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

\AtBeginDocument{{\noindent\small
{\em Electronic Journal of Differential Equations},
Vol. 2006(2006), No. 56, pp. 1--14.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu  (login: ftp)}
\thanks{\copyright 2006 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2006/56\hfil Periodic solutions]
{Periodic solutions for some partial neutral functional
differential equations}
\author[R. Benkhalti, A. Elazzouzi,  K. Ezzinbi\hfil EJDE-2006/56\hfilneg]
{Rachid Benkhalti, Abdelhai Elazzouzi,  Khalil Ezzinbi}  % in alphabetical order

\address{Rachid Benkhalti \newline
Pacific Lutheran University, Department of Mathematics, Tacoma,
Washington, 98447, USA} 
\email{benkhar@plu.edu}

\address{Abdelhai Elazzouzi \newline
Universit\'{e} Cadi Ayyad, Facult\'{e} des Sciences Semlalia,
D\'{e}partement de Math\'{e}matiques, B.P. 2390 Marrakesh,
Morocco} 
\email{a.elazzouzi@ucam.ac.ma}

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

\date{}
\thanks{Submitted November 14, 2005. Published April 28, 2006.}
\thanks{Research is supported by grant 03-030 RG/MATHS/AF/AC
from TWAS}
\subjclass[2000]{34C25, 34D40, 34K40, 34K60}
\keywords{Integral solutions; Hille-Yosida condition; boundedness;
\hfill\break\indent
ultimate boundedness; condensing map; Hale and Lunel's fixed point theorem}

\begin{abstract}
 In this work, we study the existence of periodic solutions for
 partial neutral functional differential equation.
 We assume that the linear part is not necessarily densely defined
 and satisfies the Hille-Yosida condition. In the nonhomogeneous
 linear case, we prove that the existence of a bounded solution on
 $\mathbb{R}^{+}$ implies the existence of a periodic solution.
 In nonlinear case, we use the concept of boundedness and ultimate
 boundedness to prove the existence of periodic solutions.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{definition}[theorem]{Definition}
\newtheorem{example}[theorem]{Example}

\section{Introduction}

 The aim of this work is to study the existence of a periodic
solution for the  partial neutral functional differential
equation
\begin{equation}
\begin{gathered}
\frac{d}{dt}\mathcal{D}(u_{t})=A\mathcal{D}(u_{t})+F(t,u_{t})
\quad \text{for }t\geq  0\\
u_{0}=\varphi, \quad \varphi\in C:=C([-r,0] ;X),
\end{gathered} \label{e0}
\end{equation}
where $A$ is not necessarily densely defined linear operator on a
Banach space
$X$. We suppose that $A$ satisfies the Hille-Yosida condition, which means
that there exist $\overline{M}\geq 1$, $\omega\in\mathbb{R}$ such that
$(\omega,+\infty)\subset\rho(A)$ and
\[
|R(\lambda,A)^{n}|\leq\frac{\overline{M}}{(\lambda-\omega)^{n}}\quad
 \text{for }n\in\mathbb{N},\; \lambda>\omega,
\]
where $\rho(A)$ is the resolvent set of $A$ and $R(\lambda,A)=(\lambda
-A)^{-1}$. Here $C$ is the space of continuous functions from $[-r,0]$
to $X$ endowed with the uniform norm topology, and
$\mathcal{D}:C\to X$ is a bounded linear operator which is given by
\[
\mathcal{D}\varphi:=\varphi(0)-\int_{-r}^{0}[d\eta(\theta)]
\varphi(\theta)\quad \text{for }\varphi\in C,
\]
for a mapping $\eta:[-r,0]  \to\mathcal{L}(X)$ of bounded
variation and non atomic at zero, which means that
\[
\mathop{\rm var}_{[-\epsilon,0]} (\eta)\to 0\quad \quad\text{as }
\epsilon\to0.
\]
$\mathcal{L}(X)$ is the space of bounded linear operators from
$X$ into $X$. For every $t\geq 0$, as usual, the history function
$u_{t}\in C$ is defined by
\[
u_{t}(\theta)=u(t+\theta)\quad\text{for }\theta\in[-r,0].
\]
$F$ is a continuous function from $\mathbb{R}^{+}\times C$ into $X$ which is
periodic in $t$.

The theory of functional differential equations of neutral type has been
developed recently by several authors, for instance we refer to
\cite{adimezzin,AdiEzz1,AdiEzz2,AdiEzz+,lakezzady++,adimbouzaezzin,Hal4,Hal5,wu,WuXia1,WuXia2}.
In \cite{WuXia1} and \cite{WuXia2}, the authors studied neutral partial
functional differential-difference equation
defined on the unit circle $S$, which is a model for a continuous circular
array of resistively coupled transmission lines with mixed initial boundary
conditions
\begin{equation}
\frac{d}{dt}[u(.,t)-qu(.,t-r)]  =k\frac{\partial^{2}}{\partial
x^{2}}[u(.,t)-qu(.,t-r)]  +\zeta(u_{t})\quad \text{for }t\geq
0, \label{unitcirc}
\end{equation}
where $x\in S$, $k$ is a positive constant, $\zeta$ is a continuous function
and $0\leq q<1$. The phase space is $C([-r,0],H^{1}(S))$. In
\cite{Hal4,Hal5}, the author studied the qualitative behavior of
solutions of equation (\ref{unitcirc}), and obtained several results about
stability, attractiveness of solutions and bifurcation of solutions near an
equilibrium. The idea of studying partial neutral functional differential
equations with operators satisfying Hille-Yosida condition, begins with
\cite{AdiEzz1}, where the authors studied the following class of equation
\[
\frac{d}{dt}[u(t)-Gu(t-r)]  =A[u(t)-Gu(t-r)]
+P(u_{t})+Qu(t-r),
\]
where $A$ satisfies the Hille-Yosida condition, $G$ and $Q$ are bounded linear
operators from $X$ into $X$ and $P\;$is a bounded linear operator from $C$
into $X$. It has been proved in particular, that the solutions generate a
locally Lipschitz continuous integrated semigroup. In \cite{adim ezzin trav},
the authors studied the existence, uniqueness and regularity of solutions of
\eqref{e0}. They obtained several results concerning dissipativeness
and existence of global attractor.

