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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2012 (2012), No. 47, 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}
\thanks{\copyright 2012 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2012/47\hfil Weighted pseudo-almost periodic solutions]
{Weighted pseudo-almost periodic solutions for some neutral partial
functional differential equations}

\author[M. Damak, K. Ezzinbi, L. Souden \hfil EJDE-2012/47\hfilneg]
{Mondher Damak, Khalil Ezzinbi, Lotfi Souden}  % in alphabetical order

\address{Mondher Damak \newline
Universit\'e de Sfax, Facult\'e des Sciences Sfax\\
Route de Soukra, Km 3.5, B.P. 802
3018 Sfax, Tunisie}
\email{Mondher.Damak@fss.rnu.tn}

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

\address{Lotfi Souden \newline
D\'epartement de Math\'ematiques, Facult\'e Des Sciences De Gafsa.
Cit\'e Zarroug 2121, Gafsa, Tunisie}
\email{ltfsdn@gmail.com}

\thanks{Submitted November 21, 2011. Published March 23, 2012.}
\subjclass[2000]{34K40, 35B15, 34C27, 34K14}
\keywords{Neutral equation; semigroup; mild solution; \hfill\break\indent 
Stepanov-weighted pseudo almost periodic functions;
 Banach fixed point theorem}

\begin{abstract}
 In this article we study the existence of weighted pseudo almost periodic 
 solutions of an autonomous neutral functional differential equation
 with Stepanov-Weighted pseudo almost periodic terms in a Banach space.
 We use the contraction mapping principle to show the existence
 and the uniqueness of  weighted pseudo almost periodic solution of 
 the equation.
\end{abstract}

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

\section{Introduction}

Let $(\mathbb{X},\| \cdot\|)$ be a Banach space and $p\geq 1$. The
concept of pseudo-almost periodicity, was introduced 
in the early nineties in \cite{Zhang,Zh} as a
natural generalization of the classical almost periodicity in the
sense of Bohr, the existence of pseudo-almost periodic solutions to
functional differential equations  has been of a great interest to
several authors. We refer the reader to \cite{Kha,Kl}. The concept
of weighted pseudo-almost periodic functions, which was introduced
by Diagana \cite{T. Diagana}, as a natural generalization of the classical
pseudo almost
 periodic functions. We refer to reader \cite{Kh} \cite{Khl}.
The concept of Stepanov pseudo almost periodic (or $S^p$-pseudo almost periodic) 
was introduced and studied, in the recent years in \cite{Diagana}. 
 In recent years Diagana \cite{T.Diagana}  study the
existence of pseudo-almost periodic solutions to some nonautonomous differential 
equations in the case when the semilinear
forcing term is both continuous and $S^p$-pseudo almost periodic for $p > 1$,
$$
\frac{d}{dt}u(t) = Au(t)+f(t,u(t)), \quad \text{for } t\in\mathbb{R}
$$
 Where $A$ is the infinitesimal generator  of a $C_0$-semigroup 
$\{T(t)\}_{t\geq 0}$, and $f:\mathbb{R}\times \mathbb{X}\to\mathbb{X}$
is Stepanov- pseudo almost periodic functions for $p>1$. 
In \cite{Ngue}, the author  introduced the concept of weighted pseudo-almost
periodicity in the sense of Stepanov, also called $S^p$-weighted pseudo-almost 
periodicity and study its properties.
and  present a result on the existence of weighted pseudo-almost periodic solutions to
the N-dimensional heat equation with $S^p$-weighted pseudo almost periodic 
coefficients of the form
$$
u'(t) = Au(t) + f (t, Bu(t))
$$
where $A : D(A) \subset \mathbb{X} \to \mathbb{X} $ is a
sectorial linear operator on a Banach space $\mathbb{X}$ whose
corresponding analytic semigroup $(T (t))_{t\geq 0}$ is hyperbolic
and $B$ is an arbitrary linear (possibly unbounded) operator on
$\mathbb{X}$, and $f$ is $S^p$-weighted pseudo almost periodic and
jointly continuous function. In \cite{MD}, the author studied the
existence of almost periodic solutions of an autonomous neutral
functional differential equation with Stepanov-almost periodic terms
in a Banach space of the form 
$$
\frac{d}{dt}[u(t)-F(t,u(t-g(t)))] =Au(t)+G(t,u(t),u(t-g(t)))
$$
 for $t\in\mathbb{R}$ and, $A$ is the
infinitesimal generator  of a $C_0$-semigroup $\{T(t)\}_{t\geq 0}$,
$F:\mathbb{R}\times\mathbb{X}\to\mathbb{X}$, and
$G:\mathbb{R}\times \mathbb{X}\times\mathbb{X}\to\mathbb{X}$ are
Stepanov almost periodic functions.

In this article,  we study the existence of  weighted pseudo almost periodic solutions
 of an autonomous neutral functional differential equation
 \begin{equation}\label{e1}
 \frac{d}{dt}[u(t)-F(t,u(t-r))] =
 A[u(t)-F(t,u(t-r))]+G(t,u(t),u(t-r)), 
 \end{equation}
for $t \in \mathbb{R}$,  Where $A$ is the infinitesimal generator  
of a $C_0$-semigroup 
$\{T(t)\}_{t \geq 0}$, and $F:\mathbb{R}\times\mathbb{X}\to\mathbb{X}$,
is Weighted pseudo almost periodic
and $G:\mathbb{R}\times \mathbb{X}\times\mathbb{X}\to\mathbb{X}$
is Stepanov-weighted pseudo almost periodic functions.

The rest of this article is organized as follows:
In Section 2, we introduce the basic notations and recall the
definitions and lemmas. In Section 3, we study the existence of weighted pseudo 
almost periodic solutions of \eqref{e1}. In section 4 we give
an example to illustrate our result.

