Mostafa Adimy
email: mostafa.adimy@inria.fr

Mohamed Alia
email: monsieuralia@yahoo.fr

Khalil Ezzinbi
email: ezzinbi@ucam.ac.ma

FUNCTIONAL DIFFERENTIAL EQUATIONS WITH UNBOUNDED DELAY IN EXTRAPOLATION SPACES

Keywords: 2000 Mathematics Subject Classification. 35R10, 47D06. Key words and phrases. Neutral differential equation, infinite delay, extrapolation space, mild solution, semigroup, stability

We study the existence, regularity and stability of solutions for nonlinear partial neutral functional differential equations with unbounded delay and a Hille-Yosida operator on a Banach space X. We consider two nonlinear perturbations: the first one is a function taking its values in X and the second one is a function belonging to a space larger than X, an extrapolated space. We use the extrapolation techniques to prove the existence and regularity of solutions and we establish a linearization principle for the stability of the equilibria of our equation.

Introduction

In this work, we study the existence, regularity and stability of solutions for the neutral functional differential equations with infinite delay d dt [x(t) -F (x t )] = A[x(t) -F (x t )] + G(x t ) for t ≥ 0,

x(t) = ϕ(t), for t ≤ 0, ϕ ∈ B, (1.1)
where A : D(A) → X is a linear operator on a Banach space X. We assume that A is not necessarily densely defined and satisfies the Hille-Yosida condition. This means that A satisfies the usual assumptions of the Hille-Yosida theorem characterizing the generator of a C 0 -semigroup except the density of the domain D(A) in X: there exist N 0 ≥ 1 and ω 0 ∈ R such that (ω 0 , +∞) ⊂ ρ(A) and sup{(λ

-ω 0 ) n |(λI -A) -n | : n ∈ N, λ > ω 0 } ≤ N 0 ,
where ρ(A) is the resolvent set of the operator A. The phase space B is a linear space of functions from (-∞, 0] into X satisfying some assumptions which they will be described in the sequel. For every t ≥ 0, the function x t ∈ B is defined by

x t (θ) = x(t + θ) for θ ∈ (-∞, 0].
F is a Lipschitz continuous function from B to X, and G is a continuous function from B with values in the space F 1 , where F 1 , larger than X, is the extrapolation space associated to the C 0 -semigroup generated by the part of the operator A in X 0 = D(A) (see Section 2).

Wu and Xia [START_REF] Wu | Self-sustained oscillations in a ring array of coupled lossless transmission lines[END_REF][START_REF] Wu | Rotating waves in partial neutral functional differental equations[END_REF] studied a system of partial neutral functional differential equations defined on the unit circle S, which is a model for a continuous circular array of identical resistively coupled transmission lines with mixed initial boundary conditions. This system is ∂ ∂t [x(., t) -qx(., t -r)] = k ∂ 2 ∂ξ 2 [x(., t) -qx(., t -r)] + f (x t ) for t ≥ 0, x 0 = ϕ ∈ C([-r, 0]; H 1 (S)), (1.2) where x t (ξ, θ) = x(ξ, t + θ), -r ≤ θ ≤ 0, t ≥ 0, ξ ∈ S, k is a positive constant, and 0 ≤ q < 1. The space of initial data was chosen to be C([-r, 0]; H 1 (S)).

Motivated by this work, Hale [START_REF] Hale | Partial Neutral Functional Differential Equations[END_REF][START_REF] Hale | Coupled oscillators on a circle[END_REF] presented the basic theory of existence, uniqueness and properties of the solution operator of equation (1.2), as well as Hopf bifurcation and conditions for the stability and instability of periodic orbits.

Adimy, Ezzinbi and their collaborators [START_REF] Adimy | A class of linear partial neutral functional differential equations with nondense domain[END_REF][START_REF] Adimy | Strict solutions of nonlinear hyperbolic neutral differential equations[END_REF][START_REF] Adimy | Existence and linearized stability for partial neutral functional differential equations with nondense domains[END_REF][START_REF] Adimy | Spectral decomposition for partial neutral functional differential equation[END_REF][START_REF] Benkhalti | Existence and stability for some partial neutral functional differential equations[END_REF] considered (1.1) with finite delay and the function G taking its values in the Banach space X. They established the basic theory of existence, uniqueness, stability and some properties of the solution operator. In the literature devoted to partial functional differential equations with finite delay r > 0, the state space is always the space of continuous functions on [-r, 0], and the variation of constants formula is the main tool for studying the properties of the solution operator. For more details, we refer to Travis and Webb [START_REF] Travis | Existence and stability for partial functional differential equations[END_REF], and Wu [START_REF] Wu | Theory and Applications of Partial Functional Differential Equations[END_REF].

When the delay is unbounded the situation is more complicated since the properties of the solutions depend on the phase space B. The choice of this space plays an important role in both quantitative and qualitative studies. A usual choice of B is a Banach space satisfying some assumptions which make the system well-posed. For the basic theory of functional differential equations with infinite delay in finite dimensional spaces, we refer to Hale and Kato [START_REF] Hale | Phase space for retarded equations with infinite delay[END_REF], and Hino, Murakami and Naito [START_REF] Hino | Functional Differential Equations with Infinite Delay[END_REF]. This theory was extended to partial functional differential equations with infinite delay by Henriquez [START_REF] Henriquez | Periodic solutions of quasi-linear partial functional differential equations with unbounded delay[END_REF] in 1994. Since then, many other authors investigated partial functional differential equations with infinite delay by considering different phase spaces B.