One of the most attractive areas of the qualitative theory of partial neutral
functional differential equations is the existence of periodic solutions.
Naturally, fixed point theorems play a significant role in the investigation
of the existence of periodic solutions. In finite dimensional spaces, many
works are devoted to this subject. In \cite{Burt} and \cite{haddo}, using
Horn's fixed point theorem, the authors proved that if the solutions of an
$n$-dimensional periodic ordinary differential equation are bounded and
ultimately bounded, then the system has a periodic solution. In
\cite{Ben-Bou-EzzJmaa}, the authors gave several criteria for the existence
of periodic solutions of functional differential equations with infinite
delay, they obtained the existence of periodic solutions by using Sadovskii's
fixed point theorem. In \cite{hal lop} and \cite{jin} the authors used Horn's
fixed point theorem to prove the existence of periodic solutions for
functional differential equations with finite delay. Recently, the authors in
\cite{lakezzady}, studied the following partial neutral functional
differential equation
\begin{equation}
\frac{d}{dt}\mathcal{D}(u_{t})=A\mathcal{D}(u_{t})+L(u_{t})+g(t)
\quad \text{for }t\geq 0, \label{equalaklach}
\end{equation}
where $A$ satisfies the Hille-Yosida condition, $L$ is a bounded linear
operator from $C$ into $X$ and $g$ is a continuous function for
$\mathbb{R}^{+}$ to $X$. They established a variation of constants
formula for equation (\ref{equalaklach}). This formula is used to
prove the existence of bounded, periodic and almost periodic solution
when the solution semigroup of equation (\ref{equalaklach}) with $g=0$
is hyperbolic. Recall that the main approach to prove the existence
of periodic solutions, is to consider the Poincar\'{e} map
$\mathcal{P}$ which is defined by
\[
\mathcal{P}\varphi=u_{\omega}(.,\varphi),
\]
where $u(.,\varphi)$ is the solution of \eqref{e0}. Then one
establishes the existence of fixed points of $\mathcal{P}$ which are the
initial values of periodic solutions.

In \cite{ezli,liu}, the authors used the Poincar\'{e} map and they
proved the existence of periodic solutions for nonlinear partial functional
differential equations of retarded type which correspond to $\mathcal{D}
\varphi=\varphi(0)$, they used the boundedness and the ultimate boundedness of
solutions to get a periodic solution by using Horn's fixed point theorem which
requires the compactness of the solution operator. For partial neutral
functional differential equations, the Poincar\'{e} map $\mathcal{P}$ is not
compact, and fixed point theorems requiring compactness couldn't be used. We
consider the case where $F$ is linear with respect to the second argument, we
show that the existence of a bounded solution on $\mathbb{R}^{+}$ implies the
existence of a periodic solution. To achieve this goal, we use Chow and Hale's
fixed point theorem for affine maps \cite{chowandhal} to prove that the
Poincar\'{e} map $\mathcal{P}$ has at least one fixed point. For the nonlinear
case, we use the boundedness and the ultimate boundedness and we prove the
existence of periodic solutions by using Hale and Lunel's fixed point theorem
which is an extension of Horn's fixed point theorem for condensing maps.

The work is organized as follows: in section 2, we give some definitions and
results about the solutions of \eqref{e0}. In section 3, we discuss
the existence of periodic solutions where $F$ is linear with respect to the
second argument. In section 4, we study the existence of periodic solutions in
the nonlinear case, we assume that solutions are bounded and ultimate bounded.
Finally, we propose some applications for some partial neutral functional
differential equations with diffusion. \vspace{0,2cm}

\section{Existence and estimation of solutions}

 Throughout this work, we suppose that
\begin{itemize}
 \item[(H0)] $A$ satisfies the Hille-Yosida condition.
\end{itemize}
 The following results concern
the existence of integral solutions of \eqref{e0}.

\begin{definition}[\cite{AdiEzz2,adim ezzin trav}] \rm
 A continuous function $u$ from $[-r,T]$ to $X$
with $T>0$, is an integral solution of \eqref{e0} if
\begin{itemize}
 \item[(i)] $\int_{0}^{t}\mathcal{D}(u_{s})ds\in D(A)$
  for $t\in[0,T]$,

\item[(ii)] $\mathcal{D}(u_{t})=\mathcal{D}\varphi
+A   \int_{0}^{t}\mathcal{D}(u_{s})ds+\int_{0}^{t}\ F(s,u_{s})ds$
for $t\in[0,T]$,

\item[(iii)]  $u_{0}=\varphi$.
\end{itemize}
\end{definition}

 From the closedness property of $A$, one can see that if $u$ is an
integral solution of \eqref{e0}, then
$\mathcal{D}(u_{t})\in\overline{D(A)}$ for all $t\in[0,T]$. In
particular, $\mathcal{D}\varphi\in\overline{D(A)}$. It has been
proved in \cite{AdiEzz2}, that the condition $\mathcal{D}\varphi\in
\overline{D(A)}$ is enough for the existence of integral
solutions of \eqref{e0}. The part $A_{0}$ of the operator $A$ in
$\overline{D(A)}$ is defined by
\begin{gather*}
D(A_{0})=\{  x\in D(A):Ax\in\overline{D(A)}\},\\
A_{0}x=Ax\quad\text{\ for }x\in D(A_{0}).
\end{gather*}


\begin{lemma} \cite{arendt}
$A_{0}$ generates a strongly continuous semigroup
$(T_{0}(t))_{t\geq 0}$ on $\overline{D(A)}$.
\end{lemma}

 For the existence of the integral solutions, we assume
that
\begin{itemize}
\item[(H1)]  $F$ is continuous and
Lipschitzian with respect to the second argument:
There exists a positive constant $\mu$ such that
\[
|F(t,\phi)-F(t,\psi)|\leq\mu|\phi-\psi|\quad
 \text{for } \phi, \psi \in C,\;  t\geq  0 .
\]
\end{itemize}

\begin{theorem}[{\cite[Theorem 2]{AdiEzz2}}] \label{thex}
Assume that (H0) and (H1) hold. Then, for all
$\varphi\in C$ such that $\mathcal{D}\varphi\in\overline{D(A)}$,
there exists a unique integral solution $u$ of \eqref{e0} on
$[0,+\infty)$. Moreover, $u$ is given by
\begin{equation}
\mathcal{D}(u_{t})=T_{0}(t)\mathcal{D}\varphi+\lim_{\lambda\to+\infty
}\int_{0}^{t}T_{0}(t-s)B_{\lambda}F(s,u_{s})ds\quad
\text{for }t\geq 0,
\label{var const}
\end{equation}
where $B_{\lambda}=\lambda R(\lambda,A)$ for $\lambda>\omega$.
\end{theorem}

