\documentclass[reqno]{amsart}

\AtBeginDocument{{\noindent\small 
{\em Electronic Journal of Differential Equations},
Vol. 2003(2003), No. 116, pp. 1--13.\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 2003 Texas State University-San Marcos.} 
\vspace{9mm}}

\begin{document}

\title[\hfilneg EJDE--2003/116\hfil Existence and stability]
{Existence and stability for some partial functional differential equations
with infinite delay}

\author[Khalil Ezzinbi \hfil EJDE--2003/116\hfilneg]
{Khalil Ezzinbi}

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


\date{}
\thanks{Submitted May 29, 2003. Published November 26, 2003.}
\thanks{Partially supported by grant 00-412 RG/MATHS/AF/AC from TWAS}
\subjclass[2000]{34K20, 34K30, 34K40, 45N05}
\keywords{Hille-Yosida operator, extrapolation spaces, Favard class,
regularity, \hfill\break\indent
 partial functional differential equations,   infinite delay,
 mild solution,  linearized stability}



\begin{abstract}
  We study the existence, regularity, and stability of solutions for 
  some partial functional differential equations with infinite delay.
  We assume that the linear part is not necessarily densely defined and 
  satisfies the  Hille-Yosida condition on a Banach space $X$. 
  The nonlinear term takes its values in space larger than $X$, namely 
  the extrapolated Favard class of the extrapolated semigroup 
  corresponding to the linear part. Our approach is based on the theory 
  of the extrapolation spaces.
\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 consider the partial functional differential equation with
infinite delay: 
\begin{equation}
\begin{gathered} \frac{d}{dt}x(t)=Ax(t)+F(x_{t}),\quad\text{for }t\geq 0, \\
x_0=\varphi \in \mathcal{B}, \end{gathered}  \label{a}
\end{equation}
where $A$ is a nondensely defined linear operator on a Banach space $X$ and
satisfies the Hille-Yosida condition, this means that $A$ satisfies the
usual assumption of Hille-Yosida's theorem characterizing the generators of
strongly continuous semigroups except the density of $D(A)$ in $X$: there
exist $N_{0}\geq 0$ and $\omega _{0}\in \mathbb{R}$ such that $(\omega
_{0},+\infty )\subset \rho (A)$ and 
\[
\sup \left\{ (\lambda -\omega _{0})^{n}|(\lambda -A)^{-n}|:n\in \mathbb{N}%
,\;\lambda >\omega _{0}\right\} \leq N_{0}, 
\]
where $\rho (A)$ is the resolvent set of $A$. $\mathcal{B}$ is a linear
space of functions from $(-\infty ,0]$ into $X$ satisfying some axioms which
will be described in the sequel. For every $t\geq 0$, the history function $%
x_{t}\in \mathcal{B}$ is defined by 
\[
x_{t}(\theta )=x(t+\theta ),\quad \text{for }\theta \in (-\infty ,0]. 
\]
$F$ is a continuous function from $\mathcal{B}$ with values in $F_{0}$
larger space than $X$, namely the extrapolated Favard class of the
extrapolated semigroup corresponding to the linear part $A$, (see section
2). Note that the non-density occurs in many situations due to the
restrictions on the space where the equation is considered (for example,
periodic continuous functions, H\"{o}lder continuous functions) or due to
boundary conditions (for example, the space $C^{1}$ with null value on the
boundary is non dense in the space of continuous functions) (see examples in 
\cite{Dap}). In the literature devoted to equations with finite delay, the
state space is the space of continuous functions on $[-r,0]$ with values in $%
X$, for more details we refer to the book by Wu \cite{Wu1}. When the delay
is unbounded, the selection of the state space $\mathcal{B}$ plays an
important role in the study of both quantitative and qualitative studies. A
usual choice is a semi-normed space satisfying suitable axioms, which was
introduced by Hale and Kato \cite{HalKat1}. Concerning to the theory of
functional differential equations with infinite delay, we refer to the book
by Hino, Murakami and Naito \cite{Hin1}. In recent years, the theory of
partial functional differential equations with infinite delay have been the
subject of considerable activity. In \cite{Hen3}, it has been proved the
existence and regularity of solutions of Equation (\ref{a}) when $F$ takes
its values in $X$ and $A$ is the infinitesimal generator of analytic
semigroup on $X$. This in particular contains the density of the domain $D(A%
\mathbf{)}$ in $X$ by Hille-Yosida's theorem. More recently, it has been
shown in \cite{ez10} that the density condition is not necessary to deal
with partial functional differential equations. In \cite{ez2} and \cite{ez1}%
, it has been proved the existence, regularity of solutions and stability
for Equation (\ref{a}) when $F$ takes its values in $X$ and $A$ is
nondensely defined and satisfies the Hille-Yosida condition. There are many
approaches to deal with partial differential equations with non dense
domain, one of them is based on the extrapolation approach. For more details
about the extrapolation approach, we cite the book by Engel and Nagel \cite
{nage2} and \cite{nage1}. This work was motivated by \cite{gab} where the
authors have proved the existence and regularity of solutions for the
following partial functional differential equation 
\begin{gather*}
\frac{d}{dt}x(t)=Ax(t)+G(t,x_{t}),\text{{\ for} }t\geq 0 \\
x_{0}=\varphi \in C([-r,0];X),
\end{gather*}
where $A$ generates a strongly continuous semigroup on $X$, $C([-r,0];X)$ is
the space of continuous function from $[-r,0]$ into $X$ endowed with the
uniform norm topology and $G$ is a continuous function on $\mathbb{R}%
^{+}\times C([-r,0];X)$ with values in $F_{0}$.

The purpose of this work is to discuss the existence, regularity of the mild
solutions of Equation (\ref{a}) and the asymptotic behavior of solutions
near an equilibrium. In our context we will use the extrapolation approach.
The obtained results of this work would be an extension of the results in 
\cite{ez2}, \cite{ez1}, \cite{Hen3} and \cite{gab}. The organization is as
follows: In section 2, we recall some preliminary results about the
extrapolation spaces and Favard class which will be used in the whole of the
work. In section 3, we start with our main results, in which we prove the
existence and regularity of the mild solutions of Equation (\ref{a}). In
section 4, we prove the linearized stability and finally we propose an
application.

