\documentclass[reqno]{amsart} 

\AtBeginDocument{{\noindent\small 
{\em Electronic Journal of Differential Equations},
Vol. 2003(2003), No. 62, pp. 1--15.\newline
ISSN: 1072-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.swt.edu  (login: ftp)}
\thanks{\copyright 2003 Southwest Texas State University.} 
\vspace{9mm}}

\begin{document} 

\title[\hfilneg EJDE--2003/62\hfil Non-autonomous retarded differential equations]
{Non-autonomous retarded differential equations: the variation of constants
formulas and the asymptotic behaviour} 

\author[S. Boulite, L. Maniar, \& M. Moussi\hfil EJDE--2003/62\hfilneg]
{Said Boulite, Lahcen Maniar, \& Mohammed Moussi}  % in alphabetical order

\address{Said Boulite \hfill\break
Department of Mathematics\\
University Cadi Ayyad\\ 
Faculty of Sciences Semlalia\\ 
B.P. 2390, 40000 Marrakesh, Morocco}
\email{sboulite@ucam.ac.ma}

\address{Lahcen Maniar \hfill\break
Department of Mathematics\\
University Cadi Ayyad\\ 
Faculty of Sciences Semlalia\\ 
B.P. 2390, 40000 Marrakesh, Morocco}
\email{maniar@ucam.ac.ma}


\address{Mohammed Moussi\hfill\break
Department of Mathematics\\
Faculty of Sciences\\
University Mohammed I\\
60000 Oujda, Morocco}
\email{moussi@sciences.univ-oujda.ac.ma}


\date{}
\thanks{Submitted November 4, 2002. Published May 29, 2003.}
\subjclass[2000]{47D06, 47A55, 34D05, 34G10}
\keywords{Non-autonomous retarded differential equations, \hfill\break\indent
       variation of constants formula, Dyson-Phillips series,
       asymptotic almost periodicity}

\begin{abstract}
 This paper is devoted to show a variation of constants formula for the
 operator solution to the non-autonomous retarded differential equation
 $$
 x'(t)=A(t)x(t)+L(t)x_t+f(t),\quad x_s=\varphi, \quad t\geq s\geq 0,
 $$
 in terms of the inhomogeneous term $f$, which will allow
 us to study the asymptotic behaviour of this solution. We treat
 also the existence of fundamental solutions  and the stability of
 semi-linear retarded differential equations.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{thm}{Theorem}[section]
\newtheorem{cor}[thm]{Corollary}
\newtheorem{lem}[thm]{Lemma}
\newtheorem{prop}[thm]{Proposition}
\newtheorem{defn}[thm]{Definition}


\section{Introduction}

In this paper we study the retarded non-autonomous differential
equation
\begin{equation} \label{IHRD}
 \begin{gathered} x'(t)=A(t)x(t)+L(t)x_t+f(t) ,\quad t\geq s\geq0,\\
 x_s=\varphi\in \mathcal{C}_r:=C([-r,0],E),
\end{gathered}
\end{equation}
where $(A(t),D(A(t)))_{t\geq 0}$ generates the strongly continuous
evolution family \break
 $(V(t,s))_{t\geq s\geq 0}$ on a Banach space $E$, and
$(L(t)) _{t\geq 0}$ is a family of bounded linear operators from
$\mathcal{C}_{r}$ into $E$.

In the autonomous case ($A(t)=A$, $L(t)=L$, $t\geq 0$), 
the retarded differential equation \eqref{IHRD} has been studied by many
authors using various techniques; see for example 
\cite{Ba-Ma-Rh, Desch-Schappacher, Halle, Ma-Rh, Thieme, Tr-We, Wu}. 

The case $A(t)=A$, $t\geq 0$,  has been treated recently in \cite{ Gu-Ra},
\cite{Ma1} and \cite{Rh} using extrapolation theory. In \cite{Rh}, A.
Rhandi showed recently that the solution of the homogeneous retarded
differential equation ($f\equiv0$) is given by a Dyson-Phillips series. Also in
the same case, in \cite{ De-Gu-Gy, Gu-Ra}, the authors proved that the solution
of the inhomogeneous retarded equation \eqref{IHRD} is given in terms of the
inhomogeneous term $f$ by a variation of constants formula, which was used to
study the asymptotic behaviour of the solutions of \eqref{IHRD}. Recently, in
\cite{Gu-Ra-Sh} the authors studied the asymptotic behaviour to \eqref{IHRD}
on $\mathbb R$ using the evolution semigroups and the characteristic equation.
Also, R. Schnaubelt \cite {Sch} showed the first (to our knowledge) variation
of constants formula for the inhomogeneous retarded equation \eqref{IHRD} by
using the ideas of evolution semigroups.

Our aim in this paper is to extend the results of \cite{De-Gu-Gy,Gu-Ra, Ma1,
Rh} to the fully non-autonomous equation \eqref{IHRD}. More precisely, after
the preliminary section, we establish in Section 3 the existence of mild
solutions of the homogeneous retarded differential equations, and that
solutions are provided by some evolution families. Moreover, we show that these
evolution families are given by Dyson-Phillips series and that the term retard
do not affect the asymptotic behaviour, as boundedness and asymptotic almost
periodicity, of the solutions to the non retarded Cauchy problem
\begin{equation}\label{CP}
\begin{gathered} x'(t)=A(t)x(t) ,\quad t\geq s,\\
x(s)=x\in E. \end{gathered}.
\end{equation}
Namely, we give conditions under which the evolution families solutions of the
homogeneous retarded equations  and the Cauchy problem  \eqref{CP} have the
same asymptotic behaviour (see Theorem \ref{3.5}).

In section 4, we show that the  solution $x_{t}$ of \eqref{IHRD} satisfies a
variation of constants formula, different from the one of \cite{Sch}. Using our
variation of constant formula, we show that these solutions inherit the same
asymptotic behaviour of the inhomogeneous term $f$. In the last section, we
show the existence of fundamental solutions for the homogeneous retarded
equations, which has been assumed in \cite{De-Gu-Gy}, and obtain the asymptotic
behaviour of semi-linear retarded differential equations. Before ending this
introduction, we shall mention that we can consider also the above differential
retarded equations on $\mathbb R$ and we can obtain, using the same technics,
the same results and all results of \cite{Gu-Ra}.

\section{Preliminaries}

In this section we recall some definitions and fix notations which will be
used in the sequel.

Let $X$ be a Banach space and denote by $\mathcal{L}(X) $ the
Banach algebra of all bounded linear operators on $X$.

