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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2014 (2014), No. 191, pp. 1--17.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2014 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2014/191\hfil Weighted pseudo periodic solutions]
{Weighted pseudo periodic solutions of neutral functional differential equations}

\author[Z. Xia \hfil EJDE-2014/191\hfilneg]
{Zhinan Xia}  % in alphabetical order

\address{Zhinan Xia \newline
Department of Applied Mathematics,
Zhejiang University of Technology,
Hangzhou, Zhejiang, 310023, China}
\email{xiazn299@zjut.edu.cn}

\thanks{Submitted March 6, 2014. Published September 16, 2014.}
\subjclass[2000]{35R10, 35B40}
\keywords{Weighted pseudo periodicity;
weighted Stepanov-like pseudo periodic; \hfill\break\indent
neutral functional differential equation; Banach contraction mapping principle}

\begin{abstract}
 In this article, we introduced and explore the properties of
 two sets of functions: weighted pseudo periodic functions of class $r$,
 and weighted Stepanov-like pseudo periodic functions of class $r$.
 We show the existence and uniqueness of weighted pseudo periodic solution
 of class $r$ that are solutions to neutral functional
 differential equations. Other applications to partial differential equations
 and scalar reaction-diffusion equations with delay are also given.
\end{abstract}

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

\section{Introduction}

The existence periodic solutions is one of the most
interesting and important topics in the qualitative theory of
differential equations. Many authors have made important
contributions to this theory. Recently, in \cite{alaldi12,xia}, the
concept of weighted pseudo periodicity, weighted Stepanov-like
pseudo periodicity, is introduced and studied, respectively. On the
other hand, to study issues related to delay differential equations,
Diagana \cite{diaher07} introduce the functions called pseudo almost
periodic of class $r$, for more on this topic and related
applications in differential equations, we refer the reader to
\cite{agacuesotelg11,agadiaher07,cueher09,dinlongmn13,ezztouzab}.


Motivated by the above mentioned papers, in this paper, we introduce
new class of functions called weighted pseudo periodic of class $r$,
weighted Stepanov-like pseudo periodic of class $r$, respectively.
We systematically explore the properties of these functions in
general Banach space including composition results and its
applications in differential equations.

In recent years, neutral functional differential equations have
attracted a great deal of attention of many mathematicians due to
their significance and applications in physics, mathematical
biology, control theory, and so on. The general asymptotic behavior
of solutions have been one of the most attracting topics in the
context of neutral functional differential equations
\cite{diaaga09,dimgmn12,herhen98,herhen08,herpel05,henpietab08,pierol13}.
However, to the best of our knowledge, the studies on the weighted
pseudo periodic solutions of neutral functional differential
equations is quite new and an untreated topic. This is one of the
key motivations of this study.


The paper is organized as follows. In Section 2, some notations and
preliminary results are presented. Next, we propose new class of
functions called weighted pseudo periodic functions of class $r$,
weighted Stepanov-like pseudo periodic functions of class $r$,
explore the properties of these functions and establish the
composition theorems. Sections 3 is devoted to the existence and
uniqueness of weighted pseudo periodic solutions of class of $r$ to
neutral functional differential equations. In section 4, we present
applications to partial differential equations and scalar
reaction-diffusion equations with delay.

\section{Preliminaries and basic results}

Let $(X, \|\cdot\|)$, $(Y, \|\cdot\|)$ be two Banach spaces and
$\mathbb{N}, \mathbb{Z}, \mathbb{R}$ be the stand sets of natural
numbers, integers, real numbers, respectively. To
facilitate the discussion below, we further introduce the following
notation:

\begin{itemize}
\item{$C(\mathbb{R},X)$ (resp. $C(\mathbb{R}\times
Y,X)$):~the set of continuous functions from $\mathbb{R}$ to $X$
(resp. from $\mathbb{R}\times Y$ to $X$).}

\item $BC(\mathbb{R}, X)$ (resp.
$BC(\mathbb{R}\times Y, X)$:~the Banach space of
bounded continuous functions from ${\mathbb{R}}$ to $X$ (resp. from
$\mathbb{R}\times Y$ to $X$) with the supremum
norm;


\item $L^{p}(\mathbb{R},X)$:~the space of all classes
of equivalence (with respect to the equality almost everywhere on
$\mathbb{R}$) of measurable functions $f: \mathbb{R}\to
X$ such that $\|f\|\in L^{p}(\mathbb{R},\mathbb{R})$;


\item $L^{p}_{loc}(\mathbb{R},X)$: stand for the space of all
classes of equivalence of measurable functions $f:
\mathbb{R}\to X$ such that the restriction of $f$
to every bounded subinterval of $\mathbb{R}$ is in
$L^{p}(\mathbb{R},X)$.

\item $\mathcal{C}$: the space $C([-r, 0],X)$ endowed with the sup norm
$\|\psi\|_{\mathcal{C}}$ on $[-r, 0]$.

\item $[D(A)]$: the domain of $A$ when it is endowed
with graph norm, $\|x\|_{[D(A)]}=\|x\|+\|Ax\|$ for each $x\in D(A)$.

\end{itemize}

\subsection{Weighted pseudo periodic of class $r$}

In this subsection, we introduce the new class of functions called
weighted pseudo anti-periodic of class $r$, weighted
pseudo periodic functions of class $r$, and investigate the
properties of these functions.

\begin{definition} \rm
A function $f\in C({\mathbb{R}}, X)$ is said to be anti-periodic if there
exists a $\omega\in {\mathbb{R}}\backslash\{0\}$ with the property that
 $f(t+\omega)=-f(t)$ for all $t\in{\mathbb{R}}$. The least
positive $\omega$ with this property is called the anti-periodic of
$f$. The collection of those functions is denoted by $P_{\omega
ap}({\mathbb{R}}, X)$.
\end{definition}

\begin{definition} \rm
A function $f\in C({\mathbb{R}}, X)$ is said to be periodic if there exists a
$\omega\in {\mathbb{R}}\backslash\{0\}$ with the property that $f(t+\omega)=f(t)$ for
all $t\in {\mathbb{R}}$. The least positive $\omega$ with this property is
called the periodic of $f$. The collection of those
$\omega$-periodic functions is denoted by $P_{\omega}({\mathbb{R}}, X)$.
\end{definition}

Note that if $f\in P_{\omega ap}({\mathbb{R}}, X)$, then $f\in P_{2\omega}({\mathbb{R}},X)$.


Let $U$ be the set of all functions $\rho:\mathbb{R}\to
(0,\infty)$ which are positive and locally integrable over
$\mathbb{R}$. For a given $T>0$ and each $\rho\in U$, set
$$
\mu(T,\rho):=\int^{T}_{-T}\rho(t)\mathrm{d}t.
$$
Define
\begin{gather*}
U_{\infty}:=\{\rho\in U: \lim_{T\to\infty}\mu(T,\rho)=\infty\},\\
U_{B}:=\{\rho\in U_{\infty}:\rho\,\, \text{is bounded and}
\inf_{x\in\mathbb{R}}\rho(x)>0\}.
\end{gather*}
It is clear that
$U_{B}\subset U_{\infty}\subset U$.


\begin{definition} \label{def2.3}\rm
Let $\rho_1, \rho_2\in U_{\infty}$. The function $\rho_1$ is said to be
equivalent to $\rho_2$ (i.e., $\rho_1 \sim \rho_2$) if
$\frac{\rho_1}{\rho_2}\in U_{B}$.
\end{definition}

It is trivial to show that $``\sim"$ is a binary equivalence
relation on $U_{\infty}$. The equivalence class of a given weight
$\rho\in U_{\infty}$ is denoted by
$cl(\rho)=\{ \varrho\in U_{\infty}:\rho\sim \varrho \}.$
It is clear that $U_{\infty}=\bigcup_{\rho\in
U_{\infty}}cl(\rho)$.

Let $\rho\in U_{\infty}$, $\tau\in {\mathbb{R}}$ be given, and defined
$\rho^{\tau}$ by $\rho^{\tau}(t)=\rho(t+\tau)$ for $t\in {\mathbb{R}}$. Define
\cite{zhli10}
$$
U_T=\{\rho\in U_{\infty}: \rho \sim \rho^{\tau} \text{ for each } \tau\in {\mathbb{R}}\}.
$$
It is easy to see that $U_T$ contains many of weights, such as $1$,
$(1+t^{2})/(2+t^{2})$, $e^{t}$, and $1+|t|^{n}$ with $n\in {\mathbb{N}}$ etc.
For $\rho_1,\rho_2\in U_{\infty}$, define
\begin{gather*}
WPP_{0}(\mathbb{R},X,\rho_1,\rho_2):=\Big\{f\in
BC(\mathbb{R},X):
\lim_{T\to\infty}\frac{1}{\mu(T,\rho_1)}\int^{T}_{-T}\|f(t)\|\rho_2(t)dt=0\Big\},
\\
\begin{aligned}
&WPP_{0}(\mathbb{R},X,r,\rho_1,\rho_2)\\
&:=\Big\{f\in BC(\mathbb{R},X):\lim_{T\to\infty}\frac{1}{\mu(T,\rho_1)}\int^{T}_{-T}
\Big(\sup_{\theta\in[t-r,t]}\|f(\theta)\|\Big)\rho_2(t)
 dt=0\Big\},
\end{aligned}
\\
\begin{aligned}
& WPP_{0}(\mathbb{R}\times Y,X,r,\rho_1,\rho_2)\\
&:=\Big\{f\in BC(\mathbb{R}\times Y,X):
\lim_{T\to\infty}\frac{1}{\mu(T,\rho_1)}\int^{T}_{-T}
\Big(\sup_{\theta\in[t-r,t]}\|f(\theta,u)\|\Big)\rho_2(t)
 dt=0 \\
&\quad\text{uniformly for } u\in Y\Big\}.
\end{aligned}
\end{gather*}

