Theory and Modern Applications

# Second-order neutral impulsive stochastic evolution equations with infinite daelay

## Abstract

In this paper, we study a class of second-order neutral impulsive stochasticevolution equations with infinite delay (SNISEEIs in short), in which theinitial value belongs to the abstract space ${\mathcal{B}}_{h}$. Sufficient conditions for the existence of themild solutions for SNISEEIs are derived by means of the Krasnoselskii-Schaeferfixed point theorem. Two examples are given to illustrate the obtainedresults.

## 1 Introduction

In this paper, we consider the second-order neutral impulsive stochastic evolutionequations with infinite delay (SNISEEIs in short) of the following form:

$\mathrm{d}\left[{x}^{\prime }\left(t\right)-g\left(t,{x}_{t}\right)\right]=\left[Ax\left(t\right)+f\left(t,{x}_{t}\right)\right]\phantom{\rule{0.2em}{0ex}}\mathrm{d}t+\sigma \left(t,{x}_{t}\right)\phantom{\rule{0.2em}{0ex}}\mathrm{d}w\left(t\right),\phantom{\rule{1em}{0ex}}t\in J:=\left[0,T\right],$
(1)
${x}_{0}=\varphi \in {\mathcal{B}}_{h},$
(2)
${x}^{\prime }\left(0\right)=\psi \in H,$
(3)
$\mathrm{△}x\left({t}_{k}\right)={I}_{k}\left({x}_{{t}_{k}}\right),\phantom{\rule{2em}{0ex}}\mathrm{△}{x}^{\prime }\left({t}_{k}\right)={\stackrel{˜}{I}}_{k}\left({x}_{{t}_{k}}\right),\phantom{\rule{1em}{0ex}}k=1,2,\dots ,m.$
(4)

Here, the state $x\left(\cdot \right)$ takes values in a separable real Hilbert spaceH with inner product $\left(\cdot ,\cdot \right)$ and norm $\parallel \cdot \parallel$, where $A:D\left(A\right)\subset H\to H$ is the infinitesimal generator of a stronglycontinuous cosine family $C\left(t\right)$ on H. The history ${x}_{t}:\left(-\mathrm{\infty },0\right]\to H$, ${x}_{t}\left(\theta \right)=x\left(t+\theta \right)$, for $t\ge 0$, belongs to the phase space ${\mathcal{B}}_{h}$. Now, we present the abstract phase space${\mathcal{B}}_{h}$. Assume that $h:\left(-\mathrm{\infty },0\right]\to \left(0,\mathrm{\infty }\right)$ is a continuous function with $l={\int }_{-\mathrm{\infty }}^{0}h\left(t\right)\phantom{\rule{0.2em}{0ex}}\mathrm{d}t<\mathrm{\infty }$. For any $a>0$, define

We endow ${\mathcal{B}}_{h}$ with the norm

then $\left({\mathcal{B}}_{h},{\parallel \cdot \parallel }_{{\mathcal{B}}_{h}}\right)$ is a Banach space . Let K be another separable Hilbert space with inner product${\left(\cdot ,\cdot \right)}_{K}$ and norm ${\parallel \cdot \parallel }_{K}$. Suppose $\left\{w\left(t\right):t\ge 0\right\}$ is a given K-valued Wiener process with afinite trace nuclear covariance operator $Q\ge 0$ defined on a complete probability space$\left(\mathrm{\Omega },\mathcal{F},P\right)$ equipped with a normal filtration${\left\{{\mathcal{F}}_{t}\right\}}_{t\ge 0}$, which is generated by the Wiener process w.We are also employing the same notation $\parallel \cdot \parallel$ for the operator norm $L\left(K;H\right)$, where $L\left(K;H\right)$ denotes the space of all bounded linear operatorsfrom K into H. Assume that $g,f:J×{\mathcal{B}}_{h}\to H$ ($i=1,2$) and $\sigma :J×{\mathcal{B}}_{h}\to {L}_{Q}\left(K,H\right)$ are appropriate mappings specified later. Here,${L}_{Q}\left(K,H\right)$ denotes the space of all Q-Hilbert-Schmidtoperators from K into H, which will be defined in the nextsection. The initial data $\varphi =\left\{\varphi \left(t\right):-\mathrm{\infty } is an ${\mathcal{F}}_{0}$-adapted, ${\mathcal{B}}_{h}$-valued stochastic process independent of the Wienerprocess w with finite second moment. ψ is an${\mathcal{F}}_{0}$-adapted, H-valued random variableindependent of the Wiener process w with finite second moment.${I}_{k}$ and ${\stackrel{˜}{I}}_{k}:{\mathcal{B}}_{h}\to H$ are appropriate functions. Moreover, let$0={t}_{0}<{t}_{1}<\cdots <{t}_{m}<{t}_{m+1}=T$, be given time points and the symbol$\mathrm{\Delta }\xi \left(t\right)$ represents the jump of the function ξat t, which is defined by $\mathrm{\Delta }\xi \left(t\right)=\xi \left({t}^{+}\right)-\xi \left({t}^{-}\right)$.

Stochastic partial differential equations (SPDEs in short) with delay have attractedgreat interest due to their applications in describing many sophisticated dynamicalsystems in physical, biological, medical and social sciences. One can see  and the references therein for details. Moreover, to describe the systemsinvolving derivatives with delay, Hale and Lunel  introduced the deterministic neutral functional differential equations,which are of great interest in theoretical and practical applications. Taking theenvironmental disturbances into account, Kolmanovskii and Myshkis  introduced the neutral stochastic functional differential equations(NSFDEs in short) and gave its applications in chemical engineering and aeroelasticity. The investigation of qualitative properties such as existence,uniqueness and stability for NSFDEs has received much attention. One can see [2, 5, 812] and the references therein. In addition, impulsive effects exist in manyevolution processes in which states are changed abruptly at certain moments of time,involved in such fields as medicine and biology, economics, bioengineering, chemicaltechnology etc. (see [13, 14] and the references therein).

On the other hand, the study of abstract deterministic second-order evolutionsequations governed by the generator of a strongly continuous cosine family wasinitiated by  and subsequently studied by [16, 17]. The second-order stochastic differential equations are the right modelin continuous time to account for integrated processes that can be made stationary.For instance, it is useful for engineers to model mechanical vibrations or charge ona capacitor or condenser subjected to white noise excitation through a second-orderstochastic differential equations. There are some interesting works that have beendone on the second-order stochastic differential equations. For example, McKibben  investigated the second-order damped functional stochastic evolutionequations. For further work on this topic, one can see Mahmudov and McKibben . Moreover, McKibben  established the existence and uniqueness of mild solutions for a class ofsecond-order neutral stochastic evolution equations with finite delay.Balasubramaniam and Muthukumar  gave the sufficient conditions for the approximate controllability of thesecond-order neutral stochastic evolution equations with infinite delay. For moredetails of second-order stochastic differential equations, we refer the reader to DaPrato  and the references therein.

To the best of our knowledge, there is no work reported in the literature aboutSNISEEIs and the aim of this paper is to close this gap. We aim to establish theexistence of the mild solutions for SNISEEIs by means of the Krasnoselskii-Schaeferfixed point theorem. Two types of stochastic nonlinear wave equations with infinitedelay and impulsive effects are provided to illustrate the obtained results.

The paper is organized as follows. In Section 2, we introduce somepreliminaries. In Section 3, we prove the existence of the mild solutions forSNISEEIs by means of the Krasnoselskii-Schaefer fixed point theorem. InSection 4, we study the continuous dependence of solutions on the initialvalues. Two examples are provided in the last section to illustrate the theory.

