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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2013 (2013), No. 206, pp. 1--29.\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-2013/206\hfil Asymptotic stability]
{Asymptotic stability of fractional  impulsive  neutral
 stochastic partial integro-differential equations with state-dependent delay}

\author[Z. Yan, H. Zhang \hfil EJDE-2013/206\hfilneg]
{Zuomao Yan, Hongwu Zhang}  % in alphabetical order

\address{Zuomao Yan \newline
Department of Mathematics, Hexi University, 
Zhangye, Gansu 734000, China}
\email{yanzuomao@163.com}

\address{Hongwu Zhang \newline
Department of Mathematics, Hexi University, 
Zhangye, Gansu 734000, China}
\email{zh-hongwu@163.com}

\thanks{Submitted  April 28, 2013. Published September 18, 2013.}
\subjclass[2000]{34A37, 60H15, 35R60, 93E15, 26A33}
\keywords{Asymptotic stability;
impulsive  neutral integro-differential equations;
\hfill\break\indent
stochastic  integro-differential equations; $\alpha$-resolvent operator}

\begin{abstract}
 In this article, we study the asymptotical stability in
 $p$-th moment of mild solutions to  a class of fractional  impulsive
 partial neutral stochastic  integro-differential equations
 with state-dependent delay in Hilbert spaces.
 We assume that the linear part of this equation generates an
 $\alpha$-resolvent operator and transform it into an integral equation.
 Sufficient conditions for the existence and asymptotic stability of
 solutions are derived by means of the Krasnoselskii-Schaefer type
 fixed point theorem and properties of the $\alpha$-resolvent operator.
 An illustrative example is also provided.
\end{abstract}

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

\section{Introduction}

 Partial stochastic differential equations
 have attracted the considerable attention of researchers
and many qualitative theories for the solutions of this kind have
been derived; see \cite{c3,d1} and the references therein.
In particular, the stability theory of stochastic differential
equations has been popularly applied in variety fields of science
and technology.
Several authors have established the stability results of mild
solutions for these equations by using various techniques; see, for
example,   Govindan \cite{g1} considered the existence and stability for
mild solution of stochastic partial differential equations   by
applying the comparison theorem. Caraballo and Liu \cite{c1} proved the
exponential stability for mild solution to stochastic partial
differential equations with delays by utilizing the well-known
Gronwall inequality. The exponential stability of the mild solutions
to the semilinear stochastic delay evolution equations have been
discussed by using Lyapunov functionals in Liu \cite{l1}.
The author \cite{l2}
considered the exponential stability for stochastic partial
functional differential equations by means of the Razuminkhin-type
theorem. Liu and Truman \cite{l3} investigated the almost sure
exponential stability of mild solution for stochastic partial
functional differential equation by using the analytic technique.
Taniguchi \cite{t1} discussed the exponential stability for stochastic
delay differential equations by the energy inequality. Using fixed
point approach,  Luo \cite{l5} studied the asymptotic stability of mild
solutions of stochastic partial differential equations with finite
delays. Further, Sakthivel et al. \cite{r1,s3,s4} established the
asymptotic stability and  exponential stability of second-order
stochastic evolution  equations in Hilbert spaces.


Impulsive differential and integro-differential systems are
occurring in the field of physics where it has been very intensive
research topic since the theory provides a natural framework for
mathematical modeling of many physical phenomena \cite{b2}. Moreover,
various mathematical models in the study of population dynamics,
biology, ecology and epidemic can be expressed as impulsive
stochastic differential  equations. In recent years, the qualitative
dynamics such as the existence and uniqueness, stability for
first-order impulsive partial  stochastic differential equations
 have been extensively studied by many authors; for instance,
 Sakthivel and Luo \cite{s1,s2} studied the
existence and asymptotic stability in $p$-th moment of mild
solutions to impulsive stochastic partial differential equations
through fixed point theory. Anguraj and Vinodkumar \cite{a1} investigated
the existence, uniqueness and stability of mild solutions of
impulsive stochastic semilinear neutral functional differential
equations without a Lipschitz condition and with a Lipschitz
condition. Chen \cite{c2}, Long et al. \cite{l4} discussed the exponential
$p$-stability  of impulsive stochastic  partial functional
differential equations. He and  Xu \cite{h1} studied 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.


On the other hand, fractional differential  equations play an
important role in describing some real world problems. This is
caused both by the intensive development of the theory of fractional
calculus itself and by applications of such constructions in various
domains of science, such as physics, mechanics, chemistry,
engineering, etc. For details, see  \cite{p1} and references therein. The
existence of solutions of fractional semilinear differential and
integrodifferential equations are one of the theoretical fields that
investigated by many authors \cite{e1,s8,y1,y2}.
Recently, much attention has been paid to the differential
  systems involving the  fractional
derivative and impulses. This is due to the fact that most problems
in a real life situation to which mathematical models are applicable
are basically fractional order differential equations rather than
integer order differential equations.  Consequently, there are many
contributions relative to  the solutions of various impulsive
semilinear fractional differential and integrodifferential systems
 in Banach spaces; see \cite{b1,d2,s9}.
The qualitative properties of
fractional stochastic differential equations have been considered
only in few publications \cite{c3,e2,s5,s6,y3}. More recently,
Sakthivel et al. \cite{s7} studied the existence and asymptotic stability
in $p$-th moment of a mild solution to a class of nonlinear
fractional neutral stochastic differential equations with infinite
delays in Hilbert spaces. However, up to now the existence and
asymptotic stability of mild solutions for fractional impulsive
neutral partial stochastic integro-differential equations with
state-dependent delay
  have not been considered in the
literature. In order to fill this gap, this paper studies the
existence and asymptotic stability of the following nonlinear
impulsive fractional stochastic integro-differential equation of the
form
\begin{gather}
\begin{aligned} ^{c}D^{\alpha}N(x_t)
&= A N(x_t)+\int_0^tR(t-s)N(x_{s})ds +h(t,x(t-\rho_2(t)))dt\\
&\quad + f(t,x(t-\rho_{3}(t)))\frac{dw(t)}{dt},\quad
  t\geq0, t\neq t_k,
\end{aligned} \label{e1.1}\\
x_{0}(\cdot)=\varphi\in\mathfrak{B}_{\mathcal{F}_{0}}([\tilde{m}(0),
0],H), \quad  x'(0)=0, \label{e1.2}\\
\Delta x(t_k)=I_k(x(t^{-}_k)), \quad  t =t_k,\; k=1,\ldots, m,
\label{e1.3}
\end{gather}
where the state $x(\cdot)$ takes values in a separable real Hilbert
space $H$ with inner product $\langle\cdot,\cdot\rangle_H$ and
norm $\|\cdot\|_H$, $ ^{c}D^{\alpha}$ is the
Caputo fractional derivative of order $\alpha\in(1,2)$,  $A$,
$(R(t))_{t\geq0}$ are closed linear operators defined on a common
domain which is dense in $(H,\|\cdot\|_H)$, and
$D^{\alpha}_t \sigma(t)$ represents the Caputo derivative of order
$\alpha> 0$ defined by
\[
D^{\alpha}_t \sigma(t)=
\int_0^t\eta_{n-\alpha}(t-s)\frac{d^{n}}{ds^{n}}\sigma(s)ds,
\]
where $n $ is the smallest integer greater than or equal
 to $\alpha$ and
$\eta_{\beta} (t) := t^{\beta-1}/\Gamma(\beta)$,
 $t > 0$, $\beta \geq 0$.
Let $K$ be another separable Hilbert space with inner product
$\langle\cdot,\cdot\rangle_K$  and norm $\|\cdot\|_K$.
Suppose $\{w(t):t\geq0\}$ is a given $K$-valued
Wiener process with a covariance operator $Q > 0$ defined on a
complete probability space $(\Omega,\mathcal{F},P)$ equipped with a
normal filtration $\{\mathcal{F}_t\}_{t\geq0}$, which is generated
by the Wiener process $w$;  and
   $  N(x_t) =x(0) + g(t,x(t-\rho_1(t))),x \in H$, and
$g,h : [0,\infty) \times H\to H, f:[0,\infty)\times H\to L(K, H)$, are all
Borel measurable; $ I_k:H\to H (k=1,\ldots,m)$, are given
functions. Moreover, the fixed moments of time $t_k$ satisfies
$ 0<t_1 < \dots< t_{m}< \lim_{k\to\infty} t_k =\infty$,
 $x(t_k^{+})$ and $x(t_k^{-})$ represent the right and left limits of
$x(t)$ at $t =t_k$, respectively; $\Delta x(t_k)=x(t_k^{+})-x(t_k^{-})$,
represents the jump in the state $x$ at time $t_k$ with $I_k$ determining the
size of the jump; let $\rho_{i}(t)\in
C(\mathbb{R}^{+},\mathbb{R}^{+})(i=1,2,3)$ satisfy
$t -\rho_{i}(t) \to\infty$ as
$t\to\infty$, and $ \tilde{m}(0) =\max\{\inf_{s\geq0}(s - \rho_{i}(s)), i=1,2,3\}$. Here $
\mathfrak{B}_{\mathcal{F}_{0}}([\tilde{m}(0), 0],H)$ denote the
family of all almost surely bounded, $\mathcal{F}_{0}$- measurable,
continuous random variables $\varphi(t) : [\tilde{m}(0), 0]
\to H$ with norm $\| \varphi\|_{\mathfrak{B}}
= \sup_{\tilde{m}(0)\leq t\leq 0} E\|\varphi(t)\|_H$.


To the best of our knowledge, most of the previous research on the
existence and stability investigation for impulsive stochastic
systems was based upon a Lipschitz condition. This condition turns
out to be restrictive.  In this paper, we establish sufficient
conditions for the existence and asymptotic stability in $p$-th
moment of mild for  problem \eqref{e1.1}--\eqref{e1.3}
by using  Krasnoselskii-Schaefer
type fixed point theorem \cite{b3} with the $\alpha$-resolvent operator.
The obtained result can be seen as a contribution to this emerging
field.

This article is organized as follows. In Section 2, we give
 some preliminaries. Section 3 aims to prove the main results. An
example is presented in the last section.


\section{Preliminaries}

Let $K$ and $H$ be two real separable Hilbert spaces with inner
product $\langle\cdot,\cdot\rangle_K$ and
$\langle\cdot,\cdot\rangle_H$, their inner products and by
$\|\cdot\|_K,\|\cdot\|_H$ their
vector norms, respectively.

Let $(\Omega,\mathcal{F},P;\mathbb{F}
)(\mathbb{F}=\{\mathcal{F}\}_{t\geq0})$
  be a complete probability space  satisfying that $\mathcal{F}_{0}$
contains all $P$-null sets.
 Let $\{e_{i}\}_{i=1}^{\infty}$ be a complete
orthonormal basis of $K$. Suppose that $\{w(t):t\geq0\}$ is a
cylindrical $K$-valued Brownian motion with a trace class operator
$Q$, denote Tr$(Q)=\sum_{i=1}^{\infty}\lambda_{i}=\lambda< \infty$,
which satisfies that $Qe_{i} =\lambda_{i}e_{i}$. So, actually,
$w(t)=\sum_{i=1}^{\infty}\sqrt{\lambda_{i}}w_{i}(t)e_{i}$, where
$\{w_{i}(t)\}_{i=1}^{\infty}$ are mutually independent
one-dimensional standard Brownian motions. Then, the above
$K$-valued stochastic process $w(t)$ is called a $Q$-Wiener process.
Let $L(K,H)$ denote the space of all bounded linear operators from
$K$ into $H$ equipped with the usual operator norm $\|
\cdot\|_{L(K,H)}$ and we abbreviate this notation to $L(H)$
when $H= K$. For $\varsigma\in L(K,H)$ we define
\[
\| \varsigma\|_{L_2^0}^2=\operatorname{Tr}(\varsigma Q\varsigma^{*})
=\sum_{n=1}^{\infty}\| \sqrt{\lambda_n}\varsigma
e_n\|^2.
\]
If $\|\varsigma\|^2 _{L_2^0} < \infty$, then
$\varsigma$ is called a $Q$-Hilbert-Schmidt operator, and let
$L_2^0 (K,H)$ denote the space of all $Q$-Hilbert-Schmidt
operators $\varsigma : K\to H$. For a basic reference, the
reader is referred to \cite{d1}.

Let $\mathbb{Y}$ be the space of all $\mathcal{F}_{0}$-adapted
process
$\psi(t,\tilde{w}):[\tilde{m}(0),\infty)\times\Omega\to\mathbb{R}$
which is almost certainly continuous in $t$ for fixed
$\tilde{w}\in\Omega$. Moreover $\psi(s,\tilde{w})=\varphi(s)$ for $s
\in [\tilde{m}(0),0]$ and
$E\|\psi(t,\tilde{w})\|^{p}_H\to0$ as
$t\to\infty$. Also $\mathbb{Y}$ is a Banach space when it is
equipped with a norm defined by
$$\|\psi\|_{\mathbb{Y}}=\sup_{t\geq0}E\|\psi(t)\|^{p}_H.$$

 Now, we give knowledge on the $\alpha$-resolvent operator which
appeared in \cite{s8}.

\begin{definition} \label{def2.1}\rm
 A one-parameter family of bounded
linear operators $(\mathcal{R}_{\alpha}(t))_{t\geq 0}$ on $H$ is
called an $\alpha$-resolvent operator for
\begin{gather} \label{e2.1}
^{c}D^{\alpha}x(t)=A x(t)+\int_0^tR(t-s)x(s)ds,\\
\label{e2.2}
x(0)=x_{0}\in H, \quad  x'(0)=0,
\end{gather}
 if the following conditions are satisfied
\begin{itemize}
\item[(a)] The function $\mathcal{R}_{\alpha}(\cdot) : [0,\infty) \to
L(H)$ is strongly continuous and $\mathcal{R}_{\alpha}(0)x = x$ for
all $x \in H$ and $\alpha \in (1, 2)$;

\item[(b)] For $x \in D(A)$, we have
$\mathcal{R}_{\alpha}(\cdot)x \in C([0,\infty), [D(A)])
\cap C^1((0,\infty), H)$,
\begin{gather*}
D^{\alpha}_t \mathcal{R}_{\alpha}(t)x = A\mathcal{R}_{\alpha}(t)x + \int_0^tR(t -
s)\mathcal{R}_{\alpha}(s)x\,ds,
\\
D^{\alpha}_t \mathcal{R}_{\alpha}(t)x = \mathcal{R}_{\alpha}(t)Ax +
\int_0^t\mathcal{R}_{\alpha}(t - s)R(s)x\,ds
\end{gather*}
for every $t \geq 0$.
 \end{itemize}
\end{definition}

 In this work we use the following assumptions:
\begin{itemize}
\item[(P1)] The operator $A : D(A) \subseteq H\to H$
 is a closed linear operator with $[D(A)]$ dense in $H$.
  Let $\alpha\in(1, 2)$. For some $\phi_{0}\in (0,\frac{\pi}{2}]$, for each
  $\phi < \phi_{0}$  there is a positive constant $C_{0}= C_{0}(\phi)$
   such that $\lambda \in \rho(A)$ for each
$$
\lambda\in \Sigma_{0,\alpha\vartheta}=\{\lambda\in \mathbb{C},\lambda\neq0,
|\arg(\lambda)|<\alpha\vartheta\},
$$
 where
$\vartheta=\phi+\frac{\pi}{2}$ and $\|
R(\lambda,A)\|\leq\frac{C_{0}}{|\lambda|}$ for all
$\lambda\in \Sigma_{0,\alpha\vartheta}$.

\item[(P2)] For all $t \geq0, R(t) : D(R(t)) \subseteq H\to H$
is a closed linear operator, $D(A) \subseteq D(R(t))$ and
$R(\cdot)x$ is strongly measurable on $(0,\infty)$ for each $x \in
D(A)$. There exists $b(\cdot) \in L^1_{loc}(\mathbb{R}^{+})$ such
that $\widehat{b}(\lambda)$ exists for $Re(\lambda)
> 0$ and $\| R(t)x\|_H \leq b(t)\| x\|_1$
 for all $t > 0$ and $x \in D(A)$.
Moreover, the operator valued function $\widehat{R} :
\Sigma_{0,\pi/2}\to L([D(A)], H)$ has an analytical
extension (still denoted by $\widehat{R}$) to $\Sigma_{0,\vartheta}$
such that $\| \widehat{R}(\lambda)x\|_H \leq
\| \widehat{R}(\lambda)\|_H \| x\|_1$
for all $x \in D(A)$, and $\|
\widehat{R}(\lambda)\|_H = O(1/|\lambda|)$, as
$|\lambda| \to\infty$.