\begin{definition} \label{def2.4}\rm
 Let $\rho_1,\rho_2\in U_{\infty}$. A function $f\in
C(\mathbb{R},X)$ is called weighted pseudo anti-periodic for
$\omega\in {\mathbb{R}}\backslash\{0\}$ if it can be decomposed as $f=g+\varphi$, where
$g\in P_{\omega ap}(\mathbb{R},X)$ and $\varphi\in
WPP_{0}(\mathbb{R}, X,\rho_1,\rho_2)$. Denote by $WPP_{\omega
ap}(\mathbb{R},X,\rho_1,\rho_2)$ the set of such functions.
\end{definition}


\begin{definition} \label{def2.5} \rm
 Let $\rho_1,\rho_2\in U_{\infty}$. A function $f\in
C(\mathbb{R},X)$ is called weighted pseudo periodic for $\omega\in
{\mathbb{R}}\backslash\{0\}$ if it can be decomposed as $f=g+\varphi$, where $g\in
P_{\omega}(\mathbb{R},X)$ and $\varphi\in WPP_{0}(\mathbb{R},
X,\rho_1,\rho_2)$. Denote by
$WPP_{\omega}(\mathbb{R},X,\rho_1,\rho_2)$ the set of such
functions.
\end{definition}

If $\rho_1 \sim \rho_2$, $WPP_{\omega ap}(\mathbb{R},X,\rho_1,\rho_2)$, and
$WPP_{\omega}(\mathbb{R},X,\rho_1,\rho_2)$ coincide with the
weighted pseudo anti-periodic, and the weighted pseudo periodic function
respectively, introduce by \cite{alaldi12}.


\begin{definition} \label{def2.6} \rm
 Let $\rho_1,\rho_2\in U_{\infty}$. A function $f\in
C(\mathbb{R},X)$ is called weighted pseudo anti-periodic of class
$r$ for $\omega\in {\mathbb{R}}\backslash\{0\}$ if it can be decomposed as
$f=g+\varphi$, where $g\in P_{\omega ap}(\mathbb{R},X)$ and
$\varphi\in WPP_{0}(\mathbb{R}, X,r,\rho_1,\rho_2)$.
The set of these functions is denote by
$WPP_{\omega ap}(\mathbb{R},X,r,\rho_1,\rho_2)$.
\end{definition}


\begin{definition} \label{def2.7} \rm
 Let $\rho_1,\rho_2\in U_{\infty}$. A function $f\in
C(\mathbb{R},X)$ is called weighted pseudo periodic of class $r$
for $\omega\in {\mathbb{R}}\backslash\{0\}$ if it can be decomposed as $f=g+\varphi$,
where $g\in P_{\omega}(\mathbb{R},X)$ and $\varphi\in
WPP_{0}(\mathbb{R}, X,r,\rho_1,\rho_2)$. Denote by
$WPP_{\omega}(\mathbb{R},X,r,\rho_1,\rho_2)$ the set of such
functions.
\end{definition}

\begin{remark} \label{rmk2.3} \rm
If $r=0$, then the weighted pseudo anti-periodic function of class r
reduces to the weighted pseudo anti-periodic function, the weighted
pseudo periodic function of class r reduces to the weighted pseudo
periodic function. That is, $WPP_{\omega
ap}(\mathbb{R},X,0,\rho_1,\rho_2)=WPP_{\omega
ap}(\mathbb{R},X,\rho_1,\rho_2)$, and
$WPP_{\omega}(\mathbb{R},X,0,\rho_1,\rho_2)
=WPP_{\omega}(\mathbb{R},X,\rho_1,\rho_2)$.
\end{remark}

Next, we show some properties of the space
$WPP_{\omega}(\mathbb{R},X,r,\rho_1,\rho_2)$. Similarly results
 hold for $WPP_{\omega ap}(\mathbb{R},X,r,\rho_1,\rho_2)$.

\begin{lemma}\label{ifonlyif}
Let $f\in BC({\mathbb{R}}, X)$, then $f\in WPP_{0}({\mathbb{R}},X,r,\rho_1,\rho_2)$,
$\rho_1,\rho_2\in U_{\infty}$,
$\sup_{T>0}\frac{\mu(T,\rho_2)}{\mu(T,\rho_1)}<\infty$ if and only if for every
$\varepsilon>0$,
$$
\lim_{T\to \infty}\frac{1}{\mu(T,\rho_1)}\int_{ M(T,\varepsilon, f)}\rho_2(t)dt=0,
$$
where $M(T,\varepsilon, f) :=\{t\in [-T, T]: \sup_{\theta\in[t-r,t]}\|f(\theta)\|\geq
\varepsilon \}$.
\end{lemma}

\begin{proof}
The proof is similar as \cite{dinlongmn13}.

Sufficiency: From the statement of the lemma it is clear that for any
$\varepsilon>0$, there exists $T_{0}>0$ such that for $T>T_{0}$,
$$
\frac{1}{\mu(T,\rho_1)}\int_{ M(T,\varepsilon, f)}\rho_2(t)dt
< \frac{\varepsilon}{\|f\|}.
 $$
Then
\begin{align*}
&\frac{1}{\mu(T,\rho_1)}\int^{T}_{-T}\Big(\sup_{\theta\in[t-r,t]}\|f(\theta)\|\Big)
\rho_2(t) dt \\
&=\frac{1}{\mu(T,\rho_1)}\int_{M(T,\varepsilon, f)}
\Big(\sup_{\theta\in[t-r,t]}\|f(\theta)\|\Big)\rho_2(t)dt\\
&\quad +\frac{1}{\mu(T,\rho_1)}\int_{[-T, T]\backslash M(T,\varepsilon, f)}
 \Big(\sup_{\theta\in[t-r,t]}\|f(\theta)\|\Big)\rho_2(t)dt\\
& \leq \frac{\|f\|}{\mu(T,\rho_1)}\int_{M(T,\varepsilon, f)}\rho_2(t)dt
 + \frac{\varepsilon}{\mu(T,\rho_1)}\int^{T}_{-T}\rho_2(t)dt \\
& \leq \varepsilon + \sup_{T>0}\frac{\mu(T,\rho_2)}{\mu(T,\rho_1)} \varepsilon,
\end{align*}
so
$$
\lim_{T\to \infty}\frac{1}{\mu(T,\rho_1)}\int^{T}_{-T}
\Big(\sup_{\theta\in[t-r,t]}\|f(\theta)\|\Big)\rho_2(t) dt=0.
 $$
That is, $f\in WPP_{0}({\mathbb{R}},X,r,\rho_1,\rho_2)$.


Necessity: Suppose on the contrary that there exists $\varepsilon_{0}>0$
such that
$$
\frac{1}{\mu(T,\rho_1)}\int_{ M(T,\varepsilon_{0}, f)}\rho_2(t)dt
$$
does not converge to $0$ as $T\to \infty$.
Then there exists $\delta>0$ such that for each $n$,
$$
\frac{1}{\mu(T_{n},\rho_1)}\int_{ M(T_{n},\varepsilon_{0}, f)}\rho_2(t)dt\geq \delta
\quad \text{for some }T_{n}\geq n.
$$
Then
\begin{align*}
&\frac{1}{\mu(T_{n},\rho_1)}\int^{T_{n}}_{-T_{n}}\rho_2(t)
\Big(\sup_{\theta\in[t-r,t]}\|f(\theta)\|\Big)dt \\
&\geq \frac{1}{\mu(T_{n},\rho_1)}\int_{M(T_{n},\varepsilon_{0}, f)}\rho_2(t)
\Big(\sup_{\theta\in[t-r,t]}\|f(\theta)\|\Big)dt\\
&\geq \frac{\varepsilon_{0}}{\mu(T_{n},\rho_1)}\int_{M(T_{n},\varepsilon_{0}, f)}
\rho_2(t)dt
\geq \varepsilon_{0} \delta,
\end{align*}
which contradicts the fact that
$f\in WPP_{0}({\mathbb{R}},X,r,\rho_1,\rho_2)$, and the proof is complete.
\end{proof}

