\documentclass[reqno]{amsart}
\usepackage{hyperref}

\AtBeginDocument{{\noindent\small
{\em Electronic Journal of Differential Equations},
Vol. 2006(2006), No. 88, pp. 1--16.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu  (login: ftp)}
\thanks{\copyright 2006 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2006/88\hfil Existence and regularity of local solutions]
{Existence and regularity of local solutions \\ to partial neutral
functional differential equations with infinite delay}

\author[H. Bouzahir\hfil EJDE-2006/88\hfilneg]
{Hassane Bouzahir}

\address{Hassane Bouzahir \newline
 LAMA, Universit\'{e} Ibn Zohr,\newline
 Ecole Nationale des Sciences Appliqu\'{e}es,\newline
 P. O. Box 1136 Agadir, 80 000 Morocco}
\email{hbouzahir@yahoo.fr}
\urladdr{www.geocities.com/hbouzahir}

\date{}
\thanks{Submitted Janaury 6, 2005. Published August 9, 2006.}
\subjclass[2000]{34K30, 34K40, 35R10, 45K05}
\keywords{Infinite delay; integrated semigroup; neutral type;
 \hfill\break\indent  phase space; regularity}

\begin{abstract}
 In this paper, we establish results concerning, existence,
 uniqueness, global continuation, and regularity of integral
 solutions to some partial neutral functional differential
 equations with infinite delay. These equations find their origin
 in the description of heat flow models, viscoelastic and
 thermoviscoelastic materials, and lossless transmission lines
 models; see for example \cite{DesGriSch1} and \cite{Wu1}.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{definition}[theorem]{Definition}
\newtheorem{remark}[theorem]{Remark}

\section{Introduction}

In this article, we consider the following  nonlinear partial
neutral functional differential equations with infinite delay
\begin{equation}
\begin{gathered}
\frac{\partial }{\partial t}\mathcal{D}u_{t}=A\mathcal{D}u_{t}+F(t,u_{t}),
\quad t\geq 0, \\
u_{0}=\phi \in \mathcal{B},
\end{gathered} \label{11f1}
\end{equation}
where $A:D(A)\subseteq E\to E$ is a linear operator on a Banach
space $(E,|\cdot|)$, $\mathcal{B}$ is the phase space of
functions mapping $(-\infty ,0]$ into $E$, which will be
specified later, $\mathcal{D}$ is a bounded linear operator
 from $\mathcal{B}$ to $E$ defined by
\[
\mathcal{D}\varphi =\varphi (0)-\mathcal{D}_{0}\varphi\quad \text{ for }\varphi \in
\mathcal{B}.
\]
The operator $\mathcal{D}_{0}$ is a bounded and linear from
$\mathcal{B}$ to $E$ and for each $u:(-\infty ,b]\to E$, $b>0$,
and $t\in [0,b ]$, $u_{t}$ represents, as usual, the mapping
defined from $( -\infty ,0]$ to $E$ by
\[
u_{t}(\theta )=u(t+\theta )\quad \text{for }\theta \in (
-\infty ,0].
\]
The operator $F$ is an $E$-valued nonlinear continuous mapping on
$\mathbb{R}_{+}\times \mathcal{B}$.

Throughout this paper, we suppose that
$(\mathcal{B},\|\cdot\|_{\mathcal{B }})$ is a (semi)normed abstract
linear space of functions mapping $( -\infty ,0]$ to $E$, and satisfies the
following fundamental axioms which were introduced in
\cite{HalKat1} and widely discussed in \cite{HinMurNai1}.
\begin{itemize}
\item[(A1)] There exist a positive constant $H$ and functions $K(.)$,
$M(.)$ form $\mathbb{R}^{+}\to \mathbb{R}^{+}$, with $K$ continuous and
$M$ locally bounded, such that for any $\sigma \in \mathbb{R}$
and $a>0$, if $x:(-\infty ,\sigma +a]\to E$,
$x_{\sigma }\in \mathcal{B}$ and $x(.)$ is continuous on
$[\sigma ,\sigma +a]$,
then for every $t$ in $[\sigma ,\sigma +a]$ the following conditions hold:
\begin{itemize}
\item[(i)] $x_{t}\in \mathcal{B}$,
\item[(ii)] $|x(t)|\leq H\,\| x_{t}\| _{\mathcal{B}}$,
which is equivalent to
\item[(ii')] $|\varphi (0)|\leq H\|\varphi \|_{ \mathcal{B}}$,
for every $\varphi \in \mathcal{B}$
\item[(iii)] $\| x_{t}\| _{\mathcal{B}}\leq
K(t-\sigma )\sup_{\sigma \leq s\leq t}|x(s)| +M(t-\sigma )\|
x_{\sigma }\| _{\mathcal{B}}$.
\end{itemize}

\item[(A2)] For the function $x(.)$ in (A1), $t\mapsto x_{t}$ is a
$\mathcal{B}$-valued continuous function for $ t$ in $[\sigma
,\sigma +a]$.

\item[(B1)] The space $\mathcal{B}$ is complete.
\end{itemize}

\subsection*{Example} Define for a constant $\gamma $ the following
standard space
\[
C_{\gamma }:=\{ \phi :(-\infty ,0]\to E\text{
continuous such that}\lim_{\theta \to -\infty }
e^{\gamma \theta }\phi (\theta )\text{ exists in }E\} .
\]
It is known from \cite{HinMurNai1} that $C_{\gamma }$ with the
norm $\| \phi \| _{\gamma }=\,\sup_{\theta \leq
 0}e^{\gamma \theta }|\phi (\theta
)|$, $\phi \in C_{\gamma }$, satisfies the axioms $({A1})$,
$({A2})$ and $({B})$ with $H=1$, $K(t)=\max (1,e^{-\gamma t})$ and
$M(t)=e^{-\gamma t}$ for all $t\geq 0$.

 Throughout, we also assume that the operator $A$
satisfies the Hille-Yosida condition:
\begin{itemize}
\item[(H1)] There exist $\bar{M}\geq 0$ and
$\overline{\omega } \in \mathbb{N}$
such that $] \overline{\omega },+\infty [ \subset \rho (A)$ and
\begin{equation}
\sup \{ (\lambda -\overline{\omega })^{n}\| (\lambda
I-A)^{-n}\| :n\in \mathbb{N},\,\lambda >\overline{\omega }\} \leq \bar{M}.
 \label{11f002}
\end{equation}
\end{itemize}
Let $A_{0}$ be the part of the operator $A$ in $\overline{D(A)}$,
which is defined by
\[
\begin{gathered}
D(A_{0})=\{ x\in D(A):Ax\in \overline{D(A)}\} , \\
A_{0}x=Ax\text{, for\ }x\in D(A_{0}).
\end{gathered}
\]
It is well known that $\overline{D(A_{0})}=\overline{D(A)}$ and the
operator $A_{0}$ generates a strongly continuous semigroup
$(T_{0}(t))_{t\geq 0}$ on $\overline{D(A)}$.

 From  \cite{Paz1}, we recall that for all $x\in \overline{D(A)}$ and
$t\geq 0$, one has $\int_{0}^{t}T_{0}(s)x\in D(A_{0})$ and
\begin{equation}
(A\int_{0}^{t}T_{0}(s)xds)+x=T_{0}(t)x.  \label{11f02}
\end{equation}
We also recall that $(T_{0}(t))_{t\geq 0}$ coincides on
$\overline{D(A_{0})}$ with the derivative of the locally Lipschitz
integrated semigroup $(S(t))_{t\geq 0}$ generated by $A$ on $E$.
Which is, according to \cite{Are1} and \cite{KelHie1}, a family of
bounded linear operators on $E$, that satisfies
\begin{itemize}
\item[(i)] $S(0)=0$,
\item[(ii)] for any $y\in E$, $t\to S(t)y$ is strongly continuous
with values in $E$,
\item[(iii)] $S(s)S(t)=\int_{0}^{s}(S(t+r)-S(r))dr$ for all
$ t,s\geq 0$, and for any $\tau >0$ there exists a
constant $l(\tau )>0$ such that
\[
\| S(t)-S(s)\| \leq l(\tau )|t-s|\text{
    for all }t,s\in [
0,\tau ].
\]
\end{itemize}
This integrated semigroup is exponentially bounded, that is, there
exist two constants $\bar{M}$ and $\overline{\omega }$ such that
$\| S(t)\| \leq \bar{M}e^{\overline{\omega }t}$ for all $t\geq 0$.

As stated in Hale \cite{Hal3}, Hale and Lunel \cite{HalLun1} and the
references therein, very much attention has been given to differential
difference equations of neutral type. The reason was applications on
lossless transmission lines. The development has concerned the general
theory of partial neutral functional differential equations. The origin of
the special form \eqref{11f1} is the description of heat flow models and of
the viscoelastic and thermoviscoelastic materials dynamics; see \cite
{DesGriSch1} and the references therein. The recent study of  \eqref{11f1}
has been initiated in the case of finite delay by Hale in
\cite{Hal4} and \cite{Hal5}. The motivation was a model for a
continuous circular array of resistively coupled transmission lines
with mixed initial boundary
conditions introduced by Wu and Xia (\cite{WuXia1}, \cite{WuXia2}). In
addition, Magal and Ruan have stated in \cite{MagRua1} that  \eqref{11f1}
is also a special case of age structured populations model.

