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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2015 (2015), No. 185, pp. 1--18.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2015 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2015/185\hfil Pseudo almost periodic solutions]
{Piecewise weighted pseudo almost periodic solutions of
impulsive integro-differential equations via fractional operators}

\author[Z. Xia, D. Wang \hfil EJDE-2015/185\hfilneg]
{Zhinan Xia, Dingjiang Wang}

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

\address{Dingjiang Wang (corresponding author) \newline
 Department of Applied Mathematics,
 Zhejiang University of Technology,
 Hangzhou, Zhejiang  310023,  China}
\email{wangdingj@126.com}

\thanks{Submitted October 7, 2014. Published July 10, 2015.}
\subjclass[2010]{35R12, 35B15}
\keywords{Impulsive integro-differential equation;
  pseudo almost periodicity;
\hfill\break\indent   fractional powers of operators; classical solution}

\begin{abstract}
 In this article, we show sufficient conditions for the existence,
 uniqueness and attractivity  of piecewise weighted pseudo almost
 periodic classical solution of nonlinear impulsive
 integro-differential equations. The working tools are
 based on the fixed point theorem and fractional powers of operators.
 An application to impulsive integro-differential equations is
 presented.
\end{abstract}

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


\section{Introduction}

It is well known that impulsive integro-differential equations or
impulsive differential equations create an important subject of
numerous mathematical investigations and constitute a significant
branch of differential equations.  It has great applications in
actual modeling such as population dynamics, epidemic, engineering,
optimal control, neural networks, economics, etc. With the help of
several tools of functional analysis, topology and fixed point
theorems, many authors have made important contributions to this
theory
\cite{baisimbook95,che13,hendearab11,liuzha12,liuzha13-2,stabook12}.


The concept of  pseudo almost periodic function,
which was introduced by Zhang \cite{zhang94,zhang95}, is a natural
and good generalization of the classical almost periodic functions
in the sense of Bohr. Recently,   weighted pseudo almost periodic
function is investigated in \cite{diagana2006} by the weighted
function, which was more tricky and changeable than those of the
classical functions. Many authors have made important contributions
to this function. For more  details on weighted pseudo almost
periodic function and related topics, one can see
\cite{ab11,chazhagmn12,dia08,diligmnxi07,liwan12} and the references
therein.

For the integro-differential equations, the asymptotic properties of
mild solutions have been studied from differential points, such as
almost periodicity, almost automorphy, asymptotic stability,
oscillation and so on. However, for the weighted pseudo almost
periodicity of classical solutions, it is rarely investigated,
particularly for the integro-differential equations with impulsive
effects. The existence, uniqueness and attractivity  of piecewise
weighted pseudo almost periodic classical solutions for impulsive
integro-differential equations is an untreated topic in the
literature and this fact is the motivation of the present work.

This article is organized as follows. In Section 2, we recall some
fundamental results about the notion of piecewise almost periodic
function. In Section 3, we introduce the concept of piecewise
weighted pseudo almost periodic function and explore its properties.
Sections 4 is devoted to the existence, uniqueness and attractivity
of classical solution of \eqref{nonimpulsiveintegro} by fixed point
theorem and fractional powers of operators. In Section 5, an
application to impulsive integro-differential equations is presented
to illustrate the main findings.


\section{Preliminaries}

Let $(X, \|\cdot\|)$, $(Y, \|\cdot\|)$  be Banach spaces, $\Omega$
be a subset of $X$ and $\mathbb{N}$, $\mathbb{Z}$ and $\mathbb{R}$
stand for the set of natural numbers, integers and real numbers,
respectively. For $A$ being a linear operator on $X$,
$\mathcal{D}(A)$ stands for the domain of $A$. Let $T$ be the set
consisting of all real sequences $\{t_i\}_{i\in\mathbb{Z}}$ such that
$\kappa=\inf_{i\in\mathbb{Z}}(t_{i+1}-t_i)>0$. It is immediate that
this condition implies that
$\lim_{i\to \infty}t_i=\infty$ and
$\lim_{i\to  -\infty}t_i= -\infty$.

To facilitate the discussion below, we further introduce
the following notation:

\begin{itemize}

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

\item $PC(\mathbb{R},X)$ :~the space formed by all piecewise  continuous functions
$f:\mathbb{R}\to  X$ such that $f(\cdot)$ is continuous at $t$ for
any $t\notin \{t_i\}_{i\in\mathbb{Z}}$, $f(t^{+}_i)$, $f(t^{-}_i)$
exist, and $f(t^{-}_i)=f(t_i)$ for all $i\in \mathbb{Z}$.


\item $PC(\mathbb{R}\times\Omega, X)$ :~the space formed by all piecewise  continuous functions
$f:\mathbb{R}\times\Omega \to  X$ such that for any $x\in \Omega$,
$f(\cdot, x)\in PC(\mathbb{R}, X)$ and for any $t\in\mathbb{R}$, $f(t,\cdot)$ is
continuous at $x\in \Omega$.
\end{itemize}


It is possible to define fractional powers of $A$ if $-A$ is the
infinitesimal generator of an analytic semigroup $T(t)$ in a Banach
space and $0\in \rho(A)$. For $\alpha>0$, define the fractional
power $A^{-\alpha}$ of  $A$ by
$$
A^{-\alpha}=\frac{1}{\Gamma(\alpha)} \int_{0}^{\infty}t^{\alpha-1}T(t)dt.
$$
Operator $A^{-\alpha}$ is bounded, bijective and
$A^{\alpha}=(A^{-\alpha})^{-1}$ is a closed linear operator,
$\mathcal{D}(A^{\alpha})=\mathcal{R}(A^{-\alpha})$, $A^{0}$ is the
identity operator in $X$. For $0\leq\alpha\leq 1$, $X_{\alpha}=
\mathcal{D}(A^{\alpha})$ with norm $\|x\|_{\alpha}=\|A^{\alpha}x\|$
is a Banach space.


\begin{lemma}[\cite{pazbook83}] \label{fundalemma}
 Let $-A$ be an infinitesimal operator of an
analytic semigroup $T(t)$, then
\begin{itemize}
\item [(i)]  $T(t): X\to  \mathcal{D}(A^{\alpha})$ for every
$t>0$ and $\alpha\geq 0$.

\item [(ii)] For every $x\in \mathcal{D}(A^{\alpha})$, it follows
that $T(t)A^{\alpha} x= A^{\alpha} T(t)x$.

\item [(iii)] For every $t>0$, the operator $A^{\alpha} T(t)$ is
bounded and
\begin{equation}
\|A^{\alpha}T(t)\|\leq
M_{\alpha}t^{-\alpha}e^{-\lambda t},\quad M_{\alpha}>0, ~\lambda>0.
\end{equation}

\item [(iv)] For $0<\alpha\leq 1$ and $x\in
\mathcal{D}(A^{\alpha})$, we have
$$
\|T(t)x-x\|\leq C_{\alpha}t^{\alpha}\|A^{\alpha}x\|,\quad C_{\alpha}>0.
$$
\end{itemize}
\end{lemma}

Now, we recall the concepts of discrete  almost periodic function,
discrete weighted pseudo almost periodic function, piecewise almost
periodic function.

\begin{definition}[\cite{finkbook74}] \rm
 A function $f\in C(\mathbb{R}, X)$ is said to be
almost periodic if for each $\varepsilon>0$, there exists an
$l(\varepsilon)>0$, such that every interval $J$ of length
$l(\varepsilon)$ contains a number $\tau$ with the property that
$\|f(t+\tau)-f(t)\|<\varepsilon$ for all $t\in\mathbb{R}$. Denote by
$AP(\mathbb{R}, X)$ the set of such functions.
\end{definition}

\begin{definition}[\cite{samperbook95}] \rm
  A sequence $\{ x_n\}$ is called almost
periodic if for any $\varepsilon>0$, there exists a relatively dense
set of its $\varepsilon$-periods, i.e.\ there exists a natural
number $l=l(\varepsilon)$, such that for $k\in\mathbb{Z}$, there is at least
one number $p$ in $[k, k+l]$, for which inequality
$\|x_{n+p}-x_n\|< \varepsilon$ holds for all $n\in\mathbb{N}$. Denote by
$AP(\mathbb{Z}, X)$ the set of such sequences.
\end{definition}

Let $U_{d}$ denote the collection of functions (weights)
$\rho: \mathbb{Z}\to  (0, +\infty)$. For $\rho\in U_{d}$ and
$m\in \mathbb{Z}^{+}=\{ n\in\mathbb{Z}, n\geq 0 \}$, set
$\mu(m,\rho):=\sum_{k=-m}^{m} \rho_{k}$. Denote
$ U_{d,\infty}:=\{ \rho \in U_{d}:
\lim_{m\to  \infty}\mu(m,\rho)= \infty \}$.