\section{Extrapolation spaces and Favard class}

Here and hereafter we assume that

\begin{itemize}
\item[(H1)]  $A$ is a Hille-Yosida operator on a Banach space $X$.
\end{itemize}
Let $A_{0}$ be the part of $A$ in $X_{0}=\overline{D(A)}$ which is defined
by 
\[
D(A_{0})=\left\{ x\in D(A):Ax\in \overline{D(A)}\right\} \quad
A_{0}x=Ax,\quad \text{for }x\in D(A_{0}). 
\]

\begin{lemma}[\cite{nage2}] \label{lm1} 
$A_{0}$ generates a strongly continuous semigroup $%
(T_{0}(t))_{t\geq 0}$ on $X_{0}$ and $|T_{0}(t)|\leq N_{0}e^{\omega t}$, for 
$t\geq 0$. Moreover $\rho (A)\subset \rho (A_{0})$ and $R(\lambda
,A_{0})=R(\lambda ,A)/X_{0}$, for $\lambda \in \rho (A)$.
\end{lemma}

For a fixed $\lambda _0\in \rho (A)$, we introduce on $X_0$ a new norm
defined by 
\[
\left\| x\right\| _{-1}=| R(\lambda _0,A_0)x| \quad \text{for }x\in
D(A_0)\,. 
\]
The completion $X_{-1}$ of $(X_0,\| .\|_{-1})$ is called the extrapolation
space of $X$ associated with $A$ Note that $\| .\| _{-1}$ and the norm on $%
X_0$ given by $| R(\lambda,A_0)x| $ for $\lambda \in \rho (A)$ are
equivalent. The operator $T_0(t)$ has a unique bounded linear extension $%
T_{-1}(t)$ to the Banach space $X_{-1}$ and $( T_{-1}(t)) _{t\geq 0}$ is a
strongly continuous semigroup on $X_{-1}$. $( T_{-1}(t)) _{t\geq 0}$ is
called the extrapolated semigroup of $( T_0(t))_{t\geq 0}$, we denote its
generator by $(A_{-1},D(A_{-1})$. We have some fundamental results.

\begin{lemma}[\cite{gab}]
\label{lm2} The following properties hold:

\begin{itemize}
\item[(i)]  $|T_{-1}(t)|_{L(X_{-1})}=|T_{0}(t)|_{L(X_{0})}$

\item[(ii)]  $D(A_{-1})=X_{0}$

\item[(iii)]  $A_{-1}:X_{0}\to X_{-1}$ is the unique continuous extension of 
$A_{0}$: $D(A_{0})\subset (X_{0},|.|)\to (X_{-1},\Vert .\Vert _{-1})$ and $%
(\lambda _{0}-A_{-1})$ is an isometry from $(X_{0},|.|)$ to $(X_{-1},\Vert
.\Vert _{-1})$,

\item[(iv)]  If $\lambda \in \rho (A_{0})$, then ($\lambda -A_{-1})$ is
invertible and $(\lambda -A_{-1})^{-1}\in L(X_{-1})$. In particular $\lambda
\in \rho (A_{-1})$ and $R(\lambda ,A_{-1})/X_{0}=R(\lambda ,A_{0})$.

\item[(v)]  The space $X_{0}=\overline{D(A)}$ is dense in $(X_{-1},\Vert
.\Vert _{-1})$. Hence the extrapolation space $X_{-1}$ is also the
completion of $(X,\Vert .\Vert _{-1})$ and $X\hookrightarrow X_{-1}$.

\item[(vi)]  The operator $A_{-1}$ is an extension of $A$. In particular if $%
\lambda \in \rho (A)$, then $R(\lambda ,A_{-1})/X=R(\lambda ,A)$ and $%
(\lambda ,A_{-1})X=D(A)$.
\end{itemize}
\end{lemma}

Next we introduce the Favard class corresponding to semigroup.

\begin{definition}[\cite{gab}] \label{def1}  \rm
Let $(S(t))_{t\geq 0}$ be a strongly continuous
semigroup with generator $(B,D(B))$ on a Banach space $Z$ such that $%
|S(t)|\leq Ne^{\nu t}$ for some $N\geq 1$ and $\nu \in \mathbb{R}$. The
Favard class of $(S(t))_{t\geq 0}$ is the Banach space 
\[
\mathbb{F}=\Big\{ x\in Z:\sup_{t>0}\frac{1}{t}|e^{-\nu t}S(t)x-x|<\infty 
\Big\}
\]
equipped with the norm 
\[
|x|_{\mathbb{F}}=|x|+\sup_{t>0}\frac{1}{t}|e^{-\nu t}S(t)x-x|.
\]
\end{definition}

We can see that $\mathbb{F}$ is invariant under $( S(t)) _{t\geq 0}$ and $%
D(B)\subset \mathbb{F}$. If $Z$ is reflexive then $\mathbb{F}=D(B)$.
Furthermore if we denote by $|.| _{B}$ the graph norm of $B$, then $|.| _{B}$
and $|.| _{\mathbb{F}}$ are equivalent norms on $D(B)$.

For the rest of the paper we denote by $F_1\subset X_0 $ the Favard class of
the $C_0$ semigroup $( T_0(t)) _{t\geq 0}$ and $F_0\subset X_{-1}$ the
Favard class of $( T_{-1}(t)) _{t\geq 0}$.

\begin{lemma}[\cite{gab}]
\label{lm3} For the Favard classes $F_{0}$ and $F_{1}$ the following hold:

\begin{itemize}
\item[(i)]  $(\lambda _{0}-A_{-1})F_{1}=F_{0}$

\item[(ii)]  $T_{-1}(t)F_{0}\subset F_{0}$, $t\geq 0$,

\item[(iii)]  $D(A_{0})\hookrightarrow D(A)\hookrightarrow
F_{1}\hookrightarrow X_{0}\hookrightarrow X\hookrightarrow
F_{0}\hookrightarrow X_{-1}$, where $D(A)$ is equipped with the graph norm.
\end{itemize}
\end{lemma}

\begin{proposition}[\cite{nage1}]
\label{nag1} For $f\in L_{loc}^{1}(\mathbb{R}^{+},F_{0})$, we define 
\[
(T_{-1}*f)(t)=\int_{0}^{t}T_{-1}(t-s)f(s)ds,\text{ }t\geq 0\,.
\]
Then

\begin{itemize}
\item[(i)]  $(T_{-1}*f)(t)\in \overline{D(A)},$ for all $t\geq 0,$

\item[(ii)]  $|(T_{-1}*f)(t)|\leq \overline{M}%
\int_{0}^{t}e^{w(t-s)}|f(s)|_{F_{0}}ds$, for some $\overline{M}$ independent
of $f$ and $t,$

\item[(iii)]  $\lim_{t\to 0}|(T_{-1}*f)(t)|=0$.
\end{itemize}
\end{proposition}

\begin{remark}
\label{rmk1} \textrm{Condition (iii) in Proposition \ref{nag1} implies that
the function $t\to \int_{0}^{t}T_{-1}(t-s)f(s)ds$ is continuous from $%
\mathbb{R}^{+}$ to $X_{0}$. }
\end{remark}