\item[(P1)] There exists a subspace $D \subseteq D(A)$
dense in $[D(A)]$ and a positive
constant $\widetilde{C}$ such that $ A(D) \subseteq
D(A),\widehat{R}(\lambda)(D) \subseteq D(A)$, and $\|
A\widehat{R}(\lambda)x\|_H\leq \widetilde{C}\|
x\|_H$ for every $x \in D$ and all $\lambda\in
\Sigma_{0,\vartheta}$.

\end{itemize}
In the sequel, for $r > 0 $ and $\theta\in(\frac{\pi}{2},\vartheta)$,
$$
\Sigma_{r,\theta}=\{\lambda\in \mathbb{C},|\lambda|>r,
|\arg(\lambda)|<\theta\},
$$
for $\Gamma_{r,\theta}, \Gamma^{i}_{r,\theta}$, $i= 1, 2, 3$, are the paths
\[
\Gamma^1_{r,\theta}=\{te^{i\theta}:t\geq r\}, \quad
\Gamma^2_{r,\theta}=\{te^{i\xi}:|\xi|\leq \theta\}, \quad
\Gamma^3_{r,\theta}=\{te^{-i\theta}:t\geq r\},
\]
and $\Gamma_{r,\theta}=\cup_{i=1}^3\Gamma^{i}_{r,\theta}$
oriented counterclockwise. In addition, $\rho_{\alpha}(G_{\alpha})$
are the sets
$$
\rho_{\alpha}(G_{\alpha})=\{\lambda\in\mathbb{C}
:G_{\alpha}(\lambda):=\lambda^{\alpha-1}
(\lambda^{\alpha}I-A-\widehat{R}(\lambda))^{-1}\in L(X)\}.
$$
We now define the operator family
$(\mathcal{R}_{\alpha}(t))_{t\geq 0}$  by
\[
\mathcal{R}_{\alpha}(t):= \begin{cases}
\frac{1}{2\pi i}\int_{\Gamma_{r,\theta}}e^{\lambda t}G_{\alpha}(\lambda)d
\lambda, &  t>0,\\
I,&  t=0.
\end{cases}
\]

\begin{lemma}[\cite{s8}] \label{lem2.1}
 Assume that conditions {\rm (P1)--(P3)} are fulfilled.
Then there exists a unique $\alpha$-resolvent
operator for problem \eqref{e2.1}-\eqref{e2.2}.
\end{lemma}

\begin{lemma}[\cite{s8}] \label{lem2.2}
 The function $\mathcal{R}_{\alpha}: [0,\infty) \to L(H)$
is strongly continuous and
$\mathcal{R}_{\alpha} : (0,\infty) \to L(H)$ is uniformly
continuous.
\end{lemma}

\begin{definition}[\cite{s8}] \label{def2.2} \rm
 Let $\alpha \in (1, 2)$, we
define the family $(\mathcal{S}_{\alpha}(t))_{t\geq 0}$ by
$$
\mathcal{S}_{\alpha}(t)x:=\int_0^{t}g_{\alpha-1}(t-s)\mathcal{R}_{\alpha}(s)ds
 $$
for each $t \geq 0$.
\end{definition}

\begin{lemma}[\cite{s8}] \label{lem2.3}
 If the function
$\mathcal{R}_{\alpha}(\cdot)$ is exponentially bounded in $L(H)$,
then $\mathcal{S}_{\alpha}(\cdot) $ is exponentially bounded in
$L(H)$.
\end{lemma}

\begin{lemma}[\cite{s8}] \label{lem2.4}
 If the function $\mathcal{R}_{\alpha}(\cdot)$ is exponentially bounded in
$L([D(A)]), $ then $\mathcal{S}_{\alpha}(\cdot)$ is exponentially
bounded in $L([D(A)])$.
\end{lemma}

\begin{lemma}[\cite{s8}] \label{lem2.5}
 If $R(\lambda_{0}^{\alpha} , A)$ is
compact for some $\lambda_{0}^{\alpha} \in \rho(A)$, then
$\mathcal{R}_{\alpha}(t)$ and $\mathcal{S}_{\alpha}(t)$ are compact
for all $ t > 0$.
\end{lemma}

\begin{definition} \label{def2.3}\rm
 A stochastic process $\{x(t), t \in[0, T ]\} (0 \leq T <\infty)$
is called a mild solution of \eqref{e1.1}-\eqref{e1.3} if
\begin{itemize}
\item[(i)] $x(t) $ is adapted to $\mathcal{F}_t , t \geq0$.
\item[(ii)] $x(t) \in H$ has c\`{a}dl\`{a}g paths on $t \in [0, T ] $ a.s
and for each $t \in [0, T ] $, $x(t)$ satisfies the integral
equation
\begin{equation} \label{e2.3}
 x(t)= \begin{cases}
 \mathcal{R}_{\alpha}(t)[\varphi(0)- g(0,\varphi(-\rho_1(0)))]
 + g(t,x(t-\rho_1(t)))\\
 +\int_0^t \mathcal{S}_{\alpha}(t-s)h(s,x(s-\rho_2(s)))ds
 \\
 +\int_0^t \mathcal{S}_{\alpha}(t-s)f(s,x(s-\rho_{3}(s)))dw(s),
   & t\in [0,t_1], \\[4pt]
\mathcal{R}_{\alpha}(t-t_1)[x(t_1^{-})+I_1(x(t^{-}_1))
 -  g(t_1,x(t_1^{+}-\rho_1(t_1^{+})))]\\
+g(t,x(t-\rho_1(t)))+\int_{t_1}^t
\mathcal{S}_{\alpha}(t-s)h(s,x(s-\rho_2(s)))ds\\
+\int_{t_1}^t
\mathcal{S}_{\alpha}(t-s)f(s,x(s-\rho_{3}(s)))dw(s),
& t\in (t_1,t_2], \\
\dots\\
\mathcal{R}_{\alpha}(t-t_{m})[x(t_{m}^{-})+I_{m}(x(t^{-}_{m}))
-  g(t_{m},x(t_{m}^{+}-\rho_1(t_{m}^{+})))]\\
+g(t,x(t-\rho_1(t)))+\int_{t_{m}}^t
\mathcal{S}_{\alpha}(t-s)h(s,x(s-\rho_2(s)))ds\\
+\int_{t_{m}}^t
\mathcal{S}_{\alpha}(t-s)f(s,x(s-\rho_{3}(s)))dw(s),
 & t\in (t_{m},T].
\end{cases}
\end{equation}
\end{itemize}
\end{definition}

\begin{definition} \label{def2.4} \rm
 Let $p \geq2$ be an integer. Equation
\eqref{e2.3} is said to be stable in the $p$-th moment, if for
any $\varepsilon > 0 $, there exists a
 $\tilde{\delta} > 0$ such that $\|\varphi\|_{\mathfrak{B}} <\tilde{\delta}$
guarantees that
\[
E\big[\sup_{t\geq0}\| x(t)\|_H^{p}\big]<\varepsilon.
\]
\end{definition}

\begin{definition} \label{def2.5} \rm
 Let  $p \geq2$ be an integer. Equation \eqref{e2.3} is said to be
asymptotically stable in $p$-th  moment if it stable in the
 $p$-th moment and for any
$\varphi\in\mathfrak{B}_{\mathcal{F}_{0}}([\tilde{m}(0),0],H)$,
\[
\lim_{T\to+\infty}E\big[\sup_{t\geq T}\| x(t)\|_H^{p}\big]=0.
\]
\end{definition}

\begin{lemma}[\cite{d1}] \label{lem2.6}
For any $p \geq 1$ and for arbitrary $L_2^0(K,H)$-valued predictable
process $\phi(\cdot)$ such that
\[
\sup_{s\in[0,t]}E \big\| \int_{0}^{s}\phi(v)dw(v)\big\|_H^{2p}
\leq C_{p}\Big(\int_{0}^{t}(E\|\phi(s)\|^{2p}_{L_2^0})^{1/p}
ds\Big)^{p}, \quad t\in [0, \infty),
\]
where $C_{p}=(p(2p- 1))^{p}$.
Next, we state a Krasnoselskii-Schaefer type fixed point
theorem.
\end{lemma}

\begin{lemma}[\cite{b3}] \label{lem2.7}  Let $\Phi_1,\Phi_2$ be two operators
such that:
\begin{itemize}
\item[(a)] $\Phi_1$ is a contraction, and

\item[(b)] $\Phi_2$  is completely continuous.
\end{itemize}
Then either
\begin{itemize}
\item[(i)]  the operator equation $\Phi_1x+\Phi_2x$ has a solution, or

\item[(ii)]  the set $\Upsilon= \{x \in H :\lambda \Phi_1 (\frac{x}{\lambda})
+\lambda\Phi_2x=x\}$ is unbounded for $\lambda\in(0,1)$.
\end{itemize}
\end{lemma}

\section{Main results}

In this section we present our  result on asymptotic stability
in the $p$-th moment of mild solutions of system \eqref{e1.1}-\eqref{e1.3}. for
this,  we state the following hypotheses:
\begin{itemize}
\item[(H1)] The operator families $\mathcal{R}_{\alpha}(t)$ and
$\mathcal{S}_{\alpha}(t)$ are compact for all $ t > 0$, and there
exist constants $M>0,\delta >0$ such that $\|
\mathcal{R}_{\alpha}(t)\|_{L(H)}\leq Me^{-\delta t}$ and
$\| \mathcal{S}_{\alpha}(t)\|_{L(H)}\leq Me^{-\delta
t}$ for every $t\geq0$.

\item[(H2)]   The function $g:[0,\infty)\times H\to H$
is continuous and there exists $L_{g}>0$ such that
\begin{gather*}
E\| g(t,\psi_1)- g(t,\omega_2)\|^{p}_H\leq
L_{g}\| \psi_1-\psi_2\|^{p}_H,  \quad t\geq0,
\omega_1,\psi_2\in H;
\\
E\| g(t,\psi)\|^{p}_H\leq L_{g}\|
\psi\|^{p}_H,  \quad t\geq0, \psi\in H.
\end{gather*}

\item[(H3)] The function  $h : [0,\infty)\times H\to H$ satisfies the following
conditions:
\begin{itemize}
\item[(i)]  The function $h: [0,\infty)\times H\to H$ is
continuous.
\item[(ii)]  There exist a continuous function $m_{h} : [0,\infty)\to [0,\infty)$
 and a continuous nondecreasing function
 $\Theta_{h}:[0,\infty)\to (0,\infty)$ such that
\[
E\| h (t,\psi)\|^{p}_H
 \leq m_{h}(t)\Theta_{h}(E\| \psi\|^{p}_H), \quad  t\geq0, \psi\in  H.
\]
\end{itemize}

\item[(H4)] The function  $f : [0,\infty)\times H\to L(K,H)$
satisfies the following conditions:
\begin{itemize}
\item[(i)]  The function $f : [0,\infty)\times H\to L(K,H)$ is
continuous.
\item[(ii)]  There exist a continuous function $m_{f} : [0,\infty)\to [0,\infty)$
 and a continuous nondecreasing function
 $\Theta_{f}:[0,\infty)\to (0,\infty)$ such that
\[
E\| f (t,\psi)\|^{p}_H
 \leq m_{f}(t)\Theta_{f}(E\| \psi\|^{p}_H), \  t\geq0, \psi\in
 H,
\]
with
\begin{equation} \label{e3.1}
\int_1^\infty \frac{1}{\Theta_{h}(s)+\Theta_{f}(s)}ds=\infty.
\end{equation}
\end{itemize}


\item[(H5)]  The functions $I_k: H\to H$ are completely continuous
and that there are constants $d^{j}_k$, $k = 1,2,\ldots m$,
$j=1,2$, such that $E\| I_k(x)\|^{p}_H\leq d^1_kE\| x\|^{p}_H+d^2_k$,
 for every $x \in H$.
\end{itemize}

In the proof of the existence theorem, we need the following
lemmas.

\begin{lemma} \label{lem3.1}
 Assume that conditions {\rm (H1), (H3)} hold.
Let $\Phi_1$ be the operator defined by: for each $x \in \mathbb{Y}$,
\begin{equation} \label{e3.2}
 (\Phi_1 x)(t)=\begin{cases}
\int_0^t\mathcal{S}_{\alpha}(t-s)h(s,x(s-\rho_2(s)))ds, & t\in[0,t_1],\\
\int_{t_1}^t\mathcal{S}_{\alpha}(t-s)h(s,x(s-\rho_2(s)))ds, & t\in(t_1,t_2],\\
\dots \\
\int_{t_m}^t\mathcal{S}_{\alpha}(t-s)h(s,x(s-\rho_2(s)))ds, & t\in(t_m,t_{m+1}],\\
\dots
\end{cases}
\end{equation}
Then $\Phi_1$ is continuous and maps $\mathbb{Y}$ into $\mathbb{Y}$.
\end{lemma}

\begin{proof}
We first prove that $\Phi_1$ is continuous
in the $p$-th moment on $[0,\infty)$. Let $x \in \mathbb{Y}$,
$\tilde{t} \geq 0$ and $|\xi|$ be sufficiently small.  Then for
$\tilde{t}\in[0,t_1]$, by using H\"{o}lder's inequality, we have
\begin{align*}
&E\|(\Phi_1 x)(\tilde{t}+\xi)-(\Phi_1
x)(\tilde{t})\|^{p}_H\\
&\leq 2^{p-1}E\big\|\int_0^{\tilde{t}}
[\mathcal{S}_{\alpha}(\tilde{t}+\xi-s)-\mathcal{S}_{\alpha}(\tilde{t}-s)]
h(s,x(s-\rho_2(s)))ds\big\|^{p}_H\\
\\
&\quad+2^{p-1}E\big\|\int_{\tilde{t}}^{\tilde{t}+\xi}
\mathcal{S}_{\alpha}(\tilde{t}+\xi-s)h(s,x(s-\rho_2(s)))ds\big\|^{p}_H\\
&\leq 2^{p-1}E\Big[\int_0^{\tilde{t}}\|
[\mathcal{S}_{\alpha}(\tilde{t}+\xi-s)-\mathcal{S}_{\alpha}(\tilde{t}-s)]
h(s,x(s-\rho_2(s)))\|_Hds\Bigr]^{p}\\
&\quad+2^{p-1}M^{p}E\Big[\int_{\tilde{t}}^{\tilde{t}+\xi}e^{-\delta(\tilde{t}+\xi-s)}
\|
h(s,x(s-\rho_2(s)))\|_Hds\Bigr]^{p}\\
&\leq
2^{p-1}\Big[\int_0^{\tilde{t}}\|
\mathcal{S}_{\alpha}(\tilde{t}+\xi-s)-\mathcal{S}_{\alpha}
(\tilde{t}-s)\|_{L(H)}^{(p/p-1)}ds\Bigr]^{p-1}\\
&\quad\times\int_0^{\tilde{t}}E\|
h(s,x(s-\rho_2(s)))\|^{p}_Hds\\
&\quad +2^{p-1}M^{p}\Big[\int_{\tilde{t}}^{\tilde{t}+\xi}e^{-(p\delta/p-1)
(\tilde{t}+\xi-s)}ds\Bigr]^{p-1}\\
&\quad\times \int_{\tilde{t}}^{\tilde{t}+\xi}E\|
h(s,x(s-\rho_2(s)))\|^{p}_Hds\to0 \quad \text{as } \xi\to\infty.
\end{align*}
Similarly, for any $\tilde{t}\in (t_k, t_{k+1}]$, $k = 1,2, \ldots$,
we have
\begin{align*}
&E\|(\Phi_1 x)(\tilde{t}+\xi)-(\Phi_1x)(\tilde{t})\|^{p}_H\\
&\leq
2^{p-1}\Big[\int_{t_k}^{\tilde{t}}\|
\mathcal{S}_{\alpha}(\tilde{t}+\xi-s)-\mathcal{S}_{\alpha}
(\tilde{t}-s)\|_{L(H)}^{-(p/p-1)}ds\Bigr]^{p-1}\\
&\quad\times\int_{t_k}^{\tilde{t}}E\|
h(s,x(s-\rho_2(s)))\|^{p}_Hds\\
&\quad+2^{p-1}M^{p}\Big[\int_{\tilde{t}}^{\tilde{t}+\xi}e^{-(p\delta/p-1)
(\tilde{t}+\xi-s)}ds\Bigr]^{p-1}\\
&\quad\times \int_{\tilde{t}}^{\tilde{t}+\xi}E\|
h(s,x(s-\rho_2(s)))\|^{p}_Hds\to0 \quad \text{as}
\quad \xi\to\infty.
\end{align*}
Then, for all $x(\tilde{t})\in\mathbb{Y}, \tilde{t}\geq0$, we have
\[
\begin{aligned}
E\|(\Phi_1 x)(\tilde{t}+\xi)-(\Phi_1
x)(\tilde{t})\|^{p}_H \to0 \quad \text{as }
\xi\to\infty.
\end{aligned}
\]
Thus $\Phi_1$ is continuous in the $p$-th moment on $[0,\infty)$.


