Nonlocal stochastic integro-differential equations driven by fractional Brownian motion

Abstract

In this paper, we study the existence of mild solutions for nonlocal stochastic integro-differential equations driven by fractional Brownian motions with Hurst parameter $H>\frac{1}{2}$ in a Hibert space. Sufficient conditions for the existence of mild solutions are derived by means of the Leray-Schauder nonlinear alternative. A special case of this result is given and an example is provided to illustrate the effectiveness of the proposed result.

1 Introduction

It is well known that fractional Brownian motion (fBm, for short) is a family of centered Gaussian processes with continuous sample paths indexed by the Hurst parameter $H\in(0,1)$. fBm admits stationary increments and self-similarity, and it has a long-memory when $H>\frac{1}{2}$. These compact and significant properties make fBm a natural candidate as a model for noise in a wide variety of physical phenomena, such as biological physics, condensed matter physics, telecommunication networks, mathematical finance, and so on (see [1] and the references therein). Therefore, it is interesting and important to investigate stochastic calculus with respect to fBm and related topics (we refer to [2–5] and references therein for a complete presentation of this subject).

Recently, stochastic differential equations driven by fBm have attracted a lot of attentions of works and the theory has been developed in different settings. Some interesting results of finite-dimensional stochastic differential equations with fractional noise have been investigated by Hu et al. [6], Fan [7], Liu and Yan [8, 9], and references therein; the case of stochastic differential equations driven by fBm in infinite-dimensional Hilbert spaces has also been studied extensively, for example, Boufoussi and Hajji [10] investigated the existence of neutral stochastic functional differential equations driven by fBm, Caraballo et al. [11] proved the existence and exponential behavior of mild solutions to stochastic delay evolution equations with fractional noise, and Duncan et al. [12] established the weak, strong and mild solutions to stochastic equations with multiplicative fractional noise.

Since it was introduced in 1990 by Byszewski and Lakshmikantham [13], the nonlocal Cauchy problems have been extensively studied in differential equations and dynamical systems [14–17]. It is demonstrated that the corresponding differential equations with nonlocal conditions more accurately describe the phenomena. For example, it is discovered in [16] that the nonlocal initial condition $x(0)+g(t_{1},\ldots,t_{n},x(t_{1}),\ldots,x(t_{n}))=x_{0}$, $0< t_{1}< t_{2}<\cdots <t_{n}\leq T$, has better effects in characterizing the diffusion phenomenon of a small amount of gas in a transparent tube than the classical Cauchy condition $x(0)=x_{0}$. In the infinite-dimensional framework, stochastic differential equations with nonlocal conditions driven by Brownian motion (i.e., the case $H=\frac{1}{2}$) have received a lot of attention during the last years. For example, Muthukumar et al. [18] studied the controllability of fractional stochastic integro-differential equations with nonlocal conditions, the existence of solutions for stochastic functional differential equations has been discussed in [19, 20], the second-order stochastic functional differential equations with nonlocal conditions have been investigated in [21, 22] and references therein. For recent important results of stochastic differential equations in Hilbert spaces, we refer to [23, 24] and the references therein.

In contrast, for $H\neq\frac{1}{2}$, to the best of our knowledge, there is no work concerning the existence of mild solutions for stochastic evolution equations with nonlocal conditions. Therefore, the main objective of this paper is to fill this gap. Further, many existence results of stochastic differential equations with nonlocal conditions are valid only for the Lipschitz or compact assumptions on nonlocal items. The main purpose of this manuscript is to investigate the existence of nonlocal stochastic integro-differential equations driven by fractional Brownian motion, for which the nonlocal items are valid for non-Lipschitz and noncompact assumptions.