 In the sequel, integral solutions will be called solutions.

\begin{proposition} \label{pro de depo cont}
Assume that (H0) and (H1) hold. Let $u$ and $v$ be solutions of
\eqref{e0} on $[-r,T]$ for $T>0$. Then, there exist positive constants
$N$ and $\widetilde{N}$ such that
\begin{equation}
|u_{t}-v_{t}|\leq Ne^{\widetilde{N}t}|u_{0}-v_{0}|
\quad \text{for }t\in[0,T].
\label{majj}
\end{equation}
\end{proposition}

 This is an immediate consequence of the following fundamental lemma.

\begin{lemma}[{\cite[Lemma 5]{AdiEzz2}}] \label{thwu}
There are positive constants $a,b$ and $c$
such that for any continuous function $h:\mathbb{R}^{+}\to X$, the
solution $w$ of the difference equation
\begin{gather*}
\mathcal{D}(w_{t})=h(t)\quad \text{for }t\geq 0\\
w_{0}=\varphi.
\end{gather*}
satisfies the estimate
\begin{equation}
|w_{t}(.,\varphi)|\leq\exp(at)\Big[b|w_{0}|
+c\sup_{s\in[0,t]}|h(s)|\Big] \quad\text{for }t\geq 0.
\label{inver tot}
\end{equation}
\end{lemma}

\begin{proof}[Proof of Proposition \ref{pro de depo cont}]
Let $u$ and $v$ be two solutions of \eqref{e0} on $[-r,T]$, for some
$T>0$. Then, for $t\in[0,T]$
\begin{equation}
\mathcal{D}(u_{t}-v_{t})=T_{0}(t)\mathcal{D}(u_{0}-v_{0})+\lim_{\lambda
\to+\infty}\int_{0}^{t}T_{0}(t-s)B_{\lambda}(F(s,u_{s})-F(s,v_{s}))ds.
\label{fong}
\end{equation}
Let $g$ be defined by the right hand side of \eqref{fong}. Then, by
assumption (H1), we deduce that there exist
positive constants $k_1 $ and $k_2 $ such that
\[
|g(t)|\leq k_1 |u_{0}-v_{0}|+k_2 \int_{0}
^{t}|u_{\xi}-v_{\xi}|d\xi\quad \text{for }t\in[0,T].
\]
Using estimate (\ref{inver tot}), we obtain that
\[
|u_{t}-v_{t}|\leq\widetilde{k_1 }|u_{0}-v_{0}|
+\widetilde{k_2 }\int_{0}^{t}|u_{\xi}-v_{\xi}|d\xi\quad
\text{for }t\in[0,T],
\]
for some positive constants $\widetilde{k_1 }$ and $\widetilde{k_2 }$.
Using Gronwall's Lemma, one obtains the estimate (\ref{majj}).
\end{proof}

 Consequently, we have the local boundedness of the solutions.

\begin{corollary}\label{cor de loca bou}
Assume that (H0) and (H1) hold. Then, the solutions of \eqref{e0}
are locally bounded, in the sense that for each $B_{0}>0$ and
$T_{0}>0$, there exists a constant $\overline{B}_{0}>0$, such that
$|\varphi|\leq B_{0}$ implies that $|u(t,\varphi)|\leq\overline{B}_{0}$
for $t\in[0,T_{0}]$.
\end{corollary}

 To study the qualitative behavior of solutions, we need to make
additional assumptions on the following difference equation
\begin{equation}
\begin{gathered}
\frac{d}{dt}\mathcal{D}(w_{t})=0\quad\text{for }t\geq 0\\
w_{0}=\varphi.
\end{gathered}  \label{investab}
\end{equation}
The following definition was given for neutral functional
differential equation in finite dimensional spaces, for more
details we refer to \cite{HalLun1}.

\begin{definition} \cite{adim ezzin trav} \rm
The operator $\mathcal{D}$ is stable if there exist
positive constants $\beta$ and $\gamma$ such that the solution of the
homogeneous difference equation \eqref{investab} with
$w_{0}=\varphi \in\{  \psi\in C;\quad\text{}\mathcal{D}\psi=0\}  $,
satisfies the following estimate
\[
|w_{t}(.,\varphi)|\leq\gamma\exp(-\beta t)|
\varphi|\quad\text{\ for }t\geq 0.
\]
\end{definition}

\begin{example} \rm
The operator $\mathcal{D}$ defined by
\[
\mathcal{D}\varphi=\varphi(0)-q\varphi(-r)
\]
 is stable if and only if $|q|<1$.
\end{example}

\begin{theorem}[{\cite[Lemma 2.9]{adim ezzin trav}}]  \label{thwu2}
If the operator $\mathcal{D}$ is stable. Then, there are positive
constants $a$, $b$, $c$ and $d$ such that for
any continuous function $h:\mathbb{R}^{+}\to X$, the solution $w$ of
the difference equation
\begin{gather*}
\mathcal{D}(w_{t})=h(t)\quad \text{for }t\geq 0\\
w_{0}=\varphi\in C,
\end{gather*}
satisfies the estimate
\[
|w_{t}(.,\varphi)|\leq e^{-at}\Big( b|\varphi|
+c\sup_{s\in[0,t]}|h(s)|\Big)+d\sup_{s\in[
\max\{0,t-r\},t]}|h(s)|\quad\text{for  }t\geq 0.
\]
\end{theorem}

 The Kuratowski's measure of noncompactness.
of bounded sets $K$\ on a Banach space $Y$ is defined by
\[
\alpha(K)=\inf\{  \epsilon>0:K\text{ has a finite cover of ball of
diameter less than }\epsilon\}  .
\]

\begin{lemma} \cite{LakshLeela}
Let $A_1 $ and $A_2 $ be bounded sets of a  Banach space
$Y$. Then
\begin{itemize}
\item[(i)] $\alpha(A_1 )\leq \mathop{\rm dia}(A_1 )$,
where $\mathop{\rm dia}(A_1 )=\sup_{x,y\in A_1 }|x-y|$,
\item[(ii)] $\alpha(A_1 )=0$ if and only if $A_1 $
is relatively compact in $Y$,
\item[(iii)] $\alpha(A_1 \cup A_2 )=\max\{\alpha(A_1 ),\alpha(A_2 )\}$.
\end{itemize}
\end{lemma}

 Let $\mathcal{K}:Y\to Y$ be a closed linear operator with a
dense domain $D(\mathcal{K})$ in a Banach space $Y$. We denote by
$\sigma(\mathcal{K})$ the spectrum of $\mathcal{K}$.