Next we show that $\Phi_1(\mathbb{Y}) \subset \mathbb{Y}$. By
using  (H1), (H3) and  H\"{o}lder's inequality, we have for
$t\in[0,t_1]$
\begin{align*}
E\|(\Phi_1 x)(t)\|^{p}_H
&\leq E\big\|\int_0^t \mathcal{S}_{\alpha}(t-s)h(s,x(s-\rho_2(s)))ds\big\|^{p}_H\\
&\leq E\Big[\int_0^t\|
\mathcal{S}_{\alpha}(t-s)h(s,x(s-\rho_2(s)))\|_Hds\Bigr]^{p}\\
& \leq M^{p}E\Big[\int_0^te^{-\delta(t-s)}\|
h(s,x(s-\rho_2(s)))\|_Hds\Bigr]^{p}\\
& = M^{p}E\Big[\int_0^te^{-(\delta(p-1)/p)(t-s)}e^{-(\delta/p)(t-s)}\|
h(s,x(s-\rho_2(s)))\|_Hds\Bigr]^{p}\\
& \leq M^{p}\Big[\int_0^te^{-\delta(t-s)}ds\Big]^{p-1}\int_0^te^{-\delta(t-s)}E\|
h(s,x(s-\rho_2(s)))\|^{p}_Hds\\
& \leq M^{p}\delta^{1-p}\int_0^t e^{-\delta(t-s)}m_{h}(s)\Theta_{h}(E\|
x(s-\rho_2(s))\|^{p}_H)ds.
\end{align*}
Similarly, for any $t\in (t_k, t_{k+1}]$, $k = 1,2, \ldots $, we
have
\begin{align*}
E\|(\Phi_1 x)(t)\|^{p}_H
&\leq E\big\|\int_{t_k}^t
\mathcal{S}_{\alpha}(t-s)h(s,x(s-\rho_2(s)))ds\big\|^{p}_H\\
& \leq M^{p}\delta^{1-p}\int_{t_k}^t e^{-\delta(t-s)}m_{h}(s)\Theta_{h}(E\|
x(s-\rho_2(s))\|^{p}_H)ds.
\end{align*}
Then, for all $x(t)\in\mathbb{Y}, t\in[\tilde{m}(0),\infty)$, we have
\begin{equation} \label{e3.3}
E\|(\Phi_1 x)(t)\|^{p}_H  \leq
 M^{p}\delta^{p-1}\int_0^te^{-\delta(t-
s)}m_{h}(s)\Theta_{h}(E\|
x(s-\rho_2(s))\|^{p}_H)ds.
\end{equation}
However, for any  any $\varepsilon > 0$, there exists a
$\tilde{\tau}_1 > 0$ such that
$ E\| x(s-\rho_2(s))\|^{p}_H< \varepsilon$ for
$t \geq \tilde{\tau}_1$. Thus,  we obtain
\begin{align*}
&E\|(\Phi_1 x)(t)\|^{p}_H \\
& \leq   M^{p}\delta^{1-p}e^{-\delta t}\int_0^te^{\delta
s}m_{h}(s)\Theta_{h}(E\| x(s-\rho_2(s))\|^{p}_H)ds\\
& \leq   M^{p}\delta^{1-p}e^{-\delta
t}\int_0^{\tilde{\tau}_1}e^{\delta s}m_{h}(s)\Theta_{h}(E\|
x(s-\rho_2(s))\|^{p}_H)ds
+ M^{p}\delta^{1-p}L_{h}\Theta_{h}(\varepsilon),
 \end{align*}
where $L_{h}=\sup_{t\geq0}\int_{\tilde{\tau}_1}^{t}e^{-\delta(t-
s)}m_{h}(s)ds$. As $e^{-\delta t}\to 0 $ as
$ t \to\infty$ and, there exists
$\tilde{\tau}_2\geq \tilde{\tau}_1$ such that for any
$t \geq \tilde{\tau}_2 $ we have
\[
M^{p}\delta^{p-1}e^{-\delta t}\int_0^{\tilde{\tau}_1}e^{\delta
s}m_{h}(s)\Theta_{h}(E\|
x(s-\rho_2(s))\|^{p}_H)ds<\varepsilon-
M^{p}\delta^{p-1}L_{h}\Theta_{h}(\varepsilon).
\]
From the above inequality,  for any $t \geq \tilde{\tau}_2$, we obtain
$E\|(\Phi_1 x)(t)\|^{p}_H <\varepsilon$. That is
to say $E\|(\Phi_1 x)(t)\|^{p}_H\to0$ as
$t\to\infty$. So we conclude that
$\Phi_1(\mathbb{Y})\subset\mathbb{Y}$.
\end{proof}

\begin{lemma} \label{lem3.2}
Assume that conditions {\rm (H1), (H4)} hold.
Let $\Phi_2$ be the operator defined by:
for each $x \in \mathbb{Y}$,
\[
 (\Phi_2 x)(t)= \begin{cases}
\int_0^t \mathcal{S}_{\alpha}(t-s)f(s,x(s-\rho_{3}(s)))dw(s),
  & t\in [0,t_1], \\[4pt]
\int_{t_1}^t
\mathcal{S}_{\alpha}(t-s)f(s,x(s-\rho_{3}(s)))dw(s), & t\in (t_1,t_2], \\
\dots\\
\int_{t_{m}}^t
\mathcal{S}_{\alpha}(t-s)f(s,x(s-\rho_{3}(s)))dw(s), &  t\in (t_{m},t_{m+1}],\\
 \dots
\end{cases}
\]
Then $\Phi_2$ is  continuous on $[0,\infty)$ in the $p$-th mean and
maps $\mathbb{Y}$ into itself.
\end{lemma}

\begin{proof}
 We first prove that $\Phi_2$ is continuous
in the $p$-th moment on $[0,\infty)$. Let $x \in \mathbb{Y}$,
 $\tilde{\theta} \geq 0$ and $|\xi|$ be sufficiently small.  Then for
$\tilde{\theta} \in [0,t_1]$, by using H\"{o}lder's inequality and
Lemma \ref{lem2.6}, we have
\begin{align*}
&E\|(\Phi_2 x)(\tilde{\theta}+\xi)-(\Phi_2
x)(\tilde{\theta})\|^{p}_H\\
&\leq 2^{p-1}E\big\|\int_0^{\tilde{\theta}}[
\mathcal{S}_{\alpha}(\tilde{\theta}+\xi-s)-\mathcal{S}_{\alpha}(\tilde{\theta}-s)]
f(s,x(s-\rho_{3}(s)))dw(s)\big\|^{p}_H\\
&\quad+2^{p-1}E\big\|\int_{\tilde{\theta}}^{\tilde{\theta}+\xi}
\mathcal{S}_{\alpha}(\tilde{\theta}+\xi-s)f(s,x(s-\rho_{3}(s)))dw(s)\big\|^{p}_H\\
&\leq  2^{p-1}C_{p}\Big[\int_0^{\tilde{\theta}}(E\|
[\mathcal{S}_{\alpha}(\tilde{\theta}+\xi-s)
-\mathcal{S}_{\alpha}(\tilde{\theta}-s)]f(s,x(s-\rho_{3}(s)))
\|^{p}_H)^{2/p}ds\Bigr]^{p/2}\\
&\quad+2^{p-1}C_{p}\Big[\int_{\tilde{\theta}}^{\tilde{t}+\xi}
(E\|\mathcal{S}_{\alpha}(\tilde{\theta}+\xi-s)
f(s,x(s-\rho_{3}(s)))\|^{p}_H)^{2/p}ds\Bigr]^{p/2}\to 0
\end{align*}
as $\xi\to\infty$.
Similarly, for any $\tilde{\theta}\in (t_k, t_{k+1}]$, $k = 1,2, \ldots $, we
have
\begin{align*}
&E\|(\Phi_2 x)(\tilde{\theta}+\xi)-(\Phi_2
x)(\tilde{\theta})\|^{p}_H\\
&\leq
2^{p-1}C_{p}\Big[\int_{t_k}^{\tilde{\theta}}(E\|
[\mathcal{S}_{\alpha}(\tilde{\theta}+\xi-s)
-\mathcal{S}_{\alpha}(\tilde{\theta}-s)]f(s,x(s-\rho_{3}(s)))
\|^{p}_H)^{2/p}ds\Bigr]^{p/2}\\
&\quad +2^{p-1}C_{p}\Big[\int_{\tilde{\theta}}^{\tilde{t}+\xi}
(E\|\mathcal{S}_{\alpha}(\tilde{\theta}+\xi-s)
f(s,x(s-\rho_{3}(s)))\|^{p}_H)^{2/p}ds\Bigr]^{p/2}\to 0
\end{align*}
as $\xi\to\infty$.
Then, for all $x(\tilde{\theta})\in\mathbb{Y}, \tilde{\theta}\geq0$, we have
\[
E\|(\Phi_2 x)(\tilde{\theta}+\xi)-(\Phi_2
x)(\tilde{\theta})\|^{p}_H \to0 \quad \text{as }\xi\to\infty.
\]
Thus $\Phi_2$ is continuous in the $p$-th moment on $[0,\infty)$.


Next we show that $\Phi_2(\mathbb{Y}) \subset \mathbb{Y}$. By
using (H1), (H4) and H\"{o}lder's inequality, for $t \in
[0,t_1]$,  we have
\begin{align*}
&E\|(\Phi_2 x)(t)\|^{p}_H\\
&\leq E\big\|\int_0^t
\mathcal{S}_{\alpha}(t-s)f(s,x(s-\rho_{3}(s)))dw(s)\big\|^{p}_H\\
&\leq C_{p}\Big[\int_0^t(E\|
\mathcal{S}_{\alpha}(t-s)f(s,x(s-\rho_{3}(s)))\|^{p}_H)^{(2/p)}ds\Bigr]^{p/2}\\
& \leq C_{p}M^{p}\Big[\int_0^t[e^{-p\delta(t-s)}(E\|
f(s,x(s-\rho_{3}(s)))\|^{p}_H)]^{2/p}ds\Bigr]^{p/2}\\
& \leq C_{p}M^{p}\Big[\int_0^t [e^{-p\delta(t-
s)}m_{f}(s)\Theta_{f}(E\|
x(s-\rho_{3}(s))\|^{p}_H)]^{2/p}ds\Bigr]^{p/2}\\
&=C_{p}M^{p}\Big[\int_0^t[e^{-(p-1)\delta(t- s)}e^{-\delta(t-
s)}m_{f}(s)\Theta_{f}(E\|
x(s-\rho_{3}(s))\|^{p}_H)]^{2/p}ds\Bigr]^{p/2}\\
&\leq  C_{p}M^{p}\Big[\int_0^te^{-[\frac{2(p-1)}{p-2}]\delta(t-
s)}ds\Bigr]^{p/2-1}\\
&\quad\times\int_0^te^{-\delta(t- s)}m_{f}(s)\Theta_{f}(E\|
x(s-\rho_{3}(s))\|^{p}_H)ds\\
&\leq C_{p}M^{p}
\Big[\frac{2\delta(p-1)}{p-2}\Bigr]^{1-p/2}\int_0^te^{-\delta(t-
s)}m_{f}(s)\Theta_{f}(E\|
x(s-\rho_{3}(s))\|^{p}_H)ds.
\end{align*}
Similarly, for any $t\in (t_k, t_{k+1}]$, $k = 1,2, \ldots $, we
have
\begin{align*}
&E\|(\Phi_2 x)(t)\|^{p}_H\\
&\leq E\big\|\int_{t_k}^t
\mathcal{S}_{\alpha}(t-s)f(s,x(s-\rho_{3}(s))dw(s)\big\|^{p}_H\\
& \leq  C_{p}M^{p}
\Big[\frac{2\delta(p-1)}{p-2}\Bigr]^{1-p/2}\int_{t_k}^te^{-\delta(t-
s)}m_{f}(s)\Theta_{f}(E\|
x(s-\rho_{3}(s))\|^{p}_H)ds.
\end{align*}
Then, for all $x(t)\in\mathbb{Y},t\in[\tilde{m}(0),\infty)$, we have
\begin{equation} \label{e3.4}
\begin{aligned}
&E\|(\Phi_2 x)(t)\|^{p}_H \\
& \leq  C_{p}M^{p} \Big[\frac{2\delta(p-1)}{p-2}\Bigr]^{1-p/2}
\int_0^te^{-\delta(t-s)}m_{f}(s)\Theta_{f}(E\|
x(s-\rho_{3}(s))\|^{p}_H)ds.
\end{aligned}
\end{equation}
However, for any  any $\varepsilon > 0$, there exists a
$\tilde{\theta}_1 > 0$ such that
$ E\| x(s-\rho_{3}(s))\|^{p}_H<\varepsilon$ for
$t \geq \tilde{\theta}_1$. Thus from \eqref{e3.4} we
obtain
\begin{align*}
&E\|(\Phi_2 x)(t)\|^{p}_H \\
& \leq  C_{p}M^{p}
\Big[\frac{2\delta(p-1)}{p-2}\Bigr]^{1-p/2}e^{-\delta
t}\int_0^te^{\delta
s}m_{f}(s)\Theta_{f}(E\| x(s-\rho_{3}(s))\|^{p}_H)ds\\
& \leq  C_{p}M^{p}
\Big[\frac{2\delta(p-1)}{p-2}\Bigr]^{1-p/2}e^{-\delta
t}\int_0^{\tilde{t}_1}e^{\delta s}m_{f}(s)\Theta_{f}(E\|
x(s-\rho_{3}(s))\|^{p}_H)ds\\
&\quad+C_{p}M^{p}
\Big[\frac{2\delta(p-1)}{p-2}\Bigr]^{1-p/2}L_{f}\Theta_{f}(\varepsilon),
 \end{align*}
where $L_{f}=\sup_{t\geq0}\int_{\tilde{t}_1}^{t}e^{-\delta(t-
s)}m_{f}(s)ds$. As $e^{-\delta t}\to 0 $ as
$ t \to\infty$ and, there exists $\tilde{\theta}_2\geq
\tilde{\theta}_1$ such that for any $t \geq \tilde{\theta}_2 $
we have
\begin{align*}
 &C_{p}M^{p}
[\frac{2\delta(p-1)}{p-2}]^{1-p/2}\int_0^{\tilde{t}_1}e^{-\delta(t-
s)}m_{f}(s)\Theta_{f}(E\|
x(s-\rho_{3}(s))\|^{p}_H)ds\\
&<\varepsilon-C_{p}M^{p}
[\frac{2\delta(p-1)}{p-2}]^{1-p/2}L_{f}\Theta_{f}(\varepsilon).
 \end{align*}
From the above inequality,  for any $t \geq \tilde{\theta}_2$, we obtain
$E\|(\Phi_2 x)(t)\|^{p}_H <\varepsilon$. That is
to say $E\|(\Phi_2 x)(t)\|^{p}_H\to0$ as
$t\to\infty. $ So we conclude that
$\Phi_2(\mathbb{Y})\subset\mathbb{Y}$.
\end{proof}

Now, we are ready to present our main result.