Let
\[
WPP_{0}({\mathbb{R}},{\mathbb{R}}^{+},r,\rho_1,\rho_2)
=\{ f\in WPP_{0}({\mathbb{R}},{\mathbb{R}},r,\rho_1,\rho_2): f(t)\geq 0,\; \forall t\in{\mathbb{R}}\}.
\]

\begin{lemma}\label{ifonlyif2}
Let $\alpha>0$, then $f\in WPP_{0}({\mathbb{R}},{\mathbb{R}}^{+},r,\rho_1,\rho_2)$ if and only if
$f^{\alpha}\in WPP_{0}({\mathbb{R}},{\mathbb{R}}^{+},r,\rho_1,\rho_2)$,
where $f^{\alpha}(t)=[f(t)]^{\alpha}$, $\rho_1,\rho_2\in U_{\infty}$,
$\sup_{T>0}\frac{\mu(T,\rho_2)}{\mu(T,\rho_1)}<\infty$.
\end{lemma}

\begin{proof}
By Lemma \ref{ifonlyif}, $f\in WPP_{0}({\mathbb{R}},{\mathbb{R}}^{+},r,\rho_1,\rho_2)$
if and only if for every $\varepsilon>0$,
$$
\lim_{T\to \infty}\frac{1}{\mu(T,\rho_1)}\int_{ M(T,\varepsilon, f)}\rho_2(t)dt=0,
$$
where $M(T,\varepsilon, f) :=\{t\in [-T, T]: \sup_{\theta\in[t-r,t]}
f(\theta)\geq \varepsilon\}$.
It is equivalent to for every $\varepsilon>0$,
$$
\lim_{T\to \infty}\frac{1}{\mu(T,\rho_1)}\int_{ M(T,\varepsilon,
f^{\alpha})}\rho_2(t)dt=0,
$$
where
\[
M(T,\varepsilon, f^{\alpha})
:=\{t\in [-T, T]: \sup_{\theta\in[t-r,t]}f^{\alpha}(\theta)\geq \varepsilon\}.
\]
So $f^{\alpha}\in WPP_{0}({\mathbb{R}},{\mathbb{R}}^{+},r,\rho_1,\rho_2)$.
\end{proof}


\begin{lemma}\label{paauniform}
Let $\varphi_{n}\to \varphi$ uniformly on $\mathbb{R}$ where
each $\varphi_{n}\in WPP_{0}(\mathbb{R},X,r,\rho_1,\rho_2)$,
$\rho_1,\rho_2 \in U_{\infty}$, if
$\sup_{T>0}\frac{\mu(T,\rho_2)}{\mu(T,\rho_1)}<\infty$, then
$\varphi\in WPP_{0}(\mathbb{R},X,r,\rho_1,\rho_2)$.
\end{lemma}

\begin{proof}
For $T>0$,
\begin{align*}
&\frac{1}{\mu(T,\rho_1)}\int^{T}_{-T}\Big(\sup_{\theta\in[t-r,t]}\|\varphi(\theta)\|
\Big)\rho_2(t) dt \\
&\leq \frac{1}{\mu(T,\rho_1)}\int^{T}_{-T}
\Big(\sup_{\theta\in[t-r,t]}\|\varphi_{n}(\theta)-\varphi(\theta)\|\Big)\rho_2(t)
 dt \\
&\quad +\frac{1}{\mu(T,\rho_1)}\int^{T}_{-T}
 \Big(\sup_{\theta\in[t-r,t]}\|\varphi_{n}(\theta)\|\Big)\rho_2(t) dt \\
&\leq \frac{\mu(T,\rho_2)}{\mu(T,\rho_1)}
\|\varphi_{n}-\varphi\| +\frac{1}{\mu(T,\rho_1)}\int^{T}_{-T}
\Big(\sup_{\theta\in[t-r,t]}\|\varphi_{n}(\theta)\|\Big)\rho_2(t) dt\\
&\leq \sup_{T>0}\frac{\mu(T,\rho_2)}{\mu(T,\rho_1)} \|\varphi_{n}-\varphi\|
 +\frac{1}{\mu(T,\rho_1)}\int^{T}_{-T}
\Big(\sup_{\theta\in[t-r,t]}\|\varphi_{n}(\theta)\|\Big)\rho_2(t) dt,
\end{align*}
Let $T\to \infty$ and then $n\to \infty$ in the
above inequality, it follows that
$\varphi\in WPP_{0}(\mathbb{R},X,r,\rho_1,\rho_2)$.
\end{proof}

By carrying out similar arguments as those in the proof of
\cite[Lemma 4.1]{zhli10}, we conclude the following.

\begin{lemma}\label{trans}
Let $\rho_1, \rho_2\in U_T$, $\varphi \in
WPP_{0}(\mathbb{R},X,r,\rho_1,\rho_2)$, then
$\varphi(\cdot-\tau)$ belongs to $WPP_{0}(\mathbb{R},X,r,\rho_1,\rho_2)$
for $\tau \in {\mathbb{R}}$.
\end{lemma}

Using similar ideas as in \cite{di11, di112}, one can easily show
the following result.

\begin{lemma}\label{decom}
If $\rho_1,\rho_2\in U_T$ and
$\inf_{T>0}\frac{\mu(T,\rho_2)}{\mu(T,\rho_1)}=\delta_{0}>0$,
then the decomposition of weighted pseudo periodic function of class $r$ is
unique.
\end{lemma}

By Lemma \ref{decom}, it is obvious that $(WPP_{\omega}(\mathbb{R},
X, r, \rho_1,\rho_2),\|\cdot\|)$,
$\rho_1,\rho_2\in U_T$ and
$\inf_{T>0}\frac{\mu(T,\rho_2)}{\mu(T,\rho_1)}=\delta_{0}>0$
is a Banach space when endowed with the sup norm.

\begin{lemma}\label{delay}
Let $\rho_1,\rho_2 \in U_T$,
$u\in WPP_{\omega}(\mathbb{R},
X,r,\rho_1,\rho_2)$, then $u_t$ belongs to
$WPP_{\omega}(\mathbb{R}, \mathcal{C},r,\rho_1,\rho_2)$.
\end{lemma}

\begin{proof}
Suppose that $u=\alpha+\beta$, where $\alpha\in
P_{\omega}(\mathbb{R},X)$ and $\beta\in WPP_{0}(\mathbb{R},
X,r,\rho_1,\rho_2)$, then $u_t=\alpha_t+\beta_t$ and
$\alpha_t \in P_{\omega}(\mathbb{R},
\mathcal{C},r,\rho_1,\rho_2)$. On the other hand, for $T>0$, we
see that
\begin{align*}
&\frac{1}{\mu(T,\rho_1)}\int^{T}_{-T}\Big[\sup_{\theta\in[t-r,t]}
\Big(\sup_{\tau\in[-r,0]}\|\beta(\theta+\tau)\|\Big)\Big]\rho_2(t)dt\\
&\leq \frac{1}{\mu(T,\rho_1)}\int^{T}_{-T}
 \Big(\sup_{\theta\in[t-2r,t]}\|\beta(\theta)\|\Big)\rho_2(t)dt \\
&\leq \frac{1}{\mu(T,\rho_1)}\int^{T}_{-T}
 \Big(\sup_{\theta\in[t-2r,t-r]}\|\beta(\theta)\|
 +\sup_{\theta\in[t-r,t]}\|\beta(\theta)\|\Big)\rho_2(t)dt \\
&\leq \frac{1}{\mu(T,\rho_1)}\int^{T-r}_{-T-r}
 \Big(\sup_{\theta\in[t-r,t]}\|\beta(\theta)\|\Big)\rho_2(t+r)dt \\
&\quad +\frac{1}{\mu(T,\rho_1)}\int^{T}_{-T}
 \Big(\sup_{\theta\in[t-r,t]}\|\beta(\theta)\|\Big)\rho_2(t)dt \\
&\leq \frac{\mu(T+r,\rho_1)}{\mu(T,\rho_1)} \frac{1}{\mu(T+r,\rho_1)}
\int^{T+r}_{-T-r}\Big(\sup_{\theta\in[t-r,t]}\|\beta(\theta)\|\Big)
 \rho_2(t)\frac{\rho_2(t+r)}{\rho_2(t)}dt\\
&\quad +\frac{1}{\mu(T,\rho_1)}\int^{T}_{-T}
\Big(\sup_{\theta\in[t-r,t]}\|\beta(\theta)\|\Big)\rho_2(t)dt.
\end{align*}
Since $\rho_1,\rho_2\in U_T$ implies that there exists $\eta>0$
such that $\rho_1(t+r)/\rho_1(t)\leq \eta$,
$\rho_1(t-r)/\rho_1(t)\leq \eta$,
$\rho_2(t+r)/\rho_2(t)\leq \eta$. For $T>r$,
\begin{align*}
\mu(T+r,\rho_1)&=\int^{T-r}_{-T-r}\rho_1(t)dt
+\int^{T+r}_{T-r}\rho_1(t)dt
\leq \int^{T-r}_{-T-r}\rho_1(t)dt+\int^{T+r}_{-T+r}\rho_1(t)dt \\
& =\int^{T}_{-T}\rho_1(t-r)dt+\int^{T}_{-T}\rho_1(t+r)dt \leq
2\eta \mu(T,\rho_1),
\end{align*}
then
\begin{align*}
&\frac{1}{\mu(T,\rho_1)}\int^{T}_{-T}
\Big[\sup_{\theta\in[t-r,t]}\Big(\sup_{\tau\in[-r,0]}\|\beta(\theta+\tau)\|\Big)\Big]
\rho_2(t)dt \\
&\leq \frac{2\eta^{2}}{\mu(T+r,\rho_1)}\int^{T+r}_{-T-r}
\Big(\sup_{\theta\in[t-r,t]}\|\beta(\theta)\|\Big)\rho_2(t)dt \\
&\quad +\frac{1}{\mu(T,\rho_1)}\int^{T}_{-T}
 \Big(\sup_{\theta\in[t-r,t]}\|\beta(\theta)\|\Big)\rho_2(t)dt.
\end{align*}
Note that $\beta\in WPP_{0}(\mathbb{R}, X,r,\rho_1,\rho_2),
\rho_1, \rho_2\in U_T$, then $\beta_t\in
WPP_{0}(\mathbb{R},\mathcal{C},r,\rho_1, \rho_2)$. Therefore,
$u_t\in WPP_{\omega}(\mathbb{R},\mathcal{C},r,\rho_1,
\rho_2)$.
\end{proof}