\section{Existence and regularity of solutions}

We assume that the phase space $\mathcal{B}$ is a linear space of functions
mapping $(-\infty ,0]$ into $X$, endowed with a norm $|.|_{\mathcal{B}}$ and
satisfying the following fundamental axioms introduced at first by Hale and
Kato in \cite{HalKat1}:

\begin{itemize}
\item[(A1)]  There exist a positive constant $H$ and functions $%
K,M:[0,+\infty )\to [0,+\infty )$, with $K$ is continuous and $M$ is locally
bounded, such that for any $\sigma \in \mathbb{R}$ and $a>0$, if $x:(-\infty
,\sigma +a]\to X$, $x_{\sigma }\in \mathcal{B}$ and $x$ is continuous on $%
[\sigma ,\sigma +a]$, then for all $t$ in $[\sigma ,\sigma +a]$ the
following conditions hold:

\begin{itemize}
\item[(i)]  $x_{t}\in \mathcal{B}$,

\item[(ii)]  $|x(t)|\leq H\,|x_{t}|_{\mathcal{B}}$,

\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 a function $x$ satisfying (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}

For the remaining of this work, we use the notations: for $a>0$, we define $%
K_{a}=\max_{0\leq t\leq a}K(t)$ and $M_{a}=\max_{0\leq t\leq a}M(t)$.

\begin{itemize}
\item[(H2)]  We assume that $F$ takes its values in $F_{0}$ and satisfies
the Lipschitz condition $|F(\varphi _{1})-F(\varphi _{2})|_{F_{0}}\leq
L|\varphi _{1}-\varphi _{2}|_{\mathcal{B}}$, for $\varphi _{1},\varphi
_{2}\in \mathcal{B}$.
\end{itemize}

\begin{definition}
\label{def2} \textrm{A function $x:(-\infty ,\infty )\to X$ is called a mild
solution of (\ref{a}) if $x$ is continuous on $[0,\infty )$ and satisfies 
\begin{gather*}
x(t)=T_{0}(t)\varphi (0)+\int_{0}^{t}T_{-1}(t-s)F(x_{s})ds,\quad t\geq 0 \\
x_{0}=\varphi \,.
\end{gather*}
}
\end{definition}

When $F$ maps into $X$, the mild solution coincides with the integral
solution given in \cite{ez2} and \cite{ez1}. By Proposition \ref{nag1}, the
mild solution (if it exists) takes values in $\overline{D(A)}$ only if $%
\varphi (0)\in \overline{D(A)}$.

\begin{theorem}
\label{T1} Assume that (H1)--(H2) hold. Then for $\varphi \in \mathcal{B}$
such that $\varphi (0)\in \overline{D(A)}$, Equation (\ref{a}) has a unique
mild solution $x(.,\varphi )$ which is defined for all $t\geq 0$. Moreover
for every $a>0$, there exists $\beta >0$ such that for $\varphi _{1}$, $%
\varphi _{2}\in \mathcal{B}$ with $\varphi _{1}(0)$, $\varphi _{2}(0)\in 
\overline{D(A)}$, we have 
\begin{equation}
|x_{t}(.,\varphi _{1})-x_{t}(.,\varphi _{2})|_{\mathcal{B}}\leq \beta
|\varphi _{1}-\varphi _{2}|_{\mathcal{B}},\quad \text{for }t\in [0,a].
\label{333}
\end{equation}
\end{theorem}

\begin{proof}
Let $\varphi \in \mathcal{B}$ such that $\varphi (0)\in
\overline{D(A)}$ and for $a>0$, we introduce the set $Z_{a}(\varphi )$  by
\[
Z_{a}(\varphi ):=\left\{ y\in C([0,a] ;X): y(0)=\varphi
(0)\right\} ,
\]
provided with the uniform norm topology which will be denoted by
\[
| y| _{\infty }=\sup_{s\in [ 0,a]}|y(s)| ,
\quad\text{for }y\in Z_{a}(\varphi ).
\]
For $x\in Z_{a}(\varphi )$, the function $\widetilde{x}:( -\infty ,a] \to X$
is given by
\[
\widetilde{x}(t)=\begin{cases}
x(t), & t\in [0,a], \\
\varphi (t), & -\infty <t\leq 0.
\end{cases}
\]
By virtue of conditions (H2) and (A2), we deduce that the mapping
$s\to F(\widetilde{x}_{s})$ is continuous from $[ 0,a]$ to
$F_0$. Let $\mathcal{P}$ be defined on $Z_{a}(\varphi )$
 by
\[
(\mathcal{P}x)(t)=T_0(t)\varphi (0)
+\int_0^{t}T_{-1}(t-s)F(\widetilde{x}_{s})ds,\quad
t\in [0,a].
\]
By Proposition \ref{nag1}, we get that the function
$t\to \int_0^{t}T_{-1}(t-s)F(\widetilde{x}_{s})ds$ is
continuous from $[0,a]$ to $X$, which
implies that $\mathcal{P}x\in Z_{a}(\varphi )$,  if $x\in Z_{a}(\varphi )$.
Therefore, by Proposition \ref{nag1}, we have
\[
\Big| \int_0^{t}T_{-1}(t-s)(F(\widetilde{x}_{s})-F(\widetilde{y}_{s}))ds\Big|
\leq \overline{M}\int_0^{t}e^{w(t-s)}| F(\widetilde{x%
}_{s})-F(\widetilde{y}_{s})| _{F_0}ds,
\]
 this implies by assumption (H2) that
\[
\Big| \int_0^{t}T_{-1}(t-s)(F(\widetilde{x}_{s})-F(\widetilde{y}_{s}))ds\Big|
 \leq \overline{M}L\int_0^{t}e^{w(t-s)}
 | \widetilde{x}_{s}-\widetilde{y}_{s}| _{\mathcal{B}}ds.
\]
Without loss of generality we assume that $\omega >0$ and
then by (A1) part (iii), we obtain
\begin{align*}
| (\mathcal{P}x)(t)-(\mathcal{P}y)(t)|
&\leq \overline{M}Le^{\omega a}\int_0^{t}K(s)\sup_{0\leq \xi \leq s}
| x(\xi)-y(\xi )| \,ds \\
&\leq \overline{M}Le^{\omega a}K_{a}\int_0^{t}\sup_{0\leq \xi \leq s}
| x(\xi )-y(\xi )| ds.
\end{align*}
Arguing as above we can see that
\begin{align*}
| (\mathcal{P}^{2}x)(t)-(\mathcal{P}^{2}y)(t)|
&\leq \overline{M}Le^{\omega a}K_{a}\int_0^{t}\sup_{0\leq \xi \leq s}
| (\mathcal{P}x)(\xi )-(\mathcal{P}y)(\xi )| ds, \\
&\leq (\overline{M}Le^{\omega a}K_{a})^{2}\int_0^{t}\sup_{0\leq \xi \leq s}
\int_0^{\xi }\sup_{0\leq \alpha\leq p}| x(\alpha )-y(\alpha )|
dp\,ds, \\
&\leq (\overline{M}Le^{\omega a}K_{a})^{2}\int_0^{t}\int_0^{s}dp\,ds
|x-y|_{\infty}, \\
&\leq \frac{(\overline{M}Le^{\omega a}K_{a})^{2}}{2}a^{2}
| x-y|_{\infty }.
\end{align*}
Then for every $n\in \mathbb{N}^{*}$ we have
\[
| (\mathcal{P}^{n}x)(t)-(\mathcal{P}^{n}y)(t)| \leq \frac{(\overline{M}%
Le^{\omega a}K_{a})^{n}}{n!}a^{n}| x-y| _{\infty }.
\]
Since there exists $n\in \mathbb{N}$ such that
$\frac{(\overline{M}Le^{\omega a}K_{a})^{n}}{n!}a^{n}<1$,
 it follows that $\mathcal{P}^{n}$ is a strict contraction
on the closed subset $Z_{a}(\varphi )$ of the Banach space
$C([0,a] ;X)$. Consequently $\mathcal{P}$ has a
unique fixed point in $Z_{a}(\varphi )$, this holds for every $a>0$.
We conclude that Equation (\ref{a}) has a unique mild solution $
x(.,\varphi )$ on $( -\infty ,\infty )$.
Moreover the solution depends continuously on the initial data. In fact, if
we consider two solutions $x:=x(.,\varphi _1)$ and
$y:=y(.,\varphi _2)$ for $\varphi _1,\varphi _2\in \mathcal{B}$
such that $\varphi _1(0),\varphi _2(0)\in \overline{D(A)}$,
then for every $t\in [0,a]$  with $a>0$ fixed, we have
\begin{align*}
| x(t)-y(t)|
&\leq | T_0(t)( \varphi _1(0)-\varphi
_2(0)) | +\overline{M}Le^{\omega
a}\int_0^{t}| x_{s}-y_{s}| _{\mathcal{B}}ds, \\
&\leq N_0e^{\omega a}| \varphi _1(0)-\varphi _2(0)|\\
&\quad +\overline{M}Le^{\omega a}\int_0^{t}\Big( K(s)
\max_{0\leq \xi \leq s}| x(\xi )-y(\xi )|
+M(s)| \varphi _1-\varphi _2| _{\mathcal{B}}\Big) ds, \\
&\leq HN_0e^{\omega a}| \varphi _1-\varphi _2| _{\mathcal{B}
} +\overline{M}Le^{\omega a}K_{a}\int_0^{t}\max_{0\leq \xi \leq s}
| x(\xi )-y(\xi )| ds\\
&\quad +a\overline{M}Le^{\omega a}M_{a}
| \varphi _1-\varphi _2| _{\mathcal{B}}.
\end{align*}
By Gronwall's lemma, it follows that
\[
\max_{0\leq s\leq t}| x(s)-y(s)| \leq \beta_0
| \varphi _1-\varphi _2| _{\mathcal{B}},
\quad \text{for }t\in [0,a],
\]
where $\beta _0=e^{\omega a}(HN_0+a\overline{M}LM_{a})
\exp (a\overline{M}LK_{a}e^{\omega a})$. Therefore, by (A1) part (iii),
\[
| x_{t}(.,\varphi _1)-x_{t}(.,\varphi _2)| _{\mathcal{B}}
\leq K(t)\sup_{0\leq s\leq t}| x(s,\varphi _1)-x(s,\varphi
_2)| +M( t) | \varphi _1-\varphi _2
| _{\mathcal{B}},
\]
which implies the desired estimate (\ref{333}).
\end{proof}