\begin{theorem} \label{thm3.1}
  Assume the conditions {\rm (H1)-(H5)} hold. Let $p\geq2$  be an integer.
Then the fractional impulsive stochastic differential equations
\eqref{e1.1}--\eqref{e1.3} is asymptotically stable in the $p$-th moment,
provided that
\begin{equation} \label{e3.5}
\max_{1\leq k\leq m}\{12^{p-1}
 M^{p}(1+d^1_k +2^{p-1}L_{g})+8^{p-1}L_{g}\}<1.
\end{equation}
\end{theorem}

\begin{proof}
We define the nonlinear operator
$\Psi:\mathbb{Y}\to \mathbb{Y}$  as
$(\Psi x)(t)=\varphi(t)$ for $t \in[\tilde{m}(0),0]$
and for $t \geq 0$,
\begin{equation} \label{e3.6}
(\Psi x)(t)= \begin{cases}
 \mathcal{R}_{\alpha}(t)[\varphi(0)- g(0,\varphi(-\rho_1(0)))]
 + g(t,x(t-\rho_1(t)))\\
+\int_0^t \mathcal{S}_{\alpha}(t-s)h(s,x(s-\rho_2(s)))ds
 \\
+\int_0^t \mathcal{S}_{\alpha}(t-s)f(s,x(s-\rho_{3}(s)))dw(s),\\
  \quad t\in [0,t_1], \\[4pt]
 \mathcal{R}_{\alpha}(t-t_1)[x(t_1^{-})+I_1(x(t^{-}_1))
-  g(t_1,x(t_1^{+}-\rho_1(t_1^{+})))]\\
+g(t,x(t-\rho_1(t)))+\int_{t_1}^t
\mathcal{S}_{\alpha}(t-s)h(s,x(s-\rho_2(s)))ds\\
+\int_{t_1}^t
\mathcal{S}_{\alpha}(t-s)f(s,x(s-\rho_{3}(s)))dw(s), \\
\quad t\in (t_1,t_2], \\
\dots\\
 \mathcal{R}_{\alpha}(t-t_{m})[x(t_{m}^{-})+I_{m}(x(t^{-}_{m}))
-  g(t_{m},x(t_{m}^{+}-\rho_1(t_{m}^{+})))]\\
+g(t,x(t-\rho_1(t)))+\int_{t_{m}}^t
\mathcal{S}_{\alpha}(t-s)h(s,x(s-\rho_2(s)))ds\\
+\int_{t_{m}}^t
\mathcal{S}_{\alpha}(t-s)f(s,x(s-\rho_{3}(s)))dw(s),  \\
\quad  t\in (t_{m},t_{m+1}], \\
\dots
\end{cases}
\end{equation}
Using (H2)--(H5),  and  the proofs of   Lemmas \ref{lem3.1} and
\ref{lem3.2},
 it is clear that the nonlinear operator $\Psi$ is well
defined and  continuous in $p$-th moment on $[0,\infty)$.
  Moreover,
for all $t \in [0,t_1]$  we have
\begin{align*}
&E\|(\Psi x)(t)\|^{p}_H\\
&\leq 4^{p-1}E \|\mathcal{R}_{\alpha}(t)
[\varphi(0)-g(0,\varphi(-\rho_1(0)))]\|^{p}_H
 + 4^{p-1}E\| g(t,x(t-\rho_1(t)))\|^{p}_H\\
&\quad+ 4^{p-1}E\big\|\int_0^t
\mathcal{S}_{\alpha}(t-s)h(s,x(s-\rho_2(s)))dw(s)\big\|^{p}_H\\
&\quad+ 4^{p-1}E\big\|\int_0^t
\mathcal{S}_{\alpha}(t-s)f(s,x(s-\rho_{3}(s)))dw(s)\big\|^{p}_H.
\end{align*}
Similarly, for any $t\in (t_k, t_{k+1}]$, $k = 1,2, \ldots $, we
have
\begin{align*}
E\|(\Psi x)(t)\|^{p}_H
&\leq 4^{p-1}E
\| \mathcal{R}_{\alpha}(t-t_k)[x(t_k^{-})
+I_k(x(t_k^{-}))-  g(t_k,x(t^{+}_k-\rho_1(t^{+}_k)))]\|^{p}_H\\
&\quad  + 4^{p-1}E\| g(t,x(t-\rho_1(t)))\|^{p}_H\\
 &\quad+ 4^{p-1}E\big\|\int_{t_k}^t
\mathcal{S}_{\alpha}(t-s)h(s,x(s-\rho_2(s)))ds\big\|^{p}_H\\
&\quad+ 4^{p-1}E\big\|\int_{t_k}^t
\mathcal{S}_{\alpha}(t-s)f(s,x(s-\rho_{3}(s)))dw(s)\big\|^{p}_H.
\end{align*}
Then, for all $t \geq0$, we have
\begin{align*}
E\|(\Psi x)(t)\|^{p}_H
&\leq 4^{p-1}E
\|\mathcal{R}_{\alpha}(t)[\varphi(0)-g(0,\varphi(-\rho_1(0)))]\|^{p}_H\\
&\quad+4^{p-1}E
\| \mathcal{R}_{\alpha}(t-t_k)[x(t_k^{-})
+I_k(x(t_k^{-}))-  g(t_k,x(t^{+}_k-\rho_1(t^{+}_k)))]\|^{p}_H\\
&\quad
 + 4^{p-1}E\| g(t,x(t-\rho_1(t)))\|^{p}_H\\
 &\quad+ 4^{p-1}E\big\|\int_{0}^t
\mathcal{S}_{\alpha}(t-s)h(s,x(s-\rho_2(s)))ds\big\|^{p}_H\\
&\quad+ 4^{p-1}E\big\|\int_{0}^t
\mathcal{S}_{\alpha}(t-s)f(s,x(s-\rho_{3}(s)))dw(s)\big\|^{p}_H.
\end{align*}
By (H1)--(H5), Lemmas \ref{lem3.1} and \ref{lem3.2} again, we obtain
\begin{gather*}
\begin{aligned}
&4^{p-1}E
\|\mathcal{R}_{\alpha}(t)[\varphi(0)-g(0,\varphi(-\rho_1(0)))]\|^{p}_H\\
&\leq  8^{p-1}M^{p}e^{-p\delta t}[E
\| \varphi(0)\|^{p}_H+L_{g}E\|\varphi(-\rho_1(0))\|^{p}_H]\to0
\quad \text{as} \quad t\to\infty,
\end{aligned}
\\
\begin{aligned}
 &4^{p-1}E
\| \mathcal{R}_{\alpha}(t-t_k)[x(t_k^{-})
+I_k(x(t_k^{-}))-  g(t_k,x(t^{+}_k-\rho_1(t^{+}_k)))]\|^{p}_H\\
&\leq  12^{p-1}M^{p}e^{-p\delta t}[E
\| x(t_k^{-})\|^{p}_H
+E\| I_k(x(t_k^{-}))\|^{p}_H\\
&\quad+L_{g}E\| x(t^{+}_k-\rho_1(t^{+}_k)))\|^{p}_H]\to 0
\quad \text{as } t\to\infty,
\end{aligned}
\\
 4^{p-1}E\| g(t,x(t-\rho_1(t)))\|^{p}_H
\leq 4^{p-1}L_{g}E\| x(t-\rho_1(t)))\|^{p}_H\to0 \quad \text{as } t\to\infty,
\\
4^{p-1}E\big\|\int_0^t
\mathcal{S}_{\alpha}(t-s)h(s,x(t-\rho_2(t)))ds\big\|^{p}_H\to0
\quad \text{as } t\to\infty,
\\
4^{p-1}E\big\|\int_0^t
\mathcal{S}_{\alpha}(t-s)f(s,x(t-\rho_{3}(t)))dw(s)\big\|^{p}_H\to0
\quad \text{as} \quad t\to\infty.
\end{gather*}
That is to say $E\|(\Psi x)(t)\|^{p}_H\to0$ as
$t\to\infty$. So $\Psi$ maps $\mathbb{Y}$ into itself.

Next we prove that the operator $\Psi$ has a fixed point, which is a
mild solution of the problem  \eqref{e1.1}-\eqref{e1.3}.
To see this, we decompose $\Psi$ as $\Psi_1+\Psi_2$ for $t\in[0,T]$, where
\[
(\Psi_1 x)(t)= \begin{cases}
-\mathcal{R}_{\alpha}(t)g(0,\varphi(-\rho_1(0)))
 + g(t,x(t-\rho_1(t))),  \quad \quad &t\in [0,t_1], \\[4pt]
 -\mathcal{R}_{\alpha}(t-t_1) g(t_1,x(t^{+}_1-\rho_1(t^{+}_1)))
+g(t,x(t-\rho_1(t))),
 & t\in (t_1,t_2], \\
\dots\\
 -\mathcal{R}_{\alpha}(t-t_{m})g(t_{m},x(t^{+}_{m}-\rho_1(t^{+}_{m})))
+ g(t,x(t-\rho_1(t))),
& t\in  (t_{m},T],
\end{cases}
\]
and
\[
(\Psi_2 x)(t)= \begin{cases}
 \mathcal{R}_{\alpha}(t)\varphi(0)+\int_0^t
\mathcal{S}_{\alpha}(t-s)h(s,x(s-\rho_2(s)))ds\\
+\int_0^t \mathcal{S}_{\alpha}(t-s)f(s,x(s-\rho_{3}(s)))dw(s),
    &t\in [0,t_1], \\[4pt]
 \mathcal{R}_{\alpha}(t-t_1)[x(t_1^{-})+I_1(x(t^{-}_1))]\\
+\int_{t_1}^t
\mathcal{S}_{\alpha}(t-s)h(s,x(s-\rho_2(s)))ds\\
+\int_{t_1}^t
\mathcal{S}_{\alpha}(t-s)f(s,x(t-\rho_{3}(t)))dw(s), &  t\in (t_1,t_2], \\
\dots\\
 \mathcal{R}_{\alpha}(t-t_{m})[x(t_{m}^{-})+I_{m}(x(t^{-}_{m}))]\\
+\int_{t_{m}}^t \mathcal{S}_{\alpha}(t-s)h(s,x(s-\rho_2(s)))ds\\
+\int_{t_{m}}^t \mathcal{S}_{\alpha}(t-s)f(s,x(s-\rho_{3}(s)))dw(s),
& t\in (t_{m},T].
\end{cases}
\]
To use Lemma \ref{lem2.7}, we will verify that $\Psi_1$ is a
contraction while $\Psi_2$ is a completely continuous operator.
For better readability, we break the proof into a sequence of steps.
\medskip

\noindent\textbf{Step1.} $\Psi_1$ is a contraction on $\mathbb{Y}$.
Let $t \in [0,t_1]$ and  $x, y \in \mathbb{Y}$. From (H2), we have
\begin{align*}
E\|(\Psi_1x)(t)-(\Psi_1y)(t)\|^{p}_H
& \leq E\| g(t,x(t-\rho_1(t)))
-g(t,y(t-\rho_1(t)))\|^{p}_H \\
& \leq L_{g}E\|
x(t-\rho_1(t))-y(t-\rho_1(t))\|^{p}_H \\
&\leq L_{g}\| x-y\|_{\mathbb{Y}}.
\end{align*}
Similarly, for any $t\in (t_k, t_{k+1}]$, $k = 1, \ldots ,m$, we
have
\begin{align*}
&E\|(\Psi_1x)(t)-(\Psi_1y)(t)\|^{p}_H \\
&\leq 2^{p-1}E\|\mathcal{R}_{\alpha}(t-t_k)[
-g(t_k,x(t^{+}_k-\rho_1(t^{+}_k)))+
g(t_k,y(t^{+}_k-\rho_1(t^{+}_k)))]\|^{p}_H \\
&\quad +2^{p-1}E\| g(t,x(t-\rho_1(t)))
-g(t,y(t-\rho_1(t)))\|^{p}_H \\
&\leq 2^{p-1} M^{p}L_{g}E\|
x(t^{+}_k-\rho_1(t^{+}_k))-y(t^{+}_k-\rho_1(t^{+}_k))\|^{p}_H\\
&\quad + 2^{p-1}L_{g}E\| x(t-\rho_1(t))-y(t-\rho_1(t))\|^{p}_H\\
&\leq  2^{p-1}L_{g}(M^{p}+1)\|x-y\|_{\mathbb{Y}}.
\end{align*}
Thus, for all $t \in[0,T]$,
\[
E\|(\Psi_1x)(t)-(\Psi_1y)(t)\|^{p}_H
\leq 2^{p-1}L_{g}(M^{p}+1)\|x-y\|_{\mathbb{Y}}.
\]
Taking supremum over $t$,
\[
\| \Psi_1x-\Psi_1y\|_{\mathbb{Y}}\leq L_{0}\|
x-y\|_{\mathbb{Y}},
\]
where $L_{0}=2^{p-1}L_{g}(M^{p}+1)< 1$. By \eqref{e3.5}, we see that
$L_{0}<1$. Hence, $\Psi_1$ is a contraction on $\mathbb{Y}$.
\medskip

