\documentclass[reqno]{amsart}
\usepackage{hyperref}
\usepackage{mathrsfs,amssymb}

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2013 (2013), No. 84, pp. 1--15.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2013 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2013/84\hfil Existence and stability of mild solutions]
{Existence and stability of mild solutions to impulsive
stochastic neutral partial functional differential equations}

\author[D. He, L. Xu \hfil EJDE-2013/84\hfilneg]
{Danhua He, Liguang Xu}  % in alphabetical order

\address{Danhua He \newline
Department of Mathematics, Zhejiang International Studies
University, Hangzhou, \newline 310012,  China}
\email{danhuahe@126.com}

\address{Liguang Xu \newline
Department of Applied Mathematics, Zhejiang University of Technology,
Hangzhou, 310023, China}
\email{xlg132@126.com}

\thanks{Submitted September 4, 2010. Published April 5, 2013.}
\subjclass[2000]{35R60, 60H15, 35B35, 35A01}
\keywords{Existence and uniqueness; exponential stability; mild solution;
\hfill\break\indent impulsive stochastic neutral equations}

\begin{abstract}
 In this article, we study a class of impulsive stochastic neutral
 partial functional differential equations in a real separable Hilbert space.
 By using Banach fixed point theorem, we give sufficient conditions for
 the existence and uniqueness of a mild solution. Also the exponential
 $p$-stability of a mild solution and its sample paths are obtained.
\end{abstract}

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

\section{Introduction}

Since deterministic neutral functional differential equations were
originally proposed by Hale and Meyer \cite{H1}, many researchers
including Hale and Verduyn Lunel \cite{H2}, Kolmanovskii and Nosov
\cite{K} have done extensive works on this subject  and 
their applications. However, a system is usually affected by external 
perturbations which in many
cases are of great random uncertainties such as stochastic forces on
the physical systems and noisy measurements caused by environmental
uncertainties \cite{Sakthivel2,Sakthivel3,Sakthivel4}, a stochastic
neutral functional differential equations should be produced instead
of a deterministic one. Correspondingly, a number of interesting
results on the stochastic neutral functional differential equations
have been reported \cite{5,6,3,1,2,4}. 
Sakthivel, Ren and Kim \cite{Sakthivel5} studied the existence and
asymptotic stability in pth moment of mild solutions to second-order
nonlinear neutral stochastic differential equations. 
Ren and Sakthivel \cite{Ren} studied the existence, uniqueness, 
and stability of mild solutions for second-order neutral stochastic 
evolution equations with infinite delay and Poisson jumps.
More recently, the existence, uniqueness and exponential stability 
of a mild solution of the stochastic neutral partial functional 
differential equations have been considered by Luo \cite{L1}.

However, in addition to stochastic effects, impulsive effects
exist in many evolution processes in which states are
changed abruptly at certain moments of time, involved in such fields
as medicine and biology, economics, mechanics, electronics. 
Impulsive effects often make the system properties decline or even
cause instability. Therefore, impulsive effects should be taken
into account in researching the exponential stability of the
stochastic systems. Some significant progress has been made in the
techniques and methods of studying the existence and stability for
impulsive stochastic difference equations, impulsive stochastic
differential equations with delays and impulsive stochastic partial
functional differential equations \cite{L2,18,16,17,Xu1,14,15}.
However, so far no work has been reported on the corresponding
problems for impulsive stochastic neutral partial functional
differential equations and the aim of this paper is to close this
gap.

Motivated by the above discussions, we will study the existence,
uniqueness and exponential $p$-stability of a mild solution of the
impulsive stochastic neutral partial functional differential
equations. By using Banach fixed point theorem, we give some
sufficient conditions for the existence and uniqueness of a
mild solution of this class of equations. Also the exponential
$p$-stability of a mild solution as well as its sample paths are
obtained.

\section{Preliminaries}

Throughout this paper, unless otherwise specified,  $X$ and $Y$
are two separable Hilbert spaces. with norms $\|\cdot\|_X$,
$\|\cdot\|_Y$, respectively. Let $\mathcal {L}(Y,X)$
be the space of all bounded linear operators from $Y$ into $X$
equipped with the usual norm $\|\cdot\|$.

Let $(\Omega, \mathcal {F}, \{\mathcal {F}_t\}_{t\geq0}, P)$  be a
complete probability space with a filtration 
$\{\mathcal{F}_t\}_{t\geq0}$ satisfying the usual conditions 
of complete sub-$\sigma$-fields of $\mathcal {F}$. 
Let $w(t)$, $t\geq 0$, be a $Y$-valued, $Q$-Wiener process which is 
assumed to be adapted to
$\{\mathcal {F}_t\}_{t\geq0}$ and for every $t>s$ the increments
$w(t)-w(s)$ are independent of $\mathcal {F}_s$.
 Hence, $w(t),t\geq 0$, is a continuous martingale relative to 
$\{\mathcal{F}_t\}_{t\geq0}$ and we have the following representation of
$w(t)$:
\begin{align*}
w(t)=\sum_{i=1}^{\infty}B^{i}_{t}e_i,
\end{align*}
where $\{e_i\}$ is an orthonormal set of eigenvectors of $Q$,
$\{B^{i}_{t}\}$, $t\geq0$, is a family of mutually independent real
Wiener processes with incremental covariance $\lambda_i>0$,
$Qe_i=\lambda_ie_i$ and $tr Q=\sum_{i=1}^{\infty}\lambda_i<\infty$.
Let $h(t)$ be an $\mathcal{L}(Y,X)$-valued function and $\lambda$ be
an arbitrary sequence $\{\lambda_1,\lambda_2,\dots\}$ of positive
numbers, we may often use the following notation
\begin{align*}
\|h(t)\|_{\lambda}:=\Big\{\sum_{n=1}^{\infty}|\sqrt{\lambda_n}h(t)e_n|^2\Big\}^{1/2}
\end{align*}
whenever this series is convergent.
\begin{align*}
PC[\mathbb{J},  X]=\Big\{&\psi :\mathbb{J} \to X
: \psi(s) \text{ is continuous for all but at most countably}\\
 &\text{many points $s \in \mathbb{J}$ and at these points
  $\psi(s^{+})$ and $\psi(s^{-})$}\\
&\text{exist and $\psi(s)=\psi(s^{+})$}
\Big\},
\end{align*}
where $\mathbb{J}\subset \mathbb{R}$ is an interval, $\psi(s^{+})$
and $\psi(s^{-})$ denote the right-hand and left-hand
 limits of the function $\psi(s)$ at time $s$, respectively.
Let $PC:=PC[[-\tau,0],X]$ with the norm 
$\|\varphi\|_{PC}=\sup_{-\tau\leq\theta\leq
0}\|\varphi(\theta)\|_{X}$.
 Let $PC_{\mathscr F_{0}}^b[[-\tau,0],X]$ be the Banach space
 of all bounded $\mathscr F_{0}$-measurable,
$PC[[-\tau,0],X]$-valued random variables $\phi$, satisfying
$\sup_{-\tau \leq\theta\leq 0} E\|\phi(\theta)\|_X^p<\infty$ for
$p>0$, where $E$ denote the expectation of stochastic process.

Assume that $\{S(t), t\geq 0\}$ is an analytic semigroup with its
infinitesimal generator $A$, then it is possible under some
circumstances (see \cite{P}) to define the fractional power
$(-A)^{\alpha}$ for any $\alpha\in [0,1]$ which is a closed linear
operator with its domain $\mathcal {D}((-A)^{\alpha})$.