For the regularity of the mild solution, we need to compute the integral in $%
\mathcal{B}$ from the integral in $X$. For that we suppose that $\mathcal{B}$
satisfies the condition:

\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 $.
\end{itemize}

\begin{lemma}[\cite{nai}] \label{111} 
Let $\mathcal{B}$ satisfy (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}

One can obtain the similar result, by assuming that $\mathcal{B}$ satisfies

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

\begin{lemma}
\label{112} Let $\mathcal{B}$ satisfy (D1) and $f:[0,a]\to \mathcal{B}$ be a
continuous function. Then for all $\theta \in (-\infty ,0]$, the function 
$f(.)(\theta )$ is continuous and 
\[
\Big[ \int_{0}^{a}f(t)dt\Big] (\theta )=\int_{0}^{a}f(t)(\theta )dt,\quad
\theta \in (-\infty ,0].
\]
\end{lemma}

\begin{proof}
Since the function $f$ is continuous, it follows that
\[
\int_0^{a}f(t)dt=\lim_{n\to +\infty }\frac{a}{n}
\sum{k=1}^n f(\frac{ka}{n})\quad \text{in }\mathcal{B}.
\]
By assumption (D1),
\[
\Big[ \int_0^{a}f(t)dt\Big] (\theta )
=\lim_{n\to +\infty } \frac{a}{n}\sum_{k=1}^n f(\frac{kn}{a})(\theta ),
\quad \theta \in ( -\infty,0] .
\]
Moreover by (D1), the function $f(.)(\theta )$ is continuous on $[0,a]$,
from what we infer that for all $\theta \in ( -\infty ,0]$,
the function $f(.)(\theta )$ is integrable on $[0,a] $ and
\[
\int_0^{a}f(t)(\theta )dt=\lim_{n\to +\infty } \frac{a}{n}
\sum_{k=1}^n f(\frac{kn}{a})(\theta ),\quad \theta \in ( -\infty ,0] ,
\]
 which implies
\[
\Big[ \int_0^{a}f(t)dt\Big] (\theta)=\int_0^{a}f(t)(\theta )dt,\quad
\theta \in ( -\infty,0].
\]
\end{proof}

\begin{theorem}
\label{T2} Assume that (H1) and (H2) hold and $\mathcal{B}$ satisfies (C1)
or (D1). Furthermore we assume that $F:\mathcal{B}\to F_{0}$ is continuously
differentiable and $F^{\prime }$ is locally Lipschitz. Let $\varphi \in 
\mathcal{B}$ be continuously differentiable such that 
\[
\varphi ^{\prime }\in \mathcal{B},\quad \varphi (0)\in F_{1},\quad \varphi
^{\prime }(0)\in \overline{D(A)},\quad \text{and}\quad \varphi ^{\prime
}(0)=A_{-1}\varphi (0)+F(\varphi ).
\]
Then the mild solution $x$ of (\ref{a}) belongs to $C^{1}(\mathbb{R}%
^{+},X)\cap C(\mathbb{R}^{+},F_{1})$ and satisfies 
\begin{equation}
\begin{gathered} \frac{d}{dt}x(t)=A_{-1}x(t)+F(x_{t}),\quad t\geq 0 \\
x_0=\varphi . \end{gathered}  \label{123}
\end{equation}
\end{theorem}

The proof of this theorem is based on the following Lemma.