Chen  \cite{Che1} proved some results concerning the
existence, uniqueness, and asymptotic behavior of (local and
global) solutions of  \eqref{11f1} in the case where
the delay is finite and $A$ generates a compact C$_{0}$-semigroup
on $E$. Based mainly on a detailed discussion in the book by Wu
\cite{Wu1}, Adimy and Ezzinbi have published some other
interesting results about  \eqref{11f1} but also with
finite delay (cf. \cite{AdiEzz3}-\cite{AdiEzz8}).

This work (such as \cite{AdiBouEzz1}, \cite{AdiBouEzz2} and
\cite{NagHuy1}) contributes to the construction of a complete
theory about the infinite delay case. It can be seen as an
extension to the case of neutral type of some earlier results
about functional differential equations with infinite delay in
\cite{AdiBouEzz3}. We do not suppose a global Lipschitz condition
as in \cite{AdiBouEzz1} or \cite{NagHuy1} nor a compact condition
as in \cite {AdiBouEzz2}. Under a local Lipschitz condition on
$F$, we state the local existence, uniqueness, continuation and
regularity.

We recall that in general, neutral functional differential equations with
infinite delay are functional differential equations depending on all past
and present values, which involve derivatives with infinite delay as well as
the unknown function itself. In \cite{Her2} and \cite{HerHen1}, existence
and regularity of solutions were established to the following neutral
functional differential equations with infinite delay
\begin{equation}
\begin{gathered}
\frac{d}{dt}[x(t)-G(t,x_{t})]=Ax(t)+F(t,x_{t}),\quad t\geq 0, \\
x_{0}=\varphi \in \mathcal{B},
\end{gathered} \label{5f0}
\end{equation}
where $A$ generates a strongly continuous semigroup on $E$. $G$
and $F$ are appropriate continuous functions from
$[0,+\infty)\times \mathcal{B}$ to $E$. The authors have essentially used the
analytic semigroup theory. More recently, in \cite{Her1} the
same theory was used to prove existence of mild solutions for the
so-called partial neutral functional integrodifferential equations
with infinite delay using the Leray-Schauder alternative. Finally,
more discussion about the comparison between the study of
 \eqref{11f1} and  of (\ref{5f0}) can be found in
\cite{AdiBouEzz1,AdiEzz3,Bou1}.

\section{Preliminaries}

Consider the  system
\begin{equation}
\begin{gathered}
\frac{\partial }{\partial t}\mathcal{D}u_{t}=A\mathcal{D}u_{t}\quad
\text{if }t\geq 0, \\
u(\theta )=\varphi (\theta )\quad \text{if }\theta \in (-\infty ,0]
\text{ with }\varphi \in \mathcal{B}.
\end{gathered} \label{11f2}
\end{equation}
Using (\ref{11f02}), we can see that a necessary
condition for $ u:(-\infty ,b)\to E$, $b>0$, to be a solution of
 (\ref{11f2}) is that it verifies the following
integrated equation on $(-\infty ,b)$
\begin{equation}
\begin{gathered}
\mathcal{D}u_{t}=T_{0}(t)\mathcal{D}\varphi ,\quad t\geq 0, \\
u_{0}=\varphi ,
\end{gathered}  \label{11f3}
\end{equation}
where
$\varphi \in \mathcal{Y}:=\{ \varphi \in \mathcal{B}:\mathcal{D}\varphi \in
\overline{D(A)}\}$.

The following result is only the combination of
\cite[Lemma 3]{AdiBouEzz2} and  \cite[Proposition 11]{AdiBouEzz1} which
are proved in a general framework. Precisely, here it suffices to
take $h(t):=T_{0}(t)\mathcal{D} \varphi $.

\begin{proposition}
\label{11p1} Assume that Condition (H1) is satisfied and
$\| \mathcal{D}_{0}\| K(0)<1$. Then, for given $\varphi \in \mathcal{Y}$
there exists a unique function $u$ which is continuous on
$[0,T)$ and solves  \eqref{11f3} on $(-\infty ,T)$.
Moreover, the family of operators $(\mathcal{T}(t))_{t\geq 0}$
defined on $\mathcal{Y}$ by $\mathcal{T}(t)\varphi =u_{t}(.,\varphi )$
is a $C_{0}$-semigroup on $\mathcal{Y}$.
\end{proposition}

We now define a fundamental integral solution $Z(t)$ associated to
 \eqref{11f1}. Consider for given $c\in E$ the following equation
\begin{equation}
\begin{gathered}
\mathcal{D}z_{t}=S(t)c\,\,\,\,\text{if}\,\,t\geq 0, \\
z(t)=0\,\,\,\,\,\text{if}\,\,t\in (-\infty ,0].
\end{gathered} \label{11f9}
\end{equation}
To our purpose, we make the following condition
\begin{itemize}
\item[(H2)]
There exists a continuous nondecreasing function $
\delta :[0,+\infty )\to [0,+\infty [$, $\delta (0)=0$ and a family
of continuous linear operators $W_{\varepsilon }:\mathcal{B}\to E$,
$\varepsilon \in [0,+\infty )$, such that
\[
|\mathcal{D}_{0}\varphi -\mathcal{D}_{\varepsilon }\varphi |\leq
\delta (\varepsilon )\| \varphi \| _{\mathcal{B}}\quad \text{for }
\varepsilon \in [0,+\infty )\text{ and }\varphi \in \mathcal{B} ,
\]
where the linear operator $\mathcal{D}_{\varepsilon }:\mathcal{B}\to E$ is
defined, for $\varepsilon \in [0,+\infty )$, by
\begin{gather*}
\mathcal{D}_{\varepsilon }=W_{\varepsilon }\circ \tau _{\varepsilon },
\\
\tau _{\varepsilon }(\varphi )(\theta )=\varphi (\theta -\varepsilon )
\quad  \text{for }\varphi \in \mathcal{B}\text{ and }\theta \in (-\infty ,0
].
\end{gather*}
\end{itemize}
Note that Assumption (H2) implies that the operator
$\mathcal{D} _{0}$ does not depend very strongly upon
$\varphi (0)$. It is the infinite delay version of the one introduced in
\cite{AdiEzz7,AdiEzz8}.

\begin{proposition}
\label{11p3} Assume that Conditions (H1) and (H2) are satisfied such
that $K(0)\| \mathcal{D}_{0}\|<1$. Then, for given $c\in E$,  \eqref{11f9}
 has a unique integral solution $ z:=z(.)c:(-\infty ,+\infty )\to E$.
Moreover, the operator $Z(t):E\to \mathcal{B}$ defined by
\[
Z(t)c=z_{t}(.)c
\]
satisfies, for any continuous function $f:[0,+\infty ) \to E$, the
following properties
\begin{itemize}
\item[(i)] For each $T>0$, there exists a function
$\alpha (.)\in L^{\infty }([0,T], \mathbb{R}^{+})$ and
$\beta \in \mathbb{R}$,
such that $\| Z(t)\| \leq \alpha (t)e^{t\beta }$ for all
$t\in [0,T]$;
\item[(ii)]  $Z(t)(E)\subseteq \mathcal{Y}$, for all $t\geq 0$;

\item[(iii)] For all $\tau >0$ there exists a constant $k(\tau )>0$
such that
\[
\| Z(t)c-Z(s)c\| _{\mathcal{B}}\leq k(\tau )|t-s\| c|\quad
\text{for all $t,s\in [ 0,\tau ]$ and }c\in E.
\]

\item[(iv)] For any continuous function $f:[0,+\infty ) \to E$, the
functions
\[
t\mapsto \int_{0}^{t}Z(t-s)f(s)\,ds\quad \text{and}\quad
t\mapsto \int_{0}^{t}S(t-s)f(s)\,ds
\]
are continuously differentiable for all $t\geq 0$ and satisfy
\[
\frac{d}{dt}(\int_{0}^{t}Z(t-s)f(s)\,ds)
=\lim_{h\to 0^{+}}\frac{1}{h}\int_{0}^{t}\mathcal{T}(t-s)Z(h)f(s)ds
\quad \text{for all }t\geq 0.
\]
\begin{align*}
\mathcal{D}(\frac{d}{dt}\int_{0}^{t}Z(t-s)f(s)\,ds)
&=\lim_{h\to 0^{+}}\frac{1}{h}\int_{0}^{t}S'(t-s)S(h)f(s)ds \\
&=\frac{d}{dt}\int_{0}^{t}S(t-s)f(s)ds.
\end{align*}
\end{itemize}
\end{proposition}

\begin{proof}[Sketch of Proof]
 Recalling that $\| S(t)\| \leq
\bar{M}e^{ \bar{\omega}t}$ for all $t\geq 0$, the proof of
existence, uniqueness and  (i) is only a particular case of
\cite[Lemma 3]{AdiBouEzz2} where $ h(t)=S(t)c$ and $v_{0}=0$. To
prove (ii), it suffices to remark that for any $c\in E$,
$S(t)c\in \overline{D(A) }$ for all $t\geq 0$ and
$\mathcal{D}(Z(t)c)=S(t)c$. Then $Z(t)c\in \mathcal{Y}$ for all
$t\geq 0$. We infer  (iii) from the fact that $S(.)$ is
locally Lipschitz continuous. Finally, the proof of (iv) is
exactly the same as in \cite{AdiEzz8}. Note that  (iv) also
ensures that $\int_{0}^{t}Z(t-s)f(s)ds$ is differentiable with
respect to $t$.
\end{proof}

