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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2008(2008), No. 91, pp. 1--19.\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 2008 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2008/91\hfil Strong solutions]
{Strong solutions for some nonlinear partial functional
differential equations with \\ infinite delay}

\author[M. Alia, K. Ezzinbi \hfil EJDE-2008/91\hfilneg]
{Mohamed Alia, Khalil Ezzinbi}  % in alphabetical order

\address{Mohamed Alia \newline
Universit\'{e} Cadi Ayyad\\
Facult\'{e} des Sciences Semlalia\\
D\'{e}partement de Math\'{e}matiques\\
BP. 2390, Marrakech, Morocco}
\email{monsieuralia@yahoo.fr}

\address{Khalil Ezzinbi \newline
Universit\'{e} Cadi Ayyad\\
Facult\'{e} des Sciences Semlalia\\
D\'{e}partement de Math\'{e}matiques\\
BP. 2390, Marrakech, Morocco}
\email{ezzinbi@ucam.ac.ma}


\thanks{Submitted October 25, 2007. Published June 21, 2008.}
\subjclass[2000]{34K30, 37L05, 47H06, 47H20}
\keywords{Partial functional differential equations;
infinite delay;\hfill\break\indent
 $m$-accretive operator; Kato approximation;
mild solution in the sense of Evans; strong solution; \hfill\break\indent
Radon-Nikodym property}

\begin{abstract}
 In this work, we use the Kato approximation to prove the
 existence of strong solutions for partial functional differential
 equations with infinite delay. We assume that the undelayed part
 is $m$-accretive in Banach space and the delayed part is Lipschitz
 continuous. The phase space is axiomatically defined. Firstly, we
 show the existence of the mild solution in the sense of Evans.
 Secondly, when the Banach space has the Radon-Nikodym property, we
 prove the existence of strong solutions. Some applications are
 given for parabolic and hyperbolic equations with delay. The
 results of this work are extensions of the Kato-approximation
 results of Kartsatos and Parrot \cite{kp,kp2}.
\end{abstract}

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


\section{Introduction}

In this work, we study the existence and the regularity of solutions
for the following partial functional differential equation with infinite
delay
\begin{equation}
\begin{gathered}
 u'(t)+Au(t)\ni F(u_{t})\quad \text{for }t\geq0\\
u_{0}=\phi\in\mathcal{B},
\end{gathered}  \label{1}
\end{equation}
where $A$ is a nonlinear multivalued operator with domain
$D(A)$ in a Banach space $X$, $\mathcal{B}$ is the space of
functions defined on $]-\infty,0]$ with values in
$X$, satisfying the Hale and Kato's assumptions \cite{hk}. For
$t\geq0$, the history function $u_{t} \in\mathcal{B}$ is defined
by
\[
u_{t}(\theta)=u(t+\theta)\quad \text{for }\theta \in]-\infty,0],
\]
$F:\mathcal{B}\to X$ is a continuous function. Note that the
difference between the finite and infinite delay lies in the fact that in
general the function
\begin{equation}
t\to u_{t} \label{2}
\end{equation}
is not continuous from $[0,T]$ into $\mathcal{B}$. In finite
delay, usually the phase space is $C([-r,0];X)$
the space of continuous functions from $[-r,0]$ to $X$,
consequently the history function $(\ref{2})$ is continuous.
The main problem of differential equations involving infinite delay is the
choice of the phase space for which the history function (\ref{2}) is
continuous. For more details, about this topics we refer to Hale and Kato
\cite{hk} and Hino, Murakami and Naito \cite{hn}. In \cite{tk}, Kato proposed
a new approach to prove the existence of solution for the  evolution
equation
\begin{equation}
\begin{gathered}
x'(t)+Ax(t)=0\\
x(0)=x_{0}
\end{gathered} \label{rt}
\end{equation}
where $A$ is $m$-accretive in a Banach space $X$ such that the dual
$X^{\ast}$ is uniformly convex.
 The author proposed the following approximation which
called the Kato approximation
\begin{equation}
\begin{gathered}
x_{n}'(t)+A_{n}x_{n}(t)=0\\
x_{n}(0)=x_{0}
\end{gathered}  \label{rto}
\end{equation}
where $A_{n}$ is the Yosida approximation of $A$ to show the existence of
solutions for equation (\ref{rt}).

Kartsatos and Parrott \cite{kp} employed the Kato approximation to prove
the existence of strong solutions for the  partial functional
differential equation
\begin{equation}
\begin{gathered}
 u'(t)+B(t)u(t) =F(u_{t}) \quad\text{for }t\geq0\\
u_{0}=\varphi\in C([-r,0];X),
\end{gathered}  \label{dr}
\end{equation}
where $B(t)$ is $m$-accretive on $X$, the authors proved, the existence of
strong solution if the dual space $X^{\ast}$ is uniformly convex. In
\cite{kp2}, Kartsatos and Parrott considered equation (\ref{dr}) in general
Banach space and proved the existence of a Lipschitz mild solution which
becomes a strong solution when the phase space $X$ is reflexive.  In
\cite{w1}, Ruess studied the existence of solutions for the following
multivalued partial functional differential equation
\begin{gather*}
 u'(t)+B(t)u(t)\ni G(t,x_{t})\quad\text{for }t\geq0\\
u_{0}=\varphi\in C([-r,0];X)\quad \text{or}\quad
 \varphi\in BUC((-\infty,0];X),
\end{gather*}
where $BUC((-\infty,0];X)$ is the space of bounded uniformly
continuous functions from $(-\infty,0]$ to $X$, for every $t\geq0$, the
operator $B(t)$ is $m$-accretive in a Banach space $X$, the authors proved the
existence of strong solutions when $X$ is reflexive and its norm is
differentiable at any $x\neq0$. In \cite{rues}, Ruess studied also the
existence of solution for the following equation
\begin{equation}
\begin{gathered}
 u'(t)+\alpha u(t)+Bu(t) \ni G(x_{t})\quad\text{for }t\geq0\\
u_{0}=\varphi\in\mathcal{M},
\end{gathered}  \label{rue}
\end{equation}
where the phase space $\mathcal{M}=C([-r,0];X)$
or $\in\mathcal{B}$, $\alpha\in\mathbb{R}$ \ and $B$ is $m$-accretive
operator, $G:\mathcal{M}\to X$ is Lipschitz continuous, the authors
proved the existence of strong solution of equation (\ref{rue})
if one of the following conditions holds:
\begin{itemize}

\item[(a)] $X$ is reflexive and its norm is differentiable at any $x\neq0$
and $\varphi\in\hat{D}(\mathcal{A})$, where $\hat{D}(\mathcal{A})$ denotes
the  generalized domain of the operator
\begin{gather*}
D(\mathcal{A})=\{
\varphi\in\mathcal{M}:\varphi'\in \mathcal{M},\;
\varphi(0)\in D(B),\;
\varphi'(0)\in G(\varphi)-\alpha\varphi(0)-B\varphi(0)\} \\
\mathcal{A}\varphi=-\varphi'.
\end{gather*}


\item[(b)] $X$ has the Radon-Nikodym property, $D(B)$ is closed, $B$ is
single valued with $B:D(B)\to X$ norm weakly continuous and
$\varphi\in\hat{D}(\mathcal{A})$.

\item[(c)] $X$ is any Banach space, $D(B)$ is closed, $B$ is single valued
with $B:D(B)\to X$ is continuous and either:\\
(c1) $\varphi\in\hat{D}(\mathcal{A})$\\
(c2) $\varphi\in\overline{D(\mathcal{A})}$ and B maps bounded sets into
bounded sets.

\item[(d)] $X$ is reflexive, $B:D(B)\to X$ is single valued and
demiclosed, namely, the graph of $B$ \ is norm-weakly closed in $X\times X$
and $\varphi\in\hat{D}(\mathcal{A})$.

\end{itemize}
More details can be found in the book K. S. Ha \cite{ki} where an overview on
nonlinear theory of partial functional differential equations is given.

 Travis and Webb \cite{web} gave the basic theory on the existence and
stability of equation \eqref{1} when $-A$ is linear, densely defined and
satisfies the Hille-Yosida condition, more results and applications can be
found in the book Wu \cite{wu}.  Adimy, Bouzahir and Ezzinbi \cite{az} gave
the basic theory of the existence, regularity and stability of solution of
equation \eqref{1} when $-A$ is a linear operator,  not necessarily densely
defined and satisfies the well known Hille-Yosida condition, by renorming the
space $X$, the Hille-Yosida condition is equivalent to say that $A$ is
m-accretive, in this work, the authors investigated several results on the
existence of solutions and stability by using the integrated semi-group
theory. Here we propose to extend the works of Kartsatos and Parrott
\cite{kp}, \cite{kp2} and Ruess \cite{rues}. To simplify our analysis, we
consider the case where $A$ is time-independent, but the same approach still
works in general context. Here we use the Kato approximation to show the
existence of strong solutions in Banach spaces that have the Radon-Nikodym
property. The study of the existence of strong solutions requires some
hypotheses about regularity of the space $X$ and the initial data $\varphi$.
More precisely, we propose the  Kato approximation
\begin{equation}
\begin{gathered}
u_{n}'(t)+A_{n}u_{n}(t) =F(u_{_{nt}})\quad\text{for }t\geq0,\\
u_{_{n0}}=\varphi\in\mathcal{B},
\end{gathered} \label{ap}
\end{equation}
where $A_{n}$ is the Yosida approximation of $A$. Our aim is to prove
that the solution $u_{n}$ converges uniformly on $[0,T]$ to the mild
solution of equation \eqref{1}. The advantage of this
approximation is the fact that the right hand side of equation
\eqref{ap} is a Lipschitz continuous, consequently the solutions of
equation \eqref{ap} are $C^{1}$-functions on $[0,T]$.