\begin{definition}[\cite{wu}] \rm
 The essential spectrum $\sigma_{\rm ess}(\mathcal{K})$ of $\mathcal{K}$
is the set of all $\lambda\in\mathbb{C}$ such that at least one of the
following holds:
\begin{itemize}
\item[(i)] The range $\mathop{\rm Im}(\lambda I-\mathcal{K})$ is not
closed,
\item[(ii)] the generalized eigenspace
$M_{\lambda}(\mathcal{K})= \cup_{n\geq 1} \ker(\lambda I-\mathcal{K})^{n}$
of $\lambda$ is infinite dimensional,
\item[(iii)] $\lambda$ is a limit point of $\sigma(\mathcal{K})$,
that is $\lambda\in\overline {\sigma(\mathcal{K})/\{  \lambda\}}$.
\end{itemize}
\end{definition}

 For a bounded linear operator $\mathcal{K}$ on $Y$, the
Kuratowski measure of non-compact\-ness of $\mathcal{K}$ is defined
by
\[
|\mathcal{K}|_{\alpha}=\inf\{\epsilon>0: \alpha(\mathcal{K}(B))
\leq\epsilon\alpha(B)\text{ for every bounded subset }B\text{ of }Y\}.
\]
The essential radius $r_{\rm ess}(\mathcal{K})$ is given by
\[
r_{\rm ess}(\mathcal{K})=\sup\{|\lambda|:\lambda\in
\sigma_{\rm ess}(\mathcal{K})\}.
\]
The computation of essential radius is given by the following
Nussbaum's formula.

\begin{lemma}[\cite{Nus1}] \label{nusb}
\[
r_{\rm ess}(\mathcal{K})=\lim_{n\to+\infty}
(|\mathcal{K}^{n}|_{\alpha})^{1/n}.
\]
\end{lemma}

\begin{definition}[\cite{HalLun1}] \rm
 A continuous mapping $P:Y\to Y$ is said to be an
$\alpha$-contraction if $P$ maps bounded sets into bounded sets
and if there exists a constant $k\in(0,1)$ such that
\[
\alpha(P(B))\leq k\alpha(B),\vspace{0,2cm}
\]
for every bounded subset $B$ of $Y$.
\end{definition}

\begin{definition}[\cite{HalLun1}] \rm
 A continuous mapping $P:Y\to Y$ is a condensing map on
$Y$ if $P$ maps bounded sets into bounded sets and
\[
\alpha(P(B))<\alpha(B),
\]
for every bounded subset $B$ of $Y$ such that $\alpha(B)>0$.
\end{definition}

 Let $C_{0}$ be the phase space of Equation \eqref{e0}
defined by
\[
C_{0}=\{  \varphi\in C:\mathcal{D}\varphi\in\overline{D(A)}\}.
\]
For each $t\geq 0$, we define the linear operator $\mathcal{U}(t)$
on $C_{0}$ by
\[
\mathcal{U}(t)\varphi=x_{t}(.,\varphi),
\]
where $x(.,\varphi)$ is the solution of the  equation
\begin{equation}
\begin{gathered}
\frac{d}{dt}\mathcal{D}(u_{t})=A\mathcal{D}(u_{t})\quad\text{for }t\geq
0\\
u_{0}=\varphi\in C.
\end{gathered} \label{0 homo}
\end{equation}
 Without loss of generality, we assume
that
\begin{itemize}
\item[(H2)] $(T_{0}(t))_{t\geq 0}$ is exponentially stable,
which means that there exist $\alpha_{0}>0$ and $M_{0}\geq 1$ such that
\[
|T_{0}(t)|\leq M_{0}e^{-\alpha_{0}t}\quad\text{for }
t\geq 0.
\]
\end{itemize}
Otherwise, we can replace $A$ by $A-\delta I$, where $\delta>0$ can be chosen
such that the semigroup generated by the part of $A-\delta I$ on
$\overline{D(A)}$ is exponentially stable.\vspace
{0cm}\newline We assume that
\begin{itemize}
\item[(H3)] $\mathcal{D}$ is stable.
\end{itemize}
  The following fundamental lemma plays a crucial role for
the existence of periodic solutions.

\begin{lemma}[{\cite[Proposition 2.11]{adim ezzin trav}}]
\label{exposta}
 Assume that (H0), (H2) and (H3) hold. Then, $(\mathcal{U}(t))_{t\geq 0}$
is an exponentially stable semigroup on $C_{0}$, that is
there exist $\eta>0$ and $M\geq 1$ such that
\[
|\mathcal{U}(t)|\leq Me^{-\eta t}\quad\text{for }t\geq 0.
\]
\end{lemma}

 For $\varphi\in C_{0}$, we introduce the new norm on $C_{0}$ by
\[
|\varphi|_{\eta}=\underset{t\geq 0}{\sup}\;{e^{\eta t}}|\mathcal{U}
(t)\varphi|,
\]
where $\eta$ is the positive constant given in Lemma \ref{exposta}.
Clearly,
\[
|\varphi|\leq|\varphi|_{\eta}\leq M|\varphi|,
\]
which implies that $|.|_{\eta}$ and $|.|$ are equivalent norms on $C_{0}$.
 As an immediate result, we have the following result.

\begin{corollary}\label{V contra}
Assume that (H0), (H2) and (H3) hold. Then
\[
|\mathcal{U}(t)|_{\eta}\leq e^{-\eta t}\quad\text{for }t\geq 0.
\]
\end{corollary}

\begin{proof} For every $t\geq 0$, one has,
\begin{align*}
|\mathcal{U}(t)\varphi|_{\eta}
& =\sup_{s\geq 0} {e^{\eta s}}|\mathcal{U}(s)\mathcal{U}(t)\varphi|,\\
& =e^{-\eta t}\sup_{s\geq 0}{e^{\eta(t+s)}}|\mathcal{U}
(s+t)\varphi|,\\
& \leq e^{-\eta t}\sup_{s\geq 0}{e^{\eta s}}|\mathcal{U}
(s)\varphi|=e^{-\eta t}|\varphi|_{\eta},
\end{align*}
which implies
$|\mathcal{U}(t)|_{\eta}\leq{e^{-\eta t}}$ for
$t\geq 0$.
\end{proof}
\begin{itemize}
\item[(H4)]  $T_{0}(t)$ is compact on
$\overline{D(A)}$ whenever $t>0$.
\end{itemize}

\begin{theorem}[{\cite[Theorem 5.2]{adim ezzin trav}}]\label{theodecom}
Assume that (H0), (H1), (H2), (H3) and (H4) hold. Then the solution
$u(.,\varphi)$ of \eqref{e0} is decomposed as follows:
\[
u_{t}(.,\varphi)=\mathcal{U}(t)\varphi+\mathcal{W}(t)\varphi
\quad\text{for } t\geq 0,
\]
where $\mathcal{W}(t)$ is a compact operator on $C_{0}$, for
each $t\geq 0$.
\end{theorem}