For convenience of the reader about the main equation \eqref{11f1}, we
recall the following definitions.

\begin{definition} \label{def2.3} \rm
Let $T>0$ and $\varphi \in \mathcal{B}$. We consider the following definitions.
We say that a function $u:=u(.,\varphi ):(-\infty ,T)\to E$,
$0<T\leq +\infty$, is an integral solution of
\eqref{11f1} if:
\begin{itemize}
\item[(1)] $u$ is continuous on $[0,T)$,
\item[(ii)] $\int_{0}^{t}\mathcal{D}u_{s}\,ds\in D(A)$  for
  $t\in [0,T)$,
\item[(iii)] $\mathcal{D}u_{t}=\mathcal{D}\varphi +A\int_{0}^{t}\mathcal{D}
u_{s}\,ds+\int_{0}^{t}F(s,u_{s})\,ds$
 for $ t\in [0,T)$, ${\bf (iv)}\,\;u(t)=\varphi
(t)$,   for all $t\in ( -\infty ,0]$.
\end{itemize}
\end{definition}

We deduce from \cite{AdiBouEzz1} and \cite{Thi2} that integral solutions of
 \eqref{11f1} are given for $\varphi \in \mathcal{B}$ such that
$\mathcal{D}\varphi \in \overline{D(A)}$ by the  system
\begin{equation}
\begin{gathered}
\mathcal{D}u_{t}=S'(t)\mathcal{D}\varphi +\frac{d}{dt}
\int_{0}^{t}S(t-s)F(s,u_{s})ds, \quad t\in [0,T), \\
u(t)=\varphi (t),\quad t\in (-\infty ,0],
\end{gathered}  \label{11f10}
\end{equation}

\begin{definition} \label{def2.4}
Let $\varphi \in \mathcal{B}$. We say that a function
$u:=u(.,\varphi ):(-\infty ,T)\to E,\;0<T\leq +\infty ,$\ is a
strict solution of Eq. $\,$\eqref{11f1} if the following
conditions hold:
\begin{itemize}
\item[(i)] $t\to \mathcal{D}u_{t}\in \mathcal{C}^{1}([ 0,T),E)\cap
\mathcal{C}([0,T),D(A) )$,
\item[(ii)] $u$ satisfies  \eqref{11f1} on $(-\infty ,T)$.
\end{itemize}
\end{definition}

\begin{remark} \label{12r1} \rm
It was proved in \cite{AdiBouEzz1} that if
$u:=u(.,\varphi ):(-\infty ,T)\to E,\;0<T\leq +\infty $, is an
integral solution of \eqref{11f1} such that $t\to
\mathcal{D}u_{t}$ belongs to $\mathcal{C}^{1}([0,T),E)$, then $t\to
\mathcal{D}u_{t}$ belongs to $\mathcal{C}([0,T),D(A))$.
\end{remark}

Since our method of proof needs computing integrals in
$\mathcal{B}$ from integrals in $E$, we suppose that $\mathcal{B}$
is normed and satisfies one of the following two extra axioms.
\begin{itemize}
\item[(C1)] If $(\phi _{n})_{n\geq 0}$ is a Cauchy
sequence in $\mathcal{B}$ and if $(\phi _{n})_{n\geq 0}$ converges
compactly to $\phi $ on $(-\infty ,0]$, then $\phi $ is in $
\mathcal{B}$ and $\| \phi _{n}-\phi \| _{\mathcal{B}}\to 0$, as
$n\to \infty $.

\item[(D1)]  For a sequence $(\varphi _{n})_{n\geq 0}$
in $\mathcal{B }$, if $\| \varphi _{n}\| _{\mathcal{B}}\to 0$, as
$n\to \infty $, then $|\varphi _{n}(\theta )|\to 0$, as
 $n\to \infty $, for each $\theta \in (-\infty ,0] $.
\end{itemize}

We remark that Axiom (D1) implies that the space $\mathcal{B}$ is normed.

\begin{lemma}[\cite{NaiShiMur2}] \label{11l2}
Let $\mathcal{B}$ be a normed space
which satisfies Axiom (C1) and $f:[0,a]
\to \mathcal{B}$, $a>0$, be a continuous function such that
$f(t)(\theta )$ is continuous for $(t,\theta )\in [0,a]\times
(-\infty ,0]$. Then,
\[
\big[\int_{0}^{a}f(t)dt\big](\theta )=\int_{0}^{a}f(t)(\theta
)dt,\quad \theta \in (-\infty ,0].
\]
\end{lemma}

In \cite{AdiBouEzz1}, we have also obtained the following similar result
using (D1).

\begin{lemma}[\cite{AdiBouEzz1,Bou1}] \label{11l3}
 Assume that
$\mathcal{B}$ satisfies Axiom (D1) and $f:[0,a] \to
\mathcal{B}$ is a continuous function. Then, for all $\theta \in
(-\infty ,0]$, the function $f(.)(\theta )$ is continuous on
$[0,a]$ and satisfies
\[
\big[\int_{0}^{a}f(t)dt\big](\theta )
=\int_{0}^{a}f(t)(\theta)dt,\quad \theta \in (-\infty ,0].
\]
\end{lemma}

\begin{proposition} \label{11p4}
Let $\mathcal{B}$ be a normed space which satisfies
Axiom (C1) or Axiom (D1) with
$K(0)\| \mathcal{D} _{0}\| <1$. If there exists an integral
solution $u:=u(.,\varphi ):(-\infty ,T)\to E$, $0<T\leq +\infty $,
of  \eqref{11f1}, then the function $[0,T)\ni t\mapsto u_{t}\in \mathcal{B}$
satisfies
\begin{equation}  \label{11f11}
\begin{aligned}
u_{t} &=\mathcal{T}(t)\varphi +\frac{d}{dt}\int_{0}^{t}Z(t-s)F(s,u_{s})ds
 \\
&=\mathcal{T}(t)\varphi +\lim_{h\to 0^{+}}\frac{1}{h}
\int_{0}^{t}\mathcal{T}(t-s)Z(h)F(s,u_{s})ds.
\end{aligned}
\end{equation}
Conversely, if there exists a function
$v\in \mathcal{C}([0,T), \mathcal{B})$ such that
\begin{equation}
v(t)=\mathcal{T}(t)\varphi +\frac{d}{dt}\int_{0}^{t}Z(t-s)F(s,v(s))ds,\quad
t\in [0,T)\label{11f12}
\end{equation}
then $v(t)=u_{t}$ for all $t\in [0,T)$, where
\[
u(t)=\begin{cases}
v(t)(0) & t\in [0,T)\\
\varphi (t) & t\in (-\infty ,0]
\end{cases}
\]
and $u(.)$ is an integral solution of \eqref{11f1}.
\end{proposition}

\begin{proof} First, by Proposition \ref{11p3}, it is
immediate that for any continuous function $f:[0,T) \to E$,
\[
W(t):=\int_{0}^{t}Z(t-s)f(s)ds
\]
is continuously differentiable and $W'(0)=0$. Set
\[
w(t)=\begin{cases}
W(t)(0)&\text{if }t\geq 0 \\
0&\text{if }t\in (-\infty ,0].
\end{cases}
\]
By Axiom (A1)(ii'), $w(t)$ is continuously differentiable. Lemma
\ref{11l2} or Lemma \ref{11l3} implies that for all $t\in [0,T)$,
\begin{align*}
w(t) &=(\int_{0}^{t}Z(t-s)f(s)ds)(0) \\
&=\int_{0}^{t}(Z(t-s)f(s))(0)ds \\
&=\int_{0}^{t}z(t-s)f(s)ds.
\end{align*}
In general, for all $t\in [0,T)$ and $\theta \in (-\infty,0]$,
\begin{align*}
(W(t))(\theta ) &=(\int_{0}^{t}Z(t-s)f(s)ds)(\theta ) \\
&=\int_{0}^{t}(Z(t-s)f(s))(\theta )ds \\
&=\int_{0}^{t}z(t+\theta -s)f(s)ds.
\end{align*}
Moreover, since $z(s)=0$ for all $s\in (-\infty ,0]$,
\[
\int_{0}^{t}z(t+\theta -s)f(s)ds=\int_{0}^{t+\theta}z(t+\theta
-s)f(s)ds
\]
and $(W(t))(\theta )=w(t+\theta )$. Which is equivalent to
$W(t)=w_{t}$. On the other hand, we can see that for all
$t\in [0,T)$ and $\theta \in (-\infty ,0]$,
\[
(W'(t))(\theta )=w'(t+\theta ).
\]
Hence $W'(t)=(w')_{t}$ for all $t\in [0,T)$.