This work is organized as follows: In section 2, we recall some results on the
existence of strong solution for evolution problem involving $m$-accretive
operators. In section 3,  we prove the existence of mild and strong solutions
for equation \eqref{1}. Finally, for illustration, we propose
to show the existence of solutions for some partial differential
equation with delay.

\section{Preliminary results}

In this section we recall some preliminary results on $m$-accretive operators
and some results on the phase space that will be used in the whole of this
work. Let $X$ be a Banach space and $A:X\to2^{X}$ be an operator on
$X$ with domain defined by
\[
D(A)=\{  x\in X:Ax\text{ is non empty in }X\}.
\]
We say that $(x,y)\in A$ if $x\in D(A)$ and $y\in Ax$.

\begin{definition} \label{def2.1} \rm
$A$ is said to be accretive if for $\lambda>0$,\ $(x_{1},y_{1})\in A$ and
$(x_{2},y_{2})\in A$ we have
\[
|x_{1}-x_{2}|\leq|x_{1}-x_{2}+\lambda(y_{1}-y_{2})|.
\]
\end{definition}

\begin{proposition}[\cite{ev}] \label{prop2.2}
 If $A$ is an accretive operator, then for all $\lambda>0$,
$I+\lambda A$ is a bijection from $D(A)$ into $R(I+\lambda A)$. Moreover,
$(I+\lambda A)^{-1}$ is nonexpansive on $R(I+\lambda A)$.
\end{proposition}

\begin{definition} \label{def.2.3} \rm
Let $A:D(A)\subset X\to2^{X}$. Then $A$ is said to be $m$-accretive if
$A$ is accretive and for some \ $\lambda>0$, we have
\[
R(I+\lambda A)=X.
\]
\end{definition}

\begin{remark} \label{rmk2.4} \rm
If $A$ is $m$-accretive, then for all $\lambda>0$, we have
$R(I+\lambda A)=X $.
\end{remark}


\begin{definition} \label{def2.5} \rm
The duality mapping $J:X\to2^{X^{\ast}}$ is defined by
\[
J(x)=\{x^{\ast}\in X^{\ast}:<x^{\ast},x>=|x^{\ast}|
^{2}=|x|^{2}\}.
\]
\end{definition}

By the Hahn-Banach Theorem, $J(x)$ is a non empty set for all $x\in X$. For a
general Banach space, the duality mapping $J$ is multi-valued. If the dual
$X^{\ast}$ is strictly convex, $J$ is single-valued. Moreover, if $X^{\ast}$
is uniformly convex, then $J$ is uniformly continuous on bounded sets.

\begin{definition} \label{def2.6} \rm
For every $(x,y)\in X$, we define the bracket $[.,.]$ by
\[
[x,y]=\underset{h\to0}{\lim}\frac{|x+hy|
-|x|}{h}.
\]
\end{definition}

 The following results are well known.

\begin{proposition}[\cite{ev}] \label{prop2.7}
Let $x,y,z\in X$ and $\alpha,\beta\in\mathbb{R}$. Then the following
statements hold:
\begin{itemize}
\item[(i)] $[\alpha x,\beta y]=|\beta|[x,y]$ for $\alpha\beta>0$.

\item[(ii)] $[x,\alpha x+y]=\alpha|x|+[x,y]$.

\item[(iii)] $[x,y]\geq0$ if and only if $|x+hy|\geq|x|$ for $h\geq0$.

\item[(iv)] $|[x,y]|\leq|y|$.

\item[(v)] $[x,y+z]\leq[x,y]+[x,z]$.

\item[(vi)] $[x,y]\geq-[x,-y]$.

\item[(vii)] $[x,y]=\max_{x^{\ast}\in\frac{1}{|x|}J(x)}
\langle x^{\ast},y\rangle$ for $x\neq0$.

\item[(viii)] Let $u$ be a function from a real interval $J$ to $X$ such that
$u'(t_{0})$ exists for an interior point $t_{0}$ of $J$. Then
$D_{+}|u(t_{0})|$ exists and
\[
D_{+}|u(t_{0})|=[u(t_{0}),u'(t_{0})],
\]
where $D_{+}|u(t_{0})|$ denotes the right derivative of
$|u(t)|$ at $t_{0}$.
\end{itemize}
\end{proposition}

\begin{proposition}[\cite{kp}] \label{prop2.8}
 Let $A:X\to2^{X}$ be an operator in $X$. Then the following
statements are equivalent
\begin{itemize}
\item[(i)] $A$ is accretive,

\item[(ii)] $(I+\lambda A)^{-1}$ is nonexpansive on $R(I+\lambda A)$,

\item[(iii)] $\ [x_{1}-x_{2},y_{1}-y_{2}]\geq0$ for any
$(x_{1} ,y_{1}),(x_{2},y_{2})\in A$,

\item[(iv)] for all  $(x_{1},y_{1}),(x_{2},y_{2})\in A$, there exists
$x^{\ast}\in J(x_{1}-x_{2})$ such that
\[
<x^{\ast},y_{1}-y_{2}>\geq0.
\]
\end{itemize}
\end{proposition}

Consider the  Cauchy problem
\begin{equation}
\begin{gathered}
u'(t)+Au(t)\ni f(t)\quad\text{for }t\in[0,T]\\
u(0)=u_{0}.
\end{gathered} \label{a}
\end{equation}

\begin{definition} \label{def2.9} \rm
A function $u:[0,T]\to X$ is said to be a strong
solution of \eqref{a} if
\begin{itemize}
\item[(i)] $u$ is absolutely continuous on $[0,T]$.
\item[(ii)] $u$ is differentiable on $[0,T]$ almost everywhere.
\item[(iii)] $u'(t)+Au(t)\ni f(t)$ for a.e. $t\in[0,T]$.
\item[(iv)] $\ u(0)=u_{0}$.
\end{itemize}
\end{definition}

\begin{definition}\cite{ev} \label{def2.10} \rm
 For a given $\varepsilon>0$, a partition $t_{0}<t_{1}<\dots<t_{n}$ of
$[0,t_{n}]$, and a  finite sequence $f_{0},f_{1},\dots,f_{n} $
 in $X$, the equation
\[
\frac{u_{k}-u_{k-1}}{t_{k}-t_{k-1}}+Au_{k}\ni f_{k}\quad
\text{for }k=1,2,\dots,n.
\]
is called a $\varepsilon$-discretization of $u'(t)+Au(t)\ni f(t)$, on
$[0,T]$ if,
\[
0\leq t_{0}\leq\varepsilon,\quad
0\leq T-t_{n}<\varepsilon,\quad
t_{k}-t_{k-1}<\varepsilon, \quad
\sum_{k=1}^n \int_{t_{k-1}}^{t_{k}}\|  f(\tau)-f_{k}\|
d\tau<\varepsilon.
\]
Moreover, the step function
\[
u_{\varepsilon}(t)=\begin{cases}
u_{0}&\text{for }t=0\\
u_{k}&\text{for }t\in ]t_{k-1},t_{k}]
\end{cases}
\]
is called $\varepsilon$-solution of this discretization.
\end{definition}

\begin{definition}[\cite{ev}] \label{def2.11}\rm
 A continuous function $u:[0,T]\to X$
satisfying $u(0)=u_{0}$ is called a mild solution (in the sense of Evans) of
equation \eqref{a}, if, for all $\varepsilon>0$ there exists
$u_{\varepsilon}$ an $\varepsilon-$solution of an
$\varepsilon$-discretization on  $[0,T]$ such that
\[
|u(t)-u_{\varepsilon}(t)|<\varepsilon\quad\text{for }t\in [0,T].
\]
\end{definition}

\begin{proposition}[\cite{ev}] \label{prop2.12}
If $A$ is accretive, then the following results hold
\begin{itemize}
\item[(i)] the mild solution of equation \eqref{a} if it exists, is unique.