Consider the following impulsive stochastic neutral partial
functional differential equations
\begin{equation}\label{i1}
\begin{gathered}
d[x(t)+G(t,x_t)]=[Ax(t)+f(t,x_t)]dt+g(t,x_t)dw(t), \quad t\geq0,\; t\neq t_k,\\
\Delta x(t_k)=x(t^+_k)-x(t^-_k)=I_k(x(t^-_k)),\quad t=t_k,\; k=1,2,\dots,N,\\
x_i(t)=\phi_i(t),\quad -\tau\leq t\leq 0,
 \end{gathered}
\end{equation}
where $x_t(\theta)=x(t+\theta)$ for $\theta\in[-\tau,0]$, 
$G,f:\mathbb{R}_+\times PC\to X$, and 
$g:\mathbb{R}_+\times PC\to \mathcal {L}(Y,X)$ are all Borel measurable.
 $I_k$ shows the jump in the state $x$ at time $t_k$, and $t_k$ satisfies
$0<t_1<\dots<t_N<\lim_{k\to\infty}t_k=\infty$.

\begin{definition} \label{def1}\rm
 A process $\{x(t),t\in[0,T]\}$, $0\leq
T<\infty$, is called a mild solution of \eqref{i1} if
\begin{itemize}
 \item[(i)] $x(t)$ is $\mathcal {F}_t$-adapted;

\item[(ii)] $x(t)\in X$ has c\`adl\`ag paths on $t\in[0,T]$ a.s
and for each $t\in[0,T]$, $x(t)$ satisfies the integral equation
\begin{equation}
\begin{aligned}
x(t)&=S(t)[\phi(0)+G(0,\phi)]-G(t,x_t)-\int^{t}_0AS(t-s)G(s,x_s)ds\\
&\quad +\int^{t}_{0}S(t-s)f(s,x_s)ds
+\int^{t}_{0}S(t-s)g(s,x_s)dw(s)\\
&\quad +\sum_{0<t_k<t}S(t-t_k)I_k(x(t^-_k)),
\end{aligned}
\end{equation}
for any $\phi\in PC_{\mathscr F_{0}}^b[[-\tau,0], X]$ almost surely.
\end{itemize}
\end{definition}

\section{Main results}

We assume the following hyptheses:
\begin{itemize}
 \item[(H1)] $A$ is the infinitesimal generator of an analytic semigroup
 of bounded linear operators $\{S(t),t\geq 0\}$ in $X$ satisfying
 $\|S(t)\|_{X}\leq\gamma e^{v t}$, $t\geq 0$,
 for some constants $\gamma\geq1$ and $v\in\mathbb{R}$.

\item[(H2)] There exist constants $C_1>0$ and $\alpha\in[0,1]$ such that for
any $x,y\in PC$ and $t\geq 0$, $G(t,x)\in\mathcal
{D}((-A)^{\alpha})$,
\[
\|(-A)^{\alpha}G(t,x)-(-A)^{\alpha}G(t,y)\|_{X}\leq C_1\|x-y\|_{PC}.
\]

\item[(H3)] The functions $f$ and $g$ satisfy the Lipschitz
and linear growth conditions; that is, there exist positive
constants $C_2$, $C_3$, $C_4$ such that
\begin{gather*}
\|f(t,x)-f(t,y)\|_{X}\leq C_2\|x-y\|_{PC},\\
\|g(t,x)-g(t,y)\|_{\lambda}\leq C_3\|x-y\|_{PC},\\
\|f(t,x)\|_{X}+\|g(t,x)\|_{\lambda}\leq C_4(1+\|x\|_{PC}),
\end{gather*}
for any $x,y\in PC$, $t\geq 0$.

\item[(H4)] There exist nonnegative constants $q_k$ such that for any 
$x,y\in PC$,
\begin{equation}\label{15}
\|I_k(x)-I_k(y)\|_{X}\leq q_k\|x-y\|_{PC},\quad  k=1,2,\dots,N.
\end{equation}
\end{itemize}

\begin{lemma}[{\cite[Lemma 2.1]{L1}}] \label{lem1}
Suppose that the assumption (H1) holds, then for any $\beta\geq0$,
\begin{itemize}
 \item[(i)] for each $x\in\mathcal
{D}((-A)^{\beta})$,
\[
S(t)(-A)^{\beta}x=(-A)^{\beta}S(t)x;
\]

\item[(ii)] There exist constants $M_{\beta}>0$ and $v\in\mathbb{R}$ such that
\[
\|(-A)^{\beta}S(t)\|_{X}\leq M_{\beta}t^{-\beta}e^{vt},\quad t>0.
\]
\end{itemize}
\end{lemma}

\begin{theorem} \label{thm1} 
 Under assumptions {\rm (H1)--(H4)}, system \eqref{i1} has a unique mild 
solution $x(t)$, $0\leq t\leq T$, if the following condition holds
\begin{align*}
L&:=5^{p-1}\Big[\|(-A)^{-\alpha}\|^{p}C_1^{p}+M_{1-\alpha}^{p}\tau^{p}(T)T^{p-1}
+\gamma^{p}e^{p|v|T}C_2^{p}T^{p-1} \\
&\quad +\gamma^{p}e^{p|v|T}C_3^{p}T^{\frac{p-2}{2}}
+N^{p-1}\gamma^{p}e^{p|v|T}(\sum_{k=1}^{N}q_k^{p})\Big]<1,
\end{align*}
where $M_{1-\alpha}$ and $v$ are the constants in Lemma \ref{lem1} (ii),
and
\[
p\geq 2,\quad \tau(T)=\int_{0}^{T}t^{-(1-\alpha)}e^{vt}dt<\infty.
\]
\end{theorem}