Now, suppose that $v(.,\varphi )$ is a solution of
(\ref{11f12}). The function $\mathcal{T}(t)\varphi =x_{t}$ with
$x:(-\infty ,T) \to E$ is the integral solution of
$\mathcal{D}x_{t}=S'(t) \mathcal{D}\varphi $ such that
$x_{0}=\varphi $. Set
\[
w(t)=\int_{0}^{t}z(t-s)F(s,v(s))ds.
\]
Then
\[
v(t)=x_{t}+(w')_{t}=(x+w')_{t}\,.
\]
If we set $u(t)=x(t)+w'(t)$, we obtain $v(t)=u_{t}$ and
\begin{align*}
u_{t} &=\mathcal{T}(t)\varphi +\frac{d}{dt}\int_{0}^{t}Z(t-s)F(s,v(s))ds \\
&=\mathcal{T}(t)\varphi +\frac{d}{dt}\int_{0}^{t}Z(t-s)F(s,u_{s})ds.
\end{align*}
Since $\mathcal{D}(\mathcal{T}(t)\varphi )=S'(t)\mathcal{D}\varphi $ and by
Proposition \ref{11p3},
\[
\mathcal{D}\Big(\frac{d}{dt}\int_{0}^{t}Z(t-s)F(s,u_{s})ds\Big)
=\frac{d}{dt}\int_{0}^{t}S(t-s)F(s,u_{s})ds,
\]
so that $u(t)$ is an integral solution of  \eqref{11f1}.
Conversely, let $u(.,\varphi )$ be an integral solution of
\eqref{11f1} on $(-\infty ,T)$. Then
\[
\mathcal{D}u_{t}=S'(t)\mathcal{D}\varphi +\frac{d}{dt}
\int_{0}^{t}S(t-s)F(s,u_{s})ds.
\]
By the definition of $\mathcal{T}(t)$,
\begin{align*}
\mathcal{D}u_{t}
&=\mathcal{D}(\mathcal{T}(t)\varphi +\mathcal{D}(\frac{d}{dt}
\int_{0}^{t}Z(t-s)F(s,u_{s})ds \\
&=\mathcal{D}\Big(\mathcal{T}(t)\varphi+\frac{d}{dt}
\int_{0}^{t}Z(t-s)F(s,u_{s})ds\Big)\\
&=\mathcal{D}(x_{t}+(w')_{t}),
\end{align*}
where $x:(-\infty ,T)\to E$ is the integral solution of
$\mathcal{D}x_{t}=S'(t)\mathcal{D}\varphi $, and $w(t)$ is defined by
\[
w(t)=\int_{0}^{t}z(t-s)F(s,v(s))ds.
\]
We deduce that, $\mathcal{D}[(u-(x+w'))_{t}]=0$, and
hence $u-(x+w')=0$. Consequently,
\[
u_{t} = x_{t}+(w')_{t}
= \mathcal{T}(t)\varphi +\frac{d}{dt}\int_{0}^{t}Z(t-s)F(s,u_{s})ds.
\]
Which completes the proof.
 \end{proof}

\section{Existence and regularity of local solutions}

To obtain our results on existence, uniqueness and regularity of
solutions to  \eqref{11f1}, we add an extra condition
\begin{itemize}
\item[(H3)] $F:[0,+\infty [ \times \mathcal{B}$
is Lipschitz continuous with respect to $\varphi $ on the balls of
$\mathcal{B}$; i.e., for each $r>0$ there exists a constant
$c_{0}(r)>0$ such that if $ t\geq 0$, $\varphi _{1},\varphi
_{2}\in \mathcal{B}$ and
$\| \varphi _{1}\| _{\mathcal{B}},\| \varphi _{2}\| _{\mathcal{B}}\leq r$
then
\[
|F(t,\varphi _{1})-F(t,\varphi _{2})|\leq c_{0}(r)\| \varphi
_{1}-\varphi _{2}\| _{\mathcal{B}}.
\]
\end{itemize}

\begin{theorem} \label{12t1}
Let $\mathcal{B}$ be a normed space which satisfies
Axiom (C1) or Axiom (D1) with
$K(0)\| \mathcal{D} _{0}\| <1$. Assume that (H1)
(H2) and (H3) hold. Let $\varphi \in \mathcal{B}$ such that
$\mathcal{D} \varphi \in \overline{D(A)}$.
Then, there exists a maximal interval of existence
$(-\infty ,b_{\varphi }),\,b_{\varphi }>0$, and a unique mild solution
$u(.,\varphi )$ of  \eqref{11f1}, defined on
$(-\infty ,b_{\varphi })$ and either $b_{\varphi }=+\infty $
or
\[
\limsup_{t\to b_{\varphi }^{-}}|u(t,\varphi )|=+\infty .
\]
Moreover, $u(t,\varphi )$ is a continuous function of $\varphi $,
in the sense that if $\varphi \in \mathcal{B}$,
$\mathcal{D}\varphi \in \overline{D(A)}$ and $t\in [0,b_{\varphi
})$, then there exist positive constants $\beta $ and $\alpha $
such that, for $\psi \in \mathcal{B}$, $\mathcal{D }\psi \in
\overline{D(A)}$ and $\| \varphi -\psi \| _{\mathcal{B} }<\alpha
$, we have $t\in [0,b_{\psi })$ and
\[
|u(s,\varphi )-u(s,\psi )|\leq \beta \| \varphi -\psi \|
_{\mathcal{B}}\text{ \ for all }s\in [0,t].
\]
\end{theorem}

\begin{proof} The first part of the proof is contained in
\cite{Bou2}. We prove that the solution depends continuously on
the initial data. Let $\varphi \in \mathcal{B}$ such that
$\mathcal{D}\varphi \in \overline{D(A)}$ and $t\in [0,b_{\varphi})$ be fixed.
 Set
\begin{gather*}
r=1+\sup_{0\leq s\leq t}\| u_{s}(.,\varphi )\| _{\mathcal{B}}, \\
c(t)=\overline{M}e^{\omega t}\exp (\overline{M}e^{\omega
t}c_{0}(r)kt).
\end{gather*}
Let $\alpha \in (0,1)$ be such that $c(t)\alpha <1$ and
$\psi \in \mathcal{B}$, $\mathcal{D}\psi \in \overline{D(A)}$ such that
$\|\varphi -\psi \| _{\mathcal{B}}<\alpha $. We have
\[
\| \psi \| _{\mathcal{B}}\leq \| \varphi \| _{\mathcal{B} }+\alpha <r.
\]
Let
\[
b_{0}=\sup \{ s\in (0,b_{\psi }):\| u_{\sigma }(.,\psi )\|
_{\mathcal{B}}\leq r\text{ for all }\sigma \in [0,s ]\} .
\]
Suppose that $b_{0}<t$. We can see similarly as in \cite{Bou2}
that for $ s\in [0,b_{0}]$,
\[% \begin{align*}
\| u_{s}(.,\varphi )-u_{s}(.,\psi )\| _{\mathcal{B}}
\leq \overline{M} e^{\omega t}(\| \varphi -\psi \| _{\mathcal{B}
}+c_{0}(r)k\int_{0}^{s}\| u_{\sigma }(.,\varphi )-u_{\sigma
}(.,\psi )\| _{\mathcal{B}}\,d\sigma ).
\]%\end{align*}
By Gronwall's lemma, we deduce that
\begin{equation}
\| u_{s}(.,\varphi )-u_{s}(.,\psi )\| _{\mathcal{B}}\leq c(t)\|
\varphi -\psi \| _{\mathcal{B}}.  \label{12f5}
\end{equation}
This implies
\[
\| u_{s}(.,\psi )\| _{\mathcal{B}}\leq c(t)\alpha +r-1<r\quad
\text{for all }s\in [0,b_{0}].
\]
By continuity, there exists $\delta >0$ such that
\[
\| u_{s}(.,\psi )\| _{\mathcal{B}}\leq c(t)\alpha +r-1<r\,\,\,\,\,
\text{for all }s\in [0,b_{0}+\delta ].
\]
It follows that $b_{0}$ cannot be the largest number $s>0$ such
that $ \| u_{\sigma }(.,\psi )\| _{\mathcal{B}}\leq r$, for all
$\sigma \in [0,s]$. Thus, $b_{0}\geq t$ and $t<b_{\psi }$.
Furthermore, $ \| u_{s}(.,\psi )\| _{\mathcal{B}}\leq r$, for
$s\in [0,t ]$. Then, using inequality (\ref{12f5}), we deduce the
continuous dependence on the initial data. This completes the
proof of Theorem \ref{12t1}.
\end{proof}