\item[(ii)] If $u$ is a strong solution  of equation \eqref{a},
then $u$ is a mild solution.
\end{itemize}
\end{proposition}


\begin{theorem}[\cite{ev}] \label{extsf}
Let $A$ be a $m$-accretive operator and $f\in
L^{1}(0,T;X)\smallskip$. Suppose that  $u_{0}\in
\overline{D(A)}$, then equation \eqref{a} has
a unique mild solution.
\end{theorem}

\begin{theorem}[{\cite[p.102]{bcp}}] \label{thm2.14}
 Let $A$ be an $m$-accretive operator on $X$ and take
$f\in L^{1}(0,T;X)$, then the function $u$ is a strong solution of equation
\eqref{a} if and only if $u$ is a mild solution which is absolutely
continuous and almost everywhere differentiable on $[0,T]$.
\end{theorem}

\begin{definition} \label{def2.15} \rm
A Banach space $X$ is said to have the Radon-Nikodym property if and only if
every absolutely continuous function $g:[a,b]\to X$ is
almost everywhere differentiable.
\end{definition}

\begin{definition}[{\cite[p.194]{ev}}] \label{2.16} \rm
 The generalized domain $\hat{D}(A)$ of $A$ is defined by
\[
\hat{D}(A)=\{x\in X:|x|_{A}=\underset{\lambda\to0}{\lim
}|A_{\lambda}x|<\infty\}.
\]
\end{definition}

\begin{proposition} \label{prop2.17}
Let $A:X\to2^{X}$ be $m$-accretive operator in $X$. Then
$D(A)\subset\hat{D}(A)\subset\overline{D(A)}$.
\end{proposition}

As a consequence of \cite[Theorem 5]{ev}, we deduce the following result.

\begin{theorem}[\cite{ev}] \label{fbfr}
Assume that $A$ is $m$-accretive, $f\in C([0,T];X)$ and
$u_{0}\in\overline{D(A)}$. If $X$ has the Radon-Nikodym,
then every absolutely continuous mild solution of \eqref{a} becomes a strong
solution of \eqref{a}.
\end{theorem}

\begin{definition}[{\cite[p.32]{bcp}}] \label{prop2.19} \rm
 Let $A_{n}:D(A_{n})\subset X\to2^{X}$ be a sequence
of multivalued operators  on $X$. We define the $\underset{n\to
+\infty}{\lim\inf}A_{n}$ by the operator $A_{\infty}:D(A_{\infty})\subset
X\to2^{X}$ such that
\begin{quote}
$y_{\infty}  \in A_{\infty}x_{\infty}$ if and only if
there exist $x_{n}\in D(A_{n})$ and $y_{n}\in A_{n}x_{n}$
such that $x_{n}  \to x_{\infty}$  and $y_{n}\to y_{\infty}$
as $n\to+\infty$.
\end{quote}
\end{definition}

 For $\lambda>0$, we define the resolvent of $A$ by
\[
J_{\lambda}=(I+\lambda A)^{-1}\text{.}
\]
The Yosida approximation of $A$ is defined for $\lambda>0$ by
\[
A_{\lambda}=\frac{1}{\lambda}(I-J_{\lambda}).
\]

\begin{proposition}[\cite{tk}] \label{prop2.20}
If $A$ is an accretive operator, then for $\lambda>0$,
the following statements hold
\begin{itemize}
\item[(i)] $A_{\lambda}$ is accretive and if \ $A$ is $m$-accretive, so is
$A_{\lambda}$.

\item[(ii)] $A_{\lambda}$ is a Lipschitz mapping on $R(I+\lambda A)$
with coefficient $\frac{2}{\lambda}$.
\end{itemize}
\end{proposition}

\begin{theorem}[{\cite[p.164]{bcp}}]  \label{limAn}
 Let $A$ be a $m$-accretive operator on $X$,
then
\[
A=\liminf_{\lambda\to0^{+}} A_{\lambda}.
\]
where $A_{\lambda}$ is the Yosida approximation of $A$.
\end{theorem}

\begin{theorem}[{\cite[p.159]{bcp}}]\label{cvbcp}
Let $T>0$, $\omega\in\mathbb{R}$, $(A_{n}+\omega I)_{n\geq1}$
be a sequence of \ $m$-accretive operators,
$x_{n}\in\overline{D(A_{n})}$ and $f_{n}\in L^{1}(0,T;X)$ for $n\geq1$. Let
$u_{n}$ be the mild solution of
\begin{equation}
\begin{gathered}
 u_{n}'(t)+A_{n}u_{n}(t)\ni f_{n}(t)\quad\text{for }t\in[0,T]\\
u_{n}(0)=x_{n}.
\end{gathered}  \label{b}
\end{equation}
If $f_{n}\to f_{\infty}$ in $L^{1}(0,T;X)$,
$x_{n}\to x_{\infty}$ and $A_{\infty}=\liminf_{n\to+\infty} A_{n}$,
then
\[
\lim_{n\to+\infty}u_{n}(t)=u_{\infty}(t)\quad\text{uniformly on }
[0,T],
\]
 where $u_{\infty}$ is the mild solution of the
 equation
\begin{gather*}
 u_{\infty}'(t)+A_{\infty}u_{\infty}(t)\ni
f_{\infty}(t)\quad\text{for }t\in[0,T]\\
u_{\infty}(0)=x_{\infty}.
\end{gather*}
\end{theorem}

\begin{proposition}[{\cite[p.90]{bcp}}] \label{estima}
 Let $A$ be such that $A+\omega I$ is
$m$-accretive for some $\omega\in\mathbb{R}$. Let $f$, $g$ be two functions in
$L^{1}(0,T;X)$. If $u_{1}$ and $u_{2}$ are respectively mild solutions of
$u'(t)+Au(t)\ni f(t)$ and $v'(t)+Av(t)\ni g(t)$ for
$t\in[0,T]$. Then for $0\leq s\leq t\leq T$, the following estimate
holds
\[
|u_{1}(t)-u_{2}(t)|\leq e^{\omega(t-s)}|
u_{1} (s)-u_{2}(s)|+\int_{s}^{t}e^{\omega(t-\tau)}|
f(\tau )-g(\tau)|d\tau.
\]
\end{proposition}

 In the following, we assume that the phase space $\mathcal{B}$
satisfies the the following assumptions which were
introduced by Hale and Kato \cite{hk}:
\begin{itemize}
\item[(A1)]
There exist constant $H>0$ and functions
$K,M:\mathbb{R}^{+}\to\mathbb{R}^{+}$ with
$K$ continuous and $M\in L_{\rm loc}^{\infty}(\mathbb{R} ^{+})$
 such that for all $\sigma\in\mathbb{R}$ and for any
$a>0$ if $x:(-\infty,\sigma+a]\to X$ is
such that $x_{\sigma}\in\mathcal{B}$  and
$\mathit{x:}[\sigma,\sigma+a]\to X$ is continuous, then for all
$t\in[\sigma,\sigma+a]$ we have
\begin{itemize}
\item[(i)] $x_{t}\in\mathcal{B}$
\item[(ii)] $|x(t)|\leq H|x_{t}|_{\mathcal{B}}$
(in other words $|\varphi(0)|\leq H|\varphi|_{\mathcal{B}}$, for any
$\varphi\in\mathcal{B})$,
\item[(iii)] $|x_{t}|_{\mathcal{B}}\leq K(t-\sigma)
\sup_{\sigma\leq s\leq t} |x(t)|+M(t-\sigma)|x_{\sigma}|_{\mathcal{B}}$.
\end{itemize}

\item[(A2)]
The function $t\to x_{t}$ is continuous from $[\sigma,\sigma+a]$
to $\mathcal{B}$.

\item[(B)]
$\mathcal{B} $ is complete.

\end{itemize}
Let $C_{00}$ be the space of continuous functions from
 $(-\infty,0]\ $ into $X$ with compact supports.
In the sequel we suppose that $\mathcal{B}$ satisfies
\begin{itemize}

\item[(C)] If a uniformly bounded sequence
$(\varphi _{n})_{n\geq0}$ in $C_{00}$ converges compactly to
a function $\varphi$ in $(-\infty,0]$, then
$\varphi\in \mathcal{B} $ and
$|\varphi_{n}-\varphi|_{\mathcal{B}} \to 0$
as $n\to+\infty$.
\end{itemize}

 Let $B_{0}=\{\varphi\in\mathcal{B}:\varphi(0)=0\}$.
Consider the family of the linear operators defined on $B_{0}$ by
\[
(S_{0}(t)\varphi)(\theta)=\begin{cases}
0 &\text{if }-t\leq\theta\leq0.\\
\varphi(t+\theta)&\text{if }\theta<-t.
\end{cases}
\]
Then $(S_{0}(t))_{t\geq0}$ defines a strongly continuous semigroup
on $B_{0}$.

\begin{definition}[\cite{hn}] \label{def.2.25} \rm
 We say that $\mathcal{B}$ is a fading memory space if
\begin{itemize}
\item[(i)]   $\mathcal{B}$ satisfies assumption (C),

\item[(ii)] $|S_{0}(t)\varphi|_{\mathcal{B}}\to0$ as
$t\to+\infty$ for all $\varphi\in\mathcal{B}$.
\end{itemize}
\end{definition}

Let $BC(]-\infty,0];X)$ be the space of bounded continuous functions
with values in $X$ endowed with the supremum norm. Then we have the following
interesting result.

\begin{proposition}[\cite{hn}] \label{fms}
If $\mathcal{B}$\ is a fading memory space,\ then
$BC(-\infty,0];X)$ is continuously embedded in $\mathcal{B}$;
 namely, there exists a constant $c>0$ such that
\[
|\varphi|_{\mathcal{B}}\leq c|\varphi|_{\rm BC}\quad\text{for all }
\varphi\in BC((-\infty,0];X).
\]
\end{proposition}

\section{Mild and strong solution of \eqref{1}}

\begin{definition}[In the sense of Evans] \label{def3.1} \rm
 A function $u$ $:(-\infty,+\infty)\to X$ is said to be a mild solution
 of equation $\eqref{1}$ if:
 \begin{itemize}
\item[(i)] $u_{0}=\phi$
\item[(ii)] $u$ is mild solution in the sense of Evans of the
equation
\[
u'(t)+Au(t)\ni f(t)\quad\text{for }t\geq0
\]
where $f(t)=F(u_{t})$ for $t\geq0$.
\end{itemize}
\end{definition}

\begin{definition} \label{def3.2} \rm
A function $u$ $:(-\infty,T]\to X$ is said to be a strong solution of
equation $\eqref{1}$ if:
\begin{itemize}
\item[(i)] $u_{0}=\phi$
\item[(ii)] $u$ is absolutely continuous
\item[(iii)] $u$ is almost everywhere differentiable  on $[0,T]$
and
\[
u'(t)+Au(t)\ni F(u_{t})\quad \text{for a.e. }t\in [0,T].
\]
\end{itemize}
\end{definition}

Firstly, we prove the existence of the mild solution. For this
goal, we assume:
\begin{itemize}
\item[(H1)] $(A+\omega I)$ is $m$-accretive for some $\omega\in \mathbb{R}$.