Similarly as \cite[Theorem 3.9]{agadiaher07}, we have the following
composition theorem for weighted pseudo periodic function of class
$r$.

\begin{theorem}\label{comwpaa}
Assume that $\rho_1,\rho_2\in U_{\infty}, r\geq 0, f\in
WPP_{\omega}(\mathbb{R}\times Y, X,r,\rho_1,\rho_2)$ and
 there exists a function $L_{f}:{\mathbb{R}}\to [0,+\infty)$
satisfying
\begin{itemize}
\item [(A1)] $\|f(t,u)-f(t,v)\|\leq L_{f}(t)\|u-v\|$, for all
 $t\in{\mathbb{R}}$, $u,v\in Y$;

\item [(A2)]
$\limsup_{T\to\infty}\frac{1}{\mu(T,\rho_1)}\int^{T}_{-T}
\Big(\sup_{\theta\in[t-r,t]}L_{f}(\theta)\Big)\rho_2(t)dt<\infty$;

\item [(A3)]
$\lim_{T\to\infty}\frac{1}{\mu(T,\rho_1)}\int^{T}_{-T}
\Big(\sup_{\theta\in[t-r,t]}L_{f}(\theta)\Big)\xi(t)\rho_2(t)dt=0$
for each function $\xi\in WPP_{0}({\mathbb{R}},{\mathbb{R}},\rho_1,\rho_2)$.
\end{itemize}
Then $f(\cdot,h(\cdot))\in WPP_{\omega}(\mathbb{R},X,r,\rho_1,\rho_2)$ if
$h\in WPP_{\omega}(\mathbb{R},Y,r,\rho_1,\rho_2)$.
\end{theorem}

\begin{remark}\label{rem2.4}
Note that (A2) and (A3) are verified by many functions.
Concrete examples include constant functions, and functions in
$WPP_{\omega}({\mathbb{R}},{\mathbb{R}},r,\rho_1,\rho_2)$.
\end{remark}

\subsection{Weighted Stepanov-like pseudo periodic of class $r$}

In this subsection, we introduce the new class of functions called
weighted $S^{p}$-pseudo anti-periodic of class $r$, weighted
$S^{p}$-pseudo periodic functions of class $r$, and investigate the
properties of these functions.

Let $p\in[1, \infty)$. The space $BS^{p}(\mathbb{R},X)$ of all
Stepanov bounded functions, with the exponent $p$, consists of all
measurable functions $f:\mathbb{R}\to X$ such that
$f^{b}\in L^{\infty}(\mathbb{R},L^{p}([0,1];X))$, where $f^b$ is the Bochner
transform of $f$ defined by $f^{b}(t,s):=f(t+s)$, $t\in\mathbb{R}$,
$s\in[0,1]$. $BS^{p}(\mathbb{R},X)$ is a Banach space with the norm
\cite{pankov}
$$
\|f\|_{S^{p}}=\|f^{b}\|_{L^{\infty}\left(\mathbb{R},L^{p}\right)}
=\sup_{t\in\mathbb{R}}\Big(\int^{t+1}_t\|f(\tau)\|^{p}\mathrm{d}\tau\Big)^{1/p}.
$$
It is clear that $L^{p}({\mathbb{R}}, X)\subset BS^{p}(\mathbb{R},X)\subset
L_{loc}^{p}({\mathbb{R}}, X)$ and
$BS^{p}(\mathbb{R},X)\subset BS^{q}(\mathbb{R},X)$ for $p\geq q \geq 1$.


For $\rho_1,\rho_2\in U_{\infty}$, define the weighted ergodic
space in $BS^{p}(\mathbb{R},X)$
\begin{align*}
S^{p}WPP_{0}(\mathbb{R},X,r,\rho_1,\rho_2)
&:=\Big\{f\in BS^{p}(\mathbb{R},X):
\lim_{T\to\infty}\frac{1}{\mu(T,\rho_1)} \\
&\quad \times \int^{T}_{-T}\rho_2(t)
\Big(\sup_{\theta\in[t-r,t]}\Big(\int^{\theta+1}_{\theta}\|f(s)\|^{p}ds\Big)^{1/p}
\Big)dt=0\Big\}.
\end{align*}

\begin{definition}\label{paaspdef} \rm
Let $\rho_1, \rho_2\in U_{\infty}$. A function $f\in
BS^{p}(\mathbb{R},X)$ is said to be weighted Stepanov-like pseudo
anti-periodic of class $r$ (or weighted $S^{p}$-pseudo anti-periodic
of class $r$) if there exists $\varphi\in
S^{p}WPP_{0}(\mathbb{R},X,r,\rho_1,\rho_2)$ such that the
function $g=f-\varphi$ satisfies $g(t+\omega)+g(t)=0 ~a. e.$
$t\in{\mathbb{R}}$. The collection of such functions is denoted by
$S^{p}WPP_{\omega ap}(\mathbb{R},X,r,\rho_1,\rho_2)$
\end{definition}


\begin{definition}\label{paaspdefb} \rm
Let $\rho_1, \rho_2\in U_{\infty}$. A function $f\in
BS^{p}(\mathbb{R},X)$ is said to be weighted Stepanov-like pseudo
periodic of class $r$ (or weighted $S^{p}$-pseudo periodic of class
$r$) if there exists $\varphi\in
S^{p}WPP_{0}(\mathbb{R},X,r,\rho_1,\rho_2)$ such that the
function $g=f-\varphi$ satisfies $g(t+\omega)-g(t)=0$ a. e.
$t\in{\mathbb{R}}$. Denote by
$S^{p}WPP_{\omega}(\mathbb{R},X,r,\rho_1,\rho_2)$ the collection
of such functions.
\end{definition}


Next, we show some properties of the space
$S^{p}WPP_{\omega}(\mathbb{R},X,r,\rho_1,\rho_2)$. Similarly
results  hold for $S^{p}WPP_{\omega
ap}(\mathbb{R},X,r,\rho_1,\rho_2)$.

\begin{lemma}\label{inclu}
Let $\rho_1, \rho_2\in U_T$, then
\[
WPP_{\omega}(\mathbb{R},X,r,\rho_1,\rho_2)\subset
S^{p}WPP_{\omega}(\mathbb{R},X,r,\rho_1,\rho_2).
\]
\end{lemma}

\begin{proof}
If $f\in WPP_{\omega}({\mathbb{R}},X,r,\rho_1,\rho_2)$, let
$f=f_1+f_2$, where $f_1\in P_{\omega}({\mathbb{R}},X)$, $f_2\in
WPP_{0}({\mathbb{R}},X,r,\rho_1,\rho_2)$. Then $\|f_2(\cdot)\|\in
WPP_{0}({\mathbb{R}},{\mathbb{R}}^{+},r,\rho_1,\rho_2)$. By Lemma \ref{ifonlyif2},
$\|f_2(\cdot)\|^{p}\in WPP_{0}({\mathbb{R}},{\mathbb{R}}^{+},r,\rho_1,\rho_2)$.
Note that $\|f_2(\cdot+\sigma)\|^{p}\in
WPP_{0}({\mathbb{R}},{\mathbb{R}}^{+},r,\rho_1,\rho_2)$ for each $\sigma\in [0, 1]$,
then
$$
\lim_{T\to\infty}\frac{1}{\mu(T,\rho_1)}\int^{T}_{-T}
\Big(\sup_{\theta\in[t-r,t]}\|f_2(\theta+\sigma)\|^{p}\Big)\rho_2(t)
 dt=0.
$$
by Lebesgue's dominated convergence theorem, one has
$$
\int^{1}_{0}\Big(\frac{1}{\mu(T,\rho_1)}\int^{T}_{-T}
\Big(\sup_{\theta\in[t-r,t]}\|f_2(\theta+\sigma)\|^{p}\Big)\rho_2(t)
 dt\Big)d\sigma\to 0,\quad T\to\infty,
$$
i.e.,
$$
\frac{1}{\mu(T,\rho_1)}\int^{T}_{-T}\rho_2(t)
\Big(\int^{1}_{0}\sup_{\theta\in[t-r,t]}\|f_2(\theta+\sigma)\|^{p}d\sigma
\Big)dt\to 0,\quad T\to\infty,
$$
which means that
$$
\frac{1}{\mu(T,\rho_1)}\int^{T}_{-T}\rho_2(t)\sup_{\theta\in[t-r,t]}
\Big(\int^{1}_{0}\|f_2(\theta+\sigma)\|^{p}d\sigma \Big)dt\to
0,\quad T\to\infty;
$$
i.e.,
$$
\frac{1}{\mu(T,\rho_1)}\int^{T}_{-T}\rho_2(t)\sup_{\theta\in[t-r,t]}
\Big(h_2(\theta) \Big)dt\to 0,\quad T\to\infty,
$$
so $h_2\in WPP_{0}({\mathbb{R}},
{\mathbb{R}}^{+}, r, \rho_1,\rho_2)$, where
$$
h_2(t)=\int^{1}_{0}\|f_2(t+\sigma)\|^{p}d\sigma,\quad t\in{\mathbb{R}}.
$$
By Lemma \ref{ifonlyif2}, $h^{1/p}_2\in WPP_{0}({\mathbb{R}}, {\mathbb{R}}^{+},r,
\rho_1,\rho_2)$; i.e.,
$$
\frac{1}{\mu(T,\rho_1)}\int^{T}_{-T}\rho_2(t)\sup_{\theta\in[t-r,t]}\Big(
\int^{1}_{0}\|f_2(\theta+\sigma)\|^{p}d\sigma \Big)^{1/p}dt\to
0,\quad T\to\infty,
$$
which means that $f_2\in S^{p}WPP_{0}({\mathbb{R}}, X, r, \rho_1,\rho_2)$, then
$f\in S^{p}WPP_{\omega}({\mathbb{R}},X,r, \rho_1,\rho_2)$. The proof is complete.
\end{proof}