As in \cite{AdiBouEzz3}, we can obtain the strictness of the integral
solution to  \eqref{11f1} under similar restrictive conditions
on $\varphi $ and $F$; namely, (\ref{12f77}) below, (H3) and
\begin{itemize}
\item[(H4)] $F:[0,+\infty )\times \mathcal{B} \to E$ is
continuously differentiable and the derivatives $D_{t}F$ and
$D_{\varphi }F$ satisfy the locally Lipschitz condition
(H3), i.e., for each $r>0$ there exist constants
$C_{1}(r), C_{2}(r)>0$ such that if $t\geq 0$,
$\varphi ,\psi \in \mathcal{B}$ and
$\| \varphi \| _{\mathcal{B}},\| \psi \|_{\mathcal{B}}\leq r$ then
\begin{gather}
|D_{t}F(t,\varphi )-D_{t}F(t,\psi )|\leq C_{1}(r)\| \varphi -\psi
\| _{\mathcal{B}}, \\
\| D_{\varphi }F(t,\varphi )-D_{\varphi }F(t,\psi )\| \leq
C_{2}(r)\| \varphi -\psi \| _{\mathcal{B}}\,. \label{12f0006}
\end{gather}
\end{itemize}

\begin{theorem} \label{12t02}
Suppose that (H4) and the assumptions
of Theorem \ref{12t1} are satisfied. In addition, let an element
$\varphi $ of $\mathcal{B}$ be continuously differentiable such that
\begin{equation}
\varphi '\in \mathcal{B},\quad
\mathcal{D}\varphi \in D(A),\quad
\mathcal{D}\varphi '\in \overline{D(A)}, \quad
\mathcal{D}\varphi '=A\mathcal{D}\varphi +F(0,\varphi ).
 \label{12f77}
\end{equation}
Then, the integral solution asserted by Theorem \ref{12t1} is a strict
solution of  \eqref{11f1}.
\end{theorem}