## 2 Preliminaries

In this section, we mention some preliminaries needed to establish our results. Fordetails as regards this section, the reader may refer to Da Prato and Zabczyk , Fattorini  and the references therein.

Let $\left(\mathrm{\Omega },\mathcal{F},P;\mathbf{F}\right)$ ($\mathbf{F}={\left\{{\mathcal{F}}_{t}\right\}}_{t\ge 0}$) be a complete filtered probability space satisfyingthat ${\mathcal{F}}_{0}$ contains all P-null sets of . AnH-valued random variable is an -measurable function$x\left(t\right):\mathrm{\Omega }\to H$ and the collection of random variables$S=\left\{x\left(t,\omega \right):\mathrm{\Omega }\to H|t\in J\right\}$ is called a stochastic process. Generally, we justwrite $x\left(t\right)$ instead of $x\left(t,\omega \right)$ and $x\left(t\right):J\to H$ in the space of S. Let${\left\{{e}_{i}\right\}}_{i=1}^{\mathrm{\infty }}$ be a complete orthonormal basis of K.Suppose that $\left\{w\left(t\right):t\ge 0\right\}$ is a cylindrical K-valued Wiener processwith a finite trace nuclear covariance operator $Q\ge 0$, denote $Tr\left(Q\right)={\sum }_{i=1}^{\mathrm{\infty }}{\lambda }_{i}=\lambda <\mathrm{\infty }$, which satisfies $Q{e}_{i}={\lambda }_{i}{e}_{i}$, with ${e}_{i}$ being a CONS of eigenvectors, and then, w.r.t. thisspectral representation of Q the driving Q-Wiener process can berepresented as $w\left(t\right)={\sum }_{i=1}^{\mathrm{\infty }}\sqrt{{\lambda }_{i}}{w}_{i}\left(t\right){e}_{i}$, where ${\left\{{w}_{i}\left(t\right)\right\}}_{i=1}^{\mathrm{\infty }}$ are mutually independent one-dimensional standardWiener processes. We assume that ${\mathcal{F}}_{t}=\sigma \left\{w\left(s\right):0\le s\le t\right\}$, which is a σ-algebra generated byw and ${\mathcal{F}}_{T}=\mathcal{F}$. Let $\psi \in L\left(K,H\right)$ and define

${\parallel \psi \parallel }_{Q}^{2}=Tr\left(\psi Q{\psi }^{\ast }\right)=\sum _{n=1}^{\mathrm{\infty }}{\parallel \sqrt{{\lambda }_{n}}\psi {e}_{n}\parallel }^{2}.$

If ${\parallel \psi \parallel }_{Q}<\mathrm{\infty }$, then ψ is called aQ-Hilbert-Schmidt operator. Let ${L}_{Q}\left(K,H\right)$ denote the space of all Q-Hilbert-Schmidtoperators $\psi :K\to H$. The completion ${L}_{Q}\left(K,H\right)$ of $L\left(K,H\right)$ with respect to the topology induced by the norm${\parallel \cdot \parallel }_{Q}$ with ${\parallel \psi \parallel }_{Q}^{2}=\left(\psi ,\psi \right)$ is a Hilbert space with the above norm topology.

The collection of all strongly measurable, square-integrable, H-valuedrandom variables, denoted by ${L}_{2}\left(\mathrm{\Omega },\mathcal{F},P;H\right)\equiv {L}_{2}\left(\mathrm{\Omega },H\right)$, is a Banach space equipped with norm${\parallel x\left(\cdot \right)\parallel }_{{L}_{2}}={\left(E{\parallel x\left(\cdot ,\omega \right)\parallel }_{H}^{2}\right)}^{1/2}$, where the expectation E is defined by$Ex={\int }_{\mathrm{\Omega }}x\left(\omega \right)\phantom{\rule{0.2em}{0ex}}\mathrm{d}P$. Let $C\left(J,{L}_{2}\left(\mathrm{\Omega },H\right)\right)$ be the Banach space of all continuous maps fromJ into ${L}_{2}\left(\mathrm{\Omega },H\right)$ satisfying the condition ${sup}_{t\in J}E{\parallel x\left(t\right)\parallel }^{2}<\mathrm{\infty }$. An important subspace is given by.

We say that a function $x:\left[\nu ,\tau \right]\to H$ is a normalized piecewise continuous function on$\left[\nu ,\tau \right]$ if x is piecewise continuous and leftcontinuous on $\left(\nu ,\tau \right]$. We denote by $\mathcal{P}\mathcal{C}\left(\left[\nu ,\tau \right];H\right)$ the space formed by the normalized piecewisecontinuous stochastic processes from $\left\{x\left(t\right):t\in \left[\nu ,\tau \right]\right\}$. In particular, we introduce the space$\mathcal{P}\mathcal{C}$ formed by all H-valued stochastic processes$\left\{x\left(t\right):t\in \left[0,T\right]\right\}$ such that x is continuous at$t\ne {t}_{k}$, $x\left({t}_{k}^{-}\right)=x\left({t}_{k}\right)$ and $x\left({t}_{k}^{+}\right)$ exists, for all $k=1,\dots ,m$. In the sequel, we always assume that$\mathcal{P}\mathcal{C}$ is endowed with the norm ${\parallel x\parallel }_{\mathcal{P}\mathcal{C}}={\left({sup}_{s\in J}E{\parallel x\left(s\right)\parallel }^{2}\right)}^{1/2}$. It is clear that $\left(\mathcal{P}\mathcal{C},{\parallel \cdot \parallel }_{\mathcal{P}\mathcal{C}}\right)$ is a Banach space.

To simplify the notations, we put ${t}_{0}=0$, ${t}_{n+1}=T$. For $x\in \mathcal{P}\mathcal{C}$, we denote ${\stackrel{˜}{x}}_{k}\in C\left(\left[{t}_{k},{t}_{k+1}\right];{L}_{2}\left(\mathrm{\Omega },H\right)\right)$, $k=0,1,\dots ,n$, given by

(5)

Moreover, for $B\subseteq \mathcal{P}\mathcal{C}$, we denote by ${\stackrel{˜}{B}}_{k}$, $k=0,1,\dots ,n$, the set ${\stackrel{˜}{B}}_{k}=\left\{{\stackrel{˜}{x}}_{k}:x\in B\right\}$.

Lemma 1 A set$B\subseteq \mathcal{P}\mathcal{C}$is relatively compact in$\mathcal{P}\mathcal{C}$, if and only if, the set${\stackrel{˜}{B}}_{k}$is relatively compact in$C\left(\left[{t}_{k},{t}_{k+1}\right];{L}_{2}\left(\mathrm{\Omega },H\right)\right)$, for every$k=0,1,\dots ,m$.

Now, we consider the space

where ${x}_{k}$ is the restriction of x to${J}_{k}=\left({t}_{k},{t}_{k+1}\right]$. Set ${\parallel \cdot \parallel }_{b}$ be a semi-norm in ${\mathcal{B}}_{b}$ defined by

${\parallel x\parallel }_{b}={\parallel {x}_{0}\parallel }_{{\mathcal{B}}_{h}}+\underset{0\le s\le T}{sup}{\left(E{\parallel x\left(s\right)\parallel }^{2}\right)}^{1/2},\phantom{\rule{1em}{0ex}}x\in {\mathcal{B}}_{b}.$

Then we have the following useful lemma appearing in .