For $\rho\in U_{d,\infty}$, define
\begin{gather*}
 AAP_{0}(\mathbb{Z}, X)=  \{ x_n\in l^{\infty}(\mathbb{Z}, X):
\lim_{n\to  \infty}\|x_n\|=0 \}.\\
 WPAP_{0}(\mathbb{Z}, X,\rho):=\{x_n\in l^{\infty}(\mathbb{Z}, X),~
\lim_{m\to  \infty}\frac{1}{\mu(m,\rho)}\sum_{k=-m}^{m}
\|x_{k}\|\rho_{k} =0  \}.
\end{gather*}

\begin{definition}[\cite{ruepho95}]\rm
 A sequence  $\{ x_n\}_{n\in\mathbb{Z}} \in l^{\infty}(\mathbb{Z},X)$ is called discrete
asymptotically almost periodic if
$x_n=x^{1}_n+x^{2}_n$, where $x^{1}_n\in AP(\mathbb{Z}, X)$,
$x^{2}_n\in AAP_{0}(\mathbb{Z}, X)$. Denote by $AAP(\mathbb{Z}, X)$ the set of
such sequences.
\end{definition}


\begin{definition}[\cite{dingmnnie13}] \rm
Let $\rho\in U_{d,\infty}$. A sequence  $\{
x_n\}_{n\in\mathbb{Z}} \in l^{\infty}(\mathbb{Z}, X)$ is called discrete weighted
pseudo almost periodic if it can be expressed as
$x_n=x^{1}_n+x^{2}_n$, where $x^{1}_n\in AP(\mathbb{Z}, X)$ and
$x^{2}_n\in WPAP_{0}(\mathbb{Z}, X, \rho)$. The set of such functions
denoted by $WPAP(\mathbb{Z}, X,\rho)$.
\end{definition}

For $\{t_i\}_{i\in\mathbb{Z}}\in T$, $\{t^{j}_i\}$ defined by
 $\{t^{j}_i=t_{i+j}-t_i\}, i\in \mathbb{Z}, j\in\mathbb{Z}$.


\begin{definition}[\cite{samperbook95}] \label{defpcaa} \rm
  A function $f\in PC(\mathbb{R}, X)$ is said to be
piecewise almost periodic if the following conditions are fulfilled:
\begin{itemize}
\item[(1)] $\{t^{j}_i=t_{i+j}-t_i\}$, $i, j\in\mathbb{Z}$ are
equipotentially almost periodic, that is, for any $\varepsilon>0$,
there exists a relatively dense set in $\mathbb{R}$ of $\varepsilon$-almost
periods common for all of the sequences $\{t^{j}_i\}$.

\item[(2)] For any $\varepsilon>0$, there exists a positive
number $\delta=\delta(\varepsilon)$ such that if the points $t'$ and
$t''$ belong to the same interval of continuity of $f$ and
$|t'-t''|<\delta$, then $\|f(t')-f(t'')\|<\varepsilon$.

\item[(3)] For any $\varepsilon>0$, there exists a relatively
dense set $\Omega_{\varepsilon}$ in $\mathbb{R}$ such that if $\tau\in
\Omega_{\varepsilon}$, then
$$
\|f(t+\tau)-f(t)\|<\varepsilon
$$
for all $t\in\mathbb{R}$ which satisfy the condition $|t-t_i|>\varepsilon,
i\in\mathbb{Z}$.
\end{itemize}
\end{definition}

We denote by $AP_{p}(\mathbb{R},X)$ the space of all piecewise almost
periodic functions. Throughout the rest of this paper, we always
assume that $\{t^{j}_i\}$ are equipotentially almost periodic. Let
$\mathcal{U}PC(\mathbb{R}, X)$ be the space of all functions $f\in PC(\mathbb{R},
X)$ such that $f$ satisfies the condition $(2)$ in Definition
\ref{defpcaa}.

\begin{definition} \rm
$f\in PC(\mathbb{R}\times\Omega, X)$ is said to be piecewise almost periodic
in $t$ uniformly in $x\in \Omega$ if for each compact set
$K\subseteq \Omega$, $\{f(\cdot,x): x\in K \}$ is uniformly bounded,
and given $\varepsilon>0$, there exists a relatively dense set
$\Omega_{\varepsilon}$ such that $\|f(t+\tau,x)-f(t,x)\|\leq
\varepsilon$ for all $x\in K, \tau\in \Omega_{\varepsilon}$ and
$t\in \mathbb{R}$, $|t-t_i|>\varepsilon$. Denote by
$AP_{p}(\mathbb{R}\times\Omega, X)$ the set of all such functions.
\end{definition}

\begin{lemma}[\cite{samperbook95}]\label{lemmapoints}
If the sequences $\{t_i^{j}\}$ are equipotentially almost periodic,
then for each $j>0$, there exists a
positive integer $N$ such that on each interval of length $j$, there
are no more than $N$ elements of the sequence $\{t_i\}$, $i.e.$,
$$i(s,t)\leq N(t-s)+N,$$
where $i(s,t)$ is the number of the points $\{t_i\}$ in the
interval $[s, t]$.
\end{lemma}

\begin{lemma}[\cite{samperbook95}] \label{lemmarz}
 Assume that $f\in AP_{p}(\mathbb{R}, X)$,
$\{x_i\}_{i\in\mathbb{Z}} \in AP (\mathbb{Z}, X)$, and $\{t_i^{j}\}$, $j\in\mathbb{Z}$
are equipotentially almost periodic. Then for each $\varepsilon>0$,
there exist relatively dense sets $\Omega_{\varepsilon}$ of $\mathbb{R}$ and
$Q_{\varepsilon}$ of $\mathbb{Z}$ such that
\begin{itemize}
\item [(i)] $\|f(t+\tau)-f(t)\|<\varepsilon$ for all $t\in\mathbb{R}$,
$|t-t_i|>\varepsilon$, $\tau\in \Omega_{\varepsilon}$ and
$i\in\mathbb{Z}$.

\item [(ii)] $\|x_{i+q}-x_i\|<\varepsilon$ for all $q\in Q_{\varepsilon}$ and $i\in\mathbb{Z}$.

\item [(iii)] $|t_i^{q}-\tau|<\varepsilon$ for all $q\in Q_{\varepsilon}$, $\tau\in \Omega_{\varepsilon}$ and
$i\in\mathbb{Z}$.
\end{itemize}
\end{lemma}

Now, we give the generalized Gronwall-Bellman inequality which will
be used later, one can see \cite[Theorem 2.1]{iov07} for more
details.

\begin{lemma}[generalized Gronwall-Bellman inequality] \label{gronwall}
 Let a nonnegative function
$u(t)\in PC(\mathbb{R}, X)$ satisfy for $t\geq t_{0}$
$$
u(t)\leq n(t)+\int_{t_{0}}^{t}v(\tau)u(\tau)d\tau
 + \sum_{t_{0}<t_i<t}\beta_iu(\tau_i),
$$
with $n(t)$ a positive nondecreasing function for $t\geq t_{0}$,
$\beta_i\geq 0$, $v(\tau)\geq 0$ and $\tau_i$ are the first kind
discontinuity points of the functions $u(t)$.  Then the following
estimate holds for the function $u(t)$,
$$
u(t)\leq n(t) \prod_{t_{0}<t_i<t}(1+\beta_i)e^{\int^{t}_{t_{0}}v(\tau)d\tau} .
$$
\end{lemma}



\section{Piecewise weighted pseudo almost periodicity}

In this section, we introduce the concept of piecewise weighted
pseudo almost periodic function, explore its properties and
establish the composition theorem.

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 $r>0$ and each $\rho\in U$, set
$$
\mu(r,\rho):=\int^{r}_{-r}\rho(t)dt.
$$
Define
\begin{gather*}
U_{\infty}:=\{\rho\in U:\lim_{r\to \infty}\mu(r,\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} \rm
Let $\rho_{1}, \rho_{2}\in U_{\infty}$, $\rho_{1}$ 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
$U_{\infty}=\bigcup_{\rho\in U_{\infty}}cl(\rho)$.

For $\rho\in U_{\infty}$, define
\begin{gather*}
 PC_{p}^{0}(\mathbb{R}, X) =  \{ f\in PC(\mathbb{R}, X): \lim_{t\to  \infty}\|f(t)\|=0 \},  \\
 WPAP_{p}^{0}(\mathbb{R},X,\rho):=\{f\in PC(\mathbb{R},X):
  \lim_{r\to \infty}\frac{1}{\mu(r,\rho)}\int^{r}_{-r}\rho(t)\|f(t)\|dt=0\},    \\
\begin{aligned}
& WPAP^{0}_{p}(\mathbb{R}\times \Omega, X,\rho)\\
&= \Big\{ f\in PC(\mathbb{R}\times\Omega, X): \lim_{r\to
\infty}\frac{1}{\mu(r,\rho)}\int^{r}_{-r}\rho(t)\|f(t, x)\|dt=0
\text{ uniformly with }\\
&\quad \text{respect to } x\in K, \text{ where } K
\text{ is an arbitrary compact subset of } \Omega \Big\}.
\end{aligned}
\end{gather*}

\begin{definition} \rm
A function $f\in PC(\mathbb{R}, X)$ is said to be piecewise asymptotically
almost periodic if it can be decomposed as $f=g+\varphi$, where
$g\in AP_{p}(\mathbb{R}, X)$ and $\varphi\in PC_{p}^{0}(\mathbb{R}, X)$. Denote by
$AAP_{p}(\mathbb{R}, X)$ the set of all such functions.
\end{definition}


\begin{definition} \rm
 A function $f\in PC(\mathbb{R}, X)$ is said to be piecewise
weighted pseudo almost periodic if it can be decomposed as
$f=g+\varphi$, where $g\in AP_{p}(\mathbb{R}, X)$ and $\varphi\in
WPAP^{0}_{p}(\mathbb{R}, X,\rho)$. Denote by $WPAP_{p}(\mathbb{R}, X,\rho)$ the set
of all such functions.
\end{definition}

\begin{definition} \label{paptwo} \rm
 Let $WPAP_{p}(\mathbb{R}\times\Omega, X,\rho)$ consist of all
functions $f\in PC(\mathbb{R}\times\Omega, X)$ such that $f=g+\varphi$,
where $g\in AP_{p}(\mathbb{R}\times\Omega, X)$ and $\varphi\in
WPAP^{0}_{p}(\mathbb{R}\times \Omega, X,\rho)$.
\end{definition}


Let $\rho\in U_{\infty}, \tau\in \mathbb{R}$, 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 $n\in \mathbb{N}$ et al.

It is obvious that $(WPAP_{p}(\mathbb{R},
X,\rho),\|\cdot\|_{\infty})$ (resp. $(WPAP_{p}(\mathbb{R}\times Y,
X,\rho),\|\cdot\|_{\infty})$), $\rho\in U_{T}$ is a
Banach space when endowed with the sup norm.

\begin{remark}\label{remarktrans} \rm
\begin{itemize}
\item [(i)]
For $\rho\in U_{T}$, $WPAP_{p}^{0}(\mathbb{R}, X,\rho)$ is a translation
invariant set of $PC(\mathbb{R}, X)$.