\begin{proof} Let $\varphi \in \mathcal{B}$ such that
$\varphi '\in \mathcal{B}$,
$ \mathcal{D}\varphi \in D(A)$,
$\mathcal{D}\varphi '\in \overline{D(A)}$ and
$\mathcal{D}\varphi '=A\mathcal{D}\varphi +F(0,\varphi)$.
Let $ u:=u(.,\varphi )$ be the unique integral solution of
 (\ref{11f1}) on $(-\infty ,b_{\varphi })$. To prove
that $u$ is also a strict solution, by Remark \ref{12r1}, it
suffices to show that $t\mapsto Du_{t}$ is continuously
differentiable on $[0,b_{\varphi })$. For that purpose,
consider the linear equation
\begin{equation}
\begin{gathered}
\frac{\partial }{\partial t}\mathcal{D}v_{t}=A\mathcal{D}
v_{t}+D_{t}F(t,u_{t})+D_{\varphi }F(t,u_{t})\,v_{t},\quad t\geq 0, \\
v_{0}=\varphi '.
\end{gathered}  \label{12f006}
\end{equation}
Using Axiom (A2), we can set $r:=\sup_{0\leq s\leq T}\| u_{s}\|
_{\mathcal{B}}$ for each $0\leq T<b_{\varphi }$. Then the fact
that $F$ is continuously differentiable and (\ref{12f0006}) imply
that there exists $\beta _{0}>0$ such that $\| D_{\varphi
}F(t,u_{t})\| \leq \beta _{0}$ for all $t\in [0,T]$ where $0\leq
T<b_{\varphi }$. Hence for all $0\leq T<b_{\varphi }$, the
function $G:[0,T]\times \mathcal{B}\to E$ defined by $ G(t,\psi
):=D_{t}F(t,u_{t})+D_{\varphi}F(t,u_{t})\psi $ is uniformly
Lipschitzian with respect to $\psi$. Then, using the same
reasoning as in the proof  in \cite[Theorem 7]{AdiBouEzz1}, one
can show that  (\ref{12f006}) has a unique integral solution $v$
on $(-\infty ,b_{\varphi })$ given by
\[
\begin{gathered}
\mathcal{D}v_{t}=S\,'(t)\mathcal{D}\varphi '+\frac{d}{dt}
\int_{0}^{t}S(t-s)(D_{t}F(s,u_{s})+D_{\varphi }F(s,u_{s})v_{s})
ds,\text{\ \ \ }t\in [0,b_{\varphi })\\
v_{0}=\varphi '.
\end{gathered}
\]
Let $w:(-\infty ,b_{\varphi })\to E$ be the function
defined by
\[
w(t)=\begin{cases}
\varphi (t) &\text{if }t\in (-\infty ,0], \\
\varphi (0)+\int_{0}^{t}v(s)\,ds &\text{if }t\in [0,b_{\varphi}).
\end{cases}
\]
Then,  using Lemma \ref{11l2} or Lemma \ref{11l3},
\[
w_{t}=\varphi +\int_{0}^{t}v_{s}\,ds\text{\   for }t\in
[0,b_{\varphi }).
\]
Integrating the equation of $v_{t}$, we get
\begin{equation}
\int_{0}^{t}\mathcal{D}v_{s}ds=S(t)\mathcal{D}\varphi '
+\int_{0}^{t}S(t-s)(D_{t}F(s,u_{s})+D_{\varphi
}F(s,u_{s})v_{s})ds.  \label{12f06}
\end{equation}
Since
\[
\int_{0}^{t}\mathcal{D}v_{s}ds=\mathcal{D}(\int_{0}^{t}v_{s}ds)=\mathcal{
D}w_{t}-\mathcal{D}\varphi ,
\]
equality (\ref{12f06}) becomes
\[
\mathcal{D}w_{t}=\mathcal{D}\varphi +S(t)\mathcal{D}\varphi '+\int_{0}^{t}S(t-s)(D_{t}F(s,u_{s})+D_{\varphi
}F(s,u_{s})v_{s})ds.
\]
On the other hand, from the assumption,
$\mathcal{D}\varphi '=A\mathcal{D}\varphi +F(0,\varphi )$.
Then
\[
S(t)\mathcal{D}\varphi '=S(t)A\mathcal{D}\varphi +S(t)F(0,\varphi ).
\]
Since $\mathcal{D}\varphi \in D(A)$, we have
$S(t)A\mathcal{D}\varphi =S\,'(t)\mathcal{D}\varphi -\mathcal{D}\varphi$.
Hence
\[
S(t)\mathcal{D}\varphi '=S\,'(t)\mathcal{D}\varphi -\mathcal{D}
\varphi +S(t)F(0,\varphi ).
\]
Thus $w_{t}$ satisfies
\begin{equation}
\mathcal{D}w_{t}=S'(t)\mathcal{D}\varphi +S(t)F(0,\varphi
)+\int_{0}^{t}S(t-s)(D_{t}F(s,u_{s})+D_{\varphi
}F(s,u_{s})v_{s})ds.  \label{12f07}
\end{equation}
Note that
\[
\int_{0}^{t}S(t-s)F(s,w_{s})ds=\int_{0}^{t}S(s)F(t-s,w_{t-s})\,ds.
\]
Since $t\mapsto w_{t}$ is continuously differentiable and $F(t-s,\varphi )$
is also continuously differentiable, it follows that $F(t-s,w_{t-s})$ is
continuously differentiable with respect to $t$. Thus
\begin{align*}
&\frac{d}{dt}\int_{0}^{t}S(t-s)F(s,w_{s})ds \\
&=S(t)F(0,\varphi )+\int_{0}^{t}S(s)(
D_{t}F(t-s,w_{t-s})+D_{\varphi }F(t-s,w_{t-s})\frac{d}{dt}w_{t-s})ds
\\
&=S(t)F(0,\varphi )+\int_{0}^{t}S(t-s)\Big(
D_{t}F(s,w_{s})+D_{\varphi }F(s,w_{s})v_{s}\Big)ds.
\end{align*}
We deduce that
\[
S(t)F(0,\varphi )= \frac{d}{dt}\int_{0}^{t}S(t-s)F(s,w_{s})ds
 -\int_{0}^{t}S(t-s)(D_{t}F(s,w_{s})+D_{\varphi}F(s,w_{s})v_{s})ds.
\]
Therefore,  (\ref{12f07}) becomes
\begin{align*}
\mathcal{D}w_{t}
&= S\,'(t)\mathcal{D}\varphi +\frac{d}{dt}
\int_{0}^{t}S(t-s)F(s,w_{s})ds \\
&\quad -\int_{0}^{t}S(t-s)(D_{t}F(s,w_{s})+D_{\varphi
}F(s,w_{s})v_{s})ds \\
&\quad +\int_{0}^{t}S(t-s)(D_{t}F(s,u_{s})+D_{\varphi
}F(s,u_{s})v_{s})ds.
\end{align*}
Since the integral solution $u$ satisfies
\[
\mathcal{D}u_{t}=S\,'(t)\mathcal{D}\varphi +\frac{d}{dt}
\int_{0}^{t}S(t-s)F(s,u_{s})ds,
\]
we get
\begin{align*}
\mathcal{D}(u_{t}-w_{t})
&=  \frac{d}{dt}\int_{0}^{t}S(t-s)(F(s,u_{s})-F(s,w_{s}))ds
\\
&\quad -\int_{0}^{t}S(t-s)(
D_{t}F(s,u_{s})-D_{t}F(s,w_{s}))ds \\
&\quad -\int_{0}^{t}S(t-s)(D_{\varphi }F(s,u_{s})-D_{\varphi
}F(s,w_{s}))v_{s}ds.
\end{align*}
Let $0\leq T<b_{\varphi }$ and choose
$T_{1}:=\min \{\varepsilon ,T-T/2\} $ with
$\varepsilon \in (0,T]$, we
obtain for $ t\in [0,T_{1}]$ and
$\theta \in (-\infty ,0]$
\[
-\infty <t+\theta -\varepsilon \leq t-\varepsilon \leq 0.
\]
Since $u(\theta )=w(\theta )=\varphi (\theta )$ for all
$\theta \leq 0$, it
follows that
\begin{gather*}
\tau _{\varepsilon }(u_{t})(\theta )=u_{t}(\theta -\varepsilon )
=u(t+\theta -\varepsilon )=\varphi (t+\theta -\varepsilon), \\
\tau _{\varepsilon }(w_{t})(\theta )=w_{t}(\theta -\varepsilon )
=w(t+\theta -\varepsilon )=\varphi (t+\theta -\varepsilon).
\end{gather*}
Since $W_{\varepsilon }$ is linear,
\[
\mathcal{D}_{\varepsilon }(u_{t}-w_{t})=W_{\varepsilon }\circ \tau
_{\varepsilon }(u_{t}-w_{t})=0,
\]
and
\begin{align*}
\mathcal{D}(u_{t}-w_{t})
& =u(t)-w(t)-\mathcal{D}_{0}(u_{t}-w_{t}) \\
& =u(t)-w(t)-(\mathcal{D}_{0}(u_{t}-w_{t})-\mathcal{D}_{\varepsilon
}(u_{t}-w_{t})).
\end{align*}
Consequently,
\begin{equation}
\begin{aligned}
u(t)-w(t)
= &\mathcal{D}_{0}(u_{t}-w_{t})-\mathcal{D}_{\varepsilon }(u_{t}-w_{t})\\
& +\frac{d}{dt}\int_{0}^{t}S(t-s)(F(s,u_{s}) -F(s,w_{s}))ds \\
& -\int_{0}^{t}S(t-s)(D_{t}F(s,u_{s})-D_{t}F(s,w_{s}))ds \\
& -\int_{0}^{t}S(t-s)(D_{\varphi }F(s,u_{s})-D_{\varphi}F(s,w_{s}))v_{s}ds.
\end{aligned}
\label{12f070}
\end{equation}
Recall that by Proposition \ref{11p3},
\begin{align*}
&\frac{d}{dt}\int_{0}^{t}S(t-s)(F(s,u_{s})-F(s,w_{s}))\,ds\\
&=\lim_{h\to 0^{+}}\frac{1}{h}\int_{0}^{t}S\,'(t-s)S(h)(F(
s,u_{s})-F(s,w_{s}))\,ds.
\end{align*}
Since
\[
\limsup_{h\to 0^{+}}\frac{1}{h}\| S(h)\| <+\infty .
\]
Hence, for suitable constants $\overline{M},\overline{\omega }>0$ and for
all $t\in [0,T_{1}]$,
\begin{align*}
&\big|\frac{d}{dt}\int_{0}^{t}S(t-s)(F(s,u_{s})-F(s,w_{s}))\,ds\big|\\
&\leq \overline{M}e^{\overline{\omega }T_{1}}\int_{0}^{t}
|F(s,u_{s})-F(s,w_{s})|\,ds.
\end{align*}
Since $S(t)$ is assumed to be exponentially bounded, we have also for
suitable positive constants $\bar{M}$ and $\overline{\omega }$,
\begin{align*}
&\big|\int_{0}^{t}S(t-s)(D_{t}F(s,u_{s})-D_{t}F(s,w_{s}))\,ds\big|\\
&\leq \bar{M}e^{\overline{\omega }
T_{1}}\int_{0}^{t}|D_{t}F(s,u_{s})-D_{t}F(s,w_{s})|\,ds,
\end{align*}
and
\begin{align*}
&\big|\int_{0}^{t}S(t-s)(D_{\varphi }F(s,u_{s})
-D_{\varphi }F(s,w_{s}))v_{s}\,ds\big|\\
&\leq \bar{M}e^{\overline{\omega } T_{1}}\int_{0}^{t}\| D_{\varphi
}F(s,u_{s})-D_{\varphi }F(s,w_{s})\| \| v_{s}\|
_{\mathcal{B}}\,ds.
\end{align*}
Set $K_{T}:=\max_{0\leq t\leq T}K(t)$. Since $
u_{0}=w_{0}=\varphi$, by Axiom (A1)(ii), for all $0\leq t\leq
T_{1}$,
\[
\| u_{t}-w_{t}\| _{\mathcal{B}}\leq K_{T}\sup_{0\leq s\leq
t}|u(s)-w(s)|.
\]
 From (H2) and inequality (\ref{12f070}), we infer that
\begin{align*}