Lemma 2 Assume that$x\in {\mathcal{B}}_{b}$, then for$t\in J$, ${x}_{t}\in {\mathcal{B}}_{h}$. Moreover, we have

$l{\left(E{\parallel x\left(t\right)\parallel }^{2}\right)}^{1/2}\le {\parallel {x}_{t}\parallel }_{{\mathcal{B}}_{h}}\le l\underset{0\le s\le t}{sup}{\left(E{\parallel x\left(s\right)\parallel }^{2}\right)}^{1/2}+{\parallel {x}_{0}\parallel }_{{\mathcal{B}}_{h}},$

where$l={\int }_{-\mathrm{\infty }}^{0}h\left(s\right)\phantom{\rule{0.2em}{0ex}}\mathrm{d}s<\mathrm{\infty }$.

Now, let us recall some facts about cosine families of operators$C\left(t\right)$ and $S\left(t\right)$ appeared in [15, 16].

Definition 3 A one parameter family $\left\{C\left(t\right):t\in \mathbb{R}\right\}\subset L\left(H,H\right)$ satisfying that

1. (i)

$C\left(0\right)=I$,

2. (ii)

$C\left(t\right)x$ is continuous in t on , for all $x\in H$,

3. (iii)

$C\left(t+s\right)+C\left(t-s\right)=2C\left(t\right)C\left(s\right)$, for all $t,s\in \mathbb{R}$,

is called a strongly continuous cosine family.

The corresponding strongly continuous sine family $\left\{S\left(t\right):t\in \mathbb{R}\right\}\subset L\left(H,H\right)$ is defined by $S\left(t\right)x={\int }_{0}^{t}C\left(s\right)x\phantom{\rule{0.2em}{0ex}}\mathrm{d}s$, $t\in \mathbb{R}$, $x\in H$.

The generator $A:H\to H$ of $\left\{C\left(t\right):t\in \mathbb{R}\right\}$ is given by $Ax=\frac{{\mathrm{d}}^{2}}{\mathrm{d}{t}^{2}}C\left(t\right)x{|}_{t=0}$ for all $x\in D\left(A\right)=\left\{x\in H:C\left(\cdot \right)x\in {C}^{2}\left(\mathbb{R};H\right)\right\}$.

It is well known that the infinitesimal generator A is a closed, denselydefined operator on H. Such cosine and corresponding sine families andtheir generators satisfy the following properties appearing in Fattorini :

Proposition 4 Suppose that A is the infinitesimal generator of a cosine family of operators$\left\{C\left(t\right):t\in \mathbb{R}\right\}$. Then we have

1. (i)

there exist ${M}^{\ast }\ge 1$ and $\alpha \ge 0$ such that $\parallel C\left(t\right)\parallel \le {M}^{\ast }{\mathrm{e}}^{\alpha |t|}$ and hence $\parallel S\left(t\right)\parallel \le {M}^{\ast }{\mathrm{e}}^{\alpha |t|}$,

2. (ii)

$A{\int }_{s}^{\stackrel{ˆ}{r}}S\left(u\right)x\phantom{\rule{0.2em}{0ex}}\mathrm{d}u=\left[C\left(\stackrel{ˆ}{r}\right)-C\left(s\right)\right]x$ for all $0\le s\le \stackrel{ˆ}{r}<\mathrm{\infty }$,

3. (iii)

there exists ${N}^{\ast }\ge 1$ such that $\parallel S\left(s\right)-S\left(\stackrel{ˆ}{r}\right)\parallel \le {N}^{\ast }{\int }_{s}^{\stackrel{ˆ}{r}}{\mathrm{e}}^{\alpha |s|}\phantom{\rule{0.2em}{0ex}}\mathrm{d}s$, for all $0\le s\le \stackrel{ˆ}{r}<\mathrm{\infty }$.

The uniform boundedness principle, together with Proposition 4(i), implies thatboth $\left\{C\left(t\right):t\in \left[0,T\right]\right\}$ and $\left\{S\left(t\right):t\in \left[0,T\right]\right\}$ are uniformly bounded.

To prove our results, we need the following Krasnoselskii-Schaefer type fixed pointtheorem appearing in .

Theorem 5 Let ${\mathrm{\Phi }}_{1}$ and ${\mathrm{\Phi }}_{2}$ be two operators of H such that

1. (i)

${\mathrm{\Phi }}_{1}$ is a strict contraction, and

2. (ii)

${\mathrm{\Phi }}_{2}$ is completely continuous.

Then either

1. (1)

the operator equation ${\mathrm{\Phi }}_{1}x+{\mathrm{\Phi }}_{2}x=x$ has a solution, or

2. (2)

the set $G=\left\{x\in H:\lambda {\mathrm{\Phi }}_{1}\left(\frac{x}{\lambda }\right)+\lambda {\mathrm{\Phi }}_{2}x=x\right\}$ is unbounded for $\lambda \in \left(0,1\right)$.

## 3 Existence result

In this section, we aim to give the existence of mild solutions for SNISEEIs (1)-(4).Firstly, let us propose the definition of the mild solution of SNISEEIs (1)-(4).

Definition 6 An ${\mathcal{F}}_{t}$-adapted stochastic process $x:\left(-\mathrm{\infty },T\right]\to H$ is called a mild solution of SNISEEIs (1)-(4) if

1. (i)

$\left\{{x}_{t}:t\in J\right\}$ is ${\mathcal{B}}_{h}$-valued and $x\left(\cdot \right){|}_{J}\in \mathcal{P}\mathcal{C}$;

2. (ii)

$x\left(t\right)\in H$ has càdlàg paths on $t\in J$ a.s. and for each $t\in J$, $x\left(t\right)$ satisfies the following integral equation:

$\begin{array}{rcl}x\left(t\right)& =& C\left(t\right)\varphi \left(0\right)+S\left(t\right)\left[\psi -g\left(0,\varphi \right)\right]+{\int }_{0}^{t}C\left(t-s\right)g\left(s,{x}_{s}\right)\phantom{\rule{0.2em}{0ex}}\mathrm{d}s\\ +{\int }_{0}^{t}S\left(t-s\right)f\left(s,{x}_{s}\right)\phantom{\rule{0.2em}{0ex}}\mathrm{d}s+{\int }_{0}^{t}S\left(t-s\right)\sigma \left(s,{x}_{s}\right)\phantom{\rule{0.2em}{0ex}}\mathrm{d}w\left(s\right)\\ +\sum _{0<{t}_{k}
3. (iii)

${x}_{0}=\varphi$, ${x}^{\prime }\left(0\right)=\psi$.

In this paper, we need the following assumptions:

(H1) The cosine family of operators $\left\{C\left(t\right):t\in \left[0,T\right]\right\}$ on H and the corresponding sine family$\left\{S\left(t\right):t\in \left[0,T\right]\right\}$ satisfy ${\parallel C\left(t\right)\parallel }^{2}\le M$, ${\parallel S\left(t\right)\parallel }^{2}\le M$, $t\ge 0$ for a positive constant M.

(H2) The function $f:J×{\mathcal{B}}_{h}\to H$ satisfies the following properties:

1. 1.

$f\left(\cdot ,\varphi \right):J\to H$ is strongly measurable for every $\varphi \in {\mathcal{B}}_{h}$;

2. 2.