\begin{theorem}\label{comspwpaa}
Assume  $\rho_1,\rho_2\in U_{\infty}$, $f=f_1+f_2\in
S^{p}WPP_{\omega}(\mathbb{R}\times Y, X,r,\rho_1,\rho_2)$ with
$f_2\in S^{p}WPP_{0}(\mathbb{R}\times Y, X,r, \rho_1,\rho_2)$,
$f_1(t+\omega, u)-f_1(t, u)=0$ a. e. $t\in{\mathbb{R}}$, $u\in X$, and
 there exists a function $L_{f}:{\mathbb{R}}\to [0,+\infty)$
satisfying:
\begin{itemize}
\item[(A1')] $\Big(\int^{t+1}_t\|f(s,u)-f(s,v)\|^{p}ds\Big)^{1/p}
\leq L_{f}(t)\|u-v\|$, for all $ t\in{\mathbb{R}}$, $u,v\in Y$;

\item[(A2')]
$\limsup_{T\to\infty}\frac{1}{\mu(T,\rho_1)}\int^{T}_{-T}
\Big(\sup_{\theta\in[t-r,t]}L_{f}(\theta)\Big)\rho_2(t)dt<\infty$;

\item[(A3')]
$\lim_{T\to\infty}\frac{1}{\mu(T,\rho_1)}\int^{T}_{-T}
\Big(\sup_{\theta\in[t-r,t]}L_{f}(\theta)\Big)\xi(t)\rho_2(t)dt=0$
for each function $\xi^{b}\in WPP_{0}({\mathbb{R}},L^{p}([0,1],{\mathbb{R}}),\rho_1,\rho_2)$;

\item[(A4')] $f_1$ is uniform continuous on bounded set $K'\subset Y$
for any $t\in {\mathbb{R}}$.
\end{itemize}
Then $f(\cdot,h(\cdot))\in S^{p}WPP_{\omega}(\mathbb{R},X,r,\rho_1,\rho_2)$ if
$h \in S^{p}WPP_{\omega}(\mathbb{R},Y,r,\rho_1,\rho_2)$.
\end{theorem}

The proof of the above theorem is similar to the proof of \cite[Theorem 3.2]{zhchgmn13}
and it is omitted here.
It is not difficult to see that Theorem \ref{comwpaa}, Theorem
\ref{comspwpaa}  hold for $WPP_{\omega
ap}(\mathbb{R},X,r,\rho_1,\rho_2)$, $S^{p}WPP_{\omega
ap}(\mathbb{R},X,r,\rho_1,\rho_2)$ respectively.

\section{Neutral functional differential equations}

As an application, the main goal of this section is to establish
sufficient criteria for the existence, uniqueness of the weighted
pseudo periodic solution to a class of neutral functional
differential equations of the form
\begin{equation}\label{neutral}
\frac{\mathrm{d}}{\mathrm{d}t}[u(t)+f(t,u_t)]
=Au(t)+g(t,u_t),\quad t\in\mathbb{R},
\end{equation}
where $A$ is the infinitesimal generator of a semigroup of linear
operators on $X$, $u_t\in \mathcal{C}$ is defined by
$u_t(\theta)=u(t+\theta)$ for $\theta\in [-r, 0]$, where $r$ is a
nonnegative constant.

First, we recall the definition of the so called exponential
dichotomy of a semigroup.

\begin{definition} \cite{lubook} \rm
A semigroup $(T(t))_{t\geq 0}$ is said to be
exponential dichotomy if there exist projection $P$ and constants
$M,\delta>0$ such that each $T(t)$ commutes with $P$, $Ker P$ is
invariant with respect to $T(t)$, $T(t): Im Q\to Im Q$ is
invertible and
\begin{gather}
\|T(t)Px\|\leq Me^{-\delta t}\|x\|\quad \text{for } t\geq 0, \label{estip} \\
\|T(t)Qx\|\leq Me^{\delta t}\|x\| \quad \text{for } t\leq 0, \label{estiq}
\end{gather}
where $Q:=I-P$ and $T(t):= (T(-t))^{-1}$ for $t\leq 0$.
\end{definition}

To study \eqref{neutral}, we make the following assumptions:
\begin{itemize}
\item [(H1)] The operator $A: D(A)\subset X\to X$
is the infinitesimal generator of semigroup $(T(t))_{t\geq 0}$
which has an exponential dichotomy.

\item [(H2)] $f\in WPP_{\omega}({\mathbb{R}}\times \mathcal{C}, X,r,\rho_1, \rho_2)$,
 $f$ is $D(A)$-valued, there exists a positive constant
$L_{f}$ such that
$$
 |f(t, \psi_1) -f(t, \psi_2) \|_{[D(A)]}
\leq L_{f}\|\psi_1-\psi_2\|_{\mathcal{C}}, \quad \text{for all }
 t\in {\mathbb{R}},\; \psi_{i}\in \mathcal{C},\; i=1,2.
$$

\item [(H3)] $g\in WPP_{\omega}({\mathbb{R}}\times \mathcal{C}, X,r,\rho_1, \rho_2)$
 and there exists a continuous function
 $L_{g}: {\mathbb{R}}\to {\mathbb{R}}^{+}$ such that
$$
 \|g(t, \psi_1) -g(t, \psi_2) \| \leq
 L_{g}(t)\|\psi_1-\psi_2\|_{\mathcal{C}}, \quad \text{for all }
 t\in {\mathbb{R}},\; \psi_{i}\in \mathcal{C},\;  i=1,2.
$$

\item [(H3')] $g=g_1+g_2\in S^{p}WPP_{\omega}({\mathbb{R}}\times \mathcal{C}, X,r,\rho_1,
\rho_2)$ with $g_2\in S^{p}WPP_{0}({\mathbb{R}}\times \mathcal{C},
X,r,\rho_1, \rho_2)$, $g_1(t+\omega, \psi)-g(t, \psi)=0~a.e.$
$t\in{\mathbb{R}}$, $\psi\in \mathcal{C}$, satisfying

 (i) $g_1$ is uniform continuous on bounded set $K\subset\mathcal{C}$
for any $t\in {\mathbb{R}}$.

 (ii) there exists a positive constant $L_{g}$ such that
 $$
 \Big(\int^{t+1}_t\|g(t, \psi_1)-g(t, \psi_2)\|^{p}ds\Big)^{1/p}
\leq L_{g}\|\psi_1-\psi_2\|_{\mathcal{C}},
 \quad \text{for all } t\in {\mathbb{R}}, \psi_{i}\in \mathcal{C},
 i=1,2.
$$

\item [(H4)] $\rho_1, \rho_2\in U_T$,
$\inf_{T>0}\frac{\mu(T,\rho_2)}{\mu(T,\rho_1)}=\delta_{0}>0$ and
$\sup_{T>0}\frac{\mu(T,\rho_2)}{\mu(T,\rho_1)}<\infty$.
\end{itemize}

\begin{definition}[\cite{diaaga09}] \rm
 A continuous function $u$ is said to be a mild
solution of (\ref{neutral}) provided that the function $s\to
AT(t-s)Pf(s,u_{s})$ is integrable on $(-\infty,t)$, $s\to
AT(t-s)Qf(s,u_{s})$ is integrable on $(t, \infty)$ for $t\in
\mathbb{R}$ and
\begin{align*}
u(t)&=-f(t,u_t)-\int^{t}_{-\infty}AT(t-s)Pf(s,u_{s})ds
+\int^{\infty}_tAT(t-s)Qf(s,u_{s})ds\\
&\quad +\int^{t}_{-\infty}T(t-s)Pg(s,u_{s})ds-\int^{\infty}_tT(t-s)Qg(s,u_{s})ds,
\quad t\in \mathbb{R}.
\end{align*}
\end{definition}

