth bound of (V(t))t≥0 in Cα is given by

ωess(V) := inf
t>0

1

t
log ‖V(t)‖ess.

It follows from Theorem 4.5, that

ωess(V) < 0.

Let

ω0(V) := inf
t>0

1

t
log ‖V(t)‖α

be the growth bound of (V(t))t≥0 in Cα. Then, it is well known (see [14]) that

ω0(V) = max{ωess(V),s
′(AV)},

where

s′(AV) = sup{Reλ : λ ∈ σ(AV)\σess(AV)}

and σess(AV) is the essential spectrum of AV . Consequently, the stability of (V(t))t≥0 is com-

pletely determined by s′(AV). Note that σ(AV)\σess(AV) contains a finite number of eigenvalues

of AV .

We say that λ ∈ C is a characteristic value of Equation (14) if there exists a nonzero x ∈ D(∆(λ))\{0}

such that ∆(λ)x = 0, where ∆(λ) is defined by

∆(λ) := λD(eλ.I) + AD(eλ.I) − L(eλ.I),

and the domain D(∆(λ)) is given by

D(∆(λ)) := {x ∈ Xα : D(e
λ.x) ∈ D(A) and AD(eλ.x) − L(eλ.x) ∈ Xα}.

Consequently, we deduce the following theorem.

Theorem 4.6. [4] Assume that (H0), (H1), (H2), (H4), (H5),(H6) and (H7) hold. Then,

the following assertions hold



70 Taoufik Chitioui, Khalil Ezzinbi and Amor Rebey CUBO
15, 1 (2013)

(i) λ is an eigenvalue of AV iff λ is a characteristic value of Equation (14).

(ii) If s′(AV) < 0, then (V(t))t≥0 is exponentially stable and consequently, the zero equilibrium

of (U(t))t≥0 is locally exponentially stable.

(iii) If s′(AV) = 0, then there exists ϕ ∈ Cα, ϕ 6= 0, such that ‖V(t)ϕ‖α = ‖ϕ‖α, for t ≥ 0.

(iv) If s′(AV) > 0, then there exists ϕ ∈ Cα such that ‖V(t)ϕ‖α → +∞ as t → +∞ and conse-

quently, the zero equilibrium of (U(t))t≥0 is instable.

(v) Assume that s′(AV) ≤ 0 and let s0(AV) := {λ ∈ Pσ(AV) : Reλ = 0}. If each λ in s0(AV) is

a pole of order 1 of the resolvent operator of AV, then (V(t))t≥0 is stable in the sense that

there exists a positive constant M such that ‖V(t)‖α ≤ M, for all t ≥ 0.

5 Example

To apply our theoretical results, we consider the following model of partial differential equation

with delay






∂

∂t

[
v(t,x) − qv(t − r,x) + g(

∂

∂x
v(t − r,x))

]
=
∂2

∂x2

[
v(t,x) − qv(t − r,x)

+g(
∂

∂x
v(t − r,x))

]
+ f

(
v(t − r,x),

∂

∂x
[v(t,x) − qv(t − r,x)]

)

for t ≥ 0 and x ∈ [0,π],

v(t,0) − qv(t − r,0) = v(t,π) − qv(t − r,π) = 0 for t ≥ 0,

v(θ,x) = v0(θ,x) for − r ≤ θ ≤ 0 and x ∈ [0,π],

(16)

where q, r are positive constants, u0 ∈ C([−r,0]× [0,π]; R) and f, g are Lipschitz continuous func-

tions. Let X := L2([0,π]; R) equipped with the L2-norm ‖.‖2. Consider the operator A : D(A) ⊂

X → X defined by Ay = −y
′′

with domain D(A) = H2(0,π) ∩ H10(0,π). The spectrum σ(−A) of

−A is equal to the point spectrum σp(−A) and is given by σ(−A) = σp(−A) = {−n
2 : n ≥ 1}

and the associated eigenfunctions (en)n≥1 are given by en(s) =
√

2
π

sin ns,s ∈ [0,π]. Then

Ay =

∞∑

n=1

n2(y,en)en, y ∈ D(A). For each y ∈ D(A
1
2 ) := {y ∈ X :

∞∑

n=1

n(y,en)en ∈ X} the

operator A
1
2 is given by A

1
2 y =

∞∑

n=1

n(y,en)en.

It is well known that −A is the infinitesimal generator of an analytic semigroup (T(t))t≥0 on



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Existence and stability in the α-norm for nonlinear neutral ... 71

X given by T(t)x =

∞∑

n=1

e−n
2
t(x,en)en, x ∈ X. It follows that (T(t))t≥0 is a compact semigroup

on X and 0 ∈ ρ(A). This implies that the Assumption (H0) and (H1) are satisfied.

Lemma 5.1. [27] If Y ∈ D(A
1
2 ), then Y is absolutely continuous, Y ′ ∈ X and ‖Y ′‖ = ‖A

1
2 Y‖.

