Existence results for impulsive neutral stochastic functional integro-differential inclusions with infinite delays

Abstract

In this paper, we prove the existence of mild solutions for a class of impulsive neutral stochastic functional integro-differential inclusions with infinite delays in Hilbert spaces. The results are obtained by using the fixed-point theorem for multi-valued operators due to Dhage. An example is provided to illustrate the theory.

MSC:93B05, 93E03.

1 Introduction

In this paper, we shall consider the existence of mild solutions for impulsive neutral stochastic functional integro-differential inclusions with infinite delay of the following form:

$$\begin{array}{c}d[x(t)-g(t,x_t,{\int }_0^{t}a(t,s,x_s)ds)]dt\hfill \\ \in [Ax(t)+f(t,x_t)]dt+F(t,x_t)dw(t),t\in J=[0,b],t\ne t_k,\hfill \end{array}$$

(1.1)

$$\mathrm{\Delta }x(t_k)=x\left(t_k^{+}\right)-x\left(t_k^{-}\right)=I_k\left(x\left(t_k^{-}\right)\right),k=1,2,\dots ,m,$$

(1.2)

$$x(t)=\varphi (t)\in L^2(\mathrm{\Omega },{\mathcal{B}}_h)\text{for a.e. }t\in J_0=(-\mathrm{\infty },0],$$

(1.3)

where the state $x(\cdot )$ takes values in a separable real Hilbert space H with inner product $(\cdot ,\cdot )$ and norm $|\cdot |$, A is the infinitesimal generator of a compact analytic resolvent operator $S(t)$, $t\ge 0$, in the Hilbert space H. Suppose that $\{w(t):t\ge 0\}$ is a given K-valued Brownian motion or Wiener process with a finite trace nuclear covariance operator $Q\ge 0$ and $L(K,H)$ denotes the space of all bounded linear operators from K into H. Further $a:D\times {\mathcal{B}}_h\to H$, $g:J\times {\mathcal{B}}_h\times H\to H$, $f:J\times {\mathcal{B}}_h\to H$ and $F:J\times {\mathcal{B}}_h\to \mathcal{P}(L_Q(K,H))$ are given functions, where $D=\{(t,s)\in J\times J:s\le t\}$, $\mathcal{P}(L_Q(K,H))$ is the family of all nonempty subsets of $L_Q(K,H)$ and $L_Q(K,H)$ denotes the space of all Q-Hilbert-Schmidt operators from K into H, which will be defined in Section 2. Here, $I_k\in C(H,H)$ ($k=1,2,\dots ,m$) are bounded functions. Furthermore, the fixed times $t_k$ satisfies $0=t_0<t_1<t_2<\cdots <t_m<b$, $x(t_k^{+})$ and $x(t_k^{-})$ denote the right and left limits of $x(t)$ at $t=t_k$. $\mathrm{\Delta }x(t_k)=x(t_k^{+})-x(t_k^{-})=I_k(x(t_k^{-}))$ represents the jump in the state x at time $t_k$, where $I_k$ determines the size of jump. The histories $x_t:\mathrm{\Omega }\to {\mathcal{B}}_h$, $t\ge 0$, which are defined by setting $x_t=\{x(t+s):s\in (-\mathrm{\infty },0]\}$, belong to the abstract phase space ${\mathcal{B}}_h$, which will be defined in Section 2. The initial data $\varphi =\{\varphi (t):-\mathrm{\infty }<t\le 0\}$ is an ${\mathcal{F}}_0$-measurable, ${\mathcal{B}}_h$-valued random variables independent of $\{w(t):t\ge 0\}$ with finite second moment.

The theory of impulsive integro-differential inclusions has become an active area of investigation due to their applications in the fields such as mechanics, electrical engineering, medicine biology, ecology, and so on (see [1, 2] and references therein).

The existence of impulsive neutral stochastic functional integro-differential equations or inclusions with infinite delays have attracted great interest of researchers. For example, Lin and Hu [3] consider the existence results for impulsive neutral stochastic functional integro-differential inclusions with nonlocal initial conditions. Hu and Ren [4] studied the existence results for impulsive neutral stochastic functional integro-differential equations with infinite delays.

Motivated by the previous mentioned papers, we prove the existence of solutions for impulsive neutral stochastic functional integro-differential inclusions with infinite delays.

2 Preliminaries

Throughout this paper, $(H,|\cdot |)$ and $(K,{|\cdot |}_K)$ denote two real separable Hilbert spaces. Let $(\mathrm{\Omega },\mathcal{F},P;F)$ ($F={\{{\mathcal{F}}_t\}}_{t\ge 0}$) be a complete filtered probability space satisfying the requirement that ${\mathcal{F}}_0$ contains all P-null sets of ℱ. An H-valued random variable is an ℱ-measurable function $x(t):\mathrm{\Omega }\to H$ and the collection of random variables $S=\{x(t,w):\mathrm{\Omega }\to H|t\in J\}$ is called a stochastic process. Suppose that $\{w(t):t\ge 0\}$ is a cylindrical K-valued Wiener process with a finite trace nuclear covariance operator $Q\ge 0$, denote $T_{r}Q={\sum }_{i=1}^{\mathrm{\infty }}{\lambda }_i=\lambda <\mathrm{\infty }$, which satisfies $Q{e}_i={\lambda }_i{e}_i$. So, actually, $w(t)={\sum }_{i=1}^{\mathrm{\infty }}\sqrt{{\lambda }_i}{w}_i(t)e_i$, where ${\{w_i(t)\}}_{i=1}^{\mathrm{\infty }}$ are mutually independent one-dimensional standard Wiener process. We assume that ${\mathcal{F}}_t=\sigma \{w(s):0\le s\le t\}$ is the σ-algebra generated by w and ${\mathcal{F}}_T=\mathcal{F}$. Let $\psi \in L(K,H)$ and define

$${|\psi |}_Q^2=T_r\left(\psi Q{\psi }^{\ast }\right)=\sum _{n=1}^{\mathrm{\infty }}{\left|\sqrt{{\lambda }_n}\psi e_n\right|}^2.$$

If ${|\psi |}_Q<\mathrm{\infty }$, then ψ is called a Q-Hilbert-Schmidt operator. Let $L_Q(K,H)$ denote the space of all Q-Hilbert-Schmidt operator $\psi :K\to H$. The completion $L_Q(K,H)$ of $L(K,H)$ with respect to the topology induced by the norm ${|\cdot |}_Q$, where ${|\psi |}_Q^2=〈\psi ,\psi 〉$ is a Hilbert space with the above norm topology.

Let $A:D(A)\to H$ be the infinitesimal generator of a compact, analytic resolvent operator $S(t)$, $t\ge 0$. Let $0\in \rho (A)$. Then it is possible to define the fractional power ${(-A)}^{\alpha }$ for $0<\alpha \le 1$ as a closed linear operator with its domain $D({(-A)}^{\alpha })$ being dense in H. We denote by $H_{\alpha }$ the Banach space $D(-A^{\alpha })$ endowed with the norm ${\parallel x\parallel }_{\alpha }=\parallel {(-A)}^{\alpha }x\parallel$, which is equivalent to the graph norm of ${(-A)}^{\alpha }$.

Lemma 2.1 ([5])

The following properties hold:

• (i) If $0<\beta <\alpha \le 1$, the $H_{\alpha }\subset H_{\beta }$ and the embedding is continuous and compact whenever the resolvent operator of A is compact.