In [START_REF] Adimy | Existence and stability for some partial neutral functional differential equations with infinite delay[END_REF][START_REF] Chang | Existence of some neutral partial differential equations with infinite delay[END_REF][START_REF] Ezzinbi | Existence and regularity of solutions for neutral partial functional integrodifferential equations with infinite delay[END_REF][START_REF] Hernandez | Existence results for partial neutral functional differential equations with unbounded delay[END_REF], the authors studied some classes of partial neutral functional differential equations with unbounded delay. In [START_REF] Adimy | Existence and stability for some partial neutral functional differential equations with infinite delay[END_REF] Adimy, Bouzahir and Ezzinbi used the theory of integrated semigroups to study the existence and uniqueness of mild solutions for a class of partial neutral functional differential equations with unbounded delay. Chang [START_REF] Chang | Existence of some neutral partial differential equations with infinite delay[END_REF] considered a generator of an analytic compact C 0semigroup and assumed that the nonlinear part is continuous with respect to fractional powers of this generator. He studied the existence and uniqueness of solutions of partial neutral functional differential equations with unbounded delay. Ezzinbi, Ghnimi and Taoudi [START_REF] Ezzinbi | Existence and regularity of solutions for neutral partial functional integrodifferential equations with infinite delay[END_REF] introduced a new concept of the resolvent operator adapted to a class of non-autonomous partial neutral functional differential equations with unbounded delay. They gave some basic results on the existence and uniqueness of solutions. Hernandez and Henriquez [START_REF] Hernandez | Existence results for partial neutral functional differential equations with unbounded delay[END_REF] established some results of existence of periodic solutions for a class of partial neutral functional differential equations with unbounded delay and appropriate nonlinear functions defined on a phase space.

Ezzinbi [START_REF] Ezzinbi | Existence and stability for some partial functional differential equations with infinite delay[END_REF] investigated (1.1) in the particular case where F = 0, the function G is continuous from B with values in the extrapolation space F 1 and the delay is unbounded. He used an approach based on the theory of the extrapolation spaces to study the existence, uniqueness, regularity and asymptotic behavior of solutions.

The theory of extrapolation spaces was introduced by Da Prato and Grisvard [START_REF] Da Prato | On extrapolation spaces[END_REF] in 1982 (see also Engel and Nagel [10]). It was used by Nagel and Sinestrari [START_REF] Nagel | Inhomogeneous Volterra integrodifferential equations for Hille-Yosida operators[END_REF] for a class of Volterra Integrodifferential equations with Hille-Yosida operators, and by Maniar and Rhandi [START_REF] Maniar | Extrapolation and inhomogeneous retarded differential equations in infinite dimensional Banach spaces via extrapolation spaces[END_REF] for retarded differential equations in infinite dimensional spaces. The use of this theory allows to consider nonlinear perturbations belonging to a class of spaces, larger than the space in which the unperturbed system is defined.

The main tools used to investigate (1.1) are based on the variation of constants formula. The nonlinear functions F and G are not defined in the same space. Then, we cannot consider the classical variation of constants formula introduced in our previous works. We use the extrapolation methods introduced in [START_REF] Da Prato | On extrapolation spaces[END_REF] to construct a new variation of constants formula adapted to (1.1). Then, we study the existence and regularity of mild solutions of (1.1). We establish a linearization principle for the stability of the equilibria. For the regularity of mild solutions, we adapt the method developed in [START_REF] Travis | Existence and stability for partial functional differential equations[END_REF] for partial functional differential equations with finite delay and to establish the linearization principle, we use an approach developed in [START_REF] Parrot | Positivity and a principle of linearized stability for delay differential equations[END_REF] and [START_REF] Ruess | Linearized stability of abstract differential equations with delay[END_REF]. This work is an extension of [START_REF] Ezzinbi | Existence and stability for some partial functional differential equations with infinite delay[END_REF] to partial neutral functional differential equations with infinite delay.

Extrapolation spaces and Favard class

Throughout this article, we assume that the operator A : D(A) ⊂ X → X satisfies the Hille-Yosida condition on a Banach space X:

(H1) there exist N 0 ≥ 1 and ω 0 ∈ R such that (ω 0 , +∞) ⊂ ρ(A) and sup{(λ

-ω 0 ) n |(λI -A) -n | : n ∈ N, λ > ω 0 } ≤ N 0 .
Let A 0 be the part of A in X 0 := D(A) which is defined by

D(A 0 ) = {x ∈ D(A) : Ax ∈ D(A)}, A 0 x = Ax for x ∈ D(A 0 ).
Then, we have the following classical result.

Lemma 2.1 ([10]). A 0 generates a C 0 -semigroup (T 0 (t)) t≥0 on X 0 with |T 0 (t)| ≤ N 0 e ω0t , for t ≥ 0. Moreover, ρ(A) ⊂ ρ(A 0 ) and R(λ, A 0 ) = R(λ, A) X0 for λ ∈ ρ(A), where R(λ, A) X0 is the restriction of R(λ, A) to X 0 .
For a fixed λ 0 ∈ ρ(A), we introduce on X 0 the norm

|x| -1 = |R(λ 0 , A 0 )x| for x ∈ X 0 . The completion X -1 of (X 0 , | • | -1
) is called the extrapolation space of X associated with the operator A. The norm |•| -1 , associated with λ 0 ∈ ρ(A) and any other norm on X 0 given for λ ∈ ρ(A) by |R(λ, A 0 )x| are equivalent. The operator T 0 (t) has a unique bounded linear extension T -1 (t) to the Banach space X -1 and (T -1 (t)) t≥0 is a C 0 -semigroup on X -1 . (T -1 (t)) t≥0 is called the extrapolated semigroup of (T 0 (t)) t≥0 . We denote by (A -1 , D(A -1 )) the generator of (T -1 (t)) t≥0 on the space X -1 .

For a Banach space Y , we denote by L(Y ) the space of bounded linear operators on Y . We have the following fundamental results.