\section{Existence of periodic solutions in nonhomogeneous linear
case}

 In this section, we assume that $F$ takes the form
\[
F(t,\varphi)=L(t,\varphi)+f(t)\quad\text{for }t\geq 0,\;
\varphi\in C,
\]
where $L$ is a continuous function from $\mathbb{R}^{+}\times C$ into $X$,
linear with respect to the second argument and $f$ is a continuous function
from $\mathbb{R}$ into $X$. Equation \eqref{e0} becomes
\begin{equation}
\begin{gathered}
\frac{d}{dt}\mathcal{D}(u_{t})=A\mathcal{D}(u_{t})+L(t,u_{t})+f(t)
\quad\text{for }t\geq 0,\\
u_{0}=\varphi\in C,
\end{gathered}   \label{linear}
\end{equation}
 For the existence of periodic solutions, we assume that
 \begin{itemize}
\item[(H5)] $L$ and $f$ are $\omega$-periodic in $t$.
\end{itemize}

\begin{theorem} \label{colat}
Assume that (H0), (H2), (H3), (H4)  and (H5) hold. If Equation
(\ref{linear}) has a bounded solution on $\mathbb{R}^{+}$, then it
has an $\omega$-periodic solution.
\end{theorem}

 For the proof, we use Chow and Hale's fixed point theorem which
gives sufficient conditions for affine maps to have fixed points.

\begin{theorem}[\cite{chowandhal}]\label{chohal}
Let $Y$ be a Banach space and $P:Y\to Y$ be an affine map which
is defined by
\[
Px=Sx+y,
\]
where $S$ is a bounded linear operator on $Y$ and $y$ is given in $Y$. If
$\mathop{\rm Im}(I-S)$ is closed and there exists $x_{0}\in Y$ such
that $(P^{n}(x_{0}))_{n\geq 0}$ is bounded, then $P$ has at least
one fixed point.
\end{theorem}

\begin{proof}[Proof of Theorem \ref{colat}]
Define the Poincar\'{e} map $\mathcal{P}:C_{0}\to C_{0}$ by
\[
\varphi\to u_{\omega}(.,\varphi)=u_{\omega}(.,0,\varphi ,L,f),
\]
where $u(.,0,\varphi,L,f)$ is the solution of (\ref{linear}). By the
uniqueness of solutions with respect to the initial data,
$u_{t}(.,0,\varphi,L,f)$ is decomposed as follows
\[
u_{t}(.,0,\varphi,L,f)=u_{t}(.,0,\varphi,L,0)+u_{t}(.,0,0,L,f)\quad\text{for
}t\geq 0.
\]
Therefore, the Poincar\'{e} map $\mathcal{P}$ is affine,
$\mathcal{P}\varphi=\mathcal{P}_{0}\varphi+\psi$,
where $\mathcal{P}_{0}\varphi=u_{\omega}(.,0,\varphi,L,0)$ and $\psi
=u_{\omega}(.,0,0,L,f)$.
We claim that $r_{\rm ess}(\mathcal{P}_{0})<1$. In fact, by
Theorem \ref{theodecom}, $\mathcal{P}_{0}$ is decomposed as follows
\[
\mathcal{P}_{0}\varphi=\mathcal{U}(\omega)\varphi+\mathcal{W}(\omega
)\varphi,
\]
where $\mathcal{W}(\omega)$ is a compact operator on $C_{0}$. We deduce
that
$ \alpha(\mathcal{P}_{0})\leq\alpha(\mathcal{U}(\omega))$.
By Corollary \ref{V contra}, we have
\[
\alpha(\mathcal{P}_{0})\leq\exp(-\eta\omega)<1.
\]
Using Lemma \ref{nusb}, we obtain that
$r_{\rm ess}(\mathcal{P}_{0})<1$
which implies that $1$ is not in the essential spectrum of
$\mathcal{P}_{0}$.
Consequently, $\mathop{\rm Im}(I-\mathcal{P}_{0})$ is closed.
Let $y$ be the bounded
solution of Equation (\ref{linear}) on $\mathbb{R}^{+}$. Then,
\[
\{\mathcal{P}^{n}y_{0},n\in\mathbb{N}\}=\{y_{n\omega},n\in\mathbb{N}
\},
\]
which gives that $(\mathcal{P}^{n}y_{0})_{n\geq 0}$ is bounded in $C_{0}$.
By Theorem \ref{chohal}, we deduce that $\mathcal{P}$ has at least
one fixed point, which gives an $\omega$-periodic solution
of (\ref{linear}).
\end{proof}

\section{Boundedness, ultimate boundedness and periodicity}

  In this section, we study the existence of periodic
solutions where the solutions are bounded and ultimate bounded.

\begin{definition} \rm
The solutions of \eqref{e0} are bounded if for each $B_1 >0$, there
exists a constant $\overline{B}_1 >0$, such that $|\varphi|\leq B_1 $
implies that $|u(t,\varphi)|\leq\overline{B}_1 $, for $t\geq 0$.
\end{definition}

\begin{definition} \rm
The solutions of \eqref{e0} are ultimate bounded if there is a bound
$B>0$ such that for each $B_2 >0$, there exists a constant $k>0$ such that
$|\varphi|\leq B_2 $ and $t\geq  k$ imply that $|u(t,\varphi)|\leq B$.
\end{definition}

 Recall that in \cite{ezli}, the authors have used the concept of
boundedness and ultimate boundedeness to prove the existence of a periodic
solution for partial functional differential equations of retarded
type which correspond to $\mathcal{D}\varphi=\varphi(0)$.
The relationship between the local boundedness, the boundedness and
the ultimate boundedeness is given below.