\section{Preliminaries}

In this section we give some basic results that will be used in
the next. In the rest of this paper, $(\mathbb{X},\|\cdot\|)$
stands for a complex Banach space.

\begin{definition} \label{def2.1}\rm
A continuous function $f : \mathbb{R} \to \mathbb{X}$ is said to be  almost periodic  
if for every $\epsilon>0$ there exists a positive number $l$
such that every interval of length $l$ contains a number $\tau$ such that
$$
\|f(t+\tau)-f(t)\|<\epsilon, \quad  \text{for }  t\in \mathbb{R}.
$$
\end{definition}

Let $AP(\mathbb{R};\mathbb{X})$ be the set of all almost periodic
functions from $\mathbb{R}$ to $\mathbb{X}$.
Then $(AP(\mathbb{R};\mathbb{X}),\|\cdot\|_{\infty})$ is a Banach space
with supremum norm given by
$$
\|u\|_{\infty} = \sup_{t\in \mathbb{R}}\|u(t)\|.
$$

\begin{definition} \label{def2.1b}\rm
A continuous function $f : \mathbb{R} \times \mathbb{Y} \to \mathbb{X}$ 
is said to be  almost periodic in $t$ uniformly for $y \in \mathbb{Y}$, if
for every $\epsilon>0$ and any compact subset $K$ of $\mathbb{Y}$, 
there exists a positive number $l$  such that every interval of length $l$ contains a
number $\tau$ such that
$$
\|f(t+\tau,y)-f(t,y)\|<\epsilon, \quad  \text{for }  (t,y) \in
\mathbb{R}\times K.
$$
we denote the set of such functions as $AP(\mathbb{R}\times \mathbb{Y}; \mathbb{X})$.
\end{definition}

\begin{lemma}[\cite{Amir}] \label{prop2.7}
If $f\in AP(\mathbb{R}\times \mathbb{Y};\mathbb{X})$ and $h\in
AP(\mathbb{R};\mathbb{Y})$, then the function $f(.,h(.))\in
AP(\mathbb{R};\mathbb{X})$.
\end{lemma}

Now, let $\mathbb{U}$ be the collection of function (weights)
 $\rho :\mathbb{R} \to (0,\infty)$, which are locally integrable over $\mathbb{R}$
such that $\rho(x)>0$ almost everywhere. Set For $T>0$, 
\begin{gather*}
\operatorname{meas}(T,\rho):=\int_{-T}^T \rho(t) dt, \\
\mathbb{U}_{\infty};=\{\rho \in \mathbb{U}: \lim_{T \to
\infty} m(T,\rho)=\infty \text{ and }  \liminf_{t \in \mathbb{R}} \rho(t)>0 \},\\
\mathbb{U}_B:=\{\rho \in \mathbb{U}_{\infty}: \rho \text{ is  bounded} \}.
\end{gather*}
Obviously, $\mathbb{U}_B \subset \mathbb{U}_{\infty} \subset \mathbb{U}$, 
with strict inclusions.

For each $\rho \in \mathbb{U}_{\infty}$, define
$$ 
PAP_0(X,\rho)=\{\phi \in BC(\mathbb{R},\mathbb{X}):
  \lim_{T \to +\infty} \frac{1}{\operatorname{meas}(T,\rho)} 
\int_{-T}^{T} \|\phi(s)\|\rho(s)ds=0\}
$$
similarly, $PAP_0(\mathbb{R}\times \mathbb{Y};\mathbb{X},\rho)$ denote the collection 
of all function, $\phi: \mathbb{R} \times \mathbb{Y} \to \mathbb{X}$,
 jointly continuous, and $\phi(.,y)$, bounded for each $y \in \mathbb{Y}$, and
$$
\lim_{T \to +\infty} \frac{1}{\operatorname{meas}(T,\rho)} 
\int_{-T}^{T} \|\phi(s,y)\|\rho(s)ds=0
$$ 
uniformly for any $y$ in any compact subset of $\mathbb{Y}$.

\begin{definition} \label{def2.8}\rm
 Let $\rho \in \mathbb{U}_{\infty}$. A function $f \in BC(\mathbb{R}; \mathbb{X})$ 
is called weighted pseudo almost periodic or $\rho$-pseudo almost periodic if it can
 be expressed as $f=g+h$ where $g \in AP(\mathbb{R};\mathbb{X})$ and 
$h \in PAP_0(\mathbb{R};\mathbb{X},\rho)$. The collection of such function will be 
denoted by $PAP(\mathbb{R};\mathbb{X},\rho)$.
\end{definition}

\begin{definition} \label{def2.9}\rm
 Let $\rho \in \mathbb{U}_{\infty}$. 
A function $f \in BC(\mathbb{R} \times \mathbb{Y}; \mathbb{X})$ is called weighted
pseudo almost periodic or $\rho$-pseudo
 almost periodic if it can be expressed as $f=g+h$ where 
$g \in AP(\mathbb{R} \times \mathbb{Y}; \mathbb{X})$ and
  $h \in PAP_0(\mathbb{R} \times \mathbb{Y}; \mathbb{X},\rho)$. 
The collection of such function will be denoted by
$PAP(\mathbb{R} \times \mathbb{Y}; \mathbb{X},\rho)$
\end{definition}

\begin{definition} \label{def2.9b}\rm
Let $\rho \in \mathbb{U}_{\infty}$. A function 
$f \in BC(\mathbb{R} \times \mathbb{W} \times \mathbb{Y}; \mathbb{X})$ 
is called weighted pseudo almost periodic
 or $\rho$-pseudo almost periodic if it can be expressed as $f=g+h$ where 
$g \in AP(\mathbb{R}  \times \mathbb{W} \times \mathbb{Y}; \mathbb{X})$
 and $h \in PAP_0(\mathbb{R}  \times \mathbb{W} \times \mathbb{Y}; \mathbb{X},\rho)$. 
The collection of such function will be denoted by
$PAP(\mathbb{R}  \times \mathbb{W} \times \mathbb{Y}; \mathbb{X},\rho)$
\end{definition}

\begin{remark} \rm
(1)  The weight $\rho(t)=1$, is called pseudo-almost periodic functions.\\
(2) Clearly that $PAP_0(\mathbb{R};\mathbb{X},\rho)$ is a linear
 subspace of $BC(\mathbb{R}, \mathbb{X})$.\\
(3) Let $\rho \in \mathbb{U}_{\infty}$, and assume that
$$ 
\limsup_{t \to \infty}\frac{\rho(t+\tau)}{\rho(t)}<\infty, \quad
\limsup_{T \to \infty}\frac{\operatorname{meas}(T+\tau,\rho)}{\operatorname{meas}(T,\rho)}
<\infty
$$
for all $\tau \in \mathbb{R}$. 
In that case, the space $PAP(\mathbb{R};\mathbb{X}, \rho)$ is translation invariant. 
In this article, all weights $\rho \in \mathbb{U}_{\infty}$ 
for which $PAP(\mathbb{R};\mathbb{X}, \rho)$ is translation invariant will be 
denoted $\mathbb{U}_{\rm inv}$.
\end{remark}

\begin{theorem}[\cite{J. Liang}]
Fix $\rho \in U_{\rm inv}$, the decomposition of weighted pseudo almost
periodic function $f=g+h$, where $g \in AP(\mathbb{R};\mathbb{X})$
and $h \in PAP_0(\mathbb{R};\mathbb{X},\rho)$,  is unique.
\end{theorem}