A family of operators $\mathcal{U}:=(U(t,s))_{t\geq s\geq
0}\subset\mathcal{L}(X)$ is called \textit{ a strongly continuous evolution
family} if

\begin{enumerate}
\item  $U(t,s)=U(t,r)U(r,s)$ and $U(s,s)=Id\quad $ for all $\quad t\ge r\ge
s\ge 0$,

\item  the mapping $\{(t,s)\,:t\geq s\geq0\}\ni
(t,s)\mapsto U(t,s)$ is strongly continuous.
\end{enumerate}

An evolution family $(U(t,s))_{t\geq s\geq 0}$ is said to have an
\textit{
exponential dichotomy } if there exists a projection-valued function
$P : \mathbb R_+\to  \mathcal{L}(X)$ such that the function
$P(\cdot)x$ is continuous and bounded for each $x \in X$, and
constants $\delta > 0, N = N(\delta) \geq1$ such that
\begin{itemize}
\item[(i)]  $P(t)U(t,s)=U(t,s)P(s)$;
\item[(ii)]  $U_{Q}(t,s)$, the restriction of $U(t,s)$ on $ImQ(s)$,  is invertible as an operator from $ImQ(s)$ to $%
ImQ(t)$, with $Q(\cdot):=I-P(\cdot)$;
 \item[(iii)]  $\Vert U(t,s)P(s)\Vert \leq Ne^{-\delta
(t-s)}$, and $\Vert U_{Q}(s,t)Q(t)\Vert \leq Ne^{-\delta (t-s)}$
\end{itemize}
for $t\geq s$ and $t,s\in \mathbb{R}_{+}$.  The  family of
operators $(\Gamma(t,s))_{t\geq s\geq0}\subseteq\mathcal{L}(X)$ given by
$$
\Gamma(t,s):=\begin{cases}
    U(t,s)P(s), &  t\geq s,\\
    -U_{Q}(t,s)Q(s), &  t<s, \\
\end{cases}
$$
is called the corresponding \textit{Green's} operator function.

For evolution families and well-posedness of the non-autonomous Cauchy
problems we refer to \cite{engel-nagel,Ni,Pazy,schnaubelt1}.

Let $BC(\mathbb{R}_{+},X)$ be the Banach space of all bounded continuous
functions from $\mathbb{R}_{+}$ to $X$, endowed with the uniform norm.
The closed subspace of bounded uniformly continuous functions will be
denoted by $%
BUC(\mathbb{R}_{+},X)$.

If $f:\mathbb{R}_{+}\to X$, the set of all translates, called the hull of
$f$, is $H(f):=\{f(\cdot +t): t\in\mathbb{R}_{+}\}$.

A function $f\in BC(\mathbb{R}_{+}, X)$ is said to be \textit{%
asymptotically almost periodic} if $H(f)$ is relatively compact in $BC(%
\mathbb{R}_{+}, X)$.

If $H(f)$ is weakly relatively compact in $BC(\mathbb{R}_{+}, X)$, the
bounded continuous function $f:\mathbb{R}_{+}\, \to X$ is called
\textit{Eberlein weakly asymptotically almost periodic}.

We recall also that a closed subspace ${\mathcal{E}}$ of
$BUC(\mathbb{R}_{+},X)$ is said to be \textit{translation
bi-invariant} if for all $t\geq 0$
\[
f\in {\mathcal{E}}\Longleftrightarrow f(\cdot+t)\in{\mathcal{E}},
\]
and \textit{operator invariant} if $M\circ f\in{\mathcal{E}}$ for every
$f\in{\mathcal{E}}$ and $M\in \mathcal{L}(X) $, where $M\circ f$ is defined
by $(M\circ f)(t)=M(f(t))$, $t\geq 0$. A closed subspace ${\mathcal{E}}$ of
$BUC(\mathbb{R}_{+},X)$ is said to be \textit{homogeneous } if it is
translation bi-invariant and operator invariant.

From \cite{batty-chill} the following classes of $X$-valued functions are
homogeneous closed  subspaces of $BUC(\mathbb{R}_{+},X)$:
\begin{itemize}
\item  The space $C_{0}(\mathbb{R}_{+},X)$ of all continuous functions
vanishing at infinity;

\item  The space $AAP(\mathbb{R}_{+},X)$ of asymptotically almost
periodic functions;

\item  The space $W(\mathbb{R}_{+},X)$ of Eberlein weakly asymptotically
almost periodic functions.
\end{itemize}
For more details on almost periodic functions, we refer to \cite
{amerio-prouse, kreulich1}. For the almost periodicity of solutions of
Cauchy problems, see, e.g., \cite{arendt-batty,ballotti-goldstein-parrott,
batty-hutter-raebiger,kreulich2,ruess-summers1}.

For the sequel, we need also the following fundamental lemma, see \cite
{Maniar} for more details and \cite{batty-chill} for an autonomous version.

\begin{lem} \label{man}
Let $(U(t,s))_{t\geq s\geq 0}$ be a bounded evolution family on
$X$. Let $\mathcal{E}$ be a homogeneous closed subspace of $BUC(\mathbb
R_{+},X)$. Assume that $\mathbb R_{+}\ni t\mapsto U(t+s,s)x$ belongs to
$\mathcal{E}$ for every
$x\in X$ and $s\geq 0$. If $h\in L^{1}(\mathbb R_{+},X)$, then $\mathbb %
R_{+}\ni t\mapsto\int_{0}^{t}U(t+s,s+\sigma )h(\sigma )d\sigma $ belongs
to $\mathcal{E}$ for all $s\geq 0$.
\end{lem}

\section{The homogeneous case: Dyson-Phillips series and asymptotic
behaviour}

Let $(A(t),D(A(t))_{t \geq0}$ be a stable family and generate an evolution
family \break
$(V(t,s))_{t\geq s\geq 0}$, on a Banach space $E$, such that $%
\|V(t,s)\|\leq Me^{\omega(t-s)}$, for some constants $\omega\in\mathbb R$ and
$M\geq1$. Consider also the family $\left(L(t)\right)_{t\geq 0}$  of bounded
linear operators from $\mathcal{C}_r$ into $E$, with $L(\cdot)\in BC(\mathbb
R_+, \mathcal{L}_s (\mathcal{C}_r,E ))$, i.e., $t\mapsto L(t)$ is a bounded and
strongly continuous function.

The well-posedness of the homogeneous non-autonomous retarded differential
equation
\begin{equation} \label{HRDE}
\begin{gathered} x'(t)=A(t)x(t)+L(t)x_t ,\quad t\geq s,\\
x_s=\varphi\in\mathcal{C}_r,
\end{gathered}
\end{equation}
was treated recently, e.g., in \cite{De-Gu-Gy,Gu-Ra-Sh, Ng, Sch}. In these
papers, the authors showed the existence of a unique mild solution to
\eqref{HRDE}, i.e., a continuous function $x:[s-r, \infty)\to E$
satisfying
\begin{equation}  \label{mild}
x(t)=\begin{cases}
V(t,s)\varphi(0)+\int_s^{t}V(t,\sigma)L(\sigma)x_{\sigma}d\sigma,
& t\geq s, \\
\varphi(t-s), & s-r\leq t\leq s,\end{cases}
\end{equation}
and the operator solutions $(x_t)$ are evolution families on $\mathcal{C}_r$.
In this section, we show, by a different way, the existence of mild solutions
to \eqref{HRDE}. Our aim exactly here is to show that the evolution family
solution of the equation \eqref{HRDE} is given by a Dyson-Phillips series,
and satisfies a variation of constants formula. These formulas are used to show
that the mild solutions of the retarded equations have the same asymptotic
behaviour of the trajectories $\mathbb{R}_{+}\ni t\mapsto V(t+s,s)x$,
$s\geq0$, $x\in E$.