\begin{lemma}[\cite{nage1}]
\label{lenag} For $u_{0}\in F_{1}$ and $h\in W^{1,1}(\mathbb{R}^{+},F_{0})$
such that $A_{-1}u_{0}+h(0)\in \overline{D(A)}$, the equation 
\begin{equation}
\begin{gathered} \frac{d}{dt}u(t)=A_{-1}u(t)+h(t),\text{ }t\geq 0 \\
u(0)=u_0 \end{gathered}  \label{dd}
\end{equation}
has a unique solution $u\in C^{1}(\mathbb{R}^{+},X)\cap C(\mathbb{R}%
^{+},F_{1})$.
\end{lemma}

\begin{remark}
\label{rmk2} \textrm{The mild solution of (\ref{dd}) is given by 
\[
u(t)=T_{0}(t)u_{0}+\int_{0}^{t}T_{-1}(t-s)h(s)ds,\quad t\geq 0
\]
Using Lemma \ref{lenag} we can deduce that $u\in C^{1}(\mathbb{R}^{+},X)\cap
C(\mathbb{R}^{+},F_{1})$ and satisfies (\ref{dd}) for every $t\geq 0$. }
\end{remark}

\begin{proof}[Proof of Theorem \ref{T2}]
 Set $a>0$  and let $x$  be the mild solution of (\ref{a}) on
 $[0,a] $. Consider the equation
\begin{equation}
\begin{gathered}
y(t)=T_0( t) \varphi '(0)+\int_0^{t}T_{-1}(t-s) F'( x_{s}) y_{s}ds,
\quad \text{for }t\in [0,a] \\
y_0=\varphi '.
\end{gathered} \label{8}
\end{equation}
 Then using the same reasoning as in the proof of Theorem \ref{T1} we
get that Equation (\ref{8}) has a unique solution $y$  on $(-\infty ,a] $.
Let $z:( -\infty ,a] \to X$  be defined by
\[
z(t)=\begin{cases} \varphi (0)+\int_0^{t}y(s)\,ds,& \text{for }t\in [0,a]\\
\varphi (t), &\mbox{for }t\in ( -\infty ,0].
\end{cases}
\]
By Lemma \ref{111} or Lemma \ref{112}, we deduce that
\begin{equation}
z_{t}=\varphi +\int_0^{t}y_{s}\,ds, \quad \text{for } t\in [0,a] .  \label{9}
\end{equation}
We will show that $x=z$  on $[0,a]$. From  (\ref{8}),
\begin{equation}
\int_0^{t}y(s)ds=  \int_0^{t}T_0( s) \varphi '(0)ds
+\int_0^{t}\int_0^{s}T_{-1}( s-\sigma ) F'( x_{\sigma }) y_{\sigma }d\sigma ds.
\label{10}
\end{equation}
 On the other hand $t\to z_{t}$  is differentiable
from $[0,a]$  to $\mathcal{B}$,  then by
differentiability of $F$, we get that $t\to F(z_{t})$
is differentiable from $[0,a]$  to $F_0$,
since $F_0\hookrightarrow X_{-1}$, it follows that $t\to F(z_{t})$
  is differentiable from $[0,a] $
to $X_{-1}$  and then for $t\in [0,a]$,
we have
\[
\frac{d}{dt}\int_0^{t}T_{-1}( t-s) F( z_{s}) ds=
T_{-1}( t) F(\varphi )+\int_0^{t}T_{-1}( t-s)
F'(z_{s})y_{s}ds,
\]
which implies
\begin{equation}
\int_0^{t}T_{-1}( s) F(\varphi )ds=\int_0^{t}T_{-1}(
t-s) F( z_{s}) ds+\int_0^{t}\int_0^{s}T_{-1}(
s-\sigma ) F'(z_{\sigma })y_{\sigma }d\sigma ds.  \label{13}
\end{equation}
It follows that
\begin{align*}
&z(t) \\
&=\varphi (0)+\int_0^{t}T_0( s) ( A_{-1}\varphi
(0)+F(\varphi )) ds+\int_0^{t}\int_0^{s}T_{-1}( s-\sigma
) F'( x_{\sigma }) y_{\sigma }d\sigma ds \\
&=\varphi (0)+\int_0^{t}T_{-1}( s) ( A_{-1}\varphi
(0)+F(\varphi )) ds+\int_0^{t}\int_0^{s}T_{-1}( s-\sigma
) F'( x_{\sigma }) y_{\sigma }d\sigma ds \\
&=T_{-1}(t)\varphi (0)+\int_0^{t}T_{-1}( s) F(\varphi
)ds+\int_0^{t}\int_0^{s}T_{-1}( s-\sigma ) F'
( x_{\sigma }) y_{\sigma }d\sigma ds
\end{align*}
Using (\ref{13}) we obtain
\begin{align*}
z(t)
&=T_{-1}(t)\varphi (0)+\int_0^{t}T_{-1}( t-s) F(z_{s}) ds\\
&\quad +\int_0^{t}\int_0^{s}T_{-1}( s-\sigma )
( F'( x_{\sigma }) -F'(z_{\sigma }))y_{\sigma }d\sigma ds.
\end{align*}
On the other hand $\varphi (0)\in F_1$  which gives that
$T_{-1}(t)\varphi (0)=T_0(t)\varphi (0)$,  for all $t\geq 0$.
Then
\begin{align*}
&x(t)-z(t)\\
&=\int_0^{t}T_{-1}( t-s) ( F( x_{s})-F( z_{s}) ) ds-\int_0^{t}
\int_0^{s}T_{-1}(s-\sigma ) ( F'( x_{\sigma }) -F'(z_{\sigma })) y_{\sigma }
d\sigma ds.
\end{align*}
Using the local Lipschitz condition on $F'$  we
get that there exists a positive constant $b_0$  such that
\[
\sup_{0\leq s\leq t} | x(s)-z(s)| \leq
b_0\int_0^{t}| x_{s}-z_{s}| _{\mathcal{B}}ds.
\]
 Since $x_0=z_0=\varphi $, axiom  (A1) part (iii)
implies
\[
| x_{t}-z_{t}| _{\mathcal{B}}\leq K_{a}\sup_{0\leq s\leq t}\sup | x(s)-z(s)| .
\]
Then
\[
| x_{t}-z_{t}| _{\mathcal{B}}\leq
K_{a}b_0\int_0^{t}| x_{s}-z_{s}| _{\mathcal{B}}\,ds.
\]
Using Gronwall's lemma, we conclude that
$| x_{t}-z_{t}|_{\mathcal{B}}=0$ for $t\in [0,a]$.  Consequently,
$x(t)=z(t)$ for all $t\in ( -\infty ,a] $. Hence,
$t\to x_{t}$ is continuously differentiable on $[0,a] $, for every
$a>0$. We deduce that $F(x.)\in C^{1}( \mathbb{R}^{+},F_0)$ and by
Lemma \ref{lenag} we deduce that $x$
belongs to $C^{1}( \mathbb{R}^{+},X) \cap C(\mathbb{R}^{+},F_1) $
  and satisfies (\ref{123}).
\end{proof}