\begin{proof}
 Define a nonlinear operator $\Psi:PC_{\mathscr
F_{0}}^b[[-\tau,0], X]\to PC_{\mathscr F_{0}}^b[[-\tau,0],X]$ by
\begin{equation} \label{1}
\begin{aligned}
\Psi x(t)&=S(t)\phi(0)+S(t)(-A)^{-\alpha}(-A)^{\alpha}G(0,\phi)
-(-A)^{-\alpha}(-A)^{\alpha}G(t,x_t) \\
&\quad +\int^{t}_0(-A)^{1-\alpha}S(t-s)(-A)^{\alpha}G(s,x_s)ds
+\int^{t}_{0}S(t-s)f(s,x_s)ds \\
&\quad +\int^{t}_{0}S(t-s)g(s,x_s)dw(s)
+\sum_{0<t_k<t}S(t-t_k)I_k(x(t^-_k)),\quad 0\leq t\leq T.
\end{aligned}
\end{equation}
From Lemma \ref{lem1}, (H1), (H2) and (H4), we obtain that for any $0\leq
t\leq T$,
\begin{equation} \label{11}
\begin{aligned}
\|x(t)\|_{X}
&\leq \gamma e^{|v|T}\|\phi(0)\|_{X}+\gamma
e^{|v|T}C\|(-A)^{-\alpha}\|(1+\|\phi\|_{PC})\\
&\quad +C \|(-A)^{-\alpha}\|(1+\|x_t\|_{PC})\\
&\quad +\int^{t}_0CM_{1-\alpha}(t-s)^{-(1-\alpha)}e^{v(t-s)}(1+\|x_s\|_{PC})ds\\
&\quad +\|\int^{t}_{0}S(t-s)f(s,x_s)ds\|_{X}
+\|\int^{t}_{0}S(t-s)g(s,x_s)dw(s)\|_{X}\\
&\quad +\|\sum_{0<t_k<t}S(t-t_k)I_k(x(t^-_k))\|_{X} \\
&\leq \gamma e^{|v|T}\|\phi(0)\|_{X}+\gamma
e^{|v|T}C\|(-A)^{-\alpha}\|(1+\|\phi\|_{PC})\\
&\quad +C \|(-A)^{-\alpha}\|(1+\|x_t\|_{PC})
 +CM_{1-\alpha}\tau(T)(1+\sup_{0\leq s\leq t}\|x_s\|_{PC})\\
&\quad +\gamma e^{|v|T}\sum_{k=1}^{N}q_k\|x(t^-_{k})\|_{PC}
+\|\int^{t}_{0}S(t-s)f(s,x_s)ds\|_{X}\\
&\quad +\|\int^{t}_{0}S(t-s)g(s,x_s)dw(s)\|_{X},
\end{aligned}
\end{equation}
where $C$ is a positive constant.
Since
\[
\sup_{0\leq t\leq T}\|x_t\|_{PC}\leq\sup_{0\leq t\leq
T}\|x(t)\|_{X}+\|\phi\|_{PC},
\]
 using \eqref{11} we have
\begin{equation}  \label{12}
\begin{aligned}
&\sup_{0\leq t\leq T}\|x(t)\|_{X}\\
&\leq \gamma e^{|v|T}\|\phi(0)\|_{X}+\gamma
e^{|v|T}C\|(-A)^{-\alpha}\|(1+\|\phi\|_{PC}) \\
&\quad +C \|(-A)^{-\alpha}\|(1+\sup_{0\leq t\leq
T}\|x(t)\|_{X}+\|\phi\|_{PC}) \\
&\quad +CM_{1-\alpha}\tau(T)(1+\sup_{0\leq t\leq
T}\|x(t)\|_{X}+\|\phi\|_{PC})+\gamma
e^{|v|T}\sum_{k=1}^{N}q_k\sup_{0\leq t\leq
T}\|x(t)\|_{X} \\
&\quad +\sup_{0\leq t\leq T}\|\int^{t}_{0}S(t-s)f(s,x_s)ds\|_{X}
+\sup_{0\leq t\leq T}\|\int^{t}_{0}S(t-s)g(s,x_s)dw(s)\|_{X};
\end{aligned}
\end{equation}
that is,
\begin{equation} \label{13}
\begin{aligned}
&[1-C\|(-A)^{-\alpha}\|-CM_{1-\alpha}\tau(T)-\gamma
e^{|v|T}\sum_{k=1}^{N}q_k]\sup_{0\leq t\leq
T}\|x(t)\|_{X} \\
&\leq \gamma e^{|v|T}\|\phi(0)\|_{X}+C(1+\|\phi\|_{PC})[(\gamma
e^{|v|T}+1)\|(-A)^{-\alpha}\|+M_{1-\alpha}\tau(T)] \\
&\quad +\sup_{0\leq t\leq T}\|\int^{t}_{0}S(t-s)f(s,x_s)ds\|_{X}
+\sup_{0\leq t\leq T}\|\int^{t}_{0}S(t-s)g(s,x_s)dw(s)\|_{X}.
\end{aligned}
\end{equation}
So, by the linear growth conditions in (H3), H\"older's inequality
and Burkholder-Davis-Gundy type of inequality for stochastic
convolutions \cite{G}, for any $p\geq 2$, there exist a number
$c(p,T)>0$ such that
\begin{equation} \label{10}
\begin{aligned}
&\sup_{0\leq t\leq T}E\|\Psi x(t)\|^p_{X} \\
&\leq 3^{p-1}[1-C\|(-A)^{-\alpha}\|-CM_{1-\alpha}\tau(T)-\gamma
e^{|v|T}\sum_{k=1}^{N}q_k]^{-p} \\
&\quad \times\Big\{(\gamma
e^{|v|T}\|\phi(0)\|_{X}+C(1+\|\phi\|_{PC})[(\gamma
e^{|v|T}+1)\|(-A)^{-\alpha}\|+M_{1-\alpha}\tau(T)])^{p} \\
&\quad +2^p\gamma^pe^{p|v|T}(T^{p-1}+c(p,T))C^p_4
 \int_{0}^{T}(1+E\|x_s\|_{PC}^{p})ds\Big\}.
\end{aligned}
\end{equation}
This means that $\Psi x(t)\in PC_{\mathscr F_{0}}^b[[-\tau,0], X]$,
if $x(t)\in PC_{\mathscr F_{0}}^b[[-\tau,0], X]$. Further, for
$x(t),y(t)\in PC$, $0\leq t\leq T$, we have
\begin{align*} %\label{2}
&\sup_{t\in[0,T]}E\|\Psi x(t)-\Psi y(t)\|^p_{X}\\
&\leq 5^{p-1}\sup_{t\in[0,T]}E\|G(t,x_{t})-G(t,y_{t})\|^p_{X} \\
&\quad +5^{p-1}\sup_{t\in[0,T]}E\|\int_{0}^{t}AS(t-s)G(s,x_s)ds-\int_{0}^{t}AS(t-s)G(s,y_s)ds\|^p_{X} \\
&\quad +5^{p-1}\sup_{t\in[0,T]}E\|\int_{0}^{t}S(t-s)f(s,x_s)ds-\int_{0}^{t}S(t-s)f(s,y_s)ds\|^p_{X} \\
&\quad +5^{p-1}\sup_{t\in[0,T]}E\|\int_{0}^{t}S(t-s)g(s,x_s)dw(s)-\int_{0}^{t}S(t-s)g(s,y_s)dw(s)\|^p_{X} \\
&\quad +5^{p-1}\sup_{t\in[0,T]}E\|\sum_{0<t_k<t}S(t-t_k)I_k(x(t^-_k))-\sum_{0<t_k<t}S(t-t_k)I_k(y(t^-_k))\|^p_{X} \\
&\leq 5^{p-1}\|(-A)^{-\alpha}\|^{p}C_1^{p}\sup_{t\in[0,T]}E\|x_t-y_t\|_{PC}^{p}+5^{p-1}(M_{1-\alpha}^{p}\tau^{p}(T)T^{p-1} \\
&\quad +\gamma^{p}e^{p|v| T}C_2^{p}T^{p-1}+\gamma^{p}e^{p|v| T}C_3^{p}T^{\frac{p-2}{2}})\sup_{t\in[0,T]}E\int_{0}^{t}\|x_t-y_t\|_{PC}^{p}ds \\
&\quad +5^{p-1}N^{p-1}\gamma^{p}e^{p|v|
T}\sup_{t\in[0,T]}\sum_{k=1}^NE\|I_k(x(t^-_k))-I_k(y(t^-_k))\|^p_{X} \\
&\leq 5^{p-1}\|(-A)^{-\alpha}\|^{p}C_1^{p}\sup_{t\in[0,T]}E\|x_t-y_t\|_{PC}^{p}+5^{p-1}(M_{1-\alpha}^{p}\tau^{p}(T)T^{p-1} \\
&\quad +\gamma^{p}e^{p|v| T}C_2^{p}T^{p-1}+\gamma^{p}e^{p|v| T}C_3^{p}T^{\frac{p-2}{2}})\sup_{t\in[0,T]}E\int_{0}^{t}\|x_t-y_t\|_{PC}^{p}ds \\
&\quad +5^{p-1}N^{p-1}\gamma^{p}e^{p|v|
T}(\sum_{k=1}^{N}q_k^{p})\sup_{t\in[0,T]}E\|x(t)-y(t)\|_{PC}^{p} \\
&\leq 5^{p-1}\Big[\|(-A)^{-\alpha}\|^{p}C_1^{p}+M_{1-\alpha}^{p}\tau^{p}(T)T^{p-1}
+\gamma^{p}e^{p|v| T}C_2^{p}T^{p-1} \\&\quad +\gamma^{p}e^{p|v|
T}C_3^{p}T^{\frac{p-2}{2}}+N^{p-1}\gamma^{p}e^{p|v|
T}(\sum_{k=1}^{N}q_k^{p})\Big]\sup_{t\in[0,T]}E\|x_t-y_t\|_{PC}^{p}.
\end{align*}
Hence, we obtain
\[
\sup_{t\in[0,T]}E\|\Psi x_t-\Psi y_t\|^p_{PC}\leq
L\sup_{t\in[0,T]}E\|x_t-y_t\|_{PC}^{p}.
\]
 For $L<1$, the mapping $\Psi$ is a contraction
mapping. By Banach fixed point theorem, there exists a unique fixed
point, which implies system \eqref{i1} has a unique mild solution.
The proof is complete.
\end{proof}