Let G : C1
2
→ X be defined by

G(ϕ)(x) = qϕ(−r)(x) − g(
∂

∂x
ϕ(−r)(x)) for ϕ ∈ C1

2
and x ∈ [0,π],

and F : C1
2
→ X be defined by

F(ϕ)(x) = f
(
ϕ(−r)(x),

∂

∂x
[ϕ(0)(x) − qϕ(−r)(x)]

)
for ϕ ∈ C1

2
and x ∈ [0,π].

Lemma 5.2. [4, 27] F and G are Lipschitz continuous from C1
2
into X.

Let x(t) = v(t, .) for t ≥ 0 and ϕ(θ) = v0(θ,.) for θ ∈ [−r,0]. Then, Eq. (16) takes the

following abstract form





d

dt

(
x(t) − G(t,xt)

)
= −A

(
x(t) − G(t,xt)

)
+ F(t,xt) for t ≥ 0,

x0 = ϕ.
(17)

Consequently, we have the existence and uniqueness of the mild solution of Eq.(16). Let v0 ∈ C1
2

such that

(a) v0(0, .) − qv0(−r, .) + g(
∂

∂x
v0(−r, .)) ∈ H

2[0,π] ∩ H10[0,π] and
∂

∂θ
v0 ∈ C1

2
,

(b)
∂

∂θ
v0(0,x) − q

∂

∂θ
v0(−r,x) + g

′(
∂

∂x
v0(−r,x))

∂2

∂x∂θ
v0(−r,x)

= −A
[
v0(0,x) − qv0(−r,x) + g(

∂

∂x
v0(−r,x))

]

+f
(
v0(−r,x),

∂

∂x
[v0(0,x) − qv0(−r,x)]

)
for x ∈ [0,π]

We deduce that all assumptions of Theorem 3.1 are satisfied. Hence every mild solution of

Eq. (16) is a strict solution.

In the sequel, we assume that 0 < q < 1: This means that the operator D is stable. We also

assume that f and g are continuously differentiable and f(0,0) = 0, g(0) = 0 and g′(0) = 0. Which

implies that zero is a solution of 16 and the linearized equation at zero of Equation (16) has the

following form






∂

∂t

[
v(t,x) − qv(t − r,x)

]
=
∂2

∂x2

[
v(t,x) − qv(t − r,x)

]

+av(t − r,x) + b
∂

∂x
[v(t,x) − qv(t − r,x)] for t ≥ 0 and x ∈ [0,π],

v(t,0) − qv(t − r,0) = v(t,π) − qv(t − r,π) = 0 for t ≥ 0,

v(θ,x) = v0(θ,x) for − r ≤ θ ≤ 0 and x ∈ [0,π],

(18)



72 Taoufik Chitioui, Khalil Ezzinbi and Amor Rebey CUBO
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We obtain a region of stability of Equation (18) as a function of parameters a, b and q.

Lemma 5.3. [4] The spectrum σ(Ã) of the operator à =
∂2

∂x2
+b

∂

∂x
is equal to the point spectrum

σp(Ã) and is given by {−n
2 − b

2

4
: n ≥ 1}.

Theorem 5.1. Suppose that

a < 0 and 1 +
b2

4
+
a

q
≥ 0.

Then, for every r > 0, all characteristic values of Eq. (18) have negative real parts.

Proof. Suppose that a < 0. Then, the characteristic values of Eq. (18) are determined by the

expression

λ −
ae−λr

1 − qe−λr
= −n2 −

b2

4
, n ≥ 1. (19)

Let Kn = n
2 + b

2

4
, n ≥ 1. Then, Eq. (19) becomes

eλr(λ + kn) = λq + Knq + a.

This implies that

e2Re(λ)r
(
(Re(λ) + Kn)

2 + (Im(λ))2 = q2
(
(Re(λ) + Kn +

a

q
) + (Im(λ))2

)
.

On the other hand, under the conditions

a < 0 and 1 +
b2

4
+
a

q
≥ 0,

we have, for all n ≥ 1 and λ ∈ C,

Re(λ) + Kn > Re(λ) + Kn +
a

q
≥ Re(λ) + 1 +

b2

4
+
a

q
≥ Re(λ).

Then, if we assume that Re(λ) ≥ 0, we obtain that

e2Re(λ)r < q2,

which is a contradiction. Then, Re(λ) < 0.

Remark that the stability result is independent of the delay. Finally, as an immediate conse-

quence of the last theorem, we have the local stability of the zero equilibrium of Equation (16).

Proposition 5.4. Under the same assumptions as in Theorem 5.1, zero equilibrium of Equation

(16) is locally exponentially stable.

Received: October 2012. Revised: February 2013.



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Existence and stability in the α-norm for nonlinear neutral ... 73

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