\item [(ii)]  $PC_{p}^{0}(\mathbb{R}, X) \subset WPAP^{0}_{p}(\mathbb{R}, X,\rho)$ and  $AAP_{p}(\mathbb{R}, X)\subset WPAP_{p}(\mathbb{R}, X,\rho)$
\end{itemize}
\end{remark}

Similarly as the proof of  \cite[Lemma 2.5]{dia08}, one has the following
lemma.

\begin{lemma}\label{lemmauniform}
Let $\{f_n\}_{n\in\mathbb{N}} \subset WPAP^{0}_{p}(\mathbb{R}, X,\rho)$ be a
sequence of functions. If $f_n$ converges uniformly to $f$, then
$f\in WPAP^{0}_{p}(\mathbb{R}, X,\rho)$.
\end{lemma}

Similarly as the proof of  \cite{liuzha13}, the following results
and the composition theorems are hold for piecewise weighted pseudo
almost periodic function.


\begin{theorem}\label{papiu}
Suppose the sequence of vector-valued functions $\{I_i\}_{i\in
\mathbb{Z}}$ is weighted pseudo almost periodic, i.e, for any $x\in\Omega$,
$\{I_i(x), i\in\mathbb{Z} \}$ is a weighted pseudo almost periodic
sequence. Assume that the following conditions hold:
\begin{itemize}
\item[(i)] $\{I_i(x), i\in\mathbb{Z}, x\in K \}$ is bounded for every
bounded subset $K\subset \Omega$.

\item[(ii)] $I_i(x)$ is uniformly continuous in $x\in \Omega$
uniformly in $i\in \mathbb{Z}$.
\end{itemize}
If $\phi\in WPAP_{p}(\mathbb{R}, X,\rho)\cap UPC(\mathbb{R}, X)$ such that
$\mathcal{R}(\phi)\subset \Omega$, then $I_i(\phi(t_i))$ is weighted
pseudo almost periodic.
\end{theorem}

\begin{corollary}\label{corollaryi}
Assume that the sequence of vector-valued functions $\{I_i\}_{i\in
\mathbb{Z}}$ is weighted pseudo almost periodic, and there exists a constant
$L_{1}>0$ such that
$$
\|I_i(u)-I_i(v)\|\leq L_{1}\|u-v\|,\quad \text{for all }u,v\in\Omega,~i\in\mathbb{Z}.
$$
if $\phi\in WPAP_{p}(\mathbb{R}, X,\rho)\cap UPC(\mathbb{R}, X)$ such that
$\mathcal{R}(\phi)\subset \Omega$,
then $I_i(\phi(t_i))$ is weighted pseudo almost periodic.
\end{corollary}


\begin{theorem}\label{compopap}
Suppose $f\in WPAP_{p}(\mathbb{R}\times\Omega, X,\rho)$. Assume that the
following conditions hold:
\begin{itemize}
\item [(i)] $\{f(t, u): t\in\mathbb{R}, u\in K\}$ is
bounded for every bounded subset $K\subseteq \Omega$.

\item [(ii)] $f(t,\cdot)$ is uniformly continuous in each
bounded subset of $\Omega$ uniformly in $t\in \mathbb{R}$.
\end{itemize}
If $\varphi\in WPAP_{p}(\mathbb{R}, X,\rho)$  such that $\mathcal{R}(\varphi)\subset
\Omega$, then $f(\cdot,\varphi(\cdot))\in WPAP_{p}(\mathbb{R}, X,\rho)$.
\end{theorem}

\begin{corollary}\label{compolip}
 Let $f\in WPAP_{p}(\mathbb{R}\times\Omega, X,\rho)$,
$\varphi\in WPAP_{p}(\mathbb{R}, X,\rho)$ and $\mathcal{R}(\varphi)\subset \Omega$.
Assume that there exists a constant $L_{f}>0$ such that
$$
\|f(t, u)-f(t,  v) \| \leq L_{f}\|u-v\|, \quad t\in\mathbb{R},\; u, v\in \Omega,
$$
then $f(\cdot,\varphi)\in WPAP_{p}(\mathbb{R}, X,\rho)$.
\end{corollary}

\section{Impulsive integro-differential equations}

In this section, we investigate the existence, uniqueness and
attractivity  of piecewise weighted pseudo almost periodic classical
solution of nonlinear impulsive integro-differential equations:
\begin{equation}\label{nonimpulsiveintegro}
\begin{gathered}
\frac{du(t)}{dt}+Au(t)=f(t, u(t), (Ku)(t)), \quad t\in\mathbb{R},\; t\neq t_i, \; i\in\mathbb{Z}, \\
(Ku)(t)=\int^{t}_{-\infty}k(t-s)g(s,u(s))ds,\\
\Delta u(t_i):=u(t^{+}_i)-u(t^{-}_i)=I_i(u(t_i)),
\end{gathered}
\end{equation}

First, we make the following assumptions:
\begin{itemize}
\item [(H1)] $-A$ is the infinitesimal generator of an analytic semigroup $T(t)$
such that $\|T(t)\|\leq Me^{-\omega t}$ for $t\geq 0$ and $0\in
\rho(A)$.

\item [(H2)] $k\in C(\mathbb{R}^{+}, \mathbb{R})$ and $|k(t)|\leq C_{k} e^{-\eta
t}$ for some positive constants $C_{k}, \eta$.

\item [(H3)] $g\in WPAP_{p}(\mathbb{R}\times X_{\alpha}, X,\rho)$, $\rho\in U_{T}$ and there exists a constant $L_{g}>0$ such that
$$\|g(t,u)-g(t,v)\|
\leq L_{g}\|u-v\|_{\alpha}, \quad t\in\mathbb{R},~~u ,v\in
X_{\alpha}.$$

\item [(H4)] $f\in WPAP_{p}(\mathbb{R}\times X_{\alpha}\times X_{\alpha}, X,\rho)$, $\rho\in U_{T}$ and there exists  constants $L_{f}>0$, $0<\theta<1$ such that
$$\|f(t_{1}, u_{1},v_{1})-f(t_{2}, u_{2},v_{2}) \| \leq L_{f}(|t_{1}-t_{2}|^{\theta}+\|u_{1}-u_{2}\|_{\alpha}+\|v_{1}-v_{2}\|_{\alpha}),
$$
for each $(t_i,u_i,v_i)\in \mathbb{R}\times X_{\alpha}\times
X_{\alpha},~~ i=1,2$.

\item [(H5)] $I_i\in WPAP (\mathbb{Z}, X,\rho)$  and
there exists a constant $L_{1}>0$ such that
$$\|I_i(u)-I_i(v)\|\leq L_{1}\|u-v\|_{\alpha}, \quad  t\in\mathbb{R},~~u ,v\in X_{\alpha}, ~i\in\mathbb{Z}.$$

\end{itemize}


Before starting our main results, we recall the definition of the
mild solution of \eqref{nonimpulsiveintegro}.

\begin{definition}[\cite{samperbook95}] \rm
 A function $u:\mathbb{R}\to  X$ is called a mild solution of \eqref{nonimpulsiveintegro}
if for any $t\in\mathbb{R}$, $t>\sigma$, $\sigma\neq t_i$, $i\in\mathbb{Z}$,
\begin{equation} \label{mild}
u(t)=T(t-\sigma)u(\sigma)+\int^{t}_{\sigma}T(t-s)f(s,u(s),(Ku)(s))ds
+\sum_{\sigma<t_i<t}T(t-t_i)I_i(u(t_i)),
\end{equation}
\end{definition}

Note that if (H1) holds, then \eqref{mild} can be rewritten as
\[
u(t)=\int^{t}_{-\infty}T(t-s)f(s,u(s),(Ku)(s))ds +
\sum_{t_i<t}T(t-t_i)I_i(u(t_i)).
\]


\begin{lemma}\label{papku}
Assume that {\rm (H1)--(H3)} hold, if $u\in WPAP_{p}(\mathbb{R}, X_{\alpha},\rho)$, then
$$
K(A^{-\alpha}u)(t):=\int^{t}_{-\infty}k(t-s)g(s,A^{-\alpha}u(s))ds
\in WPAP_{p}(\mathbb{R}, X,\rho).
$$
\end{lemma}

\begin{proof}
Since $A^{-\alpha}$ is bounded,
$\phi(\cdot)=g(s,A^{-\alpha}u(s))\in WPAP_{p}(\mathbb{R}, X,\rho)$
 by Corollary \ref{compolip}. Let
$\phi=\phi_{1}+\phi_{2}$, where $\phi_{1}\in AP_{p}(\mathbb{R}, X)$,
$\phi_{2}\in WPAP_{p}^{0}(\mathbb{R}, X,\rho)$, then
\begin{align*}
\int^{t}_{-\infty}k(t-s)g(s,A^{-\alpha}u(s))ds
&=\int^{t}_{-\infty}k(t-s)\phi_{1}(s)ds +\int^{t}_{-\infty}k(t-s)\phi_{2}(s)ds\\
&:=\Psi_{1}(t)+\Psi_{2}(t),
\end{align*}
where
\[
\Psi_{1}(t)= \int^{t}_{-\infty}k(t-s)\phi_{1}(s)ds,\quad
\Psi_{2}(t)= \int^{t}_{-\infty}k(t-s)\phi_{2}(s)ds.
\]

(i)  $\Psi_{1}\in AP_{p}(\mathbb{R}, X)$.
It is not difficult to see that  $\Psi_{1}\in \mathcal{U}PC(\mathbb{R}, X)$.
Since $\phi_{1}\in AP_{p}(\mathbb{R}, X)$, for $\varepsilon>0$, let
$\Omega_{\varepsilon}$ be a relatively dense set of $\mathbb{R}$ formed by
$\varepsilon$-periods of $\phi_{1}$. If $\tau\in
\Omega_{\varepsilon}$, $t\in\mathbb{R}$, $|t-t_i|>\varepsilon$, $i\in\mathbb{Z}$,
then
$$
\|\phi_{1}(t+\tau)-\phi_{1}(t)  \| <\varepsilon.
$$
Hence, by (H2), for  $t\in\mathbb{R}$, $|t-t_i|>\varepsilon$,
$i\in\mathbb{Z}$, one has
\begin{align*}
\|\Psi_{1}(t+\tau)-\Psi_{1}(t)\|
&=\|\int^{t+\tau}_{-\infty}k(t+\tau-s)\phi_{1}(s)ds-\int^{t}_{-\infty}k(t-s)\phi_{1}(s) ds \| \\
&=\|\int^{t}_{-\infty}k(t-s)(\phi_{1}(s+\tau)-\phi_{1}(s))ds  \| \\
& \leq \int^{t}_{-\infty} C_{k}e^{-\eta(t-s)} \| \phi_{1}(s+\tau)-\phi_{1}(s) \|ds \\
&< \frac{C_{k}}{\eta} \varepsilon,
\end{align*}
which implies that $\Psi_{1}\in AP_{p}(\mathbb{R}, X)$.