• (ii) For every $0<\alpha <1$, there exists a positive constant $c_{\alpha }$ such that

  $$\parallel {(-A)}^{\alpha }S(t)\parallel \le \frac{C_{\alpha }}{t^{\alpha }},t>0.$$

Now, we define the abstract phase space ${\mathcal{B}}_h$. Assume that $h:(-\mathrm{\infty },0]\to (0,\mathrm{\infty })$ is a continuous function with $l={\int }_{-\mathrm{\infty }}^{0}h(t)dt<\mathrm{\infty }$. For any $a>0$ we define

$$\begin{array}{rl}{\mathcal{B}}_h=& \{\psi :(-\mathrm{\infty },0]\to H:{\left(E{|\psi (\theta )|}^2\right)}^{\frac{1}{2}}\mathit{\text{is a bounded and measurable}}\\ \mathit{\text{function on}}[-a,0]\mathit{\text{and}}{\int }_{-\mathrm{\infty }}^{0}h(s)\underset{s\le \theta \le 0}{sup}{\left(E{|\psi (\theta )|}^2\right)}^{\frac{1}{2}}ds<\mathrm{\infty }\}.\end{array}$$

If ${\mathcal{B}}_h$ is endowed with the norm

$${\parallel \psi \parallel }_{{\mathcal{B}}_h}={\int }_{-\mathrm{\infty }}^{0}h(s)\underset{s\le \theta \le 0}{sup}{\left(E{|\psi (\theta )|}^2\right)}^{\frac{1}{2}}ds\mathit{\text{for all}}\psi \in {\mathcal{B}}_h,$$

then $({\mathcal{B}}_h,{\parallel \cdot \parallel }_{{\mathcal{B}}_h})$ is a Banach space [6]. Now, we consider the space

$$\begin{array}{rl}{\mathcal{B}}_b=& \{x:(-\mathrm{\infty },b]\to H\mathit{\text{such that}}{x}_k\in C(J_k,H)\mathit{\text{and there exist}}\\ x\left(t_k^{+}\right)\mathit{\text{and}}x\left(t_k^{-}\right)\mathit{\text{with}}x(t_k)=x\left(t_k^{-}\right),x_0=\varphi \in L^2(\mathrm{\Omega },{\mathcal{B}}_h)\mathit{\text{on}}\\ (-\mathrm{\infty },0],k=1,2,\dots ,m\},\end{array}$$

where $x_k$ is the restriction of x to $J_k=(t_k,t_{k+1}]$, $k=0,1,\dots ,m$. Let ${\parallel \cdot \parallel }_b$ be a seminorm in ${\mathcal{B}}_b$ defined by

$${\parallel x\parallel }_b={\parallel x_0\parallel }_{{\mathcal{B}}_h}+\underset{0\le s\le b}{sup}{\left(E{|x(s)|}^2\right)}^{\frac{1}{2}},x\in {\mathcal{B}}_b.$$

Lemma 2.2 ([7])

Assume that $x\in {\mathcal{B}}_b$, then for $t\in J$, $x_t\in {\mathcal{B}}_h$. Moreover

$$l{\left(E{|x(t)|}^2\right)}^{\frac{1}{2}}\le {\parallel x_t\parallel }_{{\mathcal{B}}_h}\le {\parallel x_0\parallel }_{{\mathcal{B}}_h}+l\underset{0\le s\le t}{sup}{\left(E{|x(t)|}^2\right)}^{\frac{1}{2}},$$

where $l={\int }_{-\mathrm{\infty }}^{0}h(s)ds<\mathrm{\infty }$.

We use the notation $\mathcal{P}(H)$ for the family of all subsets H and denote

$$\begin{array}{r}{\mathcal{P}}_{cl}(H)=\{Y\in \mathcal{P}(H):Y\text{ is closed}\},\\ {\mathcal{P}}_{bd}(H)=\{Y\in \mathcal{P}(H):Y\text{ is bounded}\},\\ {\mathcal{P}}_{cv}(H)=\{Y\in \mathcal{P}(H):Y\text{ is convex}\},\\ {\mathcal{P}}_{cp}(H)=\{Y\in \mathcal{P}(H):Y\text{ is compact}\}.\end{array}$$

A multi-valued mapping $\mathrm{\Gamma }:H\to \mathcal{P}(H)$ is called upper semicontinuous (u.s.c) if for any $x\in H$, the set $\mathrm{\Gamma }(x)$ is a nonempty closed subset of H and if for each open set G of H containing $\mathrm{\Gamma }(x)$, there exists an open neighborhood N of x such that $\mathrm{\Gamma }(N)\subseteq G$. Γ is said to be completely continuous if $\mathrm{\Gamma }(B)$ is relatively compact for every bounded subset of $B\subseteq H$. If the multi-valued mapping Γ is completely continuous with nonempty compact values, then Γ is u.s.c. if and only if Γ has a closed graph, i.e., $x_n\to x$, $y_n\to y$, $y_n\in \mathrm{\Gamma }(x_n)$ imply $y\in \mathrm{\Gamma }(x)$.

Definition 2.1 The multi-valued mapping $F:J\times {\mathcal{B}}_h\to \mathcal{P}(H)$ is said to be $L^2$-Carathéodory if

• (i) $t\mapsto F(t,v)$ is measurable for each $v\in {\mathcal{B}}_h$,

• (ii) $v\mapsto F(t,v)$ is u.s.c. for almost all $t\in J$ and $v\in {\mathcal{B}}_h$,

• (iii) for each $q>0$, there exists $h_q\in L^1(J,{\mathbb{R}}^{+})$ such that

  $${\parallel F(t,v)\parallel }^2=\underset{f\in F(t,v)}{sup}E\left({|f|}^2\right)\le h_q(t),$$

for all ${\parallel v\parallel }_{{\mathcal{B}}_h}^2\le q$ and for a.e. $t\in J$.

The following lemma is crucial in the proof of our main result.

Lemma 2.3 ([8])

Let I be a compact interval and H be a Hilbert space. Let F be an $L^2$-Carathéodory multi-valued mapping with $N_{F,x}\ne \varphi$ and let Γ be a linear continuous mapping from $L^2(I,H)$ to $C(I,H)$. Then the operator

$$\mathrm{\Gamma }\circ N_F:C(I,H)\to {\mathcal{P}}_{cp,cv}(H),x\mapsto (\mathrm{\Gamma }\circ N_F)(x)=\mathrm{\Gamma }(N_{F,x})$$

is a closed graph operator in $C(I,H)\times C(I,H)$, where $N_{F,x}$ is known as the selectors set from F; it is given by

$$\sigma \in N_{F,x}=\{\sigma \in L^2(L(K,H)):\sigma (t)\in F(t,x)\mathit{\text{for a.e.}}t\in J\}.$$

Theorem 2.1 ([9])

Let X be a Banach space, ${\mathrm{\Phi }}_1:X\to {\mathcal{P}}_{cl,cv,bd}(X)$ and ${\mathrm{\Phi }}_2:X\to {\mathcal{P}}_{cp,cv}(X)$ be two multi-valued operators satisfying:

• (a) ${\mathrm{\Phi }}_1$ is a contraction,

• (b) ${\mathrm{\Phi }}_2$ is u.s.c. and completely continuous.

Then either

• (i) the operator inclusion $\lambda x\in {\mathrm{\Phi }}_{1}x+{\mathrm{\Phi }}_{2}x$ has a solution for $\lambda =1$, or

• (ii) the set $G=\{x\in X:\lambda x\in {\mathrm{\Phi }}_{1}x+{\mathrm{\Phi }}_{2}x,\lambda >1\}$ is unbounded.

Lemma 2.4 ([10])

Let $v,w:[0,b]\to [0,\mathrm{\infty })$ be continuous functions. If w is nondecreasing and there are constants $\theta >0$, $0<\alpha <1$ such that

$$v(t)\le w(t)+\theta {\int }_0^t\frac{v(s)}{{(t-s)}^{1-\alpha }}ds,t\in J,$$

then

$$v(t)\le e^{\frac{{\theta }^n\mathrm{\Gamma }{(\alpha )}^n{t}^{n\alpha }}{\mathrm{\Gamma }(n\alpha )}}\sum _{j=0}^{n-1}{\left(\frac{\theta b^{\alpha }}{\alpha }\right)}^{j}w(t)$$

for every $t\in J$ and every $n\in N$ such that $n\alpha >1$ and $\mathrm{\Gamma }(\cdot )$ is the Gamma function.

3 Main result

Let $J_1=(-\mathrm{\infty },b]$. First, we present the definition of the mild solution of problem (1.1)-(1.3).

Definition 3.1 A stochastic process $x:J_1\times \mathrm{\Omega }\to H$ is called a mild solution of problem (1.1)-(1.3) if

• (i) $x(t)$ is measurable and ${\mathcal{F}}_t$-adapted for each $t\ge 0$,

• (ii) $\mathrm{\Delta }x(t_k)=x(t_k^{+})-x(t_k^{-})$, $k=1,2,\dots ,m$,

• (iii) $x(t)\in H$ has càdlàg paths on $t\in J$ a.e. and there exists a function $\sigma \in N_{F,x}$ such that

  $$\begin{array}{rl}x(t)=& S(t)[\varphi (0)-g(0,\varphi ,0)]+g(t,x_t,{\int }_0^{t}a(t,s,x_s)ds)\\ +{\int }_0^{t}AS(t-s)g(s,x_s,{\int }_0^{s}a(s,\tau ,x_{\tau })d\tau )ds+{\int }_0^{t}AS(t-s)f(s,x_s)ds\\ +{\int }_0^{t}S(t-s)\sigma (s)dw(s)+\sum _{0<t_k<t}S(t-t_k)I_k\left(x\left(t_k^{-}\right)\right),t\in J,\end{array}$$

• (iv) $x_0(\cdot )=\varphi \in L^2(\mathrm{\Omega },{\mathcal{B}}_h)$ on $J_0=(-\mathrm{\infty },0]$ satisfies ${\parallel \varphi \parallel }_{{\mathcal{B}}_h}^2<\mathrm{\infty }$.

