\documentclass[reqno]{amsart}
\usepackage{hyperref}
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\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2013 (2013), No. 104, 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}
\thanks{\copyright 2013 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2013/104\hfil Existence of pseudo almost periodic solutions]
{Existence of pseudo almost periodic solutions for a class
of partial functional differential equations}

\author[H.-S. Ding, W. Long, G. M. N'Gu\'er\'ekata \hfil EJDE-2013/104\hfilneg]
{Hui-Sheng Ding, Wei Long, Gaston M. N'Gu\'er\'ekata}  

\address{Hui-Sheng Ding \newline
College of Mathematics and Information Science,
Jiangxi Normal University\\
Nanchang, Jiangxi 330022, China}
\email{dinghs@mail.ustc.edu.cn}

\address{Wei Long \newline
College of Mathematics and Information Science,
Jiangxi Normal University\\
Nanchang, Jiangxi 330022, China}
\email{hopelw@126.com}

\address{Gaston M. N'Gu\'er\'ekata \newline
Department of Mathematics, Morgan State University \\
1700 E. Cold Spring Lane, Baltimore, MD 21251, USA}
\email{nguerekata@aol.com, Gaston.N'Guerekata@morgan.edu}

\thanks{Submitted  March 28, 2013. Published April 24, 2013.}
\subjclass[2000]{34K14, 45G10}
\keywords{Pseudo almost periodic; 
 abstract functional differential equation; \hfill\break\indent 
almost periodic}

\begin{abstract}
 In this paper, we first introduce a new class of pseudo almost
 periodic type functions and investigate some properties of pseudo
 almost periodic type functions; and then we discuss the existence of
 pseudo almost periodic solutions to the class of abstract partial
 functional differential equations $x'(t)=Ax(t)+f(t,x_t)$ with finite
 delay in a Banach space $X$.
\end{abstract}

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

\section{Introduction}


Let $(X,\|\cdot\|))$ be a Banach space. The main goal in this paper
is to study the existence of pseudo almost periodic type
solutions to the  abstract partial functional differential
equation
\begin{equation}\label{equation}
x'(t)=Ax(t)+f(t,x_t),\quad t\in \mathbb{R}
\end{equation}
in $X$, where $A$ is the infinitesimal generator of an exponential
stable $C_0$-semigroup, $f$ is some class of pseudo almost periodic
type solution, and $x_t(s)=x(t+s)$, $s\in [-\delta,0]$ with
$\delta>0$ be a fixed constant.

The study of the existence of periodic type solutions, almost
periodic type solutions, pseudo almost periodic type solutions to
equation \eqref{equation} and its variants has been of great
interest for many authors. We refer the reader to
\cite{Cuevas09,Cuevas10,diagana07,diagana,Ezzinbi1,enriquez1,
enriquez2,enriquez3,Hern1,Hern2,Hern3}
and references therein for some of recent developments on these
topics. In addition, we would like to note that in a recent work
\cite{zheng-ding-gaston}, in the sense of category, the ``amount" of
almost periodic functions (not periodic) is far more than the
``amount" of continuous periodic functions. Thus, studying the
existence of almost periodic solutions for differential equations is
necessary.

The direct impetus of this paper comes from three sources. The first
source is a paper by Cuevas and Hern\'{a}ndez \cite{Cuevas09}, in
which the authors studied the existence and uniqueness of pseudo
almost periodic solutions for \eqref{equation} with a linear
part dominated by a Hill-Yosida type operator with a non-dense
domain. The second source is a paper by Diagana and Hern\'{a}ndez
\cite{diagana07}, in which the authors investigate the existence and
uniqueness of pseudo almost periodic solutions for the following
neutral functional-differential equation:
\begin{equation}\label{equation2}
\frac{d}{dt}[u(t)+f(t,u_t)]=Au(t)+g(t,u_t),\quad t\in \mathbb{R}.
\end{equation}
The third source is a paper by Diagana \cite{diagana}, in which the
author discuss the existence of almost automorphic solutions for 
\eqref{equation2} with $S^p$-almost automorphic coefficients.

Motivated by the above three papers, in this work, we discuss
the existence of pseudo almost periodic solutions for 
\eqref{equation} with $S^p$-pseudo almost periodic type
coefficients. Throughout the rest of this paper, if there is no
special statement, then $\mathbb{N}$ will be the set of positive integers, $\mathbb{Z}$
the set of integers, $\mathbb{R}$ the set of real numbers,
$BC(\mathbb{R},X)$ the Banach space of all bounded continuous
functions from $\mathbb{R}$ to $X$ with the supremum norm, and $Y$ the
Banach space  $C([-\delta,0];X)$ with the supremum norm.

\section{Pseudo almost periodic type functions}

In this section, we will recall several important and interesting
pseudo almost periodic type functions; and we will also introduce a
new class of pseudo almost periodic type functions. Throughout this
section, let $Z$ and $W$ be two arbitrary Banach spaces.

\subsection{Classical pseudo almost periodic functions}

First, let us recall some notions about almost periodic functions
and pseudo almost periodic functions (for more details, see
\cite{Dianaga-book}).

\begin{definition} \label{def2.1} \rm
 A set $E\subset \mathbb{R}$ is said to be relatively dense if
 there exists a number $l>0$ such that
$$
(a,a+l)\cap E\neq \emptyset
$$
for every $a\in\mathbb{R}$.
\end{definition}


\begin{definition} \label{def2.2} \rm 
A continuous function $f:\mathbb{R}\to Z$ is said to be almost periodic 
if for every $\varepsilon>0$ there exists a relatively dense set
$P(\varepsilon,f)\subset \mathbb{R}$ such that
$$
\sup_{t\in\mathbb{R}}\|f(t+\tau)-f(t)\|<\varepsilon,
$$
for every $\tau \in P(\varepsilon,f)$. We denote the set of all such
functions by $AP(\mathbb{R},Z)$ or $AP(Z)$.
\end{definition}

\begin{definition} \label{uap} \rm
A continuous function $f:\mathbb{R}\times W\to Z$  is called almost
periodic in $t$ uniformly for $x\in W$ if for every $\varepsilon>0$
and every compact subset $K\subset W$, there exists a relatively
dense set $P(\varepsilon,f,K)\subset \mathbb{R}$
$$
\sup_{t\in\mathbb{R}}\|f(t+\tau,x)-f(t,x)\|<\varepsilon, 
$$
for every $\tau \in P(\varepsilon,f,K)$ and $x\in K$. We denote by
$AP(\mathbb{R}\times W,Z)$ the set of all such functions.
\end{definition}