\section{The solution semigroup and linearized stability}

Let $\mathcal{H}$ be the phase space of Equation (\ref{a}) which is given by 
\[
\mathcal{H}=\Big\{ \varphi \in \mathcal{B}:\varphi (0)\in \overline{D(A)}%
\Big\}.
\]
For $t\geq 0$, we define the continuous operator $U(t)$ on $\mathcal{H}$ by $%
U(t)\varphi =x_{t}(.,\varphi )$, where $x(.,\varphi )$ is the mild solution
of (\ref{a}). Then we can see that $(U(t))_{t\geq 0}$ is a strongly
continuous semigroups on $\mathcal{H}$. We are interested in studying the
behavior of solutions of Equation (\ref{a}) near an equilibrium . We mean by
an equilibrium a constant mild solution $x^{*}$ of (\ref{a}). Without loss
of generality we suppose that $x^{*}=0$ and $F(0)=0$.

\begin{itemize}
\item[(H3)]  $F$ is differentiable at zero.
\end{itemize}
Then the linearized equation at zero is given by 
\begin{equation}
\begin{gathered} \frac{d}{dt}y(t)=Ay(t)+\mathcal{L}(y_{t}),\quad \text{for
}t\geq 0, \\ y_0=\varphi \in \mathcal{B}, \end{gathered}  \label{28}
\end{equation}
where $\mathcal{L}=F^{\prime}(0)$. Let $( T(t))_{t\geq 0}$ be the solution
semigroup associated to Equation (\ref{28}) that is defined by $T(t)\varphi
=y_{t}(.,\varphi )$, $\varphi \in \mathcal{H}$, where $y(.,\varphi )$ is the
mild solution of Equation (\ref{28}).

\begin{theorem}
\label{T4} Assume that conditions (H1)--(H3) hold. Then for $t\geq 0$, the
derivative at zero of $U(t)$ is $T(t)$.
\end{theorem}

\begin{proof}
Let $x(.,\varphi )$  and $y(.,\varphi )$  be respectively
the mild solutions of  (\ref{a}) and  (\ref{28}).
By assumption (A1) part (iii), we have for all $t\geq 0$
\begin{align*}
| U(t)\varphi -T(t)\varphi | _{\mathcal{B}}
&\leq K(t)\sup_{0\leq \sigma \leq t}| x(\sigma ,\varphi )-y(\sigma ,\varphi)| \\
&\leq K(t)\sup_{0\leq \sigma \leq t} \Big| \int_0^{\sigma}T_{-1}(\sigma -s)
( F(U(s)\varphi )-\mathcal{L}(T(s)\varphi ))ds\Big| , \\
&\leq K(t)\overline{M}e^{\omega t}\int_0^{t}e^{-\omega s}|
F(U(s)\varphi )-\mathcal{L}(T(s)\varphi )| _{F_0}ds, \\
&\leq K(t)\overline{M}e^{\omega t}\Big(\int_0^{t}e^{-\omega s}|
F(U(s)\varphi )-\mathcal{L}(U(s)\varphi )| _{F_0}ds \\
&\quad +\int_0^{t}e^{-\omega s}| \mathcal{L}(U(s)\varphi )-\mathcal{L}
(T(s)\varphi )| _{F_0}ds\Big).
\end{align*}
 By virtue of the differentiability of $F$  at $0$
and from (\ref{333}) of Theorem \ref{T1}, we deduce that for
$\varepsilon >0$, there exists $\delta >0$  such that
\[
\int_0^{t}e^{-\omega s}| F(U(s)\varphi )-\mathcal{L}(U(s)\varphi
)| _{F_0}ds\leq \varepsilon | \varphi | _{\mathcal{B}},\quad
\text{for }| \varphi | _{\mathcal{B}}\leq \delta .
\]
On the other hand, one has
\[
\int_0^{t}e^{-\omega s}| \mathcal{L}(U(s)\varphi )-\mathcal{L}(T(s)\varphi
)| _{F_0}ds\leq | \mathcal{L}| _{\mathcal{L}(\mathcal{B},
F_0)}\int_0^{t}e^{-\omega s}| U(s)\varphi -T(s)\varphi | _{\mathcal{B}}ds.
\]
 Consequently,
\[
| U(t)\varphi -T(t)\varphi | _{\mathcal{B}}\leq K(t)\overline{M}%
e^{\omega t}\Big( \varepsilon | \varphi | _{\mathcal{B}}+|
\mathcal{L}| _{\mathcal{L}(\mathcal{B},F_0)}\int_0^{t}e^{-\omega s}|
U(s)\varphi -T(s)\varphi | _{\mathcal{B}}ds\Big) .
\]
 By Gronwall's lemma,
\[
| U(t)\varphi -T(t)\varphi | _{\mathcal{B}}\leq K(t)\overline{M}%
\varepsilon | \varphi | _{\mathcal{B}}\,\exp \Big( (| \mathcal{L}%
| _{\mathcal{L}(\mathcal{B},F_0)}K(t)\overline{M}+\omega )t\Big),
\quad t\geq 0.
\]
 Hence, we conclude that $U(t)$  is differentiable at $0$
 and $(D_{\varphi }U(t))(0)=T(t)$.
\end{proof}

\begin{theorem}
\label{thm11} Assume that conditions (H1)--(H3) hold. If the zero
equilibrium of $(T(t))_{t\geq 0}$ is exponentially stable, then the zero
equilibrium of $(U(t))_{t\geq 0}$ is locally exponentially stable in the
sense that there exist $\delta >0$ $\mu >0$, $k\geq 1$ such that 
\[
|U(t)\varphi |\leq ke^{-\mu t}|\varphi |,\quad \text{for }|\varphi |\leq
\delta \;t\geq 0. 
\]
{\ Moreover if }$\mathcal{H}$ can be decomposed as $\mathcal{H}=\mathcal{H}%
_{1}\oplus \mathcal{H}_{2}$ where $\mathcal{H}_{i}$ are $T$-invariant
subspaces of $\mathcal{H}$ and $\mathcal{H}_{1}$ is finite-dimensional and
with $\omega _{0}=\lim_{h\to \infty }\frac{1}{h}\log |T(h)/\mathcal{H}_{2}|$
we have 
\[
\inf \left\{ |\lambda |:\lambda \in \sigma (T(t)/\mathcal{H}_{1})\right\}
>e^{\omega _{0}t}. 
\]
Then zero is not stable in the sense that there exist $\varepsilon >0$ and
sequence $(\varphi _{n})_{n}$ converging to $0$ and $(t_{n})_{n}$ of
positive reals such that $|U(t_{n})\varphi _{n}|>\varepsilon $.
\end{theorem}