\noindent\textbf{Step 2.}  $\Psi_2$  maps bounded sets into bounded sets in
$\mathbb{Y}$.
Indeed, it is sufficient to show that there exists a positive constant
$\mathcal{L}$ such that for each
$x\in B_{r}=\{x:\|x\|_{\mathbb{Y}}\leq r\}$ one has
$\|\Psi_2x\|_{\mathbb{Y}}\leq\mathcal{L} $. Now, for $t\in [0,t_1]$ we have
\begin{equation} \label{e3.7}
\begin{aligned}
 (\Psi_2x)(t)
&=\mathcal{R}_{\alpha}(t)\varphi(0)
+\int_0^t\mathcal{S}_{\alpha}(t-s)h(s,x(t-\rho_2(t)))ds \\
&\quad  +\int_0^t\mathcal{S}_{\alpha}(t-s)f(s,x(t-\rho_{3}(t)))dw(s).
\end{aligned}
\end{equation}
If $x \in B_{r}$, from the definition of $\mathbb{Y}$, it follows that
\begin{align*}
E\| x(s-\rho_{i}(s))\|^{p}_H &\leq 2^{p-1}\|
\varphi\|^{p}_{\mathfrak{B}}+2^{p-1}\sup_{s\in [0,T]}E\| x(s)\|^{p}_H\\
&\leq2^{p-1} \| \varphi\|^{p}_{\mathfrak{B}}+2^{p-1}r:=r^{*},\quad i=1,2,3.
\end{align*}
By  (H1)-(H4), from \eqref{e3.7} and H\"{o}lder's inequality,   for
$t \in [0,t_1]$, we have
\begin{align*}
&E\|(\Psi_2x)(t)\|^{p}_H\\
&\leq 3^{p-1}E\|\mathcal{R}_{\alpha}(t)\varphi(0)\|^{p}_H
+3^{p-1}E\big\|\int_0^t\mathcal{S}_{\alpha}(t-s)
 h(s,x(t-\rho_2(t)))ds\big\|^{p}_H\\
&\quad+3^{p-1}E\big\|\int_0^t\mathcal{S}_{\alpha}(t-s)
 f(s,x(t-\rho_{3}(t)))dw(s)\big\|^{p}_H\\
&\leq 3^{p-1}M^{p}E\| \varphi(0)\|^{p}_H +
3^{p-1}M^{p}E\Big[\int_0^t e^{-\delta(t- s)}\|
h(s,x(t-\rho_2(t)))\|_Hds\Bigr]^{p}\\
&\quad+ 3^{p-1}C_{p}M^{p}\Big[\int_0^t[ e^{-p\delta(t-
s)}(E\|
f(s,x(t-\rho_{3}(t)))\|^{p}_H)]^{2/p}ds\Bigr]^{p/2}\\
&\leq 3^{p-1}M^{p}E\| \varphi(0)\|^{p}_H  +3^{p-1}M^{p}\Big[\int_0^t e^{-\delta(t-
s)}ds\Bigr]^{p-1}\\
&\quad \times\int_0^te^{-\delta(t- s)}E\|
h(s,x(t-\rho_2(t)))\|^{p}_Hds\\
&\quad+ 3^{p-1}C_{p}M^{p}\Big[\int_0^t [e^{-p\delta(t-
s)}m_{f}(s)\Theta_{f}(E\|
x(s))\|^{p}_H)]^{2/p}ds\Bigr]^{p/2}\\
&\leq 3^{p-1}M^{p}E\| \varphi(0)\|^{p}_H + 3^{p-1}
M^{p}\delta^{1-p}\\
&\quad \times\int_0^te^{-\delta(t- s)}m_{h}(s)\Theta_{h}(E\|
x(t-\rho_2(t))\|^{p}_H)ds+ 3^{p-1}C_{p}M^{p}\\
&\quad \times \Big[\frac{2\delta(p-1)}{p-2}\Bigr]^{1-p/2}\int_0^te^{-\delta(t-
s)}m_{f}(s)\Theta_{f}(E\| x(t-\rho_{3}(t))\|^{p}_H)ds\\
&\leq 3^{p-1}M^{p}E\| \varphi(0)\|^{p}_H+ 3^{p-1}
M^{p}\delta^{1-p}\Theta_{h}(r^{*}) \int_0^{t_1}e^{-\delta(t-s)}m_{h}(s)ds\\
&\quad + 3^{p-1} C_{p}M^{p}\Theta_{f}(r^{*})
\Big[\frac{2\delta(p-1)}{p-2}\Bigr]^{1-p/2}
\int_0^{t_1}e^{-\delta(t-s)}m_{f}(s)ds:=\mathcal{L}_{0}.
\end{align*}
Similarly, for any $t\in (t_k, t_{k+1}]$, $ k = 1, \ldots ,m$, we
have
\begin{equation} \label{e3.8}
\begin{aligned}
&(\Psi_2x)(t)\\
&=\mathcal{R}_{\alpha}(t-t_k)[x(t_k^{-})+I_k(x(t_k))]
+\int_{t_k}^t\mathcal{S}_{\alpha}(t-s)h(s,x(t-\rho_2(t)))ds\\
&\quad +\int_{t_k}^t\mathcal{S}_{\alpha}(t-s)f(s,x(t-\rho_{3}(t)))dw(s).
\end{aligned}
\end{equation}
By  {\rm (H1)--(H5)}, from \eqref{e3.8} and H\"{o}lder's inequality,
we have for $t \in (t_k, t_{k+1}]$, $k = 1, \ldots ,m$,
\begin{align*}
&E\| (\Psi_2x)(t)\|^{p}_H\\
&\leq 3^{p-1}E\|\mathcal{R}_{\alpha}(t-t_k)
[x(t_k^{-})+I_k(x(t^{-}_k))]\|^{p}_H\\
&\quad +3^{p-1}E\big\|\int_{t_k}^t\mathcal{S}_{\alpha}(t-s)
h(s,x(t-\rho_2(t)))ds\big\|^{p}_H\\
&\quad +3^{p-1}E\big\|\int_{t_k}^t\mathcal{S}_{\alpha}(t-s)
f(s,x(t-\rho_{3}(t)))dw(s)\big\|^{p}_H\\
&\leq 6^{p-1}M^{p}[E\|
x(t_k^{-})\|^{p}_H+E\| I_k(x(t^{-}_k))\|^{p}_H]\\
&\quad + 3^{p-1}M^{p}E\Big[\int_{t_k}^t e^{-\delta(t-s)}\|
h(s,x(t-\rho_2(t)))\|_Hds\Bigr]^{p}\\
&\quad + 3^{p-1}C_{p}M^{p}\Big[\int_{t_k}^t[ e^{-p\delta(t-
s)}(E\|
f(s,x(t-\rho_{3}(t)))\|^{p}_H)]^{2/p}ds\Bigr]^{p/2}\\
&\leq 6^{p-1}M^{p}(r+d^1_kE\|x(t^{-}_k)\|^{p}_H+d^2_k)+
3^{p-1}M^{p}\Big[\int_{t_k}^t e^{-\delta(t-s)}ds\Bigr]^{p-1}\\
&\quad \times\int_{t_k}^te^{-\delta(t- s)}E\|
h(s,x(t-\rho_2(t)))\|^{p}_Hds\\
&\quad+ 3^{p-1}C_{p}M^{p}\Big[\int_{t_k}^t [e^{-p\delta(t-
s)}m_{f}(s)\Theta_{f}(E\|
x(t-\rho_{3}(t))\|^{p}_H)]^{2/p}ds\Bigr]^{p/2}\\
&\leq 6^{p-1}M^{p}(r+d^1_kr+d^2_k)+ 3^{p-1}M^{p}
\delta^{p-1}\\
&\quad\times\int_{t_k}^te^{-\delta(t-
s)}m_{h}(s)\Theta_{h}(E\|
x(t-\rho_2(t))\|^{p}_H)ds\\
&\quad + 3^{p-1}C_{p}M^{p}
\Big[\frac{2\delta(p-1)}{p-2}\Bigr]^{1-p/2}\int_{t_k}^te^{-\delta(t-
s)}m_{f}(s)\Theta_{f}(E\| x(t-\rho_{3}(t))\|^{p}_H)ds\\
&\leq 6^{p-1}M^{p}(r+d_k)+
3^{p-1}M^{p}\delta^{1-p}\Theta_{h}(r^{*})\int_{t_k}^{t_{k+1}}e^{-\delta(t-
s)}m_{h}(s)ds\\
&\quad + 3^{p-1}C_{p}M^{p}\Theta_{f}(r^{*})
\Big[\frac{2\delta(p-1)}{p-2}\Bigr]^{1-p/2}\int_{t_k}^{t_{k+1}}e^{-\delta(t-
s)}m_{f}(s)ds:=\mathcal{L}_k.
\end{align*}
Take $\mathcal{L} = \max_{0\leq k\leq m}\mathcal{L}_k$, for all
$t \in[0,T]$, we have $ E\| (\Psi_2x)(t)\|^{p}_H \leq\mathcal{L}$.
 Then for each $x \in B_{r}$, we have
$\| \Psi_2x\|_{\mathbb{Y}}\leq \mathcal{L}$.
\medskip

\noindent\textbf{Step 3.}
 $\Psi_2:\mathbb{Y}\to\mathbb{Y}$ is continuous.
Let $\{x_n(t)\}_{n=0}^{\infty}\subseteq \mathbb{Y}$ with
$x_n\to x(n\to\infty)$ in $\mathbb{Y}$. Then there
is a number $r> 0$ such that $E\|
x_n(t)\|^{p}_H\leq r$ for all $n$ and a.e. $t \in[0,T]$,
so $x_n \in B_{r}$ and $x\in B_{r}$. By the assumption (H3) and
$I_k,k=1,2,\ldots,m$, are completely continuous, we have
\begin{gather*}
E\|h(s,x_n(s-\rho_2(s)))-h(s,x(s-\rho_2(s)))\|^{p}_H\to0\quad
\text{as } n\to\infty, \\
E\|f(s,x_n(s-\rho_{3}(s)))-f(s,x(s-\rho_{3}(s)))\|^{p}_H\to0\quad
\text{as }  n\to\infty
\end{gather*}
for each $s\in  [0, t]$, and since
\begin{gather*}
E\|h(s,x_n(s-\rho_2(s)))-h(s,x(s-\rho_2(s)))\|^{p}_H\leq2m_{h}(t)
\Theta_{h}(r^{*}), \\
E\| f(s,x_n(s-\rho_{3}(s)))-f(s,x(s-\rho_{3}(s)))\|^{p}_H
\leq2m_{f}(t)\Theta_{f}(r^{*}).
\end{gather*}
Then by the dominated convergence theorem,  for  $t \in [0,t_1]$, we have
\begin{align*}
&E\| (\Psi_2 x_n)(t)-(\Psi_2 x)(t)\|^{p}_H\\
& \leq 2^{p-1}E\big\|\int_0^t
\mathcal{S}_{\alpha}(t-s)[h(s,x_n(s-\rho_2(s)))-h(s,x(s-\rho_2(s)))]ds\big\|^{p}_H\\
& \quad+2^{p-1}E\big\|\int_0^t
\mathcal{S}_{\alpha}(t-s)[f(s,x_n(s-\rho_{3}(s)))-f(s,x(s-\rho_{3}(s)))]dw(s)\big\|^{p}_H\\
& \leq 2^{p-1}M^{p}E\Big[\int_0^t
 e^{-\delta(t-s)}\| h(s,x_n(s-\rho_2(s)))-h(s,x(s-\rho_2(s)))\|_Hds\Bigr]^{p}\\
& \quad +2^{p-1}C_{p}M^{p}\Big[\int_0^t(E\|
\mathcal{S}_{\alpha}(t-s)[f(s,x_n(s-\rho_{3}(s)))\\
&\quad -f(s,x(s-\rho_{3}(s)))]\|^{p}_H)^{2/p}ds\Bigr]^{p/2}\\
& \leq2^{p-1}M^{p}\delta^{1-p}\int_0^t
 e^{-\delta(t-s)}E\| h(s,x_n(s-\rho_2(s)))-h(s,x(s-\rho_2(s)))\|^{p}_Hds\\
& \quad + 2^{p-1}C_{p}M^{p}\Big[\int_0^te^{-2\delta(t-s)}(E\|
f(s,x_n(s-\rho_{3}(s)))\\
&\quad -f(s,x(s-\rho_{3}(s)))\|^{p}_H)^{2/p}ds\Bigr]^{p/2}
\to0 \quad \text{as } n\to\infty.
\end{align*}
Similarly, for any $t\in (t_k, t_{k+1}], k = 1,2, \ldots ,m$, we
have
\begin{align*}
&E\| (\Psi_2 x_n)(t)-(\Psi_2 x)(t)\|^{p}_H\\
&\leq   3^{p-1}E\|
\mathcal{R}_{\alpha}(t-t_k)[x_n(t_k^{-})-x(t_k^{-})
+I_k(x_n(t^{-}_k))-I_k(x(t^{-}_k))]\|^{p}_H\\
&\quad +3^{p-1}E\big\|\int_{t_k}^t
\mathcal{S}_{\alpha}(t-s)[h(s,x_n(s-\rho_2(s)))-h(s,x(s-\rho_2(s)))]ds\big\|^{p}_H\\
&\quad +3^{p-1}E\big\|\int_{t_k}^t
\mathcal{S}_{\alpha}(t-s)[f(s,x_n(s-\rho_{3}(s)))-f(s,x(s-\rho_{3}(s)))]dw(s)\big\|^{p}_H\\
& \leq6^{p-1}M^{p}[E\|
x_n(t_k^{-})-x(t_k^{-})\|^{p}_H
+E\| I_k(x_n(t^{-}_k))-I_k(x(t^{-}_k))\|^{p}_H]\\
& \quad+3^{p-1}M^{p}\delta^{1-p}\int_0^t
 e^{-\delta(t-s)}E\| h(s,x_n(s-\rho_2(s)))-h(s,x(s-\rho_2(s)))\|^{p}_Hds\\
&\quad +
3^{p-1}C_{p}M^{p}\Big[\int_{t_k}^te^{-2\delta(t-s)}(E\|
f(s,x_n(s-\rho_{3}(s)))\\
&\quad -f(s,x(s-\rho_{3}(s)))\|^{p}_H)^{2/p}ds\Bigr]^{p/2}
 \to 0 \quad \text{as }   n\to\infty.
\end{align*}
Then, for all $t\in[0,T]$ we have
\[
\| \Psi_2 x_n-\Psi_2
x\|_{\mathbb{Y}}\to0 \quad \text{as } n\to\infty.
\]
Therefore, $\Psi_2$ is continuous on $B_{r} $.
\medskip

\noindent\textbf{Step 4.} $\Psi_2$ maps bounded sets into
equicontinuous sets of $\mathbb{Y}$.