Lemma 2.2 ([10]

). The following properties hold:

(i) |T -1 (t)| L(X-1) = |T 0 (t)| L(X0) ; (ii) D(A -1 ) = X 0 ; (iii) A -1 : X 0 → X -1 is the unique continuous extension of the operator A 0 : D(A 0 ) ⊆ (X 0 , |.|) → (X -1 , |.| -1 ) and (λ 0 I -A -1 ) is an isometry from (X 0 , | • |) to (X -1 , | • | -1 ); (iv) If λ ∈ ρ(A), then (λI -A -1 ) is invertible and (λI -A -1 ) -1 ∈ L(X -1 ). In particular, λ ∈ ρ(A -1 ) and R(λ, A -1 ) X0 = R(λ, A 0 ); (v) The space X 0 = D(A) is dense in (X -1 , | • | -1 ). Hence, the extrapolation space X -1 is also the completion of (X, | • | -1 ) and we have X ֒→ X -1 ; (vi) The operator A -1 is an extension of the operator A. In particular, if λ ∈ ρ(A) then R(λ, A -1 ) X0 = R(λ, A) and R(λ, A -1 )(X) = D(A).
Next we introduce the Favard class of the C 0 -semigroup (T 0 (t)) t≥0 .

Definition 2.3 ([10]

). Let (S(t)) t≥0 be a C 0 -semigroup on a Banach space Y such that |S(t)| ≤ N e νt for some N ≥ 1 and ν ∈ R. The Favard class of (S(t)) t ≥0 is the space

F = {x ∈ Y : sup t>0 1 t |e -νt S(t)x -x| < +∞}.
This space equipped with the norm

|x| F = |x| + sup t>0 1 t |e -νt S(t)x -x| , is a Banach space.
For the rest of this article, we denote by F 0 ⊂ X 0 the Favard class of the C 0semigroup (T 0 (t)) t≥0 and by F 1 the Favard class of the C 0 -semigroup (T -1 (t)) t≥0 .

Lemma 2.4 ([10]

). For the Banach spaces F 0 and F 1 the following properties hold:

(i) (λ 0 I -A -1 )(F 0 ) = F 1 ; (ii) T -1 (t)(F 1 ) ⊂ F 1 for t ≥ 0; (iii) D(A 0 ) ֒→ D(A) ֒→ F 0 ֒→ X 0 ֒→ X ֒→ F 1 ֒→ X -1
, where D(A) is equipped with the graph norm.

Proposition 2.5 ([20]). For f ∈ L 1 loc (R + , F 1 ), we define (T -1 * f )(t) = t 0 T -1 (t -s)f (s)ds for t ≥ 0. Then (i) (T -1 * f )(t) ∈ X 0 for all t ≥ 0; (ii) |(T -1 * f )(t)| ≤ M e ωt t 0 e -ωs |f (s)| F1 ds, where M is a constant independent of f ; (iii) lim t→0 |(T -1 * f )(t)| = 0.
Remark 2.6. Assertion (iii) in Proposition 2.5 implies that the function

T -1 * f : t → t 0 T -1 (t -s)f (s)ds is continuous from R + to X 0 .

Existence, uniqueness and regularity of solutions

Let B be the phase space of (1.1). That is a linear space of functions from (-∞, 0] into X satisfying the following two assumptions (see [START_REF] Hale | Phase space for retarded equations with infinite delay[END_REF]).

(A1) There exist a constant H > 0 and functions K, M : R + → R + with K continuous and M ∈ L ∞ loc (R + ) such that for all σ ∈ R and any a > 0 if

x : (-∞, σ + a] → X is such that x σ ∈ B and x : [σ, σ + a] → X is continuous, then for all t ∈ [σ, σ + a] we have (i) x t ∈ B, (ii) |x(t)| ≤ H|x t | B , (iii) |x t | B ≤ K(t -σ) sup σ≤s≤t |x(s)| + M (t -σ)|x σ | B . (A2) the function t → x t is continuous from [σ, σ + a] to B. (B1) B is complete.
For the nonlinear functions F and G, we assume that they are Lipschitz continuous.

(H2) F is a Lipschitz continuous function from B to X:

|F (ϕ 1 ) -F (ϕ 2 )| ≤ L 0 |ϕ 1 -ϕ 2 | B for ϕ 1 , ϕ 2 ∈ B. (H3) G is a Lipschitz continuous function from B to F 1 : |G(ϕ 1 ) -G(ϕ 2 )| F1 ≤ L 1 |ϕ 1 -ϕ 2 | B for ϕ 1 , ϕ 2 ∈ B.
All the results in this work are obtained by assuming that the function K satisfies (A1), and the Lischitz constant L 0 in (H2) satisfies

(H4) L 0 K(0) < 1.
This assumption implies that there exists a > 0 such that L 0 K a < 1 where K a = sup 0≤t≤a (K(t)).

We need the following fundamental prior estimation.

Lemma 3.1. Assume that (H2) and (H4) hold and let a > 0 be such that L 0 K a < 1.

Let ψ ∈ B and h ∈ C([0, a]; X) be such that ψ(0) -F (ψ) = h(0). Then, there exists a unique continuous function x on [0, a] such that

x(t) -F (x t ) = h(t) for t ∈ [0, a], x(t) = ψ(t) for t ∈ (-∞, 0]. (3.1)
Moreover, there exist α a > 0 and β a > 0 such that

|x t | B ≤ α a |ψ| B + β a sup 0≤s≤t |h(s)| for t ∈ [0, a]. (3.2)
Proof. We introduce the space

Y = {x ∈ C([0, a]; X) : x(0) = ψ(0)}
endowed with the uniform norm topology. For x ∈ Y , we define its extension x over (-∞, 0] by

x(t) = x(t) for t ∈ [0, a], ψ(t) for t ∈ (-∞, 0].
Then, by (A2), the function t → xt is continuous from [0, a] to B. Let us now define the operator K by (K(x))(t) = F (x t ) + h(t) for t ≥ 0.