Now, we assume the following hypotheses:

(H1) A is the infinitesimal generator of a compact analytic resolvent operator $S(t)$, $t\ge 0$, in the Hilbert space H and there exist positive constants M and $M_1$ such that

$${\parallel S(t)\parallel }^2\le M,\parallel A^{-\beta }\parallel \le M_1,t\in J.$$

(H2) $a:D\times {\mathcal{B}}_h\to H$, $D=\{(t,s)\in J\times J:t\ge s\}$ is a continuous function and there exists a constant $M_a$ such that

$$E{\left|{\int }_0^t[a(t,s,x)-a(t,s,y)]ds\right|}^2\le M_a{\parallel x-y\parallel }_{{\mathcal{B}}_h}^2\text{for all }t\in J,x,y\in {\mathcal{B}}_h.$$

(H3) There exist constants $0<\beta <1$ and $M_g$ such that g is $H_{\beta }$-valued, ${(-A)}^{\beta }g$ is continuous and

$$E{|{(-A)}^{\beta }g(t,x_1,y_1)-{(-A)}^{\beta }g(t,x_2,y_2)|}^2\le M_g[{\parallel x_1-x_2\parallel }_{{\mathcal{B}}_h}^2+E{|y_1-y_2|}^2].$$

(H4) The function $f:J\times {\mathcal{B}}_h\to H$ satisfies the following conditions:

• (i) $t\mapsto f(t,s)$ is measurable for each $x\in {\mathcal{B}}_h$;

• (ii) $x\mapsto f(t,x)$ is continuous for almost all $t\in J$;

• (iii) There exists a constant $M_f$ such that

  $$E{|{(-A)}^{\beta }f(t,x)-{(-A)}^{\beta }f(t,y)|}^2\le M_f{\parallel x-y\parallel }_{{\mathcal{B}}_h}^2$$

  for all $x,y\in {\mathcal{B}}_h$, $t\in J$ and

  $$E{|f(t,x)|}^2\le p(t)\psi \left({\parallel x\parallel }_{{\mathcal{B}}_h}^2\right)$$

  for almost all $t\in J$, where $p\in L^1(J,\mathbb{R})$, $\psi :{\mathbb{R}}_{+}\to (0,\mathrm{\infty })$ is continuous and increasing with

  $$\begin{array}{c}{\int }_0^b\overline{\mu (s)}ds\le {\int }_{B_0{k}_1}^{\mathrm{\infty }}\frac{1}{\psi (s)}ds,\hfill \\ \overline{\mu }(t)=B_0{k}_{3}p(t),\hfill \\ k_1=\frac{4{\parallel \varphi \parallel }_{{\mathcal{B}}_h}^2+l^{2}F}{1-96l^2{\parallel {(-A)}^{-\beta }\parallel }^2{M}_g(1+2M_a)},\hfill \\ k_2=\frac{96b{l}^2{M}_g(1+2M_a)c_{1-\beta }^2}{1-96l^2{\parallel {(-A)}^{-\beta }\parallel }^2{M}_g(1+2M_a)},\hfill \\ k_3=\frac{48Mb{l}^2}{1-96l^2{\parallel {(-A)}^{-\beta }\parallel }^2{M}_g(1+2M_a)},\hfill \\ L_0=3l^2[M_g(1+M_a)({\parallel {(-A)}^{-\beta }\parallel }^2+\frac{{(C_{1-\beta }{b}^{\beta })}^2}{2\beta -1})+M_f\frac{{(C_{1-\beta }{b}^{\beta })}^2}{2\beta -1}]<1,\hfill \\ B_0=e^{k_2^n\mathrm{\Gamma }{(\beta )}^n{b}^{n\beta }/\mathrm{\Gamma }(n\beta )}\sum _{j=1}^{n-1}{\left(\frac{k_2{b}^{\beta }}{\beta }\right)}^j,\hfill \\ c_1=b^2\underset{(t,s)\in D}{sup}{a}^2(t,s,0),c_2={\parallel {(-A)}^{\beta }\parallel }^2\underset{t\in J}{sup}{\parallel g(t,0,0)\parallel }^2\hfill \end{array}$$

  and

  $$\begin{array}{rl}\mathcal{F}=& 4M{|\varphi (0)|}^2+96(M+{\parallel {(-A)}^{-\beta }\parallel }^2)c_2+192{\parallel {(-A)}^{-\beta }\parallel }^2{M}_g{c}_1\\ +\frac{192b^{2\beta }{C}_{1-\beta }^2}{2\beta -1}(c_2+2M_g{c}_1)+48M{\parallel \mu \parallel }_{L_{\mathrm{loc}}^1(J,{\mathbb{R}}^{+})}{b}^{2}Tr(Q)\\ +48M{m}^2\sum _{k=1}^m{d}_k+96M{\parallel {(-A)}^{-\beta }\parallel }^2{M}_g{\parallel \varphi \parallel }_{{\mathcal{B}}_h}^2.\end{array}$$

  (H5) The multi-valued mapping $F:J\times {\mathcal{B}}_h\to {\mathcal{P}}_{bd,cl,cv}(L(K,H))$ is an $L^2$-Carathéodory function that satisfies the following conditions:

  • (i) For each $t\in J$, the function $F(t,\cdot ):{\mathcal{B}}_h\to {\mathcal{P}}_{bd,cl,cv}(L(K,H))$ is u.s.c. and for each fixed $x\in {\mathcal{B}}_h$, the function $F(\cdot ,x)$ is measurable. For each $x\in {\mathcal{B}}_h$, the set

    $$N_{F,x}=\{\sigma \in L^2(K,H):\sigma (t)\in F(t,x)\text{ for a.e. }t\in J\}$$

  is nonempty.

• (ii) There exists a positive function $\mu \in L_{\mathrm{loc}}^1(J,{\mathbb{R}}^{+})$ such that

  $${\parallel F(t,x)\parallel }^2=\underset{\sigma \in F(t,x)}{sup}E{|\sigma |}^2\le \mu (t).$$

(H6) $I_k\in C(H_{\alpha },H_{\alpha })$ and there exist positive constants $d_k$ such that for each $x\in H_{\alpha }$,

$${|I_k(x)|}^2\le d_k,k=1,2,\dots ,m.$$

We consider the mapping $\mathrm{\Phi }:{\mathcal{B}}_h\to \mathcal{P}({\mathcal{B}}_h)$ defined by

$$\mathrm{\Phi }x(t)=\{\begin{array}{c}\varphi (t),t\in (-\mathrm{\infty },0],\hfill \\ S(t)[\varphi (0)-g(0,\varphi ,0)]+g(t,x_t,{\int }_0^{t}a(t,s,x_s)ds)\hfill \\ +{\int }_0^{t}AS(t-s)g(s,x_s,{\int }_0^{s}a(s,\tau ,x_{\tau })d\tau )ds\hfill \\ +{\int }_0^{t}AS(t-s)f(s,x_s)ds+{\int }_0^{t}S(t-s)\sigma (s)dw(s)\hfill \\ +{\sum }_{0<t_k<t}S(t-t_k)I_k(x(t_k^{-})),t\in J,\hfill \end{array}$$

where $\sigma \in N_{F,x}$. For each $\varphi \in {\mathcal{B}}_h$, we define

$$\tilde{\varphi }(t)=\{\begin{array}{cc}\varphi (t),\hfill & t\in (-\mathrm{\infty },0],\hfill \\ S(t)\varphi (0),\hfill & t\in J,\hfill \end{array}$$

and then $\tilde{\varphi }\in {\mathcal{B}}_h$. Let $x(t)=y(t)+\tilde{\varphi }(t)$, $t\in (-\mathrm{\infty },b]$. Then it is easy to see that x satisfies (1.1)-(1.3) if and only if y satisfies $y_0=0$ and