(ii)  $\Psi_{2}\in WPAP_{p}^{0}(\mathbb{R}, X,\rho)$.
In fact, for $r>0$, one has
\begin{align*}
\frac{1}{\mu(r,\rho)}\int^{r}_{-r}\rho(t)\|\Psi_{2}(t)\|dt
&=\frac{1}{\mu(r,\rho)}\int^{r}_{-r}\rho(t)\|\int^{t}_{-\infty}k(t-s)\phi_{2}(s)ds\|dt \\
&=\frac{1}{\mu(r,\rho)}\int^{r}_{-r}\rho(t)\|\int^{\infty}_{0}k(s)\phi_{2}(t-s)ds\|dt\\
&\leq \frac{1}{\mu(r,\rho)}\int^{r}_{-r}\int^{\infty}_{0}C_{k}e^{-\eta s}\rho(t) \|\phi_{2}(t-s)\|dsdt\\
&\leq  \int^{\infty}_{0}C_{k}e^{-\eta s}\Phi_{r}(s)ds,
\end{align*}
where
$$
\Phi_{r}(s)=\frac{1}{\mu(r,\rho)} \int^{r}_{-r}\rho(t)\|\phi_{2}(t-s)\|dt.
$$
Since $\phi_{2}\in WPAP_{p}^{0}(\mathbb{R}, X,\rho)$, $\rho\in U_{T}$ it
follows that $\phi_{2}(\cdot-s)\in WPAP_{p}^{0}(\mathbb{R}, X,\rho)$ for
each $s\in\mathbb{R}$ by Remark \ref{remarktrans}, hence
$\lim_{r\to \infty}\Phi_{r}(s) =0$ for all $s\in \mathbb{R}$.
By using the Lebesgue dominated convergence theorem, we have
$\Psi_{2}\in WPAP_{p}^{0}(\mathbb{R}, X,\rho)$. This completes the proof.
\end{proof}

\begin{lemma}\label{lemmatwo}
Assume that {\rm (H1)--(H4)} hold, if $u\in WPAP_{p}(\mathbb{R},X,\rho)$, then
$$
(\Lambda u)(t):=\int^{t}_{-\infty}A^{\alpha}T(t-s)f(s,A^{-\alpha}u(s),
K(A^{-\alpha}u(s)))ds\in WPAA_{p}(\mathbb{R}, X,\rho).
$$
\end{lemma}

\begin{proof}
We first show that $\Lambda u$ is well defined. In fact, if
$u\in WPAA_{p}(\mathbb{R}, X,\rho)$, one has
$K(A^{-\alpha}u) \in WPAA_{p}(\mathbb{R}, X,\rho)$ by Lemma \ref{papku},
 and $f(\cdot,A^{-\alpha}u(\cdot),
K(A^{-\alpha}u(\cdot)))\in WPAA_{p}(\mathbb{R}, X,\rho)$ by Corollary
\ref{compolip}. Hence $h(\cdot)=f(\cdot,A^{-\alpha}u(\cdot),
K(A^{-\alpha}u(\cdot)))\in WPAA_{p}(\mathbb{R}, X,\rho)$, then
$\|h\|:=\sup_{t\in \mathbb{R}}\|h(t)\|<\infty$. By Lemma
\ref{fundalemma}, one has
\begin{align*}
\|A^{\alpha}T(t-s)f(s,A^{-\alpha}u(s), K(A^{-\alpha}u(s)))\|  
&\leq M_{\alpha}(t-s)^{-\alpha}e^{-\lambda (t-s)} \|h(s)\| \\
&\leq M_{\alpha}(t-s)^{-\alpha}e^{-\lambda (t-s)}\|h\|,
\end{align*}
since
$$
\int^{t}_{-\infty}(t-s)^{-\alpha}e^{-\lambda (t-s)} ds
=\lambda^{\alpha-1}\Gamma(1-\alpha),
$$
where $\Gamma$ is the classical Gamma function. Hence
\[
A^{\alpha}T(t-s)f(s,A^{-\alpha}u(s), K(A^{-\alpha}u(s)))
\]
 is integrable over $(-\infty, t)$ for $t\in \mathbb{R}$.

Now, let $h=h_{1}+h_{2}$,  where $h_{1}\in AP_{p}(\mathbb{R}, X)$,
$h_{2}\in WPAP_{p}^{0}(\mathbb{R}, X,\rho)$, then
\begin{align*}
(\Lambda u)(t)
&= \int^{t}_{-\infty}A^{\alpha}T(t-s)h_{1}(s)ds +\int^{t}_{-\infty}A^{\alpha}T(t-s)h_{2}(s)ds\\
&:=\Lambda_{1}(t)+\Lambda_{2}(t),
\end{align*}
 where
\[
\Lambda_{1}(t)=\int^{t}_{-\infty}A^{\alpha}T(t-s)h_{1}(s)ds,\quad
\Lambda_{2}(t)=\int^{t}_{-\infty}A^{\alpha}T(t-s)h_{2}(s)ds.
\]


(i) $\Lambda_{1}\in \mathcal{U}PC(\mathbb{R}, X)$.
Let $t', t''\in (t_i, t_{i+1})$, $i\in\mathbb{Z}$, $t''<t'$, then
\begin{align*} %\label{ineq0}
&\Lambda_{1} (t') -\Lambda_{1} (t'')\\
&=\int_{-\infty}^{t'}A^{\alpha}T(t'-s)h_{1}(s)ds
 -\int_{-\infty}^{t''}A^{\alpha}T(t''-s)h_{1}(s)ds \\
&=\int_{-\infty}^{t''}A^{\alpha}(T(t'-s)-T(t''-s))h_{1}(s)ds
 + \int^{t'}_{t''}A^{\alpha}T(t'-s)h_{1}(s)ds \\
&=\int_{-\infty}^{t''}[T(t'-t'')-I]A^{\alpha}T(t''-s)h_{1}(s)ds +
\int^{t'}_{t''}A^{\alpha}T(t'-s)h_{1}(s)ds,
\end{align*}
It is easy to see that for any $\varepsilon>0$, there exists
\[
0<\delta< \Big( \frac{(1-\alpha)\varepsilon}{2M_{\alpha}\|h_{1}\|}
\Big)^{1/(1-\alpha)}
\]
 such that if $t', t''$ belongs to a same
continuity and $0<t'-t''<\delta$, then
$$
\|T(t'-t'')-I \| \leq \frac{ \varepsilon}{2M_{\alpha} \|h_{1}\|
 \lambda^{\alpha-1}\Gamma(1-\alpha)}.
$$
So
\begin{align*}
&\|\Lambda_{1} (t') -\Lambda_{1} (t'')\|\\
&\leq \int_{-\infty}^{t''}\|[T(t'-t'')-I]A^{\alpha}T(t''-s)h_{1}(s)\|ds
 +\int_{t''}^{t'}\|A^{\alpha}T(t'-s)h_{1}(s)\|ds  \\
&\leq  \int_{-\infty}^{t''}\frac{ \varepsilon}{2M_{\alpha} \lambda^{\alpha-1}
 \Gamma(1-\alpha) \|h_{1}\| }  M_{\alpha}(t''-s)^{-\alpha}
 e^{-\lambda (t''-s)}\|h_{1}\| ds  \\
&\quad +\int_{t''}^{t'} M_{\alpha}(t'-s)^{-\alpha} e^{-\lambda (t'-s)} \|h_{1}\| ds  \\
&\leq  \frac{\varepsilon}{2}+\frac{M_{\alpha} \|h_{1}\| \delta^{1-\alpha}}{1-\alpha}  \\
&\leq  \frac{\varepsilon}{2}+\frac{\varepsilon}{2}=\varepsilon,
\end{align*}
which implies that $\Lambda_{1}\in \mathcal{U}PC(\mathbb{R}, X)$.



(ii)  $\Lambda_{1}\in AP_{p}(\mathbb{R}, X)$.
Since $h_{1}\in AP_{p}(\mathbb{R}, X)$, for $\varepsilon>0$, there exists a
relatively dense set $\Omega_{\varepsilon}$ such that for
$\tau\in \Omega_{\varepsilon}$, $t\in\mathbb{R}$, $|t-t_i|>\varepsilon$, $i\in\mathbb{Z}$,
$$
\|h_{1}(t+\tau)-h_{1}(t)  \| <\varepsilon.
$$

Hence, by Lemma \ref{fundalemma}, for  $t\in\mathbb{R}$,
$|t-t_i|>\varepsilon$, $i\in\mathbb{Z}$, one has
\begin{align*}
&\|\Lambda_{1}(t+\tau)-\Lambda_{1}(t)\|\\
&=\|\int^{t+\tau}_{-\infty}A^{\alpha}T(t+\tau-s)h_{1}(s)ds
 -\int^{t}_{-\infty}A^{\alpha}T(t-s)h_{1}(s)ds \| \\
&=\|\int^{t}_{-\infty}A^{\alpha}T(t-s)(h_{1}(s+\tau)-h_{1}(s))ds\| \\
& \leq\int^{t}_{-\infty}\|A^{\alpha}T(t-s)(h_{1}(s+\tau)-h_{1}(s))\| ds\\
& \leq \int^{t}_{-\infty} M_{\alpha}(t-s)^{-\alpha}e^{-\lambda(t-s)}
 \| h_{1}(s+\tau)-h_{1}(s) \|ds \\
&< M_{\alpha}\lambda^{\alpha-1}\Gamma(1-\alpha) \varepsilon,
\end{align*}
that is $\Lambda_{1}\in AP_{p}(\mathbb{R}, X)$.



