REGULARITY OF SOLUTIONS OF PARTIAL NEUTRAL FUNCTIONAL DIFFERENTIAL EQUATIONS WITH UNBOUNDED DELAY ∗

We prove the existence of regular solutions for a class of quasi-linear partial neutral functional differential equations with unbounded delay that can be described as the abstract retarded functional differential equation(x(t) + F (t,xₜ)) = Ax(t)+G(t,xₜ), where A is the infinitesimal generator of a strongly continuous semigroup of bounded linear operators on a Banach space X and F , G are appropriated functions.


Introduction
The purpose of this paper is to establish some results of regularity, in a sense to be specified later, for solutions of a class of quasi-linear neutral functional differential equations with unbounded delay that can be described in the form d dt (x(t) + F (t, x t )) = Ax(t) + G(t, x t ), t > σ, x σ = ϕ ∈ B, (1.1) where A is the infinitesimal generator of an uniformly bounded analytic semigroup of bounded linear operators, (T (t)) t≥0 , on a Banach space X, the history x t : (−∞, 0] → X, x t (θ) = x(t + θ), belongs to some abstract phase space B defined axiomatically, Ω ⊂ B is open, 0 ≤ σ < a and F, G : [σ, a] × Ω → X are appropriate continuous functions.
Neutral differential equations arise in many areas of applied mathematics and such equations have received much attention in recent years.A good guide to the literature for neutral functional differential equations is the Hale & lunel book [3] and the references therein.The work in partial neutral functional differential equations with unbounded delay was initiated by Hernández & Henríquez in [4,5].In these papers, Hernández & Henríquez proved the existence of mild, strong and periodic solutions for the neutral equation (1.1).In general, the results were obtained using the semigroup theory and the Sadovskii fixed point theorem ( see [11] ).
The results obtained in this paper are the continuation of papers [4], [5] on the existence of mild, strong and periodical solutions for the neutral system (1.1) and generalization of the results reported by Henriquez in [6].
Throughout this paper, X will be a Banach space provided with norm • and A : D(A) → X will be the infinitesimal generator of an uniformly bounded analytic semigroup, T = (T (t)) t≥0 , of linear operators on X.For the theory of strongly continuous semigroup, we refer to Pazy [10] and Krein [9].We will point out here some notations and properties that will be used in this work.It is well know that there exist constants M and w ∈ IR such that If T is a uniformly bounded and analytic semigroup such that 0 ∈ ρ(A), then it is possible to define the fractional power (−A) α , for 0 < α ≤ 1, as a closed linear operator on its domain D(−A) α .Furthermore, the subspace D(−A) α is dense in X, and the expression defines a norm in D(−A) α .If X α represents the space D(−A) α endowed with the norm • α , then the following properties are well known ( [10], pp.74 ): Lemma 1.If the previous conditions hold: 1. Let 0 < α ≤ 1.Then X α is a Banach space.
3. For every constant a > 0, there exists C a > 0 such that 4. For every a > 0 there exists a positive constant C a such that (T (t) − I)(−A) −α ≤ C a t α , 0 < t ≤ a.
In this work we will employ an axiomatic definition of the phase space B, introduced by Hale and Kato [2].To establish the axioms of the space B we follows the terminology used in Hino-Murakami-Naito [8], and thus, B will be a linear space of functions mapping (−∞, 0] into X, endowed with a seminorm • B .We will assume that B satisfies the following axioms: Where (B) The space B is complete.
For the literature on phase space, we refer the reader to [8].While noting here that from the axiom (A − 1), it follows, that the operator function is an strongly continuous semigroup of bounded linear operators on B. In this paper, A w with domain D (A w ) will be the infinitesimal generator of W (•).
To obtain some of our results we will require additional properties for the phase space B, in particular we consider the following axiom (see [6], pp.526 for details) ; → X be a continuous function such that x σ ≡ 0and the right derivative, denoted x (σ + ), exists.If the function ψ defined by ψ (θ) = 0 for θ < 0 and ψ (0 On the other hand, for a linear map P : D (P ) ⊂ X → X and ϕ ∈ B such that ϕ (θ) ∈ D (P ) for every θ ≤ 0, we denote by P ϕ : (−∞, 0] → X, defined by P ϕ = P (ϕ(θ)).For any 0 < α ≤ 1 we use the notation B α for the vector space It is easily to prove that B α , endowed with the seminorm defined by is a phase space of functions with values in X α .
The paper is organized as follows.In section 2, we define the different concepts used in this work and establish the existence of N-classical and classical solutions for the initial value problem (1.1) .Our results are based on the properties of analytic semigroups and the ideas contained in Pazy [10] and Henríquez [6].
Throughout this work we assume that X is an abstract Banach space.The terminology and notations are those generally used in operator theory.In particular, if X and Y are Banach spaces, we indicate by L (X : Y ) the Banach space of the bounded linear operators of X in Y and we abbreviate this notation to L (X) when ever X = Y .In addition B r (x : X) will denote the closed ball in the space X with center at x and radius r.
For some bounded function ξ : [σ, a] → X and σ ≤ s < t ≤ a we employ the notation and we will write simply ξ t for ξ(•) [σ,t] when no confusion arises.
If x ∈ X, we will use the notation X x for the function X x : (−∞, 0] → X where X x = 0 for θ < 0 and X x (0) = x.
Finally, a function f : I ⊂ IR → X is α-Hölder continuous, 0 < α ≤ 1, if there exists a constant L > 0 such that We represent by C 0, α (I; X) the space of α-Hölder continuous function from I into X.Similarly, C k, α (I; X) consist of those functions from I into X, that are k-times continuously differentiable and whose k th -derivative is α-Hölder continuous.