\begin{theorem}[\cite{Diag}]
Fix $\rho \in U_{\rm inv}$, then the space
$(PAP(\mathbb{R};\mathbb{X},\rho),\|\cdot\|_{\infty})$ is a Banach
space.
\end{theorem}

\begin{lemma}[\cite{Xiaoxing Chen}] \label{prop65}
Let $\rho \in U_{\rm inv}$. If $f \in  PAP_0(\mathbb{R};\mathbb{X},
\rho)$, then for all $r \in \mathbb{R}$,
 $f(. - r) \in PAP_0(\mathbb{R};\mathbb{X}, \rho)$.
\end{lemma}

\begin{lemma}[\cite{T.Diagana}] \label{lemma23}
Let $\{f_n\}_{n \in \mathbb{N}} \subset PAP(\mathbb{R};\mathbb{X},\rho)$
 be a sequence of functions. If $f_n$ converges uniformly to some $f$, then
$f \in PAP(\mathbb{R};\mathbb{X},\rho)$.
\end{lemma}

\begin{theorem}[\cite{Tok}] \label{thm2.12}
Let $\rho \in \mathbb{U}_{\infty}$,  $F \in PAP(\mathbb{R}\times \mathbb{Y};\mathbb{X},\rho)$  and $h \in PAP(\mathbb{R};\mathbb{Y},\rho)$. Assume that
there exists  $L$  such that
$$
\|F(t,x)-F(t,y)\|\leq L\|x-y\|
$$
for all $t \in \mathbb{R}$ and for each $x, y \in \mathbb{X}$.
Then $F(.,h(.)) \in PAP(\mathbb{R};\mathbb{X},\rho)$
\end{theorem}

\begin{definition} \label{def2.10}\rm
 The Bochner transform $f^b(t,s), t \in \mathbb{R}, s \in [0,1]$, of a function
 $f(t)$ on $\mathbb{R}$, with value in $\mathbb{X}$, is defined by 
$$f^b(t,s)=f(t+s).$$
\end{definition}

\begin{definition} \label{def2.11}\rm
Let $1 \leq p <\infty$. The space $BS^p(\mathbb{R};\mathbb{X})$ of
all Stepanov bounded functions, with the exponent $p$, consists of
all measurable functions $f$ on $\mathbb{R}$ with value in
$\mathbb{X}$ such that $f^b \in L^{\infty}(\mathbb{R},
L^p(0,1);\mathbb{X})$. This is a Banach space with the norm
$$
\|f\|_{S^{p}} = \|f^b\|_{L^{\infty}(\mathbb{R}, L^p)}= \sup_{t\in\mathbb{R}}
\Big(\int_{t}^{t+1}\|f(s)\|^{p}ds\Big)^{1/p}<\infty.
$$
A function, $f\in L^{p}_{\rm loc}(\mathbb{R};\mathbb{X})$ is
$p$-Stepanov bounded ($S^{p}$-bounded) if $ \|f\|_{S^{p}}<\infty$.
It is obvious that $L^p(\mathbb{R}; \mathbb{X}) \subset
BS^p(\mathbb{R};\mathbb{X}) \subset L^{p}_{\rm loc}(\mathbb{R};\mathbb{X})$
\end{definition}

\begin{definition} \label{stepanovap}\rm
A function $f\in BS^p(\mathbb{R};\mathbb{X})$ is said to be
almost periodic in the sense of Stepanov ($S^{p}$-almost periodic)
if for every $\epsilon>0$ there exists a positive number $l$  such
that every interval of length $l$ contains a number $\tau$ such that
$$
\sup_{t\in\mathbb{R}}\Big(\int_{t}^{t+1}\|f(s+\tau)-f(s)\|^{p}ds
\Big)^{1/p}<\epsilon.
$$
\end{definition}

Let $S^{p}-AP(\mathbb{R};\mathbb{X})$ be the set of all
$S^{p}$-almost periodic functions.
It is clear that, if $f$ is almost periodic implies
$f$ is $S^{p}$-almost periodic; that is,
 $AP(\mathbb{R};\mathbb{X})\subset S^{p}-AP(\mathbb{R};\mathbb{X})$.
Moreover, if $1\leq m<p$, then $f(t)$ is $S^{p}$-almost periodic
implies $f(t)$ is $S^{m}$-almost periodic.

\begin{definition}[\cite{Ngue}] \rm
 Let $\rho \in \mathbb{U}_{\infty}$. A function $f \in BS^p$ is said to be a $S^p$
-weighted pseudo almost periodic (or Stepanov-
like weighted pseudo almost periodic) if it can be expressed as $f = g+ \phi$, where 
$g^b \in AP(L^p ((0, 1); \mathbb{X}))$ and $\phi^b \in PAP_0(L^p((0, 1); \mathbb{X}),\rho)$.
 The collection of such functions will be denoted by $PAP(\mathbb{X}, \rho, p)$ or 
$S^p-PAP(\mathbb{X},\rho)$.
 \end{definition}

 \begin{definition}[\cite{Ngue}] \rm
 Let $\rho \in \mathbb{U}_{\infty}$. A function $f:\mathbb{R}\times \mathbb{Y} \to \mathbb{X}$ with $f(.,x) \in L_{loc}^p(\mathbb{R},\mathbb{X})$
 for each $x \in \mathbb{X}$ is said to be $S^p$
-weighted pseudo almost periodic  if it can be expressed as $f = g+
\phi$, where $g^b \in AP(\mathbb{R} \times L^p (0, 1)); \mathbb{X})$
and $\phi^b \in PAP_0(\mathbb{R}\times L^p((0, 1));
\mathbb{X},\rho)$.
 The collection of such functions will be denoted by $PAP(\mathbb{X}, \rho, p)$ or $S^p-PAP(\mathbb{X},\rho)$.
  \end{definition}

 \begin{theorem}[\cite{Ngue}]
  Let $\rho \in \mathbb{U}_{\rm inv}$ and let $p \geq 1$, if $f \in  PAP(\mathbb{X}, \rho)$ then $f \in PAP(\mathbb{X}, \rho, p)$.
 \end{theorem}

 \begin{theorem}[\cite{Ngue}]
Let $\rho \in \mathbb{U}_{\infty}$ and let $f: \mathbb{R}\times
\mathbb{Y} \to \mathbb{X}$ be a
$S^{p}-PAP(\mathbb{R}\times\mathbb{Y};\mathbb{X},\rho))$, suppose that $F$ 
is lipschitz in $y \in \mathbb{Y}$ uniformly in $t \in \mathbb{R}$.\\
If $g\in PAP(\mathbb{R};\mathbb{Y},\rho)$  then $f(.,g(.))$ belong
to $S^{p}-PAP(\mathbb{R};\mathbb{X},\rho)$.
\end{theorem}