It is known that the evolution family solution to
the non-retarded equation ($L(t)\equiv0$) is given by
\[
U(t,s)\varphi(\tau)=\begin{cases}
V(t+\tau,s)\varphi(0), & t+\tau\geq s, \\
\varphi(t+ \tau-s), & s-r\leq t+\tau\leq s\,;
\end{cases}
\]
see for example \cite{Gu-Ra, Sch}. To obtain our aim, we need the following
fundamental result.

\begin{lem} \label{conv}
Let $g\in C(\mathbb R_{+},E)$. Then
\[
\lim_{\lambda \to  +\infty }\int_{s}^{t}U(t,\sigma
)\lambda e^{\lambda \cdot }R(\lambda ,A(0))g(\sigma )d\sigma
\]
exists in $\mathcal{C}_{r}$ uniformly in compact sets of
$\{(t,s):t\geq s\geq 0\}$.
\end{lem}

\begin{proof} Set, for $\lambda\geq\lambda_0>\max(\omega,0)$ and
$0\leq s\leq t\leq
T$ (for some $T>0$),
\[
\mathcal{W}_{\lambda}(t,s):=\int_s^tU(t,\sigma)\lambda
e^{\lambda\cdot}R(\lambda,A(0))g(\sigma)d\sigma.
\]
For $\tau \in [-r,0]$ and $t+\tau\geq s$, we have
\begin{align*}
&\mathcal{W}_{\lambda}(t,s)(\tau)\\
&=\int_s^{t+\tau}U(t,\sigma)%
\lambda e^{\lambda\cdot}R(\lambda,A(0))g(\sigma)(\tau)d\sigma
+\int_{t+\tau}^tU(t,\sigma)\lambda e^{\lambda\cdot}R(\lambda,A(0))
g(\sigma)(\tau)d\sigma \\
&=\int_s^{t+\tau}V(t+\theta,\sigma)\lambda
R(\lambda,A(0))g(\sigma)d\sigma +\int_{t+\tau}^t\lambda
e^{\lambda(t+\tau-\sigma)}R(\lambda,A(0))g(\sigma)d\sigma.
\end{align*}
Let $\lambda,\mu\geq\lambda_0$. We have then,
\begin{align*}
&\mathcal{W}_{\lambda}(t,s)(\tau)-\mathcal{W}_{\mu}(t,s)(\tau)\\
&=\int_s^{t+\tau}V(t+\theta,\sigma)\left[\lambda R(\lambda,A(0))-\mu
R(\mu,A(0))\right]g(\sigma)d\sigma \\
&\quad +\int_{t+\tau}^t\big[\lambda
e^{\lambda(t+\tau-\sigma)}R(\lambda,A(0))-\mu
e^{\mu(t+\tau-\sigma)}R(\mu,A(0))\big]g(\sigma)d\sigma,
\end{align*}
and
\begin{align*}
&\mathcal{W}_{\lambda}(t,s)(\tau)-\mathcal{W}_{\mu}(t,s)(\tau)\\
&=\int_{s}^t\big[\lambda e^{\lambda(t+\tau-\sigma)}R(\lambda,A(0))-\mu
e^{\mu(t+\tau-\sigma)}R(\mu,A(0))\big]g(\sigma)d\sigma
\end{align*}
for $t+\tau\leq s$.
Thus,
\begin{align*}
&\|\mathcal{W}_{\lambda}(t,s)(\tau)-\mathcal{W}_{\mu}(t,s)(\tau)\|\\
&\leq\tilde{M}(T)\int_0^T\|\left[\lambda R(\lambda,A(0))-\mu
R(\mu,A(0))\right]g(\sigma)\|d\sigma +\tilde{M} (\frac1{\lambda}+
\frac1{\mu})\sup_{0\leq \sigma\leq T}\|g(\sigma)\|.
\end{align*}
Hence, as
\[
\lim_{\lambda,\mu\to +\infty}\|\left[\lambda
R(\lambda,A(0))-\mu R(\mu,A(0))\right]g(\sigma)\|=0\quad\mbox{for all
} \sigma\in[0,T],
\]
by Lebesgue dominated convergence theorem, we have
\[
\lim_{\lambda,\mu\to +\infty}\int_0^T\|\left[\lambda
R(\lambda,A(0))-\mu R(\mu,A(0))\right]g(\sigma)\|d\sigma=0.
\]
Consequently,
\[
\sup_{\tau\in[-r,0]}\|\mathcal{W}_{\lambda}(t,s)(\tau)-\mathcal{W}%
_{\mu}(t,s)(\tau)\| \to 0 \quad\text{as }
\lambda,\mu\to +\infty
\]
uniformly for $0\leq s\leq t\leq T$. This completes the proof.
\end{proof}

 From this lemma, we can define the operators $U_n(t,s)$ as
\begin{gather*}
U_0(t,s)\varphi=U(t,s)\varphi,   \\
U_n(t,s)\varphi=\lim_{\lambda
\to  +\infty } {\int_{s}^t U(t,\sigma)e^{\lambda\cdot}\lambda
R(\lambda,A(0))L(\sigma)U_{n-1}(\sigma,s)\varphi\,d\sigma}
\end{gather*}
for all $\varphi\in \mathcal{C}_r, \,n\geq1,$ and $0\leq s\leq t$.

The first main result of this section can now be stated.

\begin{thm} \begin{itemize}
\item[(i)]  The expansion
$U_{L}(t,s):=\sum_{n\geq 0}U_{n}(t,s)$, $t\ge s \ge 0$,
converges in $\mathcal{L}(\mathcal{C}_{r})$ uniformly on
$\{(t,s):0\leq s\leq t\leq T\} $ (for all $T>0$), and
$(U_{L}(t,s))_{t\leq s\leq 0}$ is an evolution family on
$\mathcal{C}_{r}$. Further, this variation of constants
formula
\begin{equation}
U_{L}(t,s)\varphi =U(t,s)\varphi +\lim_{\lambda \to
\infty }\int_{s}^{t}U(t,\sigma )e^{\lambda \cdot }\lambda R(\lambda
,A(0))L(\sigma )U_{L}(\sigma ,s)\varphi \,d\sigma  \label{vari}
\end{equation}
holds for all $\varphi \in \mathcal{C}_{r}$ and $t\ge s\ge0$.

\item[(ii)]  For every $\varphi \in \mathcal{C}_{r}$ and $s\geq 0$ the
function defined by
\begin{equation}
x(t,s,\varphi ):=\begin{cases}
U_{L}(t,s)\varphi (0), & t\geq s, \\ \varphi (t-s), &s-r\leq t\leq s,
\end{cases} \label{x}
\end{equation}
is the unique mild solution of $\eqref{HRDE}$, and 
$$
x_t=U_{L}(t ,s)\varphi, \quad t\geq s\geq0.$$
\end{itemize}
\end{thm}

