\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 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 $00$, we deduce that the set $U_{\varepsilon}(t)=\{(\Gamma_1^{\varepsilon} x)(t) : x\in B_{r}\}$ is relatively compact in $H$ for every $\varepsilon, 0<\varepsilon0$, 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 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. 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