\item[(H2)] There exists a constant $L>0$ such that
\[
|F(\phi)-F(\psi)|\leq L|\phi-\psi|_{\mathcal{B}}
\quad \text{for }\phi,\psi\in \mathcal{B}.
\]
\end{itemize}

\begin{theorem}\label{sfpc}
Assume that {\rm (H1), (H2)} hold. Let $\phi\in\mathcal{B}$ be
such that $\phi (0)\in\overline{D(A)}$. Then equation \eqref{1}
has a unique mild solution defined on $[0,+\infty)$.
\end{theorem}

\begin{proof} Without loss of generality we assume that $\omega=0$.
Let $T>0$. Consider the set
\[
Y=\{v:[0,T]\to X\text{ is continuous and }v(0)=\phi(0)\}.
\]
 For $v\in Y$, we consider the equation
\begin{equation}
\begin{gathered}
u'(t)+Au(t)\ni F(\tilde{v}_{t})\quad\text{for }t\in[0,T]\\
u(0)=\phi(0)
\end{gathered}   \label{c}
\end{equation}
where
\[
\tilde{v}=\begin{cases}
\phi &\text{on }(-\infty,0]\\
v &\text{on }[0,T]
\end{cases}
\]
 From assumption (A2) the mapping
$t\mapsto \tilde{v}_{t}$  is continuous.
Consequently, the mapping
$t\mapsto F(\tilde{v}_{t})$ is continuous.

In virtue of Theorem \ref{extsf}, equation \eqref{c} has a unique mild
solution $u(v)$ on $[0,T]$. Let us now define the operator
\begin{align*}
\mathbb{K}:  Y&\to Y\\
 v&\to u(v)
\end{align*}
and show that $\mathbb{K}$ has an unique fixed point on $Y$. Notice that
$\mathbb{K}$ is well defined  and
$\mathbb{K}(Y) \subset Y$.

Let $v_{1}$ and $v_{2}$ be in $Y$. Set
$u_{1}=\mathbb{K}(v_{1})$  and $u_{2}=\mathbb{K}(v_{2})$.
Then
\begin{gather*}
u_{1}'(t)+Au_{1}(t)\ni F(\tilde{v}_{1_{t}})\\
u_{2}'(t)+Au_{2}(t)\ni F(\tilde{v}_{2_{t}}).
\end{gather*}
By Proposition \ref{estima}, we deduce that
\[
|u_{1}(t)-u_{2}(t)|\leq L\int_{0}^{t}|\tilde{v}_{1_{s} }-\tilde{v}_{2_{s}}
|_{\mathcal{B}}ds.
\]
 From assumption (A1)(iii) and using the fact that
$\tilde{v}_{1_{0}}=\tilde{v}_{2_{0}}=\phi$,
we deduce that
\begin{align*}
|\tilde{v}_{1_{s}}-\tilde{v}_{2_{s}}|_{\mathcal{B}}
& \leq K(s)\sup_{0\leq\tau\leq s} |v_{1}(\tau)-v_{2} (\tau)|\\
& \leq K(s)\sup_{0\leq\tau\leq T}|v_{1}(\tau)-v_{2} (\tau)|.
\end{align*}
Set
\[
K_{T}=\sup_{t\in[0,T]} K(t).
\]
Hence
\[
|u_{1}(t)-u_{2}(t)|\leq K_{T}T\sup_{\tau\in[0,T]}|v_{1}(\tau)-v_{2}(\tau)|.
\]
Thus
\[
\sup_{t\in[0,T]} |u_{1}(t)-u_{2}(t)|
\leq LK_{T}T \sup_{t\in[0,T]}|v_{1}(t)-v_{2}(t)|.
\]
Finally for $T$ appropriately small, $\mathbb{K}$ is strictly contractive. By
the Banach fixed point theorem we have the existence and uniqueness of $u$
which is a mild solution of equation $\eqref{1}$ on $[0,T]$. We
proceed by steep and we can extend continuously the solution on
$[ 0,T]$ for every $T>0$.
\end{proof}

 As a consequence of Theorem \ref{fbfr}, we deduce the following result.

\begin{theorem}\label{sfpc0}
Assume that $X$ has the Radon-Nikodym property and $u$ is a mild
solution of equation equation \eqref{1}. If $u$ is lipschitz continuous on
$[0,T]$, then $u$ becomes a strong solution.
\end{theorem}

 For the regularity of the mild solution we suppose the following
 hypotheses:
\begin{itemize}
\item[(H3)]  $X$ has Radon-Nikodym property.

\item[(H4)] $\mathcal{B}$ is a fading memory space.

\item[(H5)] $\phi\in C^{1}((-\infty,0];X)\cap{\mathcal{B}}$,
$\phi'\in{\mathcal{B}\ }$such that $\phi' $ is
bounded and $\phi(0)\in\hat{D}(A)$.

\end{itemize}

Consider the  Kato approximation
\begin{equation}
\begin{gathered}
u_{n}'(t)+A_{n}u_{n}(t)=F(u_{n_{t}})\quad\text{for }t\geq0\\
u_{n_{0}}=\phi
\end{gathered}  \label{d}
\end{equation}
where for $n\geq1$,
\[
J_{n}=(I+(\frac{1}{n})A)^{-1}
\]
is the resolvent of $A$ and
$A_{n}=n(I-J_{n})$  is the Yosida approximation of $A$.

Now, We  state our main result of this work on the existence
of strong solutions.

\begin{theorem} \label{thp}
Assume that {\rm (H1)--(H5)} hold. Then there exists a unique
strong solution $u$ of equation \eqref{1} on $[0,+\infty)$ such that
\[
u(t)=\lim_{n\to+\infty} u_{n}(t)
\]
uniformly on each compact subset of $[0,+\infty)$,
where $u_{n}$ is the solution of equation \eqref{d}.
Moreover, $u(t)\in \hat{D}(A)$ for $t\geq0$.
\end{theorem}


Let $T>0$. The proof will be done in the
following steps:
\\
(i) The approximate equation \eqref{d} with second
term $-A_{n}u_{n}(t)+F(u_{n_{t}})$ is Lipschitz with respect to the second
variable. Hence by a fixed point argument we show that equation \eqref{d}
has a unique solution $u_{n}$ on $[0,T]$ which is of class
$C^{1}$ on $[0,T]$.
\\
(ii)  We prove that $u_{n}$ and $u_{n}'$ are uniformly bounded on $[0,T]$.
\\
(iii) We prove that the strong limit of $u_{n}$ exists uniformly in
$[0,T]$ as $n\to+\infty$ which is denoted by $u$ .
\\
(iv) We prove that $u$ is a strong solution of equation
\eqref{1}.



\begin{lemma} \label{lem3.6}
Suppose that {\rm (H1), (H2)} are satisfied and
$\phi\in \mathcal{B}$ is such that $\phi(0)\in\hat{D}(A)$.
Then for every $T>0$, there exists $\varrho>0$ such that
$|u_{n}(t)|\leq\varrho$ for all $n$, and for $t\in[0,T]$.
\end{lemma}