\begin{proposition}
\label{relatbondandlocall} The local boundedness and ultimate boundedness of
solutions of \eqref{e0} imply the boundedness of the solutions.
\end{proposition}

\begin{proof}
Let $B$ be given by the ultimate
boundedness, then for any $B_1 >0$, there exists a constant $k>0$ such that
$|\varphi|\leq B_1 $ and $t\geq  k$ imply that $|u(t,\varphi)|\leq B$. Local
boundedness of solutions gives that there exists a constant $B_2 >B$ such
that $|\varphi|\leq B_1 $ implies that $|u(t,\varphi)|<B_2 $, for
$t\in[0,k]$. It follows that for any positive constant $B_1 $, there exists a
constant $B_2 >B$ such that $|\varphi|\leq B_1 $ implies that $|u(t,\varphi
)|<B_2 $, for $t\geq 0$.
\end{proof}

\begin{proposition} \label{P con}
Under assumptions  (H0)--(H4), the
Poincar\'{e} map $\mathcal{P}$ is an $\alpha$-contraction on $C_{0}$.
\end{proposition}

\begin{proof} By Theorem \ref{theodecom}, $\mathcal{P}$
is decomposed as
\[
\mathcal{P}\varphi=\mathcal{U}(\omega)\varphi+\mathcal{W}(\omega
)\varphi,
\]
where $\mathcal{W}(\omega)$ is a compact operator on $C_{0}$.
Let $\Omega$ a bounded set in $C_{0}$. Then
\[
\alpha(\mathcal{P}(\Omega))\leq\alpha(\mathcal{U}(\omega)(\Omega)).
\]
Corollary \ref{V contra} implies
\[
\alpha(\mathcal{P}(\Omega))<\exp(-\eta\omega)\alpha(\Omega)
\quad\text{for  any  bounded  set $\Omega$ in }C_{0},
\]
which gives that $\mathcal{P}$ is an $\alpha$-contraction map on $C_{0}$.
\end{proof}

 In the following, we assume that
\begin{itemize}
\item[(H6)] $F$ is $\omega$-periodic in $t$.
\end{itemize}

\begin{theorem}\label{theo de perio nonline}
Assume that (H0), (H1), (H2), (H3), (H4) and (H6) hold.
If the solutions of \eqref{e0} are ultimately bounded, then \eqref{e0}
has an $\omega$-periodic solution.
\end{theorem}

 For the proof, we use Hale and Lunel's fixed point theorem which is
an extension of  the well known Horn's fixed point theorem for
condensing maps.

\begin{theorem}[{\cite[Hale and Lunel's fixed point theorem]{HalLun1}}]
 Suppose $S_{0}\subseteq S_1 \subseteq S_2 $ are convex bounded
subsets of a Banach space $Y$, such that $S_{0}$, $S_2 $ are closed
and $S_1 $ is open in $S_2 $. Let $P$ be a
condensing map on $Y$ such that $P^{j}(S_1 )\subseteq S_2 $, for $j\geq 0$,
and there is a number $N(S_1 )$ such that $P^{k}(S_1 )\subseteq S_{0}$, for
$k\geq  N(S_1 )$, then $P$ has a fixed point.\label{hornfix}
\end{theorem}

\begin{proof}[Proof of Theorem \ref{theo de perio nonline}]
By Corollary \ref{cor de loca bou} and Proposition \ref{relatbondandlocall}, we
know that the solutions of \eqref{e0} are bounded and ultimate
bounded. Let $B$ be the bound in the definition of ultimate boundedness. By
the boundedness of solutions, there exists a constant $B_1 >B$ such that for
$|\varphi|\leq B$ and $t\geq 0$, one has $|u(t,\varphi)|<B_1 $. Moreover,
there exists a constant $B_2 >B_1 $ such that for $|\varphi|\leq B_1 $ and
$t\geq 0$, then $|u(t,\varphi)|<B_2 $. By using the ultimate boundedness of
solutions of \eqref{e0}, we can see that there exists a positive
integer $m=m(B_1 )$ such that for $|\varphi|\leq B_1 $ and $t\geq  m\omega$,
we have $|u(t,\varphi)|<B$. On the other hand,
\[
\mathcal{P}^{j}\varphi=u_{j\omega}(.,\varphi)\quad
\text{for } j\in\mathbb{N}.
\]
Let $k=\big[\frac{r}{\omega}\big]  +m+1$, where $[t]$ denotes the
integer part of $t$. Then for $j\geq  k\ $and$\ |\varphi|\leq B_1 $,
one has
\begin{equation}
|\mathcal{P}^{j}(\varphi)|=|u_{j\omega}(.,\varphi)|\leq B, \label{ss}
\end{equation}
and for $j\in\{1,2,\dots,k-1\}\ $and$\ |\varphi|\leq B_1 $,
\begin{equation}
|\mathcal{P}^{j}(\varphi)|=|u_{j\omega}(.,\varphi)|\leq B_2 . \label{ss1}
\end{equation}
We define the sets
\begin{gather*}
S_{0}  =\{\varphi\in C_{0}:|\varphi|\leq B\},\\
S_1   =\{\varphi\in C_{0}:|\varphi|<B_1 \},\\
S_2   =\{\varphi\in C_{0}:|\varphi|\leq B_2 \}.
\end{gather*}
Clearly, $S_{0},\ S_1 \ $and$\ S_2 $ are convex bounded subsets of the
Banach space $C_{0}$. Moreover $S_{0}\subseteq S_1 \subseteq S_2 $, $S_{0}$
and $S_2 $ are closed and $S_1 $ is open in $S_2 $, and
\[
P^{j}(S_1 )\subseteq S_2 \quad \text{for all } j\geq 0.
\]
In fact, inequality (\ref{ss}) gives that there exists a positive integer
$k=k(S_1 )$ such that
\[
P^{j}(S_1 )\subseteq S_{0}\subseteq S_2 \quad \text{for } j\geq  k,
\]
and from inequality (\ref{ss1}), we deduce that
\[
P^{j}(S_1 )\subseteq S_2 \quad \text{for } j\in\{1,2,\dots,k-1\}.
\]
By Proposition \ref{P con}, $\mathcal{P}$ is an $\alpha$-contraction map on
$C_{0}$. Consequently, Theorem \ref{hornfix} gives that the Poincar\'{e} map
$\mathcal{P}$ has at least one fixed point which gives an $\omega$-periodic
solution of \eqref{e0}.
\end{proof}