|u(t)-w(t)|\leq  & K_{T}\delta (\varepsilon )\sup_
{0\leq s\leq t}|u(s)-w(s)|\\
& +\overline{M}e^{\overline{\omega }T}k\int_{0}^{t}|F(
s,u_{s})-F(s,w_{s})|ds \\
& +\bar{M}e^{\overline{\omega }T}\int_{0}^{t}|
D_{t}F(s,u_{s})-D_{t}F(s,w_{s})|ds \\
& +\bar{M}e^{\overline{\omega }T}\int_{0}^{t}\| D_{\varphi
}F(s,u_{s})-D_{\varphi }F(s,w_{s})\| \| v_{s}\| _{\mathcal{B}
}\,ds.
\end{align*}
Choose $\varepsilon $ small enough such that $K_{T}\delta
(\varepsilon )<1$. Thus for all $t\in [0,T_{1}]$,
\begin{align*}
\| u_{t}-w_{t}\| _{\mathcal{B}}
&\leq   K_{T}\sup_{0\leq s\leq
T_{1}}|u(s)-w(s)|\\
& \leq K_{T}(1-K_{T}\delta (\varepsilon ))^{-1}\overline{M}e^{\overline{
\omega }T_{1}}k\int_{0}^{t}|F(s,u_{s})-F(
s,w_{s})|ds \\
&\quad +K_{T}(1-K_{T}\delta (\varepsilon ))^{-1}\bar{M}e^{\overline{\omega }
T_{1}}\int_{0}^{t}|D_{t}F(s,u_{s})-D_{t}F(s,w_{s})|ds \\
&\quad +K_{T}(1-K_{T}\delta (\varepsilon
))^{-1}\bar{M}e^{\overline{\omega } T_{1}}\int_{0}^{t}\|
D_{\varphi }F(s,u_{s})-D_{\varphi }F(s,w_{s})\| \| v_{s}\|
_{\mathcal{B}}\,ds.
\end{align*}
Set
\[
r:=\max \big(\sup_{0\leq s\leq T_{1}}\| u_{s}\|
_{B}\,,\,\sup_{0\leq s\leq T_{1}}\| v_{s}\| _{\mathcal{B}},
\sup_{0\leq s\leq T_{1}}\| w_{s}\| _{\mathcal{B} }\big).
\]
There exist $C_{0}(r),C_{1}(r),C_{2}(r)>0$ such that, for
$s\in [ 0,T_{1}]$,
\begin{gather*}
|F(s,u_{s})-F(s,w_{s})|\leq C_{0}(r)\| u_{s}-w_{s}\| _{\mathcal{B}}, \\
|D_{t}F(t,u_{s})-D_{t}F(t,w_{s})|\leq C_{1}(r)\|
u_{s}-w_{s}\| _{\mathcal{B}}, \\
\| D_{\varphi }F(t,u_{s})-D_{\varphi }F(t,w_{s})\| \leq C_{2}(r)\|
u_{s}-w_{s}\| _{\mathcal{B}}.
\end{gather*}
This implies that for suitable positive constants $\overline{M}$
and $ \overline{\omega }$, for all $t\in [0,T_{1}]$,
\[
\| u_{t}-w_{t}\| _{\mathcal{B}}\leq \frac{K_{T}\overline{M}e^{
\overline{\omega }T_{1}}}{1-K_{T}\delta (\varepsilon )}(
kC_{0}(r)+C_{1}(r)+rC_{2}(r))\int_{0}^{t}\| u_{s}-w_{s}\|
_{\mathcal{B}}ds.
\]
By the Gronwall lemma, $\| u_{t}-w_{t}\| _{\mathcal{B}}$ for any
$t\in [0,T_{1}]$. Using Axiom (A1)(ii), we deduce that
$u(t)=w(t)$ for all $t\in [0,T_{1}]$.  We can repeat the
previous argument on $[T_{1},T_{2}]$, where
$ T_{2}:=\min \{2\varepsilon ,T-T/2^{2}\} $ and $\varepsilon \in (0,T]$,
$K_{T}\delta (\varepsilon )<1$, with the initial condition
$u_{T_{1}}$. We obtain for $t\in [T_{1},T_{2}]$ and
$ \theta \in (-\infty ,0]$,
\[
-\infty <t+\theta -\varepsilon \leq t-\varepsilon \leq T_{2}-\varepsilon
\leq \varepsilon \leq T_{1}.
\]
Since $u_{T_{1}}(\theta )=w_{T_{1}}(\theta )$ for all $\theta \leq 0$, it
follows that for $t\in [T_{1},T_{2}]$,
\[
\tau _{\varepsilon }(u_{t})(\theta )=u_{t}(\theta -\varepsilon )
=u(t+\theta -\varepsilon )=w(t+\theta -\varepsilon
)=w_{t}(\theta -\varepsilon )=\tau _{\varepsilon
}(w_{t})(\theta )
\]
Since $W_{\varepsilon }$ is linear,
$\mathcal{D}_{\varepsilon }(u_{t}-w_{t})=W_{\varepsilon }\circ \tau
_{\varepsilon }(u_{t}-w_{t})=0$
and
\begin{align*}
\mathcal{D}(u_{t}-w_{t})
& =u(t)-w(t)-\mathcal{D}_{0}(u_{t}-w_{t})\\
& =u(t)-w(t)-(\mathcal{D}_{0}(u_{t}-w_{t})
-\mathcal{D}_{\varepsilon}(u_{t}-w_{t})).
\end{align*}
Consequently,
\begin{align*}
u(t)-w(t)
=& \mathcal{D}_{0}(u_{t}-w_{t})-\mathcal{D}_{\varepsilon }(u_{t}-w_{t})
\\
& +\frac{d}{dt}\int_{T_{1}}^{t}S(t-s) (F(s,u_{s})-F(s,w_{s}))ds \\
& -\int_{T_{1}}^{t}S(t-s) (D_{t}F(s,u_{s})-D_{t}F(s,w_{s}))ds \\
& -\int_{T_{1}}^{t}S(t-s) (D_{\varphi }F(s,u_{s})-D_{\varphi
}F(s,w_{s}))v_{s}ds.
\end{align*}
Recall that by Proposition \ref{11p3},
\begin{align*}
&\frac{d}{dt}\int_{T_{1}}^{t}S(t-s)(F(s,u_{s})-F(
s,w_{s}))\,ds \\
&=\lim_{h\to 0^{+}}\frac{1}{h}\int_{T_{1}}^{t}S\,'(t-s)S(h)(
F(s,u_{s})-F(s,w_{s}))\,ds,
\end{align*}
since
$\limsup_{h\to 0^{+}}\frac{1}{h}\| S(h)\| <+\infty$.
Hence, for suitable constants $\overline{M},\overline{\omega }>0$ and for
all $t\in [T_{1},T_{2}]$,
\[
|\frac{d}{dt}\int_{T_{1}}^{t}S(t-s)(F(s,u_{s})-F(s,w_{s}))\,ds|
\leq \overline{M}e^{\overline{\omega}T_{2}}\int_{T_{1}}^{t}|F(s,u_{s})
-F(s,w_{s})|\,ds.
\]
Since $S(t)$ is assumed to be exponentially bounded, we have also for
suitable positive constants $\bar{M}$ and $\overline{\omega }$,
\begin{align*}
&\big|\int_{T_{1}}^{t}S(t-s)(D_{t}F(s,u_{s})
-D_{t}F(s,w_{s}))\,ds\big|\\
&\leq \bar{M}e^{\overline{\omega }
T_{2}}\int_{T_{1}}^{t}|D_{t}F(s,u_{s})-D_{t}F(s,w_{s})|\,ds,
\end{align*}
and
\begin{align*}
&\big|\int_{T_{1}}^{t}S(t-s)(D_{\varphi }F(s,u_{s})
-D_{\varphi }F(s,w_{s}))v_{s}\,ds\big|\\
&\leq \bar{M}e^{\overline{\omega } T_{2}}\int_{T_{1}}^{t}\| D_{\varphi
}F(s,u_{s})-D_{\varphi }F(s,w_{s})\| \| v_{s}\|
_{\mathcal{B}}\,ds.
\end{align*}
Note that $\max_{T_{1}\leq t\leq T}K(t-T_{1})\leq K_{T}$. Since
$u_{T_{1}}=w_{T_{1}}$, by Axiom (A1)(iii), for all
$T_{1}\leq t\leq T_{2}$,
\[
\| u_{t}-w_{t}\| _{\mathcal{B}}\leq K_{T}\sup_{T_{1}\leq s\leq
t}|u(s)-w(s)|,
\]
and
\begin{align*}
|u(t)-w(t)|\leq
& K_{T}\delta (\varepsilon )\sup_{
T_{1}\leq s\leq t}|u(s)-w(s)|\\
& +\overline{M}e^{\overline{\omega }T}k\int_{T_{1}}^{t}|F(
s,u_{s})-F(s,w_{s})|ds \\
& +\bar{M}e^{\overline{\omega }T}\int_{T_{1}}^{t}|
D_{t}F(s,u_{s})-D_{t}F(s,w_{s})|ds \\
& +\bar{M}e^{\overline{\omega }T}\int_{T_{1}}^{t}\| D_{\varphi
}F(s,u_{s})-D_{\varphi }F(s,w_{s})\| \| v_{s}\| _{\mathcal{B}
}\,ds.
\end{align*}
Recall that $K_{T}\delta (\varepsilon )<1$. Thus for all
$t\in [ 0,T_{1})$,
\begin{align*}
\| u_{t}-w_{t}\| _{\mathcal{B}}
&\leq K_{T} \sup_{T_{1}\leq s\leq T_{2}}|u(s)-w(s)|\\
&\leq K_{T}(1-K_{T}\delta (\varepsilon ))^{-1}\overline{M}e^{\overline{
\omega }T_{2}}k\int_{0}^{t}|F(s,u_{s})-F(
s,w_{s})|ds \\
&\quad +K_{T}(1-K_{T}\delta (\varepsilon ))^{-1}\bar{M}e^{\overline{\omega }
T_{2}}\int_{0}^{t}|D_{t}F(s,u_{s})-D_{t}F(s,w_{s})|ds \\
&\quad +K_{T}(1-K_{T}\delta (\varepsilon
))^{-1}\bar{M}e^{\overline{\omega } T_{2}}\int_{0}^{t}\|
D_{\varphi }F(s,u_{s})-D_{\varphi }F(s,w_{s})\| \| v_{s}\|
_{\mathcal{B}}\,ds.
\end{align*}
Set
\[
r:=\max \Big(\sup_{T_{1}\leq s\leq T_{2}}\| u_{s}\|
_{B},\sup_{T_{1}\leq s\leq T_{2}}\| v_{s}\|
_{\mathcal{B}},\sup_{T_{1}\leq s\leq T_{2}} \| w_{s}\|
_{\mathcal{B}}\Big),
\]
There exist $C_{0}(r),\,C_{1}(r)$,$\,C_{2}(r)>0$ such that,
for $s\in [T_{1},T_{2}]$,
\begin{gather*}
|F(s,u_{s})-F(s,w_{s})|\leq
C_{0}(r)\| u_{s}-w_{s}\| _{\mathcal{B}}, \\
|D_{t}F(t,u_{s})-D_{t}F(t,w_{s})|\leq C_{1}(r)\|
u_{s}-w_{s}\| _{\mathcal{B}}, \\
\| D_{\varphi }F(t,u_{s})-D_{\varphi }F(t,w_{s})\| \leq C_{2}(r)\|
u_{s}-w_{s}\| _{\mathcal{B}}.
\end{gather*}
This implies that for suitable positive constants $\overline{M}$
and $ \overline{\omega }$, and all $t\in [T_{1},T_{2}]$,
\[
\| u_{t}-w_{t}\| _{\mathcal{B}}\leq \frac{K_{T}\overline{M}e^{
\overline{\omega }T_{2}}}{1-K_{T}\delta (\varepsilon )}(
kC_{0}(r)+C_{1}(r)+rC_{2}(r))\int_{T_{1}}^{t}\| u_{s}-w_{s}\|
_{\mathcal{B}}ds.
\]
By the Gronwall lemma, $\| u_{t}-w_{t}\| _{\mathcal{B}}=0$ for any
$ t\in [T_{1},T_{2}]$. Using Axiom (A1)(ii), we deduce that
$u(t)=w(t)$ for all $t\in [T_{1},T_{2}]$. Proceeding inductively
we obtain $u(t)=w(t)$ for all $t\in [0,T]$ for any $T$ in
$[0,b_{\varphi })$. Finally, since
\[
t\mapsto \mathcal{D}w_{t}=\mathcal{D}\varphi +\mathcal{D}\Big(
\int_{0}^{t}v_{s}ds\Big)
=\mathcal{D}\varphi +\int_{0}^{t}\mathcal{D}v_{s}ds
\]
is continuously differentiable, the function $t\mapsto \mathcal{D}u_{t}$
is continuously differentiable. This completes the proof of
Theorem \ref{12t02}.
\end{proof}