(iii)  $\Lambda_{2}\in WPAP_{p}^{0}(\mathbb{R}, X)$.
In fact, for $r>0$, one has
\begin{align*}
\frac{1}{\mu(r,\rho)}\int^{r}_{-r}\rho(t)\|\Lambda_{2}(t)\|dt
&=\frac{1}{\mu(r,\rho)}\int^{r}_{-r}\rho(t)\|\int^{t}_{-\infty}
 A^{\alpha}T(t-s)h_{2}(s)ds\|dt \\
&=\frac{1}{\mu(r,\rho)}\int^{r}_{-r}\rho(t)\|\int^{\infty}_{0}
 A^{\alpha}T(s)h_{2}(t-s)ds\|dt\\
&\leq\frac{1}{\mu(r,\rho)}\int^{r}_{-r}\int^{\infty}_{0}\rho(t)
 \|A^{\alpha}T(s)h_{2}(t-s)\|dsdt\\
&\leq \frac{1}{\mu(r,\rho)}\int^{r}_{-r}\int^{\infty}_{0}M_{\alpha}
 s^{-\alpha}e^{-\lambda s}\rho(t) \|h_{2}(t-s)\|dsdt\\
&\leq  M_{\alpha}\int^{\infty}_{0}s^{-\alpha}e^{-\lambda s}H_{r}(s)ds,
\end{align*}
where
$$
H_{r}(s)=\frac{1}{\mu(r,\rho)} \int^{r}_{-r}\rho(t)\|h_{2}(t-s)\|dt.
$$
Since $h_{2}\in WPAP_{p}^{0}(\mathbb{R}, X,\rho)$, $\rho\in U_{T}$,  it
follows that $h_{2}(\cdot-s)\in WPAP_{p}^{0}(\mathbb{R}, X,\rho)$ for each
$s\in\mathbb{R}$ by Remark \ref{remarktrans}, hence
$\lim_{r\to \infty}H_{r}(s) =0$ for all $s\in \mathbb{R}$. By
using the Lebesgue dominated convergence theorem, we have
$\Lambda_{2}\in WPAP_{p}^{0}(\mathbb{R}, X,\rho)$.
\end{proof}

\begin{theorem}\label{theoremlip}
Assume that {\rm (H1)--(H5)} hold and if $\Theta<1$, where
$$
\Theta:= M_{\alpha}\lambda^{\alpha-1}\Gamma(1-\alpha)
L_{f}\left(  L_{g}C_{k} \eta^{-1}  + 1\right)+ 2M_{\alpha} N
L_{1}\left( m^{-\alpha} + (e^{\lambda}-1)^{-1}  \right),
$$
 then
\eqref{nonimpulsiveintegro} has a unique classical solution $u\in
WPAP_{p}(\mathbb{R}, X,\rho)$ and $u(t)$ is an attractor.
\end{theorem}

\begin{proof}
Let $\mathcal{F}: WPAP_{p}(\mathbb{R}, X,\rho)\cap \mathcal{U}PC(\mathbb{R},
X)\to  PC(\mathbb{R}, X)$ be the operator defined by
\begin{equation} \label{operator}
\begin{aligned}
(\mathcal{F} u) (t)
&=\int^{t}_{-\infty}A^{\alpha}T(t-s)f(s,A^{-\alpha}u(s),
K(A^{-\alpha}u(s)))ds\\
&\quad  +\sum_{t_i<t}A^{\alpha}T(t-t_i)I_i(A^{-\alpha}u(t_i)).
\end{aligned}
\end{equation}
We will show that $\mathcal{F}$ has a fixed point in $WPAP_{p}(\mathbb{R},
X,\rho)\cap \mathcal{U}PC(\mathbb{R}, X)$ and divide the proof into several
steps.



(i)  $\mathcal{F}u \in WPAP_{p}(\mathbb{R}, X,\rho)\cap \mathcal{U}PC(\mathbb{R}, X)$.
As in the proof of Lemma \ref{lemmatwo}, it is not difficult
to see that $\mathcal{F}u \in \mathcal{U}PC(\mathbb{R}, X)$. Next, we show
that $\mathcal{F}u \in WPAP_{p}(\mathbb{R}, X,\rho)$. For $u\in WPAP_{p}(\mathbb{R},
X,\rho)$, By Lemma \ref{lemmatwo}, one has
\[
(\Lambda
u)(t)=\int^{t}_{-\infty}A^{\alpha}T(t-s)f(s,A^{-\alpha}u(s),
K(A^{-\alpha}u(s)))ds \in WPAP_{p}(\mathbb{R}, X,\rho),
\]
It remains to show that
\begin{equation}
\sum_{t_i<t}A^{\alpha}T(t-t_i)I_i(A^{-\alpha}u(t_i))
\in  WPAP_{p}(\mathbb{R}, X,\rho). \label{ineqxiao}
\end{equation}

By Corollary \ref{corollaryi},
$I_i(A^{-\alpha}u(t_i))\in WPAP (\mathbb{Z}, X,\rho)$, then let
$I_i(A^{-\alpha}u(t_i))=\beta_i+\gamma_i $, where
$\beta_i\in AP (\mathbb{Z}, X)$ and $\gamma_i\in WPAP_{0}(\mathbb{Z}, X,\rho)$,
so
\begin{align*}
\sum_{t_i<t}A^{\alpha}T(t-t_i)I_i(A^{-\alpha}u(t_i))
&=\sum_{t_i<t}A^{\alpha}T(t-t_i)\beta_i
+\sum_{t_i<t}A^{\alpha}T(t-t_i)\gamma_i \\
&:=\Phi_{1}(t) +\Phi_{2}(t).
\end{align*}

Since $\{t^{j}_i\}, i, j\in\mathbb{Z}$ are equipotentially almost
periodic, then by Lemma \ref{lemmarz},  for any $\varepsilon>0$,
there exists relative dense sets of real numbers
$\Omega_{\varepsilon}$ and integers $Q_{\varepsilon}$, such that for
$t_i<t\leq t_{i+1}$, $\tau\in \Omega_{\varepsilon}$, $q\in
Q_{\varepsilon}$, $|t-t_i|>\varepsilon$,
$|t-t_{i+1}|>\varepsilon$, $j\in\mathbb{Z}$, one has
\begin{gather*}
t+\tau>t_i+\varepsilon+\tau>t_{i+q}, \\
t_{i+q+1}>t_{i+1}+\tau-\varepsilon >t+\tau,
\end{gather*}
that is,
$t_{i+q}<t+\tau< t_{i+q+1}$; then
\begin{align*}
&\|\Phi_{1}(t+\tau)-\Phi_{1}(t)\| \\
& = \| \sum_{t_i<t+\tau}A^{\alpha}T(t+\tau-t_i)\beta_i
 -\sum_{t_i<t}A^{\alpha}T(t-t_i)\beta_i\|\\
&\leq \sum_{t_i<t} \|A^{\alpha}T(t-t_i)(\beta_{i+q}-\beta_i)\|   \\
&\leq\sum_{t_i<t} M_{\alpha} (t-t_i)^{-\alpha}e^{-\lambda(t-t_i)}\|\beta_{i+q}
 -\beta_i\| \\
&\leq M_{\alpha}\varepsilon\sum_{t_i<t}(t-t_i)^{-\alpha}e^{-\lambda(t-t_i)}\\
& \leq M_{\alpha}\varepsilon \Big( \sum_{0<t-t_i\leq 1}
(t-t_i)^{-\alpha}e^{-\lambda(t-t_i)} +
\sum_{j=1}^{\infty} \sum_{j<t-t_i\leq j+1}
(t-t_i)^{-\alpha}e^{-\lambda(t-t_i)} \Big) \\
& \leq M_{\alpha}\varepsilon \Big( \sum_{0<t-t_i\leq 1}
(t-t_i)^{-\alpha} + \sum_{j=1}^{\infty}
\sum_{j<t-t_i\leq j+1} e^{-\lambda(t-t_i)} \Big) \\
& \leq 2M_{\alpha}N\varepsilon ( m^{-\alpha} +
(e^{\lambda}-1)^{-1}),
\end{align*}
where $m= \min\{ t-t_i, 0<t-t_i\leq 1 \}$, $N$ is the constant in
the Lemma \ref{lemmapoints}.  Hence $\Phi_{1}\in AP_{p}(\mathbb{R}, X)$.