Let $0 < \tau_1 < \tau_2 \leq t_1$. Then, by using
H\"{o}lder's inequality and Lemma \ref{lem2.6},  for each $x\in B_{r}$, we have
\begin{align*}
&E\|(\Psi_2x)(\tau_2)- (\Psi_2x)(\tau_1)\|^{p}_H\\
& \leq 7^{p-1}E
\|[\mathcal{R}_{\alpha}(\tau_2)-\mathcal{R}_{\alpha}(\tau_1)]
\varphi(0)\|^{p}_H\\
&\quad +7^{p-1}E\big\| \int_0^{\tau_1-\varepsilon}[
\mathcal{S}_{\alpha}(\tau_2-s)-\mathcal{S}_{\alpha}(\tau_1-s)]
h(s,x(s-\rho_2(s)))ds\big\|^{p}_H
\\
&\quad  +7^{p-1}E\big\|
\int_{\tau_1-\varepsilon}^{\tau_1}
[\mathcal{S}_{\alpha}(\tau_2-s)-\mathcal{S}_{\alpha}(\tau_1-s)]
h(s,x(s-\rho_2(s)))ds\big\|^{p}_H\\
&\quad +7^{p-1}E\big\| \int_
{\tau_1}^{\tau_2}\mathcal{S}_{\alpha} (\tau_2-s)
h(s,x(s-\rho_2(s)))ds\big\|^{p}_H\\
&\quad +7^{p-1}E\big\| \int_0^{\tau_1-\varepsilon}[
\mathcal{S}_{\alpha}(\tau_2-s)-\mathcal{S}_{\alpha}(\tau_1-s)]
f(s,x(s-\rho_{3}(s)))dw(s)\big\|^{p}_H
\\
&\quad  +7^{p-1}E\big\|
\int_{\tau_1-\varepsilon}^{\tau_1}
[\mathcal{S}_{\alpha}(\tau_2-s)-\mathcal{S}_{\alpha}(\tau_1-s)]
f(s,x(s-\rho_{3}(s)))dw(s)\big\|^{p}_H\\
&\quad +7^{p-1}E\big\| \int_
{\tau_1}^{\tau_2}\mathcal{S}_{\alpha} (\tau_2-s)
f(s,x(s-\rho_{3}(s)))dw(s)\big\|^{p}_H\\
& \quad \leq7^{p-1}
E\|[\mathcal{R}_{\alpha}(\tau_2)-\mathcal{R}_{\alpha}(\tau_1)]
\varphi(0)\|^{p}_H\\
&\quad + 7^{p-1}E\Big[\int_0^{\tau_1-\varepsilon}\|
\mathcal{S}_{\alpha}(\tau_2-s)-\mathcal{S}_{\alpha}(\tau_1-s)\|_{L(H)}
\| h(s,x(s-\rho_1(s)))\|_H
ds\Big]^{p}\\
&\quad  +7^{p-1}E\Big[\int_{\tau_1-\varepsilon}^{\tau_1}
\|
\mathcal{S}_{\alpha}(\tau_2-s)-\mathcal{S}_{\alpha}(\tau_1-s)\|_{L(H)}\|
h(s,x(s-\rho_1(s)))\|_H
ds\Big]^{p}\\
&\quad +7^{p-1}E\Big[\int_ {\tau_1}^{\tau_2}\|
\mathcal{S}_{\alpha}(\tau_2-s)\|_{L(H)}\|
h(s,x(s-\rho_1(s)))\|_H
ds\Big]^{p}\\
&\quad + 7^{p-1}C_{p}\Big[\int_0^{\tau_1-\varepsilon}[\|
\mathcal{S}_{\alpha}(\tau_2-s)-\mathcal{S}_{\alpha}(\tau_1-s)\|_{L(H)}^{p}\\
&\quad\times\big(E\| f(s,x(s-\rho_{3}(s)))\|^{p}_H\big)]^{2/p}
ds\Big]^{p/2}
\\
&\quad +7^{p-1}C_{p}\Big[\int_{\tau_1-\varepsilon}^{\tau_1}[ \|
\mathcal{S}_{\alpha}(\tau_2-s)-\mathcal{S}_{\alpha}(\tau_1-s)\|_{L(H)}^{p}\\
&\quad\times \big(E\| f(s,x(s-\rho_{3}(s)))\|^{p}_H\big)]^{2/p}
ds\Big]^{p/2}\\
&\quad +7^{p-1}C_{p}\Big[\int_ {\tau_1}^{\tau_2}[\|
\mathcal{S}_{\alpha}(\tau_2-s)\|_{L(H)}^{p}(E\|
f(s,x(s-\rho_{3}(s)))\|^{p}_H)]^{2/p}
ds\Big]^{p/2}
\\
& \leq 7^{p-1} E\|[\mathcal{R}_{\alpha}(\tau_2)-\mathcal{R}_{\alpha}(\tau_1)]
\varphi(0)\|^{p}_H \\
&\quad +7^{p-1}T^{p}\int_0^{\tau_1-\varepsilon}\|\mathcal{S}_{\alpha}(\tau_2-s)
-\mathcal{S}_{\alpha}(\tau_1-s)\|_{L(H)}^{p}
m_{h}(s)\Theta_{h}(E\| x(s-\rho_1(s)))\|^{p}_H)ds
\\
&\quad +14^{p-1}M^{p}\Big[\int_{\tau_1-\varepsilon}^{\tau_1}
e^{-\delta (\tau_1-s)}ds\Bigr]^{p-1}
\int_{\tau_1-\varepsilon}^{\tau_1} e^{-\delta
(\tau_1-s)}m_{h}(s)\\
&\quad\times \Theta_{h}(E\| x(s-\rho_1(s)))\|^{p}_H)ds 
\\
&\quad + 7^{p-1}M^{p} \Big[\int_
{\tau_1}^{\tau_2}e^{-\delta (\tau_2-s)}ds\Bigr]^{p-1}\int_
{\tau_1}^{\tau_2}e^{-\delta (\tau_2-s)}m_{h}(s)
\Theta_{h}(E\| x(s-\rho_2(s)))\|^{p}_H)ds
\\
&\quad +4^{p-1}C_{p}
\Big[\int_0^{\tau_1-\varepsilon}[\|
\mathcal{S}_{\alpha}(\tau_2-s)-\mathcal{S}_{\alpha}(\tau_1-s)\|_{L(H)}^{p}
\\
&\quad \times m_{f}(s)\Theta_{f}(E\| x(s-\rho_{3}(s))\|^{p}_H)]^{2/p}ds\Bigr]^{p/2}\\
&\quad  +14^{p-1}C_{p}M^{p}
\Big[\int_{\tau_1-\varepsilon}^{\tau_1} [e^{-p\delta
(\tau_1-s)}m_{f}(s)
\Theta_{f}(E\| x(s-\rho_{3}(s))\|^{p}_H)]^{2/p}ds\Bigr]^{p/2} \\
&\quad + 7^{p-1}C_{p}M^{p} \Big[\int_
{\tau_1}^{\tau_2}[e^{-p\delta (\tau_2-s)}m_{f}(s)
\Theta_{f}(E\| x(s-\rho_{3}(s))\|^{p}_H)]^{2/p}ds\Bigr]^{p/2}\\
&  \leq7^{p-1}
E\|[\mathcal{R}_{\alpha}(\tau_2)-\mathcal{R}_{\alpha}(\tau_1)]
\varphi(0)\|^{p}_H\\
&\quad +7^{p-1}T^{p}
\Theta_{h}(r^{*})\int_0^{\tau_1-\varepsilon}\|\mathcal{S}_{\alpha}(\tau_2-s)
-\mathcal{S}_{\alpha}(\tau_1-s)\|_{L(H)}^{p}
m_{h}(s)ds\\
&\quad  +14^{p-1}M^{p} \Theta_{h}(r^{*})\delta^{1-p}
\int_{\tau_1-\varepsilon}^{\tau_1} e^{-\delta(\tau_1-
s)}m_{h}(s)ds \\
&\quad + 7^{p-1}M^{p} \Theta_{h}(r^{*})\delta^{1-p}\int_
{\tau_1}^{\tau_2}e^{-\delta (\tau_2-s)}m_{h}(s)ds\\
&\quad +7^{p-1}C_{p}
\Theta_{f}(r^{*})\Big[\int_0^{\tau_1-\varepsilon}[\|
\mathcal{S}_{\alpha}(\tau_2-s)-\mathcal{S}_{\alpha}(\tau_1-s)\|_{L(H)}^{p}
m_{f}(s)]^{2/p}ds\Bigr]^{p/2}\\
&\quad  +14^{p-1}C_{p}M^{p}
\Theta_{f}(r^{*})\Big[\frac{2\delta(p-1)}{p-2}\Bigr]^{1-p/2}
\int_{\tau_1-\varepsilon}^{\tau_1} e^{-\delta(\tau_1-
s)}m_{f}(s)ds \\
&\quad + 4^{p-1}C_{p}M^{p}
\Theta_{f}(r^{*})\Big[\frac{2\delta(p-1)}{p-2}\Bigr]^{1-p/2}\int_
{\tau_1}^{\tau_2}e^{-\delta (\tau_2-s)}m_{f}(s)ds.
\end{align*}
Similarly, for any $\tau_1, \tau_2 \in (t_k, t_{k+1}]$,
$\tau_1 < \tau_2$, $k = 1, \ldots ,m$, we have
\begin{equation} \label{e3.9}
\begin{aligned}
&(\Psi_2x)(t)\\
&=\mathcal{R}_{\alpha}(t-t_k)[\bar{x}(t_k^{-})+I_k(x(t_k))]
+\int_{t_k}^t\mathcal{S}_{\alpha}(t-s) h(s,x(s-\rho_2(s)))ds\\
&\quad +\int_{t_k}^t\mathcal{S}_{\alpha}(t-s) f(s,x(s-\rho_{3}(s)))dw(s).
\end{aligned}
\end{equation}
Then
\begin{align*}
&E\| (\Psi_2x)(\tau_2)- (\Psi_2x)(\tau_1)\|^{p}_H\\
& \leq 7^{p-1}E\|[\mathcal{R}_{\alpha}(\tau_2)-\mathcal{R}_{\alpha}(\tau_1)]
[x(t_k^{-})+I_k(x(t^{-}_k))]\|^{p}_H\\
&\quad +7^{p-1}E\big\|
\int_{t_k}^{\tau_1-\varepsilon}[
\mathcal{S}_{\alpha}(\tau_2-s)-\mathcal{S}_{\alpha}(\tau_1-s)]
h(s,x(s-\rho_2(s)))ds\big\|^{p}_H
\\
&\quad  +7^{p-1}E\big\|
\int_{\tau_1-\varepsilon}^{\tau_1}
[\mathcal{S}_{\alpha}(\tau_2-s)-\mathcal{S}_{\alpha}(\tau_1-s)]
h(s,x(s-\rho_2(s)))ds\big\|^{p}_H\\
&\quad +7^{p-1}E\big\| \int_
{\tau_1}^{\tau_2}\mathcal{S}_{\alpha} (\tau_2-s)
h(s,x(s-\rho_2(s)))ds\big\|^{p}_H\\
&\quad +7^{p-1}E\big\|
\int_{t_k}^{\tau_1-\varepsilon}[
\mathcal{S}_{\alpha}(\tau_2-s)-\mathcal{S}_{\alpha}(\tau_1-s)]
f(s,x(s-\rho_{3}(s)))dw(s)\big\|^{p}_H
\\
&\quad  +7^{p-1}E\big\|
\int_{\tau_1-\varepsilon}^{\tau_1}
[\mathcal{S}_{\alpha}(\tau_2-s)-\mathcal{S}_{\alpha}(\tau_1-s)]
f(s,x(s-\rho_{3}(s)))dw(s)\big\|^{p}_H\\
&\quad +7^{p-1}E\big\| \int_
{\tau_1}^{\tau_2}\mathcal{S}_{\alpha} (\tau_2-s)
f(s,x(s-\rho_{3}(s)))dw(s)\big\|^{p}_H\\
& \leq
4^{p-1}E\|[\mathcal{R}_{\alpha}(\tau_2)-\mathcal{R}_{\alpha}(\tau_1)]
[x(t_k^{-})+I_k(x(t^{-}_k))]\|^{p}_H\\
&\quad +7^{p-1}T^{p}
\Theta_{h}(r^{*})\int_{t_k}^{\tau_1-\varepsilon}\|\mathcal{S}_{\alpha}(\tau_2-s)
-\mathcal{S}_{\alpha}(\tau_1-s)\|_{L(H)}^{p}
m_{h}(s)ds\\
&\quad  +14^{p-1}M^{p} \Theta_{h}(r^{*})\delta^{1-p}
\int_{\tau_1-\varepsilon}^{\tau_1} e^{-\delta(\tau_1-
s)}m_{h}(s)ds \\
&\quad + 7^{p-1}M^{p} \Theta_{h}(r^{*})\delta^{1-p}\int_
{\tau_1}^{\tau_2}e^{-\delta (\tau_2-s)}m_{h}(s)ds\\
&\quad +4^{p-1}C_{p}
\Theta_{f}(r^{*})\Big[\int_{t_k}^{\tau_1-\varepsilon}[\|
\mathcal{S}_{\alpha}(\tau_2-s)-\mathcal{S}_{\alpha}(\tau_1-s)\|_{L(H)}^{p}
m_{f}(s)]^{2/p}ds\Bigr]^{p/2}\\
&\quad  +8^{p-1}C_{p}M^{p}
\Theta_{f}(r^{*})\Big[\frac{2\delta(p-1)}{p-2}\Bigr]^{1-p/2}
\int_{\tau_1-\varepsilon}^{\tau_1} e^{-\delta(t-
s)}m_{f}(s)ds \\
&\quad + 4^{p-1}C_{p}M^{p}
\Theta_{f}(r^{*})\Big[\frac{2\delta(p-1)}{p-2}\Bigr]^{1-p/2}\int_
{\tau_1}^{\tau_2}e^{-\delta s}m_{f}(s)ds.
\end{align*}
The fact of $I_k$, $k = 1, 2,\ldots,m$, are completely continuous in
$H$ and the compactness of $\mathcal{R}_{\alpha}(t)$,
$\mathcal{S}_{\alpha}(t)$ for $ t> 0$ imply the continuity in the
uniform operator topology.  So, as $\tau_2-\tau_1\to0$,
with $\varepsilon$ is sufficiently small, the right-hand side of the
above inequality is independent of $x\in B_{r}$ and tends to zero.
 The equicontinuities
for the cases $ \tau_1 < \tau_2 \leq 0$ or
$ \tau_1\leq 0 \leq \tau_2 \leq T$ are very simple.
Thus the set $\{\Psi_2x: x\in B_{r}\}$ is equicontinuous.
\medskip

\noindent\textbf{Step 5.} The set $ W(t)=\{(\Psi_2x) (t) : x\in B_{r}\}$ is
relatively compact in $H$.
To this end, we decompose  $\Psi_2$ by  $\Psi_2 = \Gamma_1 +
\Gamma_2$, where
\[
(\Gamma_1 x)(t)= \begin{cases}
  \int_0^tS_{\alpha}(t-s)h(s,x(s-\rho_2(s)))ds\\
 +\int_0^tS_{\alpha}(t-s)f(s,x(s-\rho_{3}(s)))dw(s),
& t\in [0,t_1], \\[4pt]
\int_{t_1}^tS_{\alpha}(t-s)h(s,x(s-\rho_2(s)))ds \\
+\int_{t_1}^tS_{\alpha}(t-s)f(s,x(s-\rho_{3}(s)))dw(s),
&   t\in (t_1,t_2], \\
\dots\\
\int_{t_{m}}^tS_{\alpha}(t-s)h(s,x(s-\rho_2(s)))ds\\
+\int_{t_{m}}^tS_{\alpha}(t-s)f(s,x(s-\rho_{3}(s)))dw(s),
&  t\in (t_{m},T],
\end{cases}
\]
and
\[
(\Gamma_2 x)(t)= \begin{cases}
 \mathcal{R}_{\alpha}(t)\varphi(0), & t\in [0,t_1], \\
\mathcal{R}_{\alpha}(t-t_1)[x(t_1^{-})+I_1(x(t^{-}_1))], & t\in (t_1,t_2], \\
\dots\\
\mathcal{R}_{\alpha}(t-t_{m})[x(t_{m}^{-})+I_{m}(x(t^{-}_{m}))],
&t\in (t_{m},T].
\end{cases}
\]

We now prove that $\Gamma_1(B_{r})(t)=\{(\Gamma_1x)(t): x\in
B_{r}\}$ is relatively compact for every $t\in [0,T]$.
 Let $0<t\leq s \leq t_1$  be fixed and let
$\varepsilon$ be a real number satisfying $0 < \varepsilon < t$. For
$x\in B_{r}$, we define
\begin{align*}
(\Gamma_1^{\varepsilon} x)(t)(t)
&= \int_0^{t-\varepsilon}
\mathcal{S}_{\alpha}(t-s)h(s,x(s-\rho_2(s)))ds\\
&\quad +\int_0^{t-\varepsilon}
\mathcal{S}_{\alpha}(t-s)f(s,x(s-\rho_{3}(s)))dw(s).
\end{align*}
Using the compactness of $ \mathcal{S}_{\alpha}(t)$ for
$t >0$, we deduce that  the set
 $U_{\varepsilon}(t)=\{(\Gamma_1^{\varepsilon} x)(t) : x\in B_{r}\}$
is relatively compact in $H$ for every $\varepsilon, 0<\varepsilon<t$.
Moreover, by using H\"{o}lder's inequality, we have for every $x\in B_{r}$
\begin{align*}
&E\|(\Gamma_1 x)(t)(t)-(\Gamma_1^{\varepsilon}
x)(t)(t)\|^{p}_H \\
&\leq 2^{p-1}E\big\|\int_{t-\varepsilon}^{t}
\mathcal{S}_{\alpha}(t-s)h(s,x(s-\rho_2(s)))ds\big\|^{p}_H\\
&\quad +2^{p-1}E\big\|\int_{t-\varepsilon}^{t}
\mathcal{S}_{\alpha}(t-s)f(s,x(s-\rho_{3}(s)))dw(s)\big\|^{p}_H\\
&\leq 2^{p-1}M^{p}E\Big[\int_{t-\varepsilon}^{t}
e^{-\delta(t-s)}\| h(s,x(s-\rho_2(s)))\|_Hds\Big]^{p}\\
&\quad + 2^{p-1}C_{p} M^{p}\Big[\int_{t-\varepsilon}^{t}
 [e^{-p\delta (t-s)}E\| f(s,x(s-\rho_{3}(s)))\|^{p}_H]^{2/p}ds\Bigr]^{p/2}\\
&\leq 2^{p-1}M^{p}\delta^{1-p}\int_{t-\varepsilon}^{t}
e^{-\delta(t-s)}m_{h}(s)\Theta_{h}(E\|
x(s-\rho_2(s))\|^{p}_H)ds\\
&\quad +2^{p-1}C_{p} M^{p}\Big[\int_{t-\varepsilon}^{t} [
e^{-p\delta (t-s)}m_{f}(s)\Theta_{f}(E\|
x(s-\rho_{3}(s))\|^{p}_H)]^{2/p}ds\Bigr]^{p/2}\\
 &\leq 2^{p-1}M^{p}\delta^{1-p}\Theta_{h}(r^{*})\int_{t-\varepsilon}^{t}
e^{-\delta(t-s)}m_{h}(s)ds\\
&\quad +2^{p-1}C_{p} M^{p}\Theta_{f}(r^{*})
\Big[\frac{2\delta(p-1)}{p-2}\Bigr]^{1-p/2}\int_{t-\varepsilon}^te^{-\delta(t-
s)}m_{f}(s)ds.
\end{align*}
Similarly, for any $t\in (t_k, t_{k+1}]$, $k = 1, \ldots ,m$. Let
$t_k<t\leq s \leq t_{k+1}$  be fixed and let $\varepsilon$ be a
real number satisfying $0 < \varepsilon < t$. For $x\in B_{r}$, we
define
\[
(\Gamma_1^{\varepsilon} x)(t)=\int_{t_k}^{t-\varepsilon}
\mathcal{S}_{\alpha}(t-s)h(s,x(s-\rho_2(s)))ds+
\int_{t_k}^{t-\varepsilon}
\mathcal{S}_{\alpha}(t-s)f(s,x(s-\rho_{3}(s)))dw(s).
\]
 Using the compactness of
$ \mathcal{S}_{\alpha}(t)$ for $t>0$, we deduce that  the set
$U_{\varepsilon}(t)=\{(\Gamma_1^{\varepsilon} x)(t) : x\in B_{r}\}$
 is relatively compact in $H$ for every $\varepsilon, 0<\varepsilon<t$.
 Moreover, for every $x\in B_{r}$ we have
\begin{align*}
&E\|(\Gamma_1 x)(t)(t)-(\Gamma_1^{\varepsilon}
x)(t)(t)\|^{p}_H \\
&\leq 2^{p-1}E\big\|\int_{t-\varepsilon}^{t}
\mathcal{S}_{\alpha}(t-s)h(s,x(s-\rho_2(s)))ds\big\|^{p}_H\\
&\quad +2^{p-1}E\big\|\int_{t-\varepsilon}^{t}
\mathcal{S}_{\alpha}(t-s)f(s,x(s-\rho_{3}(s)))dw(s)\big\|^{p}_H\\
& \leq 2^{p-1}M^{p}\delta^{1-p}\Theta_{h}(r^{*})\int_{t-\varepsilon}^{t}
e^{-\delta(t-s)}m_{h}(s)ds\\
&\quad +2^{p-1}C_{p} M^{p}\Theta_{f}(r^{*})
\Big[\frac{2\delta(p-1)}{p-2}\Bigr]^{1-p/2}\int_{t-\varepsilon}^te^{-\delta(t-
s)}m_{f}(s)ds.
\end{align*}
 There are relatively compact sets arbitrarily close to the set
$W(t)=\{(\Gamma_1x) (t) : x\in B_{r}\}$, and $W(t)$ is a
relatively compact in $H$. It is easy to see that
$\Gamma_1(B_{r})$ is uniformly bounded.
Since we have shown $\Phi_1( B_{r})$
is equicontinuous collection, by the Arzel\'{a}-Ascoli theorem it
suffices to show that $\Gamma_1$ maps $ B_{r}$ into a relatively
compact set in $H$.