$f\left(t,\cdot \right):{\mathcal{B}}_{h}\to H$ is continuous for each $t\in J$;

3. 3.

there exist an integrable function $m:J\to \left[0,\mathrm{\infty }\right)$ and a continuous nondecreasing function $\mathrm{\Psi }:\left[0,\mathrm{\infty }\right)\to \left(0,\mathrm{\infty }\right)$ such that for every $\left(t,\varphi \right)\in J×{\mathcal{B}}_{h}$, we have

$E{\parallel f\left(t,\varphi \right)\parallel }^{2}\le m\left(t\right)\mathrm{\Psi }\left({\parallel \varphi \parallel }_{{\mathcal{B}}_{h}}^{2}\right),\phantom{\rule{2em}{0ex}}\underset{\zeta \to \mathrm{\infty }}{lim inf}\frac{\mathrm{\Psi }\left(\zeta \right)}{\zeta }=\mathrm{\Lambda }<\mathrm{\infty }.$

(H3) The function $\sigma :J×{\mathcal{B}}_{h}\to {L}_{Q}\left(K,H\right)$ satisfies the following properties:

1. 1.

$\sigma \left(t,\cdot \right):{\mathcal{B}}_{h}\to {L}_{Q}\left(K,H\right)$ is continuous for almost all $t\in J$;

2. 2.

$\sigma \left(\cdot ,x\right):J\to {L}_{Q}\left(K,H\right)$ is strongly ${\mathcal{F}}_{t}$-measurable for each $x\in {\mathcal{B}}_{h}$;

3. 3.

there exists a positive constant ${L}_{\sigma }$ such that

$\begin{array}{c}E{\parallel \sigma \left(t,{x}_{1}\right)-\sigma \left(t,{x}_{2}\right)\parallel }^{2}\le {L}_{\sigma }{\parallel {x}_{1}-{x}_{2}\parallel }_{{\mathcal{B}}_{h}}^{2},\phantom{\rule{1em}{0ex}}\left(t,{x}_{i}\right)\in J×{\mathcal{B}}_{h},i=1,2;\hfill \\ E{\parallel \sigma \left(t,x\right)\parallel }^{2}\le {L}_{\sigma }\left({\parallel x\parallel }_{{\mathcal{B}}_{h}}^{2}+1\right),\phantom{\rule{1em}{0ex}}\left(t,x\right)\in J×{\mathcal{B}}_{h}.\hfill \end{array}$

(H4) The function $g:J×{\mathcal{B}}_{h}\to H$ is continuous and there exists a positive constant${L}_{g}$ such that

$\begin{array}{c}E{\parallel g\left(t,{x}_{1}\right)-g\left(t,{x}_{2}\right)\parallel }^{2}\le {L}_{g}{\parallel {x}_{1}-{x}_{2}\parallel }_{{\mathcal{B}}_{h}}^{2},\phantom{\rule{1em}{0ex}}\left(t,{x}_{i}\right)\in J×{\mathcal{B}}_{h},i=1,2;\hfill \\ E{\parallel g\left(t,x\right)\parallel }^{2}\le {L}_{g}\left({\parallel x\parallel }_{{\mathcal{B}}_{h}}^{2}+1\right),\phantom{\rule{1em}{0ex}}\left(t,x\right)\in J×{\mathcal{B}}_{h}.\hfill \end{array}$

(H5) The functions ${I}_{k}$ and ${\stackrel{˜}{I}}_{k}:{\mathcal{B}}_{h}\to H$ are continuous and there are positive constants${L}_{{I}_{k}}$, ${L}_{{\stackrel{˜}{I}}_{k}}$, $k=1,2,\dots ,m$ such that

$\begin{array}{c}E{\parallel {I}_{k}\left(x\right)-{I}_{k}\left(y\right)\parallel }^{2}\le {L}_{{I}_{k}}{\parallel x-y\parallel }_{{\mathcal{B}}_{h}}^{2},\phantom{\rule{1em}{0ex}}x,y\in {\mathcal{B}}_{h},k=1,2,\dots ,m,\hfill \\ E{\parallel {\stackrel{˜}{I}}_{k}\left(x\right)-{\stackrel{˜}{I}}_{k}\left(y\right)\parallel }^{2}\le {L}_{{\stackrel{˜}{I}}_{k}}{\parallel x-y\parallel }_{{\mathcal{B}}_{h}}^{2},\phantom{\rule{1em}{0ex}}x,y\in {\mathcal{B}}_{h},k=1,2,\dots ,m.\hfill \end{array}$

The main result of this section is the following theorem.

Theorem 7 Assume the conditions (H1)-(H5) hold and assume that$S\left(t\right)$is compact. Then there exists a mild solution of SNISEEIs (1)-(4)provided that

$12M{l}^{2}\left[T\mathrm{\Lambda }{\int }_{0}^{T}m\left(s\right)\phantom{\rule{0.2em}{0ex}}\mathrm{d}s+2\sum _{k=1}^{m}\left({L}_{{I}_{k}}+{L}_{{\stackrel{˜}{I}}_{k}}\right)+2T\left(T{L}_{g}+Tr\left(Q\right){L}_{\sigma }\right)\right]<1$
(6)

and

${L}_{0}=8M{l}^{2}\left[T\left(T{L}_{g}+Tr\left(Q\right){L}_{\sigma }\right)+\sum _{k=1}^{m}\left({L}_{{I}_{k}}+{L}_{{\stackrel{˜}{I}}_{k}}\right)\right]<1.$
(7)

Proof In the sequel, the notation ${B}_{r}\left(x,Z\right)$ stands for the closed ball with center at xand radius $r>0$ in Z, where $\left(Z,{\parallel \cdot \parallel }_{Z}\right)$ is a Banach space. Let $y:\left(-\mathrm{\infty },T\right]\to H$ be defined by

$y\left(t\right)=\left\{\begin{array}{cc}\varphi \left(t\right),\hfill & t\in \left(-\mathrm{\infty },0\right],\hfill \\ C\left(t\right)\varphi \left(0\right)+S\left(t\right)\psi ,\hfill & t\in J.\hfill \end{array}$

On the space $Y=\left\{x\in \mathcal{P}\mathcal{C}:x\left(0\right)=\varphi \left(0\right)\right\}$ endowed with the uniform convergence topology, wedefine the operator $\mathrm{\Phi }:Y\to Y$ by

$\begin{array}{rl}\mathrm{\Phi }x\left(t\right)=& C\left(t\right)\varphi \left(0\right)+S\left(t\right)\left[\psi -g\left(0,\varphi \right)\right]+{\int }_{0}^{t}C\left(t-s\right)g\left(s,{\overline{x}}_{s}\right)\phantom{\rule{0.2em}{0ex}}\mathrm{d}s\\ +{\int }_{0}^{t}S\left(t-s\right)f\left(s,{\overline{x}}_{s}\right)\phantom{\rule{0.2em}{0ex}}\mathrm{d}s+{\int }_{0}^{t}S\left(t-s\right)\sigma \left(s,{\overline{x}}_{s}\right)\phantom{\rule{0.2em}{0ex}}\mathrm{d}w\left(s\right)\\ +\sum _{0<{t}_{k}

where $\overline{x}$ is such that ${\overline{x}}_{0}=\varphi$ and $\overline{x}=x$ on J. From Lemma 2 and the assumptionon ϕ, we infer that $\mathrm{\Phi }x\in \mathcal{P}\mathcal{C}$. Our proof will be split into the following threesteps.