\begin{proof}
For $n=0$, we have
\[
\|U_0(t,s)\|\leq Me^{\omega(t-s)},\quad\quad t\geq s\geq0.
\]
For $n=1$, $t\geq s\geq0$ and $\varphi\in\mathcal{C}_r$
\[
U_1(t,s)\varphi=\lim_{\lambda \to  +\infty } {\int_{s}^t
U(t,\sigma)e^{\lambda\cdot}\lambda
R(\lambda,A(0))L(\sigma)U_0(\sigma,s)\varphi\,d\sigma}.
\]
For all $\tau\in[-r,0]$, we have
\begin{align*}
\|U_1(t,s)\varphi\|&=\lim_{\lambda \to  +\infty } \|{%
\int_{s}^t U(t,\sigma)e^{\lambda\cdot}\lambda
R(\lambda,A(0))L(\sigma)U_0(\sigma,s)\varphi \,d\sigma}\| \\
&\leq M^2{\int_{s}^t
e^{\omega(t-\sigma)}\|L(\sigma)\|Me^{\omega(\sigma-s)}\|\varphi\| \,d\sigma}
\\
&\leq M^2\|L(\cdot)\|_{\infty}Me^{\omega(t-s)}(t-s)\|\varphi\|.
\end{align*}
Hence
\[
\|U_1(t,s)\|\leq M^2\|L(\cdot)\|_{\infty}Me^{\omega(t-s)}(t-s)\|\varphi\|.
\]
By induction, one can see
\[
\|U_n(t,s)\|\leq \frac{(M^2\|L(\cdot)\|_{\infty})^n}{n!}
Me^{\omega(t-s)},\quad t\geq s\geq0.
\]
Therefore, the expansion $\sum_{n\geq0}U_n(t,s)$ converges in
$\mathcal{L}(\mathcal{C}_r)$ uniformly for $0\leq s\leq t\leq T$. The strong
continuity of $(U_L(t,s))_{t\geq s\geq0}$ can be obtained from Lemma (\ref{conv}) and the
uniform convergence of the series. The rest of
(i) is easy to see.\newline
For (ii), from the variation of constants formula (\ref{vari}), we have
\begin{equation}
U_L(t,s)\varphi(\tau)=\begin{cases}
V(t+\tau,s)\varphi(0)\\
+\int_s^{t+\tau}V(t+\tau,
\sigma)L(\sigma)U_L(\sigma,s) \varphi\,d\sigma, & t+\tau\geq s, \\
\varphi(t+\tau-s),& s-r\leq t+\tau\leq s. \end{cases}  \label{UL}
\end{equation}
Hence, the translation property
\[
U_L(t,s)\varphi(\tau)= \begin{cases} U_L(t+\tau,s)\varphi(0),
&t+\tau\geq s,\\
\varphi(t+\tau-s),& s-r\leq t+\tau\leq s, \end{cases}
\]
holds, and then the function defined by (\ref{x}) satisfies
$x_t=U_L(t,s)\varphi$. Consequently, it is now clear that $x$ is a 
mild solution of \eqref{HRDE}.
\end{proof}

Now we deal with the question of the robustness of some asymptotic behaviour
of the non-retarded equation \eqref{CP} under the introduction of the term
retard. More precisely if we assume that the trajectories
$t\mapsto V(t+s,s)x$, $s\geq 0$, $x\in E$ belong to some  homogeneous closed
subspace $\mathcal{E}$ of $BUC(\mathbb{R}_{+},E)$, and there exist  constants
$0<q<\frac{1}{M}$ and $s_0\geq 0$ such that
\begin{equation}
\label{(H)}\quad \int_0^{+\infty}\|L(\tau+s)U(\tau+s,s)\varphi \|d\tau\leq
q\|\varphi\|
\end{equation}
for all $\varphi\in \mathcal{C}_r$ and $s\geq s_0$, we can obtain the
following main result.

\begin{thm} \label{3.5}
Assume \eqref{(H)} and that the trajectories $t\mapsto V(t+s,s)x$,
$s\geq0$, $x\in E$ belong to some homogeneous closed  subspace
$\mathcal{E}$ of $BUC( \mathbb{R}_{+},E)$. Then, the solution $t\mapsto
x(t+s,s,\varphi) $ of
\eqref{HRDE} belongs to $\mathcal{E}$ for all $\varphi \in \mathcal{C}_{r}$ and
 $s\geq 0$.
\end{thm}

\begin{proof} Take $t\geq0, s\geq s_0$ and $\varphi\in \mathcal{C}_r$. By the above
results, we have that
\begin{align*}
x(t+s,s,\varphi)&=U_L(t+s,s)\varphi(0)\\
&=V(t+s,s)\varphi(0)+\int_0^{t}V(t+s,
\sigma+s)L(\sigma+s) U_L(\sigma+s,s)\varphi\,d\sigma\,.
\end{align*}
According to Lemma \ref{man}, we have to show only that
$L(\cdot+s)U_L(\cdot+s)\varphi$ is in $L^1(\mathbb{R}_{+},E)$. For this, let
$t\geq0$.
The variation of constants formula (\ref{vari}) leads to
\begin{align*}
&L(t+s)U_L(t+s,s)\varphi\\
&=L(t+s)U(t+s,s)\varphi \\
&\quad+\lim_{\lambda \to  +\infty }
\int_s^{t+s}L(t+s)U(t+s,\sigma)e^{\lambda\cdot}\lambda
R(\lambda,A(0))L(\sigma)U_L(\sigma,s)\varphi\,d\sigma.
\end{align*}
Therefore,
\begin{align*}
&\int_0^tL(\tau+s)U_L(\tau+s,s)\varphi d\tau\\
&=\int_0^tL(\tau+s)U(\tau+s,s)\varphi d\tau \\
&\quad +\lim_{\lambda \to  +\infty }
\int_0^t\int_s^{\tau+s}L(\tau+s)U(\tau+s,\sigma)e^{\lambda\cdot}\lambda
R(\lambda,A(0))L(\sigma)U_L(\sigma,s)\varphi\,d\sigma d\tau \\
&=\int_0^tL(\tau+s)U(\tau+s,s)\varphi d\tau \\
&\quad+\lim_{\lambda \to  +\infty }
\int_s^{t+s}\int_0^{t+s-\sigma}L(\tau+\sigma)U(\tau+\sigma,\sigma)
e^{\lambda\cdot}
\lambda R(\lambda,A(0))L(\sigma)U_L(\sigma,s)\varphi\,d\tau d\sigma,
\end{align*}
and the estimate $\eqref{(H)}$ implies
\begin{align*}
&\int_0^t\|L(\tau+s)U_L(\tau+s,s)\varphi \|d\tau\\
&\leq \int_0^t\|L(\tau+s)U(\tau+s,s)\varphi \|d\tau \\
& \quad +\lim_{\lambda \to  +\infty }\int_s^{t+s}\int_0^{t+s-\sigma}
\|L(\tau+\sigma)U(\tau+\sigma,\sigma)e^{\lambda\cdot}
\lambda R(\lambda,A(0))L(\sigma)U_L(\sigma,s)\varphi\|\,d\tau d\sigma \\
&\leq q\|\varphi\| +q\lim_{\lambda \to  +\infty }
\int_s^{t+s}\|e^{\lambda\cdot}\lambda
R(\lambda,A(0))L(\sigma)U_L(\sigma,s)\varphi\|\,d\sigma \\
&\leq q\|\varphi\| +qM\int_0^{t}\|L(\sigma+s)U_L(\sigma+s,s)
\varphi\|\,d\sigma.
\end{align*}
This proves our claim.
For $0\leq s\leq s_0$ and $t\geq 0$, one can write
\[
U_{L}(t+s_0+s,s)\varphi(0)=U_{L}(t+s+s_0,s+s_0)U_{L}(s+s_0,s)\varphi(0).
\]
As $s+s_0\geq s_0$, then as shown above $t\mapsto
U_{L}(t+s_0+s,s)\varphi(0)$ belongs to $\mathcal{E}$ and by the translation
bi-invariance of $\mathcal{E}$, $t\mapsto U_{L}(t+s,s)x$ belongs to ${%
\mathcal{E}}$. This completes the proof.
\end{proof}

\section{The inhomogeneous case: the variation of constants formula and the
asymptotic behaviour}

Consider again the inhomogeneous retarded differential equation
\begin{equation}\label{IHRDE}
 \begin{gathered} x'(t)=A(t)x(t)+L(t)x_t+f(t),\quad
 t\geq s\geq0,\\
 x_s=\varphi\in \mathcal{C}_r, \end{gathered}
\end{equation}
where $(A(t),D(A(t)))_{ t \geq0}$ is a stable family and generates a strongly
continuous evolution family $(V(t,s))_{t\geq s\geq 0}$, on a Banach space $E$,
 such that $\|V(t,s)\|\leq Me^{\omega(t-s)}$,
with $\omega\in\mathbb R$ and $M\geq1$,
$L(\cdot)\in BC(\mathbb R_+, \mathcal{L}_s(\mathcal{C}_r,E))$, and
$f\in L_{\rm loc}^1(\mathbb R_+,E)$.