Next, we show that $\Gamma_2(B_{r})(t)=\{(\Gamma_2x)(t): x\in
B_{r}\}$ is relatively compact for every $t\in [0,T]$.
For all $t \in [0, t_1]$, since $(\Gamma_2x)(t) =
\mathcal{R}_{\alpha}(t)\varphi(0)$, by (H1), it follows that
$\{(\Gamma_2x)(t) : t \in [0,t_1], x \in B_{r} \}$ is a compact
subset of $H$. On the other hand, for $t\in (t_k, t_{k+1}], k= 1,
\ldots ,m$, and $x \in B_{r}$, and that the interval $[0, T]$ is
divided into finite subintervals by $t_k$, $k= 1, 2, \ldots , m$,
so that we  need to prove that
\[
 W=\{\mathcal{R}_{\alpha}(t-t_k)[x(t_k^{-})+I_k(x(t^{-}_k))],
 \quad   t\in [t_k,t_{k+1}], \; x \in B_{r}\}
\]
is relatively compact in $C([t_k, t_{k+1}], H)$. In
fact,  from (H1) and (H5), it follows that the set
$\{\mathcal{R}_{\alpha}(t-t_k)[x(t_k^{-})+I_k(x(t^{-}_k))],
\; x \in B_{r}\}$   is relatively compact in $H$, for all
$t\in [t_k, t_{k+1}]$, $ k= 1, \ldots ,m$. Also, we see that the
functions in $W$ are equicontinuous due to the compactness of
$I_k$ and the strong continuity of the  operator
$\mathcal{R}_{\alpha}(t)$, for all $t \in [0, T]$. Now an
application of the Arzel\'{a}-Ascoli theorem justifies the
relatively compactness of $W$. Therefore,  we conclude that operator
$\Gamma_2$  is also a compact map.
\medskip

\noindent\textbf{Step 6.} We shall show the set
$\Upsilon=\{x\in\mathbb{Y}:\lambda
\Psi_1(\frac{x}{\lambda})+\lambda\Psi_2(x)=x \text{ for some }
 \lambda\in (0, 1) \}$ is bounded on $[0,T]$.
To do this, we consider the  nonlinear operator equation
\begin{equation} \label{e3.10}
x(t)=\lambda \Psi x(t), \ 0 < \lambda < 1,
\end{equation}
where $\Psi$ is already defined.
Next  we  gives a priori estimate for the solution of the above
equation. Indeed,  let $x \in\mathbb{Y}$ be a possible solution of
$x=\lambda \Psi(x)$ for some $0 < \lambda < 1$. This implies by
\eqref{e3.10} that for each $t\in [0,T]$
 we have
\begin{equation} \label{e3.11}
 x(t)= \begin{cases}
 \lambda\mathcal{R}_{\alpha}(t)[\varphi(0)- g(0,\varphi(-\rho_1(0)))]
 + \lambda g(t,x(t-\rho_1(t)))\\
+\lambda\int_0^t
\mathcal{S}_{\alpha}(t-s)h(s,x(s-\rho_2(s)))ds
 \\
+\lambda\int_0^t
\mathcal{S}_{\alpha}(t-s)f(s,x(s-\rho_{3}(s)))dw(s),
    & t\in [0,t_1], \\[4pt]
 \lambda\mathcal{R}_{\alpha}(t-t_1)[x(t_1^{-})+I_1(x(t^{-}_1))-  g(t_1,x(t_1^{+}-\rho_1(t_1^{+})))]\\
+\lambda g(t,x(t-\rho_1(t)))+\lambda\int_{t_1}^t
\mathcal{S}_{\alpha}(t-s)h(s,x(s-\rho_2(s)))ds\\
+\lambda\int_{t_1}^t
\mathcal{S}_{\alpha}(t-s)f(s,x(s-\rho_{3}(s)))dw(s),
& t\in (t_1,t_2], \\
\dots\\
 \lambda\mathcal{R}_{\alpha}(t-t_{m})[x(t_{m}^{-})+I_{m}(x(t^{-}_{m}))-  g(t_{m},x(t_{m}^{+}-\rho_1(t_{m}^{+})))]\\
+\lambda g(t,x(t-\rho_1(t)))+\lambda\int_{t_{m}}^t
\mathcal{S}_{\alpha}(t-s)h(s,x(s-\rho_2(s)))ds\\
+\lambda\int_{t_{m}}^t
\mathcal{S}_{\alpha}(t-s)f(s,x(s-\rho_{3}(s)))dw(s),
 & t\in (t_{m},T].
\end{cases}
\end{equation}
 By using H\"{o}lder's
inequality and Lemma \ref{lem2.6},  we have for $t \in [0,t_1]$
\begin{align*}
&E\|x(t)\|^{p}_H\\
&\leq4^{p-1}E\|\mathcal{R}_{\alpha}(t)[\varphi(0)-g(0,\varphi(-\rho_1(0)))]\|^{p}_H
 + 4^{p-1}E\| g(t,x(t-\rho_1(t)))\|^{p}_H\\
&\quad+ 4^{p-1}E\big\|\int_0^t
\mathcal{S}_{\alpha}(t-s)h(s,x(s-\rho_2(s)))ds\big\|^{p}_H\\
&\quad+ 4^{p-1}E\big\|\int_0^t
\mathcal{S}_{\alpha}(t-s)f(s,x(s-\rho_{3}(s)))dw(s)\big\|^{p}_H\\
&\leq8^{p-1}
 M^{p}[E\| \varphi(0)
\|^{p}_H
+E\| g(0,\varphi(-\rho_1(0)))\|^{p}_H]
+ 4^{p-1}E\| g(t,x(t-\rho_1(t)))\|^{p}_H\\
&\quad+ 4^{p-1}M^{p}E\Big[\int_{0}^t e^{-\delta(t-
s)}\| h(s,x(s-\rho_2(s)))\|_Hds\Bigr]^{p}\\
&\quad+ 4^{p-1}\Big[\int_{0}^t[ e^{-p\delta(t-
s)}(E\| f(s,x(s-\rho_{3}(s)))\|^{p}_H)]^{2/p}ds\Bigr]^{p/2}\\
 &\leq 8^{p-1}M^{p}[E\| \varphi(0)
\|^{p}_H+L_{g}E\| \varphi(-\rho_1(0))
\|^{p}_H]+4^{p-1}L_{g}E\| x(t-\rho_1(t)))
\|^{p}_H\\
&\quad + 4^{p-1}M^{p}\Big[\int_0^t e^{-(p\delta/p-1)(t-
s)}ds\Bigr]^{p-1}\int_0^tm_{h}(s)\Theta_{h}(E\|
x(s-\rho_2(s))
\|^{p}_H)ds\\
&\quad + 4^{p-1}C_{p}M^{p}\Big[\int_0^t [e^{-p\delta(t-
s)}m_{f}(s)\Theta_{f}(E\| x(s-\rho_{3}(s))
\|^{p}_H)]^{2/p}ds\Bigr]^{p/2}.
 \end{align*}
Similarly, for any $t\in (t_k, t_{k+1}]$, $k = 1, \ldots ,m$, we
have
\begin{align*}
&E\| x(t)\|^{p}_H\\
&\leq4^{p-1}E\| \mathcal{R}_{\alpha}(t-t_k)[x(t_k^{-})+I_k(x(t^{-}_k))-
g(t_k,x(t_k^{+}-\rho_1(t_k^{+})))]\|^{p}_H\\
&\quad + 4^{p-1}E\| g(t,x(t-\rho_1(t)))\|^{p}_H\\
 &\quad+ 4^{p-1}E\big\|\int_{t_k}^t
\mathcal{S}_{\alpha}(t-s)h(s,x(s-\rho_2(s)))ds\big\|^{p}_H\\
&\quad+ 4^{p-1}E\big\|\int_{t_k}^t
\mathcal{S}_{\alpha}(t-s)f(s,x(s-\rho_{3}(s)))dw(s)\big\|^{p}_H\\
&\leq 12^{p-1}  M^{p}[E\| x(t_k^{-})
\|^{p}_H +E\| I_k(x(t^{-}_k)\|^{p}_H
+E\| g(t_k,x(t_k^{+}-\rho_1(t_k^{+})))\|^{p}_H]\\
&\quad + 4^{p-1}E\| g(t,x(t-\rho_1(t)))\|^{p}_H\\
&\quad+ 4^{p-1}M^{p}\Big[\int_{t_k}^t e^{-\delta(t- s)}\|
h(s,x(s-\rho_2(s)))\|_Hds\Bigr]^{p}\\
&\quad+ 4^{p-1}\Big[\int_{t_k}^t [e^{-p\delta(t-
s)}(E\| f(s,x(s-\rho_{3}(s)))\|^{p}_H)]^{2/p}ds\Bigr]^{p/2}\\
&\leq 12^{p-1}  M^{p}[E\| x(t_k^{-})
\|^{p}_H +d^1_kE\|
x(t^{-}_k)\|^{p}_H+d^2_k +L_{g}E\|
x(t_k^{+}-\rho_1(t_k^{+})) \|^{p}_H]\\
&\quad+4^{p-1}L_{g}E\| x(t-\rho_1(t))
\|^{p}_H\\
&\quad+ 4^{p-1}M^{p}\Big[\int_{t_k}^t e^{-(p\delta/p-1)(t-
s)}ds\Bigr]^{p-1}\int_{t_k}^t m_{h}(s)\Theta_{h}(E\|
x(s-\rho_2(s)) \|^{p}_H)ds\\
&\quad+ 4^{p-1}C_{p}M^{p}\Big[\int_{t_k}^t [e^{-p\delta(t-
s)}m_{f}(s)\Theta_{f}(E\| x(s-\rho_{3}(s)))
\|^{p}_H)]^{2/p}ds\Bigr]^{p/2}.
 \end{align*}
Then, for all $t\in[0,T]$, we have
\begin{align*}
&E\| x(t)\|^{p}_H\\
&\leq\widetilde{M}+ 12^{p-1}
 M^{p}[E\| x(t_k^{-})
\|^{p}_H+d^1_kE\|x(t^{-}_k)\|^{p}_H\\
&\quad +L_{g}E\| x(t_k^{+}-\rho_1(t_k^{+}))
\|^{p}_H]+4^{p-1}L_{g}E\| x(s-\rho_1(s)) \|^{p}_H \\
&\quad + 4^{p-1}M^{p}\Big[\int_0^t e^{-(p\delta/p-1)(t-
s)}ds\Bigr]^{p-1}\int_0^tm_{h}(s)\Theta_{h}(E\|
x(s-\rho_2(s)) \|^{p}_H)ds\\
&\quad+ 4^{p-1}C_{p}M^{p}\Big[\int_0^t [e^{-p\delta(t-
s)}m_{f}(s)\Theta_{f}(E\| x(s-\rho_{3}(s))
\|^{p}_H)]^{2/p}ds\Bigr]^{p/2},
 \end{align*}
where 
\[
\widetilde{M}=\max\{8^{p-1}M^{p}[E\| \varphi(0)
\|^{p}_H+L_{g}E\| \varphi(-\rho_1(0))\|^{p}_H],12^{p-1}
 M^{p}\tilde{d}\},
\]
$\tilde{d}=\max_{1\leq k\leq m}d^2_k$.
By the definition of $\mathbb{Y}$, it follows that
\[
E\| x(s-\rho_{i}(s))\|^{p}_H \leq 2^{p-1}\|
\varphi\|^{p}_{\mathfrak{B}}+2^{p-1}\sup_{s\in
[0,t]}\| x(s)\|^{p}_H, i=1,2,3.
\]
If $\mu(t) = 2^{p-1}\|
\varphi\|^{p}_{\mathfrak{B}}+2^{p-1}\sup_{s\in
[0,t]}E\| x(s)\|^{p}_H$, we obtain that
\begin{align*}
\mu(t)&\leq  2^{p-1}\|
\varphi\|^{p}_{\mathfrak{B}}+2^{p-1}\widetilde{M}+ 12^{p-1}
 M^{p}[\mu(t) +d^1_k\mu(t)+2^{p-1}
 L_{g}\mu(t)]+8^{p-1}L_{g}\mu(t)\\
&\quad + 8^{p-1}M^{p}\Big[\int_0^t e^{-(p\delta/p-1)(t-
s)}ds\Bigr]^{p-1}\int_0^tm_{h}(s)\Theta_{h}(\mu(s))ds\\
&\quad + 8^{p-1}C_{p}M^{p}\Big[\int_0^t [e^{-p\delta(t-
s)}m_{f}(s)\Theta_{f}(\mu(s))]^{2/p}ds\Bigr]^{p/2}.
 \end{align*}
Note that
\begin{align*}
&\Big[\int_0^t [e^{-p\delta(t-
s)}m_{f}(s)\Theta_{f}(\mu(s))]^{2/p}ds\Bigr]^{p/2} \\
&\leq \Big[\int_0^te^{-[\frac{2p}{p-2}]\delta(t-
s)}ds\Bigr]^{p/2-1}\int_0^tm_{f}(s)\Theta_{f}(\mu(s))ds\\
&\leq [\frac{2\delta p}{p-2}]^{1-p/2}
 \int_0^tm_{f}(s)\Theta_{f}(\mu(s))ds.
\end{align*}
So, we obtain
\begin{align*}
\mu(t)&\leq 2^{p-1}\|
\varphi\|^{p}_{\mathfrak{B}}+2^{p-1}\widetilde{M}+ 12^{p-1}
 M^{p}[\mu(t) +d^1_k\mu(t) +2^{p-1}L_{g}\mu(t)]\\
&\quad+8^{p-1}L_{g}\mu(t)+ 8^{p-1}M^{p}[\frac{p\delta
}{p-1}]^{1-p}\int_0^tm_{h}(s)\Theta_{h}(\mu(s))ds\\
&\quad+ 8^{p-1}C_{p}M^{p}[\frac{2\delta p}{p-2}]^{1-p/2}
\int_0^tm_{f}(s)\Theta_{f}(\mu(s))ds.
 \end{align*}
Since $\widetilde{L}=\max_{1\leq k\leq m}\{12^{p-1}
 M^{p}(1+d^1_k +2^{p-1}L_{g})+8^{p-1}L_{g}\}<1$, we obtain
\begin{align*}
\mu(t)
&\leq \frac{1}{1-\widetilde{L}}\Big[2^{p-1}\|
\varphi\|^{p}_{\mathfrak{B}}+2^{p-1}\widetilde{M}+8^{p-1}M^{p}
[\frac{p\delta}{p-1}]^{1-p}\int_0^tm_{h}(s)\Theta_{h}(\mu(s))ds\\
&\quad+ 8^{p-1}C_{p}M^{p}[\frac{2\delta
p}{p-2}]^{1-p/2}\int_0^tm_{f}(s)\Theta_{f}(\mu(s))ds\Bigr].
 \end{align*}
Denoting by $\zeta(t )$ the right-hand side of the above inequality,
we have
\begin{gather*}
\mu(t)\leq \zeta(t) \quad \text{for all } t\in [0,T],
\\
\zeta(0) =\frac{1}{1-\widetilde{L}}[2^{p-1}\|
\varphi\|^{p}_{\mathfrak{B}}+2^{p-1}\widetilde{M}],
\\
\begin{aligned}
\zeta'(t)
&=\frac{1}{1-\widetilde{L}}\Big[8^{p-1}M^{p}[\frac{p\delta }{p-1}]^{1-p}m_{h}(t)\Theta_{h}(\mu(t))\\
&\quad+ 8^{p-1}C_{p}M^{p}[\frac{2\delta p}{p-2}]^{1-p/2}m_{f}(t)\Theta_{f}(\mu(t))\Bigr]\\
&\leq \frac{1}{1-\widetilde{L}}
\Big[8^{p-1}M^{p}[\frac{p\delta }{p-1}]^{1-p}m_{h}(t)\Theta_{h}(\zeta(t))\\
&\quad+8^{p-1}C_{p}M^{p}[\frac{2\delta p}{p-2}]^{1-p/2}m_{f}(t)
 \Theta_{f}(\zeta(t))\Bigr]\\
&\leq m^{*}(t)[\Theta_{h}(\zeta(t))+\Theta_{f}(\zeta(t))],
 \end{aligned}
\end{gather*}
where
\begin{align*}
m^{*}(t)=\max\Big\{& \frac{1}{1-\widetilde{L}}
8^{p-1}M^{p}[\frac{p\delta }{p-1}]^{1-p}m_{h}(t),\\
&\frac{1}{1-\widetilde{L}}8^{p-1}C_{p}M^{p}
[\frac{2\delta p}{p-2}]^{1-p/2}m_{f}(t)\Bigr\}.
\end{align*}
This implies that
\[
\int_{\zeta(0)}^{\zeta(t)}\frac{du}{\Theta_{h}(u)+\Theta_{f}(u)}\leq\int_{0}^{T}
m^{*}(s)ds<\infty.
\]
This inequality shows that there is a constant $\widetilde{K}$ such that
$\xi(t)\leq \widetilde{K}, t\in[0,T]$, and hence $\|
x\|_{\mathbb{Y}}\leq\zeta(t)\leq \widetilde{K}$, where
$\widetilde{K}$ depends only on $M,\delta,p,C_{p}$ and on the
functions $m_{f}(\cdot),\Theta_{f}(\cdot)$. This indicates that
$\Upsilon$ is bounded on $[0,T]$. Consequently, by  Lemma \ref{lem2.7}, we
deduce that $\Psi_1+\Psi_2$ has a fixed point
$x(\cdot)\in\mathbb{Y}$, which is a mild solution of the system
\eqref{e1.1}-\eqref{e1.3} with $x(s)=\varphi(s)$ on $[\tilde{m}(0), 0]$  and
$E\| x(t)\|_H^{p} \to 0$ as $t \to\infty$.
This shows that the asymptotic stability of the
mild solution of \eqref{e1.1}-\eqref{e1.3}. In fact, let $\varepsilon > 0 $ be
given and choose $\tilde{\delta} > 0$ such that $\tilde{\delta}  <
\varepsilon$ and satisfies
\[
[16^{p-1}M^{p}+8^{p-1}M^{p}[\delta^{1-p}L_{h}
+C_{p}(\frac{2\delta(p-1)}{p-2})^{1-2/p}L_{f}]]\tilde{\delta}
+\widetilde{L}\varepsilon <\varepsilon.
\]
If $x(t) = x(t, \varphi)$
is mild solution of \eqref{e1.1}-\eqref{e1.3}, with
$\|\varphi\|^{p}_{\mathcal{B}}+L_{g}E\|\varphi(-\rho_1(0))\|^{p}_H
<\tilde{\delta}$, then $(\Psi x)(t) = x(t) $ and satisfies
$E\| x(t)\|_H^{p} < \varepsilon$ for every $t \geq
0. $ Notice that $E\| x(t)\|_H^{p} < \varepsilon$ on
$t \in[\tilde{m}(0), 0]$. If there exists $\tilde{t}$ such that
$E\| x(\tilde{t})\|_H^{p} = \varepsilon$ and
$E\| x(s)\|_H^{p} < \varepsilon$ for $s
\in[\tilde{m}(0), \tilde{t}]$. Then \eqref{e3.4} show that
\begin{align*}
&E\| x(t)\|^{p}_H\\
&\leq \Big[16^{p-1}M^{p}e^{-p\delta\tilde{t}}
+8^{p-1}M^{p}\big[\delta^{1-p}L_{h}+C_{p}
\big(\frac{2\delta(p-1)}{p-2}\bigr)^{1-2/p}L_{f}\bigr]\Bigr]\tilde{\delta}
+\widetilde{L}\varepsilon <\varepsilon,
\end{align*}
which contradicts the definition of $\tilde{t}$. Therefore,
 the mild solution of \eqref{e1.1}-\eqref{e1.3} is asymptotically
stable in $p$-th moment. The proof is complete.
\end{proof}