Step 1. In what follows, we prove that there exists $r>0$ such that $\mathrm{\Phi }\left({B}_{r}\left(y{|}_{J},Y\right)\right)\subseteq {B}_{r}\left(y{|}_{J},Y\right)$. In fact, if it is not true, then for each$r>0$ there exist ${x}^{r}\in {B}_{r}\left(y{|}_{J},Y\right)$ and ${t}^{r}\in J$ such that $r. Therefore, from Lemma 2 and the assumptions, wehave

$\begin{array}{rcl}{r}^{2}& <& E{\parallel \mathrm{\Phi }\left({x}^{r}\left({t}^{r}\right)\right)-y\left({t}^{r}\right)\parallel }^{2}\\ \le & 6E{\parallel S\left({t}^{r}\right)g\left(0,\varphi \right)\parallel }^{2}+6E{\left({\int }_{0}^{{t}^{r}}C\left({t}^{r}-s\right)g\left(s,{\overline{x}}_{s}^{r}\right)\phantom{\rule{0.2em}{0ex}}\mathrm{d}s\right)}^{2}\\ +6E{\left({\int }_{0}^{{t}^{r}}S\left({t}^{r}-s\right)f\left(s,{\overline{x}}_{s}^{r}\right)\phantom{\rule{0.2em}{0ex}}\mathrm{d}s\right)}^{2}+6MTr\left(Q\right)E{\int }_{0}^{{t}^{r}}{\parallel \sigma \left(s,{\overline{x}}_{s}^{r}\right)\parallel }^{2}\phantom{\rule{0.2em}{0ex}}\mathrm{d}s\\ +6E{\left(\sum _{0<{t}_{k}

where ${\parallel y\parallel }_{T}={sup}_{0\le s\le T}E\parallel y\left(s\right)\parallel$. Dividing both sides by ${r}^{2}$ and taking the limit as $r\to \mathrm{\infty }$, we obtain

$1\le 12M{l}^{2}\left[T\mathrm{\Lambda }{\int }_{0}^{T}m\left(s\right)\phantom{\rule{0.2em}{0ex}}\mathrm{d}s+2\sum _{k=1}^{m}\left({L}_{{I}_{k}}+{L}_{{\stackrel{˜}{I}}_{k}}\right)+2T\left(T{L}_{g}+Tr\left(Q\right){L}_{\sigma }\right)\right],$

which contradicts (6). Thus, for some positive number r,$\mathrm{\Phi }\left({B}_{r}\left(y{|}_{J},Y\right)\right)\subseteq {B}_{r}\left(y{|}_{J},Y\right)$. In what follows, we aim to show that the operatorΦ has a fixed point on ${B}_{r}\left(y{|}_{J},Y\right)$, which implies that (1)-(4) has a mild solution. Tothis end, we decompose Φ as $\mathrm{\Phi }={\mathrm{\Phi }}_{1}+{\mathrm{\Phi }}_{2}$, where ${\mathrm{\Phi }}_{1}$, ${\mathrm{\Phi }}_{2}$ are defined on ${B}_{r}\left(y{|}_{J},Y\right)$, respectively, by

$\begin{array}{rcl}\left({\mathrm{\Phi }}_{1}x\right)\left(t\right)& =& C\left(t\right)\varphi \left(0\right)+S\left(t\right)\left[\psi -g\left(0,\varphi \right)\right]\\ +{\int }_{0}^{t}C\left(t-s\right)g\left(s,{\overline{x}}_{s}\right)\phantom{\rule{0.2em}{0ex}}\mathrm{d}s+{\int }_{0}^{t}S\left(t-s\right)\sigma \left(s,{\overline{x}}_{s}\right)\phantom{\rule{0.2em}{0ex}}\mathrm{d}w\left(s\right)\\ +\sum _{0<{t}_{k}

and

$\left({\mathrm{\Phi }}_{2}x\right)\left(t\right)={\int }_{0}^{t}S\left(t-s\right)f\left(s,{\overline{x}}_{s}\right)\phantom{\rule{0.2em}{0ex}}\mathrm{d}s,$

for $t\in J$. We will show that ${\mathrm{\Phi }}_{1}$ is a contraction and ${\mathrm{\Phi }}_{2}$ is completely continuous.

Step 2. ${\mathrm{\Phi }}_{1}$ is a contraction. Let $x,y\in {B}_{r}\left(y{|}_{J},Y\right)$. Then, for each $t\in J$, we have

$\begin{array}{c}E{\parallel \left({\mathrm{\Phi }}_{1}x\right)\left(t\right)-\left({\mathrm{\Phi }}_{1}y\right)\left(t\right)\parallel }^{2}\hfill \\ \phantom{\rule{1em}{0ex}}\le 4E{\parallel {\int }_{0}^{t}C\left(t-s\right)\left[g\left(s,{\overline{x}}_{s}\right)-g\left(s,{\overline{y}}_{s}\right)\right]\phantom{\rule{0.2em}{0ex}}\mathrm{d}s\parallel }^{2}\hfill \\ \phantom{\rule{2em}{0ex}}+4E{\parallel {\int }_{0}^{t}S\left(t-s\right)\left[\sigma \left(s,{\overline{x}}_{s}\right)-\sigma \left(s,{\overline{y}}_{s}\right)\right]\phantom{\rule{0.2em}{0ex}}\mathrm{d}w\left(s\right)\parallel }^{2}\hfill \\ \phantom{\rule{2em}{0ex}}+4E{\parallel \sum _{0<{t}_{k}

Therefore, we get

${\parallel \left({\mathrm{\Phi }}_{1}x\right)\left(t\right)-\left({\mathrm{\Phi }}_{1}y\right)\left(t\right)\parallel }_{\mathcal{P}\mathcal{C}}^{2}\le {L}_{0}{\parallel \overline{x}-\overline{y}\parallel }_{\mathcal{P}\mathcal{C}}^{2},$

where ${L}_{0}=8M{l}^{2}\left[T\left(T{L}_{g}+Tr\left(Q\right){L}_{\sigma }\right)+{\sum }_{k=1}^{m}\left({L}_{{I}_{k}}+{L}_{{\stackrel{˜}{I}}_{k}}\right)\right]$. Thus, we obtain

${\parallel {\mathrm{\Phi }}_{1}x-{\mathrm{\Phi }}_{1}y\parallel }_{\mathcal{P}\mathcal{C}}^{2}\le {L}_{0}{\parallel x-y\parallel }_{\mathcal{P}\mathcal{C}}^{2}.$

By (7), we see that ${\mathrm{\Phi }}_{1}$ is a contraction on ${B}_{r}\left(y{|}_{J},Y\right)$.

Step 3. ${\mathrm{\Phi }}_{2}$ is completely continuous on ${B}_{r}\left(y{|}_{J},Y\right)$.

Claim 1${\mathrm{\Phi }}_{2}$maps bounded sets to bounded sets in${B}_{r}\left(y{|}_{J},Y\right)$.

In the sequel, ${r}^{\ast }$, ${r}^{\ast \ast }$ are the numbers defined by ${r}^{\ast }:=2{l}^{2}{sup}_{0\le s\le t}\left(E{\parallel x\left(s\right)\parallel }^{2}\right)+2{\parallel \varphi \parallel }_{{\mathcal{B}}_{h}}^{2}$ and ${r}^{\ast \ast }:=M\mathrm{\Psi }\left({r}^{\ast }\right){\int }_{0}^{t}m\left(s\right)\phantom{\rule{0.2em}{0ex}}\mathrm{d}s$, respectively.