$$\begin{array}{rl}y(t)=& -S(t)g(0,\varphi ,0)+g(t,y_t+{\tilde{\varphi }}_t,{\int }_0^{t}a(t,s,y_s+{\tilde{\varphi }}_s)ds)\\ +{\int }_0^{t}AS(t-s)g(s,y_s+{\tilde{\varphi }}_s,{\int }_0^{s}a(s,\tau ,y_{\tau }+{\tilde{\varphi }}_{\tau })d\tau )ds\\ +{\int }_0^{t}AS(t-s)f(s,y_s+{\varphi }_s)ds+{\int }_0^{t}S(t-s)\sigma (s)dw(s)\\ +\sum _{0<t_k<t}S(t-t_k)I_k(y\left(t_k^{-}\right)+\tilde{\varphi }\left(t_k^{-}\right)),t\in J,\end{array}$$

where $\sigma \in N_{F,y}$. Let ${\mathcal{B}}_h^{\prime }=\{y\in {\mathcal{B}}_h:y_0=0\in {\mathcal{B}}_h\}$. For any $y\in {\mathcal{B}}_h^{\prime }$,

$$\begin{array}{rl}{\parallel y\parallel }_b& ={\parallel y_0\parallel }_{{\mathcal{B}}_h}+\underset{0\le s\le b}{sup}{\left(E{|y(s)|}^2\right)}^{\frac{1}{2}}\\ =\underset{0\le s\le b}{sup}{\left(E{|y(s)|}^2\right)}^{\frac{1}{2}}\end{array}$$

and thus $({\mathcal{B}}_h^{\prime },{\parallel \cdot \parallel }_b)$ is a Banach space. Set ${\mathcal{B}}_q=\{y\in {\mathcal{B}}_h^{\prime }:{\parallel y\parallel }_b^2\le q\}$ for some $q\ge 0$. Then ${\mathcal{B}}_q\subseteq {\mathcal{B}}_h^{\prime }$ is uniformly bounded and for any $y\in {\mathcal{B}}_q$, from Lemma 2.2, we see that

$$\begin{array}{rcl}{\parallel y_t+{\tilde{\varphi }}_t\parallel }_{{\mathcal{B}}_h}^2& \le & 2{\parallel y_t\parallel }_{{\mathcal{B}}_h}^2+2{\parallel {\tilde{\varphi }}_t\parallel }_{{\mathcal{B}}_h}^2\\ \le & 4l^2\underset{0\le s\le t}{sup}E{|y(s)|}^2+4{\parallel y_0\parallel }_{{\mathcal{B}}_h}^2\\ +4l^2\underset{0\le s\le t}{sup}{\parallel \tilde{\varphi }(s)\parallel }^2+4{\parallel {\tilde{\varphi }}_0\parallel }_{{\mathcal{B}}_h}^2\\ \le & 4l^2(q+M{|\varphi (0)|}^2)+4{\parallel \tilde{\varphi }\parallel }_{{\mathcal{B}}_h}^2\\ :=& q^{\prime }.\end{array}$$

Define the operator $\tilde{\mathrm{\Phi }}:{\mathcal{B}}_h^{\prime }\to \mathcal{P}({\mathcal{B}}_h^{\prime })$ by

$$\tilde{\mathrm{\Phi }}y(t)=\{\begin{array}{c}0,t\in (-\mathrm{\infty },0],\hfill \\ -S(t)g(0,\varphi ,0)+g(t,y_t+{\tilde{\varphi }}_t,{\int }_0^{t}a(t,s,y_s+{\tilde{\varphi }}_s)ds)\hfill \\ +{\int }_0^{t}AS(t-s)g(s,y_s+\tilde{\varphi },{\int }_0^{s}a(s,\tau ,y_{\tau }+{\tilde{\varphi }}_{\tau })d\tau )ds\hfill \\ +{\int }_0^{t}AS(t-s)f(s,y_s+{\tilde{\varphi }}_s)ds+{\int }_0^{t}S(t-s)\sigma (s)dw(s)\hfill \\ +{\sum }_{0<t_k<t}S(t-t_k)I_k(y(t_k^{-})+\tilde{\varphi }(t_k^{-})),t\in J,\hfill \end{array}$$

where $\sigma \in N_{F,y}$. Obviously, the operator Φ has a fixed point is equivalent to proving that $\tilde{\mathrm{\Phi }}$ has a fixed point. Now, we decompose $\tilde{\mathrm{\Phi }}$ as ${\tilde{\mathrm{\Phi }}}_1+{\tilde{\mathrm{\Phi }}}_2$, where

$$\begin{array}{rl}{\tilde{\mathrm{\Phi }}}_{1}y(t)=& -S(t)g(0,\varphi ,0)+g(t,y_t+{\tilde{\varphi }}_t,{\int }_0^{t}a(t,s,y_s+{\tilde{\varphi }}_s)ds)\\ +{\int }_0^{t}AS(t-s)g(s,y_s+{\tilde{\varphi }}_s,{\int }_0^{s}a(s,\tau ,y_{\tau }+{\tilde{\varphi }}_{\tau })d\tau )ds\\ +{\int }_0^{t}AS(t-s)f(s,y_s+{\tilde{\varphi }}_s)ds\end{array}$$

and

$${\tilde{\mathrm{\Phi }}}_{2}y(t)={\int }_0^{s}S(t-s)\sigma (s)dw(s)+\sum _{0<t_k<t}S(t-t_k)I_k(y\left(t_k^{-}\right)+\tilde{\varphi }\left(t_k^{-}\right)),t\in J,$$

where $\sigma \in N_{F,y}$. In what follows, we show that the operators ${\tilde{\mathrm{\Phi }}}_1$ and ${\tilde{\mathrm{\Phi }}}_2$ satisfy all the conditions of Theorem 2.1.

Lemma 3.1 Assume that the assumptions (H1)-(H6) hold. Then ${\tilde{\mathrm{\Phi }}}_1$ is a contraction and ${\tilde{\mathrm{\Phi }}}_2$ is u.s.c. and completely continuous.

Proof We give the proof in several steps:

Step 1. ${\tilde{\mathrm{\Phi }}}_1$ is a contraction.

Let $u,v\in {\mathcal{B}}_h^{\prime }$. Then we have

$$\begin{array}{c}E{|{\tilde{\varphi }}_{1}u(t)-{\tilde{\varphi }}_{1}v(t)|}^2\hfill \\ \le 3E{|g(t,u_t+{\tilde{\varphi }}_t,{\int }_0^{t}a(t,s,u_s+{\tilde{\varphi }}_s)ds)-g(t,v_t+{\tilde{\varphi }}_t,{\int }_0^{t}a(t,s,v_s+{\tilde{\varphi }}_s)ds)|}^2\hfill \\ +3bE\left({\int }_0^t\right|AS(t-s)[g(s,u_s+{\tilde{\varphi }}_s,{\int }_0^{s}a(s,\tau ,u_{\tau }+{\tilde{\varphi }}_{\tau })d\tau )\hfill \\ -g(s,v_s+{\tilde{\varphi }}_s,{\int }_0^{s}a(s,\tau ,u_{\tau }+{\tilde{\varphi }}_{\tau })d\tau )\left]{|}^{2}ds\right)\hfill \\ +3bE\left({\int }_0^t{|AS(t-s)[f(s,u_s+{\tilde{\varphi }}_s)-f(s,v_s+{\tilde{\varphi }}_s)]|}^{2}ds\right)\hfill \\ \le 3{\parallel {(-A)}^{-\beta }\parallel }^2{M}_g({\parallel u_t-v_t\parallel }_{{\mathcal{B}}_h}^2+M_a{\parallel u_t-v_t\parallel }_{{\mathcal{B}}_h}^2)\hfill \\ +3b{\int }_0^t\frac{C_{1-\beta }^2}{{(t-s)}^{2(1-\beta )}}{M}_g({\parallel u_s-v_s\parallel }_{{\mathcal{B}}_h}^2+M_a{\parallel u_s-v_s\parallel }_{{\mathcal{B}}_h}^2)ds\hfill \\ +3b{\int }_0^t\frac{C_{1-\beta }^2}{{(t-s)}^{2(1-\beta )}}{M}_f{\parallel u_s-v_s\parallel }_{{\mathcal{B}}_h}^{2}ds\hfill \\ \le 3{\parallel {(-A)}^{-\beta }\parallel }^2{M}_g(1+M_a){\parallel u_t-v_t\parallel }_{{\mathcal{B}}_h}^2\hfill \\ +3M_g(1+M_a)\frac{{(C_{1-\beta }{b}^{\beta })}^2}{2\beta -1}{\parallel u_t-v_t\parallel }_{{\mathcal{B}}_h}^2\hfill \\ +3M_f\frac{{(C_{1-\beta }{b}^{\beta })}^2}{2\beta -1}{\parallel u_t-v_t\parallel }_{{\mathcal{B}}_h}^2\hfill \\ \le 3[M_g(1+M_a)({\parallel {(-A)}^{-\beta }\parallel }^2+\frac{{(C_{1-\beta }{b}^{\beta })}^2}{2\beta -1})+M_f\frac{{(C_{1-\beta }{b}^{\beta })}^2}{2\beta -1}]\hfill \\ \times [l^2\underset{s\in [0,b]}{sup}E{|u(s)-v(s)|}^2+{\parallel u_0\parallel }_{{\mathcal{B}}_h}^2+{\parallel v_0\parallel }_{{\mathcal{B}}_h}^2]\hfill \\ =3l^2[M_g(1+M_a)({\parallel {(-A)}^{-\beta }\parallel }^2+\frac{{(C_{1-\beta }{b}^{\beta })}^2}{2\beta -1})+M_f\frac{{(C_{1-\beta }{b}^{\beta })}^2}{2\beta -1}]\underset{s\in [0,b]}{sup}E{|u(s)-v(s)|}^2\hfill \\ =L_0\underset{s\in [0,b]}{sup}E{|u(s)-v(s)|}^2,\hfill \end{array}$$