The proof of this theorem is based on Theorem \ref{T4} and on the following
result.

\begin{theorem}[\cite{Des}]
\label{thm12} Let $(V(t))_{t\geq 0}$ be a nonlinear strongly continuous
semigroup on a subset $\Omega $ of a Banach space $Z$ and assume that $%
x_{0}\in \Omega $ is an equilibrium of $(V(t))_{t\geq 0}$ such that $V(t)$
is differentiable at $x_{0}$ for each $t\geq 0$, with $W(t)$ the derivative
at $x_{0}$ of $V(t),\,t\geq 0$. Then $(W(t))_{t\geq 0}$ is a strongly
continuous semigroup of bounded linear operators on $Z$. If the zero
equilibrium of $(W(t))_{t\geq 0}$ is exponentially stable, then $x_{0}$ is
locally exponentially stable of $(V(t))_{t\geq 0}$. Moreover if $Z$ can be
decomposed as $Z=Z_{1}\oplus Z_{2}$ where $Z_{i}$ are $W$-invariant
subspaces of $Z$ and $Z_{1}$ is finite-dimensional and with $\omega
_{0}=\lim_{h\to \infty }\frac{1}{h}\log |W(h)/Z_{2}|$ we have 
\[
\inf \big\{ |\lambda |:\lambda \in \sigma (W(t)/Z_{1})\big\} >e^{\omega
_{0}t}. 
\]
Then the equilibrium $x_{0}$ is not stable in the sense that there exist 
$\varepsilon >0$ and sequence $(x_{n})_{n}$ converging to $x_{0}$ and 
$(t_{n})_{n}$ of positive reals such that $|V(t_{n})x_{n}-x_{0}|>\varepsilon $.
\end{theorem}

\section{Applications}

To apply our abstract result, we consider the partial functional
differential equations with infinite delay, 
\begin{equation}
\begin{gathered} \frac{\partial }{\partial t}v(t,\xi ) =-\frac{\partial
}{\partial \xi }v(t,\xi )+m(\xi )\int_{-\infty }^{0} K(\theta ,v(t+\theta
,\xi ))d\theta ,\quad \text{for }\xi \in [0,1] ,\;t\geq 0, \\ v(t,0)=0,\quad
t\geq 0, \\ v(\theta ,\xi )=v_0(\theta ,\xi ),\quad \text{for }\xi \in [0,1]
\; \theta \in ( -\infty ,0] , \end{gathered}  \label{F10}
\end{equation}
where $K$ is a continuous function from $( -\infty,0] \times \mathbb{R}$
into $\mathbb{R}$ and $v_0:( -\infty ,0] \times [0,1] \to \mathbb{R}$ is an
appropriate function. Here and hereafter we suppose that $m $ is not
necessarily continuous on $[0,1]$ and $m\in L^{\infty }(0,1)$. Let $A$ be
the operator defined on $X=C( [0,1] ;\mathbb{R}) $ by 
\[
D(A)=\{ g\in C^{1}( [0,1] ;\mathbb{R}):g(0)=0\}, \quad Ag=-g^{\prime}. 
\]
Then $\overline{D(A)}=C_0( [0,1] ;\mathbb{R}) =\{ g\in C( [0,1] ;\mathbb{R})
:g(0)=0\} $.

\begin{lemma}[\cite{nage1}]
The operator $A$ is a Hille-Yosida operator on $X$. The part $A_{0}$ of $A$
in $C_{0}([0,1];\mathbb{R})$ generates a strongly continuous semigroup 
$(T_{0}(t))_{t\geq 0}$ on $C_{0}([0,1];\mathbb{R})$ which is given for 
$g\in C_{0}([0,1];\mathbb{R})$ by 
\[
(T_{0}(t)g)(\xi )=
\begin{cases}
g(\xi -t),&\text{if }t\leq \xi  \\
0,&\text{if }t>\xi .
\end{cases}
\]
\end{lemma}

Let $\mathop{\rm Lip}_0[0,1] $ be the space of Lipschitz continuous function
on $[0,1] $ vanishing at zero.

\begin{lemma}[\cite{nage1}]
The following properties hold:

\begin{itemize}
\item[(i)]  The Favard class $F_{0}$ of the extrapolated semigroup 
$(T_{-1}(t))_{t\geq 0}$ is given by $F_{0}=L^{\infty }(0,1)$.

\item[(ii)]  The Favard class $F_{1}$ of semigroup $(T_{0}(t))_{t\geq 0}$ is
given by $F_{1}=\mathop{\rm Lip}_{0}[0,1]$, where the norm is given by 
\[
|g|_{\mathrm{Lip}}=\sup_{0\leq x_{1}<x_{2}\leq 1}
\frac{|g(x_{1})-g(x_{2})|}{x_{1}-x_{2}}.
\]

\item[(iii)]  The extrapolated operator $A_{-1}$ coincides on $F_{1}$ with
the a.e. derivative.
\end{itemize}
\end{lemma}

Set $\gamma >0$. For the phase space, we choose $\mathcal{B}$ to be defined
by 
\[
\mathcal{B}=C_{\gamma }=\big\{ \phi \in C(( -\infty ,0] ;X): \lim_{\theta
\to -\infty } e^{\gamma \theta }\phi (\theta )\text{ exists in }X\big\}
\]
with the norm $| \phi | _{\gamma }=\sup_{\theta\leq 0} e^{\gamma \theta }|
\phi (\theta )|$, $\phi \in C_{\gamma }$.

\begin{lemma}[\cite{Hin1}]
The space $C_{\gamma }$ satisfies Assumptions (A1), (A2), (B1), (C1), and
(D1).
\end{lemma}

We assume the following:

\begin{itemize}
\item[(a)]  $K$ is measurable in $(\theta ,z)$, $K(.,0)$ is integrable on $%
(-\infty ,0]$ and there exists a positive function $G$ such that 
\[
|K(\theta ,z_{1})-K(\theta ,z_{2})|\leq G(\theta )|z_{1}-z_{2}|,\quad \text{%
for }\theta \in (-\infty ,0]\;z_{1},z_{2}\in \mathbb{R}.
\]