Denote
$$
PAP_{0}(Z):=\{f\in BC(\mathbb{R},Z):\lim_{r\to+\infty}\frac{1}{2r}
\int_{-r}^{r}\|f(t)\|dt=0\}.
$$

\begin{definition} \label{def2.4} \rm
A continuous function $f:\mathbb{R}\to Z$ is called pseudo almost
periodic if it can be expressed as $f=g+h$, where $g\in AP(Z)$ and
$h \in PAP_{0}(Z)$. The set of such functions will be denoted by
$PAP(Z)$.
\end{definition}

\begin{definition} \label{pap-2} \rm
A continuous function $f:\mathbb{R}\times W \to Z$ is called pseudo
almost periodic if
$$
f(t,x)=g(t,x)+h(t,x),\quad t\in \mathbb{R},\ x\in W,
$$ 
where $g\in AP(\mathbb{R}\times W,Z)$ and
for each $x\in W$, $h(\cdot,x) \in PAP_{0}(Z)$. We denote by
$PAP(\mathbb{R}\times W ,Z)$ the set of all such functions.
\end{definition}

\begin{remark}\rm
Note that Definition \ref{pap-2} is slightly
different from the notion used in many previous papers.
\end{remark}

\subsection{Stepanov-like pseudo almost periodic functions}
In this subsection, if there is no special statement, we assume
that $p\geq 1$. Next, let us recall some notions about Stepanov-like
pseudo almost periodicity (for more details, see
\cite{Dianaga08,gaston-pankov}).

\begin{definition} \label{def2.7} \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 values in $Z$, is defined by
$$ f^b(t,s):=f(t+s).$$
\end{definition}

\begin{definition} \label{def2.8} \rm
The space $BS^p(Z)$ of all Stepanov bounded functions, with the
exponent $p$, consists of all measurable functions $f$ on
$\mathbb{R}$ with values in $Z$ such that
$$
\|f\|_{S^p}:=\sup_{t\in\mathbb{R}}\Big(\int_t^{t+1}\|f(\tau)\|^p\,d\tau\Big)^{1/p}<+\infty.
$$
\end{definition}

It is obvious that $L^p(\mathbb{R};Z)\subset BS^p(Z) \subset L^p_{loc}(\mathbb{R};Z)$ and 
$BS^p(Z)\subset BS^q(Z)$ whenever $p\geq q\geq 1$.

\begin{definition} \label{def2.9} \rm
A function $f\in BS^p(Z)$ is said to be Stepanov almost periodic if
$f^b\in AP\big(L^p(0,1;Z)\big)$; that is, for every $\varepsilon>0$,
there exists a relatively dense set $P(\varepsilon,f)\subset \mathbb{R}$
such that
$$
\sup_{t\in\mathbb{R}}\Big(\int_0^1\|f(t+s+\tau)-f(t+s)\|^pds\Big)^{1/p}
<\varepsilon, 
$$
for every $\tau \in P(\varepsilon,f)$. We denote the set of all such
functions by $S^pAP(\mathbb{R},X)$ or $S^pAP(X)$.
\end{definition}

\begin{remark}  \rm
It is clear that $AP(X)\subset S^pAP(X) \subset S^qAP(X)$ for $p\geq
q\geq 1$.
\end{remark}


\begin{definition} \label{def2.11} \rm
A function $f:\mathbb{R}\times W \to Z$, $(t,u)\mapsto f(t,u)$ with
$f(\cdot,u)\in BS^p(Z)$ for every $u\in W$, is said to be Stepanov
almost periodic in $t\in \mathbb{R}$ uniformly for $u\in W$, if for every
$\varepsilon>0$ and every compact set $K\subset W$, there exists a
relatively dense set $P(\varepsilon,f,K)\subset \mathbb{R}$ such that
\begin{equation*}
\sup_{t\in\mathbb{R}}\Big(\int_0^1\|f(t+s+\tau,u)-f(t+s,u)\|^pds\Big)^{1/p}
<\varepsilon,
\end{equation*}
for every $\tau \in P(\varepsilon,f,K)$ and every $u\in K$. We
denote by $S^pAP(\mathbb{R}\times W , Z)$ the set of all such functions.
\end{definition}

It is also easy to show that $ S^pAP(\mathbb{R}\times W , Z) \subset
S^qAP(\mathbb{R}\times W , Z) $ for $p\geq q\geq 1$.

Next, we denote $S^pPAP_0(Z)$ be the set of all functions 
$f\in BS^p(Z)$ with $f^b\in PAP_{0}(L^p(0,1;Z))$; i.e.,
$$
\lim_{r\to+\infty}\frac{1}{2r} \int_{-r}^{r}
\Big(\int^1_0\|f(t+s)\|^pds\Big)^{1/p}dt=0.
$$


\begin{definition} \label{def2.12} \rm
A function $f\in BS^p(Z)$ is said to be Stepanov-like pseudo almost
periodic if it can be decomposed as $f=g+h$ with $g\in S^pAP(Z)$ and
$h\in S^pPAP_0( Z)$. We denote the set of all such functions by
$S^pPAP(\mathbb{R},Z)$ or $S^pPAP(Z)$.
\end{definition}

\begin{definition} \label{def2.13} \rm
A function $F:\mathbb{R}\times W \to Z, (t,u)\mapsto F(t,u)$ with
$F(\cdot,u)\in BS^p(Z)$ for each $u\in W$, is called Stepanov-like
pseudo almost periodic in $t\in \mathbb{R}$ uniformly for $u\in W$ if it can
be decomposed as $F=G+H$ with $G\in S^pAP(\mathbb{R}\times W , Z)$ and
$H(\cdot,u)\in S^pPAP_0(Z)$ for each $u\in W$. We denote by
$S^pPAP(\mathbb{R}\times W , Z)$ the set of all such functions.
\end{definition}