\begin{theorem} \label{thm2} 
 Under assumptions {\rm (H1)--(H4)},
 the mild solution of \eqref{i1} is exponentially
$p$-stable $(p\geq2)$ provided
\[
\|I_k(x(t^-_k))\|^{p}_{X}\leq d_k, \quad d_k\geq 0,\; k=1,2,\dots,N,
\]
and
\begin{equation} \label{newe16}
\frac{[6^{p-1}C^p_1M^p_{1-\alpha}\Gamma^{p/q}(q\alpha-q+1)/r^{p\alpha-1}
+6^{p-1}C^p_2\gamma^p/r^{p-1}
+6^{p-1}\gamma^pC^p_3/r^{\frac{p}{2}-1}]}{(1-6^{p-1}C^p_1\|(-A)^{-\alpha}\|^p)}>0,
\end{equation}
where $M_{1-\alpha}>0$, $r=-v>0$ are the constants in Lemma \ref{lem1} (ii)
and
$\Gamma(\cdot)$ is the gamma function.
\end{theorem}

\begin{proof} 
Note that in this situation,  for any $\beta\geq0$, there exist constants
 $M_{\beta}>0$ and $r>0$ such that
\[
\|(-A)^{\beta}S(t)\|_{X}\leq M_{\beta}t^{-\beta}e^{-rt},\quad t>0.
\]
It is known that
\begin{align*}
x(t)&=S(t)[\phi(0)+G(0,\phi)]-G(t,x_t)-\int^{t}_0AS(t-s)G(s,x_s)ds \\
&\quad +\int^{t}_{0}S(t-s)f(s,x_s)ds+\int^{t}_{0}S(t-s)g(s,x_s)dw(s)\\
&\quad +\sum_{0<t_k<t}S(t-t_k)I_k(x(t^-_k)).
\end{align*}
By  Lemma \ref{lem1}, (H2), and the fact that $f(t,0)\equiv0$
almost surely in $t$, we obtain that for any $t\geq\tau$,
$-\tau\leq\theta\leq0$,
\begin{equation}  \label{h1}
\begin{aligned}
&\|x(t+\theta)\|_{X}\\
&\leq \gamma
e^{-r(t+\theta)}(\|\phi(0)\|+C_1\|\phi\|_{PC}\|(-A)^{-\alpha}\|)
+C_1\|(-A)^{-\alpha}\|\|x_{t+\theta}\|_{PC} \\
&\quad +\int^{t+\theta}_0C_1\frac{M_{1-\alpha}e^{-r(t+\theta-s)}}{(t+\theta-s)^{(1-\alpha)}}\|x_s\|_{PC}ds
+\|\int^{t+\theta}_{0}S(t+\theta-s)f(s,x_s)ds\|_{X} \\
&\quad +\|\int^{t+\theta}_{0}S(t+\theta-s)g(s,x_s)dw(s)\|_{X}
+\sum_{k=1}^{N}\gamma e^{-r(t+\theta-t_k)}\|I_k(x(t_k^-))\|_{X}.
\end{aligned}
\end{equation}
Thus, for any $t\geq\tau$ and $-\tau\leq\theta\leq0$, we have
\begin{equation} \label{h2}
\begin{aligned}
E\|x(t+\theta)\|^p_{X}
&\leq 6^{p-1}\gamma^p
e^{-pr(t+\theta)}(\|\phi(0)\|+C_1\|(-A)^{-\alpha}\|\|\phi\|_{PC})^p \\
&\quad +6^{p-1}C^p_1\|(-A)^{-\alpha}\|^pE\|x_{t+\theta}\|^p_{PC} \\
&\quad +6^{p-1}E(\int^{t+\theta}_0C_1\frac{M_{1-\alpha}e^{-r(t+\theta-s)}}{(t+\theta-s)^{(1-\alpha)}}\|x_s\|_{PC}ds)^p \\
&\quad +6^{p-1}E\|\int^{t+\theta}_{0}S(t+\theta-s)f(s,x_s)ds\|^p_{X} \\
&\quad +6^{p-1}E\|\int^{t+\theta}_{0}S(t+\theta-s)g(s,x_s)dw(s)\|^p_{X} \\
&\quad +6^{p-1}N^{p-1}\gamma^p
\sum_{k=1}^{N}e^{-pr(t+\theta-t_k)}E\|I_k(x(t_k^-))\|^p_{X}.
\end{aligned}
\end{equation}
By using H\"older's inequality and Burkholder-Davis-Gundy type of
inequality, we have that for any $t\geq\tau$,
$-\tau\leq\theta\leq0$,
% \label{h}
\begin{align*}
&E\Big(\int^{t+\theta}_0C_1\frac{M_{1-\alpha}
 e^{-r(t+\theta-s)}}{(t+\theta-s)^{(1-\alpha)}}\|x_s\|_{PC}ds\Big)^p \\
&=E\Big(\int^{t+\theta}_0C_1\frac{M_{1-\alpha}
e^{-(\frac{1}{q}+\frac{1}{p})r(t+\theta-s)}}
{(t+\theta-s)^{(1-\alpha)}}\|x_s\|_{PC}ds\Big)^p \\
&\leq\Big[\Big(\int^{t+\theta}_0(C_1\frac{M_{1-\alpha}
 e^{-\frac{r}{q}(t+\theta-s)}}{(t+\theta-s)^{(1-\alpha)}})^qds\Big)^{1/q}
\Big]^p\\
&\quad\times E\Big[\Big(\int^{t+\theta}_0(e^{-\frac{r}{p}(t+\theta-s)}
 \|x_s\|_{PC})^pds\Big)^{\frac{1}{p}}\Big]^p \\
&=C^p_1M^p_{1-\alpha}\Big(\int^{t+\theta}_0\frac{e^{-rs}}
 {s^{q(1-\alpha)}}ds\Big)^{p/q}
\int^{t+\theta}_0e^{-r(t+\theta-s)}E\|x_s\|^p_{PC}ds \\
&=C^p_1M^p_{1-\alpha}\Gamma^{p/q}(q\alpha-q+1)/r^{p\alpha-1}
\int^{t+\theta}_0e^{-r(t+\theta-s)}E\|x_s\|^p_{PC}ds,
\end{align*}
where $p,q>1$, $\frac{1}{q}+\frac{1}{p}=1$.
\begin{equation} \label{h3}
\begin{aligned}
E\|\int^{t+\theta}_{0}S(t+\theta-s)f(s,x_s)ds\|^p_{X}
&\leq \gamma^pE\|\int^{t+\theta}_{0}e^{-r(t+\theta-s)}f(s,x_s)ds\|^p_{X} \\
&=\gamma^pE\|\int^{t+\theta}_{0}e^{-(\frac{1}{q}
 +\frac{1}{p})r(t+\theta-s)}f(s,x_s)ds\|^p_{X} \\
&\leq C^p_2\gamma^p/r^{p-1}\int^{t+\theta}_{0}e^{-r(t+\theta-s)}E\|x_s\|^p_{PC}ds,
\end{aligned}
\end{equation}
and
\begin{equation} \label{h4}
\begin{aligned}
&E\|\int^{t+\theta}_{0}S(t+\theta-s)g(s,x_s)dw(s)\|^p_{X} \\
&=E[\|\int^{t+\theta}_{0}S(t+\theta-s)g(s,x_s)dw(s)\|^2_{X}]^{\frac{p}{2}} \\
&\leq \gamma^pE[\int^{t+\theta}_{0}e^{-r(t+\theta-s)}
 \|g(s,x_s)\|^2_{\lambda}ds]^{\frac{p}{2}} \\
&= \gamma^pE[\int^{t+\theta}_{0}e^{-(1-\frac{2}{p})r(t+\theta-s)}
 e^{-\frac{2}{p}r(t+\theta-s)}\|g(s,x_s)\|^2_{\lambda}ds]^{\frac{p}{2}} \\
&\leq \gamma^pC^p_3/r^{\frac{p}{2}-1}\int^{t+\theta}_{0}
 e^{-r(t+\theta-s)}E\|x_s\|^p_{PC}ds,
\end{aligned}
\end{equation}
which immediately implies
\begin{equation} \label{h5}
\begin{aligned}
E\|x(t+\theta)\|^p_{X}
&\leq 6^{p-1}\gamma^p
e^{-pr(t+\theta)}(\|\phi(0)\|+C_1\|(-A)^{-\alpha}\|\|\phi\|_{PC})^p\\
&\quad +6^{p-1}C^p_1\|(-A)^{-\alpha}\|^pE\|x_{t+\theta}\|^p_{PC} \\
&\quad +[6^{p-1}C^p_1M^p_{1-\alpha}\Gamma^{p/q}(q\alpha-q+1)/r^{p\alpha-1}+6^{p-1}C^p_2\gamma^p/r^{p-1} \\
&\quad +6^{p-1}\gamma^pC^p_3/r^{\frac{p}{2}-1}]
\int^{t+\theta}_{0}e^{-r(t+\theta-s)}E\|x_s\|^p_{PC}ds \\
&\quad +6^{p-1}N^{p-1}\gamma^p
e^{-pr(t+\theta)}\sum_{k=1}^{N}e^{prt_k}d_k;
\end{aligned}
\end{equation}
that is,
\begin{equation} \label{h6}
\begin{aligned}
E\|x(t+\theta)\|^p_{X}
&\leq 6^{p-1}\gamma^p
e^{-pr(t+\theta)}[(\|\phi(0)\|+C_1\|(-A)^{-\alpha}\|\|\phi\|_{PC})^p+d] \\
&\quad +6^{p-1}C^p_1\|(-A)^{-\alpha}\|^pE\|x_{t+\theta}\|^p_{PC} \\
&\quad +[6^{p-1}C^p_1M^p_{1-\alpha}\Gamma^{p/q}(q\alpha-q+1)/r^{p\alpha-1}+6^{p-1}C^p_2\gamma^p/r^{p-1} \\
&\quad +6^{p-1}\gamma^pC^p_3/r^{\frac{p}{2}-1}]
\int^{t+\theta}_{0}e^{-r(t+\theta-s)}E\|x_s\|^p_{PC}ds,
\end{aligned}
\end{equation}
where $d=N^{p-1}\sum_{k=1}^{N}e^{prt_k}d_k$.