The following definition is standard in the literature \cite{De-Gu-Gy,
Gu-Ra-Sh, Sch}.

\begin{defn} \rm
A continuous function $x:=x(\cdot ,s,\varphi ):[-r,\infty )\to E$
is called a mild solution of $\eqref{IHRDE}$ if
\begin{equation}
x(t)=\begin{cases}
V(t,s)\varphi (0)+\int_{s}^{t}V(t,\sigma )\left[ L(\sigma
)x_{\sigma }+f(\sigma )\right] \,d\sigma , & t\geq s, \\
\varphi (t-s), & s-r\leq t\leq s.
\end{cases} \label{sol}
\end{equation}
\end{defn}

The existence of mild solutions of $\eqref{IHRDE}$ has been treated recently by
many authors, e.g., \cite{Gu-Ra-Sh, Sch}. Our aim, in this section, is to
show that the mild solutions of these equations  are given by variation of
constants formulas in terms of the inhomogeneous term $f$. To this purpose, we
need the fundamental lemma.

\begin{lem}
\label{4.2} For every $f\in L_{\rm loc}^{1}(\mathbb R_{+},E)$, the limit
\[
\lim_{\lambda \to  +\infty }{\int_{s}^{t}U_{L}(t,\sigma
)e^{\lambda \cdot }\lambda R(\lambda ,A(0))f(\sigma )\,d\sigma }
\]
exists in $\mathcal{C}_{r}$ uniformly in compact sets of
$\{(t,s):0\leq s\leq
t\}$.
\end{lem}

\begin{proof}  Let $T>0$, $0\leq s\leq t\leq T$, and $\lambda\geq\max(0,\omega)$.
Assume first that $f\in C([0,T],E)$, and set
\[
\mathcal{Z}_{\lambda}(t,s)=\int_s^tU_L(t,\sigma)\lambda
e^{\lambda\cdot}R(\lambda,A(0))f(\sigma)d\sigma.
\]
For $\tau\in[-r,0]$ such that $t+\tau\geq s$, from the formula (\ref{UL}),
we have
\begin{equation}
\begin{aligned}
&\mathcal{Z}_{\lambda}(t,s)(\tau)\\
&=\int_s^{t+\tau}U_L(t,\sigma)
\lambda e^{\lambda\cdot}R(\lambda,A(0))f(\sigma)(\tau)d\sigma
+\int_{t+\tau}^tU_L(t,\sigma)\lambda e^{\lambda\cdot}R(\lambda,A(0))
f(\sigma)(\tau)d\sigma  \\
&=\int_s^{t+\tau}V(t+\tau,\sigma)\lambda
R(\lambda,A(0))f(\sigma)d\sigma +\int_{t+\tau}^t\lambda
e^{\lambda(t+\tau-\sigma)}R(\lambda,A(0))f(\sigma)d\sigma   \\
&\quad+\int_s^{t+\tau}\int_{\sigma}^{t+\tau}V(t+\tau,\delta)
L(\delta)U_L(\delta,\sigma)\lambda
e^{\lambda\cdot}R(\lambda,A(0))f(\sigma)\,d\delta\,d\sigma.
\end{aligned}\label{1}
\end{equation}
The last term in the right-hand side  of this equality leads to
\begin{align*}
&\int_s^{t+\tau}\int_{\sigma}^{t+\tau}V(t+\tau,
\delta)L(\delta)U_L(\delta,\sigma)\lambda
e^{\lambda\cdot}R(\lambda,A(0))f(\sigma)\,d\delta\,d\sigma \\
&=\int_s^{t+\tau}\int_s^{\delta}V(t+
\tau,\delta)L(\delta)U_L(\delta,\sigma)\lambda e^{\lambda\cdot}
R(\lambda,A(0))f(\sigma)d\sigma\,d\delta \\
&=\int_s^{t+\tau}V(t+\tau,\delta)L(\delta) \mathcal{Z}
_{\lambda}(\delta,s)d\delta.
\end{align*}
Then, if $t+\tau\geq s$ we obtain
\begin{align*}
&\mathcal{Z}_{\lambda}(t,s)(\tau)\\
&=\int_s^{t+\tau}V(t+\tau,
\sigma)\lambda R(\lambda,A(0))f(\sigma)d\sigma+%
\int_{t+\tau}^t\lambda
e^{\lambda(t+\tau-\sigma)}R(\lambda,A(0))f(\sigma)d\sigma \\
&\quad+\int_s^{t+\tau}V(t+\tau,\sigma)L(\sigma) \mathcal{Z}
_{\lambda}(\sigma,s)d\sigma.
\end{align*}
If $t+\tau\leq s$, we have
\[
\mathcal{Z}_{\lambda}(t,s)(\tau)=\int_s^t\lambda
e^{\lambda(t+\tau-\sigma)}R(\lambda,A(0))f(\sigma)d\sigma.
\]
Now, let $\lambda,\mu\geq\lambda_0>\max(\omega,0)$,
\begin{align*}
&\mathcal{Z}_{\lambda}(t,s)(\tau)-\mathcal{Z}_{\mu}(t,s)(\tau)\\
&=\begin{cases}
\int_s^{t+\tau}V(t+\tau,\sigma)\big[\lambda
R(\lambda,A(0))-\mu R(\mu,A(0))\big]f(\sigma)d\sigma   \\
+\int_{t+\tau}^t\big[\lambda e^{\lambda(t+\tau-\sigma)}R(\lambda,A(0))-\mu
e^{\mu(t+\tau-\sigma)}R(\mu,A(0))\big]f(\sigma)d\sigma   \\
+\int_s^{t+\tau}V(t+\tau,\sigma)L(\sigma)\big[ \mathcal{Z}
_{\lambda}(\sigma,s)-\mathcal{Z}_{\mu}(\sigma,s)\big]d\sigma,
&t+\tau\geq s   \\[5pt]
\int_s^t[\lambda e^{\lambda(t+\tau-\sigma)}R(\lambda,A(0))-\mu
e^{\mu(t+\tau-\sigma)}R(\mu,A(0))] f(\sigma)d\sigma, & t+\tau\leq s.
\end{cases}
\end{align*}
For $t+\tau\geq s$, we get easily
\begin{align*}
&\|\mathcal{Z}_{\lambda}(t,s)(\tau)-\mathcal{Z}_{\mu}(t,s)(\tau)\|\\
&\leq \tilde{M}(T)\int_0^T\| (\lambda R(\lambda,A(0))-\mu
R(\mu,A(0)))f(\sigma)\|d\sigma \\
&\quad +M(\frac1{\lambda}+\frac1{\mu})
\sup_{0\leq\sigma\leq T}\|f(\sigma)\|+Me^{\omega T}\|L(\cdot)\|_{\infty}
\int_0^{T}\|\mathcal{Z}_{\lambda}(\sigma,s)-\mathcal{Z}_{\mu}(\sigma,s)\|d\sigma,
\end{align*}
and for $t+\tau\leq s$
\begin{align*}
&\|\mathcal{Z}_{\lambda}(t,s)(\tau)-\mathcal{Z}_{\mu}(t,s)(\tau)\|\\
&\leq \tilde{M}(T)\int_0^T\| (\lambda R(\lambda,A(0))-\mu
R(\mu,A(0)))f(\sigma)\|d\sigma +M(\frac1{\lambda}+\frac1{\mu})
\sup_{0\leq\sigma\leq T}\|f(\sigma)\|.
\end{align*}
Hence, as
\[
\lim_{\lambda,\mu\to +\infty}\|\left[\lambda
R(\lambda,A(0))-\mu R(\mu,A(0))\right]f(\sigma)\|=0\quad\mbox{for all
} \sigma\in[0,T],
\]
by the Lebesgue dominated convergence theorem, for $\varepsilon>0$ and $%
\lambda,\mu$ sufficiently large we have
\[
\|\mathcal{Z}_{\lambda}(t,s)(\tau)-\mathcal{Z}_{\mu}(t,s)(\tau)\|\leq%
\varepsilon+ Me^{\omega T}\int_s^{t}\|L(\cdot)\|_{\infty}\|
\mathcal{Z}_{\lambda}(\sigma,s)-\mathcal{Z}_{\mu}(\sigma,s)\|d\sigma.
\]
An application of Gronwall's inequality yields
\[
\|\mathcal{Z}_{\lambda}(t,s)(\tau)-\mathcal{Z}_{\mu}(t,s)(\tau)\|\leq
\varepsilon e^{Me^{\omega T}(t-s)\|L(\cdot)\|_{\infty}}.
\]
Then, we can conclude that
\[
\sup{\tau\in[-r,0]}\|\mathcal{Z}_{\lambda}(t,s)(\tau)-\mathcal{Z}
_{\mu}(t,s)(\tau)\| \to 0 \quad \mbox{as }\lambda,\mu\to +\infty
\] uniformly on
$\{(t,s):0\leq s\leq t\leq T\}$.