\subsection{Pseudo almost periodic type functions of class $\eta$}
In this subsection, if there is no special statement, we assume
that $\eta \geq 0$ and $p\geq 1$. First, let us recall the notion of
pseudo almost periodic type functions of class $\eta$ (for more
details, see \cite{diagana07}).
Denote 
$$ 
PAP_{0}(Z,\eta):=\{f\in
BC(\mathbb{R},Z):\lim_{r\to+\infty}\frac{1}{2r}
\int_{-r}^{r}\Big(\sup_{s\in [-\eta,0]}\|f(t+s)\|\Big)dt=0\}.
$$

\begin{definition} \label{def2.14} \rm
A function $f\in BC(\mathbb{R},Z)$ \ is called pseudo almost
periodic of class $\eta$ if it can be expressed as $f=g+h$, where
$g\in AP(Z)$ and $h \in PAP_{0}(Z,\eta)$. The set of such functions
will be denoted by $PAP(Z,\eta)$.
\end{definition}

\begin{definition} \label{def2.15} \rm
A continuous function $f:\mathbb{R}\times W \to Z$ is called pseudo
almost periodic of class $\eta$ if
$$
f(t,x)=g(t,x)+h(t,x),\quad t\in \mathbb{R},\ x\in W,
$$ 
where  $g\in AP(\mathbb{R}\times W,Z)$ and
for each $x\in W$, $h(\cdot,x) \in PAP_{0}(Z,\eta)$. We denote by
$PAP(\mathbb{R}\times W ,Z,\eta)$ the set of all such functions.
\end{definition}

Next, to study pseudo almost periodicity of equation
\eqref{equation}, we introduce a new class of pseudo almost periodic
type functions; i.e., \textit{Stepanov-like pseudo almost periodic
functions of class $\eta$}. We denote $S^pPAP_0(Z,\eta)$ be the set
of all functions $f\in BS^p(Z)$ with $f^b\in
PAP_{0}(L^p(0,1;Z),\eta)$; i.e.,
 $$
\lim_{r\to+\infty}\frac{1}{2r}
\int_{-r}^{r}\Big[\sup_{\theta\in
[-\eta,0]}\Big(\int^1_0\|f(t+\theta+s)\|^pds\Big)^{1/p}\Big]dt=0.
$$


\begin{definition} \label{def2.16} \rm
A function $f\in BS^p(Z)$ \ is called $S^p$-pseudo almost periodic
of class $\eta$ if it can be expressed as $f=g+h$, where $g\in
S^pAP(Z)$ and $h \in S^pPAP_{0}(Z,\eta)$. The set of such functions
will be denoted by $S^pPAP(Z,\eta)$.
\end{definition}


\begin{definition} \label{def2.17} \rm
A function $f:\mathbb{R}\times W \to Z$ is called $S^p$-pseudo
almost periodic of class $\eta$ if
$$
f(t,x)=g(t,x)+h(t,x),\quad t\in \mathbb{R},\ x\in W,
$$ 
where $g\in S^pAP(\mathbb{R}\times W,Z)$ and
for each $x\in W$, $h(\cdot,x) \in S^pPAP_{0}(Z,\eta)$. We denote by
$S^pPAP(\mathbb{R}\times W ,Z,\eta)$ the set of all such functions.
\end{definition}

\subsection{Some properties of pseudo almost periodic type functions}

In this subsection, we  investigate some properties and
relationships of the above pseudo almost periodic type functions.

\begin{lemma}\label{lem4}
Let $f\in BC(\mathbb{R},Z)$. Then $f\in PAP_0(Z)$ if and only if for every
$\varepsilon>0$,
$$
\lim_{r\to +\infty}\frac{\operatorname{meas} M_{r,\varepsilon}(f)}{2r}=0,
$$
where $M_{r,\varepsilon}(f):=\{t\in[-r,r]:\|f(t)\|\geq
\varepsilon\}$.
\end{lemma}

\begin{proof}
It can be deduced from some earlier results (e.g, see 
\cite[Lemma 3.2]{ding}).
\end{proof}

Define $PAP_0(\mathbb{R}^+):=\{f\in PAP_0(\mathbb{R}):f(t)\geq 0,\, \forall t\in\mathbb{R}\}$.

\begin{lemma}\label{lem5}
Let $\alpha>0$. Then $f\in PAP_0(\mathbb{R}^+)$ if and only if 
$f^{\alpha}\in PAP_0(\mathbb{R}^+)$, where $f^{\alpha}(t):=[f(t)]^{\alpha}$.
\end{lemma}

\begin{proof}
By using Lemma \ref{lem4}, we have $f\in PAP_0(\mathbb{R}^+)$ if
and only if for every  $\varepsilon>0$,
\[
\lim_{r\to +\infty}\frac{\operatorname{meas}\{t\in[-r,r]:f(t)\geq
\varepsilon\}}{2r}=0,
\]
which is equivalent to:  for every $\varepsilon>0$,
\[
\lim_{r\to +\infty}\frac{\operatorname{meas} \{t\in[-r,r]:f^{\alpha}(t)\geq
\varepsilon\}}{2r}=0;
\] 
i.e., $f^{\alpha}\in PAP_0(\mathbb{R}^+)$.
\end{proof}


\begin{lemma}\label{lem6}
Let $p\geq 1$. Then
$$
PAP_0(Z)\subsetneqq  S^pPAP_0(Z),
$$
and thus
$$
PAP(Z)\subsetneqq  S^pPAP(Z) .
$$
\end{lemma}

\begin{proof}
It suffices to show that $PAP_0(Z)\subsetneqq  S^pPAP_0(Z) $. 
First, let us show that $PAP_0(Z)\subset  S^pPAP_0(Z)$. 
Note that the proof of $PAP_0(Z)\subset  S^pPAP_0(Z)$ is given
in \cite{diagana07}. But, here we give a different proof.

Let $f\in PAP_0(Z) $. Then $\|f(\cdot)\|\in PAP_0(\mathbb{R}) $.
 By Lemma \ref{lem5}, $\|f(\cdot)\|^p\in PAP_0(\mathbb{R}) $. 
Then, by using Lebesgue's dominated convergence theorem, 
noting that $\|f(\cdot+s)\|^p\in PAP_0(\mathbb{R}) $ for each $s\in [0,1]$, we obtain
$$
\int^1_0\Big[\frac{1}{2r}\int^r_{-r}\|f(t+s)\|^pdt\Big]ds\to 0, \quad 
r\to +\infty;
$$
i.e.,
$$
\frac{1}{2r}\int^r_{-r}\Big[\int^1_0\|f(t+s)\|^pds\Big]dt\to 0, \quad 
r\to +\infty,
$$
which means that $g\in PAP_0(\mathbb{R})$, where
$$
g(t)=\int^1_0\|f(t+s)\|^pds,\quad t\in\mathbb{R}.
$$
Again by Lemma \ref{lem5}, $g^{1/p}\in PAP_0(\mathbb{R})$; i.e.,
$$
\frac{1}{2r}\int^r_{-r}\Big[\int^1_0\|f(t+s)\|^pds\Big]^{1/p}dt\to 0, \quad 
r\to +\infty,
$$
which means that $f\in S^pPAP_0(Z)$.
In addition, it is easy to see that $PAP_0(Z)\neq  S^pPAP_0(Z)$.
\end{proof}