We have to show that K has a unique fixed point on Y . Since h ∈ C([0, a]; X) and

ψ(0) -F (ψ) = h(0), then K(Y ) ⊂ Y . Moreover, |(K(x) -K(y))(t)| ≤ L 0 |x t -ỹt | B
By the property L 0 K a < 1, we obtain that K is a strict contraction. By a Banach fixed point theorem, we deduce the existence and uniqueness of x, solution of Problem (3.1) on the interval (-∞, a]. By (A1)-(iii), for t ∈ [0, a] we have

|x t | B ≤ K a sup 0≤s≤t |x(s)| + M a |x 0 | B , where M a = sup 0≤s≤a M (s).
It follows that

|x t | B ≤ K a sup 0≤s≤t (|F (x s )| + |h(s)|) + M a |ψ| B , ≤ K a (L 0 sup 0≤s≤t (|x s -ψ| B ) + |F (ψ)| + sup 0≤s≤t |h(s)|) + M a |ψ| B . Since ψ(0) -F (ψ) = h(0), we deduce that (1 -K a L 0 )|x t | B ≤ (K a L 0 + M a )|ψ| B + K a sup 0≤s≤t |h(s)| + K a (|ψ(0)| + |h(0)|).
By (A1)-(ii), we obtain

(1 -K a L 0 )|x t | B ≤ (K a L 0 + M a + K a H)|ψ| B + 2K a sup 0≤s≤t |h(s)|.
Finally, we arrive to

|x t | B ≤ α a |ψ| B + β a sup 0≤s≤t |h(s)| for t ∈ [0, a],
where

α a = K a (L 0 + H) + M a 1 -K a L 0 and β a = 2K a 1 -K a L 0 .
Definition 3.2. Let a > 0. A function x : (-∞, a] → X is called a mild solution of (1.1) on (-∞, a] if x is continuous and satisfies

x(t) -F (x t ) = T 0 (t)[ϕ(0) -F (ϕ)] + t 0 T -1 (t -s)G(x s )ds for t ∈ [0, a], x(t) = ϕ(t) for t ∈ (-∞, 0]. (3.3)
Theorem 3.3. Assume that (H1), (H2), (H3), (H4) hold. Let a > 0 be fixed such that L 0 K a < 1. Then, for ϕ ∈ B such that

ϕ(0) -F (ϕ) ∈ X 0 , Equation (1.1) has a unique mild solution x(., ϕ) defined on (-∞, a]. Moreover, if L 0 K ∞ < 1, where K ∞ = sup t≥0 (K(t)), then the unique mild solution x(., ϕ) is defined on (-∞, ∞).
Proof. As in the proof of Lemma 3.1, consider the set

Y = {x ∈ C([0, a]; X) : x(0) = ϕ(0)},
and the extension x of x ∈ Y over (-∞, 0] defined by

x(t) = x(t) for t ∈ [0, a], ϕ(t) for t ∈ (-∞, 0].
Then by (A2), the function t → xt is continuous. Let us now define the operator H by

(H(x))(t) = F (x t ) + T 0 (t)[ϕ(0) -F (ϕ)] + t 0 T -1 (t -s)G(x s )ds for t ∈ [0, a].
We have to show that H has a unique fixed point on Y . In fact, by Proposition 2.5, H(Y ) ⊂ Y . Moreover, for t ∈ [0, a], we have

|(H(x) -H(y))(t)| ≤ L 0 |x t -ỹt | B + N 0 t 0 e ω0(t-s) |G(x s ) -G(ỹ s )| F1 ds.
Hence

|(H(x) -H(y))(t)| ≤ L 0 |x t -ỹt | B + N 0 L 1 t 0 e ω0(t-s) |x s -ỹs | B ds.
Without loss of generality, we suppose that

ω 0 > 0. Let b ∈ (0, a]. Then, for t ∈ [0, b] |(H(x) -H(y))(t)| ≤ L 0 K a sup 0≤s≤a |x(s) -y(s)| + N 0 L 1 bK a e ω0a sup 0≤s≤a |x(s) -y(s)|.
Consequently,

|(H(x) -H(y))(t)| ≤ (L 0 K a + N 0 L 1 bK a e ω0a ) sup 0≤s≤a |x(s) -y(s)|.
We choose b ∈ (0, a] such that

K a L 0 + N 0 L 1 bK a e ω0a < 1.
Then, H is a strict contraction for t ∈ We study now the regularity of the solution. We give a sufficient condition for the mild solution of (1.1) to be continuously differentiable and to satisfy an abstract differential equation. We need some preliminary results on the space B. To this end, we suppose the additionally assumption.

(C1) If (ϕ n ) n≥0 is a Cauchy sequence in B and if (ϕ n ) n≥0 converges compactly to ϕ on (-∞, 0], then ϕ is in B and |ϕ n -ϕ| B → 0 as n → +∞. We recall the following result.

Lemma 3.4 ([21]

). Let B be satisfy (C1) and f : [0, a] → B be a continuous function such that the function

(t, θ) → f (t)(θ) is continuous on [0, a] × (-∞, 0]. Then a 0 f (t)dt (θ) = a 0 f (t)(θ)dt for θ ∈ (-∞, 0].
For the regularity of the mild solutions, we add the following hypotheses on F , G and the initial condition.

(H5) F : B → X is continuously differentiable and F ′ is locally Lipschitz continuous. (H6) G : B → F 1 is continuously differentiable and G ′ is locally Lipschitz continuous.