 It is clear that the space $(PAP(L^p((0, 1), \mathbb{X}),\rho),\|\cdot\|_{S^p})$ 
is a Banach space.
In other words, a function $f \in  L^{p}_{\rm loc}(\mathbb{R};\mathbb{X})$ is said 
to be $S^p$-weighted pseudo-almost periodic relatively to the weight
 $\rho \in  \mathbb{U}_{\infty},$
if its Bochner transform $f^b:\mathbb{R} \to L^p ((0, 1),
\mathbb{X})$ is weighted pseudo-almost periodic in the sense that
there exist two functions $g$ and $\phi$ for $\mathbb{R}$ into
$\mathbb{X}$ such that $g^b \in AP(L^p ((0, 1), \mathbb{X}))$ and
$\phi^b \in PAP_0(\mathbb{R},L^p((0, 1); \mathbb{X}),\rho)$, that is
$\phi^b \in BC(\mathbb{R},(L^p((0, 1); \mathbb{X}))$ and
$$
\lim_{T \to +\infty} \frac{1}{\operatorname{meas}(T,\rho)} 
\int_{-T}^{T} (\int_{t}^{t+1}\|\phi(s)\|^p ds)^{1/p}\rho(t)dt=0
$$

We define the set $S^{p}-PAP((\mathbb{R}\times
\mathbb{Y}; \mathbb{X}), \rho)$ which consists of all functions
$f:\mathbb{R}\times\mathbb{Y}\to\mathbb{X}$ such that $f(.,
y)\in S^p-PAP((\mathbb{R}\times \mathbb{Y};\mathbb{X}, \rho)$ uniformly for each $y\in
K$, where $K$ is any compact subset of $\mathbb{Y}$.

We define the set $S^p-PAP((\mathbb{R}\times
\mathbb{W}\times\mathbb{Y}; \mathbb{X}), \rho)$ which consists of all
functions
$f:\mathbb{R}\times\mathbb{W}\times\mathbb{Y}\to\mathbb{X}$
such that $f(., w,y)\in S^{p}-PAP(\mathbb{R};\mathbb{X}, \rho)$
uniformly for each $(w,y)\in K$, where $K$ is any compact subset
of $\mathbb{W}\times\mathbb{Y}$.

 \section{Main results}

 In this section we prove the existence and uniqueness of weighted pseudo almost
periodic mild solution for \eqref{e1}.
For the rest of this article,  we consider the following assumptions.
\begin{itemize}

\item[(H0)] Assume that $A$ is the infinitesimal generator of
an exponentially stable $c_0$-semigroup $\{T(t)\}_{t\geq 0}$ acting
on $\mathbb{X}$; that is, there exists constants $\omega>0$ and
$M\geq1$ such that
\begin{equation}
\|T(t)\|\leq Me^{-\omega t}\quad \text{for } t \in \mathbb{R} \label{e3.1}
\end{equation}
\item[(H1)] The function $F$ belong to $PAP(\mathbb{R}\times
\mathbb{X};\mathbb{X},\rho)$ satisfy the property that there exists
$L_F>0$ such that
$$
\|F(t,u)-F(t,v)\|\leq L_F\|u-v\|
$$
for all $t \in\mathbb{R}$ and for each $u,v\in
\mathbb{X}$.

\item[(H2)]  The function $G$ belong $S^{p}-PAP(\mathbb{R}\times
\mathbb{X} \times \mathbb{X};\mathbb{X},\rho)$ and satisfy the
followings property that there exists $L_G>0$ such that
$$
\|G(t,x_1,y_1)-G(t,x_2,y_2)\| \leq L_G(\|x_1-x_2\|+\|y_1-y_2\|)
$$
for all $t\in\mathbb{R}$ and for $(x_{1},y_1),(x_2,y_2)\in
\mathbb{X} \times \mathbb{X}$.
\end{itemize}

\begin{definition}  \label{prop1} \rm
Let $u: \mathbb{R} \to \mathbb{X}$ be an integral solution
of \eqref{e1}. Then for any $t \geq \sigma$ and any
$\sigma \in \mathbb{R}$,
\begin{equation}
\begin{split}
&u(t)-F(t,u(t-r))\\
&=T(t-\sigma)[u(\sigma)-F(\sigma,u(\sigma-r))]+\int_{\sigma}^tT(t-s)G(s,u(s),u(s-r))ds,
\quad t\in\mathbb{R}.   
\end{split} \label{sol1}
\end{equation}
\end{definition}

\begin{theorem}
Assume that {\rm (H0)--(H2)} hold. Let $u$ be a bounded integral solution 
of  \eqref{e1} on $\mathbb{R}$ then, for all $t \in \mathbb{R}$
\begin{equation}
u(t)=F(t,u(t-r))+ \int_{-\infty}^tT(t-s)G(s,u(s),u(s-r))ds.
\end{equation}
\end{theorem}

\begin{proof}
For any $\sigma \in \mathbb{R}$, we have for all
$t > \sigma$,
 $$
u(t)-F(t,u(t-r))=T(t-\sigma)[u(\sigma)-F(\sigma,u(\sigma-r))]
+\int_{\sigma}^tT(t-s)G(s,u(s),u(s-r))ds
$$
Since $u$ is bounded and $F$ is lipschitz continuous with respect the second argument, 
then there exists a constant $M$ such
that $\|u(t)\|\leq M$ for all $t \in \mathbb{R}$, we have
$$ 
\|T(t-\sigma)[u(\sigma)-F(\sigma,u(\sigma-r))]\|
\leq Me^{-\omega(t-\sigma)}(M+2ML_F+\|F(0,u(-r))\|) 
$$ 
and $\lim_{\sigma \to -\infty}T(t-\sigma)[u(\sigma)-F(\sigma,u(\sigma-r))]=0$.
Hence we have 
$$
u(t)=F(t,u(t-r))+ \int_{-\infty}^tT(t-s)G(s,u(s),u(s-r))ds,
\quad t\in\mathbb{R}.
$$
Conversely if $u$ belongs to $BC(\mathbb{R},\mathbb{X})$, it is easy
to see that the operator $\Gamma u(t)= F(t,u(t-r))+
\int_{-\infty}^tT(t-s)G(s,u(s),u(s-r))ds$ defined on
$BC(\mathbb{R},\mathbb{X})$ into itself, if  $u$ is given by
$u(t)=F(t,u(t-r))+ \int_{-\infty}^tT(t-s)G(s,u(s),u(s-r))ds, \quad
t\in\mathbb{R} $ then for any $t \geq \sigma$,
\begin{align*}
u(t)
&=F(t,u(t-r))+ \int_{-\infty}^{\sigma}T(t-s)G(s,u(s),u(s-r))ds \\
&\quad +
\int_{\sigma}^tT(t-s)G(s,u(s),u(s-r))ds\\
&=F(t,u(t-r))
+T(t-\sigma)[u(\sigma)-F(\sigma,u(\sigma-r))]\\
&\quad +\int_{\sigma}^tT(t-s)G(s,u(s),u(s-r))ds
\end{align*}
\end{proof}