\begin{lemma}\label{lem7}
Let $p\geq 1$. Then
$$
PAP_0(Z,1)\subsetneqq  S^pPAP_0(Z,1),
$$
and thus
$$
PAP(Z,1)\subsetneqq  S^pPAP(Z,1) .
$$
\end{lemma}

\begin{proof}
It suffices to show that $PAP_0(Z,1)\subset S^pPAP_0(Z,1)$. Let 
$f\in PAP_0(Z,1)$. Then $g\in  PAP_0(\mathbb{R})$, where
$$
g(t)=\sup_{\theta\in [-1,0]}\|f(t+\theta)\|,\quad t\in\mathbb{R}.
$$
It follows from Lemma \ref{lem6} that $g\in  S^pPAP_0(\mathbb{R})$. Thus, we
have
\begin{align*}
&\frac{1}{2r}\int_{-r}^{r}\Big[\sup_{\theta\in
[-1,0]}\Big(\int^1_0\|f(t+\theta+s)\|^pds\Big)^{1/p}\Big]dt\\
&\leq \frac{1}{2r}\int_{-r}^{r}\Big[\int^1_0\Big(\sup_{\theta\in
[-1,0]}\|f(t+\theta+s)\|\Big)^pds\Big]^{1/p}dt\\
&\leq \frac{1}{2r}\int_{-r}^{r}\Big[\int^1_0|g(t+s)|^pds\Big]^{1/p}dt\to
0,\quad r\to +\infty,
\end{align*}
which means that $f\in S^pPAP_0(Z,1)$.
\end{proof}

Next, we present an example of function, which belongs to
$S^pPAP_0(\mathbb{R},1)$.

\begin{example}\label{example-SpPAP0}\rm
Let $f$ be defined on $\mathbb{R}$ by
$$
f(t)=\begin{cases}
1 & t\in [2^n-1,2^n],\; n\in\mathbb{N},\\
0 & \text{otherwise}.
\end{cases}
$$
Then
$$
g(t):=\sup_{\theta\in [-1,0]}|f(t+\theta)|
=\begin{cases}
1 & t\in [2^n-1,2^n+1],\ n\in\mathbb{N},\\
0 & \text{otherwise}.
\end{cases}
$$
For each $r>0$, denote $n_r=[\log_2 r]+1$. Then
$$
2^{n_r-1}\leq r\leq 2^{n_r}.
$$
By a direct calculation, we obtain
\begin{align*}
&\frac{1}{2r}\int_{-r}^{r}\Big[\int^1_0|g(t+s)|^pds\Big]^{1/p}dt\\
&\leq \frac{1}{2r}\int_{-2^{n_r}}^{2^{n_r}}
\Big[\int^1_0|g(t+s)|^pds\Big]^{1/p}dt\\
&\leq \frac{3n_r}{2r}\to 0, \quad r\to +\infty.
\end{align*}
Then, it follows from the proof of Lemma \ref{lem7} that 
$f\in S^pPAP_0(\mathbb{R},1)$. In addition, it is obvious that 
$f \notin PAP_0(\mathbb{R},1)$.
\end{example}


\begin{lemma}
Let $\eta>0$ and $p\geq 1$. The following properties hold:
\begin{itemize}
\item[(a)] $PAP_{0}(Z,\eta)$ is
translation invariant; i.e., for each $\alpha \in\mathbb{R}$, $f\in
PAP_{0}(Z,\eta)$ implies that $f_{\alpha}\in PAP_{0}(Z,\eta)$, where
$f_{\alpha}(t)=f(t+\alpha)$, $t\in \mathbb{R}$;

\item[(b)] $PAP_{0}(Z,\eta)=PAP_{0}(Z,1)$;

\item[(c)] $S^pPAP_{0}(Z,\eta)$ is translation invariant, 
$S^pPAP_{0}(Z,\eta)=S^pPAP_{0}(Z,1)$;

\item[(d)] $S^pPAP(Z,\eta)$ is translation invariant, 
$S^pPAP(Z,\eta)=S^pPAP_{0}(Z,1)$;

\item[(e)] $S^pPAP_{0}(Z,1)$ is a closed linear subspace of $BS^p(Z)$.
\end{itemize}
\end{lemma}

\begin{proof}
Let $\alpha \in\mathbb{R}$ and $f\in PAP_{0}(Z,\eta)$. Noting that
\begin{align*}\frac{1}{2r}
\int_{-r}^{r}\Big(\sup_{s\in
[-\eta,0]}\|f_{\alpha}(t+s)\|\Big)dt
\leq \frac{1}{2r} \int_{-r-\alpha}^{r+\alpha}
\Big(\sup_{s\in [-\eta,0]}\|f(t+s)\|\Big)dt,
\end{align*}
we know that $f_{\alpha}\in PAP_{0}(Z,\eta)$.

To prove (b), it suffices to show that 
$PAP_{0}(Z,1)\subset PAP_{0}(Z,n)$ for each $n\in\mathbb{N}$. Fix $n\in\mathbb{N}$ 
and $f\in PAP_{0}(Z,1)$. In view of (a) and
\begin{align*}
&\frac{1}{2r}
\int_{-r}^{r}\Big(\sup_{s\in [-n,0]}\|f(t+s)\|\Big)dt\\
&\leq \sum_{k=1}^n \frac{1}{2r} \int_{-r}^{r}\Big(\sup_{s\in
[-k,-k+1]}\|f(t+s)\|\Big)dt\\
&= \sum_{k=1}^n \frac{1}{2r} \int_{-r}^{r}\Big(\sup_{s\in
[-1,0]}\|f_{1-k}(t+s)\|\Big)dt,
\end{align*}
we conclude that $f\in PAP_{0}(Z,n)$.

By noting the definition of $S^pPAP_{0}(Z,\eta)$, one can deduce (c)
from (a) and (b). In addition, (d) follows from (c) and translation
invariance of $S^pAP(Z)$.