\begin{lemma}\label{lemma}
Assume that {\rm (H1), (H2), (H4)} hold, if $u\in
WPP_{\omega}({\mathbb{R}}, X,r,\rho_1, \rho_2)$, then
\begin{gather*}
 (\Lambda_1 f)(t)=\int^{t}_{-\infty}AT(t-s)Pf(s, u_{s})ds
 \in WPP_{\omega}({\mathbb{R}}, X,r,\rho_1, \rho_2).\\
 (\Lambda_2 f)(t)=\int^{\infty}_tAT(t-s)Qf(s, u_{s})ds \in
WPP_{\omega}({\mathbb{R}}, X,r,\rho_1, \rho_2).
\end{gather*}
\end{lemma}

\begin{proof}
By Lemma \ref{delay} and Theorem \ref{comwpaa},
$f(s, u_{s}):=h(s)\in WPP_{\omega}(\mathbb{R},X,r,\rho_1,\rho_2)$ and $h$ is
$D(A)$-valued. Let $h(s)=h_1(s)+h_2(s)$ where $h_1\in
P_{\omega}({\mathbb{R}}, X)$ and $h_2\in WPP_{0}({\mathbb{R}}, X,r,\rho_1,\rho_2)$.
Then
\begin{align*}
(\Lambda_1 f)(t)
&=\int^{t}_{-\infty}AT(t-s)Ph_1(s)ds+\int^{t}_{-\infty}AT(t-s)Ph_2(s)ds\\
&:=(\Lambda_{11} h_1)(t)+(\Lambda_{12} h_2)(t),
\end{align*}
where
$$
(\Lambda_{11} h_1)(t)=\int^{t}_{-\infty}AT(t-s) Ph_1(s)ds,
\quad (\Lambda_{12} h_2)(t)=\int^{t}_{-\infty}AT(t-s) Ph_2(s)ds,
$$
for $t\in{\mathbb{R}}$.
From $h_1\in P_{\omega}({\mathbb{R}}, X)$,
$$
(\Lambda_{11} h_1)(t+\omega)=\int^{t+\omega}_{-\infty}AT(t+\omega-s)Ph_1(s)ds
=(\Lambda_{11} h_1)(t);
$$
then $\Lambda_{11} h_1\in P_{\omega}(\mathbb{R},X)$.

Next, we show that $\Lambda_{12} h_2\in
WPP_{0}(\mathbb{R},X,r,\rho_1,\rho_2)$; that is,
$$
\lim_{T\to\infty}\frac{1}{\mu(T,\rho_1)}\int^{T}_{-T}
\Big(\sup_{\theta\in[t-r,t]}\|(\Lambda_{12} h_2)
(\theta)\|\Big)\rho_2(t)dt=0.
$$
In fact, for $T>0$, one has
\begin{align*}
&\frac{1}{\mu(T,\rho_1)}\int^{T}_{-T}
\Big(\sup_{\theta\in[t-r,t]}\|(\Lambda_{12} h_2)
(\theta)\|\Big)\rho_2(t)dt \\
&=\frac{1}{\mu(T,\rho_1)}\int^{T}_{-T}\Big(\sup_{\theta\in[t-r,t]}
 \|\int^{\theta}_{-\infty}AT(\theta-s)Ph_2(s)ds\|\Big)\rho_2(t)dt \\
&=\frac{1}{\mu(T,\rho_1)}\int^{T}_{-T}\Big(\sup_{\theta\in[t-r,t]}
 \|\int^{\infty}_{0}AT(s)Ph_2(\theta-s)ds\|\Big)\rho_2(t)dt\\
&\leq
\frac{1}{\mu(T,\rho_1)}\int^{T}_{-T}\Big(\sup_{\theta\in[t-r,t]}
 \int^{\infty}_{0}Me^{-\delta s}\|Ah_2(\theta-s)\|ds\Big)\rho_2(t)dt\\
&\leq
\frac{1}{\mu(T,\rho_1)}\int^{T}_{-T}\Big(\sup_{\theta\in[t-r,t]}
 \int^{\infty}_{0}Me^{-\delta s}\|h_2(\theta-s)\|_{[D(A)]}ds\Big)\rho_2(t)dt\\
&\leq \int^{\infty}_{0}Me^{-\delta s}\Phi_T(s)ds,
\end{align*}
where
$$
\Phi_T(s)=\frac{1}{\mu(T,\rho_1)} \int^{T}_{-T}
 \Big(\sup_{\theta\in[t-r,t]}\|h_2(\theta-s)\|_{[D(A)]}\Big)\rho_2(t)dt.
$$
Since $\rho_1,\rho_2\in U_T$, by Lemma \ref{trans}, we have
$h_2(\cdot-s)\in WPP_{0}(\mathbb{R},
[D(A)],r,\rho_1,\rho_2)$ for each $s\in\mathbb{R}$; hence
$\lim_{T\to\infty}\Phi_T(s) =0$ for all $s\in {\mathbb{R}}$.
Then $\Lambda_{12} h_2$ belongs to $WPP_{0}(\mathbb{R},X,r,\rho_1,\rho_2)$
by using the Lebesgue 
dominated convergence theorem, so $\Lambda_1 f\in
WPP_{\omega}(\mathbb{R},X,r,\rho_1,\rho_2)$.

The proof of $\Lambda_2 f$ is similar to that of $\Lambda_1 f$,
one makes use of \eqref{estiq} rather than \eqref{estip}. This
completes the proof.
\end{proof}


\begin{lemma}\label{lemma2}
Assume that {\rm (H1), (H4)} hold, if
$\phi\in S^{p}WPP_{\omega}({\mathbb{R}}, X,r,\rho_1, \rho_2)$, then
\begin{gather*}
 (\Gamma_1 \phi)(t)=\int^{t}_{-\infty}T(t-s)P\phi(s)ds
\in WPP_{\omega}({\mathbb{R}}, X,r,\rho_1, \rho_2), \\
 (\Gamma_2 \phi)(t)=\int^{\infty}_tT(t-s)Q\phi(s)ds \in
WPP_{\omega}({\mathbb{R}}, X,r,\rho_1, \rho_2).
\end{gather*}
\end{lemma}

\begin{proof}
By $\phi\in S^{p}WPP_{\omega}(\mathbb{R},X,r,\rho_1,\rho_2)$,
we let $\phi(s)=\phi_1(s)+\phi_2(s)$, where $\phi_2\in
S^{p}WPP_{0}({\mathbb{R}}, X,r,\rho_1,\rho_2)$ and
$\phi_1(t+\omega)-\phi_1(t)=0$ a.e. $t\in{\mathbb{R}}$, then
$$
(\Gamma_1 \phi)(t)=\int^{t}_{-\infty}T(t-s)P\phi_1(s)ds
+\int^{t}_{-\infty}T(t-s)P\phi_2(s)ds:=(\Gamma_{11}
\phi_1)(t)+(\Gamma_{12} \phi_2)(t).
$$
First, we show that $\Gamma_{12} \phi_2\in
WPP_{0}(\mathbb{R},X,r,\rho_1,\rho_2)$. Consider the integrals
$$
Y_{n}(t)=\int^{t-n+1}_{t-n}T(t-s)P\phi_2(s)ds.
$$
Fix $n\in \mathbb{N}$ and $t\in \mathbb{R}$, we have
\begin{align*}
\|Y_{n}(t+h)-Y_{n}(t)\|
&\leq \int^{n}_{n-1}\|T(s)P(\phi_2(t+h-s)-\phi_2(t-s))\|ds\\
&\leq  M\int^{t-n+1}_{t-n}\|\phi_2(s+h)-\phi_2(s)\|ds\\
&\leq M\left(\int^{t-n+1}_{t-n}\|\phi_2(s+h)-\phi_2(s)\|^{p}ds\right)^{1/p}.
\end{align*}
In view of $\phi_2\in L^{p}_{loc}(\mathbb{R},X)$, we get
$$
\lim_{h\to 0}\int^{t-n+1}_{t-n}\|\phi_2(s+h)-\phi_2(s)\|^{p}ds=0,
$$
which yields
$\lim_{h\to 0}\|Y_{n}(t+h)-Y_{n}(t)\|=0$.
This means that $Y_{n}(t)$ is continuous.