\begin{theorem}
Assume that {\rm (H0)--(H2)} hold. If
$(L_F+\frac{2ML_G}{\omega}<1)$ then there exists a unique bounded
solution of \eqref{e1} on $\mathbb{R}$.
\end{theorem}

\begin{proof}
We consider $\Gamma: BC(\mathbb{R};\mathbb{X}) \to
BC(\mathbb{R};\mathbb{X})$ defined by, $\Gamma u(t)= F(t,u(t-r))+
\int_{-\infty}^tT(t-s)G(s,u(s),u(s-r))ds$. Let $u,v\in
BC(\mathbb{R};\mathbb{X})$. We observed that
\begin{align*}
&\|(\Gamma u)(t)-(\Gamma v)(t)\|\\
&\leq \|F(t,u(t-r))-F(t-v(t-r))\|\\
&\quad +\int_{-\infty}^t\|T(t-s)\|\|G(s,u(s),u(s-r))-G(s,v(s),v(s-r))\|ds\\
&\leq L_F\|u(t-r)-v(t-r)\|\\
&\quad + ML_G\int_{-\infty}^te^{-\omega(t-s)}(\|u(s)-v(s)\| +\|u-v\|_{\infty})ds\\
&\leq L_F\|u-v\|_{\infty}+2ML_G
\Big(\int_{-\infty}^te^{-\omega(t-s)}ds\Big)\|u-v\|_{\infty}\\
&\leq \big(L_F+\frac{2M}{\omega}L_G\big)\|u-v\|_{\infty}.
\end{align*}
Thus
\[
\|\Gamma u-\Gamma v)\|_{\infty}\leq \big(L_F+\frac{2M}{\omega}L_G\big) \|u-v\|_{\infty}.
\]
Thus $\Gamma$ is a contraction map on $BC(\mathbb{R};\mathbb{X})$.
Therefore,  $\Gamma$ has unique fixed
point in $BC(\mathbb{R};\mathbb{X})$, therefore the equation
\eqref{e1} has unique mild solution.
\end{proof}

\begin{theorem}[\cite{Tok}] \label{thm3.4}
Let $\rho \in \mathbb{U}_{\infty}$. 
If $G \in PAP^p(\mathbb{R}\times \mathbb{W}\times \mathbb{Y}; 
\mathbb{X},\rho)$  satisfies the
Lipschitz condition 
$$
\|G(t,x_1,y_1)-G(t,x_2,y_2) \|_X \leq L_G(\|x_1-x_2 \|_W+\|y_1-y_2 \|_Y)
$$ 
for all $t \in \mathbb{R}$ and $ x_1,x_2 \in \mathbb{W}$ and 
$y_1,y_2 \in \mathbb{Y}$. 
 If $h\in PAP(\mathbb{R}; \mathbb{Y},\rho)$ and 
$\phi \in PAP(\mathbb{R};\mathbb{W},\rho)$,
then $G(.,\phi(.), h(.))\in PAP^p(\mathbb{R};\mathbb{X},\rho)$.
\end{theorem}

We define two mappings $\Gamma$ and $\Lambda$ by
\begin{gather}
(\Gamma u)(t)= F(t,u(t-r))+\int_{-\infty}^tT(t-s)G(s,u(s),
u(s-r))ds,   \label{3.1}\\
(\Lambda f)(t)= \int_{-\infty}^tT(t-s)f(s)ds,\quad t\in\mathbb{R}.
\label{3.2}
\end{gather}


\begin{proposition} \label{prop3.5}
Let $\rho \in \mathbb{U}_{\rm inv}$. If $u \in
PAP(\mathbb{R};\mathbb{X},\rho)$ and $r \in \mathbb{R}$, then
$u(.-r)\in PAP(\mathbb{R};\mathbb{X},\rho)$.
\end{proposition}

\begin{proof}
We have $u(.)=x(.)+y(.)$, where $x(.) \in AP(\mathbb{R};\mathbb{X})$
and $y(.) \in PAP_0(\mathbb{R};\mathbb{X},\rho)$. It is easy to see
that $x(t-r)$ belong to $AP(\mathbb{R};\mathbb{X})$ and from lemma
\ref{prop65}, we have  $y(.-r)\in PAP_0(\mathbb{R};\mathbb{X},\rho)$
\end{proof}


\begin{lemma}  \label{lemma31}
Let $\rho \in \mathbb{U}_{\infty}$. If $f$ is an $S^{p}$-weighted
pseudo almost periodic function, then the function $(\Lambda f) \in
PAP(\mathbb{R};\mathbb{X},\rho)$.
\end{lemma}

\begin{proof}
Since $f \in S^pPAP(\mathbb{R};\mathbb{X},\rho)$, then $f=g+h$ where
$g \in S^p AP(\mathbb{R};\mathbb{X})$ and $h \in
S^pPAP_0(\mathbb{R};\mathbb{X},\rho)$. We consider
\[
(\Lambda g)(t)= \int_{-\infty}^tT(t-s)g(s)ds,\quad
(\Lambda h)(t)= \int_{-\infty}^tT(t-s)h(s)ds,\quad
t\in\mathbb{R}.
\]
The conjugate  of $p$ is denoted by  $q$; that is,
$\frac{1}{p} + \frac{1}{q} = 1$.
 We divide the proof onto several steps:

\noindent\textbf{Step 1:}  $\bullet$  If $p>1$ then $1<q<+\infty$.
we prove that  $(\Lambda g)(t) \in AP(\mathbb{R};\mathbb{X})$.
 we consider
$$
(\Lambda  g)_{n}(t)= \int_{t-n}^{t-n+1}T(t-s)g(s)ds,\quad
n\in\mathbb{N},\;t\in\mathbb{R}.
$$
Using the Holder inequality and the estimate \ref{e3.1}, it follows that
\begin{equation}
\begin{split}
\|(\Lambda g)_n(t)\|
&=\| \int_{t-n}^{t-n+1}T(t-s)g(s)ds \| \\
& \leq  \int_{t-n}^{t-n+1}\|T(t-s)\|\|g(s)\|ds\\
&\leq  M [\int_{t-n}^{t-n+1}e^{-q\omega(t-s)}ds]^{1/q}
[\int_{t-n}^{t-n+1}\|g(s)\|^pds]^{1/p}\\
& \leq \frac{M(e^{q \omega}-1)}{\sqrt[q]{q \omega}}e^{-n \omega} \|g\|_{S^p}.
\end{split}
\end{equation}
Using the assumption that $\sum_{n=1}^{\infty}e^{-\omega n}$ is convergent, 
we then deduce from the well-known Weirstrass theorem that the series
 $\sum_{n=1}^{\infty} (\Lambda g)_n(t)$ is uniformly convergent on $\mathbb{R}$, 
furthermore, $\sum_{n=1}^{\infty} (\Lambda g)_n(t)=(\Lambda g)(t)$ then
 $(\Lambda g)(.)$ is continuous and
$$
\|(\Lambda g)(t)\|\leq \sum_{n=1}^{\infty} \|(\Lambda g)_n(t)\| 
\leq \frac{M(e^{q \omega}-1)}{\sqrt[q]{q \omega}} \|g\|_{S^p}
 \sum_{n=1}^{\infty}e^{-\omega n}
\quad \text{for each } t \in \mathbb{R}
$$
$\bullet$  If $p=1$ then $q=\infty$. Using the  Holder inequality and the estimate 
\ref{e3.1}, we have 
\begin{equation}
\begin{split}
\|(\Lambda g)_n(t)\|
&=\| \int_{t-n}^{t-n+1}T(t-s)g(s)ds \| \\
& \leq  \int_{t-n}^{t-n+1}\|T(t-s)\|\|g(s)\|ds\\
&\leq  M \sup_{t-n\leq s \leq t-n+1}e^{-\omega(t-s)} \int_{t-n}^{t-n+1}\|g(s)\|ds\\
& \leq  M e^{-\omega(n-1)}\|g\|_{S^1}.
\end{split}
\end{equation}
Using the assumption that $\sum_{n=1}^{\infty}e^{-\omega (n-1)}$ is convergent,
 we then deduce from the well-known Weirstrass theorem that the series
 $\sum_{n=1}^{\infty} (\Lambda g)_n(t)$ is uniformly convergent on $\mathbb{R}$, 
furthermore $\sum_{n=1}^{\infty} (\Lambda g)_n(t)=(\Lambda g)(t)$ then
 $(\Lambda g)(.)$ is continuous and
$$ 
\|(\Lambda g)(t)\|\leq \sum_{n=1}^{\infty} \|(\Lambda g)_n(t)\| 
\leq M  \|g\|_{S^1} \sum_{n=1}^{\infty}e^{-\omega (n-1)}
\ \text{for each }  t \in \mathbb{R}
$$

\noindent\textbf{Step 2:}
$\bullet$  If $p>1$, We prove
that $(\Lambda g)(.) \in AP(\mathbb{R};\mathbb{X})$. Since $g \in
S^p AP(\mathbb{R};\mathbb{X})$, then for each $\epsilon > 0$ there exists
 $l(\epsilon) > 0$ such that every interval of length $l(\epsilon)$ contains 
a $\tau$ with the property
$$
\sup_{t \in \mathbb{R}}[\int_{t}^{t+1}\|g(s+\tau)-g(s)\|^pds^{1/p}]
<\epsilon_1 \epsilon
$$
 where $\epsilon_1= \sqrt[q]{q \omega}(e^{\omega}-1)/\big(M(e^{q \omega}-1)\big)$,
\begin{equation}
\begin{split}
&\|(\Lambda g)_n(t+\tau)-(\Lambda_1g)_n(t)\|\\
&=\| \int_{t+\tau-n}^{t+\tau-n+1}T(t+\tau-s)g(s)ds - \int_{t-n}^{t-n+1}T(t-s)g(s)ds\| \\
& \leq  \int_{t-n}^{t-n+1}\|T(t-s)\|\|g(s+\tau)-g(s)\|ds\\
&\leq  M [\int_{t-n}^{t-n+1}e^{-q\omega(t-s)}ds]^{1/q}[\int_{t-n}^{t-n+1}\|g(s+\tau)-g(s)\|^pds]^{1/p}\\
& \leq \frac{M(e^{q \omega}-1)}{\sqrt[q]{q \omega}}e^{-n \omega} \epsilon_1 \epsilon.
\end{split}
\end{equation}
Therefore,
$$\sum_{n=1}^{\infty}\|(\Lambda g)_n(t+\tau)-(\Lambda
g)_n(t)\| < \epsilon \epsilon_1 \frac{M(e^{q \omega}-1)}{\sqrt[q]{q
\omega}} \sum_{n=1}^{\infty}e^{-n \omega}=\epsilon ,
$$
hence
$ \sum_{n=1}^{\infty}(\Lambda g)_n(.) \in AP(\mathbb{R},\mathbb{X})$
for any $n \in \mathbb{N}$ and
$(\Lambda g)(.) \in AP(\mathbb{R};\mathbb{X})$.


$\bullet$  If $p=1$ then $q=\infty$. We prove
that $(\Lambda g)(.) \in AP(\mathbb{R};\mathbb{X})$. 
Since $g \in S^1 AP(\mathbb{R};\mathbb{X})$, then for each $\epsilon > 0$ there
exists  $l(\epsilon) > 0$ such that every interval of length $l(\epsilon)$ contains 
a $\tau$ with the property
$$
\sup_{t \in \mathbb{R}}\int_{t}^{t+1}\|g(s+\tau)-g(s)\|ds<\epsilon_1 \epsilon 
$$ 
where
$\epsilon_1= (1-e^{-\omega})/M$,
\begin{equation}
\begin{split}
&\|(\Lambda g)_n(t+\tau)-(\Lambda_1g)_n(t)\|\\
&=\| \int_{t+\tau-n}^{t+\tau-n+1}T(t+\tau-s)g(s)ds - \int_{t-n}^{t-n+1}T(t-s)g(s)ds\| \\
& \leq  \int_{t-n}^{t-n+1}\|T(t-s)\|\|g(s+\tau)-g(s)\|ds\\
&\leq  M \sup_{t-n \leq s \leq t-n+1}e^{\omega(t-s)} \int_{t-n}^{t-n+1}\|g(s+\tau)-g(s)\|ds\\
& \leq Me^{ -\omega (n-1)} \epsilon_1 \epsilon.
\end{split}
\end{equation}
Therefore,
$$ 
\sum_{n=1}^{\infty}\|(\Lambda g)_n(t+\tau)-(\Lambda g)_n(t)\| 
< \epsilon \epsilon_1 M \sum_{n=1}^{\infty}e^{ -\omega (n-1)}=\epsilon ,
$$ 
hence
$ \sum_{n=1}^{\infty}(\Lambda g)_n(.) \in AP(\mathbb{R},\mathbb{X})$
for any $n \in \mathbb{N}$ and
$(\Lambda g)(.) \in AP(\mathbb{R};\mathbb{X})$.