Next, we show that $\Phi_{2} \in WPAP_{p}^{0}(\mathbb{R}, X,\rho)$. For a
given $i\in\mathbb{Z}$, define the function $\eta(t)$  by
$$
\eta(t)=A^{\alpha}T(t-t_i)\gamma_i,\quad t_i<t\leq t_{i+1},
$$
then
$$
\lim_{t\to  \infty} \|\eta(t)\|=
\lim_{t\to  \infty} \|A^{\alpha}T(t-t_i)\gamma_i\|
\leq \lim_{t\to  \infty} M_{\alpha}
(t-t_i)^{-\alpha} e^{-\lambda(t-t_i)} \|\gamma_i\|=0,
$$
then
$\eta\in PC_{p}^{0}(\mathbb{R}, X)\subset WPAP_{p}^{0}(\mathbb{R}, X,\rho)$. Define
$\eta_n:\mathbb{R}\to  X$ by
$$
\eta_n(t)=A^{\alpha}T(t- t_{i-n})\gamma_{i-n},\quad t_i<t\leq t_{i+1},~~n\in \mathbb{N}^{+},
$$
so $\eta_n\in WPAP_{p}^{0}(\mathbb{R}, X,\rho)$. Moreover,
\begin{align*}
\|\eta_n(t)\|
&= \| A^{\alpha}T(t- t_{i-n})\gamma_{i-n}  \|\\
& \leq  M_{\alpha} \sup_{i\in\mathbb{Z}} \|\gamma_i\|
(t-t_{i-n})^{-\alpha}
e^{-\lambda(t-t_{i-n})} \\
& \leq  M_{\alpha} \sup_{i\in\mathbb{Z}}
\|\gamma_i\|(t-t_i+n\kappa)^{-\alpha} e^{-\lambda(t-t_i)}
e^{-\lambda\kappa n}\\
& \leq  M_{\alpha} \sup_{i\in\mathbb{Z}}
\|\gamma_i\|\kappa^{-\alpha}n^{-\alpha} e^{-\lambda\kappa n},
\end{align*}
therefore, the series $\sum_{n=1}^{\infty}\eta_n$ is
uniformly convergent on $\mathbb{R}$. By Lemma \ref{lemmauniform}, one has
$$
\Phi_{2}(t)= \sum_{t_i<t}A^{\alpha}T(t-t_i)\gamma_i
= \sum_{n=0}^{\infty}\eta_n \in WPAP^{0}_{p}(\mathbb{R}, X,\rho).
$$
So \eqref{ineqxiao} holds.

(ii)  $\mathcal{F}$ is a contraction.
For $u, v\in WPAP_{p}(\mathbb{R}, X,\rho)\cap \mathcal{U}PC(\mathbb{R}, X)$,
\begin{align*}
&\|(\mathcal{F} u) (t) -(\mathcal{F} v) (t)\| \\
& \leq \int_{-\infty}^{t}\|A^{\alpha}T(t-s)
 [f(s,A^{-\alpha}u(s),K(A^{-\alpha}u(s)))-f(s,A^{-\alpha}v(s),
 K(A^{-\alpha}v(s)))]\|ds\\
&\quad +\sum_{t_i<t}\|A^{\alpha}T(t-t_i)[I_i(A^{-\alpha}u(t_i))
 -I_i(A^{-\alpha}v(t_i))]\|\\
& \leq \int_{-\infty}^{t}M_{\alpha}(t-s)^{-\alpha}
 e^{-\lambda(t-s)}\big\|f(s,A^{-\alpha}u(s),K(A^{-\alpha}u(s)))\\
&\quad -f(s,A^{-\alpha}v(s),K(A^{-\alpha}v(s)))\big\|ds\\
&\quad +\sum_{t_i<t}M_{\alpha}(t-t_i)^{-\alpha}
 e^{-\lambda(t-t_i)}\|I_i(A^{-\alpha}u(t_i))-I_i(A^{-\alpha}v(t_i))\| \\
& \leq \Big[ M_{\alpha}\lambda^{\alpha-1}\Gamma(1-\alpha)
L_{f}\left(  L_{g}C_{k} \eta^{-1}  + 1\right)\\
&\quad + 2M_{\alpha} N
L_{1}\left( m^{-\alpha} + (e^{\lambda}-1)^{-1}  \right)
\Big]\|u-v\|,
\end{align*}
Since $\Theta<1$, $\mathcal{F}$ is a contraction.

By  the Banach contraction mapping principle, $\mathcal{F}$ has a
unique fixed point $u_{0}\in WPAP_{p}(\mathbb{R}, X, \rho)$ such that
\begin{equation} \label{fixedpoint}
\begin{aligned}
u_{0}&=\int^{t}_{-\infty}A^{\alpha}T(t-s)f(s,A^{-\alpha}u_{0}(s),
K(A^{-\alpha}u_{0}(s)))ds \\
&\quad + \sum_{t_i<t}A^{\alpha}T(t-t_i)I_i(A^{-\alpha}u_{0}(t_i)).
\end{aligned}
\end{equation}
 Since $A^{\alpha}$ is closed,
$$
A^{-\alpha}u_{0}=\int^{t}_{-\infty}T(t-s)f(s,A^{-\alpha}u_{0}(s),
K(A^{-\alpha}u_{0}(s)))ds +
\sum_{t_i<t}T(t-t_i)I_i(A^{-\alpha}u_{0}(t_i)).
$$
Which implies that $A^{-\alpha}u_{0}$ is a mild solution  of
\eqref{nonimpulsiveintegro}. Next, we show that it is a classical
solution.


(iii)  $u_{0}$ is H\"{o}lder continuous.
Note that for every $0<\beta<1-\alpha$ and $h\in (0, \kappa)$, $t\in
(t_i, t_{i+1}-h]$, by Lemma \ref{fundalemma},  one has
$$
\|(T(h)-I)A^{\alpha}T(t-s)\|\leq C_{\beta}h^{\beta} \|A^{\alpha+\beta}T(t-s)\|
$$
and
\begin{align*}
&\|u_{0}(t+h)-u_{0}\|\\
&\leq \| \int^{t}_{-\infty}(T(h)-I)A^{\alpha}T(t-s)f(s,A^{-\alpha}u_{0}(s),
K(A^{-\alpha}u_{0}(s)))ds       \| \\
&\quad +\|
\int^{t+h}_{t}A^{\alpha}T(t+h-s)f(s,A^{-\alpha}u_{0}(s),
K(A^{-\alpha}u_{0}(s)))ds       \|\\
& \leq M_{\alpha+\beta} M C_{\beta} h^{\beta}
\int^{t}_{-\infty}(t-s)^{-(\alpha+\beta)}e^{-\lambda(t-s)}ds+M_{\alpha}M
\int^{t+h}_{t}T(t+h-s)^{-\alpha}ds\\
& \leq M_{\alpha+\beta} M C_{\beta} h^{\beta}
\int^{t}_{-\infty}(t-s)^{-(\alpha+\beta)}e^{-\lambda(t-s)}ds+M_{\alpha}M \frac{h^{1-\alpha}}{1-\alpha},\\
\end{align*}
where
\[
M=\sup_{(t,u,v)\in \mathbb{R}\times X_{\alpha} \times
X_{\alpha}}\|f(t,u,v)\|.
\]
 It follows that there is a constant $C>0$
such that
$$
\|u_{0}(t+h)-u_{0}\|\leq Ch^{\beta}
$$
and therefore $u_{0}$ is H\"{o}lder continuous on $\mathbb{R}$.

Finally, it remains to prove that
$t\to f(t,A^{-\alpha}u_{0}(t), K(A^{-\alpha}u_{0}(t)))$ is H\"{o}lder
continuous on $\mathbb{R}$. By (H4), one has
\begin{align*}
&\|f(t,A^{-\alpha}u_{0}(t), K(A^{-\alpha}u_{0}(t)))-
f(s,A^{-\alpha}u_{0}(s), K(A^{-\alpha}u_{0}(s)))\|\\
& \leq L_{f}(|t-s|^{\theta}+\|u_{0}(t)-u_{0}(s)\|+\|K(A^{-\alpha}u_{0}(t))
 -K(A^{-\alpha}u_{0}(s))\|_{\alpha}).
\end{align*}
Hence $f(t,A^{-\alpha}u_{0}(t), K(A^{-\alpha}u_{0}(t)))$ is
H\"{o}lder continuous on $\mathbb{R}$. Let $u_{0}$ be the solution of
\eqref{fixedpoint} and consider the equation
\begin{gather*}
\frac{du(t)}{dt}+Au(t)=f(t,A^{-\alpha}u_{0}(t), K(A^{-\alpha}u_{0}(t))), \quad
t\in\mathbb{R},\; t\neq t_i, \; i\in\mathbb{Z}, \\
 \Delta u(t_i)=I_i(A^{-\alpha}u_{0}(t_i)).
\end{gather*}
Then this equation  has a  unique classical solution given by
\cite{staalz10}
$$
u(t)= \int^{t}_{-\infty}T(t-s)f(s,A^{-\alpha}u_{0}(s),
K(A^{-\alpha}u_{0}(s)))ds +
\sum_{t_i<t}T(t-t_i)I_i(A^{-\alpha}u_{0}(t_i)).
$$
Moreover, we have $u(t)\in \mathcal{D}(A)$ and $u(t)\in
\mathcal{D}(A^{\alpha})$. Therefore, it follows that
\begin{align*}
A^{\alpha}u(t)
&= \int^{t}_{-\infty}A^{\alpha}T(t-s)f(s,A^{-\alpha}u_{0}(s),
K(A^{-\alpha}u_{0}(s)))ds \\
&\quad + \sum_{t_i<t}A^{\alpha}T(t-t_i)I_i(A^{-\alpha}u_{0}(t_i))\\
&=u_{0}(t),
\end{align*}
which implies that $u(t)=A^{-\alpha}u_{0}(t)$ is the classical
solution of \eqref{nonimpulsiveintegro}, which is a piecewise
weighted almost periodic solution.