(H7) ϕ ∈ C 1 ((-∞, 0]; X) ∩ B, ϕ ′ ∈ B, ϕ(0) -F (ϕ) ∈ F 0 , ϕ ′ (0) -F ′ (ϕ)ϕ ′ ∈ D(A) and ϕ ′ (0) -F ′ (ϕ)ϕ ′ = A -1 [ϕ(0) -F (ϕ)] + G(ϕ).
We have the following theorem. Theorem 3.5. Assume that (H1)-(H7) hold and let a > 0 be such that L 0 K a < 1. Then, the mild solution x of (1.1) on (-∞, a] with x 0 = ϕ ∈ B, belongs to

C 1 ([0, a], X) ∩ C([0, a], F 0 ) and satisfies d dt [x(t) -F (x t )] = A -1 [x(t) -F (x t )] + G(x t ) for t ∈ [0, a]. (3.4)
The proof of this theorem is based on the following fundamental lemma.

Lemma 3.6 ([20, Corollary 3.5]). Let u : R + → X be defined by

u(t) = T 0 (t)u 0 + t 0 T -1 (t -s)f (s)ds for t ≥ 0. (3.5) If u 0 ∈ F 0 and f ∈ W 1,1 (R + , F 1 ) such that A -1 u 0 + f (0) ∈ D(A), then u ∈ C 1 (R + , X) ∩ C(R + , F 0 ) and satisfies d dt u(t) = A -1 u(t) + f (t) for t ≥ 0.
Proof of Theorem 3.5. Let x be the mild solution of (1.1) on [0, a]. Consider the function

y(t) =        F ′ (x t )y t + T 0 (t)[ϕ ′ (0) -F ′ (ϕ)ϕ ′ ] + t 0 T -1 (t -s)G ′ (x s )y s ds for t ∈ [0, a], ϕ ′ (t), for t ∈ (-∞, 0]. (3.6)
Using the strict contraction principle, we show that (3.6) has a unique solution y on (-∞, a]. Let z : (-∞, a] → X be defined by

z(t) = ϕ(0) + t 0 y(s) ds for t ∈ [0, a], ϕ(t) for t ∈ (-∞, 0].
As a consequence of Lemma 3.4, we see that

z t = ϕ + t 0 y s ds for t ∈ [0, a].
To complete the proof, we have to show that

x = z on [0, a]. Since s → G(z s ) is continuously differentiable with the X -1 -norm, d dt t 0 T -1 (t -s)G(z s )ds = T -1 (t)G(ϕ) + t 0 T -1 (t -s)G ′ (z s )y s ds.
This implies that

t 0 T -1 (s)G(ϕ)ds = t 0 T -1 (t-s)G(z s )ds- t 0 s 0 T -1 (s-τ )G ′ (z τ )y τ dτ ds. (3.7)
Let t ∈ [0, a] and define

z 1 (t) = x(t) -F (x t ), z 2 (t) = z(t) -F (z t ).
We have

z 2 (t) -z 2 (0) = t 0 z ′ 2 (s)ds = t 0 (z ′ (s) -F ′ (z s )y s )ds, = t 0 y(s)ds - t 0 F ′ (z s )y s ds.
It follows that

z 2 (t) = ϕ(0) -F (ϕ) + t 0 F ′ (x s )y s ds + t 0 T -1 (s)(A -1 [ϕ(0) -F (ϕ)] + G(ϕ))ds + t 0 s 0 T -1 (s -τ )G ′ (x τ )y τ dτ ds - t 0 F ′ (z s )y s ds. Since t 0 T -1 (s)(A -1 [ϕ(0) -F (ϕ)])ds = T -1 (t)(ϕ(0) -F (ϕ)) -(ϕ(0) -F (ϕ)),
we obtain

z 2 (t) = ϕ(0) -F (ϕ) + t 0 (F ′ (x s ) -F ′ (z s ))y s ds + T -1 (t)(ϕ(0) -F (ϕ)) -(ϕ(0) -F (ϕ)) + t 0 T -1 (s)G(ϕ)ds + t 0 s 0 T -1 (s -τ )G ′ (x τ )y τ dτ ds.
Using (3.7), we obtain

z 2 (t) = T 0 (t)(ϕ(0) -F (ϕ)) + t 0 (F ′ (x s ) -F ′ (z s ))y s ds + t 0 T -1 (s)G(ϕ)ds + t 0 s 0 T -1 (s -τ )G ′ (x τ )y τ dτ ds, and
z 2 (t) = T 0 (t)(ϕ(0) -F (ϕ)) + t 0 (F ′ (x s ) -F ′ (z s ))y s ds + t 0 T -1 (t -s)G(z s )ds + t 0 s 0 T -1 (s -τ )(G ′ (x τ ) -G ′ (z τ ))y τ dτ ds. Since z 1 (t) = x(t) -F (x t ) = T 0 (t)[ϕ(0) -F (ϕ)] + t 0 T -1 (t -s)G(x s )ds,
we deduce that

z 2 (t) -z 1 (t) = t 0 T -1 (t -s)(G(z s ) -G(x s ))ds + t 0 (F ′ (x s ) -F ′ (z s ))y s ds + t 0 s 0 T -1 (s -τ )(G ′ (x τ ) -G ′ (z τ ))y τ dτ ds.
The local Lipschitz conditions on F ′ and G ′ imply that there is a positive constant k 0 such that

|z 2 (t) -z 1 (t)| ≤ k 0 t 0 |x s -z s | B ds.
Consequently,

|x(t) -z(t)| ≤ L 0 |x t -z t | B + k 0 t 0 |x s -z s | B ds, ≤ L 0 K a sup 0≤s≤t |x(s) -z(s)| + k 0 t 0 |x s -z s | B ds. It follows that sup 0≤s≤t |x(s) -z(s)| ≤ k 0 1 -L 0 K a t 0 |x s -z s | B ds.
Then, we obtain

|x t -z t | B ≤ K a sup 0≤s≤t |x(s) -z(s)| ≤ k 0 K a 1 -L 0 K a t 0 |x s -z s | B ds.
By Gronwall's lemma, we deduce that

x t = z t on [0, a]. Consequently, x(t) = z(t) for all t ∈ (-∞, a]. Hence, t → x t is continuously differentiable on [0, a]. We deduce that t → G(x t ) ∈ C 1 ([0, a], F 1 )
. By Lemma 3.6, we conclude that the function t → x(t) -F (x t ) belongs to C 1 ([0, a], X) ∩ C([0, a], F 0 ) and satisfies (3.4) for all t ∈ [0, a].