\noindent\textbf{Step 3:}
We show that $(\Lambda h)(.) \in PAP_0(\mathbb{R};\mathbb{X},\rho)$. 

$\bullet$  If $p>1$. Let $T>0$, $h \in S^pPAP_0(\mathbb{R};\mathbb{X},\rho)$ we have that
\begin{equation}
\lim_{T \to +\infty}\frac{1}{\operatorname{meas}(T,\rho)}
\int^T_{-T}(\int_s^{s+1} \| h(\sigma)\|^pd\sigma)^{1/p}\rho(s)ds=0
\end{equation}
First we prove that $\Lambda h \in BC(\mathbb{R};\mathbb{X})$.
 Indeed it is similar to previous works of $\Lambda g$. 
Next we prove that $\Lambda h \in PAP_0(\mathbb{R};\mathbb{X},\rho)$.
we consider 
$$
(\Lambda  h)_{n}(t)= \int_{t-n}^{t-n+1}T(t-s)h(s)ds,\quad n\in\mathbb{N},\;t\in\mathbb{R}.
$$
Using the Holder inequality and the estimate \ref{e3.1}, it follows that
\begin{equation}
\begin{split}
\|(\Lambda h)_n(t)\|
&=\| \int_{t-n}^{t-n+1}T(t-s)h(s)ds \| \\
& \leq  \int_{t-n}^{t-n+1}\|T(t-s)\|\|h(s)\|ds\\
&\leq  M [\int_{t-n}^{t-n+1}e^{-q\omega(t-s)}ds]^{1/q}[\int_{t-n}^{t-n+1}\|h(s)\|^pds]^{1/p}\\
& \leq \frac{M(e^{q \omega}-1)}{\sqrt[q]{q \omega}}e^{-n \omega} \|h\|_{S^p}.
\end{split}
\end{equation}
It follows that
\begin{align*}
&\frac{1}{\operatorname{meas}(T,\rho)} \int^T_{-T}\| (\Lambda h)_n(t)\|\rho(t)dt \\
&\leq \frac{M(e^{q \omega}-1)}{\sqrt[q]{q \omega}}e^{-n \omega}
 \frac{1}{\operatorname{meas}(T,\rho)}
\int^T_{-T}\|h(t)\|_{S^p}\rho(t)dt
\end{align*}
and 
\begin{align*}
&\lim_{T \to +\infty}\frac{1}{\operatorname{meas}(T,\rho)} 
\int^T_{-T}\| (\Lambda h)_n(t)\|\rho(t)dt\\
&\leq \frac{M(e^{q \omega}-1)}{\sqrt[q]{q \omega}}e^{-n \omega}.
\lim_{T \to +\infty}\frac{1}{\operatorname{meas}(T,\rho)} 
\int^T_{-T}\|h(t)\|_{S^p}\rho(t)dt=0
\end{align*}
hence $(\Lambda h)_n \in PAP_0(\mathbb{R},X,\rho)$. On the other hand, using the 
assumption that $ \sum_{n=1}^{\infty}e^{-\omega n}$ is convergent,
we then deduce from the well-known Weirstrass theorem that the series
 $\sum_{n=1}^{\infty} (\Lambda h)_n(t)$ is uniformly convergent on $\mathbb{R}$, 
furthermore $ \sum_{n=1}^{\infty} (\Lambda h)_n(t)=(\Lambda h)(t)$.
 Consequently $ \sum_{n=1}^{\infty} (\Lambda h)_n(t) \in PAP_0(\mathbb{R},X,\rho)$ 
and so $(\Lambda h)(t)$ from  lemma \ref{lemma23}.

$\bullet$  If $p=1$, Let $T>0$, $h \in S^pPAP_0(\mathbb{R};\mathbb{X},\rho)$ we have that
\begin{equation}
\lim_{T \to +\infty}\frac{1}{\operatorname{meas}(T,\rho)} \int^T_{-T}
\Big(\int_s^{s+1} \| h(\sigma)\|d\sigma\Big) \rho(s)ds=0
\end{equation}
Using the Holder inequality and the estimate \ref{e3.1}, it follows that
\begin{equation}
\begin{split}
\|(\Lambda h)_n(t)\|
&=\| \int_{t-n}^{t-n+1}T(t-s)h(s)ds \| \\
& \leq  \int_{t-n}^{t-n+1}\|T(t-s)\|\|h(s)\|ds\\
&\leq  M \sup_{t-n \leq s \leq t-n+1}e^{-\omega(t-s)}\int_{t-n}^{t-n+1}\|h(s)\|ds\\
& \leq Me^{-\omega(n-1)} \|h\|_{S^1}.
\end{split}
\end{equation}
We have
\begin{align*}
&\lim_{T \to +\infty} \frac{1}{\operatorname{meas}(T,\rho)} \int^T_{-T}\| 
(\Lambda h)_n(t)\|\rho(t)dt \\
&\leq   Me^{-\omega(n-1)}\lim_{T \to +\infty}\frac{1}{\operatorname{meas}(T,\rho)}
\int^T_{-T}\|h(t)\|_{S^1}\rho(t)dt=0
\end{align*}
hence $(\Lambda h)_n \in PAP_0(\mathbb{R},X,\rho)$. On the other hand, using the
 assumption that $ \sum_{n=1}^{\infty}e^{-\omega (n-1)}$ is convergent,
we then deduce from the well-known Weirstrass theorem that the series
 $\sum_{n=1}^{\infty} (\Lambda h)_n(t)$ is uniformly convergent on $\mathbb{R}$, 
furthermore $ \sum_{n=1}^{\infty} (\Lambda h)_n(t)=(\Lambda h)(t)$.
 Consequently $ \sum_{n=1}^{\infty} (\Lambda h)_n(t) \in PAP_0(\mathbb{R},X,\rho)$ and 
so $(\Lambda h)(t)$ from  lemma \ref{lemma23}.
\end{proof}