\begin{remark} \label{rmk3.1} \rm
It is well known that the study on nonlocal problems are motivated
by physical problems. For example, it is used to determine the
unknown physical parameters in some inverse heat conduction problems
\cite{d3}. Due to the importance of nonlocal  conditions in different
fields, there has been an increasing interest in study of the
fractional  impulsive stochastic differential equations involving
nonlocal conditions (see \cite{s5}). In this remark, we will try to make
some simulations about the above results and study the asymptotical
stability in $p$-th moment of mild solutions to  a class of
fractional  impulsive partial neutral stochastic
 integro-differential equations with nonlocal conditions in
Hilbert spaces
 \begin{gather}
\begin{gathered}
^{c}D^{\alpha}N(x(t))= A
N(x(t))+\int_0^tR(t-s)N(x(s))ds+ h(t,x(t)) + f(t,x(t))\frac{dw(t)}{dt}, \\
  t\geq0, t\neq t_k,
\end{gathered} \label{e3.12}\\
 \label{e3.13}
x_{0}+G(x)=x_{0}, \quad x'(0)=0, \\
 \label{e3.14}
 \Delta x(t_k)=I_k(x(t^{-}_k)), \quad t =t_k,\; k=1,\ldots, m,
\end{gather}
where  $ ^{c}D^{\alpha}, A, Q,w$ are defined as in \eqref{e1.1}-\eqref{e1.3}.
 Here
$ N(x) = x(0) + g(t,x),x \in H$, and $g : [0,\infty) \times
H\to H, f:[0,\infty)\times H\to L(K, H)$, are all
Borel measurable; $ I_k:H\to H (k=1,\ldots,m),
G:\mathbb{Y}\to H $ are given functions, where $\mathbb{Y}$
be the space of all $\mathcal{F}_{0}$-adapted process
$\psi(t,\tilde{w}):[0,\infty)\times\Omega\to\mathbb{R}$
which is almost certainly continuous in $t$ for fixed
$\tilde{w}\in\Omega$. Moreover $\psi(0,\tilde{w})=x_{0}$ and
$E\|\psi(t,\tilde{w})\|^{p}_H\to0$ as
$t\to\infty$. Also $\mathbb{Y}$ is a Banach space when it is
equipped with a norm defined by
$$
\|\psi\|_{\mathbb{Y}}=\sup_{t\geq0}E\|\psi(t)\|^{p}_H.
$$
To prove the Asymptotic stability result, we assume that the
following condition holds.
\begin{itemize}
\item[(H6)]  The functions $G: \mathbb{Y}\to H$ are completely continuous
and that there is a constant $c $ such that $E\|
G(x)\|^{p}_H\leq c $ for every $x \in \mathbb{Y}$.
\end{itemize}
Further, the mild solution of the Fractional impulsive stochastic
 system \eqref{e3.12}-\eqref{e3.14} can be written as
\[
 x(t)= \begin{cases}
 \mathcal{R}_{\alpha}(t)[x_{0}-G(x)-g(0,x(0))]
 + g(t,x(t))\\
+\int_0^t \mathcal{S}_{\alpha}(t-s)h(s,x(s))ds\\
+\int_0^t \mathcal{S}_{\alpha}(t-s)f(s,x(s))dw(s),
    & t\in [0,t_1], \\[4pt]
 \mathcal{R}_{\alpha}(t-t_1)[x(t_1^{-})+I_1(x(t^{-}_1))-  g(t_1,x(t^{+}_1))]\\
+g(t,x(t))+\int_{t_1}^t
\mathcal{S}_{\alpha}(t-s)h(s,x(s))ds\\
+\int_{t_1}^t
\mathcal{S}_{\alpha}(t-s)f(s,x(s))dw(s), \quad& t\in (t_1,t_2], \\
\dots\\
 \mathcal{R}_{\alpha}(t-t_{m})[x(t_{m}^{-})+I_{m}(x(t^{-}_{m}))
-  g(t_{m},x(t^{+}_{m}))]\\
 +  g(t,x(t)) +\int_{t_{m}}^t
\mathcal{S}_{\alpha}(t-s)h(s,x(s))ds\\
 +\int_{t_{m}}^t \mathcal{S}_{\alpha}(t-s)f(s,x(s))dw(s),
&  t\in (t_{m},T].
\end{cases}
\]
One can easily prove that by adopting and
employing the method used in Theorem \ref{thm3.1}, the fractional impulsive
stochastic differential equations \eqref{e3.12}-\eqref{e3.14}
 is asymptotically stable in $p$-th moment.
\end{remark}

\section{Example}

Consider the  fractional impulsive partial stochastic
neutral  integro-differential equation
\begin{gather}
\begin{aligned}
 \frac{\partial^{\alpha}N(z_t)(x)}{\partial t^{\alpha}}
&=\frac{\partial^2N(z_t)(x)}{\partial
x^2}+\int_0^t(t-s)^{\sigma}
e^{-\mu(t-s)}\frac{\partial^2N(z_t)(x)}{\partial
x^2}ds \\
 & \quad +\varsigma(t,x, z(t-\rho_2(t),x))
+\varpi(t,x, z(t-\rho_{3}(t),x))\frac{dw(t)}{dt},\\
&\quad  t\geq0, \quad 0\leq x\leq \pi,\quad t\neq t_k,
\end{aligned} \label{e4.1}
\\
 z(t,0)=z(t,\pi)=0, \quad t\geq0,  \label{e4.2}\\
 z_t (0, x ) = 0,  \quad 0\leq x\leq \pi, \label{e4.3} \\
z(\tau,x)=\varphi(\tau,x), \quad \tau\leq0,\;  0\leq x\leq \pi, \label{e4.4}\\
 \triangle  z(t_k,x)=z(t^{+}_k,x)-z(t^{-}_k,x)
=\int_{0}^{\pi}\eta_k(s,z(t_k,x))ds, \quad k=1,2,  \ldots,m, \label{e4.5}
 \end{gather}
where $(t_k)_k\in\mathbb{N}$ is a strictly increasing sequence
of positive numbers,
$D^{\alpha}_t=\frac{\partial^{\alpha}}{\partial t^{\alpha}}$  is a
Caputo fractional partial derivative of order $\alpha\in(1,2)$,
$\sigma$, and $\mu$ are positive numbers and $w(t)$ denotes a
standard cylindrical Wiener process in $H$ defined on a stochastic
space $(\Omega,\mathcal{F},P)$. In this system, $\rho_{i}(t)\in
C(\mathbb{R}^{+},\mathbb{R}^{+}),i=1,2,3$, and
$$
N(z_t)(x)=z(t,x)-\vartheta(t,x, z(t-\rho_1(t),x)).
$$
Let $H=L^2([0,\pi])$ with the norm $\|\cdot\|_H$
and define the operators $A:D(A)\subseteq H \to H$ by
$A\omega=\omega''$ with the domain
$$
D(A) :=\{\omega \in H : \omega,\omega'
\text{ are absolutely continuous, }  \omega'' \in H,
\omega(0) = \omega(\pi) = 0\}.
$$
Then
$$
A\omega = \sum_{n=1 }^{\infty} n^2\langle
\omega,\omega_n\rangle \omega_n, \quad \omega\in D(A),
$$
where $\omega_n(x) = \sqrt{\frac{2}{\pi}} \sin(nx)$, $n=1,2,\ldots$ is the
orthogonal set of eigenvectors of $A$. It is well known that $A$
generates a strongly continuous semigroup $T (t), t \geq0$ which is
compact, analytic and self-adjoint in $H$  and $A$ is sectorial of
type and (P1) is satisfied. The operator
$R(t) : D(A) \subseteq H \to H, t \geq 0, R(t)x
= t^{\sigma}e^{-\omega t}x'' $ for $x \in D(A)$. Moreover, it is easy to see
that conditions (P2) and (P3) in Section 2 are satisfied with $b(t)
= t^{\sigma}e^{-\mu t}$ and $D = C_{0}^{\infty} ([0, \pi])$, where
$C_{0}^{\infty} ([0, \pi])$ is the space of infinitely
differentiable functions that vanish at $x = 0$ and $x = \pi$.

Additionally, we will assume that
 \begin{itemize}
\item[(i)] The function
 $\vartheta : [0, \infty) \times [0,\pi] \times\mathbb{R}\to\mathbb{R}$
is continuous and there exists a positive constant $L_{\vartheta}$
such that
\[
|\vartheta(t,x,u)-\vartheta(t,x,v)|\leq L_{\vartheta}|u-v|, \quad
  t \geq 0, x \in [0, \pi], u,v\in\mathbb{R}.
\]

\item[(ii)] The function
$\varsigma: [0, \infty) \times [0,\pi] \times\mathbb{R}\to\mathbb{R}$
is continuous and there exists a positive continuous function
$m_{\varsigma}(\cdot) : \mathbb{R}\times[0, \pi] \to\mathbb{R}$ such that
\[
|\varsigma(t,x,u)|\leq m_{\varsigma}(t,x)|u|, \quad
  t \geq 0, x \in [0,\pi], u\in\mathbb{R}.
\]

\item[(iii)] The function $\vartheta : [0, \infty) \times [0,\pi] \times\mathbb{R}\to\mathbb{R}$
is continuous and there exists a positive continuous function
$m_{\varpi}(\cdot) : \mathbb{R}\times[0, \pi] \to\mathbb{R}$
such that
\[
|\varpi(t,x,u)|\leq m_{\varpi}(t,x)|u|, \quad  t \geq 0, x \in [0, \pi],
u\in\mathbb{R}.
\]

\item[(iv)] The functions $\eta_k : \mathbb{R}^2 \to\mathbb{R},k\in\mathbb{N}$,
are completely continuous and there are positive continuous
functions $L_k:[0,\pi]\to \mathbb{R}(k=1,2,\ldots,m)$ such
that
 $|\eta_k(s,u)|\leq L_k(s)|u|$, $s\in[0,\pi]$, $u\in\mathbb{R}$.
\end{itemize}

We can define  $N: H\to H, $
$g,h:[0,\infty)\times H\to H, f:[0,\infty)\times
H\to L(K,H) $ and $I_k :H\to H $ respectively by
 \begin{gather*}
N(z_t)(x)=\varphi(0,x)+g(t,z)(x), \\
g(t,z)(x)=\vartheta(t,x,z(t-\rho_1(t),x)), \\
h(t,z)(x)=\varsigma(t,x,z(t-\rho_2(t),x)), \\
f(t,z)(x)=\varpi(t,x,z(t-\rho_{3}(t),x)), \\
I_k(z)(x)=\int_{0}^{\pi}\eta_k(s,z(x))ds.
\end{gather*}
Then the problem \eqref{e4.1}--\eqref{e4.5} can be written as
\eqref{e1.1}--\eqref{e1.3}. Moreover,
 it is easy to see that
\begin{gather*}
E\| g(t,z_1)-g(t,z_2)\|^{p}_H\leq L_{g}\| z_1-z_2\|^{p}_H, \ z_1,z_2\in H,\\
E\| h(t,z)\|^{p}_H\leq m_{h}(t)\| z\|^{p}_H, z\in H,\\
E\| f(t,z)\|^{p}_H\leq m_{f}(t)\| z\|^{p}_H, z\in H,\\
E\| I_k(z)\|^{p}_H\leq d_k\| z\|^{p}_H,
z\in H, k=1,2,\ldots,m,
\end{gather*}
where $L_{g}=L^{p}_{\vartheta},m_{h}(t)=\sup_{x\in[0,\pi]}m^{p}_{\tau}(t,x)$,
$m_{f}(t)=\sup_{x\in[0,\pi]}m^{p}_{\varpi}(t,x)$,
$d_k=[\int_{0}^{\pi}L_k(s)ds]^{p}$, $k=1,2,\ldots,m$. Further, we
can impose some suitable conditions on the above-defined functions
to verify the assumptions on Theorem \ref{thm3.1}, we can conclude that
system \eqref{e4.1}-\eqref{e4.5} has at least one mild solution, then the mild
solutions  is asymptotically stable in the $p$-th mean.

\subsection*{Conclusions}

In this article, we are focused on the theory study on the
asymptotical stability in the $p$-th moment of mild solutions to  a
class of fractional  impulsive partial neutral stochastic
 integro-differential equations with state-dependent delay.
We derive some interesting sufficient conditions to
guarantee the asymptotical stability results for fractional
impulsive  stochastic evolution systems in infinite dimensional
spaces. Our techniques rely on the fractional calculus, properties
of  the $\alpha$-resolvent operator, and  Krasnoselskii-Schaefer
type fixed point theorem. Our methods not only present a new way to
study such problems under  the Lipschitz condition not required in
the paper,  but also provide
new theory results  appeared in paper previously are generalized to
the fractional stochastic systems settings and the case of
state-dependent delay with impulsive conditions. An application is
provided to illustrate the applicability of the new result.

  Our future work will try to make some the above results and study
the exponential stability in $p$-th moment of mild solutions to
fractional impulsive partial neutral stochastic
 integro-differential equations with state-dependent delay.

\subsection*{Acknowledgments}
 The authors would like to thank the editor and the anonymous reviewers
for their constructive comments and suggestions to improve the
the original manuscript.

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\end{document}