where $L_0=3l^2[M_g(1+M_a)({\parallel {(-A)}^{-\beta }\parallel }^2+\frac{{(C_{1-\beta }{b}^{\beta })}^2}{2\beta -1})+M_f\frac{{(C_{1-\beta }{b}^{\beta })}^2}{2\beta -1}]<1$ and we have used the fact that ${\parallel u_0\parallel }_{{\mathcal{B}}_h}^2=0$ and ${\parallel v_0\parallel }_{{\mathcal{B}}_h}^2=0$. Taking the supremum over t, we obtain

$${\parallel {\tilde{\varphi }}_{1}u-{\tilde{\varphi }}_{1}v\parallel }_b^2\le L_0{\parallel u-v\parallel }_b^2$$

and so ${\tilde{\varphi }}_1$ is a contraction.

Now, we show that the operator ${\tilde{\mathrm{\Phi }}}_2$ is completely continuous.

Step 2. ${\tilde{\mathrm{\Phi }}}_{2}y$ is convex for each $y\in {\mathcal{B}}_h^{\prime }$.

In fact, if $u_1,u_2\in {\tilde{\mathrm{\Phi }}}_2(y)$, then there exist ${\sigma }_1,{\sigma }_2\in N_{F,y}$ such that

$$u_i(t)={\int }_0^{t}S(t-s){\sigma }_i(s)dw(s)+\sum _{0<t_k<t}S(t-t_k)I_k(y\left(t_k^{-}\right)+\tilde{\varphi }\left(t_k^{-}\right))$$

for $i=1,2$ and $t\in J$. Let $\lambda \in [0,1]$. Then for each $t\in J$, we have

$$\begin{array}{rl}\lambda u_1(t)+(1-\lambda )u_2(t)=& {\int }_0^{t}S(t-s)[\lambda {\sigma }_1(s)+(1-\lambda ){\sigma }_2(s)]dw(s)\\ +\sum _{0<t_k<t}S(t-t_k)I_k(y\left(t_k^{-}\right)+\tilde{\varphi }\left(t_k^{-}\right)).\end{array}$$

Since $N_{F,y}$ is convex (because F has convex values), we obtain

$$\lambda u_1(t)+(1-\lambda )u_2(t)\in {\tilde{\mathrm{\Phi }}}_2(y).$$

Step 3. ${\tilde{\mathrm{\Phi }}}_2$ maps bounded sets into bounded sets in ${\mathcal{B}}_h^{\prime }$.

It is enough to show that there exists a positive constant Λ such that for each $u\in {\tilde{\mathrm{\Phi }}}_{2}y$, $y\in {\mathcal{B}}_q=\{y\in {\mathcal{B}}_h^{\prime }:{\parallel y\parallel }_b\le q\}$ one has ${\parallel u\parallel }_b\le \mathrm{\Lambda }$. If $u\in {\tilde{\mathrm{\Phi }}}_2(y)$, there exists $\sigma \in N_{F,y}$ such that for each $t\in J$

$$u(t)={\int }_0^{t}S(s-t)\sigma (s)dw(s)+\sum _{0<t_k<t}S(t-t_k)I_k(y\left(t_k^{-}\right)+\tilde{\varphi }\left(t_k^{-}\right))$$

and so

$$\begin{array}{rl}E{|u(t)|}^2& =E{|{\int }_0^{t}S(t-s)\sigma (s)dw(s)+\sum _{0<t_k<t}S(t-t_k)I_k(y\left(t_k^{-}\right)+\tilde{\varphi }\left(t_k^{-}\right))|}^2\\ \le 2E{|{\int }_0^{t}S(t-s)\sigma (s)dw(s)|}^2+2E{|\sum _{0<t_k<t}S(t-t_k)I_k(y\left(t_k^{-}\right)+\tilde{\varphi }\left(t_k^{-}\right))|}^2\\ \le 2Tr(Q)Mb{\int }_0^b\mu (s)ds+2M{m}^2\sum _{k=1}^m{d}_k\\ \le 2Tr(Q)M{b}^2{\parallel \mu \parallel }_{L_{\mathrm{loc}}^1(J,{\mathbb{R}}^{+})}+2M{m}^2\sum _{k=1}^m{d}_k\\ :=\mathrm{\Lambda }.\end{array}$$

Thus, for each $y\in {\mathcal{B}}_h^{\prime }$, we get ${\parallel u\parallel }_b^2\le \mathrm{\Lambda }$.

Step 4. ${\tilde{\mathrm{\Phi }}}_2$ maps bounded sets into equicontinuous sets of ${\mathcal{B}}_h^{\prime }$.

Let $0<{\tau }_1<{\tau }_2\le b$. For each $y\in {\mathcal{B}}_q=\{y\in {\mathcal{B}}_h^{\prime }:{\parallel y\parallel }_b\le q\}$ and $u\in {\tilde{\mathrm{\Phi }}}_2(y)$. Let ${\tau }_1,{\tau }_2\in J\mathrm{\setminus }\{t_1,t_2,\dots ,t_m\}$. Then there exists $\sigma \in N_{F,y}$ such that for each $t\in J$,

$$u(t)={\int }_0^{t}S(t-s)\sigma (s)dw(s)+\sum _{0<t_k<t}S(t-t_k)I_k(y\left(t_k^{-}\right)+\tilde{\varphi }\left(t_k^{-}\right)).$$

Thus we have

$$\begin{array}{c}E{|u({\tau }_2)-u({\tau }_1)|}^2\hfill \\ =E|{\int }_0^{{\tau }_2}S({\tau }_2-s)\sigma (s)dw(s)+\sum _{0<t_k<{\tau }_2}S({\tau }_2-s)I_k(y\left(t_k^{-}\right)+\tilde{\varphi }\left(t_k^{-}\right))\hfill \\ -{\int }_0^{{\tau }_1}S({\tau }_1-s)\sigma (s)dw(s)-\sum _{0<t_k<{\tau }_1}S({\tau }_1-t_k)I_k(y\left(t_k^{-}\right)+\tilde{\varphi }\left(t_k^{-}\right)){|}^2\hfill \\ \le 2E|{\int }_0^{{\tau }_1-\epsilon }(S({\tau }_2-s)\sigma (s)-S({\tau }_1-s)\sigma (s))dw(s)\hfill \\ +{\int }_{{\tau }_1-\epsilon }^{{\tau }_1}(S({\tau }_2-s)\sigma (s)-S({\tau }_1-s)\sigma (s))dw(s)+{\int }_{{\tau }_1}^{{\tau }_2}S({\tau }_2-s)\sigma (s)dw(s){|}^2\hfill \\ +2E|\sum _{0<t_k<{\tau }_1}[S({\tau }_2-t_k)-S({\tau }_1-t_k)]I_k(y\left(t_k^{-}\right)+\tilde{\varphi }\left(t_k^{-}\right))\hfill \\ +\sum _{{\tau }_1<t_k<{\tau }_2}S({\tau }_2-t_k)I_k(y\left(t_k^{-}\right)+\tilde{\varphi }\left(t_k^{-}\right)){|}^2\hfill \\ \le 6\epsilon Tr(Q){\int }_0^{{\tau }_1-\epsilon }\mu (s){\parallel S({\tau }_2-s)-S({\tau }_1-s)\parallel }^{2}ds\hfill \\ +6\epsilon Tr(Q){\int }_{{\tau }_1-\epsilon }^{{\tau }_1}\mu (s){\parallel S({\tau }_2-s)-S({\tau }_1-s)\parallel }^{2}ds\hfill \\ +6({\tau }_2-{\tau }_1)Tr(Q){\int }_{{\tau }_1}^{{\tau }_2}\mu (s){\parallel S({\tau }_2-s)\parallel }^{2}ds\hfill \\ +4m^2\sum _{0<t_k<{\tau }_1}{\parallel S({\tau }_2-s)-S({\tau }_1-s)\parallel }^2{d}_k\hfill \\ +4m^{2}M\sum _{{\tau }_1<t_k<{\tau }_2}{d}_k.\hfill \end{array}$$