Therefore, for arbitrary $0<\epsilon<r$ and $T>\tau$ large enough, we
have
\begin{equation} \label{h7}
\begin{aligned}
&\int^{T}_{\tau}e^{\epsilon
t}E\|x(t+\theta)\|^p_{X}dt\\
&\leq 6^{p-1}\gamma^p [(\|\phi(0)\|+C_1\|(-A)^{-\alpha}\|\|\phi\|_{PC})^p+d]
 \int^{T}_{\tau}e^{-pr(t+\theta)+\epsilon t}dt \\
&\quad +6^{p-1}C^p_1\|(-A)^{-\alpha}\|^p\int^{T}_{\tau}e^{\epsilon
t}E\|x_{t+\theta}\|^p_{PC}dt \\
&\quad +[6^{p-1}C^p_1M^p_{1-\alpha}\Gamma^{p/q}(q\alpha-q+1)/r^{p\alpha-1}+6^{p-1}C^p_2\gamma^p/r^{p-1} \\
&\quad +6^{p-1}\gamma^pC^p_3/r^{\frac{p}{2}-1}]
\int^{T}_{\tau}\int^{t+\theta}_{0}e^{-r(t+\theta-s)+\epsilon
t}E\|x_s\|^p_{PC}\,ds\,dt.
\end{aligned}
\end{equation}
On the other hand, note that for any $-\tau\leq\theta\leq 0$ and
$t\geq\tau$,
\begin{equation} \label{h8}
\begin{aligned}
&\int^{T}_{\tau}\int^{t+\theta}_{0}e^{-r(t+\theta-s)+\epsilon
t}E\|x_s\|^p_{PC}\,ds\,dt \\
&=\int^{\tau+\theta}_{0}\int^{T}_{\tau}e^{-r(t+\theta-s)+\epsilon
t}E\|x_s\|^p_{PC}\,dt\,ds\\
&\quad +\int^{T+\theta}_{\tau+\theta}\int^{T}_{s-\theta}e^{-r(t+\theta-s)+\epsilon
t}E\|x_s\|^p_{PC}\,dt\,ds \\
&\leq \frac{1}{r-\epsilon}\int^{\tau+\theta}_{0}e^{r(s-\theta)}E\|x_s\|^p_{PC}ds
+\frac{1}{r-\epsilon}\int^{T+\theta}_{\tau+\theta}e^{\epsilon(s-\theta)}E\|x_s\|^p_{PC}ds \\
&\leq \frac{1}{r-\epsilon}\int^{\tau}_{0}e^{r(s-\theta)}E\|x_s\|^p_{PC}ds
+\frac{1}{r-\epsilon}\int^{T}_{0}e^{\epsilon(s-\theta)}E\|x_s\|^p_{PC}ds.
\end{aligned}
\end{equation}
Therefore, substituting \eqref{h8} into \eqref{h7} yields
% \label{h9}
\begin{align*}
&\int^{T}_{\tau}e^{\epsilon
t}E\|x(t+\theta)\|^p_{X}dt\\
&\leq 6^{p-1}\gamma^p
[(\|\phi(0)\|+C_1\|(-A)^{-\alpha}\|\|\phi\|_{PC})^p+d]\int^{T}_{\tau}e^{-pr(t+\theta)+\epsilon t}dt \\
&\quad +6^{p-1}C^p_1\|(-A)^{-\alpha}\|^p\int^{T}_{\tau}e^{\epsilon
t}E\|x_{t+\theta}\|^p_{PC}dt \\
&\quad +[6^{p-1}C^p_1M^p_{1-\alpha}\Gamma^{p/q}(q\alpha-q+1)/r^{p\alpha-1}+6^{p-1}C^p_2\gamma^p/r^{p-1} \\
&\quad +6^{p-1}\gamma^pC^p_3/r^{\frac{p}{2}-1}]\frac{1}{(r-\epsilon)}
\int^{\tau}_{0}e^{r(s-\theta)}E\|x_s\|^p_{PC}ds \\
&\quad +[6^{p-1}C^p_1M^p_{1-\alpha}\Gamma^{p/q}(q\alpha-q+1)/r^{p\alpha-1}+6^{p-1}C^p_2\gamma^p/r^{p-1} \\
&\quad +6^{p-1}\gamma^pC^p_3/r^{\frac{p}{2}-1}]\frac{1}{(r-\epsilon)}\int^{T}_{0}e^{\epsilon(s-\theta)}E\|x_s\|^p_{PC}ds \\
\leq&6^{p-1}\gamma^p
[(\|\phi(0)\|+C_1\|(-A)^{-\alpha}\|\|\phi\|_{PC})^p+d]\int^{T}_{\tau}e^{-pr(t-\tau)+\epsilon t}dt \\
&\quad +6^{p-1}C^p_1\|(-A)^{-\alpha}\|^pe^{\epsilon
\tau}\int^{T}_{\tau}e^{\epsilon
s}E\|x_s\|^p_{PC}ds \\
&\quad +6^{p-1}C^p_1\|(-A)^{-\alpha}\|^pe^{\epsilon
\tau}\int^{r}_{0}e^{\epsilon
s}E\|x_s\|^p_{PC}ds \\
&\quad +[6^{p-1}C^p_1M^p_{1-\alpha}\Gamma^{p/q}(q\alpha-q+1)/r^{p\alpha-1}+6^{p-1}C^p_2\gamma^p/r^{p-1} \\
&\quad +6^{p-1}\gamma^pC^p_3/r^{\frac{p}{2}-1}]\frac{1}{(r-\epsilon)}