\begin{proof} Let $a=\phi(0)$. Then
\begin{align*}
D_{+}|u_{n}(t)-a|&  =[u_{n}(t)-a,u_{n}'(t)]\\
&  =[u_{n}(t)-a,-A_{n}u_{n}(t)+F(u_{n_{t}})]\\
&  =[u_{n}(t)-a,-A_{n}u_{n}(t)+A_{n}a-A_{n}a+F(u_{n_{t}})-F(\phi
)+F(\phi)]\\
&  \leq[u_{n}(t)-a,-A_{n}u_{n}(t)+A_{n}a]+|
A_{n}a|+|F(\phi)|+L|u_{n_{t}}-\phi|.
\end{align*}
Since $A$ is $m$-accretive, it follows that $[
u_{n}(t)-a,-A_{n} u_{n}(t)+A_{n}a]\leq0$. Consequently,
\[
D_{+}|u_{n}(t)-a|\leq|A_{n}a|+|
F(\phi)|+L|u_{n_{t}}-\phi|.
\]
 Since $\phi(0)\in\hat{D}(A)$,  $\sup_{n\geq1} |A_{n}a|<\infty$;
 and consequently
\begin{equation}
 D_{+}|u_{n}(t)-a| \leq k_{1}+L| u_{n_{t}}-\phi|_{\mathcal{B}},
\label{di}
\end{equation}
where $k_{1}=\sup_{n\geq1} |A_{n}a|+|F(\phi)|$.
By solving the differential inequality (\ref{di}), we
deduce
\[
|u_{n}(t)-a|\leq k_{1}T+L\int_{0}^{t}|
u_{n_{s}} -\phi|_{\mathcal{B}}ds\quad\text{for }t\in[0,T],
\]
consequently,
\[
\underset{s\in[0,t]}{\sup}|u_{n}(s)-a|
\leq k_{1}T+L\int_{0}^{t}|u_{n_{s}}-\phi|_{\mathcal{B}} ds.
\]
 It follows that
\[
K(t)\sup_{s\in[0,t]} |u_{n}(s)-a|\leq
K(t)k_{1}T+LK(t)\int_{0}^{t}|u_{n_{s}}-\phi|_{\mathcal{B}}ds;
\]
moreover,
\begin{align*}
&K(t)\sup_{s\in[0,t]}|u_{n}(s)-a|+M(t)|\phi-a|_{\mathcal{B}}\\
&  \leq K(t)k_{1}T+LK(t)\int_{0}^{t}|u_{n_{s}}-\phi|_{\mathcal{B}}ds+M(t)
|\phi-a|_{\mathcal{B}}\\
&  \leq K_{T}k_{1}T+LK_{T}\int_{0}^{t}|u_{n_{s}}-\phi|
_{\mathcal{B}}ds+M_{T}|\phi-a|_{\mathcal{B}},
\end{align*}
where
$M_{T}=\sup_{t\in[0,T]} M(t)$.
Let $k_{2}=K_{T}k_{1}T+m|\phi-a|_{\mathcal{B}}$.
We obtain
\[
K(t)\sup_{s\in[0,t]}|u_{n}(s)-a|+M(t)|\phi-a|
_{\mathcal{B}}\leq k_{2}+LK_{T} \int_{0}^{t}|
u_{n_{s}}-\phi|_{\mathcal{B}}ds.
\]
Applying assumption (A1)(iii), we have
\[
|u_{n_{t}}-a|_{\mathcal{B}}
 K(t) \sup_{s\in[0,t]} |u_{n}(s)-a| +M(t)|\phi-a|_{\mathcal{B}}
 \leq k_{2}+LK_{T}\int_{0}^{t}|u_{n_{s}}-\phi|_{\mathcal{B}}ds.
\]
Consequently
\begin{align*}
|u_{n_{t}}-\phi|_{\mathcal{B}}  &  \leq|u_{n_{t}
}-a|_{\mathcal{B}}+|\phi-a|_{\mathcal{B}}\\
&  \leq|\phi-a|_{\mathcal{B}}+k_{2}+LK_{T}\int_{0} ^{t}|
u_{n_{s}}-\phi|_{\mathcal{B}}ds.
\end{align*}
we set
$k_{3}=|\phi-a|_{\mathcal{B}}+k_{2}$,
we then have
\[
|u_{n_{t}}-\phi|_{\mathcal{B}}\leq
k_{3}+LK_{T} \int_{0}^{t}|u_{n_{s}}-\phi|
_{\mathcal{B}}ds.
\]
Gronwall's Lemma implies
\[
|u_{n_{t}}-\phi|_{\mathcal{B}}\leq
k_{3}e^{LK_{T} T}.
\]
Since for all $\psi\in$ $\mathcal{B}$, we have
$|\psi(0)|\leq H|\psi|_{\mathcal{B}}$, it follows that
\begin{gather*}
|u_{n}(t)-\phi(0)|\leq H|u_{n_{t}}-\phi|
_{\mathcal{B}}, \\
|u_{n}(t)-\phi(0)|\leq Hk_{3}e^{LK_{T}T}=N.
\end{gather*}
Finally, we arrive at
\[
|u_{n}(t)|\leq|\phi(0)|+N,
\]
which implies that $(u_{n})_{n\geq1}$ is uniformly bounded in $C([
0,T];X)$.
\end{proof}

To prove that $(u_{n}')_{n\geq1}$ is uniformly
bounded, we need the following two lemmas.

\begin{lemma}[\cite{kp}] \label{lm1}
Let $w\in C^{1}([0,T];X)$. Then for any $s\in[0,T)$ one has
\[
\lim_{h\to0^{+}}\sup_{\theta\in[-s,0]}
\frac{|w(s+\theta+h)-w(s+\theta)|}{h}=\sup_{\theta\in[-s,0]}
|w'(s+\theta)|.
\]
\end{lemma}

\begin{lemma}[\cite{kp}]\label{lm}
Let $w\in C^{1}([-h_{0},0];X)\cap C^{1}([0,h_{0} ];X)$. Then
\[
\limsup_{h\to0^{+}} \sup_{\theta\in[-(s+h),-s]}
\frac{|w(s+\theta+h)-w(s+\theta)|} {h}\leq|w_{+}'(0)|+|w_{-}^{' }(0)|
\]
for $s\geq0$ where $w_{+}'(0)$ and $w_{-}'(0)$ denote
respectively the right and left derivative at 0.
\end{lemma}

\begin{lemma}\label{drbo}
There exists a constant $\beta>0$ such that
$|u_{n}'(t)|\leq \beta$ for all $n\geq1$ and $t\in[0,T]$.
\end{lemma}

\begin{proof}
Let $z_{n}(t)=u_{n}(t+h)-u_{n}(t)$.
Then
\[
D_{+}|z_{n}(t)|=[
z_{n}(t),z_{n}'(t)]=[
z_{n}(t),-A_{n}u_{n}(t+h)+A_{n}u_{n}(t)+F(u_{n_{t+h}})-F(u_{n_{t}
})].
\]
Since $A_{n}$, is accretive,
\[
[ z_{n}(t),-A_{n}u_{n}(t+h)+A_{n}u_{n}(t)]\leq0.
\]
Consequently
\[
D_{+}|z_{n}(t)|\leq L|u_{n_{t+h}}-u_{n_{t}}|
_{\mathcal{B}},
\]
which implies that
\begin{gather*}
|z_{n}(t)|\leq|z_{n}(0)|+L\int_{0}^{t}|
u_{n_{s+h}}-u_{n_{s}}|_{\mathcal{B}}ds, \\
\frac{|u_{n}(t+h)-u_{n}(t)|}{h}\leq\frac{|
u_{n} (h)-u_{n}(0)|}{h}+L\int_{0}^{t}\frac{|
u_{n_{s+h}}-u_{n_{s} }|_{\mathcal{B}}}{h}ds.
\end{gather*}
It remains to estimate
\[
\int_{0}^{t}\frac{|u_{n_{s+h}}-u_{n_{s}}|_{\mathcal{B}}}{h}ds.
\]
 Using Proposition \ref{fms}, we deduce that
\[
|u_{n_{s+h}}-u_{n_{s}}|_{\mathcal{B}}
\leq c|u_{n_{s+h}}-u_{n_{s}}|_{_{_{_{BC}}}}
=c\sup_{\theta\leq0}|u_{n}(s+\theta+h)-u_{n}(s+\theta)|.
\]
We have to estimate
\[
\sup_{\theta\leq0}\frac{|u_{n}(s+\theta+h)-u_{n} (s+\theta)|}{h}.
\]
In fact one has,
\begin{align*}
\sup_{\theta\leq0}{\sup}|u_{n}(s+\theta+h)-u_{n}(s+\theta)|
&\leq\sup_{\theta\leq-(s+h)} |u_{n}(s+\theta+h)-u_{n}(s+\theta)|\\
&\quad +\sup_{\theta\in[-(s+h),-s]}|u_{n}(s+\theta+h)-u_{n}(s+\theta)|\\
&\quad +\sup_{\theta\in[-s,0]}|u_{n}(s+\theta +h)-u_{n}(s+\theta)|
\end{align*}
For $s+\theta+h\leq0$ and $s+\theta\leq0$, one has
\begin{align*}
\sup_{\theta\leq-(s+h)}
\frac{| u_{n}(s+\theta+h)-u_{n} (s+\theta)|}{h}
&=\sup_{\theta\leq-(s+h)}\frac{|\phi(s+\theta+h)-\phi(s+\theta)|}{h}\\
& \leq\sup_{\theta\leq0}|\phi'(\theta)|
=N_{1}.
\end{align*}
If $\theta\in[-(s+h),-s]$, then
$s+\theta+h\geq0$ and $s+\theta\leq0$. Since $u_{n}\in
C^{1}([0,T];X)$ and $\phi\in C^{1} (-\infty,0];X)$, hence Lemma
\ref{lm} yields
\[
\limsup_{h\to0^{+}} \sup_{\theta\in[-(s+h),-s]}
\frac{|u_{n}(s+\theta+h)-u_{n}(s+\theta )|}{h}
\leq|u_{n}'(0)|+|\phi'(0)|
\]
with $u_{n}'(0)$ denotes the right derivative of $u_{n}$ at $0$, and
$\phi'(0)$ denotes the left derivative of $\phi$ at $0$. If $\theta
\in[-s,0]$ then $s+\theta\geq0$, and Lemma \ref{lm1}
yields
\begin{align*}
&\limsup_{h\to0^{+}} \sup_{\theta\in[-s,0]}\frac{|u_{n}(s+\theta+h)
-u_{n}(s+\theta)|}{h}.\\
&=\sup_{\theta\in[-s,0]} \sup_{h\to0^{+}}
\frac{|u_{n}(s+\theta+h)-u_{n}(s+\theta)|}{h}.\\
&=\sup_{\theta\in[-s,0]}|u_{n}'(s+\theta)|.
\end{align*}
\begin{align*}
\int_{0}^{t} \frac{|u_{n_{s+h}}-u_{n_{s}}|_{BC}}{h}ds
& = \int_{0}^{t} \sup_{\theta\leq0} \frac{|u_{n}(s+\theta+h)-u_{n}
(s+\theta)|}{h}ds\\
& \leq \int_{0}^{t}
\sup_{\theta\leq-(s+h)} \frac{|u_{n}(s+\theta+h)-u_{n}(s+\theta)|}{h}ds\\
&\quad +\int_{0}^{t} \sup_{\theta\in[-(s+h),-s]}\frac{|
u_{n}(s+\theta+h)-u_{n}(s+\theta)|}{h}ds\\
&\quad + \int_{0}^{t} \sup_{\theta\in[-s,0]}
\frac{|u_{n} (s+\theta+h)-u_{n}(s+\theta)|}{h}ds.
\end{align*}
\begin{align*}
\limsup_{h\to0^{+}}\frac{|u_{n}(t+h)-u_{n}(t)|}{h}
& =\lim_{h\to0^{+}}\frac{|u_{n}(t+h)-u_{n}(t)|}{h}\\
& \leq|u_{n}'(0)|+cN_{1}TL+Lc
\int_{0}^{t}(|u_{n}'(0)|+|\phi'(0)|)ds\\
&\quad +Lc \int_{0}^{t}
\sup_{\theta\in[-s,0]} |u_{n}' (s+\theta)|ds.
\end{align*}
 Consequently,
\begin{align*}
|u_{n}'(t)|& =\lim_{h\to0^{+}}\frac{|u_{n}(t+h)-u_{n}(t)|}{h}\\
& \leq(1+cLT)|u_{n}'(0)|+cL(N_{1}+|\phi'(0)|)T\\
&\quad +Lc \int_{0}^{t}
\sup_{\theta\in[-s,0]}|u_{n}' (s+\theta)|ds.
\end{align*}
Furthermore,
\[
 |u_{n}'(0)| \leq|A_{n}\phi(0)|+|F(\phi)| \leq k_{0}+|F(\phi)|,
\]
where $k_{0}=\sup_{n\geq1}|A_{n}a|$. Hence
\begin{align*}
|u_{n}'(t)|& =\lim_{h\to0^{+}}\frac{|u_{n}(t+h)-u_{n}(t)|}{h}\\
& \leq(1+cLT)(k_{0}+|F(\phi)|)+Lc(N_{1}+|\phi'(0)|)T\\
& \quad+Lc \int_{0}^{t}\sup_{\theta\in[-s,0]}
|u_{n}' (s+\theta)|ds.
\end{align*}
Let
\[
k_{3}=(1+cLT)(k_{0}+|F(\phi)|)+Lc(N_{1}+|\phi'(0)|)T.
\]
Hence for  $\theta\leq0$  such that $-t\leq\theta$, we get
\[
 \sup_{\theta\in[-t,0]}|u_{n}' (t+\theta)|
 \leq k_{3}+Lc \int_{0}^{t} \sup_{\theta\in[-s,0]}|u_{n}' (s+\theta)|ds.
\]
Gronwall's Lemma implies
\[
\sup_{\theta\in[-t,0]}|u_{n}' (t+\theta)|\leq k_{3}e^{LcT}=\beta.
\]
Finally for $\theta=0$ we conclude
$|u_{n}'(t)|\leq\beta$ which proves $(u_{n}'(t))_{n}$ is
uniformly bounded.
\end{proof}