Next, we show the attractivity of $u(t)$. Since $u(t)\in
\mathcal{D}(A^{\alpha})$ is the $WPAP_{p}$  mild solution, so if
$t>\sigma$, $\sigma\neq t_i$, $i\in\mathbb{Z}$,
\[
u(t)=T(t-\sigma)u(\sigma)+\int^{t}_{\sigma}T(t-s)f(s,u(s),(Ku)(s))ds
 +\sum_{\sigma<t_i<t}T(t-t_i)I_i(u(t_i)).
\]
Let $u(t)=u(t,\sigma,\varphi)$ and $v(t)=v(t,\sigma,\psi)$ be two
mild solution of \eqref{nonimpulsiveintegro}, then
\begin{gather*}
u(t)=T(t-\sigma)\varphi+\int^{t}_{\sigma}T(t-s)f(s,u(s),(Ku)(s))ds
 +\sum_{\sigma<t_i<t}T(t-t_i)I_i(u(t_i)),\\
v(t)=T(t-\sigma)\psi+\int^{t}_{\sigma}T(t-s)f(s,v(s),(Kv)(s))ds
 +\sum_{\sigma<t_i<t}T(t-t_i)I_i(v(t_i)).
\end{gather*}
So
\begin{align*}
\|u(t)-v(t)\|_{\alpha}
& \leq \|A^{\alpha}T(t-\sigma)[\varphi-\psi]
\| \\
&\quad + \|\int^{t}_{\sigma}A^{\alpha}T(t-s)[f(s,u(s),(Ku)(s))-
f(s,v(s),(Kv)(s))
]ds \|\\
&\quad +\|\sum_{\sigma<t_i<t}A^{\alpha}T(t-t_i)[I_i(u(t_i))-I_i(v(t_i))]
\| \\
&\leq
M_{\alpha}(t-\sigma)^{-\alpha}e^{-\lambda(t-\sigma)}\|\varphi-\psi\|\\
&\quad +\int^{t}_{\sigma}M_{\alpha}(t-s)^{-\alpha}e^{-\lambda(t-s)}L_{f}
 (L_{g}C_{k}\eta^{-1}+1)\|u(s)-v(s)\|_{\alpha}ds\\
&\quad +\sum_{\sigma<t_i<t}M_{\alpha}(t-t_i)^{-\alpha}e^{-\lambda(t-t_i)}
 L_{1}\|u(t)-v(t)\|_{\alpha}.
\end{align*}
For
$M_{\alpha}\kappa^{-\alpha}L_{f}(L_{g}C_{k}\eta^{-1}+1)<\lambda$,
one has
$e^{[M_{\alpha}\kappa^{-\alpha}L_{f}(L_{g}C_{k}\eta^{-1}+1)
 -\lambda](t-\sigma)}\to 0$, and $(t-\sigma)^{-\alpha}\to  0$,
$(t-t_i)^{-\alpha}\to  0$ as $t\to \infty$, hence
for $\varepsilon>0$, there exist $T>\max(0, \sigma+\kappa)$, such
that for $t>T$,
$$
e^{[M_{\alpha}\kappa^{-\alpha}L_{f}(L_{g}C_{k}\eta^{-1}+1)
 -\lambda](t-\sigma)}<\varepsilon,\quad
(t-\sigma)^{-\alpha}<\varepsilon,\quad
(t-t_i)^{-\alpha}<\varepsilon.
$$
Hence for $t>T$, one has
\begin{align*}
\|u(t)-v(t)\|_{\alpha}
&\leq
M_{\alpha}(t-\sigma)^{-\alpha}e^{-\lambda(t-\sigma)}\|\varphi-\psi\|\\
&\quad +\int^{t-\kappa}_{\sigma}M_{\alpha}(t-s)^{-\alpha}e^{-\lambda(t-s)}
 L_{f}(L_{g}C_{k}\eta^{-1}+1)\|u(s)-v(s)\|_{\alpha}ds\\
&\quad +\int^{t}_{t-\kappa}M_{\alpha}(t-s)^{-\alpha}e^{-\lambda(t-s)}
 L_{f}(L_{g}C_{k}\eta^{-1}+1)\|u(s)-v(s)\|_{\alpha}ds\\
&\quad +\sum_{\sigma<t_i<t}M_{\alpha}(t-t_i)^{-\alpha}e^{-\lambda(t-t_i)}
 L_{1}\|u(t)-v(t)\|_{\alpha}\\
&\leq M_{\alpha}\varepsilon e^{-\lambda(t-\sigma)}\|\varphi-\psi\|\\
&\quad +\int^{t-\kappa}_{\sigma}M_{\alpha}\kappa^{-\alpha}e^{-\lambda(t-s)}
 L_{f}(L_{g}C_{k}\eta^{-1}+1)\|u(s)-v(s)\|_{\alpha}ds\\
&\quad +M_{\alpha}L_{f}(L_{g}C_{k}\eta^{-1}+1)(\|u\|_{\alpha,\infty}
 +\|v\|_{\alpha,\infty})\int^{t}_{t-\kappa}(t-s)^{-\alpha}ds\\
&\quad +\sum_{\sigma<t_i<t}M_{\alpha}\varepsilon
 e^{-\lambda(t-t_i)}L_{1}\|u(t)-v(t)\|_{\alpha}\\
&\leq M_{\alpha}\varepsilon e^{-\lambda(t-\sigma)}\|\varphi-\psi\|\\
&\quad +\int^{t}_{\sigma}M_{\alpha}\kappa^{-\alpha}e^{-\lambda(t-s)}L_{f}
 (L_{g}C_{k}\eta^{-1}+1)\|u(s)-v(s)\|_{\alpha}ds\\
&\quad +M_{\alpha}L_{f}(L_{g}C_{k}\eta^{-1}+1)(\|u\|_{\alpha,\infty}
 +\|v\|_{\alpha,\infty})\kappa^{1-\alpha}(1-\alpha)^{-1}\\
&\quad +\sum_{\sigma<t_i<t}M_{\alpha}\varepsilon
e^{-\lambda(t-t_i)}L_{1}\|u(t)-v(t)\|_{\alpha}.
\end{align*}
Let $y(t)=e^{\lambda t}\|u(t)-v(t)\|_{\alpha}$, then
\begin{align*}
y(t)&\leq M_{\alpha}\varepsilon
y(\sigma)+e^{\lambda t}M_{\alpha}L_{f}(L_{g}C_{k}\eta^{-1}+1)(\|u\|_{\alpha,\infty}+\|v\|_{\alpha,\infty})\kappa^{1-\alpha}(1-\alpha)^{-1}\\
&\quad +\int^{t}_{\sigma}M_{\alpha}\kappa^{-\alpha}L_{f}(L_{g}C_{k}\eta^{-1}+1)y(s)ds
+\sum_{\sigma<t_i<t}M_{\alpha}\varepsilon L_{1}y(t_i).
\end{align*}
By the generalized Gronwall-Bellman inequality (Lemma \ref{gronwall}), one has
\begin{align*}
\|y(t)\|&\leq \left[ M_{\alpha}\varepsilon y(\sigma)+e^{\lambda
t}M_{\alpha}L_{f}(L_{g}C_{k}\eta^{-1}+1)(\|u\|_{\alpha,\infty}+\|v\|_{\alpha,\infty})\kappa^{1-\alpha}(1-\alpha)^{-1}
\right]\\
&\quad \times
\prod_{\sigma<t_i<t}(1+M_{\alpha}L_{1}\varepsilon)e^{M_{\alpha}\kappa^{-\alpha}L_{f}(L_{g}C_{k}\eta^{-1}+1)(t-\sigma)}\\
&=\left[ M_{\alpha}\varepsilon y(\sigma)+e^{\lambda
t}M_{\alpha}L_{f}(L_{g}C_{k}\eta^{-1}+1)(\|u\|_{\alpha,\infty}+\|v\|_{\alpha,\infty})\kappa^{1-\alpha}(1-\alpha)^{-1}
\right]\\
&\quad \times (1+M_{\alpha}L_{1}\varepsilon)^{i(\sigma,
t)}e^{M_{\alpha}\kappa^{-\alpha}L_{f}(L_{g}C_{k}\eta^{-1}+1)(t-\sigma)},
\end{align*}
where $i(\sigma, t)$ defined in Lemma \ref{lemmapoints}. That is
\begin{align*}
&\|u(t)-v(t)\|_{\alpha} \\
&\leq \left[ M_{\alpha}\varepsilon
\|\varphi-\psi\|+M_{\alpha}L_{f}(L_{g}C_{k}\eta^{-1}+1)
 (\|u\|_{\alpha,\infty}+\|v\|_{\alpha,\infty})\kappa^{1-\alpha}(1-\alpha)^{-1}
\right]\\
&\quad \times (1+M_{\alpha}L_{1}\varepsilon)^{i(\sigma,
t)}e^{[M_{\alpha}\kappa^{-\alpha}L_{f}(L_{g}C_{k}\eta^{-1}+1)-\lambda](t-\sigma)}\\
&\leq \left[ M_{\alpha}\varepsilon
\|\varphi-\psi\|+M_{\alpha}L_{f}(L_{g}C_{k}\eta^{-1}+1)(\|u\|_{\alpha,\infty}+\|v\|_{\alpha,\infty})\kappa^{1-\alpha}(1-\alpha)^{-1}
\right]\\
&\quad \times (1+M_{\alpha}L_{1}\varepsilon)^{i(\sigma, t)}\varepsilon,
\end{align*}
so $u(t)$ is an attractor.
\end{proof}