\begin{lemma} \label{lem3.7}
Let $\rho \in \mathbb{U}_{\rm inv}$. The operator $\Gamma u$ is weighted pseudo
almost periodic for $u$ is weighted pseudo almost periodic.
\end{lemma}

\begin{proof}
For $u(t)$ being weighted pseudo almost periodic, from Proposition \ref{prop3.5},
we see that $u(t-r)$ as weighted pseudo almost periodic, and
from (H1) and theorem \ref{thm2.12}, it is easy to see that
$F(t,u(t-r))$  belong to  $PAP(\mathbb{R};\mathbb{X},\rho)$. Now we
will show that $\int_{-\infty}^tT(t-s)G(s,u(s),u(s-r))ds$
beongs to $PAP(\mathbb{R};\mathbb{X},\rho)$, indeed
from Theorem \ref{thm3.4} and assumption (H2), it is easy to see
that $G(s,u(s),u(s-r))$  belongs to
$PAP^p(\mathbb{R};\mathbb{X},\rho)$.  The proof of lemma is completed
using the previous lemma.
\end{proof}

\begin{theorem} \label{thm35}
Let $\rho \in U_{\rm inv}$ and assume that {\rm (H0)--(H2)} hold.
If $(L_F+\frac{2M}{\omega}L_G)<1$. Then
\eqref{e1} has unique  weighted pseudo almost periodic mild solution.
\end{theorem}

\begin{proof}
Let $u,v\in PAP(\mathbb{R};\mathbb{X},\rho)$. We observed that
\begin{align*}
&\|(\Gamma u)(t)-(\Gamma v)(t)\|\\
&\leq \|F(t,u(t-r))-F(t-v(t-r))\|\\
&\quad +\int_{-\infty}^t\|T(t-s)\|\|G(s,u(s),u(s-r))-G(s,v(s),v(s-r))\|ds\\
&\leq L_F\|u(t-r)-v(t-r)\|\\
&\quad + ML_G\int_{-\infty}^te^{-\omega(t-s)}(\|u(s)-v(s)\| +\|u-v\|_{\infty})ds\\
&\leq L_F\|u-v\|_{\infty}+2ML_G
\Big(\int_{-\infty}^te^{-\omega(t-s)}ds\Big)\|u-v\|_{\infty}\\
&\leq \big(L_F+\frac{2M}{\omega}L_G\big)\|u-v\|_{\infty}.
\end{align*}
Thus
\[
\|\Gamma u-\Gamma
v)\|_{\infty}\leq \big(L_F+\frac{2M}{\omega}L_G\big)
\|u-v\|_{\infty}.
\]
Then $\Gamma$ is a contraction map on
$PAP(\mathbb{R};\mathbb{X},\rho)$. Therefore,
 $\Gamma$ has unique fixed
point in $PAP(\mathbb{R};\mathbb{X},\rho)$, that is, there exist unique
$u \in PAP(\mathbb{R};\mathbb{X},\rho)$ such that $\Gamma u=u$.
Therefore, \eqref{e1} has a unique weighted pseudo almost periodic
mild solution.
\end{proof}

\section{Application}

To illustrate the above results we examine the existence of weighted pseudo almost
 periodic solution to the differential equation 
\begin{equation}  \label{eqn2}
\begin{gathered}
\begin{aligned}
&\frac{d}{dt}[u(t,x)-F(t,u(t-r,x))]\\
&=\frac{d^2}{dx^2}[u(t,x)-F(t,u(t-r,x))]+ G(t,u(t,x),u(t-r,x)),
\quad t \in \mathbb{R}, \; x \in [0,\pi]
\end{aligned}\\
u(t,0)-F(t,u(t-r,0))= u(t,\pi)-F(t,u(t-r,\pi))=0, \quad t \in \mathbb{R}
\end{gathered}\
\end{equation}
Set $(\mathbb{X}, \|.\|)=(L^2[0,\pi], \|.\|_2)$, and define
\begin{gather*}
D(A)=\{u \in L^2[0,\pi], u'' \in L^2[0,\pi], u((0)=u(\pi)=0 \},\\
Au=\Delta u=u'' \ \ \ \text{for all } \ \ t \in \mathbb{R}
\end{gather*}
It is well known that $A$ is the infinitesimal generator of an exponentially 
stable $C_0$-semigroup $\{T(t)\}_{t\geq 0}$, with $M=\omega=1$ in 
\eqref{e3.1}.
Let $\rho(t) = 1 + t^ 2$. It can be easily shown that 
$\rho \in \mathbb{U}_{\rm inv}$. Let 
$F:\mathbb{R}\times \mathbb{X} \to \mathbb{X}$ defined by
$$
F(t,x)=\sin(t)+\sin(\sqrt{2}t)+\gamma e^{-|t|}\sin(u)
$$
it is checked that $F$ belong to $PAP(\mathbb{R}\times \mathbb{X},
\rho)$ and satisfy 
$$
\|F(t,u)-F(t,v)\|\leq \mid \gamma \mid \|u-v\|,
\quad \text{for all } t \in \mathbb{R} \text{ and }  u,v \in
\mathbb{X}
$$ 
Let $G:\mathbb{R}\times \mathbb{X} \times \mathbb{X} \to \mathbb{X}$ defined by
$$
G(t,u,v)=\cos(t)+\cos(\sqrt{2}t)+\theta(u,v)+\lambda e^{-|t|}
$$
Furthermore, it can be easily checked that
$g(t)=\cos(t)+\cos(\sqrt{2}t)+\lambda e^{-|t|}$ belong to 
$S^p-PAP(\mathbb{R}, \rho)$. If we suppose that the function
$\theta$ satisfying
$$
\|\theta(u,v)-\theta(u',v')\| \leq |\beta| (\| u-u'\|+\| v-v'\|) \quad
\text{for all } u,v,u',v' \in \mathbb{X}
$$ 
then  there exists $|\beta|>0$ such that
$$
\|G(t,u,v)-G(t,u',v')\| \leq |\beta| (\| u-u'\|+\|v-v'\|) \quad
\text{for all } \ u,u',v,v' \in \mathbb{X}
$$ 
Consequently all assumption (H0), (H1) and (H2) are
satisfied then by  theorem \ref{thm35}; we deduce the following
result.
In conclusion, under the above assumption, if 
$$
|\gamma|+2|\beta |<1,
$$ 
then \eqref{eqn2} has a unique  weighted pseudo almost periodic mild
solution on $\mathbb{R}$.

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\end{document}