The solution semigroup and the principle of linearized stability

In this section, we assume that (H2') the function F is a bounded linear operator from B to X with L 0 its norm, and (H4') L 0 K ∞ < 1. Let H be the phase space of (1.1) given by

H = {ϕ ∈ B : ϕ(0) -F (ϕ) ∈ X 0 }.
Define the operator U (t) on H, for t ≥ 0, by U (t)(ϕ) = x t (., ϕ), where x(., ϕ) is the mild solution of (1.1) on R. Then, we have the following proposition.

Proposition 4.1. The family (U (t)) t≥0 is a nonlinear strongly continuous semigroup on H; that is,

(i) U (0) = I; (ii) U (t + s) = U (t)U (s), for t, s ≥ 0; (iii) for all ϕ ∈ H, U (t)(ϕ) is a continuous function of t ≥ 0 with values in H;
(iv) U (t) satisfies, for t ≥ 0, θ ∈ (-∞, 0] and ϕ ∈ H, the translation property

U (t)(ϕ)(θ) = (U (t + θ)(ϕ))(0) if t + θ ≥ 0, ϕ(t + θ) if t + θ ≤ 0, (v) for each a > 0, there exists a function m ∈ L ∞ ((0, a), R + ) such that |U (t)ϕ -U (t)ψ| B ≤ m(t)|ϕ -ψ| B for t ∈ [0, a] and ϕ, ψ ∈ H.
Proof. (i) and (ii) are a consequence of the uniqueness of the solution. (iii) comes from the fact that the solution is continuous for every t ≥ 0. (iv) is a consequence of the definition of U . To prove (v), consider ϕ, ψ ∈ H and their associated solutions x and y. Then, for t ≥ 0, we have

x(t) = F (x t ) + T 0 (t)[ϕ(0) -F (ϕ)] + t 0 T -1 (t -s)G(x s )ds, y(t) = F (y t ) + T 0 (t)[ψ(0) -F (ψ)] + t 0 T -1 (t -s)G(y s )ds.
It follows that

|x(t) -y(t)| ≤ L 0 |x t -y t | B + N 0 e ω0t (H + L 0 )|ϕ -ψ| B + N 0 L 1 e ω0t t 0 |x s -y s | B ds. Then (1-K ∞ L 0 )|x t -y t | B ≤ K ∞ N 0 e ω0t (H +L 0 )|ϕ -ψ| B +N 0 K ∞ L 1 e ω0t t 0 |x s -y s | B ds.
By Gronwall's lemma, we conclude that, for every t ≥ 0, U (t) is a Lipschitz continuous function.

By an equilibrium, we mean a constant solution x * of (1.1). Without loss of generality, we suppose that x * = 0. Then, we assume that (H8) G(0) = 0 and G is continuously differentiable at zero. Then the linearized equation at zero of (1.1) is given by

d dt [y(t) -F (y t )] = A[y(t) -F (y t )] + G ′ (0)y t for t ≥ 0, y 0 = ϕ ∈ H. (4.1)
Let (V (t)) t≥0 be the C 0 -semigroup solution on H of (4.1).

Theorem 4.2. Assume that (H1), (H2'), (H3), (H4'), (H8) hold. Then, for t ≥ 0, the derivative at zero of U (t) is V (t).

Proof. Let ϕ ∈ H. Consider the unique solution x (resp. y) on R of (

). Then, for t ≥ 0, we have

x(t) = F (x t ) + T 0 (t)[ϕ(0) -F (ϕ)] + t 0 T -1 (t -s)G(x s )ds, y(t) = F (y t ) + T 0 (t)[ϕ(0) -F (ϕ)] + t 0 T -1 (t -s)G ′ (0)y s ds,
and, for t ≤ 0, x(t) = y(t) = ϕ(t). Let t ≥ 0. Then

x(t) -y(t) = F (x t ) -F (y t ) + t 0 T -1 (t -s)(G(x s ) -G ′ (0)y s )ds.
Hence

x(t) -y(t) = F (x t ) -F (y t ) + t 0 T -1 (t -s)(G(y s ) -G ′ (0)y s )ds + t 0 T -1 (t -s)(G(x s ) -G(y s ))ds.
Using (v) of Proposition 4.1 and thanks to the differentiability property of the function G at 0, we see that for ε > 0, there exists η > 0 such that

|G(y t ) -G ′ (0)y t | F1 ≤ ε|ϕ| B for |ϕ| B ≤ η and t ≥ 0.
This implies that, by (A1)-(iii), there exist constants k 0 and k such that for t ≥ 0

|x t -y t | B ≤ K ∞ |x(t) -y(t)| ≤ L 0 K ∞ |x t -y t | B + k 0 K ∞ ε|ϕ| B + kK ∞ t 0 |x s -y s | B .
Then

|x t -y t | B ≤ k 0 K ∞ 1 -K ∞ L 0 ε|ϕ| B + kK ∞ t 0 |x s -y s | B .
By Gronwall's lemma, we obtain

|x t -y t | B ≤ ε|ϕ| B for |ϕ| B ≤ η.
We conclude that U (t) is differentiable at 0 and D ϕ U (t)(0) = V (t) for t ≥ 0.

Finally, we obtain the important result.