\begin{remark} \rm
Consider the nonlinear impulsive integro-differential equations with
delay:
\begin{equation}\label{nonimpulsiveintegrodelay}
\begin{gathered}
\frac{du(t)}{dt}+Au(t)=f(t, u(t-\tau), (Ku)(t)), \quad t\in\mathbb{R},\; t\neq t_i,\;
i\in\mathbb{Z}, \\
(Ku)(t)=\int^{t}_{-\infty}k(t-s)g(s,u(s))ds,\\
 \Delta u(t_i):=u(t^{+}_i)-u(t^{-}_i)=I_i(u(t_i)),
\end{gathered}
\end{equation}
where $\tau\in\mathbb{R}^{+}$. Note that if
$u=u_{1}+u_{2}\in WPAP_{p}(\mathbb{R}, X,\rho)$, where
$u_{1}\in AP_{p}(\mathbb{R}, X)$, $u_{2}\in WPAP^{0}_{p}(\mathbb{R}, X,\rho)$.
For given $\tau\in\mathbb{R}^{+}$, it is not difficult to see that
$u_{1}(t-\tau)\in AP_{p}(\mathbb{R}, X)$. For $r>0$, we see that
\begin{align*}
&\frac{1}{\mu(r,\rho)}\int_{-r}^{r}\rho(t)\|u_{2}(t-\tau)\|dt\\
&=\frac{1}{\mu(r,\rho)}\int_{-r-\tau}^{r-\tau}\rho(t+\tau)\|u_{2}(t)\|dt\\
&\leq \frac{\mu(r+\tau,\rho)}{\mu(r,\rho)}\times\frac{1}{\mu(r+\tau,\rho)}
\int_{-r-\tau}^{r+\tau}\frac{\rho(t+\tau)}{\rho(t)}\rho(t)\|u_{2}(t)\|dt.
\end{align*}
Since $\rho\in U_{T}$, it implies that there exists $\eta>0$ such that
$\frac{\rho(t+\tau)}{\rho(t)}\leq \eta$,
$\frac{\rho(t-\tau)}{\rho(t)}\leq \eta$, for $r>\tau$,
\begin{align*}
\mu(r+\tau,\rho)
&=\int^{r-\tau}_{-r-\tau}\rho(t)dt+\int^{r+\tau}_{r-\tau}\rho(t)dt\\
& \leq \int^{r-\tau}_{-r-\tau}\rho(t)dt+\int^{r+\tau}_{-r+\tau}\rho(t)dt \\
& =\int^{r}_{-r}\rho(t-\tau)dt+\int^{r}_{-r}\rho(t+\tau)dt \leq
2\eta \mu(r,\rho),
\end{align*}
then by $u_{2}\in WPAP^{0}_{p}(\mathbb{R}, X,\rho)$, one has
\[
\frac{1}{\mu(r,\rho)}\int_{-r}^{r}\rho(t)\|u_{2}(t-\tau)\|dt 
\leq \frac{2\eta^{2}}{\mu(r+\tau,\rho)}
 \int_{-r-\tau}^{r+\tau}\rho(t)\|u_{2}(t)\|dt\to 0
\]
as $\to \infty$.
Hence $u_{2}(t-\tau)\in WPAP^{0}_{p}(\mathbb{R}, X,\rho)$, that is
$u(t-\tau)\in WPAP_{p}(\mathbb{R}, X,\rho)$ for $\tau\in\mathbb{R}^{+}$. Thus the
conclusion of Theorem \ref{theoremlip} holds for
\eqref{nonimpulsiveintegrodelay}.
\end{remark}


\begin{remark} \rm
If $(Ku)(t)=0$, then impulsive integro-differential equations
\eqref{nonimpulsiveintegro} become nonlinear impulsive differential
equations:
\begin{equation}\label{impulsivede}
\begin{gathered}
\frac{du(t)}{dt}+Au(t)=f(t, u(t)), \quad t\in\mathbb{R},\; t\neq t_i,\; i\in\mathbb{Z}, \\
 \Delta u(t_i)=I_i(u(t_i)),
\end{gathered}
\end{equation}
By Theorem \ref{theoremlip}, one has the following corollary.
\end{remark}

\begin{corollary}\label{theoremdelip}
Assume that {\rm (H1),  (H5)} hold and satisfy the
condition:
\begin{itemize}
\item [(H4')] $f\in WPAP_{p}(\mathbb{R}\times X_{\alpha}, X,\rho)$,
$\rho\in U_{T}$ and there exists  constants $L_{f}>0$, $0<\theta<1$ such that
$$\|f(t_{1}, u)-f(t_{2}, v) \| \leq
L_{f}(|t_{1}-t_{2}|^{\theta}+\|u-v\|_{\alpha}),\quad t\in\mathbb{R}, ~u, v\in
X_{\alpha},
$$
\end{itemize}
then \eqref{impulsivede} has a unique classical solution
$u\in WPAP_{p}(\mathbb{R}, X,\rho)$ which is an attractor if
$$
\vartheta:= M_{\alpha}\lambda^{\alpha-1}\Gamma(1-\alpha)
L_{f}+ 2M_{\alpha} N L_{1}\left( m^{-\alpha} + (e^{\lambda}-1)^{-1}
\right)<1.
$$
\end{corollary}


\section{Example}

Consider the integro-differential equation with impulsive effects
\begin{equation}\label{exam2}
\begin{gathered}
\frac{\partial w(t,x)}{\partial t}-\frac{\partial^{2} w(t,x)}{\partial
x^{2}}=f(t,x,w(t,x),Kw(t,x)),  \quad t\in\mathbb{R},\; t\neq t_i,\; i\in\mathbb{Z},\; x\in(0,1),\\
Kw(t,x)=\int^{t}_{-\infty}e^{-\eta(t-s)}g(s,x,w(s,x))ds, \\
\Delta w (t_i, x)=\beta_iw(t_i, x), \quad i\in\mathbb{Z}, \; x\in[0,1], \\
 w(t,0)=w(t,1)=0,
\end{gathered}
\end{equation}
where  $t_i=i+\frac{1}{4}|\sin i + \sin\sqrt{2}i|$,
$\beta_i\in WPAP(\mathbb{Z}, \mathbb{R},\rho)$, $\rho\in U_{T}$. Note that $\{t^{j}_i\}$,
$i\in\mathbb{Z}$, $j\in\mathbb{Z}$ are equipotentially almost periodic and
$\kappa=\inf_{i\in\mathbb{Z}} (t_{i+1}-t_i)>0 $, one can see
\cite{liuzha13,samperbook95} for more details.

Define the operator $A$ by
$$
A u:=-u'',\quad  u\in \mathcal{D}(A),
$$
where
$$
\mathcal{D}(A):=\{u\in H^{1}_{0}((0,1),\mathbb{R})\cap
H^{2}((0,1),\mathbb{R}): ~u''\in H \}.
$$
The operator $A$ is the
infinitesimal generator of an analytic semigroup $(T(t))_{t\geq 0}$
and also self adjoint \cite{pazbook83}. Let $\alpha=1/2$, so
$\mathcal{D}(A^{1/2})$ is a Banach space endowed with the norm
$$
\|u\|_{1/2}=\|A^{1/2}u\|,\quad u\in \mathcal{D}(A^{1/2}).
$$
We call this space $X_{1/2}$. For more details about $X_{1/2}$, one
can see \cite{bahsid02}.

Let $u(t)x=w(t,x)$, $t\in\mathbb{R},x\in[0,1]$ and
\[
f(t, u(t), (Ku)(t))(x)=f(t,x,w(t,x),Kw(t,x)).
\]
Then \eqref{exam2} can be
rewritten as the abstract form \eqref{nonimpulsiveintegro}. Since
$I_i(u)=\beta_iu$ and $\beta_i\in WPAP(\mathbb{Z}, \mathbb{R},\rho)$, then
(H5) hold with $L_{1}=\sup_{i\in\mathbb{Z}}\|\beta_i\|$. By
Theorem \ref{theoremlip}, one has the following result.

\begin{theorem}
Assume that
\begin{itemize}
\item [(A1)] $g\in WPAP_{p}(\mathbb{R}\times X_{1/2}, X,\rho)$, $\rho\in U_{T}$
and there exists a constant $L_{g}>0$ such that
$$
\|g(t,u)-g(t,v)\|
\leq L_{g}\|u-v\|_{1/2}, \quad t\in\mathbb{R},~~u ,v\in X_{1/2}.
$$

\item [(A2)] $f\in WPAP_{p}(\mathbb{R}\times X_{1/2}\times X_{1/2}, X,\rho)$,
$\rho\in U_{T}$ and there exists  constants $L_{f}>0$, $0<\theta<1$ such that
$$
\|f(t_{1}, u_{1},v_{1})-f(t_{2}, u_{2},v_{2}) \|
\leq L_{f}(|t_{1}-t_{2}|^{\theta}+\|u_{1}-u_{2}\|_{1/2}+\|v_{1}-v_{2}\|_{1/2}),
$$
for each $(t_i,u_i,v_i)\in \mathbb{R}\times X_{1/2}\times X_{1/2}$, $i=1,2$.

\item [(A3)] $\beta_i\in WPAP(\mathbb{Z}, \mathbb{R},\rho)$, $\rho\in U_{T}$.
\end{itemize}
then \eqref{exam2} has a unique $WPAP_{p}$ solution which is an attractor  if $\vartheta<1$, where
$$
\vartheta:= M_{\alpha}\lambda^{-\frac{1}{2}}\Gamma\big(\frac{1}{2}\big)
L_{f}\left(  L_{g}\eta^{-1}  + 1\right)+ 2M_{\alpha} N \Big(
m^{-\frac{1}{2}} + (e^{\lambda}-1)^{-1}  \Big)
\sup_{i\in\mathbb{Z}}\|\beta_i\|.
$$
\end{theorem}

\subsection*{Conclusion}
The notion of almost periodic function $AP(\mathbb{R}, X)$ introduced by
Bohr in 1925. Since then, there have various important
generalization of this concept, like:
\begin{itemize}
\item [(i)] Asymptotically almost periodic function $AAP(\mathbb{R}, X)$;

\item [(ii)] Weakly almost periodic function $WAP(\mathbb{R}, X)$;

\item [(iii)] Pseudo almost periodic function $PAP(\mathbb{R}, X)$;

\item [(iv)] Weighted pseudo almost periodic function $WPAP(\mathbb{R}, X,\rho)$;
\end{itemize}
and many more. For origin references, details of these functions,
one can see \cite{xznfm2012} and the relationship between these
functions as follows:
$$
AP(\mathbb{R}, X)\subset AAP(\mathbb{R}, X) \subset WAP(\mathbb{R}, X) \subset PAP(\mathbb{R}, X)
\subset  WPAP(\mathbb{R}, X,\rho).
$$
The application of these functions in the context of various kinds
of abstract differential equations attracted many mathematicians. In
this paper, by the fixed point theorem and fractional powers of
operators, we investigate the applications of weighted pseudo almost
periodic functions to the impulsive integro-differential equations.
The existence, uniqueness and attractivity  of piecewise $WPAP$
classical solutions of nonlinear impulsive integro-differential
equations are given.

\subsection*{Acknowledgements}
The authors would like to thank the editor and referees for their
valuable comments and remarks, which led to a great improvement of
the article. This research is supported by the National Natural
Science Foundation of China (No.11426201, 11271065, 61273016) and
the Natural Science Foundation of Zhejiang Province
(No.LQ13A010015).


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\end{document}