The right-hand side of the above inequality tends to zero as ${\tau }_1\to {\tau }_2$ with ε sufficiently small, since $S(t)$ is strongly continuous and the compactness of $S(t)$ for $t>0$ implies the continuity in the uniform operator topology. Thus, the set $\{{\tilde{\mathrm{\Phi }}}_{2}y:y\in {\mathcal{B}}_q\}$ is equicontinuous. Here we consider the case $0<{\tau }_1<{\tau }_2\le b$, since the case ${\tau }_1<{\tau }_2\le 0$ or ${\tau }_1\le 0\le {\tau }_2\le b$ is simple.

Step 5. ${\tilde{\mathrm{\Phi }}}_2$ maps ${\mathcal{B}}_q$ into a precompact set in H.

Let $0<t\le b$ and $0<\epsilon <t$. For $y\in {\mathcal{B}}_q$ and $u\in {\tilde{\mathrm{\Phi }}}_2(y)$, there exists $\sigma \in N_{F,y}$ such that

$$\begin{array}{rl}u(t)=& {\int }_0^{t-\epsilon }S(t-s)\sigma (s)dw(s)+{\int }_{t-\epsilon }^{t}S(t-s)\sigma (s)dw(s)\\ +\sum _{0<t_k<t}S(t-t_k)I_k(y\left(t_k^{-}\right)+\tilde{\varphi }\left(t_k^{-}\right)).\end{array}$$

Define

$$u_{\epsilon }(t)=S(\epsilon ){\int }_0^{t-\epsilon }S(t-\epsilon -s)\sigma (s)dw(s)+\sum _{0<t_k<t-\epsilon }S(t-t_k)I_k(y\left(t_k^{-}\right)+\tilde{\varphi }\left(t_k^{-}\right)).$$

Since $S(t)$ is a compact operator, the set $V_{\epsilon }(t)=\{u_{\epsilon }(t):u_{\epsilon }\in {\tilde{\mathrm{\Phi }}}_2({\mathcal{B}}_q)\}$ is relatively compact in H for each ε, $0<\epsilon <t$. Moreover,

$$\begin{array}{c}E{|u(t)-u_{\epsilon }(t)|}^2\hfill \\ =E|{\int }_0^{t-\epsilon }S(t-s)\sigma (s)dw(s)+{\int }_{t-\epsilon }^{t}S(t-s)\sigma (s)dw(s)\hfill \\ +\sum _{0<t_k<t}S(t-t_k)I_k(y\left(t_k^{-}\right)+\tilde{\varphi }\left(t_k^{-}\right))-S(\epsilon ){\int }_0^{t-\epsilon }S(t-\epsilon -s)\sigma (s)dw(s)\hfill \\ -\sum _{0<t_k<t-\epsilon }S(t-t_k)I_k(y\left(t_k^{-}\right)+\tilde{\varphi }\left(t_k^{-}\right)){|}^2\hfill \\ \le 4MbTr(Q)\epsilon {\parallel \mu \parallel }_{L_{\mathrm{loc}}^1(J,{\mathbb{R}}^{+})}+4m^{2}M\sum _{t-\epsilon <t_k<t}{d}_k.\hfill \end{array}$$

Therefore letting $\epsilon \to 0$, we can see that there are relative compact sets arbitrarily close to the set $\{u(t):u\in {\tilde{\mathrm{\Phi }}}_2(B_q)\}$. Thus, the set $\{u(t):u\in {\tilde{\mathrm{\Phi }}}_2(B_q)\}$ is relatively compact in H. Hence, the Arzelá-Ascoli theorem shows that ${\tilde{\mathrm{\Phi }}}_2$ is a compact multi-valued mapping.

Step 6. ${\tilde{\mathrm{\Phi }}}_2$ has a closed graph.

Let $y_n\to y_{\ast }$, $u_n\in {\tilde{\mathrm{\Phi }}}_2(y_n)$ and $u_n\to u_{\ast }$. We prove that $u_{\ast }\in {\tilde{\mathrm{\Phi }}}_2(y_{\ast })$.

Indeed, $u_n\in {\tilde{\mathrm{\Phi }}}_2(y_n)$ means that there exists ${\sigma }_n\in N_{F,y_n}$ such that

$$u_n(t)={\int }_0^{t}S(t-s){\sigma }_n(s)dw(s)+\sum _{0<t_k<t}S(t-t_k)I_k(y_n\left(t_k^{-}\right)+\tilde{\varphi }\left(t_k^{-}\right)),t\in J.$$

Thus we must prove that there exists ${\sigma }_{\ast }\in N_{F,y_{\ast }}$ such that

$$u_{\ast }(t)={\int }_0^{t}S(t-s){\sigma }_{\ast }(s)dw(s)+\sum _{0<t_k<t}S(t-t_k)I_k(y_{\ast }\left(t_k^{-}\right)+\tilde{\varphi }\left(t_k^{-}\right)),t\in J.$$

Since $I_k$, $k=1,2,\dots ,m$, are continuous, we see that

$${\parallel \sum _{0<t_k<t}S(t-t_k)I_k(y_n\left(t_k^{-}\right)+\tilde{\varphi }\left(t_k^{-}\right))-\sum _{0<t_k<t}S(t-t_k)I_k(y_{\ast }\left(t_k^{-}\right)+\tilde{\varphi }\left(t_k^{-}\right))\parallel }_b^2\to 0$$

as $n\to \mathrm{\infty }$. Consider the linear continuous operator $\mathrm{\Gamma }:L^2(J,H)\to C(J,H)$ with $\mathrm{\Gamma }(\sigma )(t)={\int }_0^{t}S(t-s)\sigma (s)dw(s)$, where $\sigma \in N_{F,y}$. From Lemma 2.3, it follows that $\mathrm{\Gamma }\circ N_F$ is a closed graph operator. Moreover, we have

$$u_n(t)-\sum _{0<t_k<t}S(t-t_k)I_k(y_n\left(t_k^{-}\right)+\tilde{\varphi }\left(t_k^{-}\right))\in \mathrm{\Gamma }(N_{F,y_n}).$$

Since $y_n\to y_{\ast }$, from Lemma 2.3, we obtain

$$u_{\ast }(t)-\sum _{0<t_k<t}S(t-t_k)I_k(y_{\ast }\left(t_k^{-}\right)+\tilde{\varphi }\left(t_k^{-}\right))\in \mathrm{\Gamma }(N_{F,y_{\ast }}).$$

That is, there exists a ${\sigma }_{\ast }\in N_{F,y_{\ast }}$ such that

$$\begin{array}{rl}{u}_{\ast }(t)-\sum _{0<t_k<t}S(t-t_k)I_k(y_{\ast }\left(t_k^{-}\right)+\tilde{\varphi }\left(t_k^{-}\right))& =\mathrm{\Gamma }({\sigma }_{\ast }(t))\\ ={\int }_0^{t}S(t-s){\sigma }_{\ast }(s)dw(s).\end{array}$$

Therefore ${\tilde{\mathrm{\Phi }}}_2$ has a closed graph and ${\tilde{\mathrm{\Phi }}}_2$ is u.s.c. This completes the proof. □

Lemma 3.2 Assume that the assumptions (H1)-(H2) hold. Then there exists a constant $K>0$ such that ${\parallel y_t+{\tilde{\varphi }}_t\parallel }_{{\mathcal{B}}_h}^2\le K$ for all $t\in J$, where K is depends only on b and the functions ψ and $\overline{\mu }$.