$\begin{array}{rcl}E{\parallel \left({\mathrm{\Phi }}_{2}x\right)\left(t\right)\parallel }^{2}& \le & {\int }_{0}^{t}{\parallel s\left(t-s\right)\parallel }^{2}E{\parallel f\left(s,{\overline{x}}_{s}\right)\parallel }^{2}\phantom{\rule{0.2em}{0ex}}\mathrm{d}s\\ \le & M{\int }_{0}^{t}m\left(t\right)\mathrm{\Psi }\left({\parallel {\overline{x}}_{s}\parallel }_{{\mathcal{B}}_{h}}^{2}\right)\phantom{\rule{0.2em}{0ex}}\mathrm{d}s\\ \le & M{\int }_{0}^{t}m\left(t\right)\mathrm{\Psi }\left(2{l}^{2}\underset{0\le s\le t}{sup}\left(E{\parallel x\left(s\right)\parallel }^{2}\right)+2{\parallel \varphi \parallel }_{{\mathcal{B}}_{h}}^{2}\right)\phantom{\rule{0.2em}{0ex}}\mathrm{d}s\\ \le & M\mathrm{\Psi }\left({r}^{\ast }\right){\int }_{0}^{t}m\left(s\right)\phantom{\rule{0.2em}{0ex}}\mathrm{d}s\\ =& {r}^{\ast \ast },\end{array}$

which shows the desired result of the claim.

Claim 2 The set of functions${\mathrm{\Phi }}_{2}\left({B}_{r}\left(y{|}_{J},Y\right)\right)$is equicontinuous on J.

Let $\epsilon >0$ small enough and $0<{t}_{1}<{t}_{2}$. We get

$\begin{array}{rcl}E{\parallel \left({\mathrm{\Phi }}_{2}x\right)\left({t}_{2}\right)-\left({\mathrm{\Phi }}_{2}x\right)\left({t}_{1}\right)\parallel }^{2}& \le & 3T{\int }_{0}^{{t}_{1}-\epsilon }{\parallel S\left({t}_{2}-s\right)-S\left({t}_{1}-s\right)\parallel }^{2}E{\parallel f\left(s,{\overline{x}}_{s}\right)\parallel }^{2}\phantom{\rule{0.2em}{0ex}}\mathrm{d}s\\ +3\epsilon {\int }_{{t}_{1}-\epsilon }^{{t}_{1}}{\parallel S\left({t}_{2}-s\right)-S\left({t}_{1}-s\right)\parallel }^{2}E{\parallel f\left(s,{\overline{x}}_{s}\right)\parallel }^{2}\phantom{\rule{0.2em}{0ex}}\mathrm{d}s\\ +3\left({t}_{2}-{t}_{1}\right){\int }_{{t}_{1}}^{{t}_{2}}{\parallel S\left({t}_{2}-s\right)\parallel }^{2}E{\parallel f\left(s,{\overline{x}}_{s}\right)\parallel }^{2}\phantom{\rule{0.2em}{0ex}}\mathrm{d}s\\ \le & 3T\mathrm{\Psi }\left({r}^{\ast }\right){\int }_{0}^{{t}_{1}-\epsilon }{\parallel S\left({t}_{2}-s\right)-S\left({t}_{1}-s\right)\parallel }^{2}m\left(s\right)\phantom{\rule{0.2em}{0ex}}\mathrm{d}s\\ +3\epsilon \mathrm{\Psi }\left({r}^{\ast }\right){\int }_{{t}_{1}-\epsilon }^{{t}_{1}}{\parallel S\left({t}_{2}-s\right)-S\left({t}_{1}-s\right)\parallel }^{2}m\left(s\right)\phantom{\rule{0.2em}{0ex}}\mathrm{d}s\\ +3\left({t}_{2}-{t}_{1}\right)\mathrm{\Psi }\left({r}^{\ast }\right){\int }_{{t}_{1}}^{{t}_{2}}{\parallel S\left({t}_{2}-s\right)\parallel }^{2}m\left(s\right)\phantom{\rule{0.2em}{0ex}}\mathrm{d}s,\end{array}$

which proves that ${\mathrm{\Phi }}_{2}\left({B}_{r}\left(y{|}_{J},Y\right)\right)$ is equicontinuous on J.

Claim 3${\mathrm{\Phi }}_{2}$maps$\left({B}_{r}\left(y{|}_{J},Y\right)\right)$into a precompact set in$\left({B}_{r}\left(y{|}_{J},Y\right)\right)$. That is, for each fixed$t\in J$, the set$V\left(t\right)=\left\{{\mathrm{\Phi }}_{2}z\left(t\right):z\in \left({B}_{r}\left(y{|}_{J},Y\right)\right)\right\}$is precompact in$\left({B}_{r}\left(y{|}_{J},Y\right)\right)$.

Obviously, $V\left(0\right)=\left\{{\mathrm{\Phi }}_{2}\left(0\right)\right\}$. Let $t>0$ fixed and for $0<\epsilon , define

$\left({\mathrm{\Phi }}_{2}^{\epsilon }x\right)\left(t\right)=S\left(\epsilon \right){\int }_{0}^{t-\epsilon }S\left(t-\epsilon -s\right)f\left(s,{\overline{x}}_{s}\right)\phantom{\rule{0.2em}{0ex}}\mathrm{d}s.$

Since $S\left(t\right)$ is a compact operator, the set${V}^{\epsilon }\left(t\right)=\left\{{\mathrm{\Phi }}_{2}^{\epsilon }x\left(t\right):x\in \left({B}_{r}\left(y{|}_{J},Y\right)\right)\right\}$ is relatively compact in H for everyε, $0<\epsilon . Moreover, for each $x\in \left({B}_{r}\left(y{|}_{J},Y\right)\right)$, we have

$\begin{array}{rcl}E{\parallel \left({\mathrm{\Phi }}_{2}x\right)\left(t\right)-\left({\mathrm{\Phi }}_{2}^{\epsilon }x\right)\left(t\right)\parallel }^{2}& \le & \epsilon {\int }_{t-\epsilon }^{t}{\parallel S\left(t-s\right)\parallel }^{2}E{\parallel f\left(s,{\overline{x}}_{s}\right)\parallel }^{2}\phantom{\rule{0.2em}{0ex}}\mathrm{d}s\\ \le & M\epsilon {\int }_{t-\epsilon }^{t}m\left(t\right)\mathrm{\Psi }\left({\parallel {\overline{x}}_{s}\parallel }_{{\mathcal{B}}_{h}}^{2}\right)\phantom{\rule{0.2em}{0ex}}\mathrm{d}s\\ \le & M\mathrm{\Psi }\left({r}^{\ast }\right)\epsilon {\int }_{t-\epsilon }^{t}m\left(s\right)\phantom{\rule{0.2em}{0ex}}\mathrm{d}s.\end{array}$

Therefore, we have

and there are precompact sets arbitrary close to the set $V\left(t\right)=\left\{{\mathrm{\Phi }}_{2}x\left(t\right):x\in \left({B}_{r}\left(y{|}_{J},Y\right)\right)\right\}$. Thus, the set $V\left(t\right)=\left\{{\mathrm{\Phi }}_{2}x\left(t\right):x\in \left({B}_{r}\left(y{|}_{J},Y\right)\right)\right\}$ is precompact in $\left({B}_{r}\left(y{|}_{J},Y\right)\right)$. Therefore, from the Arzela-Ascoli theorem, theoperator ${\mathrm{\Phi }}_{2}$ is completely continuous. From Theorem 5, weinfer that there exists a mild solution for the system (1)-(4).  □

## 4 Examples

In this section, two types of stochastic nonlinear wave equations with infinite delayand impulsive effects are provided to illustrate the theory obtained.