\item[(b)]  $G(.)e^{-\gamma .}$ is integrable on $(-\infty ,0]$,

\item[(c)]  $v_{0}\in C((-\infty ,0]\times [0,1];\mathbb{R}$, $v_{0}(0,0)=0$
and $\lim_{\theta \to -\infty }e^{\gamma \theta }v_{0}(\theta ,.)$ exists in 
$X$.
\end{itemize}
By making the following change of variables: 
\begin{gather*}
x(t)(\xi )=v(t,\xi ),t\geq 0,\quad \xi \in [0,1] , \\
\varphi (\theta )(\xi )=v_0(\theta ,\xi ),\theta \leq 0,\quad \xi \in [0,1] ,
\\
F(\phi )(\xi )=m(\xi )\int_{-\infty }^{0} K(\theta ,\phi (\theta
)(\xi))d\theta , \quad \xi \in [0,1] ,\; \phi \in C_{\gamma },
\end{gather*}
Equation (\ref{F10}) takes the abstract form 
\begin{equation}
\begin{gathered} \frac{dx}{dt}(t)=Ax(t)+F(x_{t}),\quad t\geq 0, \\
x_0=\varphi \in C_{\gamma }. \end{gathered}  \label{exe2}
\end{equation}

\begin{proposition}
\label{prop16} Assume that (a)--(c) above hold. Then (\ref{exe2}) has a
unique mild solution on $(-\infty ,\infty )$.
\end{proposition}

\begin{proof} Since $m\in L^{\infty }(0,1)$, it follows that $F$
 doesn't take its values in $X$. However $F$
takes its values in $L^{\infty }(0,1)$.  In fact if 
$\phi _1\in C_{\gamma }$,  then 
\begin{align*}
| F(\phi _1)| _{L^{\infty }(0,1)}
&\leq \mathop{\rm ess\,sup}_{x\in [0,1] } | m(x)| \sup_{0\leq \xi \leq 1}
\int_{-\infty }^{0}G(\theta )| \phi _1(\theta )(\xi )| d\theta\\
&\quad +\mathop{\rm ess\,sup}_{x\in [0,1] } | m(x)|
\int_{-\infty }^{0}| K(\theta ,0)| d\theta .
\end{align*}
Moreover, 
\[
\sup_{0\leq \xi \leq 1} \int_{-\infty }^{0}G(\theta )|\phi _1(\theta )(\xi )| d\theta 
\leq \int_{-\infty }^{0}e^{-\gamma \theta }G(\theta )d\theta 
\sup_{-\infty <\theta \leq0,\; 0\leq \xi \leq 1}
e^{\gamma \theta }| \phi _1(\theta)(\xi )| .
\]
It follows that for every $\phi _1\in C_{\gamma }$,
$F(\phi _1)\in L^{\infty }(0,1)$. Moreover for 
$\phi _1,\phi_2\in C_{\gamma }$, we have 
\begin{align*}
&| F(\phi _1)-F(\phi _2)| _{L^{\infty }(0,1)} \\
&= \mathop{\rm ess\,sup}_{x\in [0,1]} | m(x)| 
\sup_{0\leq \xi \leq 1} \int_{-\infty }^{0}G(\theta )
| \phi _1(\theta )(\xi )-\phi _2(\theta )(\xi )| d\theta \\
&\leq | m| _{L^{\infty }(0,1)} \sup_{0\leq \xi \leq 1}
\int_{-\infty }^{0}e^{-\gamma \theta }G(\theta )( e^{\gamma \theta
}| \phi _1(\theta )(\xi )-\phi _2(\theta )(\xi )| )
d\theta , \\
&\leq | m| _{L^{\infty }(0,1)}\Big( \int_{-\infty
}^{0}e^{-\gamma \theta }G(\theta )d\theta \Big) 
\sup_{-\infty <\theta \leq 0,\; 0\leq \xi \leq 1} e^{\gamma \theta }|
\phi _1(\theta )(\xi )-\phi _2(\theta )(\xi )| .
\end{align*}
Then $F$  is Lipschitzian from $C_{\gamma }$  to $L^{\infty }(0,1)$. 
Assumption $({\bf c})$ implies that $\varphi \in C_{\gamma }$ and 
$\varphi (0)\in \overline{D(A)}$.
As a consequence,  (\ref{exe2}) has a unique
mild solution $v$ on $(- \infty , \infty)$.
\end{proof}

For the regularity, we assume that $F$ is continuously differentiable from $%
C_{\gamma }$ to $L^{\infty }(0,1)$ and $F^{\prime }$ is locally Lipschitz.
Let $v_{0}\in C((-\infty ,0]\times [0,1];\mathbb{R})$ be such that:

\begin{itemize}
\item[(d)]  $\frac{\partial v_{0}}{\partial \theta }$ exists and continuous
on $(-\infty ,0]\times [0,1]$, $\lim_{\theta \to -\infty }\big( e^{\gamma
\theta }\big( 
\frac{\partial }{\partial \theta }v_{0}(\theta ,.)\big)\big)$ exists in $X$
and $v_{0}(0,.)\in \mathop{\rm Lip}_{0}[0,1]$,

\item[(e)]  $\frac{\partial }{\partial \theta }v_{0}(0,\xi )=-\frac{\partial 
}{\partial \xi }v_{0}(0,\xi )+m(\xi )\int_{-\infty }^{0}K(\theta
,v_{0}(\theta ,\xi ))d\theta $, for a.e. $\xi \in [0,1]$,

\item[(f)]  $\frac{\partial }{\partial \theta }v_{0}(0,0)=0$.
\end{itemize}

\begin{proposition}
\label{prop17} Assume that (a)--(f) above hold. Let $v$ be the mild solution
of (\ref{exe2}). Then $v$ belongs to $C^{1}(\mathbb{R}^{+},C([0,1],\mathbb{R}%
))\cap C(\mathbb{R}^{+},\mathop{\rm Lip}_{0}[0,1])$ and satisfies 
\begin{equation}
\begin{gathered} \frac{\partial }{\partial t}v(t,\xi ) =-\frac{\partial }{\partial \xi }v(t,\xi )+m(\xi ) \int_{-\infty }^{0}K(\theta ,v(t+\theta ,\xi))d\theta ,\\ \text{for a.e. }\xi \in [0,1] ,\; t\geq 0 \\ v(t,0)=0,\quad t\geq 0 \\ v(\theta ,\xi )=v_0(\theta ,\xi ),\quad \mbox{for
} (\theta ,\xi )\in ( -\infty ,0] \times [0,1] . \end{gathered}  \label{qq}
\end{equation}
\end{proposition}

\begin{proof}
Conditions (d), (e), and (f) imply that $\varphi$ is continuously 
differentiable and
\[
\varphi '\in C_{\gamma },\quad \varphi (0)\in F_1,\quad 
\varphi '(0)\in \overline{D(A)}, \quad \varphi '(0)=A_{-1}\varphi (0)+F(\varphi ).
\]
Since all assumptions of Theorem \ref{T2} are satisfied, we deduce
that the mild solution $v$  belongs to 
$C^{1}(\mathbb{R}^{+},C_0([0,1] ,\mathbb{R}))\cap C(\mathbb{R}^{+},
\mathop{\rm Lip}_0[0,1] )$  and satisfies (\ref{qq}).
\end{proof}

\subsection*{Acknowledgments}

The author would like to thank the anonymous referee for his/her valuable
suggestions and comments which helped in improving the original manuscript.

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