Proof Let y be a possible solution of $y\in \lambda \tilde{\mathrm{\Phi }}(y)$ for some $0<\lambda <1$. Then there exists $\sigma \in N_{F,y}$ such that for $t\in J$ we have

$$\begin{array}{rl}y(t)=& -\lambda S(t)g(0,\varphi ,0)+\lambda g(t,y_t+{\tilde{\varphi }}_t,{\int }_0^{t}a(t,s,y_s+{\tilde{\varphi }}_s)ds)\\ +\lambda {\int }_0^{t}AS(t-s)g(s,y_s+{\tilde{\varphi }}_s,{\int }_0^{s}a(s,\tau ,y_{\tau }+{\tilde{\varphi }}_{\tau })d\tau )ds\\ +\lambda {\int }_0^{t}S(t-s)f(s,y_s+{\tilde{\varphi }}_s)ds+{\int }_0^{t}S(t-s)\sigma (s)dw(s)\\ +\lambda \sum _{0<t_k<t}S(t-t_k)I_k(y\left(t_k^{-}\right)+\tilde{\varphi }\left(t_k^{-}\right)).\end{array}$$

Then, by the assumptions, we deduce that

$$\begin{array}{rl}E{|y(t)|}^2\le & E|-S(t)g(0,\varphi ,0)+g(t,y_t+{\tilde{\varphi }}_t,{\int }_0^{t}a(t,s,y_s+{\tilde{\varphi }}_s)ds)\\ +{\int }_0^{t}AS(t-s)g(s,y_s+{\tilde{\varphi }}_s,{\int }_0^{s}a(s,\tau ,y_{\tau }+{\tilde{\varphi }}_{\tau })d\tau )ds\\ +{\int }_0^{t}S(t-s)f(s,y_s+{\tilde{\varphi }}_s)ds+{\int }_0^{t}S(t-s)\sigma (s)dw(s)\\ +\sum _{0<t_k<t}S(t-t_k)I_k(y\left(t_k^{-}\right)+\tilde{\varphi }\left(t_k^{-}\right)){|}^2\\ \le & 12\{2M({\parallel {(-A)}^{-\beta }\parallel }^2{M}_g{\parallel \varphi \parallel }_{B_h}^2+c_2)\\ +2{\parallel {(-A)}^{-\beta }\parallel }^2[M_g({\parallel y_s+{\tilde{\varphi }}_s\parallel }_{{\mathcal{B}}_h}^2+2M_a{\parallel y_s+{\tilde{\varphi }}_s\parallel }_{{\mathcal{B}}_h}^2+2c_1)+c_2]\\ +2b{\int }_0^t\frac{c_{1-\beta }^2}{{(t-s)}^{2(1-\beta )}}[M_g({\parallel y_s+{\tilde{\varphi }}_s\parallel }_{{\mathcal{B}}_h}^2+2M_a{\parallel y_s+{\tilde{\varphi }}_s\parallel }_{{\mathcal{B}}_h}^2+2c_1)+c_2]ds\\ +Mb{\int }_0^{t}p(s)\psi \left({\parallel y_s+{\tilde{\varphi }}_s\parallel }_{{\mathcal{B}}_h}^2\right)ds+M{\parallel \mu \parallel }_{L_{\mathrm{loc}}^1(J,{\mathbb{R}}^{+})}{b}^{2}Tr(Q)+M{m}^2\sum _{k=1}^m{d}_k\}\\ =& 24(M+{\parallel {(-A)}^{-\beta }\parallel }^2)c_2+48{\parallel {(-A)}^{-\beta }\parallel }^2{M}_g{c}_1+\frac{48b^{2\beta }{c}_{1-\beta }^2}{2\beta -1}(c_2+2M_g{c}_1)\\ +12M{\parallel \mu \parallel }_{L_{\mathrm{loc}}^1(J,{\mathbb{R}}^{+})}{b}^{2}Tr(Q)+12M{m}^2\sum _{k=1}^m{d}_k+24M{\parallel {(-A)}^{-\beta }\parallel }^2{M}_g{\parallel \varphi \parallel }_{{\mathcal{B}}_h}^2\\ +24{\parallel {(-A)}^{-\beta }\parallel }^2{M}_g(1+2M_a){\parallel y_s+{\tilde{\varphi }}_s\parallel }_{{\mathcal{B}}_h}^2\\ +24b{M}_g(1+2M_a)c_{1-\beta }^2{\int }_0^t\frac{{\parallel y_s+{\tilde{\varphi }}_s\parallel }_{{\mathcal{B}}_h}^2}{{(t-s)}^{2(1-\beta )}}ds\\ +12Mb{\int }_0^{t}p(s)\psi \left({\parallel y_s+{\tilde{\varphi }}_s\parallel }_{{\mathcal{B}}_h}^2\right)ds.\end{array}$$

From Lemma 2.2 we see that

$${\parallel y_t+{\tilde{\varphi }}_t\parallel }_{{\mathcal{B}}_h}^2\le 4l^2\underset{0\le s\le t}{sup}E{|y(s)|}^2+4l^{2}M{|\tilde{\varphi }(0)|}^2+4{\parallel \tilde{\varphi }\parallel }_{{\mathcal{B}}_h}^2.$$

Thus, for any $t\in J$, we have

$$\begin{array}{c}{\parallel y_t+{\tilde{\varphi }}_t\parallel }_{{\mathcal{B}}_h}^2\hfill \\ \le 4l^{2}M{|\tilde{\varphi }(0)|}^2+4{\parallel \tilde{\varphi }\parallel }_{{\mathcal{B}}_h}^2+96l^2(M+{\parallel {(-A)}^{-\beta }\parallel }^2)c_2\hfill \\ +192l^2{\parallel {(-A)}^{-\beta }\parallel }^2{M}_g{c}_1+\frac{192l^2{b}^{2\beta }{C}_{1-\beta }^2}{2\beta -1}(c_2+2M_g{c}_1)\hfill \\ +48M{\parallel \mu \parallel }_{L_{\mathrm{loc}}^1(J,{\mathbb{R}}^{+})}{b}^2{l}^{2}Tr(Q)+48M{l}^2{m}^2\sum _{k=1}^m{d}_k\hfill \\ +96M{l}^2{\parallel {(-A)}^{-\beta }\parallel }^2{M}_g{\parallel \varphi \parallel }_{{\mathcal{B}}_h}^2\hfill \\ +96l^2{\parallel {(-A)}^{-\beta }\parallel }^2{M}_g(1+2M_a){\parallel y_s+{\tilde{\varphi }}_s\parallel }_{{\mathcal{B}}_h}^2\hfill \\ +96b{l}^2{M}_g(1+2M_a)C_{1-\beta }^2{\int }_0^t\frac{{\parallel y_s+{\tilde{\varphi }}_s\parallel }_{{\mathcal{B}}_h}^2}{{(t-s)}^{2(1-\beta )}}ds\hfill \\ +48Mb{l}^2{\int }_0^{t}p(s)\psi \left({\parallel y_s+{\tilde{\varphi }}_s\parallel }_{{\mathcal{B}}_h}^2\right)ds\hfill \\ =4{\parallel \varphi \parallel }_{{\mathcal{B}}_h}^2+l^2\mathcal{F}+96l^2{\parallel {(-A)}^{-\beta }\parallel }^2{M}_g(1+2M_a)\underset{0\le s\le t}{sup}{\parallel y_s+{\tilde{\varphi }}_s\parallel }_{{\mathcal{B}}_h}^2\hfill \\ +96b{l}^2{M}_g(1+2M_a)C_{1-\beta }^2{\int }_0^t\frac{{\parallel y_s+{\tilde{\varphi }}_s\parallel }_{{\mathcal{B}}_h}^2}{{(t-s)}^{2(1-\beta )}}ds\hfill \\ +48Mb{l}^2{\int }_0^{t}p(s)\psi \left({\parallel y_s+{\tilde{\varphi }}_s\parallel }_{{\mathcal{B}}_h}^2\right)ds.\hfill \end{array}$$

Let $v(t)={sup}_{0\le s\le t}{\parallel y_s+{\tilde{\varphi }}_s\parallel }_{{\mathcal{B}}_h}^2$. Then the function $v(t)$ is nondecreasing in J. Thus, we obtain

$$\begin{array}{rl}v(t)\le & 4{\parallel \varphi \parallel }_{{\mathcal{B}}_h}^2+l^2\mathcal{F}+96l^2{\parallel {(-A)}^{-\beta }\parallel }^2{M}_g(1+2M_a)v(t)\\ +96b{l}^2{M}_g(1+2M_a)C_{1-\beta }^2{\int }_0^t\frac{v(s)}{{(t-s)}^{2(1-\beta )}}ds\\ +48Mb{l}^2{\int }_0^{t}p(s)\psi (v(s))ds.\end{array}$$