\begin{lemma}
Suppose that {\rm(H1)--(H5)} hold. Then the
sequence $(u_{n})_{n\geq1}$ converges uniformly to the mild
solution $u$ of  \eqref{1} on $[0,T]$.
\end{lemma}

\begin{proof} Let $u$ be the mild solution of \eqref{1}
and $v_{n}$ be the mild solution of the  equation
\begin{equation}
\begin{gathered}
 v_{n}'(t)+A_{n}v_{n}(t)=F(u_t)\quad\text{for }t\in[0,T] \\
v_{n}(0)=\phi(0).
\end{gathered}  \label{e}
\end{equation}
 From Theorem \ref{cvbcp}, we deduce that $v_{n}\to u$ as
$n\to\infty$ uniformly on $[0,T]$.  Setting
\[
z_{n}(t)=u_{n}(t)-v_{n}(t)\quad\text{for }t\in[0,T],
\]
we have
\[
 D_{+}|z_{n}(t)|=[z_{n}(t),z_{n}'(t)] =[
z_{n}(t),-A_{n}u_{n}(t)+A_{n}v_{n}(t)+F(u_{n_{t}})-F(u_{t})].
\]
 Thus
\[
D_{+}|z_{n}(t)|\leq L|u_{n_{t}}-u_t|_{\mathcal{B}}.
\]
 Hence
\begin{align*}
|u_{n}(t)-v_{n}(t)|
& \leq L \int_{0}^{t}|u_{n_{s}}-u_{s}|_{\mathcal{B}}ds.\\
& \leq Lc \int_{0}^{t} |u_{n_{s}}-u_{s}|_{BC}ds.\\
& \leq Lc \int_{0}^{t}
\sup_{\theta\leq0} |u_{n}(s+\theta)-u(s+\theta)|ds.\\
& \leq LcT\sup_{\tau\in[0,T]}|u_{n} (\tau)-u(\tau)|.
\end{align*}
It follows that
\begin{align*}
\sup_{t\in[0,T]}|u_{n}(t)-v_{n}(t)|
& \leq LcT\sup_{\tau\in[0,T]}|u_{n}(\tau)-u(\tau)|\\
& \leq LcT\sup_{t\in[0,T]}(|u_{n}(t)-v_{n}(t)|+|v_{n}(t)-u(t)|).
\end{align*}
 Let $T_{0}$ be such that For $LcT_{0}<1$, we deduce that
\[
\sup_{t\in[0,T_{0}]}|u_{n}(t)-v_{n}(t)|
\leq\frac{LcT_{0}}{1-LcT_{0}}\sup_{t\in[0,T_{0}]}|v_{n}(t)-u(t)|,
\]
and $v_{n}\to u$ uniformly on $[0,T_{0}]$,
which implies
\[
|u_{n}(t)-v_{n}(t)|\to0\;\text{as}\;\;n\to
\infty\text{ uniformly on }[0,T_{0}].
\]
Consequently, for $T_{0}$ small enough, we have
\[
u_{n}\to u\quad \text{uniformly on }[0,T_{0}].
\]
Since the derivation of $u_{n}'$ are uniformly bounded ,
which implies that $u$ is lipschitz continuous on $[0,T_{0}]$.
 Since $X$ has the Radon-Nikodym property, it follows that $u$ is almost
everywhere differentiable, by Theorem \ref{fbfr}, we deduce that $u$ is a
strong solution of equation \eqref{1} on \thinspace$[0,T_{0}]$,
 for $T_{0}$ small enough.  The strong solution can be extended on
$[0,+\infty)$, in fact, consider the equation
\begin{equation}
\begin{gathered}
 w'(t)+Aw(t)\ni F(w_{t})
\quad\text{for }t\in[T_{0},T_{1}]\\
w_{T_{0}}=u_{T_{0}},
\end{gathered}  \label{exd}
\end{equation}
Arguing as above, we prove for $T_{1}-T_{0}$ small enough that
(\ref{exd}) has a strong solution on $[T_{0},T_{1}]$ which
extends the strong solution of \eqref{1} on the entire interval
$[T_{0},T_{1}]$, we use the same argument to extend
continuously the strong solution in the whole interval
 $[0,+\infty )$.
To show that $u(t)\in\hat{D}(A)$ for $t\geq0$. we use the
following Lemma.

\begin{lemma}[\cite{ev}] \label{elm3.11}
Assume $A$ is $m$-accretive and $u_{0}\in\hat{D}(A)$. If $f$ is
measurable and of essentially bounded variation on $[0,T]$. Let
$u$ be the mild solution solution of equation \eqref{a}. Then $u(t)\in\hat
{D}(A)$ for $t\geq0$.
\end{lemma}

In our case, $f(t)=F(u_{t})$ for $t\geq0$. Since the initial value
$\varphi$ is a Lipschitz continuous function on $(-\infty,0]$
and the mild solution of equation \eqref{1} is  Lipschitz on
$[0,T]$, using the fact that $\mathcal{B}$ is a fading memory space, we
deduce that the function $t\to u_{t}$ is Lipschitz and consequently,
we deduce that the function $t\to F(u_{t})$ is Lipschitz and of course
is of essentially bounded variation on $[0,T]$, by Lemma, we
conclude that $u(t)\in\hat{D}(A)$ for $t\geq0$.
\end{proof}

\section{Applications}

\subsection*{Example 1: Parabolic case}

Let $\beta$ be a maximal monotone subset of
$\mathbb{R}\times\mathbb{R}$ such
that $0\in D(\beta)$ and $\beta_{p}\subset L^{p}(0,1)\times L^{p}(0,1) $,
$1<p<+\infty$, be the operator defined by
\begin{gather*}
\begin{aligned}
 D(\beta_{p})=\{&u\in L^{p}(0,1):
\text{there exists $v\in L^{p}(0,1)$  such that }\\
& v(x)\in\beta(u(x)) \text{ a.e. in }[0,1] \}
\end{aligned}\\
\beta_{p}(u)=\{v\in L^{p}(0,1):v(x)\in\beta(u(x))\text{ a.e. in
}[0,1]\}.
\end{gather*}

\begin{lemma}[\cite{bar}]
$\beta_{p}$ is $m$-accretive on $L^{p}(0,1)$.
\end{lemma}

\begin{proposition}[\cite{bar}] \label{acc}
The operator $A:L^{p}(0,1)\to L^{p}(0,1)$
defined by
\begin{gather*}
D(A)=W_{0}^{1,p}\cap W_{0}^{2,p}\cap D(\beta_{p})\\
A(u)=-\Delta u+\beta_{p}(u)
\end{gather*}
 is $m$-accretive in $L^{p}(0,1)$.
\end{proposition}