e^{r\tau}\int_{0}^{\tau}e^{rs}E\|x_s\|^p_{PC}ds \\
&\quad +[6^{p-1}C^p_1M^p_{1-\alpha}\Gamma^{p/q}(q\alpha-q+1)/r^{p\alpha-1}+6^{p-1}C^p_2\gamma^p/r^{p-1} \\
&\quad +6^{p-1}\gamma^pC^p_3/r^{\frac{p}{2}-1}]\frac{1}{(r-\epsilon)}e^{
\epsilon\tau} \int_{0}^{T}e^{\epsilon s}E\|x_s\|^p_{PC}ds;
\end{align*}
i. e.,
\begin{equation} \label{h11}
\int^{T}_{\tau}e^{\epsilon t}E\|x_t\|^p_{PC}dt\leq
L_1(\epsilon)+L_2(\epsilon)\int^{T}_{0}e^{\epsilon
s}E\|x_s\|^p_{PC}ds,
\end{equation}
where
\begin{equation} \label{h12}
\begin{aligned}
&L_1(\epsilon)\\
&=(1-6^{p-1}C^p_1\|(-A)^{-\alpha}\|^pe^{\epsilon
\tau})^{-1}\Big\{6^{p-1}\gamma^p
[(\|\phi(0)\|+C_1\|(-A)^{-\alpha}\|\|\phi\|_{PC})^p+d]\\
&\quad \times \int^{T}_{\tau}e^{-pr(t-\tau)+\epsilon t}dt
+6^{p-1}C^p_1\|(-A)^{-\alpha}\|^pe^{\epsilon
\tau}\int^{\tau}_{0}e^{\epsilon s}E\|x_s\|^p_{PC}ds
\\
&\quad +[6^{p-1}C^p_1M^p_{1-\alpha}\Gamma^{p/q}(q\alpha-q+1)/r^{p\alpha-1}
 +6^{p-1}C^p_2\gamma^p/r^{p-1} \\
&\quad +6^{p-1}\gamma^pC^p_3/r^{\frac{p}{2}-1}]
 \frac{1}{(r-\epsilon)}e^{r\tau}
\int_{0}^{\tau}e^{rs}E\|x_s\|^p_{PC}ds\Big\},
\end{aligned}
\end{equation}
\begin{equation} \label{h13}
\begin{aligned}
L_2(\epsilon)
&=(1-6^{p-1}C^p_1\|(-A)^{-\alpha}\|^pe^{\epsilon
\tau})^{-1}\Big\{[6^{p-1}C^p_1M^p_{1-\alpha}\Gamma^{p/q}
(q\alpha-q+1)/r^{p\alpha-1}\\
&\quad +6^{p-1}C^p_2\gamma^p/r^{p-1}
 +6^{p-1}\gamma^pC^p_3/r^{\frac{p}{2}-1}]\frac{1}{(r-\epsilon)}\Big\}
e^{\epsilon r}.
\end{aligned}
\end{equation}
Hence,  from \eqref{h11} we have
\begin{equation} \label{h14}
\int^{T}_{0}e^{\epsilon t}E\|x_t\|^p_{PC}dt\leq
L_1(\epsilon)+\int^{r}_{0}e^{\epsilon
t}E\|x_t\|^p_{PC}dt+L_2(\epsilon)\int^{T}_{0}e^{\epsilon
t}E\|x_t\|^p_{X}dt.
\end{equation}
On the other hand, by virtue of \eqref{newe16}, it is possible to
choose a suitable $0<\epsilon<r$ small enough such that
$L_2(\epsilon)<1$, for such an $\epsilon>0$, we may deduce that
there exists a real number $L_3(\epsilon)>0$ such that
\begin{equation} \label{h15}
\int^{T}_{0}e^{\epsilon t}E\|x_t\|^p_{PC}dt\leq
(1-L_2(\epsilon))^{-1}(L_1(\epsilon)+\int^{r}_{0}e^{\epsilon
t}E\|x_t\|^p_{PC}dt):=L_3(\epsilon)<\infty.
\end{equation}
From \eqref{h6},  for any
$-\tau\leq\theta\leq 0$, $t\geq\tau$, we have
\begin{equation} \label{h16}
\begin{aligned}
&E\|x(t+\theta)\|^p_{X}\\
&\leq 6^{p-1}\gamma^p
e^{-pr(t+\theta)}[(\|\phi(0)\|+C_1\|(-A)^{-\alpha}\|\|\phi\|_{PC})^p+d]e^{-\epsilon(t-\tau)} \\
&\quad +6^{p-1}C^p_1\|(-A)^{-\alpha}\|^pE\|x_{t+\theta}\|^p_{PC} \\
&\quad +[6^{p-1}C^p_1M^p_{1-\alpha}\Gamma^{p/q}(q\alpha-q+1)/r^{p\alpha-1}+6^{p-1}C^p_2\gamma^p/r^{p-1} \\
&\quad +6^{p-1}\gamma^pC^p_3/r^{\frac{p}{2}-1}]L_3(\epsilon)e^{-\epsilon(t-\tau)}.
\end{aligned}
\end{equation}
Now, we proceed with our arguments by considering two possible
situations for any fixed $t\geq 0$.