\section{Applications}

\subsection*{Nonhomogeneous linear case}
To illustrate our previous results, we consider the following model
 of continuous circular array of resistively coupled transmission
 lines which is taken from \cite{WuXia2}
\begin{equation}
\begin{gathered}
\begin{aligned}
\frac{\partial}{\partial t}[u(t,x)-qu(t-r,x)]
& =\frac {\partial^{2}}{\partial x^{2}}[u(t,x)-qu(t-r,x)]
 +a_1 (t)u(t-r,x)\\
&\quad +h_1 (t,x)\quad\text{for }t\geq 0,\; x\in[0,\pi],
\end{aligned}\\
[u(t,x)-qu(t-r,x)]_{_{x=0,\pi}}=0\quad\text{for }t\geq 0,
\\
u(\theta,x)=\varphi_{0}(\theta,x)\quad\text{for }\theta\in[-r,0],\;
x\in[0,\pi],
\end{gathered} \label{w1}
\end{equation}
where $a_1 :\mathbb{R}\to\mathbb{R}$,
$h_1 :\mathbb{R}\times [0,\pi]\to\mathbb{R}$ and
$\varphi_{0}:[-r,0]\times[0,\pi]\to\mathbb{R}$ are continuous
functions and $0<q<1$.
 Let $X=C([0,\pi]  ;\mathbb{R})$ be the space of continuous functions
from $[0,\pi]$ to $\mathbb{R}$ endowed with the uniform norm topology.
 In order to rewrite (\ref{w1}) in abstract form, we introduce the
linear operator $A:D(A)\subset X\to X$ defined by
\begin{gather*}
D(A)=\{  y\in C^{2}([0,\pi]  ;\mathbb{R}):y(0)=y(\pi)=0\} ,\\
Ay=y''.
\end{gather*}

\begin{lemma}[\cite{dap}] \label{hy}
$(0,+\infty)\subset\rho(A)$ and
$|(\lambda I-A)^{-1}|\leq\frac{1}{\lambda}$ for $\lambda>0$.
\end{lemma}

The above lemma implies that assumption $\mathbf{(H}_{0})$
is satisfied. Moreover, one has
\[
\overline{D(A)}=\{  y\in X:y(0)=y(\pi)=0\}  .
\]
 Let $\mathcal{D}:C\to X$ and $L:\mathbb{R}\times C\to X$ be the
 bounded linear operators defined respectively by
\begin{gather*}
\mathcal{D}\varphi:=\varphi(0)-q\varphi(-r),\\
L(t,\varphi)=a_1 (t)\ \varphi(-r)\quad\text{for }t\in\mathbb{R},\;
\varphi\in C([-r,0];X).
\end{gather*}
Let $f:\mathbb{R}\longrightarrow X$ be given by
\[
f(t)(x)=h_1 (t,x)\quad\text{for }t\in\mathbb{R},\; x\in[0,\pi].
\]
  Then, Equation (\ref{w1}) takes the abstract form
(\ref{linear}). Since $0<q<1$, the operator $\mathcal{D}$ is stable.
 The part $A_{0}$ of $A$ in $\overline{D(A)}$ is
defined by
\begin{gather*}
D(A_{0})=\{y\in C^{2}([0,\pi]  ;\mathbb{R})
:y(0)=y(\pi)=y''(0)=y''(\pi)=0\},\\
A_{0}y=y''.
\end{gather*}


\begin{lemma} \cite{Ben-Bou-EzzJmaa}
$A_{0}$ generates a compact strongly continuous
semigroup $(T_{0}(t))_{t\geq 0}$ on $\overline{D(A)}$ such that
\[
|T_{0}(t)|\leq e^{-t}\;\quad\text{for }t\geq 0.
\]
\end{lemma}

 Therefore, the assumptions (H2) and (H4) hold. Consequently,
 for any $\varphi\in C$ such that
\[
\mathcal{D}\varphi\in\{y\in X:\,y(0)=y(\pi)=0\}\,.
\]
Equation (\ref{linear}) has a unique solution $u$ on
$[-r,+\infty)$. To establish the existence of a periodic
solution of (\ref{linear}), we suppose that
\begin{itemize}
\item[(H7)] $a_1 $ and $h_1 $ are $\omega$-periodic
in $t$.

\item[(H8)] There exists a positive constant $\beta\in(0,1)$
such that
$|a_1 |_{\infty}\leq(1-q)\beta$,
where $|a_1 |_{\infty}=\sup_{s\in\mathbb{R}}{|a_1 (s)|}$.
\end{itemize}

\begin{proposition} \label{prop5.3}
Assume that (H7) and (H8) hold. Then \eqref{linear} has an
$\omega$-periodic solution.
\end{proposition}