To apply the previous abstract results, we consider the following
multivalued parabolic partial functional differential equation
\begin{equation}
\begin{gathered}
\frac{\partial u(t,x)}{\partial t}-\frac{\partial^{2}u(t,x)}{\partial x^{2}
}+\beta(u(t,x))\ni
\int_{-\infty}^{0}G(\theta,u(t+\theta,x))d\theta\quad
\text{for }t\in[0,1],\; x\in]0,1[ \\
u(t,0)=u(t,1)=0\quad\text{for }t\in[0,1],\\
u(\theta,x)=\varphi(\theta,x)\quad\text{for }\theta\in\mathbb{R}^{-},\;
x\in]0,1[\,.
\end{gathered} \label{ab}
\end{equation}
The phase space is
\[
B=C_{\gamma}=\big\{\varphi\in C(]-\infty
,0];L^{p}(0,1):\sup_{\theta\leq0} e^{\gamma\theta
}|\varphi(\theta)|_{p}<+\infty\big\},
\]
where $\gamma>0$, endowed with the norm
\[
|\varphi|_{C_{\gamma}}=\sup_{\theta\leq0}
e^{\gamma\theta}|\varphi(\theta)|_{p},
\]
where
\[
|\varphi(\theta)|_{p}=\Big(\int_{0}^{1}|\varphi
(\theta)(x)|^{p}dx\Big)^{1/p}.
\]
Let $X=L^{p}(0,1)$, with $1<p<+\infty.$\thinspace$G:]-\infty,0]\times
\mathbb{R}\to\mathbb{R}$ is such that
\begin{itemize}
\item[(i)] the mapping $\theta\mapsto G(\theta,0)$ belongs to
$L^{1}(-\infty,0)$.
\item[(ii)]
$|G(\theta,x_{1})-G(\theta,x_{2})|\leq\vartheta (\theta)|x_{1}-x_{2}|$
for all  $\theta\in]-\infty,0]$ and $x_{1},x_{2}\in\mathbb{R}$.
\end{itemize}
We assume that
$\vartheta e^{-(\gamma+\varepsilon)}\in L^{q}(]-\infty,0])$ for some
$\varepsilon>0$ and $\frac{1}{p}+\frac{1}{q}=1$.

\begin{lemma}[\cite{hn}] \label{lem4.3}
 $C_{\gamma}$ satisfies assumptions {\rm (A1), (A2)} and {\rm (B)};
moreover $C_{\gamma}$ is a fading memory space.
\end{lemma}

We introduce the function $F:C_{\gamma}\to L^{p}(0,1)$ defined by
\[
(F\varphi)(x)=\int_{-\infty}^{0} G(\theta,\varphi(\theta)(x))d\theta
\quad\text{for a.e. }x\in[0,1].
\]

\begin{lemma} \label{lem4.4}
Under the above conditions, the function $F:C_{\gamma}\to L^{p}(0,1) $
is Lipschitz continuous.
\end{lemma}

\begin{proof}
Let $\varphi\in C_{\gamma}$ and $x\in[0,1]$. Then
\begin{align*}
|(F(\varphi))(x)-(F(0))(x)|& =\big|\int_{-\infty}
^{0}G(\theta,\varphi(\theta)(x))d\theta-\int_{-\infty}^{0}G(\theta
,0)d\theta\big|\\
&  \leq\int_{-\infty}^{0}|G(\theta,\varphi(\theta)(x))-G(\theta
,0)|d\theta\\
&  \leq\int_{-\infty}^{0}\vartheta(\theta)|\varphi(\theta)(x)|
d\theta \\
&  \leq\int_{-\infty}^{0}\vartheta(\theta)e^{-(\gamma+\varepsilon)\theta
}e^{(\gamma+\varepsilon)\theta}|\varphi(\theta)(x)|d\theta.
\end{align*}
Hence
\[
|(F\varphi)(x)-(F(0))(x)|^{p}
\leq\Big(\int_{-\infty}^{0}
\vartheta(\theta)e^{-(\gamma+\varepsilon)\theta}e^{(\gamma+\varepsilon)\theta
}|\varphi(\theta)(x)|d\theta\Big)^{p}.
\]
Using Hypothesis (ii) and H\"{o}lder's inequality, we obtain
\[
|(F\varphi)(x)-(F(0))(x)|
^{p}\leq\Big(\int_{-\infty}^{0}(\vartheta(\theta))^{q}
e^{-q(\gamma+\varepsilon)\theta}d\theta\Big)^{p/q}
\int_{-\infty}^{0}e^{p(\gamma+\varepsilon)\theta}|\varphi
(\theta)(x)|^{p}d\theta)
\]
and
\begin{align*}
&\int_{0}^{1}|(F\varphi)(x)-(F(0))(x)|^{p}dx\\
&\leq \int_{-\infty}^{0}\Big((\vartheta(\theta))^{q}e^{-q(\gamma+\varepsilon)
\theta}d\theta \Big)^{p/q}\int_{0}^{1}\int_{-\infty}^{0}e^{p(\gamma+\varepsilon)\theta
}|\varphi(\theta)(x)|^{p}d\theta dx.
\end{align*}
Let
\[
\lambda=(\int_{-\infty}^{0}(\vartheta(\theta))^{q}e^{-q(\gamma
+\varepsilon)\theta}d\theta)^{p/q}<+\infty.
\]
 By hypothesis (ii),
\begin{align*}
\int_{0}^{1}|(F\varphi)(x)-(F(0))(x)|^{p}dx
&  \leq\lambda \int_{-\infty}^{0}e^{p\varepsilon\theta}
\int_{0}^{1}e^{p\gamma\theta}| \varphi(\theta)(x)|^{p}dx\,d\theta\\
&  \leq\lambda\Big(\underset{\theta\leq0}{\sup\text{ }}e^{p\gamma\theta}\int
_{0}^{1}|\varphi(\theta)(x)|^{p}dx\Big)\int_{-\infty}
^{0}e^{p\varepsilon\theta}d\theta\\
&  \leq\frac{1}{p\varepsilon}\lambda|\varphi|_{C_{\gamma}}^{p}.
\end{align*}
Hence
\[
|F(\varphi)-F(0)|_{p}\leq\big(\frac{1}{p\varepsilon}\lambda
\big)^{1/p}|\varphi|_{C_{\gamma}}.
\]
Since$|F(0)|_{p}<\infty$, $F(\varphi)\in L^{p}(0,1)$.
Now, let $\varphi,\psi\in C_{\gamma}$ and $x\in[0,1]$. Then
\begin{align*}
|(F\varphi)(x)-(F\psi)(x)|
&  =\int_{-\infty}
^{0}G(\theta,\varphi(\theta)(x))d\theta-\int_{-\infty}^{0}G(\theta,\psi
(\theta)(x))d\theta\big|\\
&  \leq\int_{-\infty}^{0}|G(\theta,\varphi(\theta)(x))-G(\theta
,\psi(\theta)(x))|d\theta\\
&  \leq\int_{-\infty}^{0}\vartheta(\theta)|\varphi(\theta
)(x)-\psi(\theta)(x)|d\theta\\
&  \leq\int_{-\infty}^{0}\vartheta(\theta)e^{-(\gamma+\varepsilon)\theta
}e^{(\gamma+\varepsilon)\theta}|\varphi(\theta)(x)-\psi(\theta
)(x)|d\theta.
\end{align*}
Hence
\[
|(F\varphi)(x)-(F\psi)(x)|^{p}
\leq\Big(\int_{-\infty}^{0}
\vartheta(\theta)e^{-(\gamma+\varepsilon)\theta}e^{(\gamma+\varepsilon)\theta
}|\varphi(\theta)(x)-\psi(\theta)(x)|d\theta\Big)^{p}.
\]
By H\"{o}lder's inequality,
\begin{align*}
&|(F\varphi)(x)-(F\psi)(x)| ^{p}\\
&\leq\Big(\int_{-\infty}^{0}
(\vartheta(\theta))^{q}e^{-q(\gamma+\varepsilon)\theta}d\theta\Big)
^{p/q}\int_{-\infty}^{0}e^{p(\gamma+\varepsilon)\theta}|\varphi
(\theta)(x)-\psi(\theta)(x)|^{p}d\theta.
\end{align*}
Thus
\begin{align*}
&\int_{0}^{1}|(F\varphi)(x)-(F\psi)(x)|^{p}dx  \\
&  \leq
\Big(\int_{-\infty}^{0}(\vartheta(\theta))^{q}e^{-q(\gamma+\varepsilon)\theta
}d\theta\Big)^{p/q}
\int_{0}^{1}\int_{-\infty}^{0}e^{p(\gamma+\varepsilon)\theta}|
\varphi(\theta)(x)-\psi(\theta)(x)|^{p}d\theta \,dx.
\end{align*}
Then
\begin{align*}
\int_{0}^{1}|(F\varphi)(x)-(F\psi)(x)|^{p}dx
&  \leq \lambda\int_{-\infty}^{0}e^{p\varepsilon\theta}
\int_{0}^{1}e^{p\gamma\theta }|\varphi(\theta)(x)
-\psi(\theta)(x)|^{p}dx\,d\theta\\
&  \leq\lambda(\underset{\theta\leq0}{\sup}\text{ }e^{p\gamma\theta}
\int_{0}^{1}|\varphi(\theta)(x)-\psi(\theta)(x)|^{p}dx)
\int_{-\infty}^{0}e^{p\varepsilon\theta}d\theta\\
& \leq\frac{1}{p\varepsilon}\lambda|\varphi-\psi|_{C_{\gamma} }^{p}.
\end{align*}
Therefore,
\[
|F(\varphi)-F(\psi)|_{p}\leq(\frac{1}{p\varepsilon}
\lambda)^{1/p}|\varphi-\psi|_{C_{\gamma}}.
\]
\end{proof}
Let function $\phi$ defined by
\[
\phi(\theta)(x)=\varphi(\theta,x)\quad\text{for }\theta\leq0,\;  x\in[0,1].
\]
Then \eqref{ab} takes the  abstract form
\begin{equation}
\begin{gathered}
u'(t)+Au(t)\ni F(u_{t})
\quad\text{for }t\geq0\\
u_{0}=\phi\in C_{\gamma}.
\end{gathered} \label{bn}
\end{equation}
Consequently, by Theorem \ref{thp}, we deduce the following result.