Example 8 We consider the following second-order stochastic Volterraintegro-differential equations with initial-boundary conditions and impulsiveeffects:

$\begin{array}{c}\mathrm{d}\left[\frac{\partial x\left(t,\xi \right)}{\partial t}-{\int }_{-\mathrm{\infty }}^{t}{K}_{1}\left(t,s\right){F}_{1}\left(x\left(s,\xi \right)\right)\phantom{\rule{0.2em}{0ex}}\mathrm{d}s\right]\hfill \\ \phantom{\rule{1em}{0ex}}=\frac{{\partial }^{2}x\left(t,\xi \right)}{\partial {\xi }^{2}}\phantom{\rule{0.2em}{0ex}}\mathrm{d}t+\left({\int }_{-\mathrm{\infty }}^{t}{K}_{2}\left(t,s\right){F}_{2}\left(x\left(s,\xi \right)\right)\phantom{\rule{0.2em}{0ex}}\mathrm{d}s\right)\phantom{\rule{0.2em}{0ex}}\mathrm{d}t\hfill \\ \phantom{\rule{2em}{0ex}}+\left({\int }_{-\mathrm{\infty }}^{t}{K}_{3}\left(t,s\right){F}_{3}\left(x\left(s,\xi \right)\right)\phantom{\rule{0.2em}{0ex}}\mathrm{d}s\right)\phantom{\rule{0.2em}{0ex}}\mathrm{d}w\left(t\right),\hfill \\ \phantom{\rule{2em}{0ex}}0<\xi <\pi ,0\le t\le T,t\ne {t}_{k},\hfill \\ \phantom{\rule{2em}{0ex}}k=1,2,\dots ,m,\hfill \end{array}$
(8)
$x\left(t,\xi \right)=\varphi \left(t,\xi \right),\phantom{\rule{1em}{0ex}}-\mathrm{\infty }
(9)
$x\left(t,0\right)=x\left(t,\pi \right)=0,\phantom{\rule{1em}{0ex}}0\le t\le T,$
(10)
$\frac{\partial x\left(0,\xi \right)}{\partial t}={x}_{1}\left(\xi \right),\phantom{\rule{1em}{0ex}}0<\xi <\pi ,$
(11)
$\begin{array}{c}x\left({t}_{k}^{+}\right)-x\left({t}_{k}^{-}\right)={I}_{k}\left(x\left({t}_{k}\right)\right),\phantom{\rule{2em}{0ex}}{x}^{\prime }\left({t}_{k}^{+}\right)-{x}^{\prime }\left({t}_{k}^{-}\right)={\stackrel{˜}{I}}_{k}\left(x\left({t}_{k}\right)\right),\hfill \\ \phantom{\rule{1em}{0ex}}k=1,2,\dots ,m,\hfill \end{array}$
(12)

where $w\left(t\right)$ is a standard cylindrical Wiener process in defined on the probability space $\left(\mathrm{\Omega },\mathcal{F},P\right)$.

Let $H={L}_{2}\left[0,\pi \right]$. The operator A is defined by

The spectrum of A consists of the eigenvalues $-{n}^{2}$ for $n\in \mathbb{N}$, with associated eigenvectors ${z}_{n}\left(\xi \right)={\left(\frac{2}{\pi }\right)}^{1/2}sin\left(n\xi \right)$. Furthermore, the set $\left\{{z}_{n};n\in \mathbb{N}\right\}$ is an orthonormal basis of H. In particular,

$Ax=\sum _{n=1}^{\mathrm{\infty }}-{n}^{2}〈x,{z}_{n}〉{z}_{n},\phantom{\rule{1em}{0ex}}x\in D\left(A\right).$

The operators $C\left(t\right)$ defined by

$C\left(t\right)x=\sum _{n=1}^{\mathrm{\infty }}cos\left(nt\right)〈x,{z}_{n}〉{z}_{n},\phantom{\rule{1em}{0ex}}t\in \mathbb{R},$

form a cosine function on H, with associated sine function

$S\left(t\right)x=\sum _{n=1}^{\mathrm{\infty }}\frac{sin\left(nt\right)}{n}〈x,{z}_{n}〉{z}_{n},\phantom{\rule{1em}{0ex}}t\in \mathbb{R}.$

From , for all $x\in H$, $t\in \mathbb{R}$, $\parallel S\left(t\right)\parallel \le 1$ and $\parallel C\left(t\right)\parallel \le 1$.

Let ${K}_{i}\left(t,s\right)\in C\left({\mathbb{R}}^{2},\mathbb{R}\right)$, $i=1,2,3$ and assume that there exists a positive continuousfunction $f\left(s\right)$ on ${\mathbb{R}}_{-}$ such that

$|{K}_{i}\left(t,t+s\right)|\le f\left(s\right),\phantom{\rule{1em}{0ex}}i=1,2,3,\phantom{\rule{2em}{0ex}}{l}_{0}={\int }_{-\mathrm{\infty }}^{0}f\left(s\right)\phantom{\rule{0.2em}{0ex}}\mathrm{d}s<\mathrm{\infty }.$

Now, we give the phase space ${\mathcal{B}}_{h}$. Assume that $h:\left(-\mathrm{\infty },0\right]\to \left(0,\mathrm{\infty }\right)$ is a continuous function with $l={\int }_{-\mathrm{\infty }}^{0}h\left(t\right)\phantom{\rule{0.2em}{0ex}}\mathrm{d}t<\mathrm{\infty }$. For any $a>0$, define

We endow ${\mathcal{B}}_{h}$ with the norm

Then $\left({\mathcal{B}}_{h},{\parallel \cdot \parallel }_{{\mathcal{B}}_{h}}\right)$ is a Banach space. Let

$\begin{array}{c}\varphi \left(\xi \right)={\int }_{-\mathrm{\infty }}^{0}h\left(s\right)\varphi \left(s,\xi \right)\phantom{\rule{0.2em}{0ex}}\mathrm{d}s,\phantom{\rule{2em}{0ex}}g\left(t,\varphi \right)\left(\xi \right)={\int }_{-\mathrm{\infty }}^{0}{K}_{1}\left(t,t+s\right){F}_{1}\left(\varphi \left(s,\xi \right)\right)\phantom{\rule{0.2em}{0ex}}\mathrm{d}s,\hfill \\ f\left(t,\varphi \right)\left(\xi \right)={\int }_{-\mathrm{\infty }}^{0}{K}_{2}\left(t,t+s\right){F}_{2}\left(\varphi \left(s,\xi \right)\right)\phantom{\rule{0.2em}{0ex}}\mathrm{d}s,\phantom{\rule{2em}{0ex}}\sigma \left(t,\varphi \right)\left(\xi \right)={\int }_{-\mathrm{\infty }}^{0}{K}_{3}\left(t,t+s\right)\left(\varphi \left(s,\xi \right)\right)\phantom{\rule{0.2em}{0ex}}\mathrm{d}s.\hfill \end{array}$

Then (8)-(12) can be rewritten in the abstract form (1)-(4). We can propose suitableconditions on the coefficients appeared in the above equation to guarantee (8)-(12)has at least one mild solution by means of Theorem 7.