\begin{proof}
We will first show that \eqref{e0}
has a bounded solution on $\mathbb{R}^{+}$. Let $\rho=
\frac{1}{1+q}\big(1+\frac{|f|_{\infty}}{1-\beta}\big)$
and $\varphi\in C$ such that $|\varphi|<\rho$. Then
$|\varphi(0)-q\varphi(-r)|<(1+q)\rho$.
We claim that
\begin{equation}
|u(t)-qu(t-r)|\leq(1+q)\rho\quad\text{for all }t\geq 0. \label{etoi}
\end{equation}
We proceed by contradiction. Let $t_{0}$ be the first time such that
(\ref{etoi}) is not true. Then,
\[
t_{0}=\inf\{  t>0:|u(t)-qu(t-r)|>(1+q)\rho\}  .
\]
By continuity, one can see that
\[
|u(t_{0})-qu(t_{0}-r)|=(1+q)\rho,
\]
and there exists a positive constant $\varepsilon>0$ such that
\[
|u(t)-qu(t-r)|>(1+q)\rho\quad\text{for }t\in(t_{0},t_{0}+\varepsilon).
\]
Using the variation-of-constants formula (\ref{var const}), we get
that
\[
|u(t_{0})-qu(t_{0}-r)|\leq e^{-t_{0}}(
1+q)\rho+     \int_{0}^{t_{0}}e^{-(t_{0}-s)}
  [|a_1 |_{\infty}|u(s+\theta)|d\theta+|f|_{\infty}]  ds.
\]
Since $|u(t)-qu(t-r)|\leq(1+q)\rho$ for $t\leq t_{0}$, then
\[
|u(t)|\leq(1+q)\rho+q| u(t-r)|\quad\text{for }t\in[-r,t_{0}]  .
\]
$|\varphi|<\rho$, then we can see that
\[
|u(t)|\leq\frac{1+q}{1-q}\rho\quad\text{for }t\in[-r,t_{0}],
\]
and
\[
|u(t_{0})-qu(t_{0}-r)|\leq e^{-t_{0}}(1+q)\rho+(1-e^{-t_{0}})[
\frac{1+q}{1-q}|a_1 |_{\infty}\rho+|f|_{\infty}]  .
\]
Using hypotheses (H8), we obtain
\[
|u(t_{0})-qu(t_{0}-r)|\leq e^{-t_{0}}(
1+q)\rho+(1-e^{-t_{0}})(\beta(1+q)
\rho+|f|_{\infty}),
\]
consequently,
\begin{gather*}
|u(t_{0})-qu(t_{0}-r)|\leq(1+q)
\rho-(1-e^{-t_{0}})((1-\beta)(1+q)\rho-|f|_{\infty}),
\\
|u(t_{0})-qu(t_{0}-r)|\leq(
1+q)\rho-(1-e^{-t_{0}})(1-\beta)<(1+q)\rho.
\end{gather*}
By continuity, there exists a positive $\varepsilon_{0}$ such that
\[
|u(t)-qu(t-r)|<(1+q)\rho\quad\text{for }t\in(t_{0},t_{0}+\varepsilon_{0}),
\]
which gives a contradiction and we deduce that
\[
|u(t)-qu(t-r)|\leq(1+q)\rho\quad\text{for }t\geq 0.
\]
Let $t\in[0,r]$. Then
\[
|u(t)|\leq(1+q)\rho+q\rho \leq(1+q)(1+q)\rho,
\]
and for $t\in[ r,2r]$,
\[
|u(t)|\leq(1+q)(
1+q+q^{2})\rho.
\]
We proceed by steps, then for $t\in[(n-1)r,nr]$, we have
\[
|u(t)|\leq(1+q)(
1+q+q^{2}+\dots+q^{n})\rho.
\]
Consequently,
\[
|u(t)|\leq(1+q)\rho
\underset{n\geq 0}{\sum}q^{n}=\frac{1+q}{1-q}\rho\quad\text{for all }t\geq
0.
\]
Then, Equation (\ref{linear}) has a bounded solution $u$ on
$\mathbb{R}^{+}$.
By Theorem \ref{colat}, we deduce that Equation (\ref{linear}) has an
$\omega$-periodic solution.
\end{proof}

\subsection*{Nonlinear case}
 We consider the nonlinear equation
\begin{equation}
\begin{gathered}
\begin{aligned}
\frac{\partial}{\partial t}[u(t,x)-qu(t-r,x)]
&=\frac {\partial^{2}}{\partial x^{2}}[u(t,x)-qu(t-r,x)]  +a_2
(t)g_1 (u(t-r,x)) \\
&\quad +h_2 (t,x) \quad\text{for }t\geq 0,\; x\in[0,\pi],
\end{aligned}\\
[u(t,x)-qu(t-r,x)]_{x=0,\pi}=0\quad\text{for }t\geq 0,
\\
u(\theta,x)=\varphi_{0}(\theta,x)\quad\text{for }\theta\in[-r,0],\;
x\in[0,\pi],
\end{gathered}   \label{exanon}
\end{equation}
where $g_1 :\mathbb{R}\to\mathbb{R}$ is a Lipschitz continuous
function and $a_2 $,
$\varphi_{0}:[-r,0]\times[0,\pi]\to \mathbb{R}$ are continuous
functions and $0<q<1$.  We
define the function $F:\mathbb{R}\times C\to X$ by
\[
F(t,\varphi)(x)=a_2 (t)g_1 (\varphi(-r)(x))+h_2 (t,x)\quad
\text{for }t\in\mathbb{R},\; x\in[0,\pi],\; \varphi\in C.
\]
Then, (\ref{exanon}) takes the abstract form \eqref{e0}.

We assume that
\begin{itemize}
\item[(H9)] $a_2 $, $h_2 $ are $\omega$-periodic in
$t$.

\item[(H10)] $g_1 $ is bounded on $\mathbb{R}$.
\end{itemize}

\begin{proposition} \label{lem ultima}
 Assume that (H9) and (H10) hold. Then, the solutions of \eqref{e0}
 are ultimately bounded.
\end{proposition}

\begin{proof}
Since $0<q<1$, then the operator $\mathcal{D}$ is stable.
By Theorem \ref{thwu}, we deduce that there
exist positive constants $\overline{a},\ \overline{b}$ and
$\overline{c}$ such that
\begin{equation}
|u_{t}(.,\varphi)|\leq\overline{a}e^{-\overline{b}t}\big(
|\varphi|+\sup_{s\in[0,t]}|h(s)|\big)
+\overline{c}\sup_{s\in[\max\{0,t-r\},t]}|h(s)|,
\label{est wu expo}
\end{equation}
where
\[
h(t)=T_{0}(t)\mathcal{D}\varphi+\lim_{\lambda\to+\infty}\int_{0}
^{t}T_{0}(t-s)B_{\lambda}F(s,u_{s})ds\quad\text{for }t\geq 0.
\]
Using Assumption (H10) and the fact that
$|T_{0}(t)|\leq e^{-t}\quad\text{for }t\geq 0$, we obtain that there
exist positive constants $\widetilde{a}$ and $\widetilde{b}$, such that

\[
|h(t)|\leq\widetilde{a}e^{-t}|\varphi|+\widetilde{b}\quad\text{for }t\geq 0,
\]
which implies for $t>r$ that
\[
\label{enq2}\underset{s\in[ t-r,t]}{\sup}|h(s)|\leq\widetilde{a}
e^{r-t}|\varphi|+\widetilde{b}\quad\text{for }\varphi\in C.\newline
\]
Using the estimate \eqref{est wu expo}, we obtain
\[
|u_{t}(.,\varphi)|\leq ae^{-bt}|\varphi|+c\quad\text{for }t>r\,\;
\varphi\in C,
\]
for some positive constants $a$, $b$ and $c$. Consequently, there exists a
positive constant $\widetilde{K}$ such that
\[
\limsup_{t\to+\infty} |u(t,\varphi)|<\widetilde
{K}\quad\text{for }\varphi\in C,
\]
and we deduce that the solutions of \eqref{e0} are ultimately bounded.
\end{proof}

  Consequently by Theorem \ref{theo de perio nonline}, we
obtain the following result.

\begin{proposition}
Assume that (H9) and (H10) hold. Then \eqref{e0} has an
$\omega$-periodic solution.
\end{proposition}

\subsection*{Acknowledgments}
 The authors would like to thank the anonymous referee for his/her
 careful reading of the original version.

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\end{document}