By H\"{o}lder's inequality, one has
\begin{align*}
\|Y_{n}(t)\|
&\leq\int^{n}_{n-1}\|T(s) P \phi_2(t-s)\|ds\\
&\leq\int^{n}_{n-1}Me^{-\delta s}\|\phi_2(t-s)\|ds\\
&\leq Me^{-\delta (n-1)}\int^{n}_{n-1}\|\phi_2(t-s)\|ds\\
&\leq Me^{-\delta (n-1)}\int^{t-n+1}_{t-n}\|\phi_2(s)\|ds\\
&\leq Me^{-\delta (n-1)}\Big(\int^{t-n+1}_{t-n}\|\phi_2(s)\|^{p}ds\Big)^{1/p}\\
&\leq Me^{-\delta (n-1)}\|\phi_2\|_{S^{p}}.
\end{align*}
Since
$$
\sum^{\infty}_{n=1}Me^{-\delta (n-1)}\|\phi_2\|_{S^{p}}\leq
\frac{M}{1-e^{- \delta}}\|\phi_2\|_{S^{p}}<+\infty,
$$
it follows that $\sum^{\infty}_{n=1}Y_{n}(t)$ converges  uniformly
on $\mathbb{R}$. Let $Y(t)=\sum_{n=1}^{\infty}Y_{n}(t)$ for
$t\in \mathbb{R}$. Then
$$
Y(t)=(\Gamma_{12} \phi_2)(t)=\int^{t}_{-\infty}T(t-s)P\phi_2(s)ds,\quad
t\in \mathbb{R}.
$$
It is obvious that $Y(t)\in BC(\mathbb{R},X)$. So, we only need to
show that
\begin{equation}\label{wpaay}
\lim_{T\to \infty}\frac{1}{\mu(T,\rho_1)}
\int^{T}_{-T}\rho_2(t)\Big(\sup_{\theta\in[t-r,t]}\|Y(\theta)\|\Big)dt=0.
\end{equation}
In fact, by H\"{o}lder inequality,
\begin{align*}
\|Y_{n}(t)\|&\leq \int^{n}_{n-1}Me^{-\delta s} \|\phi_2(t-s)\|ds\\
 &\leq  \widetilde{M} \int^{t-n+1}_{t-n}\|\phi_2(s)\|ds\\
 &\leq \widetilde{ M} \Big(\int^{t-n+1}_{t-n}\|\phi_2(s)\|^{p}ds\Big)^{1/p},
\end{align*}
for some constant $\widetilde{ M}>0$;
then
\begin{align*}
&\frac{1}{\mu(T,\rho_1)}
\int^{T}_{-T}\rho_2(t)\Big(\sup_{\theta\in[t-r,t]}\|Y_{n}(\theta)\|\Big)dt\\
&\leq \frac{\widetilde{M}}{\mu(T,\rho_1)}\int^{T}_{-T}\rho_2(t)
\Big(\sup_{\theta\in[t-r,t]}\Big(\int^{\theta-n+1}_{\theta-n}
 \|\phi_2(s)\|^{p}ds\Big)^{1/p}\Big)dt,
\end{align*}
and hence $Y_{n}\in WPP_{0}(\mathbb{R},X,r,\rho_1,\rho_2)$
since $\phi_2\in S^{p}WPP_{0}(\mathbb{R},X,r,\rho_1,\rho_2)$.
By Lemma \ref{paauniform}, \eqref{wpaay} holds, whence
$\Gamma_{12}\phi_2 \in WPP_{0}(\mathbb{R},X,r,\rho_1,\rho_2)$.

From $\phi_1(t+\omega)-\phi_1(t)=0$ a.e. $t\in{\mathbb{R}}$, one has
$$
(\Gamma_{11} \phi_1)(t+\omega)=\int^{t+\omega}_{-\infty}T(t+\omega-s)
P\phi_1(s)ds=(\Gamma_{11} \phi_1)(t),\quad \text{a.e. }t\in{\mathbb{R}}.
$$
Hence $\Gamma_1 \phi \in WPP_{\omega}(\mathbb{R},X,r,\rho_1,\rho_2)$.

The proof of $\Gamma_2 \phi$ is similar to that of $\Gamma_1 \phi$,
one  uses \eqref{estiq} rather than \eqref{estip}.
This completes the proof.
\end{proof}

\begin{theorem}\label{theorem}
Assume that {\rm (H1)--(H4)} hold and $g$ satisfy the conditions
{\rm (A2)--(A3)}, if
$$
\vartheta:=\Big( L_{f}+ \frac{2M L_{f}}{\delta} +
M\sup_{t\in{\mathbb{R}}}\int^{t}_{-\infty} e^{-\delta(t-s)} L_{g}(s) ds
+ M\sup_{t\in{\mathbb{R}}}\int^{\infty}_t e^{\delta(t-s)} L_{g}(s)
ds\Big)<1,
$$
then \eqref{neutral} has a unique mild solution of
$WPP_{\omega}$ type.
\end{theorem}

\begin{proof}
Define $\mathcal{F}:
WPP_{\omega}(\mathbb{R},X,r,\rho_1,\rho_2)\to
WPP_{\omega}(\mathbb{R},X,r,\rho_1,\rho_2)$ as
\begin{equation} \label{operator}
\begin{aligned}
(\mathcal{F} u)(t)
&=-f(t,u_t)-\int^{t}_{-\infty}AT(t-s)Pf(s,u_{s})ds
 +\int^{\infty}_tAT(t-s)Qf(s,u_{s})ds  \\
&\quad +\int^{t}_{-\infty}T(t-s)Pg(s,u_{s})ds
 -\int^{\infty}_tT(t-s)Qg(s,u_{s})ds,\quad t\in \mathbb{R}.
\end{aligned}
\end{equation}
If $u\in WPP_{\omega}(\mathbb{R},X,r,\rho_1,\rho_2)$, then
$u_t\in WPP_{\omega}(\mathbb{R},\mathcal{C},r,\rho_1,\rho_2)$
by Lemma \ref{delay}; therefore by Theorem \ref{comwpaa},
\[
g(s,u_{s}) \in WPP_{\omega}(\mathbb{R},X,r,\rho_1,\rho_2)\subset
S^{p}WPP_{\omega}(\mathbb{R},X,r,\rho_1,\rho_2).
\]
By Lemmas \ref{lemma} and  \ref{lemma2}, it is not difficult to see that
$\mathcal{F}$ is well defined.

For any $u,v\in WPP_{\omega}(\mathbb{R},X,r,\rho_1,\rho_2)$, we
have
\begin{align*}
&\|(\mathcal{F} u)(t)-(\mathcal{F} v)(t)\| \\
&\leq \|f(t,u_t)-f(t,v_t)\|
 +\int^{t}_{-\infty}\|AT(t-s)P(f(s,u_{s})-f(s,v_{s}))\|ds\\
&\quad +\int^{\infty}_t\|AT(t-s)Q(f(s,u_{s})-f(s,v_{s}))\|ds\\
&\quad +\int^{t}_{-\infty}\|T(t-s)P(g(s,u_{s})-g(s,v_{s}))\|ds \\
&\quad +\int^{\infty}_t\|T(t-s)Q(g(s,u_{s})-g(s,v_{s}))\|ds \\
&\leq L_{f}\|u_t-u_t\|_{\mathcal{C}}+M L_{f}
\int^{t}_{-\infty}e^{-\delta(t-s)}\|u_{s}-u_{s}\|_{\mathcal{C}}ds\\
&\quad +M L_{f}\int^{\infty}_te^{\delta(t-s)}\|u_{s}-u_{s}\|_{\mathcal{C}}ds
 +M\int^{t}_{-\infty} e^{-\delta(t-s)} L_{g}(s)\|u_{s}-u_{s}\|_{\mathcal{C}}ds\\
&\quad +M\int^{\infty}_t e^{\delta(t-s)} L_{g}(s)\|u_{s}-u_{s}\|_{\mathcal{C}}ds\\
&\leq \Big( L_{f}+ \frac{2M L_{f}}{\delta} +
M\sup_{t\in{\mathbb{R}}}\int^{t}_{-\infty} e^{-\delta(t-s)} L_{g}(s) ds\\
&\quad + M\sup_{t\in{\mathbb{R}}}\int^{\infty}_t e^{\delta(t-s)} L_{g}(s) ds\Big) \|u-v\|\\
&\leq \vartheta \|u-v\|,
\end{align*}
then $\mathcal{F}$ is a contraction since $\vartheta<1$. By the
Banach contraction mapping principle, $\mathcal{F}$ has a unique
fixed point in $WPP_{\omega}(\mathbb{R},X,r,\rho_1,\rho_2)$,
which is the unique $WPP_{\omega}$ solution to \eqref{neutral}.
\end{proof}

In (H3), if $L_{g}(t)\equiv L_{g}$, It is not difficult to see
that $g$ satisfy the conditions (A2)--(A3) and $\vartheta=
L_{f}+ 2M(L_{f}+L_{g})/\delta$.


\begin{theorem}\label{theoremh}
Assume that {\rm (H1), (H2), (H3'), (H4)} hold. If
$$
\Theta:=\Big( L_{f}+ \frac{2M L_{f}}{\delta} + \frac{2ML_{g}}{1-e^{-\delta}} \Big) <1,
$$
then \eqref{neutral} has a unique mild solution of $WPP_{\omega}$ type.
\end{theorem}

\begin{proof}
Define the operator $\mathcal{F}$ as in \eqref{operator}.
Let $u\in WPP_{\omega}(\mathbb{R},X,r,\rho_1,\rho_2)$, then it is not
difficult to see that $g(s, u_{s})\in S^{p}WPP_{\omega}(\mathbb{R},X,r,\rho_1,\rho_2)$
 by Theorem \ref{comspwpaa}, so $\Gamma$ is well defined by Lemma \ref{lemma}
and Lemma \ref{lemma2}.