From this we derive that

$$\begin{array}{rl}v(t)\le & \frac{4{\parallel \varphi \parallel }_{{\mathcal{B}}_h}^2+l^2\mathcal{F}}{1-96l^2{\parallel {(-A)}^{-\beta }\parallel }^2{M}_g(1+2M_a)}\\ +\frac{96b{l}^2{M}_g(1+2M_a)C_{1-\beta }^2}{1-96l^2{\parallel {(-A)}^{-\beta }\parallel }^2{M}_g(1+2M_a)}{\int }_0^t\frac{v(s)}{{(t-s)}^{1-\beta }}ds\\ +\frac{48Mb{l}^2}{1-96l^2{\parallel {(-A)}^{-\beta }\parallel }^2{M}_g(1+2M_a)}{\int }_0^{t}p(s)\psi (v(s))ds\\ \le & k_1+k_2{\int }_0^t\frac{v(s)}{{(t-s)}^{1-\beta }}ds+k_3{\int }_0^{t}p(s)\psi (v(s))ds.\end{array}$$

By Lemma 2.4, we get

$$v(t)\le B_0(k_1+k_3{\int }_0^{t}p(s)\psi (v(s))ds),$$

where

$$B_0=e^{k_2^n\mathrm{\Gamma }{(\beta )}^n{b}^{n\beta }/\mathrm{\Gamma }(n\beta )}\sum _{j=1}^{n-1}{\left(\frac{k_2{b}^{\beta }}{\beta }\right)}^j.$$

Let us take the right-hand side of the above inequality as $\mu (t)$. Then $\mu (0)=B_0{k}_1$, $v(t)\le \mu (t)$, $t\in J$ and

$${\mu }^{\prime }(t)\le B_0{k}_{3}p(t)\psi (v(t)).$$

Since ψ is nondecreasing, we have

$$\begin{array}{rl}{\mu }^{\prime }(t)& \le B_0{k}_{3}p(t)\psi (\mu (t))\\ =\overline{\mu }(t)\psi (\mu (t)).\end{array}$$

It follows that

$$\begin{array}{rl}{\int }_{\mu (0)}^{\mu (t)}\frac{1}{\psi (s)}ds& \le {\int }_0^b\overline{\mu (s)}ds\\ \le {\int }_{B_0{K}_1}^{\mathrm{\infty }}\frac{1}{\psi (s)}ds,\end{array}$$

which indicates that $\mu (t)<\mathrm{\infty }$. Thus, there exists a constant K such that $\mu (t)\le K$, $t\in J$. Furthermore, we see that ${\parallel y_t+{\tilde{\varphi }}_t\parallel }_{{\mathcal{B}}_h}^2\le v(t)\le \mu (t)\le K$, $t\in J$. □

Theorem 3.1 Assume that the assumptions (H1)-(H6) hold. The problem (1.1)-(1.3) has at least one mild solution on J.

Proof Let us take the set

$$G(\mathrm{\Phi })=\{x\in {\mathcal{B}}_h:x\in \lambda \mathrm{\Phi }(x)\text{ for some }\lambda \in (0,1)\}.$$

Then for any $x\in G(\mathrm{\Phi })$, we have

$${\parallel x_t\parallel }_{{\mathcal{B}}_h}^2={\parallel y_t+{\tilde{\varphi }}_t\parallel }_{{\mathcal{B}}_h}^2\le K,t\in J,$$

where $K>0$ is a constant in Lemma 3.2. This show that G is bounded on J. Hence from Theorem 2.1 there exists a fixed point $x(t)$ for Φ on ${\mathcal{B}}_h$, which is a mild solution of (1.1)-(1.3) on J. □

4 An example

As an application of Theorem 3.1, we consider the impulsive neutral stochastic functional integro-differential inclusion of the following form:

$$\begin{array}{c}\frac{\partial }{\partial t}(z(t,x)+g(t,z(t-h,x),{\int }_0^{t}a(t,s,z(s-h,x))ds))\hfill \\ \in \frac{{\partial }^2}{\partial x^2}z(t,x)+(f(t,z(t-h,x))+[Q_1(t,z(t-h,x)),Q_2(t,z(t-h,x))])dw(t),\hfill \end{array}$$

(4.1)

$$\begin{array}{c}0\le x\le \pi ,t\in J,t\ne t_k,\hfill \\ \mathrm{\Delta }z(t_k,x)=z(t_k^{+},x)-z(t_k^{-},x)=I_k\left(z(t_k^{-},x)\right),k=1,2,\dots ,m,\hfill \end{array}$$

(4.2)

$$z(t,0)=z(t,\pi )=0,t\in J,$$

(4.3)

$$z(t,x)=\rho (t,x),-\mathrm{\infty }<t\le 0,0\le x\le \pi ,$$

(4.4)

where $J=[0,b]$, $k=1,2,\dots ,m$, $z(t_k^{+},x)={lim}_{h\to 0^{+}}z(t_k+h,x)$, $z(t_k^{-},x)={lim}_{h\to 0^{-}}z(t_k+h,x)$, $Q_1,Q_2:J\times \mathbb{R}\to \mathbb{R}$ are two given functions and $w(t)$ is a one-dimensional standard Wiener process. We assume that for each $t\in J$, $Q_1(t,\cdot )$ is lower semicontinuous and $Q_2(t,\cdot )$ is upper semicontinuous. Let $J_1=(-\mathrm{\infty },b]$ and $H=L^2([0,\pi ])$ with norm $\parallel \cdot \parallel$. Define $A:H\to H$ by $Av=v^{″}$ with domain $D(A)=\{v\in H:v,v^{\prime }\text{ are absolutely continuous},v^{″}\in H,v(0)=v(\pi )=0\}$. Then

$$Av=\sum _{n=1}^{\mathrm{\infty }}{n}^2(v,v_n),v\in D(A),$$

where $v_n=\sqrt{\frac{2}{\pi }}sin(ns)$, $n=1,2,\dots$ , is the orthogonal set of eigenvectors in A. It is well known that A is the infinitesimal generator of an analytic semigroup $S(t)$, $t\ge 0$ in H given by

$$S(t)v=\sum _{n=1}^{\mathrm{\infty }}{e}^{-n^{2}t}(v,v_n)v_n,v\in H.$$

For every $v\in H$, ${(-A)}^{-\frac{1}{2}}v={\sum }_{n=1}^{\mathrm{\infty }}\frac{1}{n}(v,v_n)v_n$ and $\parallel {(-A)}^{-\frac{1}{2}}\parallel =1$. The operator ${(-A)}^{\frac{1}{2}}$ is given by

$${(-A)}^{\frac{1}{2}}v=\sum _{n=1}^{\mathrm{\infty }}n(v,v_n)v_n$$

on the space $D({(-A)}^{-\frac{1}{2}})=\{v\in H:{\sum }_{n=1}^{\mathrm{\infty }}n(v,v_n)v_n\in H\}$. Since the analytic semigroup $S(t)$ is compact [10], there exists a constant $M>0$ such that $\parallel S(t)\parallel \le M$ and satisfies (H1). Now, we give a special ${\mathcal{B}}_h$-space. Let $h(s)=e^{2s}$, $s<0$. Then $l={\int }_{-\mathrm{\infty }}^{0}h(s)ds=\frac{1}{2}$ and let

$${\parallel \phi \parallel }_{{\mathcal{B}}_h}={\int }_{-\mathrm{\infty }}^{0}h(s)\underset{s\le \theta \le 0}{sup}{\left(E{|\phi (\theta )|}^2\right)}^{\frac{1}{2}}ds.$$

It follows from [5] that $({\mathcal{B}}_h,{\parallel \cdot \parallel }_{{\mathcal{B}}_h})$ is a Banach space. Hence for $(t,\varphi )\in [0,b]\times {\mathcal{B}}_h$, let

$$\begin{array}{r}\varphi (\theta )x=\varphi (\theta ,x),(\theta ,x)\in (-\mathrm{\infty },0]\times [0,\pi ],\\ z(t)(x)=z(t,x)\end{array}$$

and

$$F(t,\varphi )(x)=[Q_1(t,\varphi (\theta ,x)),Q_2(t,\varphi (\theta ,x))],-\mathrm{\infty }<\theta \le 0,x\in [0,\pi ].$$

Then (4.1)-(4.4) can be rewritten as the abstract form as the system (1.1)-(1.3). If we assume that (H2)-(H6) are satisfied, then the system (4.1)-(4.4) has a mild solution on $[0,b]$.