Let $f_n$ in $C([0,T],E)$ be a subsequence converging to $f$ in
$L^1(0,T;E)$. We have
\[
\|\int_0^tU_L(t,\sigma)\lambda
e^{\lambda\cdot}R(\lambda,A(0))[f_n(\sigma)-f(\sigma)]d\sigma\|\leq
M(T,s)\|f_n-f\|_{L^1}.
\]
Hence, this provides the existence of the limit for
$f\in L^1_{\rm loc}(\mathbb{R}_{+},E)$, and this completes the proof.
\end{proof}

\begin{thm} \label{theom}
Let $\varphi\in\mathcal{C}_r$ and $s\geq 0$, then the function
$x$ defined by
\[
x(t):=\begin{cases} u(t)(0), & t\geq s,\\
\varphi(t-s),& s-r\leq t\leq s,
\end{cases}
\]
with
\[
u(t):=U_L(t,s)\varphi+\lim_{\lambda \to  +\infty }\int_s^tU_L(t,%
\sigma)e^{\lambda\cdot}\lambda R(\lambda,A(0))f(\sigma)\,d\sigma,\quad t\geq
s,
\]
is a mild solution of $\eqref{IHRDE}$. Conversely, if $x$ is a mild
solution of $\eqref{IHRDE}$, then
\begin{equation}  \label{variat}
x_t=U_L(t,s)\varphi+\lim_{\lambda \to  +\infty }\int_s^tU_L(t,
\sigma)e^{\lambda\cdot}\lambda R(\lambda,A(0))f(\sigma)\,d\sigma,\quad t\geq
s.
\end{equation}
\end{thm}

\begin{proof}
Let $\tau\in[-r,0]$, then for $t+\tau\geq s$, we have
\begin{align*}
&u(t)(\tau)\\
&=V(t+\tau,s)\varphi(0)+\int_s^{t+\tau}V(t+\tau,
\sigma)L(\sigma)U_L(\sigma,s)\varphi\,d\sigma \\
&\quad +\lim_{\lambda \to  +\infty }\int_s^{t+\tau}V(t+\tau,
\sigma)\lambda R(\lambda,A(0))f(\sigma)\,d\sigma\\
&\quad +\lim_{\lambda \to  +\infty }\int_{t+\tau}^t\lambda
e^{\lambda(t+\tau-\sigma)}R(\lambda,A(0))f(\sigma)\,d\sigma \\
&\quad +\lim_{\lambda \to  +\infty }\int_s^{t+\tau}\int_{
\sigma}^{t+\tau}V(t+\tau,\delta)L(\delta)U_L(\delta,\sigma)\lambda
e^{\lambda\cdot} R(\lambda,A(0))f(\sigma)\,d\delta d\sigma \\
&=V(t+\tau,s)\varphi(0)+\int_s^{t+\tau}V(t+\tau,\sigma)L(
\sigma)U_L(\sigma,s)\varphi\,d\sigma+\int_s^{t+\tau}V(t+\tau,
\sigma)f(\sigma)\,d\sigma \\
&\quad +\lim_{\lambda \to  +\infty }\int_s^{t+\tau}V(t+\tau,
\delta)L(\delta)\int_{s}^{\delta}U_L(\delta,\sigma)\lambda e^{\lambda\cdot}
R(\lambda,A(0))f(\sigma)\,d\sigma d\delta \\
&=V(t+\tau,s)\varphi(0)+\int_s^{t+\tau}V(t+\tau,\sigma)L(
\sigma)U_L(\sigma,s)\varphi\,d\sigma+\int_s^{t+\tau}V(t+\tau,
\sigma)f(\sigma)\,d\sigma \\
&\quad +\int_s^{t+\tau}V(t+\tau,
\delta)L(\delta)\big[\lim_{\lambda \to  +\infty
}\int_{s}^{\delta}U_L(
\delta,\sigma)\lambda e^{\lambda\cdot}
R(\lambda,A(0))f(\sigma)\,d\sigma\Big] d\delta \\
&=V(t+\tau,s)\varphi(0)+\int_s^{t+\tau}V(t+\tau,\sigma)f(\sigma)\,d\sigma \\
&\quad +\lim_{\lambda \to  +\infty }\int_s^{t+\tau}V(t+\tau,
\delta) L(\delta)[U_L(\delta,s)\varphi \\
&\quad +\lim_{\lambda \to  +\infty}\int_{s}^{\delta} U_L(\delta,\sigma)\lambda
e^{\lambda\cdot}R(\lambda,A(0))f(\sigma)\,d\sigma ]d\delta .
\end{align*}
Then,
\begin{equation}
u(t)(\tau)=V(t+\tau,s)\varphi(0)+\int_s^{t+\tau}V(t+\tau,
\delta)L(\delta)\left[u(\delta)+f(\delta)\right]\,d\delta.  \label{2}
\end{equation}
For $t+\tau\leq s$, we obtain
\[
u(t)(\tau)=\varphi(t+\tau-s)+\lim_{\lambda \to  +\infty}
\int_s^te^{\lambda(t+\tau-\sigma)}\lambda R(\lambda,A(0))f(\sigma)\,d\sigma.
\]
Therefore,
\begin{equation}  \label{3}
u(t)(\tau)=\varphi(t+\tau-s).
\end{equation}
This implies that the function $x$ satisfies $x_t=u(t), \quad t\geq 0, $ and
(\ref{sol}). Thus $x$ is a mild solution of $\eqref{IHRDE}$.