It remains to show (e). Let $f_n\to f$ in $BS^p(Z)$ and $f_n\in
S^pPAP_{0}(Z,1)$. Then
\begin{align*}
&\frac{1}{2r} \int_{-r}^{r}\Big[\sup_{\theta\in
[-1,0]}\Big(\int^1_0\|f(t+\theta+s)\|^pds\Big)^{1/p}\Big]dt\\
&=\frac{1}{2r} \int_{-r}^{r}\Big[\sup_{\theta\in
[-1,0]}\|f(t+\theta+\cdot)\|_{L^p(0,1;Z)}\Big]dt\\
&\leq \frac{1}{2r} \int_{-r}^{r}\Big[\sup_{\theta\in
[-1,0]}\|f_n(t+\theta+\cdot)\|_{L^p(0,1;Z)}\Big]dt\\
&\quad +\frac{1}{2r} \int_{-r}^{r}\Big[\sup_{\theta\in
[-1,0]}\|f_n(t+\theta+\cdot)-f(t+\theta+\cdot)\|_{L^p(0,1;Z)}\Big]dt
\\
&\leq \frac{1}{2r} \int_{-r}^{r}\Big[\sup_{\theta\in
[-1,0]}\|f_n(t+\theta+\cdot)\|_{L^p(0,1;Z)}\Big]dt
+\|f_n-f\|_{BS^p(Z)},
\end{align*}
which yields  $f\in S^pPAP_{0}(Z,1)$. In addition, it is easy to
show that $S^pPAP_{0}(Z,1)$ is a linear subspace of $BS^p(Z)$. This
completes the proof.
\end{proof}

\begin{remark}\rm
It has been noted in \cite{diagana07} that
$PAP_0(Z,1)$ is a closed subspace of $BC(\mathbb{R},Z)$ and $PAP(Z,1)$ is a
Banach space under the supremum norm.
\end{remark}

\section{Existence of pseudo almost periodic solutions}

In this section, we discuss the existence of pseudo almost
periodic solutions to equation \eqref{equation}. For convenience, we
first list some assumptions:
\begin{itemize}
\item[(A1)] $A$ generates a $C_0$-semigroup
$T(t)$ in $X$ satisfying $\|T(t)\|\leq M e^{-\omega t}$ for all
$t\geq 0$, where $M,\omega>0$ are fixed constants.

\item[(A2)] $f\in S^1PAP(\mathbb{R}\times
Y,X,1)$ with $f=g+h$, where $g\in S^1AP(\mathbb{R}\times Y,X)$ and $h\in
S^1PAP_0(\mathbb{R}\times Y,X,1)$; Moreover, there exists a constant $L>0$
such that for all $t\in\mathbb{R}$ and $u,v\in Y$, $$\|f(t,u)-f(t,v)\|\leq
L\|u-v\|,\quad \|g(t,u)-g(t,v)\|\leq L\|u-v\|.$$
\end{itemize}

If (A1) holds, one can define the mild solution of equation
\eqref{equation} as follows:

\begin{definition} \label{def3.1} \rm 
A continuous function $u:\mathbb{R}\to X$ is called a mild solution of 
 \eqref{equation} if
$$
u(t)=T(t-s)u(s)+\int^t_s T(t-\tau)f(\tau,u_{\tau})d\tau,\quad t\geq s.
$$
\end{definition}

Before establishing our main results, we need to prove two important
lemmas.

\begin{lemma}\label{composition}
Assume that $x\in PAP(X,1)$ and (A2) holds. Then $t\mapsto
f(t,x_t)$ belongs to $S^1PAP(X,1)$.
\end{lemma}

\begin{proof}
Let $x=y+z$, where $y \in AP(X)$, $z\in PAP_0(X,1)$. It follows from the 
proof of \cite[Theorem 3.3]{diagana07} that $t\mapsto y_t$ belongs 
to $AP(Y)$, $t\mapsto z_t$ belongs to $PAP_0(Y,1)$, and 
$x_t=y_t+z_t,\ t\in\mathbb{R}$. Let
$$
I(t):=g(t,y_t),\quad J(t):=f(t,x_t)-f(t,y_t),\quad H(t):=h(t,y_t).
$$
Then $f(t,x_t)=I(t)+J(t)+H(t)$. Next, we give the proof in three
steps.
\smallskip

\noindent\textbf{Step 1.} $I\in S^1AP(X)$.

First, it is easy to prove that $I\in BS^1(X)$. Let
$K=\overline{\{y_t:t\in\mathbb{R}\}}$. Then $K$ is a compact subset of $Y$.
Thus, for each $\varepsilon>0$, there exists $u_1,\ldots,u_k\in K$
such that
$$
K \subset \cup_{i=1}^k B(u_i,\varepsilon),
$$
where $k$ is a positive integer dependent on $\varepsilon$. Denote
$$
i(t)=\min\{i=1,2,\ldots,k: \|y_t-u_i\|<\varepsilon\},\quad t\in\mathbb{R}.
$$
In addition, since $t\mapsto y_t$ belongs to $AP(Y)$ and
$g(\cdot,u_i)\in S^1AP(X)$, $i=1,2,\ldots,k$, for the above
$\varepsilon>0$, there exists a relatively dense set
$P(\varepsilon)\subset \mathbb{R}$ such that
\begin{equation}\label{001}
\int^1_0 \|g(t+s+\tau,u_i)-g(t+s,u_i)\|ds
<\frac{\varepsilon}{k},\quad i=1,2,\ldots,k,
\end{equation} 
and
\begin{equation}\label{002}
\|y_{t+\tau}-y_{t}\|<\varepsilon
\end{equation}
for all $\tau \in P(\varepsilon)$ and $t\in\mathbb{R}$. Now, by using
\eqref{001}, \eqref{002}, and $\|g(t,u)-g(t,v)\|\leq L\|u-v\|$, we
obtain
\begin{align*}
& \int^1_0 \|I(t+s+\tau)-I(t+s)\|ds\\
&= \int^1_0 \|g(t+s+\tau,y_{t+\tau+s})-g(t+s,y_{t+s})\|ds\\
&\leq  \int^1_0 \|g(t+s+\tau,y_{t+\tau+s})-g(t+s+\tau,y_{t+s})\|ds\\
&\quad +\int^1_0 \|g(t+s+\tau,y_{t+s})-g(t+s,y_{t+s})\|ds\\
&\leq  L\varepsilon+\int^1_0
\|g(t+s+\tau,y_{t+s})-g(t+s,y_{t+s})\|ds\\
&\leq   L\varepsilon+\int^1_0
\|g(t+s+\tau,y_{t+s})-g(t+s+\tau,u_{i(t+s)})\|ds\\
&\quad +\int^1_0 \|g(t+s+\tau,u_{i(t+s)})-g(t+s,u_{i(t+s)})\|ds\\
&\quad +\int^1_0 \|g(t+s,u_{i(t+s)})-g(t+s,y_{t+s})\|ds\\
&\leq  3L\varepsilon+\int^1_0
\sum_{i=1}^k\|g(t+s+\tau,u_i)-g(t+s,u_i)\|ds\\
&=  3L\varepsilon+\sum_{i=1}^k\int^1_0
\|g(t+s+\tau,u_i)-g(t+s,u_i)\|ds\leq (3L+1)\varepsilon
\end{align*}
for all $\tau \in P(\varepsilon)$ and $t\in\mathbb{R}$.
\smallskip