\subsection*{Acknowledgments}
The author would like to thank Professors M. Adimy and K. Ezzinbi
for helpful discussions; thanks also to Professor S. Ruan and the
MSO at the University of Miami, for the facilities offered. This
research was supported by TWAS under contract No. 04-150
RG/MATHS/AF/AC, and by the Moroccan-American Fulbright Visiting
Scholar program.

\begin{thebibliography}{00}

\bibitem{AdiBouEzz1}  M. Adimy, H. Bouzahir, K. Ezzinbi;
\emph{Existence and stability for some partial neutral functional
differential equations with infinite delay}, J. Math. Anal. Appl.,
Vol. 294, Issue 2, June 15, 438-461 (2004).

\bibitem{AdiBouEzz2}  M. Adimy, H. Bouzahir, K. Ezzinbi;
\emph{Local existence for a class of partial neutral functional
differential equations with infinite delay}, Differential
Equations Dynam. Systems, Vol. 12, Nos. 3-4, July-October, 353-370
(2004).

\bibitem{AdiBouEzz3}  M. Adimy, H. Bouzahir, K. Ezzinbi;
\emph{Local existence and stability for some partial functional
differential equations with infinite delay},  Nonlinear Analysis
TMA, Vol. 48, N. 3, 323-348 (2002).

\bibitem{AdiEzz3}  M. Adimy, K. Ezzinbi; \emph{A class of linear partial
neutral functional differential equations with non-dense domain},
 J. Differential Equations, Vol. 147, N. 2, 285-332 (1998).

\bibitem{AdiEzz4}  M. Adimy, K. Ezzinbi; \emph{Strict solutions of
nonlinear hyperbolic neutral differential equations},  Appl. Math.
Lett., Vol. 12, N. 1, 107-112 (1999).

\bibitem{AdiEzz7}  M. Adimy, K. Ezzinbi; \emph{Existence and linearized
stability for partial neutral functional differential equations
with non dense domains},  Differential Equations and Dynam.
Systems, Vol. 7, N. 3, 371-417 (1999).

\bibitem{AdiEzz8}  M. Adimy, K. Ezzinbi; \emph{Existence and stability
for a class of partial neutral functional differential equations},
Hiroshima Math. J., Vol. 34, N. 3, 251-294 (2004).

\bibitem{Are1}  W. Arendt; \emph{Resolvent positive operators and integrated
semigroup}, Proc. London Math. Soc., (3), Vol. 54, 321-349 (1987).

\bibitem{Bou1}  H. Bouzahir; \emph{Contribution \`{a} l'\'{e}tude des
aspects quantitatifs et qualitatifs pour une classe
d'\'{e}quations diff\'{e}rentielles \`{a} retard infini, en
dimension infinie},  Ph. D. thesis, Cadi Ayyad University of
Marrakesh, N. 40, April 05, (2001).

\bibitem{Bou2}  H. Bouzahir; \emph{On neutral functional differential
equations with infinite delay},  Fixed Point Theory, Vol. 5, Nos.
1, 11-21 (2005).

\bibitem{BusWu1}  S. Busenberg, B. Wu; \emph{Convergence theorems for
integrated semigroups}, Differential Integral Equations, Vol. 5,
N. 3, 509-520 (1992).

\bibitem{Che1}  Y. Chen; \emph{Abstract partial functional differential
equations of neutral type: basic theory}, Differential Equations
Dynam. Systems, Vol. 7, N. 3, 331-348 (1999).

\bibitem{ColGur1}  B. D. Coleman, M. E. Gurtin; \emph{Equipresence and
constitutive equations for rigid heat conductors},  Z. Angew.
Math. Phys., Vol. 18, 199-208 (1967).

\bibitem{DapSin1}  G. Da Prato and E. Sinestrari; \emph{Differential
operators with non-dense domains}, Ann. Scuola Norm. Sup. Pisa Cl.
Sci. (4), N. 2, Vol. 14, 285-344 (1987).

\bibitem{DesGriSch1}  W. Desch, R. Grimmer,  W. Schappacher;
\emph{Wellposedness and wave propagation for a class of
integrodifferential equations in Banach space}, J. Differential
Equations 74, 391-411, (1988).

\bibitem{GurPip1}  M. E. Gurtin, A. C. Pipkin; \emph{A general theory of
heat conduction with finite wave speeds}, Arch. Rational Mech.
Anal. 31, 113-126, (1968).

\bibitem{Hal3}  J. K. Hale; \emph{Theory of functional differential
equations},  Springer-Verlag, New York (1977).

\bibitem{Hal4}  J. K. Hale; \emph{Partial neutral functional differential
equations},  Rev. Roumaine Math. Pure Appl., Vol. 39, 339-344
(1994).

\bibitem{Hal5}  J. K. Hale; \emph{Coupled Oscillators on a Circle},
Resenhas IME-USP, Vol. 1, N. 4, 441-457 (1994).

\bibitem{HalKat1}  J. K. Hale, J. Kato; \emph{Phase space for retarded
equations with infinite delay},  Funkcial. Ekvac., Vol. 21, 11-41
(1978).

\bibitem{HalLun1}  J. K. Hale, S. Lunel; \emph{Introduction to functional
differential equations},  Springer-Verlag, New York, (1993).

\bibitem{Her1}  E. Hernandez; \emph{Regularity of solutions of partial
neutral functional differential equations with unbounded delay},
Proyecciones, Vol. 21, N. 1, 65-95 (2002).

\bibitem{Her2}  E. Hernandez; \emph{Existence results for partial neutral
functional integrodifferential equations with unbounded delay}, J.
Math. Anal. Appl., Vol. 292 , N. 1, 194-210 (2004).

\bibitem{HerHen1}  E. Hernandez, H. R. Henriquez; \emph{Existence results
for partial neutral functional differential equations with
unbounded delay}, {\it J. Math. Anal. Appl., Vol. 221, N. 2,
452-475 (1998).

\bibitem{HinMurNai1}  Y. Hino, S. Murakami,  T. Naito; \emph{Functional
Differential Equations with Infinite Delay}, Vol. 1473, Lecture
Notes in Mathematics, Springer-Verlag, Berlin (1991).

\bibitem{Kel1}  H. Kellermann; \emph{Integrated Semigroups},
Dissertation, T\"{u}bingen, (1986).}

\bibitem{KelHie1}  H. Kellermann, M. Hieber; \emph{Integrated semigroup},
 J. Fun. Anal., Vol. 15, 160-180 (1989).

\bibitem{LiaXiaCas1}  J. Liang, T. Xiao, J. Casteren; \emph{A note on
semilinear abstract functional differential and
integrodifferential equations with infinite delay},  Appl. Math.
Lett., Vol. 17, N. 4, 473-477 (2004).

\bibitem{MagRua1}  P. Magal, S. Ruan; \emph{On integrated semigroups and
age structured models in L$^{p}$ space},  preprint.

\bibitem{NagHuy1}  R. Nagel, N. T. Huy; \emph{Linear neutral partial
differential equations: a semigroup approach},  Int. J. Math.
Math. Sci., N. 23, 1433-1445 (2003).

\bibitem{NaiShiMur2}  T. Naito, J. S. Shin, S. Murakami; \emph{The
generator of the solution semigroup for the general linear
functional-differential equation},  Bull. Univ. Electro-Comm.,
Vol. 11, N. 1, 29-38 (1998).

\bibitem{Neu1}  F. Neubrander; \emph{Integrated semigroups and their
applications to the abstract Cauchy problems},  Pacific J. Math.,
Vol. 135, N. 1, 111-155 (1988).

\bibitem{Paz1}  A. Pazy; \emph{Semigroups of linear operators and
applications to partial differential equations}, Applied
Mathematical Sciences, 44, Springer-Verlag, New York, (1983).

\bibitem{Sin1}  E. Sinestrari; \emph{On the abstract Cauchy problem of
parabolic type in spaces of continuous functions}, J. Math. Anal.
Appl., Vol. 107, 16-66 (1985).

\bibitem{Thi1}  H. Thieme; \emph{Integrated semigroups and integrated
solutions to abstract Cauchy problems},  J. Math. Anal. Appl.,
Vol. 152, 416-447 (1990).

\bibitem{Thi2}  H. Thieme; \emph{Semiflows generated by Lipschitz
perturbations of non-densely defined operators},  Differential
Integral Equations, Vol. 3, 1035-1066 (1990).

\bibitem{TraWeb1}  C. Travis, G. F. Webb; \emph{Existence and stability
for partial functional differential equations},  Trans. Amer.
Math. Soc., Vol. 200, 395-418 (1974).

\bibitem{Wu1}  J. Wu; \emph{Theory and Applications of Partial Functional
Differential Equations},  Springer-Verlag, (1996).

\bibitem{WuXia1}  J. Wu, H. Xia; \emph{Self-sustained oscillations in a
ring array of coupled lossless transmission lines},  J.
Differential Equations, Vol. 124, 247-278 (1996).

\bibitem{WuXia2}  J. Wu, H. Xia; \emph{Rotating waves in neutral partial
functional differential equations},  J. Dynam. Differential
Equations, Vol. 11, N. 2, 209-238 (1999).

\end{thebibliography}


\end{document}