Conversely, let $x(t)$ be a mild solution of $\eqref{IHRDE}$, we have to show
that $x_t=u(t)$ for all $t\geq s$. Let $t\geq s$ and $\tau\in[-r,0]$, from
(\ref {sol}), (\ref{2}) and (\ref{3}), we obtain
\[
x_t(\tau)-u(t)(\tau)=\begin{cases}
\int_s^{t+\tau}V(t+\tau,\sigma)L(\sigma)(x_{\sigma}-u(\sigma))d\sigma, &
t+\tau\geq s, \\
0, &s-r\leq  t+\tau\leq s,
\end{cases}
\]
and by the Gronwall's inequality, we conclude that $x_t=u(t)$.
\end{proof}

The variation of constants formula (\ref{variat}) will play a crucial role in
the study of the asymptotic behaviour of the solutions to the retarded
differential equation \eqref{IHRDE}. For this purpose, we begin by the
following lemma.

\begin{lem}
Assume that $(U_L(t,s))_{t\geq s\geq0}$ has an exponential dichotomy. Let
$f\in BC(\mathbb{R}_{+},X)$, then the limit
\begin{equation}
\lim_{\lambda\to +\infty}\int_0^{\infty}\Gamma(t,\sigma)
\lambda e^{\lambda\cdot}R(\lambda,A(0))f(\sigma)d\sigma
\end{equation}
exists uniformly for $t$ in compact intervals of $\mathbb{R}_{+}$.
\end{lem}

\begin{proof} 
Let $t\in[0,T]$, for some $T>0$. We have
\begin{align*}
&\int_{0}^{+\infty}\Gamma(t,\sigma) \lambda
e^{\lambda\cdot}R(\lambda,A(0))f(\sigma)d\sigma\\
&=\int_{0}^{t}\Gamma(t,\sigma)\lambda e^{\lambda\cdot}R(\lambda,A(0))f(\sigma)
d\sigma
+\int_{t}^{+\infty}\Gamma(t,\sigma)\lambda
e^{\lambda\cdot}R(\lambda,A(0))f(\sigma)d\sigma.
\end{align*}
By the definition of the Green's function, the two members of the above sum are well defined. by Lemma \ref{4.2}, 
the limit of the first member  exists. Hence, it remains 
to show that the second one 
\[
\mathcal{E}_{\lambda}(t):=\int_{t}^{+\infty}\Gamma(t,\sigma)%
\lambda e^{\lambda\cdot}R(\lambda,A(0))f(\sigma)d\sigma
\]

is a Cauchy sequence.

First, one can verify for all $t\geq s\geq 0$
\[
\mathcal{E}_{\lambda}(t)=U_L(t,s)\omega_{\lambda}(s)+
\int_s^tU_L(t,\sigma)Q(\sigma)\lambda e^{\lambda\cdot}
R(\lambda,A(0))f(\sigma)d\sigma.
\]
Then, for $r>0$ we have
\begin{align*}
Q(t+r)\mathcal{E}_{\lambda}(t+r)
&=U(t+r,t)Q(t)\mathcal{E}_{\lambda}(t)\\
&\quad +Q(t+r)\int_{t}^{t+r}U_L(t+r,s)Q(\sigma)\lambda e^{\lambda\cdot}
R(\lambda,A(0))f(\sigma)d\sigma,
\end{align*}
and as $\mathcal{E}_{\lambda}(t)\in \mathop{\rm Im}Q(t)$, we obtain
\begin{align*}
\mathcal{E}_{\lambda}(t)
&=[U_Q(t+r,t)]^{-1}Q(t+r)\mathcal{E}_{\lambda}(t+r)\\
&\quad -Q(t+r)\int_{t}^{t+r}U_L(t+r,s)Q(\sigma)\lambda
e^{\lambda\cdot} R(\lambda,A(0))f(\sigma)d\sigma.
\end{align*}
Now, for $\lambda, \mu\geq\lambda_0>\max(\omega, 0)$
\begin{align*}
\mathcal{E}_{\lambda}(t)-\mathcal{E}_{\mu}(t)
&=[U_Q(t+r,t)]^{-1}Q(t+r)[\mathcal{E}_{\lambda}(t+r)-\mathcal{E}_{\mu}(t+r)] \\
&\quad-[U_Q(t+r,t)]^{-1}Q(t+r)\int_{t}^{t+r}U_L(t+r,\sigma)[\lambda
e^{\lambda\cdot}R(\lambda,A(0))\\
&\quad -\mu e^{\mu\cdot}R(\mu,A(0))]f(\sigma)d\sigma.
\end{align*}
Passing to the norm, it follows
\begin{align*}
\|\mathcal{E}_{\lambda}(t)-\mathcal{E}_{\mu}(t)\|
&\leq \|[U_Q(t+r,t)]^{-1}Q(t+r)[\mathcal{E}_{\lambda}(t+r)-\mathcal{E}
_{\mu}(t+r)]\|\\
&\quad +\Big\|[U_Q(t+r,t)]^{-1}Q(t+r)\int_{t}^{t+r}U_L(t+r,\sigma)[
\lambda e^{\lambda\cdot}R(\lambda,A(0))\\
&\quad -\mu e^{\mu\cdot}R(\mu,A(0))]f(\sigma)d\sigma\Big\|& \\
&\leq C_1e^{-\alpha r}\Big[C_2\|f\|+\|Q(t+r)
\int_{t}^{t+r}U_L(t+r,\sigma)[\lambda e^{\lambda\cdot}R(\lambda,A(0))\\
&\quad -\mu e^{\mu\cdot}R(\mu,A(0))]f(\sigma)d\sigma\|\Big].
\end{align*}
For $r$ sufficiently large and in view of Lemma \ref{4.2}, we can
conclude that $\mathcal{E}_\lambda$ is a Cauchy sequence.
\end{proof}

\begin{prop}
Assume that $(U_L(t,s))_{t\geq s\geq0}$ has an exponential dichotomy, then
for $f\in BC(\mathbb R_+,E)$, the function $v:[0,\infty)\to
E$ defined
by
\begin{equation} \label{bsol}
v(t):=\begin{cases}[U_L(t,0)P(0)\varphi\\
+\lim_{\lambda\to +\infty}
\int_0^{\infty}\Gamma(t,\sigma)\lambda
e^{\lambda\cdot}R(\lambda,A(0))f(\sigma)d\sigma](0), &\quad t\geq0\\[3pt]
\varphi(t),& -r\leq t\leq0
\end{cases}
\end{equation}
is the unique bounded mild solution of \eqref{IHRDE}
with $\varphi$ is an initial condition which satisfies
\begin{equation}  \label{bsol1}
Q(0)\varphi=\lim_{\lambda\to +\infty}%
\int_0^{\infty}\Gamma(0,\sigma)\lambda
e^{\lambda\cdot}R(\lambda,A(0))f(\sigma)d\sigma.
\end{equation}
Moreover, if $f\in C_0({\rm I\!R_{+}},E)$ then $v$ also belongs to
$C_0(\mathbb{R}_{+},E)$.
\end{prop}