Theorem 4.3. Assume that (H1), (H2'), (H3), (H4'), (H8) hold. If the semigroup (V (t)) t≥0 on H 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 then, the equilibrium w is not stable in the sense that there exist ε > 0 and sequences (x n ) n converging to w and (t n ) n of positive real numbers such that |W (t n )x n -w| > ε.

|U (t)(ϕ)| ≤ ke -

An example

Consider the equation

∂ ∂t [u(t, x) - 0 -∞ α(θ)u(t + θ, x)dθ] = - ∂ ∂x [u(t, x) - 0 -∞ α(θ)u(t + θ, x)dθ] + 0 -∞ H(x, θ, u(t + θ, x))dθ for t > 0 and x ∈ [0, 1], u(t, 0) - 0 -∞ α(θ)u(t + θ, 0)dθ = 0 for t > 0, u(θ, x) = u 0 (θ, x) for (θ, x) ∈ (-∞, 0] × [0, 1], (5.1)
where

u 0 ∈ C((-∞, 0] × [0, 1]; R), α : (-∞, 0] → R is a continuous function and H : [0, 1] × (-∞, 0] × R → R is a function satisfying (E1) and (E2) below. We put X = C([0, 1]; R).
We use the extrapolation method to prove the well-posedness of (5.1). Let A be the operator defined on X by [START_REF] Nagel | Inhomogeneous Volterra integrodifferential equations for Hille-Yosida operators[END_REF]). The operator A satisfies the Hille-Yosida condition (H1) on X. The C 0 -semigroup (T 0 (t)) t≥0 on the space

D(A) = {h ∈ C 1 ([0, 1]; R) : h(0) = 0}, Ah = -h ′ . Then, D(A) = C 0 ([0, 1]; R) = {h ∈ C([0, 1]; R) : h(0) = 0}. Lemma 5.1 ([
D(A) = C 0 ([0, 1]; R) generated by the part A 0 of A is given for u ∈ C 0 ([0, 1]; R), by (T 0 (t)u)(x) = u(x -t) for t ≤ x, 0 for t > x.
Let Lip 0 [0, 1] be the space of Lipschitz continuous functions on [0, 1] vanishing at zero, with the norm

|g| Lip = sup 0≤x1<x2≤1 |g(x 2 ) -g(x 1 )| x 2 -x 1 .
Lemma 5.2. [START_REF] Nagel | Inhomogeneous Volterra integrodifferential equations for Hille-Yosida operators[END_REF] The Favard class of the semigroup (T 0 (t)) t≥0 is given by F 0 = Lip 0 [0, 1] and the Favard class of the extrapolated semigroup (T -1 (t)) t≥0 is given by F 1 = L ∞ (0, 1). The extrapolated operator A -1 coincides on F 0 almost everywhere with the derivative operation.

Let γ > 0. Consider the phase space |ϕ(θ)(x)| for θ ≤ 0.

B = C γ = {ϕ ∈ C((-∞, 0

Lemma 5.3 ([15]

). The space C γ satisfies the (A1), (A2), (B1) (C1).

Assume that:

(E1) ess sup x∈[0,1] ( 0 -∞ |H(x, θ, 0)|dθ) < +∞; (E2) |H(x, θ, z 1 ) -H(x, θ, z 2 )| ≤ β(θ, x)|z 1 -z 2 | for x ∈ [0, 1], θ ∈ (-∞, 0] and z 1 , z 2 ∈ R, with ess sup x∈[0,1] ( 0 -∞ e -γθ β(θ, x)dθ) < +∞; (E3) 0 -∞ e -γθ |α(θ)|dθ < 1;
Let F be the linear operator from C γ to X defined by

F (ϕ)(x) = 0 -∞ α(θ)ϕ(θ)(x)dθ for ϕ ∈ C γ and x ∈ [0, 1]. Then sup x∈[0,1] |F (ϕ)(x)| ≤ sup x∈[0,1] 0 -∞ |α(θ)| |ϕ(θ)(x)|dθ ≤ 0 -∞ e -γθ |α(θ)|dθ |ϕ| Cγ ≤ L 1 |ϕ| Cγ ,
where

L 1 = 0 -∞ e -γθ |α(θ)|dθ < +∞. Hence |F (ϕ)| ≤ L 1 |ϕ| Cγ . Then, F is a bounded linear operator from C γ to X.
We introduce the function G defined on C γ , by

(G(ϕ))(x) = 0 -∞ H(x, θ, ϕ(θ)(x))dθ a.e. x ∈ [0, 1], ϕ ∈ C γ .
Lemma 5.4. Assume that the conditions (E1) and (E2) are satisfied. Then, for all ϕ ∈ C γ , G(ϕ) ∈ L ∞ (0, 1) and G : C γ → L ∞ (0, 1) is Lipschitz continuous.

Proof. By (E1), we have Since G(0) ∈ L ∞ (0, 1), we have G(ϕ) ∈ L ∞ (0, 1). On the other hand, we conclude that G : C γ → L ∞ (0, 1) is Lipschitz continuous.

ess sup x∈[0,1] (|G(0)(x)|) ≤ ess sup x∈[0,1] 0 -∞ |H(x, θ, 0)|dθ < +∞. Consequently, G(0) ∈ F 1 = L ∞ (0, 1). Let ϕ, ψ ∈ C γ . Then |G(ϕ) -G(ψ)| F1 = ess sup x∈[0,1] (|G(ϕ)(x) -G(ψ)(x)|), ≤ ess sup x∈[0,1] 0 -∞ |H(x, θ, ϕ(θ)(x)) -H(x,
For t ≥ 0, x ∈ [0, 1] and θ ≤ 0, we make the following change of variables v(t)(x) = u(t, x), ϕ(θ)(x) = u 0 (θ, x).

Then, (5.1) takes the abstract form

∂ ∂t [v(t) -F (v t )] = A[v(t) -F (v t )] + G(v t ), v 0 = ϕ ∈ C γ .
(5.2) Proposition 5.5. Assume that (E1), (E2), (E3) hold. Let ϕ ∈ C γ be such that ϕ(0) -F (ϕ) ∈ D(A). Then (5.2) has a unique mild solution v on an interval (-∞, a], with a > 0.