Let $u,v\in WPP_{\omega}(\mathbb{R},X,r,\rho_1,\rho_2)$, one has
\begin{align*}
&\|(\mathcal{F} u)(t)-(\mathcal{F} v)(t)\| \\
&\leq L_{f}\|u_t-v_t\|_{\mathcal{C}}+M L_{f}
\int^{t}_{-\infty}e^{-\delta(t-s)}\|u_{s}-v_{s}\|_{\mathcal{C}}ds\\
&\quad +M L_{f}\int^{\infty}_te^{\delta(t-s)}\|u_{s}-v_{s}\|_{\mathcal{C}}ds
 + M\int^{t}_{-\infty} e^{-\delta(t-s)} \|g(s,u_{s})-g(s,v_{s})\| ds \\
&\quad + M\int^{\infty}_t e^{\delta(t-s)} \|g(s,u_{s})-g(s,v_{s})\| ds\\
&\leq L_{f}\|u_t-v_t\|_{\mathcal{C}}+M L_{f}
\int^{t}_{-\infty}e^{-\delta(t-s)}\|u_{s}-v_{s}\|_{\mathcal{C}}ds\\
&\quad +M L_{f}\int^{\infty}_te^{\delta(t-s)}\|u_{s}-v_{s}\|_{\mathcal{C}}ds
 +M\int^{\infty}_{0} e^{-\delta s} \|g(t-s,u_{t-s})-g(t-s,v_{t-s})\| ds \\
&\quad +M\int^{0}_{-\infty} e^{\delta s} \|g(t-s,u_{t-s})-g(t-s,v_{t-s})\| ds\\
&\leq \Big(L_{f}+\frac{2ML_{f}}{\delta}\Big)\|u-v\|
 + M\sum_{k=0}^{\infty} \int^{k+1}_{k} e^{-\delta s} \|g(t-s,u_{t-s})-g(t-s,v_{t-s})\|ds
 \\
&\quad + M\sum_{k=-\infty}^{0} \int^{k}_{k-1} e^{\delta s}
 \|g(t-s,u_{t-s})-g(t-s,v_{t-s})\| ds\\
&\leq \Big(L_{f}+\frac{2ML_{f}}{\delta}\Big)\|u-v\|
 +M\sum_{k=0}^{\infty}e^{-\delta k}
 \Big(\int^{t-k}_{t-k-1} \|g(s,u_{s})-g(s,v_{s})\|^{p}ds\Big)^{1/p}\\
&\quad + M\sum_{k=-\infty}^{0}e^{\delta k}
\Big(\int^{t-k+1}_{t-k} \|g(s,u_{s})-g(s,v_{s})\|^{p}ds\Big)^{1/p} \\
&\leq \Big( L_{f}+ \frac{2M L_{f}}{\delta}+M L_{g}\sum_{k=0}^{\infty}
 e^{-\delta k}+ M\sum_{k=-\infty}^{0}e^{\delta k} \Big) \|u-v\|\\
&\leq \Big( L_{f}+ \frac{2M L_{f}}{\delta} +\frac{2ML_{g}}{1-e^{-\delta}} \Big) \|u-v\|\\
&\leq \Theta \|u-v\|.
\end{align*}
By the Banach contraction mapping principle, $\mathcal{F}$ has a
unique fixed point in
$WPP_{\omega}(\mathbb{R},X,r,\rho_1,\rho_2)$, which is the
unique $WPP_{\omega}$ solution to \eqref{neutral}.
\end{proof}


\begin{remark} \rm
It is easy to see that similar results of Theorem \ref{theorem} and
Theorem \ref{theoremh}  hold for $WPP_{\omega ap}({\mathbb{R}},X,r,\rho_1,\rho_2)$
mild solution, that is \eqref{neutral} has a unique $WPP_{\omega ap}$
mild solution, in this case,
$f\in WPP_{\omega ap}({\mathbb{R}}\times \mathcal{C}, X,r,\rho_1,\rho_2)$ in (H2),
$g\in WPP_{\omega ap}({\mathbb{R}}\times \mathcal{C}, X,r,\rho_1, \rho_2)$ in (H3),
$g\in S^{p}WPP_{\omega ap}({\mathbb{R}}\times \mathcal{C}, X,r,\rho_1, \rho_2)$ in (H3').
\end{remark}

\section{Examples}

In this section, we provide some examples to illustrate our main
results.

\begin{example} \rm
Consider the partial differential equation
\begin{equation} \label{exameq}
\begin{gathered}
\begin{aligned}
&\frac{\partial}{\partial
t}\Big[u(t,\xi)+\int^{0}_{-r}\int_{0}^{\pi}b(s,\eta,\xi)u(t+s,\eta)d\eta ds\Big]\\
& =\frac{\partial^{2}}{\partial \xi^{2}}u(t,\xi)
 +a_{0}(t)u(t,\xi)+\int^{0}_{-r}a_1(s)u(t+s,\xi)ds,
\quad (t,\xi)\in {\mathbb{R}}\times[0,\pi],
\end{aligned} \\
u(t,0)=u(t,\pi)=0,
\end{gathered}
\end{equation}
where $a_{0}\in WPP_{\omega}({\mathbb{R}},{\mathbb{R}},r,\rho_1,\rho_2)$,
$\rho_1=e^{t}, \rho_2=1+t^{2}$ and the following conditions
hold:

 The functions $b(\cdot)$,
 $\frac{\partial^{i}}{\partial \zeta^{i}}b(\tau, \eta, \zeta)$, $i=1, 2$
are (Lebesgue) measurable, $b(\tau, \eta,0)=b(\tau, \eta,\pi)=0$
for every $(\tau, \eta)$ and
$$
N_1:=\max\Big\{ \int_{0}^{\pi}\int^{0}_{-r}\int_{0}^{\pi}
\Big(\frac{\partial^{i}}{\partial \zeta^{i}}b(\tau, \eta,
\zeta)\Big)^{2}d\eta d\tau d\zeta,: i=0, 1, 2 \Big\}<\infty.
$$
\end{example}
Under these conditions, let
$X=(L^{2}([0,\pi],{\mathbb{R}}),\|\cdot\|_{L^{2}})$ and define the operator
$A$ on $X$ by $Au=u''$
with
$$
D(A)=\{u\in X: u''\in X, u(0)=u(\pi)=0 \}.
$$
It is well known that $A$ is the infinitesimal generator of a
semigroup $(T(t))_{t\geq 0}$ on $X$ such that $\|T(t)\|\leq e^{-t}$
for every $t\geq 0$.

Define the functions $f, g: {\mathbb{R}}\times \mathcal{C}\to X$ by
\begin{gather*}
 f(t,\psi)(\xi) :=\int_{-r}^{0}\int_{0}^{\pi}b(s,\eta,\xi)\psi(s,\eta)\,d\eta \,ds, \\
 g(t,\psi)(\xi) :=a_{0}(t)\psi(0,\xi)+\int^{0}_{-r}a_1(s)\psi(s,\xi)\,ds,
\end{gather*}
then \eqref{exameq} can be rewritten as an abstract system of the
form \eqref{neutral}, where $u(t)=u(t,\cdot)$. By a straightforward
estimation that uses $(i)$, one can show that $f$ is
$D(A)$-valued, and the following hold:
\begin{gather*}
 \|Af(t,\cdot)\|\leq (N_1r)^{1/2}, \quad t\in{\mathbb{R}}, \\
\|g(t,\cdot)\|\leq \|a_{0}\|+r^{1/2}\Big(\int^{0}_{-r}a_1^{2}(s)ds\Big)^{1/2},
\quad t\in{\mathbb{R}}.
\end{gather*}
See \cite{agadiaher07,diaher07} for more details. The next
result is a consequence of Theorem \ref{theorem}.


\begin{theorem}
Under the previous assumptions, \eqref{exameq} as a unique weighted
pseudo periodic solution whenever
$$
3\sqrt{N_1r}+2\|a_{0}\|+2r^{1/2}\Big(\int^{0}_{-r}a_1^{2}(s)ds\Big)^{1/2}<1.
$$
\end{theorem}

\begin{example} \rm
Consider the following scalar reaction-diffusion equation with
delay
\begin{equation} \label{examequation}
\begin{gathered}
\frac{\partial }{\partial t}u(t,x)=\frac{\partial^{2} }{\partial x^{2}}u(t,x)
 +g(t, u(t-r,x)), \\
u(t,0)=u(t,\pi)=0, \\
u(\tau,x)=\varphi(\tau,x),\quad \tau\in[-r,0],\; x\in[0,\pi],
\end{gathered}
\end{equation}
where $g=g_1+g_2\in S^{p}WPP_{\omega}({\mathbb{R}}\times
\mathcal{C},{\mathbb{R}},r,\rho_1,\rho_2)$ with
$g_1\in P_{\omega}({\mathbb{R}}\times \mathcal{C},{\mathbb{R}})$,
$g_2\in S^{p}WPP_{0}({\mathbb{R}}\times \mathcal{C},{\mathbb{R}},r,\rho_1,\rho_2)$,
$\rho_1=e^{t}, \rho_2=1+t^{2}$.
\end{example}

By Theorem \ref{theoremh}, one has the following result.

\begin{theorem}
Assume that there exists a positive constant $L_{g}$ such that
for $i=1,2$,
$$
\Big(\int^{t+1}_t\|g(t, \psi_1)-g(t, \psi_2)\|^{p}ds\Big)^{1/p}
\leq L_{g}\|\psi_1-\psi_2\|_{\mathcal{C}},
 \quad \text{for all } t\in {\mathbb{R}},\; \psi_{i}\in \mathcal{C}\,.
$$
Then there exists a unique $WPP_{\omega}$ solution of
\eqref{examequation} if $2L_{g}<1-e^{-1}$.
\end{theorem}

\subsection*{Acknowledgements}
This research was supported by Zhejiang Provincial
Natural Science Foundation of China under Grant No. LQ13A010015.


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\end{document}