Regularity of Mild Solutions
In this section we will study the regularity of mild solutions of the abstract Cauchy problem (1.1).Henceforth we will assume that A is the infinitesimal generator of a uniformly bounded analytic semigroup, T = (T (t)) t≥0 , on X, that Ω ⊂ B is open and that F, G : [σ, a]×Ω → X are continuous functions.Further, to avoid unnecessary notation, we suppose that 0 ∈ ρ(A) and that T (t) ≤ M , for some constant M ≥ 1 and every t ≥ 0. Our regularity results are based on those of regularity of mild solutions for the abstract Cauchy problem By analogy with the abstract Cauchy problem (2.1) we adopt the following definitions: Definition 1.We will say that a function x : (−∞, σ +b) → X, σ + b ≤ a, is a mild solution of the abstract Cauchy problem (1.1) ) The existence and uniqueness of mild solution of system (1.1) was established in [5] as consequence of the contraction principle.More precisely: Theorem 1.Let ϕ ∈ Ω and assume that the following conditions hold: a) There exist β ∈ (0, 1) and L ≥ 0 such that the function F is X β -valued and satisfies the Lipschitz condition for every σ ≤ s, t ≤ a and ψ 1 , ψ 2 ∈ Ω.
Then there exists a unique mild solution x(•, ϕ) of the abstract Cauchy problem (1.1) defined on (−∞, σ + r), for some 0 < r < a − σ.Furthermore, if Ω = B then r can be chosen independent of ϕ.
Considering the concepts of mild and classical solutions adopted by Henriquez in [6], we introduce the followings definitions.

Definition 2. We will say that a function
Definition 3. We will say that a function In relation with the previous definitions, we consider the following result.
Proposition 1.The following properties hold.
which in turn implies that

The proof is complete
To prove our first regularity Theorem, we need previously some technical results.Next, we study the regularity of mild solutions of the abstract Cauchy problem To this end, for a mild solution, u(•), of (2.6) we introduce the decomposition The proofs of the following three results follow from the proofs of Theorems 4.3.2,4.3.5 and Lemma 4.3.4 in Pazy [10].However there are some differences that require special attention and we include the principal ideas of the proofs for completeness.
The property for u (•) is proved in usual form.The proof is complete.

Lemma 3. Under the assumptions of Lemma 2,
Proof.At first we observe that for t > s and ξ ∈ (0, 1), and hence Let δ > 0 and t ∈ (δ, a].Using (2.10), for h > 0 we get (2.12) Next we estimate each I i (t, h) separately.From inequality (2.11) we find that For the second term we see that Similarly, for the third term we find that The assertion is now consequence of (2.12), (2.14), (2.15) and (2.16).The proof is complete.
Proposition 2. Assume that the assumptions of Lemma 2 hold.If u(•) is the mild solution of (2.6), then the following properties are verified.
then the following properties are verified: In the rest of this paper, we always assume that the functions F, G verifies the hypothesis in Theorem 1.1.Moreover, to simplify our notations, we only consider the case σ = 0.
Now we establish a first result about the existence of regular solutions; specifically we prove existence of N-classical solutions.
Theorem 2. Assume that there exist constants 0