Proof. Lemma 5.1 and Lemma 5.4 imply that the hypotheses (H1), (H2') and (H3) hold. For the space C γ , one can see that K(0) = 1 and L 1 < 1. It follows that the hypothesis (H4) is true.

To obtain the regularity of mild solutions of (5.2), we assume that the function z → H(x, θ, z) is differentiable and satisfies the following hypothesis. Then we obtain

ϕ ∈ C 1 ((-∞, 0]; X) ∩ B, ϕ ′ ∈ B, ϕ(0) -F (ϕ) ∈ F 0 , ϕ ′ (0) -F (ϕ ′ ) ∈ D(A) ϕ ′ (0) -F (ϕ ′ ) = A -1 [ϕ(0) -F (ϕ)] + G(ϕ).
Consequently, the hypothesis (H7) is satisfied. We conclude with the following proposition.

  [0, b]. By the Banach fixed point theorem, we have the existence and uniqueness of a mild solution of (1.1) on the interval (-∞, b]. We proceed by steps on each interval [kb, (k + 1)b], k = 0, 1, . . . to extend the solution continuously on (-∞, a]. Furthermore, if we suppose L 0 K ∞ < 1 then, we can use the same method to extend the solution continuously on (-∞, +∞).

  ]; X) : sup θ≤0 (e γθ |ϕ(θ)|) < +∞}, endowed with the norm |ϕ| Cγ = sup θ≤0 (e γθ |ϕ(θ)|), where |ϕ(θ)| = sup x∈[0,1]

  θ, ψ(θ)(x))|dθ , ≤ ess sup x∈[0,1] 0 -∞ β(θ)|ϕ(θ)(x) -ψ(θ)(x)|dθ , ≤ ess sup x∈[0,1] 0 -∞ e -γθ β(θ, x)dθ supθ≤0 × e γθ ess sup x∈[0,1] (|ϕ(θ)(x) -ψ(θ)(x)|) , ≤ ess sup x∈[0,1] 0 -∞ e -γθ β(θ, x)dθ |ϕ -ψ| Cγ .

(

  E4) | ∂ ∂z H(x, θ, z)| ≤ µ(x, θ) for x ∈ [0, 1], θ ∈ (-∞, 0] and z ∈ R, with ess sup x∈[0,1] 0 -∞ e -γθ µ(x, θ)dθ < +∞. (E5) | ∂ ∂z H(x, θ, z 1 ) -∂ ∂z H(x, θ, z 2 )| ≤ ϑ(θ, x)|z 1 -z 2 | for x ∈ [0, 1], θ ∈ (-∞, 0] and z 1 , z 2 ∈ R, with ess sup x∈[0,1] ( 0 -∞ e -γθ ϑ(θ, x)dθ) < +∞.It is not difficult to see that assumptions (E4) and (E5) imply that the function G satisfies (H6).We add the following hypothesis on the regularity of the initial condition. (E6)u 0 ∈ C 1 ((-∞; 0] × [0, 1]; R) such that (i) sup θ≤0 e -γθ [ess sup x∈[0,1] | ∂u0 ∂θ (θ, x)|] < ∞, (ii) u 0 (0, .) -0 -∞ α(θ)u 0 (θ, .)dθ ∈ Lip 0 [0, 1], (iii) ∂u0 ∂θ (0, .) -0 -∞ α(θ) ∂u0 ∂θ (θ, .)dθ ∈ C 0 ([0, 1]; R),(iv) for a.e. x ∈ [0, 1] we have ∂u 0 ∂θ (0, x) -)u 0 (θ, x)dθ) + 0 -∞ H(x, θ, u 0 (θ, x))dθ .

Proposition 5 . 6 .

 56 Let (E1), (E2), (E3), (E4), (E5), (E6) be satisfied. Then the mild solution v of (5.2) belongs to C 1 ([0, a]; C([0, 1], R)) ∩ C([0, a]; Lip 0 [0, 1]) and the function u : [0, a] × [0, 1] → R defined by u(t, x) = v(t)(x), satisfies (5.1), for t ∈ [0, a] and a.e x ∈ [0, 1].

  µt |ϕ| for ϕ ∈ H with |ϕ| ≤ δ and t ≥ 0.Moreover, if H can be decomposed as H = H 1 ⊕ H 2 , where H i are V -invariant subspaces of H, H 1 is finite-dimensional and inf{|λ| : λ ∈ σ(V (t)/H 1 )} > e ωt , where ω = lim then, the zero equilibrium of (U (t)) t≥0 is not stable in the sense that there exist ε > 0, a sequence (ϕ n ) n converging to 0, and a sequence(t n ) n of positive real numbers such that |U (t n )ϕ n | > ε.The proof of this theorem is based on Theorem 4.2 and on the following result.Theorem 4.4 ([9]). Let (W (t)) t≥0 be a nonlinear C 0 -semigroup on a subset Ω of a Banach space Y . Assume that w ∈ Ω is an equilibrium of (W (t)) t≥0 and W (t) is differentiable at w for each t ≥ 0. Let Z(t) be the derivative at w of W (t), t ≥ 0. Then (Z(t)) t≥0 is a C 0 -semigroup of bounded linear operators on Y . If the semigroup (Z(t)) t≥0 is exponentially stable, then the equilibrium w of (W (t)) t≥0 is locally exponentially stable. Moreover, if Y can be decomposed as Y = Y 1 ⊕ Y 2 , where Y i are Z-invariant subspaces of Y , Y 1 is finite-dimensional and inf{|λ| : λ ∈ σ(Z(t)/Y 1 )} > e ωt with ω = lim

	h→∞	1 h	log |V (h)/H 2 |,
	h→∞	1 h	log |Z(h)/Y 2 |,