\noindent\textbf{Step 2.} $J\in S^1PAP_0(X,1)$.
Noting that
\begin{align*}
\|J(t)\|=\|f(t,x_t)-f(t,y_t)\|\leq L\|z_t\|,
\end{align*}
we conclude that $J$ is bounded, and thus $J\in BS^1(X)$. On the
other hand, we have
\begin{align*}
& \frac{1}{2r}\int^r_{-r}\sup_{\theta\in
[-1,0]}\|J^b(t+\theta)\|dt\\
&=  \frac{1}{2r}\int^r_{-r}\Big[\sup_{\theta\in
[-1,0]}\int^1_0\|J(t+\theta+s)\|ds\Big]dt\\
&=  \frac{L}{2r}\int^r_{-r}\Big[\sup_{\theta\in
[-1,0]}\int^1_0\|z_{t+\theta+s}\|ds\Big]dt\\
&=  \frac{L}{2r}\int^r_{-r}\Big[\sup_{\theta\in
[-1,0]}\int^1_0\sup_{\alpha\in[-\delta,0]}\|z(t+\theta+s+\alpha)\|ds\Big]dt\\
&\leq  \frac{L}{2r}\int^r_{-r}
\Big[\int^1_0\sup_{\theta\in[-1-\delta,0]}\|z(t+\theta+s)\|ds\Big]dt\\
&=  L\cdot \int^1_0\Big[\frac{1}{2r}\int^r_{-r}
\sup_{\theta\in[-1-\delta,0]}\|z(t+\theta+s)\|dt \Big]ds,
\end{align*}
combining this with the fact that 
$z(\cdot+s)\in PAP_0(X,1)=PAP_0(X,1+\delta)$ for each $s\in [0,1]$ and $z$ is
bounded, by using the Lebesgue's dominated convergence theorem, we
obtain that
$$
\lim_{r\to +\infty}\frac{1}{2r}\int^r_{-r}\sup_{\theta\in
[-1,0]}\|J^b(t+\theta)\|dt=0.
$$

\noindent\textbf{Step 3.} $H\in S^1PAP_0(X,1)$.
Take an arbitrary $\varepsilon>0$. Let $K$ and $u_i$,
$i=1,2,\ldots,k,$ be as in the proof of Step 1. Then, for all 
$t\in \mathbb{R}$, there holds
\begin{align*}
\|H(t)\|=\|h(t,y_t)\|
&= \|h(t,y_t)-h(t,u_{i(t)})\|+\|h(t,u_{i(t)})\|\\
&\leq  2L\varepsilon+\sum_{i=1}^k \|h(t,u_{i})\|,
\end{align*}
which yields 
\begin{equation} \label{1}
\begin{aligned}
& \frac{1}{2r}\int^r_{-r}\sup_{\theta\in
[-1,0]}\|H^b(t+\theta)\|dt \\
&=  \frac{1}{2r}\int^r_{-r}\Big[\sup_{\theta\in
[-1,0]}\int^1_0\|H(t+\theta+s)\|ds\Big]dt \\
&\leq  2L\varepsilon+\frac{1}{2r}\int^r_{-r}\Big[\sup_{\theta\in
[-1,0]}\int^1_0\sum_{i=1}^k \|h(t+\theta+s,u_{i})\|ds\Big]dt \\
&\leq 2L\varepsilon+\sum_{i=1}^k\frac{1}{2r}\int^r_{-r}\Big[\sup_{\theta\in
[-1,0]}\int^1_0 \|h(t+\theta+s,u_{i})\|ds\Big]dt.
\end{aligned}
\end{equation}
On the other hand, since $h\in S^1PAP_0(\mathbb{R}\times Y,X,1)$,
$h(\cdot,u_i)\in S^1PAP_0(X,1)$; i.e.,
$$
\lim_{r\to +\infty}\frac{1}{2r}\int^r_{-r}\Big[\sup_{\theta\in
[-1,0]}\int^1_0 \|h(t+\theta+s,u_{i})\|ds\Big]dt=0,
$$
which and \eqref{1} yields
$$
\limsup_{r\to +\infty}\frac{1}{2r}\int^r_{-r}\sup_{\theta\in
[-1,0]}\|H^b(t+\theta)\|dt\leq 2L\varepsilon.
$$
Then, by the arbitrariness of $\varepsilon$, we conclude that
$$
\lim_{r\to +\infty}\frac{1}{2r}\int^r_{-r}\sup_{\theta\in
[-1,0]}\|H^b(t+\theta)\|dt=0.
$$
In addition, it is also easy to see
that $H\in BS^1(X)$. This completes the proof.
\end{proof}

\begin{remark}\rm
It is needed to note that we establish a composition theorem of
$S^p$-almost periodic functions in Step 1 of Lemma
\ref{composition}. In fact, in some previous work (cf.
\cite{ding-jocaaa,long}), we established several composition
theorems of $S^p$-almost periodic functions. However, in the related
results of \cite{ding-jocaaa,long}, $f$ is assumed to be
$S^p$-almost periodic with $p>1$.
\end{remark}

\begin{lemma}\label{convolution}
Let $f\in S^1PAP(X,1)$ and $T(t)$ be a $C_0$-semigroup on $X$ satisfying 
$\|T(t)\|\leq M e^{-\omega t}$ for all
$t\geq 0$, where $M,\omega>0$ are fixed constants. Then
$$
t\mapsto \int^t_{-\infty}T(t-s)f(s)ds
$$
belongs to $PAP(X,1)$.
\end{lemma}

\begin{proof}
Let $f=g+h$, where $g\in S^1AP(X)$ and $h\in S^1PAP_0(X,1)$. Denote
$$
\Phi(t)=\int^t_{-\infty}T(t-s)g(s)ds,\quad
\Psi(t)=\int^t_{-\infty}T(t-s)h(s)ds,\quad t\in\mathbb{R}.
$$
 We will give the proof by two steps.
\smallskip