Example 9 We consider the following stochastic nonlinear wave equation withimpulsive effects and infinite delay:

$\begin{array}{c}\mathrm{d}\left[\frac{\partial x\left(t,\xi \right)}{\partial t}-{f}_{1}\left(t,x\left(t-r,\xi \right)\right)\right]\hfill \\ \phantom{\rule{1em}{0ex}}=\frac{{\partial }^{2}x\left(t,\xi \right)}{\partial {\xi }^{2}}\phantom{\rule{0.2em}{0ex}}\mathrm{d}t+{f}_{2}\left(t,x\left(t-r,\xi \right)\right)\phantom{\rule{0.2em}{0ex}}\mathrm{d}t\hfill \\ \phantom{\rule{2em}{0ex}}+\sigma \left(t,x\left(t-r,\xi \right)\right)\phantom{\rule{0.2em}{0ex}}\mathrm{d}w\left(t\right),\hfill \\ \phantom{\rule{2em}{0ex}}0\le \xi \le \pi ,0\le t\le T,r>0,t\ne {t}_{k},k=1,2,\dots ,m,\hfill \end{array}$
(13)
$x\left(t,\xi \right)=\varphi \left(t,\xi \right),\phantom{\rule{1em}{0ex}}-\mathrm{\infty }
(14)
$x\left(t,0\right)=x\left(t,\pi \right)=0,\phantom{\rule{1em}{0ex}}0\le t\le T,$
(15)
$\frac{\partial x\left(0,\xi \right)}{\partial t}={x}_{1}\left(\xi \right),\phantom{\rule{1em}{0ex}}0<\xi <\pi ,$
(16)
$\begin{array}{c}x\left({t}_{k}^{+}\right)-x\left({t}_{k}^{-}\right)={I}_{k}\left(x\left({t}_{k}\right)\right),\phantom{\rule{2em}{0ex}}{x}^{\prime }\left({t}_{k}^{+}\right)-{x}^{\prime }\left({t}_{k}^{-}\right)={\stackrel{˜}{I}}_{k}\left(x\left({t}_{k}\right)\right),\hfill \\ \phantom{\rule{1em}{0ex}}k=1,2,\dots ,m,\hfill \end{array}$
(17)

where ${x}_{1}\in {L}_{0}^{2}\left(\mathrm{\Omega };H\right)$, $\varphi \in {\mathcal{B}}_{h}$, ${\mathcal{B}}_{h}$ is defined as Example 8, $H={L}^{2}\left(\left[0,\pi \right]\right)$, and w is an H-valued Wienerprocess.

Let A, $C\left(t\right)$ and $S\left(t\right)$ be defined as Example 8. Then the above system(13)-(17) can be rewritten in the form of (1)-(4). Further, we assume that${f}_{i}:\left[0,T\right]×\mathbb{R}\to \mathbb{R}$ ($i=1,2$), $\sigma :\left[0,T\right]×\mathbb{R}\to BL\left(H\right)$ and ${I}_{k}$, ${\stackrel{˜}{I}}_{k}$ satisfy (H2)-(H5). Then (13)-(17) has at least onemild solution.

## Author’s contributions

The author is the only person who is responsible to this work.

## References

1. Hino Y, Murakami S, Naito T Lecture Notes in Mathematics 1473. In Functional Differential Equations with Infinite Delay. Springer, Berlin; 1991.

2. Caraballo T: Asymptotic exponential stability of stochastic partial differential equationswith delay. Stochastics 1990, 33: 27–47.

3. Da Prato G, Zabczyk J: Stochastic Equations in Infinite Dimensions. Cambridge University Press, Cambridge; 1992.

4. Mahmudov NI: Existence and uniqueness results for neutral SDEs in Hilbert spaces. Stoch. Anal. Appl. 2006, 24: 79–95. 10.1080/07362990500397582

5. Mao X: Stochastic Differential Equations and Applications. Horwood, Chichester; 1997.

6. Hale JK, Lunel SMV: Introduction to Functional Differential Equations. Springer, Berlin; 1991.

7. Kolmanovskii VB, Myshkis A: Applied Theory of Functional Differential Equations. Kluwer Academic, Norwell; 1992.

8. Govindan TE: Exponential stability in mean-square of parabolic quasilinear stochasticdelay evolution equations. Stoch. Anal. Appl. 1999, 17: 443–461. 10.1080/07362999908809612

9. Ren Y, Chen L: A note on the neutral stochastic functional differential equations withinfinite delay and Poisson jumps in an abstract space. J. Math. Phys. 2009., 50: Article ID 082704

10. Ren Y, Lu S, Xia N: Remarks on the existence and uniqueness of the solutions to stochasticfunctional differential equations with infinite delay. J. Comput. Appl. Math. 2008, 220: 364–372. 10.1016/j.cam.2007.08.022

11. Ren Y, Xia N: Existence, uniqueness and stability of the solutions to neutral stochasticfunctional differential equations with infinite delay. Appl. Math. Comput. 2009, 210: 72–79. 10.1016/j.amc.2008.11.009

12. Ren Y, Xia N: A note on the existence and uniqueness of the solution to neutral stochasticfunctional differential equations with infinite delay. Appl. Math. Comput. 2009, 214: 457–461. 10.1016/j.amc.2009.04.013

13. Nieto JJ, Rodriguez-Lopez R: New comparison results for impulsive integro-differential equations andapplications. J. Math. Anal. Appl. 2007, 328: 1343–1368. 10.1016/j.jmaa.2006.06.029

14. Samoilenko AM, Perestyuk NA: Impulsive Differential Equations. World Scientific, Singapore; 1995.

15. Travis CC, Webb GF: Cosine families and abstract nonlinear second order differentialequations. Acta Math. Acad. Sci. Hung. 1978, 32: 76–96.

16. Fattorini HO North-Holland Mathematics Studies 108. In Second Order Linear Differential Equations in Banach Spaces. North-Holland, Amsterdam; 1985.

17. Travis CC, Webb GF: Second order differential equations in Banach space. In Proceedings of an International Symposium on Nonlinear Equations in AbstractSpaces. Academic Press, New York; 1987:331–361.

18. McKibben MA: Second-order damped functional stochastic evolution equations in Hilbertspace. Dyn. Syst. Appl. 2003, 12: 467–487.

19. Mahmudov NI, McKibben MA: Abstract second-order damped McKean-Vlasov stochastic evolution equations. Stoch. Anal. Appl. 2006, 24: 303–328. 10.1080/07362990500522247

20. McKibben MA: Second-order neutral stochastic evolution equations with heredity. J. Appl. Math. Stoch. Anal. 2004, 2004: 177–192. 10.1155/S1048953304309020

21. Balasubramaniam P, Muthukumar P: Approximate controllability of second-order stochastic distributed implicitfunctional differential systems with infinite delay. J. Optim. Theory Appl. 2009, 143: 225–244. 10.1007/s10957-009-9564-x

22. Da Prato G: Non-linear stochastic partial differential equations. In Mathematics of Complexity and Dynamical Systems. Springer, Berlin; 2011:1126–1136.

23. Li Y, Liu B: Existence of solution of nonlinear neutral functional differential inclusionswith infinite delay. Stoch. Anal. Appl. 2007, 25: 397–415. 10.1080/07362990601139610

24. Burton TA, Kirk C: A fixed point theorem of Krasnoselskii-Schaefer type. Math. Nachr. 1998, 189: 23–31. 10.1002/mana.19981890103

## Acknowledgements

The author sincerely thanks the reviewers for their valuable suggestions anduseful comments. This work is partially supported by Mathematical disciplineproject of Anhui Agricultural University (Project number. XKXWD2013020;XK2013029).

## Author information

Authors

### Corresponding author

Correspondence to Chaohui Yue. 