Firstly, suppose that
$\sup_{-\tau\leq\theta\leq0}E\|x_{t+\theta}\|^p_{PC}=E\|x_t\|^p_{PC}$,
then from \eqref{h16} we have 
\begin{equation} \label{h17}
\begin{aligned}
E\|x_t\|^p_{X}
&\leq 6^{p-1}\gamma^p e^{-pr(t+\theta)}[(\|\phi(0)\|
 +C_1\|(-A)^{-\alpha}\|\|\phi\|_{PC})^p+d]e^{-\epsilon(t-\tau)} \\
&\quad +6^{p-1}C^p_1\|(-A)^{-\alpha}\|^pE\|x_t\|^p_{PC} \\
&\quad +[6^{p-1}C^p_1M^p_{1-\alpha}\Gamma^{p/q}(q\alpha-q+1)/r^{p\alpha-1}
 +6^{p-1}C^p_2\gamma^p/r^{p-1} \\
&\quad +6^{p-1}\gamma^pC^p_3/r^{\frac{p}{2}-1}]L_3(\epsilon)
 e^{-\epsilon(t-\tau)},
\end{aligned}
\end{equation}
which immediately yields
\begin{equation} \label{h18}
\begin{aligned}
E\|x_t\|^p_{PC}
&\leq(1-6^{p-1}C^p_1\|(-A)^{-\alpha}\|^p)^{-1}[6^{p-1}\gamma^p
e^{-pr(t+\theta)} \\
&\quad\times [(\|\phi(0)\|+C_1\|(-A)^{-\alpha}\|\|\phi\|_{PC})^p+d]e^{\epsilon\tau} \\
&\quad +[6^{p-1}C^p_1M^p_{1-\alpha}\Gamma^{p/q}(q\alpha-q+1)/r^{p\alpha-1}+6^{p-1}C^p_2\gamma^p/r^{p-1} \\
&\quad +6^{p-1}\gamma^pC^p_3/r^{\frac{p}{2}-1}]L_3(\epsilon)e^{\epsilon\tau}]e^{-\epsilon
t}:=L_4(\epsilon)e^{-\epsilon t}.
\end{aligned}
\end{equation}
On the other hand, for this fixed $t\geq 0$, if
$\sup_{-\tau\leq\theta\leq0}E\|x_{t+\theta}\|^p_{PC}=E\|x_{t-\tau}\|^p_{PC}$,
then  from \eqref{h16} we have
\begin{equation} \label{h19}
\begin{aligned}
E\|x_t\|^p_{PC}
&\leq6^{p-1}\gamma^p
e^{-pr(t+\theta)}[(\|\phi(0)\|+C_1\|(-A)^{-\alpha}\|\|\phi\|_{PC})^p+d]
 e^{-\epsilon(t-\tau)} \\
&\quad +6^{p-1}C^p_1\|(-A)^{-\alpha}\|^pE\|x_{t-r}\|^p_{PC} \\
&\quad +[6^{p-1}C^p_1M^p_{1-\alpha}\Gamma^{p/q}(q\alpha-q+1)/r^{p\alpha-1}
 +6^{p-1}C^p_2\gamma^p/r^{p-1} \\
&\quad +6^{p-1}\gamma^pC^p_3/r^{\frac{p}{2}-1}]L_3(\epsilon)e^{-\epsilon(t-\tau)}.
\end{aligned}
\end{equation}
Hence, in this situation we have
\begin{equation} \label{h20}
E\|x_t\|^p_{PC}\leq L_5(\epsilon)e^{-\epsilon
t}+L_6(\epsilon)E\|x_{t-\tau}\|^p_{PC},
\end{equation}
where
\begin{equation} \label{h21}
\begin{aligned}
 L_5(\epsilon)
&=6^{p-1}\gamma^p
e^{-pr(t+\theta)}[(\|\phi(0)\|+C_1\|(-A)^{-\alpha}\|\|\phi\|_{PC})^p+d]e^{\epsilon\tau} \\
&\quad +[6^{p-1}C^p_1M^p_{1-\alpha}\Gamma^{p/q}(q\alpha-q+1)/r^{p\alpha-1}+6^{p-1}C^p_2\gamma^p/r^{p-1} \\
&\quad +6^{p-1}\gamma^pC^p_3/r^{\frac{p}{2}-1}]L_3(\epsilon)e^{\epsilon\tau},
\end{aligned}
\end{equation}
and
\begin{equation} \label{h22}
L_6(\epsilon)=6^{p-1}C^p_1\|(-A)^{-\alpha}\|^p.
\end{equation}
Combining \eqref{h18} with \eqref{h20}, for any $t\geq\tau$,  we have
\begin{equation} \label{h23}
E\|x_t\|^p_{PC}\leq\max\{L_4(\epsilon)e^{-\epsilon t},
L_5(\epsilon)e^{-\epsilon t}+L_6(\epsilon)E\|x_{t-\tau}\|^p_{PC}\}.
\end{equation}
In other words,  for any $t\geq2\tau$, we have
\begin{equation} \label{h24}
E\|x_{t-\tau}\|^p_{PC}\leq\max\{L_4(\epsilon)e^{-\epsilon (t-\tau)},
L_5(\epsilon)e^{-\epsilon
(t-\tau)}+L_6(\epsilon)E\|x_{t-2\tau}\|^p_{PC}\}.
\end{equation}
In view of \eqref{h23} and \eqref{h24}, we obtain that for any
$t\geq 2\tau$,
\begin{equation}  \label{h25}
\begin{aligned}
&E\|x_t\|^p_{PC}\\
&\leq \max\Big\{L_4(\epsilon)e^{-\epsilon t},
[L_5(\epsilon)+L_6(\epsilon)L_4(\epsilon)e^{-\epsilon
\tau}]e^{-\epsilon t}, \\
&\quad\sum_{i=1}^{2}(L_5(\epsilon)L_6^{i-1}(\epsilon)e^{\epsilon(i-1)\tau})e^{-\epsilon
t}+L_6^2(\epsilon)E\|x_{t-2\tau}\|^p_{PC}\Big\} \\
&\quad:=\max\{\hat{L}_2(\epsilon)e^{-\epsilon
t},\sum_{i=1}^{2}(L_5(\epsilon)L_6^{i-1}(\epsilon)e^{\epsilon(i-1)\tau})e^{-\epsilon
t}+L_6^2(\epsilon)E\|x_{t-2\tau}\|^p_{PC}\},
\end{aligned}
\end{equation}
where
\[
\hat{L}_2(\epsilon)=\max\{L_4(\epsilon),
L_5(\epsilon)+L_6(\epsilon)L_4(\epsilon)e^{-\epsilon\tau}\}.
\]
By induction, there exists a positive number $\hat{L}_m(\epsilon)$
such that for any $t\geq\tau$,
\begin{equation} \label{h26}
\begin{aligned}
&E\|x_t\|^p_{PC}\\
&\leq\max\big\{\hat{L}_m(\epsilon)e^{-\epsilon
t},\sum_{i=1}^{m}(L_5(\epsilon)L_6^{i-1}(\epsilon)
 e^{\epsilon(i-1)\tau})e^{-\epsilon t}
 +L_6^m(\epsilon)E\|x_{t-m\tau}\|^p_{PC}\big\},
\end{aligned}
\end{equation}
where $m$ is the positive integer such that $0\leq t-m\tau<\tau$.
Obviously, to obtain the desired exponential
stability of \eqref{i1}, we  need to consider only the term
\begin{equation} \label{h27}
E\|x_t\|^p_{PC}\leq\sum_{i=1}^{m}(L_5(\epsilon)L_6^{i-1}(\epsilon)e^{\epsilon(i-1)\tau})e^{-\epsilon
t}+L_6^m(\epsilon)E\|x_{t-m\tau}\|^p_{PC}.
\end{equation}
In fact, choose a suitable $\epsilon>0$ small enough such that
$L_6(\epsilon)e^{\epsilon\tau}<1$, then we may deduce from
\eqref{h27} that
\begin{equation} \label{h28}
\begin{aligned}
&E\|x_t\|^p_{PC}\\
&\leq\Big(L_5(\epsilon)\sum_{i=1}^{m}
(L_6^{i-1}(\epsilon)e^{\epsilon(i-1)\tau})\Big)e^{-\epsilon
t}+L_6^m(\epsilon)(\|\phi\|^p_{X}+E\|x_{\tau}\|^p_{PC}) \\
&\leq \Big(L_5(\epsilon)\sum_{i=1}^{m}
(L_6^{i-1}(\epsilon)e^{\epsilon(i-1)\tau})\Big)e^{-\epsilon
t}+L_6^m(\epsilon)(L_5(\epsilon)e^{-\epsilon\tau}+(1+L_6(\epsilon))\|\phi\|^p_{PC}) \\
&\leq \Big(L_5(\epsilon)\sum_{i=1}^{m}
(L_6^{i-1}(\epsilon)e^{\epsilon(i-1)\tau})\Big)e^{-\epsilon
t}+(L_5(\epsilon)e^{-\epsilon\tau}+(1+L_6(\epsilon))\|\phi\|^p_{PC})(L_6(\epsilon))^{t/r} \\
&\leq \Big(L_5(\epsilon)\sum_{i=1}^{\infty}
(L_6^{i-1}(\epsilon)e^{\epsilon(i-1)\tau})\Big)e^{-\epsilon
t}+(L_5(\epsilon)e^{-\epsilon\tau}+(1+L_6(\epsilon))\|\phi\|^p_{PC})(L_6(\epsilon))^{t/r} \\
&=\frac{L_5(\epsilon)}{1-L_6(\epsilon)e^{\epsilon\tau}}e^{-\epsilon
t}+(L_5(\epsilon)e^{-\epsilon\tau}+(1+L_6(\epsilon))\|\phi\|^p_{PC})(L_6(\epsilon))^{t/r} \\
&=\frac{L_5(\epsilon)}{1-L_6(\epsilon)e^{\epsilon\tau}}e^{-\epsilon
t}+(L_5(\epsilon)e^{-\epsilon\tau}+(1+L_6(\epsilon))\|\phi\|^p_{PC})e^{-\epsilon_1t},
\end{aligned}
\end{equation}
where $0<\epsilon_1=-\ln L_6(\epsilon)/\tau$.