\noindent\textbf{Step 1.} $\Phi\in AP(X)$.
Let 
$$
\Phi_k(t)=\int^k_{k-1}T(s)g(t-s)ds,\quad k\in\mathbb{N},\ t\in\mathbb{R}.
$$
Then $\Phi(t)=\sum_{k=1}^{\infty}\Phi_k(t)$, and
$\sum_{k=1}^{\infty}\Phi_k(t)$ is uniformly convergent on
$\mathbb{R}$ since
$$
\|\Phi_k(t)\|\leq Me^{-\omega(k-1)}\int^k_{k-1}\|g(t-s)\|ds
\leq Me^{-\omega(k-1)}\|g\|_{S^1}.
$$
On the other hand, noting that $g\in S^1AP(X)$ and
\begin{align*}
\|\Phi_k(t_1)-\Phi_k(t_2)\|
&\leq M\int^k_{k-1}\|g(t_1-s)-g(t_2-s)\|\\
&= M\int^1_{0}\|g(t_1-k+s)-g(t_2-k+s)\|
\end{align*}
for all $t_1,t_2\in\mathbb{R}$, we deduce that $\Phi_k\in AP(X)$ for each
$k\in \mathbb{N}$. Thus, $\Phi\in AP(X)$ since $AP(X)$ is a closed subspace
of $BC(\mathbb{R},X)$.
\smallskip

\noindent\textbf{Step 2.} $\Psi\in PAP_0(X,1)$.
Similar to step 1, we denote
$$
\Psi_k(t)=\int^k_{k-1}T(s)h(t-s)ds,\quad k\in\mathbb{N},\ t\in\mathbb{R}.
$$ 
Then
$\Psi(t)=\sum_{k=1}^{\infty}\Psi_k(t)$, and
$\sum_{k=1}^{\infty}\Psi_k(t)$ is uniformly convergent on
$\mathbb{R}$. Since $PAP_0(X,1)$ is a closed subspace of $BC(\mathbb{R},X)$, it
remains to show that $\Psi_k\in PAP_0(X,1)$ for each $k\in \mathbb{N}$.
Now, fix $k\in\mathbb{N}$. We have
\begin{equation} \label{3}
\begin{aligned}
& \frac{1}{2r}\int^r_{-r}\sup_{\theta\in
[-1,0]}\|\Psi_k(t+\theta)\|dt \\
&= \frac{1}{2r}\int^r_{-r}\sup_{\theta\in
[-1,0]}\big\|\int^k_{k-1}T(s)h(t+\theta-s)ds\big\|dt \\
&\leq \frac{M}{2r}\int^r_{-r}\Big[\sup_{\theta\in
[-1,0]}\int^k_{k-1}\|h(t+\theta-s)\|ds\Big]dt \\
&= \frac{M}{2r}\int^r_{-r}\Big[\sup_{\theta\in
[-1,0]}\int^1_{0}\|h(t-k+\theta+s)\|ds\Big]dt \\
&= \frac{M}{2r}\int^r_{-r}\sup_{\theta\in
[-1,0]}\|h^b(t-k+\theta)\|dt.
\end{aligned}
\end{equation}
On the other hand, since $h\in S^1PAP_0(X,1)$,
$h^b\in PAP_0(L^1(0,1;X),1)$. Then, by the translation invariance of
$PAP_0(L^1(0,1;X),1)$, we know that
$h^b(\cdot-k)\in PAP_0(L^1(0,1;X),1)$, which and \eqref{3} yields that
$$
\lim_{r\to+\infty}\frac{1}{2r}\int^r_{-r}\sup_{\theta\in
[-1,0]}\|\Psi_k(t+\theta)\|dt=0;
$$
i.e., $\Psi_k\in PAP_0(X,1)$.
This completes the proof.
\end{proof}

Now, we are ready to present our main result.

\begin{theorem}\label{theorem}
Assume that {\rm (A1)--(A2)} hold. Then \eqref{equation} has a unique 
pseudo almost periodic mild solution
in $PAP(X,1)$ provided that $L_f< \frac{\omega}{M}$, where
$$
L_f:=\sup_{t\in\mathbb{R}}\sup_{u,v\in Y,u\neq v}\frac{\|f(t,u)-f(t,v)\|}{\|u-v\|}.
$$
\end{theorem}

\begin{proof}
Let
$$
(\mathfrak{F} x)(t)=\int^t_{-\infty}T(t-s)f(s,x_s)ds, \quad 
t\in \mathbb{R},\ x\in PAP(X,1).
$$
By Lemma \ref{composition}, we obtain that $s\mapsto f(s,x_s)$
belongs to $S^1PAP(X,1)$ for each $x\in PAP(X,1)$. Then, by Lemma
\ref{convolution}, we conclude that $\mathfrak{F} x\in PAP(X,1)$ for
each $x\in PAP(X,1)$, which means that $\mathfrak{F}$ maps
$PAP(X,1)$ into $PAP(X,1)$.

On the other hand, for all $u,v\in PAP(X,1)$, by a direct
calculation, one can get
$$
\|\mathfrak{F}  u-\mathfrak{F} v\|\leq \frac{\omega L_f}{M}\|u-v\|.
$$
Noting $L_f< \frac{\omega}{M}$ and $PAP(X,1)$ is a Banach space,
there exists a unique fixed point $x^*\in  PAP(X,1)$ of
$\mathfrak{F}$. At last, it is not difficult to show that $x^*$ is
just the unique pseudo almost periodic mild solution in $PAP(X,1)$
of equation \eqref{equation}.
\end{proof}

\begin{example}\rm
Consider the  partial functional differential equation
\begin{equation} \label{example}
u_t(t,x)= u_{xx}(t,x)+a(t)\int^0_{-1}\frac{e^s
\sin[u(t+s,x)]}{3}ds,\ t\in\mathbb{R},\ x\in (0,\pi),
\end{equation} 
where $a(t)=b(t)+f(t)$, $f(t)$ is as in Example \ref{example-SpPAP0}, and
$$
b(t)=\begin{cases}
 \sin k+\sin \pi k & t\in [k,k+\frac{1}{2}],\; k\in\mathbb{Z},\\
0 & \text{otherwise}.
\end{cases}
$$
Let $X=L^2(0,\pi)$; $(Au)(x)=u''(x)$ for $x\in (0,\pi)$ and $u\in
D(A)$, where
\begin{align*}
D(A)=\big\{&u\in C^1[0,1]: u' \text{ is absolutely continuous on } [0,1],\\
& u''\in X, u(0)=u(\pi)=0\big\};
\end{align*}
and
$$
f(t,\varphi)(x)=a(t)\int^0_{-1}\frac{e^s \sin[\varphi(s)(x)]}{3}ds,\quad 
t\in \mathbb{R},\  \varphi\in C([-1,0];X),\;x\in (0,\pi).
$$
It is well-known that $A$ generates a $C_0$ semigroup $T(t)$
satisfying
$$
\|T(t)\|\leq e^{- t},\quad t\geq 0,
$$
which means that (A1) holds with $M=\omega=1$.