continuous functions and that the following conditions hold:
(a) The function F is X β -valued and there exist positive constants L i , L i , i = 1, 2, such that (c) ϕ ∈ Ω α and there exists 0 < ξ ≤ 1 such that the function Proof: From our assumptions on the operator A, see lemma (1), we fix positive constants C α and C α+1−β such that for all t ∈ (0, T ] In the space Y = C([0, b] : X) provided with the topology of uniform convergence, we define; where ũ is the extension of u to (−∞, b] with ũ0 = (−A) α ϕ.From (2.19), it follows that A(ϕ, α, b) is a nonempty, convex and closed subset of Y .On A(ϕ, α, b) we define the operator Φ by the expression In order to use the contraction mapping principle, we now show that the range of Φ is included in A(ϕ, α, b).To this end we introduce the functions y α , z i : (−∞, b] → X, i = 1, 2, 3, where y α (t) = T (t)(−A) α ϕ for t ≥ 0, (y α ) 0 = (−A) α ϕ and (2.27) Using axiom (A) concerning the phase space, we estimate each term on the right hand side of (2.27) separately.Directly from the choice of b we get the estimate (2.28) On the other hand, for t ∈ [0, b] and for and substituting (2.30) into (2.29) Now, for the function z 2 (•) we have that Employing the last inequality in (2.32) we obtain that; (2.33) Similarly for z 3 , (2.34) Combining (2.27), (2.28), (2.31), (2.33) and (2.34), we conclude that Φ(u) ∈ A(ϕ, α, b).
We estimate each term on the right hand side of the last inequality separately.For the third term we have equivalently, for some constant C 3 , independent of t and h.
With respect to the fourth term, we get which can be abbreviated as where C 4 is a constant independent of t and h.
For I 5 we find that In a similar manner we can prove that where C 6 and C 7 are positive constants independents of t ∈ [0, b) and 0 < h < 1.
We know from Corollary (1), that the abstract Cauchy problem has a unique classical solution y ∈ C((0, b]; X 1 ) which is given by (2.51) Operating on (2.51) with (−A) α and using the ideas in the proof of Lemma 2, follows that extension of z such that z0 = ϕ.Clearly, z is a N-classical solution of the neutral problem (1.1).The proof is complete.Now we turn our attention to the problem of existence of classical solutions.In the rest of this paper, for a function j : [0, a] × B → X and h ∈ IR we use the notation ∂ h j for the function Moreover, if j is differentiable we will employ the following decomposition where R(j, t, ψ, s, ψ 1 ) → 0 as (s, ψ 1 ) → 0. (2.53) To prove Theorem 3 below, we will employ the following property.
Lemma 4. Let X, Y be Banach spaces, Ω ⊂ X open, K ⊂ Ω compact and f : Ω ⊂ X → Y be a continuously differentiable function.Then, for every > 0 there exists δ > 0 such that The next result establishes the existence of classical solutions for the neutral system (1.1), making use of usual regularity assumptions for the functions (−A) β F and G. Theorem 3. Let assumptions in Theorem 1 be satisfied.Assume that ϕ ∈ D(A W ), that (−A) β F and G are continuously differentiable on [0, a] × Ω, that F is continuous with values in X 1 and that D((−A) β F )(0, ϕ) ≡ 0. If X G(0,ϕ) ≡ 0 or X G(0,ϕ) ∈ B and B satisfies axiom C 3 , then there exists a unique classical solution of the system (1.1) defined on [0, b] for some 0 < b < a.
Proof.Let u := u(•, ϕ) the mild solution of (1.1).In the following we assume that u(•) is defined on (−∞, 2b] where 0 < 2b < a and and where The existence and uniqueness of local solution to the integral equation (2.55)-(2.56) is clear and we omit the proof.In what follows we assume that z( In order to prove the assertion, for t ∈ [0, b] and 0 < h < 1 sufficiently small we have Next we use the notations I i (t, h), i = 1, ...6, for the terms of the right hand side of the last inequality.
It is clear that On the other hand for the third term Using similar arguments, we have for I 4 that and so, from (2.58) we get Clearly u (•) = z(•) if h −1 (u h − ϕ) − z 0 → 0 as h → 0. Next we will prove this convergence.For h > 0 we consider the decomposition where we use the Lipschitz continuity of s → u s , see Proposition (3.1) in [5].Consequently then there exists a unique Nclassical solution x(•, ϕ) of the abstract Cauchy problem (1.1) defined on (−∞, b), for some 0 < b < a.