\begin{proposition} \label{prop4.5}
Under the above assumption, let $\phi\in C_{\gamma}\cap
C^{1}(]-\infty,0];X)$ be such that $\phi'\in C_{\gamma}$,
$\phi'$ bounded and $\phi(0)\in\hat{D}(A)$.
Then  \eqref{bn} has a unique strong solution $u$ and
the function $v$ defined by
\[
v(t,x)=u(t)(x)\quad\text{for a.e. }(t,x)\in[0,1]\times]0,1[
\]
satisfies \eqref{ab} for almost everywhere $(t,x)\in[0,1]\times]0,1[$.
\end{proposition}

\subsection*{Example 2: Hyperbolic case}
We consider the  hyperbolic equation
\begin{equation}
\begin{gathered}
 \frac{\partial}{\partial t}u(t,x)+\frac{\partial}{\partial x}(g(u(t,x)))
= \int_{-\infty}^{0} H(\theta,x,u(t+\theta,x))d\theta\quad
\text{for }t\geq0,\; x\in\mathbb{R}\\
u(\theta,x)=\varphi_{0}(\theta,x)\quad\text{for }\theta\leq0,\;
x\in\mathbb{R}
\end{gathered} \label{hype}
\end{equation}
where $g:\mathbb{R}\to \mathbb{R}$ is continuous and strictly monotone with
$g(\mathbb{R})=\mathbb{R}$.
$H:]-\infty,0]\times\mathbb{R}\times \mathbb{R}\to \mathbb{R}$ and
the initial value function
$\varphi_{0}:]-\infty,0]\times\mathbb{R}\to
\mathbb{R}$ will be defined in the sequel.

Let $X=L^{1}(\mathbb{R})$ and define the operator
\begin{gather*}
D(A)=\big\{v\in L^{1}(\mathbb{R)\cap}L^{\infty}(\mathbb{R)}:\frac{d}
{dx}(g(v(x))\in L^{1}(\mathbb{R)}\big\}\\
Av=\frac{d}{dx}(g(v(x)).
\end{gather*}

\begin{lemma}[\cite{kp2}] \label{lem4.6}
$A$ is $m$-accretive operator in $L^{1}(\mathbb{R)}$.
\end{lemma}

 As above, we choose the phase space
\[
\text{$\mathcal{B}$}\ =C_{\gamma}=\{\varphi\in C(]
-\infty ,0]
;L^{1}(\mathbb{R)}):\underset{\theta\leq0}{\sup}\text{ }
e^{\gamma\theta}|\varphi(\theta)|
_{1}<+\infty\},
\]
where $\gamma>0$, we provide $C_{\gamma}$ with the  norm
\[
|\varphi|_{C_{\gamma}}=\sup_{\theta\leq0}
e^{\gamma\theta}|\varphi(\theta)|_{1},
\]
where
\[
|\varphi(\theta)|_{1}=\int_{-\infty}^{\infty}|
\varphi(\theta)(x)|dx.
\]
Let $F$ be defined on $C_{\gamma}$ by
\[
F(\varphi)(x)=\int_{-\infty}^{0}H(\theta,x,\varphi(\theta,x))
d\theta\quad\text{for }t\geq0,\; x\in\mathbb{R}.
\]
And the function $\phi$ defined by
\[
\phi(\theta)(x)=\varphi_{0}(\theta,x)\quad\text{for for }
\theta\leq0,\; x\in\mathbb{R}
\]
Then equation \ref{hype} takes the  abstract form
\begin{gather*}
 u'(t)+Au(t)=F(u_{t}) \quad\text{for }t\geq0\\
u_{0}=\phi\in C_{\gamma}
\end{gather*}
We assume that $H$ satisfies
\[
|H(\theta,x,y_{1})-H(\theta,x,y_{2})|\leq\kappa
(\theta)|y_{1}-y_{2}|\quad \text{for }
\theta\in]-\infty,0]\,\; x,y_{1},y_{2} \in\mathbb{R}
\]
with
\[
\int_{-\infty}^{0}e^{-\gamma\theta}\kappa(\theta)d\theta<\infty.
\]
Moreover, we assume that
\[
H(.,.,0)\in L^{1}(]-\infty,0]\times\mathbb{R)}.
\]
Under the above condition, $F:C_{\gamma}\to L^{1}(\mathbb{R)}$ is
Lipschitz continuous. Let $\varphi\in C_{\gamma}$. Then $F(\varphi)\in
L^{1}(\mathbb{R)}$ due to the fact, that
\[
F(0)\in L^{1}(\mathbb{R)}.
\]
For the Lipschitz condition, take $\varphi,\psi\in C_{\gamma}$ and
$x\in\mathbb{R}$. Then
\[
|(F(\varphi)-F(\psi))(x)|
\leq\int_{-\infty} ^{0}\kappa(\theta)|
\varphi(\theta,x)-\psi(\theta,x)|d\theta\quad\text{for
}x\in\mathbb{R}.
\]
It follows that
\[
\int_{-\infty}^{\infty}|(F(\varphi)-F(\psi))(x)| dx
\leq\int_{-\infty}^{0}e^{-\gamma\theta}\kappa(\theta)e^{-\gamma\theta
}\int_{-\infty}^{\infty}|\varphi(\theta,x)-\psi(\theta,x)|
dx\,d\theta.
\]
Consequently,
\[
|F(\varphi)-F(\psi)|_{1}\leq\int_{-\infty}^{0}e^{-\gamma\theta
}\kappa(\theta)d\theta|\varphi-\psi|_{C_{\gamma}}.
\]
By theorem \ref{sfpc}, we deduce the following result.

\begin{proposition} \label{prop4.7}
Let the initial data function $\varphi_{0}$ be such that
$\phi\in C_{\gamma}$  and $\phi(0)\in\overline{D(A)}$.
Then  \eqref{1} has a unique mild solution  defined on
$[0,+\infty)$.
\end{proposition}

\subsection*{Acknowledgments}
The authors would like to thank the anonymous referees for their
careful reading of the original manuscript.
Their valuable suggestions made numerous improvements.

\begin{thebibliography}{00}

\bibitem{az} M. Adimy, H. Bouzahir and K. Ezzinbi;
\emph{Local Existence and Stability for Some Partial Functional
Differential Equation with Infinite Delay},
Nonlinear Analysis, Theory Methods and Applications, 48, 323-348, (2002).

\bibitem{bar} V. Barbu;
\emph{Nonlinear Semigroups and Differential Equations in
Banach Spaces}, Noordhoff Leyden, (1976).

\bibitem{bcp} P. Benilan M. G. Crandall and A. Pazy;
\emph{Nonlinear Evolution Equations in Banach Spaces}, (Unpublished book).

\bibitem{bg} B. Burch and J. Goldstein;
\emph{Nonlinear semigroups and a problem in
heat conduction}, Houston J. Math.4, 311-328, (1978).

\bibitem{ev} L. C. Evans;
\emph{Nonlinear evolution equations in arbitrary Banach
spaces}, Israel J. Math., Vol. 26, No. 1, 1-42, (1977).

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

\bibitem{hn} Y. Hino ,S. Murakam,  Naito;
\emph{Functional Differental Equations with
Infinite Delay}, Springer-Verlag, (1991).

\bibitem{kp} A. G. Kartsatos and M. E. Parrott;
\emph{Convergence of the Kato
approximants for evolution equations involving functional perturbations},
Journal of Differential Equations, 47, 358-377, (1983).

\bibitem{kp2} A. G. Kartsatos and M. E. Parrott;
\emph{The weak solution of functional differential equation
in a Banach Space}, Journal of Differential
Equation, 75, 290-302, (1988).

\bibitem{tk} T. Kato;
\emph{Nonlinear semigroups and evolution equations, Journal of
the Mathematical Society of Japan}, Vol. 19,  508-520, (1967).

\bibitem{w1} W. M. Ruess;
\emph{The evolution operator approach to functional
differential equations with delay}, Proceedings of the
American Mathematical
Society, Vo. 119, 783-791, (1993).

\bibitem{rues} W. M. Ruess and W. H. Summers;
\emph{Operator semigroups for functional differential equations
with delay}, Transactions of the American
Mathematical Society, Vol. 341, No. 2, 695-719, (1994).

\bibitem{ki} K. Sik Ha;
\emph{Nonlinear Functional Evolutions in Banach Spaces},
Kluwer Academic Publishers, $(2003)$.

\bibitem{web} C. C. Travis and G. F. Webb;
\emph{Existence and stability for
partial functional differential equations}, Transactions of the American
Mathematical Society, Vol. 200, 395-418, (1974).

\bibitem{wu} J. Wu;
\emph{Theory and Applications of Partial Functional Differential
Equations}, Graduate Texts in Applied Mathematics,
Springer-Verlag, Vol. 119, (1996).

\end{thebibliography}
\end{document}