It is not difficult to show that $b\in S^1AP(\mathbb{R})$. Combining this with
Example \ref{example-SpPAP0}, we obtain $a\in S^1PAP(\mathbb{R},1)$, which
yields that $f\in S^1PAP(\mathbb{R}\times C([-1,0];X),X,1)$. In addition, by
a direct calculation, we obtain
$$
\|f(t,\varphi)-f(t,\phi)\|\leq \frac{\sqrt{2}}{2}\|\varphi-\phi\|
$$
for all $\varphi,\phi\in C([-1,0];X)$. Then, we conclude that (A2)
holds. Moreover, we have
$$
L_f\leq\frac{\sqrt{2}}{2}<1 .
$$
Then, by using Theorem \ref{theorem}, equation \eqref{example} has a
unique pseudo almost periodic mild solution in $PAP(X,1)$.
\end{example}

\subsection*{Acknowledgements}
This work was supported by grant 11101192 from
the NSF of China, grant 211090 from the Key Project of 
Chinese Ministry of Education,
grant 20114BAB211002 from the  NSF of Jiangxi Province, and grant
GJJ12173 from the Jiangxi Provincial Education Department.


\begin{thebibliography}{99}

\bibitem{Cuevas09} C. Cuevas, E. Hern\'{a}ndez;
\emph{Pseudo-almost periodic solutions for
abstract partial functional differential equations}, Applied
Mathematics Letters 22 (2009), 534--538.


\bibitem{Cuevas10} C. Cuevas, J. C. de Souza;
\emph{Existence of $S$-asymptotically $\omega$-periodic solutions
 for fractional order functional integro-differential equations with 
infinite delay}, Nonlinear Anal. TMA 72 (2010), 1683--1689.

\bibitem{diagana07} T. Diagana, E. Hern\'{a}ndez;
\emph{Existence and uniqueness of pseudo almost
periodic solutions to some abstract partial neutral
functional-differential equations and applications}, J. Math. Anal.
Appl. 327 (2007), 776--791.


\bibitem{diagana} T. Diagana;
\emph{Existence of almost automorphic solutions to some neutral 
functional differential equations with infinite delay},
 Electronic Journal of Differential Equations, Vol.
2008(2008), No. 129, pp. 1--14.

\bibitem{Dianaga08} T. Diagana;
\emph{Stepanov-like pseudo-almost periodicity and its applications to some
nonautonomous differential equations}, Nonlinear Anal. TMA 69 (2008),
4277--4285.

\bibitem{Dianaga-book} T. Diagana;
\emph{Pseudo almost periodic functions in Banach spaces}, Nova
Science Publishers, Inc., New York, 2007.

\bibitem{ding} H. S. Ding, W. Long, G. M. N'Gu\'{e}r\'{e}kata;
\emph{A composition theorem for weighted pseudo-almost automorphic
functions and applications}, Nonlinear Anal. TMA 73 (2010),
2644--2650.


\bibitem{ding-jocaaa} H. S. Ding, W. Long, G. M. N'Gu\'{e}r\'{e}kata;
\emph{Almost periodic solutions to abstract semilinear evolution equations
with Stepanov almost periodic coefficients}, Journal of Computational
Analysis and Applications, 13 (2011), 231--242.


\bibitem{Ezzinbi1} K. Ezzinbi, G. M. N'Gu\'{e}r\'{e}kata;
\emph{Almost automorphic solutions for partial functional 
differential equations with infinite delay},
Semigroup Forum 75 (2007), 95--115.

\bibitem{enriquez1} H. R. Henr\'{\i}quez;
\emph{Asymptotically almost periodic solutions of
abstract retarded functional differential equations of first order},
Nonlinear Analysis RWA 10 (2009), 2441--2454.

\bibitem{enriquez2} H. R. Henr\'{\i}quez, C. Lizama;
\emph{Compact almost automorphic solutions to integral equations 
with infinite delay}, Nonlinear Anal.
TMA 71 (2009), 6029--6037.


\bibitem{enriquez3} H. R. Henr\'{\i}quez;
\emph{Asymptotically almost periodic solutions of abstract retarded
functional differential equations of second order}, Z. angew. Math.
Phys. 60 (2009), 797--822.


\bibitem{Hern1} E. Hern\'{a}ndez, R. Agarwal, T. Diagana;
\emph{Weighted pseudo almost periodic solutions to some partial 
neutral functional differential equations}, 
J. Nonlinear Convex Anal. 8 (2007), 397--415.


\bibitem{Hern2} E. Hern\'{a}ndez, T. Diagana, M. Rabello;
\emph{Pseudo almost periodic solutions to some non-autonomous 
neutral functional differential equations with unbounded delay}, 
Math. Comput. Model. 45 (2007), 1241--1252.

\bibitem{Hern3} E. Hern\'{a}ndez, M. Pierri, M. A. Ben\'{a};
\emph{Asymptotically almost periodic solutions for abstract neutral
integro-differential equations}, Applied Mathematics and Computation
217 (2011), 8963--8972.

\bibitem{long} W. Long, H. S. Ding;
\emph{Composition theorems of stepanov almost periodic functions and
stepanov-like pseudo almost periodic functions}, Advances in
Difference Equations Volume 2011, Article ID 654695, 12 pages,
doi:10.1155/2011/654695.

\bibitem{gaston-pankov} G. M. N'Gu\'{e}r\'{e}kata, A. Pankov;
\emph{Stepanov-like almost automorphic functions and monotone evolution 
equations}, Nonlinear Anal. TMA 68 (2008), 2658--2667.

\bibitem{zheng-ding-gaston} Z. M. Zheng, H. S. Ding, G. M. N'Gu\'{e}r\'{e}kata;
\emph{The space of continuous periodic functions is a set of first
category in $AP(X)$}, Journal of Function Spaces and Applications
Volume 2013 (2013), Article ID 275702.

\end{thebibliography}

\end{document}