\begin{proof} Let $v$ be a mild solution of \eqref{IHRDE}. From
Theorem \ref{theom}, one can verify that
\begin{align*}
v(t)&=U_L(t,0)(
\varphi+\lim_{\lambda\to +\infty}
\int_0^{\infty}\Gamma(0,\sigma)\lambda
e^{\lambda\cdot}R(\lambda,A(0))f(\sigma)d\sigma)\\
&\quad + \lim_{\lambda\to +\infty}\int_0^{\infty}\Gamma(t,\sigma)
\lambda e^{\lambda\cdot}R(\lambda,A(0))f(\sigma)d\sigma.
\end{align*}
As $t\mapsto \lim_{\lambda\to +\infty}
\int_0^{\infty}\Gamma(t,\sigma)\lambda
e^{\lambda\cdot}R(\lambda,A(0))f(\sigma)d\sigma$
is a bounded function, then $v$ is bounded if and only if
\[
t\mapsto U_L(t,0)(\varphi+\lim_{\lambda\to +\infty}%
\int_0^{\infty}\Gamma(0,\sigma)\lambda
e^{\lambda\cdot}R(\lambda,A(0))f(\sigma)d\sigma)
\]
is  bounded, and this  is equivalent to
$[\varphi+\lim_{\lambda\to +\infty}
\int_0^{\infty}\Gamma(0,\sigma)\lambda
e^{\lambda\cdot}R(\lambda,A(0))f(\sigma)d\sigma]$ belongs to
$\mathop{\rm Im} P(0)$,
i.e.,
\[
Q(0)\varphi=-\lim_{\lambda\to +\infty}
\int_0^{\infty}\Gamma(0,\sigma)\lambda
e^{\lambda\cdot}R(\lambda,A(0))f(\sigma)d\sigma.
\]
Consequently, $v$ is given by (\ref{bsol}).

Now, let us take $f\in C_0(\mathbb R_+,E)$ and show that
\[
C_0(\mathbb R_+,E)\ni\omega(\cdot):=\lim_{\lambda\to +
\infty}\int_0^{\infty}\Gamma(\cdot,\sigma)\lambda
e^{\lambda\cdot}R(\lambda,A(0))f(\sigma)d\sigma.
\]
Let
\begin{gather*}
I(t):=\lim_{\lambda\to +\infty}%
\int_0^t U_L(t,\sigma)P(\sigma)\lambda e^{\lambda\cdot}
R(\lambda,A(0))f(\sigma)d\sigma,\\
J(t):= \lim_{\lambda\to +\infty}\int_t^{\infty}
U_Q(t,\sigma)Q(\sigma)\lambda
e^{\lambda\cdot}R(\lambda,A(0))f(\sigma)d\sigma.
\end{gather*}
Then, $\omega(t)=I(t)-J(t)$.

Since $f(t)\to 0$ as $t\to \infty$, there exists
$t_0\geq0 $ such that $\|f(t)\|\leq\frac{\varepsilon}{2MN\delta}$ for all
$t\geq t_0$.
Hence, $\|I\|\leq\frac{2MN}{\delta}\|f\|_{\infty}e^{-\delta(t-t_0)}+\frac{%
\varepsilon}{2}$ and this implies that $I(t)\to 0$ as $t\to +\infty$.

For $J(\cdot)$, we have
\[
J(t)=\lim_{\lambda\to +\infty}%
\int_0^{\infty} U_Q(t,t+\sigma)Q(t+\sigma)\lambda
e^{\lambda\cdot}R(\lambda,A(0))f(t+\sigma)d\sigma.
\]
Then,
\[
\|J(t)\|\leq MN\int_0^{\infty}
e^{-\delta\sigma}\|f(t+\sigma)\|d\sigma.
\]
Therefore, letting $t$ approach infinity we obtain the result.
\end{proof}

\section{Fundamental Solutions and Stability Results}

This section is devoted to use the notion of fundamental solutions to study the
stability of the semi-linear retarded equation
\begin{equation}\label{SNRDE}
 \begin{gathered} x'(t)=A(t)x(t)+L(t)x_t+F(t,x_t),\quad t\geq s\geq0,\\ 
x_s=\varphi\in \mathcal{C}_r,
\end{gathered}
\end{equation}
where $(A(t),D(A(t)))_{t\geq 0}$ and $\left( L(t)\right) _{t\geq 0}$ are
defined as above and $F$ a nonlinear function from 
$\mathbb{R}_+\times\mathcal{C}_r$ to $E$.

In \cite{De-Gu-Gy}, the authors showed the existence of fundamental solutions
of \eqref{HRDE}, when $A(t)= A$. But to study the stability of the
non-autonomous retarded equation \eqref{SNRDE} they assumed the existence of
these fundamental solutions. Here, using our previous variation of constants
formulas, we are able to show this assumed result, and then obtain the
same stability results for \eqref{SNRDE} as in \cite[Theorems 4.6, 4.7, 4.10 ]
{De-Gu-Gy}.

\begin{defn} \rm
A family $(\Phi(s,t))_{t\geq s\geq0}$ of bounded linear operators on $E$ is
called a fundamental solution of $\eqref{HRDE}$ if

\begin{itemize}
\item[(i)]  $\Phi(t,t)=Id, \quad t\geq0$.

\item[(ii)]  $(t,s)\mapsto \Phi(t,s)$ is strongly continuous for $t\geq
s\geq0$.

\item[(iii)]  For every $g\in L^1_{\rm loc}(\mathbb R_+,E)$ 
and $t\in [s-r,\infty)$, the map
\[
 t\mapsto\begin{cases}
\int_s^t\Phi(t,\sigma)g(\sigma)d\sigma, & t\geq s\geq0, \\
\varphi(t-s), & s-r\leq t\leq s,
\end{cases}
\]
is the unique mild solution of
\begin{gather*} 
v'(t)=A(t)x(t)+L(t)x_t+g(t) ,\quad t\geq s\geq0,\\ 
v_s=\varphi\in \mathcal{C}_r. 
\end{gather*} 
\end{itemize}
\end{defn}

Let us define on $\mathcal{C}_r$ the function
\begin{equation}
[R_{\lambda,x}(t,s)](\tau)=\begin{cases}
\lambda R(\lambda,A(0))x, & t+\tau\geq s, \\
\lambda e^{\lambda(t+\tau-s)}R(\lambda,A(0))x, & t+\tau< s,
\end{cases}
\end{equation}
with $x\in E$, $\tau \in[-r,0]$ and $\lambda>\max(\omega,0)$.

\begin{thm}
Suppose that, for all $x\in E$, the limit
\begin{equation}
\lim_{\lambda \to  +\infty
}\int_{s}^{t}V(t,\sigma )L(\sigma )R_{\lambda ,x}(\sigma,s)d\sigma
\end{equation}
exists uniformly on compact sets of $\{(t,s):t\geq s\geq 0\}$. Then, there
exists a fundamental solution $(\Phi (t,s))_{t\geq s\geq 0}$ of
$\eqref{HRDE}$ given by
\begin{equation}
\Phi (t,s)x=\lim_{\lambda \to  +\infty }\lambda [U_{L}(t,s)e^{\lambda
\cdot }R(\lambda ,A(0))x)](0)
\end{equation}
for $t\geq s\geq 0$ and $x\in E$.
\end{thm}

This theorem follows fom the variation of constants formula in Theorem 
\ref{theom}, and the proof is similar to the one given in 
\cite[Theorem 3.2]{De-Gu-Gy}.



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