Hence, from \eqref{h26} and \eqref{h28}, ones may deduce that there
exist numbers $C(\phi)>0$ and $\epsilon>0$ such that
\[
E\|x(t;\phi)\|^p_{X}\leq E\|x_t\|^p_{PC}\leq C(\phi)e^{-\epsilon t},
\quad \text{for any } t\geq\tau.
\]
The proof is complete.
\end{proof}

\section{An illustrative example}

In this section, we illustrate the obtained result. Let
$X=L^2([0,\pi])$ and $A$ be defined as $Af=f''$ with domain
$D(A)=\{f(\cdot)\in L^2([0,\pi]): f''\in L^2([0,\pi]),\;f(0)=f(\pi)=0\}$.

It is well known that the analytic semigroup $S(t)(t\geq0)$,
generated by the operator $A$ on $X$ which is a separable Hilbert
spaces. Furthermore, $A$ has a discrete spectrum with eigenvalues of
the form $-n^2, n\in N$, and corresponding normalized eigenfunctions
given by $e_n(\xi):=(2/\pi)^{1/2}\sin(n\xi)$.

In addition to the above, the following conditions hold:
\begin{itemize}
 \item[(a)] $\{e_n:n\in N\}$ is an orthonormal basis of $X$.

\item[(b)] If $f\in X$, then $S(t)f=\sum_{n=1}^{\infty}e^{-n^2t}
\langle f,e_n\rangle e_n$ and
$Af=-\sum_{n=1}^{\infty}n^2\langle f,e_n\rangle e_n$ for every 
$f\in D(A)$.

\item[(c)] For $f\in X$,
 $(-A)^{-\frac{1}{2}}f=\sum_{n=1}^{\infty}\frac{1}{n}\langle f,e_n\rangle e_n$.

\item[(d)] The operator $(-A)^{1/2}:
 D((-A)^{1/2})\subseteqq X\to X$ is given by
\[
(-A)^{1/2}f=\sum_{n=1}^{\infty}n\langle f,e_n\rangle
e_n,\quad \forall f\in D((-A)^{1/2}),
\]
where $D((-A)^{1/2})=\{f(\cdot)\in
X:\;\sum_{n=1}^{\infty}n\langle f,e_n\rangle e_n\in X\}$.
\end{itemize}
Consider the  stochastic neutral partial functional
differential equation
\begin{equation} \label{e2}
\begin{gathered}
\begin{aligned}
&\frac{\partial}{\partial t}
\Big[x(t,\xi)+\int_{-\tau}^{0}\int_{0}^{\pi}b(s,\eta,\xi)x(t+s,\eta)d\eta
ds\Big]\\
&=\frac{\partial^2}{\partial\xi^2}x(t,\xi)+p_0(\xi)x(t,\xi)
+\int_{-\tau}^{0}p(s)x(t+s,\xi)ds
+p_1\cos x(t-\tau,\xi)dB_t,\\
&\quad t\neq t_k,\;\xi\in I=[0,\pi],
\end{aligned}\\
\Delta x(t_k)=x(t^+_k)-x(t^-_k)=I_k(x(t^-_k)),\quad t=t_k,\; k=1,2,\dots,N,
\\
x(t,0)=x(t,\pi)=0, \quad t\in R,\\
x(s,\xi)=\phi(s,\xi),\quad
 \phi(\cdot,\xi)\in PC_{\mathscr F_{0}}^b[[-\tau,0], R],\quad
\phi(s,\cdot)\in L^2([0,\pi]),
\end{gathered}
\end{equation}
where $B_t$ is the standard one-dimensional Wiener process, the
functions $p_0, p$ are continuous, $p_1\geq0$. And
$I_k(x(t_k))=e^{-k}x(t_k)$. We also assume that
\begin{itemize}
 \item[(i)] The function $b(\cdot)$ is (Lebesgue) measurable and
 \begin{align*}
\int_{0}^{\pi}\int_{-\tau}^{0}\int_{0}^{\pi}b^2(\theta,\eta,\xi)\,d\eta
d\theta\, d\xi<\infty.
\end{align*}

\item[(ii)] The function $\frac{\partial^i}{\partial\xi^i}b(\theta,\eta,\xi)$,
 $i=1,2$, are measurable, $b(\theta,\eta,0)=(\theta,\eta,\pi)=0$ for
 every $(\theta,\eta)$ and
\[
L_1:=\max\Big\{\int_{0}^{\pi}\int_{-\tau}^{0}\int_{0}^{\pi}
(\frac{\partial^i}{\partial\xi^i}b(\theta,\eta,\xi))^2d\eta d\theta
d\xi: i=0,1,2\Big\}<\infty.
\]
\end{itemize}
Define $G,f:C([-\tau,0];X)$ by setting
\begin{gather*}
G(t,x)(\xi):=B(x)(\xi):=\int_{-\tau}^{0}\int_{0}^{\pi}b(s,\eta,\xi)x(s,\eta)\,
d\eta\, ds, \\
f(t,x)(\xi):=p_0(\xi)x(t,\xi) +\int_{-\tau}^{0}p(s)x(s,\xi)ds.
\end{gather*}
From (i), it is clear that $B$ is a bounded linear operator on
$X$. Furthermore, from the definition of $B$ and (ii), we obtain
\begin{align}
\langle B(x),e_n\rangle
&=\int_{0}^{\pi}\Big[\int_{-\tau}^{0}\int_{0}^{\pi}b(s,\eta,\xi)x(s,\eta)d\eta
ds\Big](\frac{2}{\pi})^{1/2}\sin(n\xi)d\xi \\
&=\frac{1}{n}(\frac{2}{\pi})^{1/2}\langle\int_{-\tau}^{0}\int_{0}^{\pi}\frac{\partial}{\partial\xi}b(s,\eta,\xi)x(s,\eta)d\eta
ds,\cos(n\xi)\rangle \\
&=\frac{1}{n}(\frac{2}{\pi})^{1/2}\langle B_1(x),\cos(n\xi)\rangle,
\end{align}
where
$B_1(x)=\int_{-\tau}^{0}\int_{0}^{\pi}\frac{\partial}{\partial\xi}
b(s,\eta,\xi)x(s,\eta)d\eta\, ds$.
 By using (ii) again, we obtain that $B_1: X\to X$ is
a bounded linear operator with $\|B_1\|\leq L_1$, so
$\|(-A)^{1/2}B(x)\|=\|B_1(x)\|$. Hence $B(x)\in
D[(-A)^{1/2}]$, and $\|(-A)^{1/2}B\|\leq L_1$.
Similarly, we can prove that $f$ is a bounded linear operator on
$X$,
\[
\|f(t,\cdot)\|\leq\sup_{t\in\mathbb{R}}\|p_0(t)\|
+\sqrt{\tau(\int_{-\tau}^{0}p^2(s)ds)},\quad t\in\mathbb{R}.
\]
 By simple parameters computation, we can easily
verify that all conditions of Theorems \ref{thm1} and \ref{thm2} are satisfied.
Therefore, \eqref{e2} has a unique mild solution, which is exponential
$p$-stable.

\subsection*{Acknowledgements}
The authors thank Professor Daoyi Xu (Sichuan University) for his
useful suggestions.
This work was supported by National Natural Science Foundation of China
under Grants 11101367, 11271270 and 61074039.


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\end{document}