In this paper, we consider the existence of mild solutions for a class of stochastic integro-differential equations of the following form:

$$ \textstyle\begin{cases} dx(t)= A(t)[x(t)+\int_{0}^{t}B(t,s)x(s)\,ds+f(t,x(t))] \,dt +\sigma \,dB^{H}(t), \quad t\in J:=[0.T], \\ x(0)+g(x)=x_{0}, \end{cases} $$

(1.1)

in a real separable Hilbert space $\mathcal{H}$ with inner product $(\cdot,\cdot)$ and norm $\Vert \cdot \Vert$, where $A(t)$ is a closed linear operator with a dense domain $D(A)$, which is independent of t, $B(t,s)$ is a bounded operator in $\mathcal{H}$. $B^{H}$ is a fBm with $H\in(\frac{1}{2},1)$ on a real separable Hilbert space V. $f:J\times\mathcal{H}\rightarrow\mathcal{H}$, $\sigma\in L_{2}^{0}(V, \mathcal{H})$, $g:C(J,\mathcal{H})\rightarrow \mathcal{H}$ are appropriate functions specified later. Here, $L_{2}^{0}(V, \mathcal {H})$ denotes the space of all Q-Hilbert-Schmidt operators from V into $\mathcal{H}$ (see Section 2 below). $x_{0}$ is an $\mathcal {F}_{0}$-measurable random variable independent of $B^{H}$ with finite second moment.

A brief outline of this paper is given. In Section 2, we present some basic notations and preliminaries; in Section 3, the existence result of system (1.1) is investigated by means of the Leray-Schauder nonlinear alternative, a special case of system (1.1) is also considered. An example is presented to illustrate the effectiveness of the main result.

2 Preliminaries

Throughout this paper, we assume that $H\in(\frac{1}{2},1)$ unless otherwise specified. In this section, our goal is to introduce some useful results about fBm and the corresponding stochastic integral taking values in a Hilbert space. For more details on this section, we refer to Hu [2], Mishura [5] and the references therein.

2.1 Fractional Brownian motion

We begin by recalling the definition of a fBm. Let $(\Omega,\mathcal {F},P)$ be a complete probability space and $H\in(0,1)$ be given. A fBm with Hurst parameter H is a continuous centered Gaussian process $\{\beta^{H}(t); t\in\mathbb {R}\}$ with the covariance function

$$R_{H}(t,s)=E \bigl[\beta^{H} (s)\beta^{H} (t) \bigr] =\frac{1}{2} \bigl(s^{2H}+t^{2H}-\vert t-s\vert ^{2H} \bigr), \quad s, t\in\mathbb{R}. $$

It is well known that fBm $\beta^{H}(t)$ with $H>\frac{1}{2}$ admits the Wiener integral representation of the following form:

$$\beta^{H}(t)= \int_{0}^{t}K_{H}(t,s)\,dB(s), $$

where B is a standard Brownian motion and the kernel $K_{H}(t,s)$ is given by

$$K_{H}(t,s)=c_{H} \int_{s}^{t}(u-s)^{H-\frac{3}{2}} \biggl( \frac{u}{s} \biggr)^{H-\frac {1}{2}}\,du,\quad t>s, $$

with $c_{H}=\sqrt{\frac{H(2H-1)}{\beta(2-2H,H-\frac{1}{2})}}$, here $\beta (\cdot,\cdot)$ denotes the Beta function.

Let $T>0$ be an arbitrary fixed horizon and $\phi\in L^{2}(0,T)$ be a deterministic function. It is well known from [25] that the Wiener integral of ϕ with respect to $\beta^{H}$ is given by

$$\int_{0}^{T}\phi(s)\,d\beta^{H}(s)= \int_{0}^{T}K_{H}^{\star} \phi(s)\,dB(s), $$

where $K_{H}^{\star}\phi(s)=\int_{s}^{T}\phi(r)\frac{\partial K_{H}}{\partial r}(r,s)\,dr$.

2.2 Stochastic integral in Hilbert space

We now introduce fBms with values in a Hilbert space and the corresponding stochastic integral. Let $V=(V,(\cdot,\cdot)_{V}, \Vert \cdot \Vert _{V} )$ be a real, separable Hilbert space. $\mathcal{L}(V,\mathcal{H})$ represents the space of all bounded linear operators from V to $\mathcal{H}$. Let $\{e_{n}, n\in N\}$ be a complete orthonormal basis in V and $Q\in \mathcal{L}(V,\mathcal{H})$ be an operator defined by $Qe_{n}=\lambda_{n}e_{n}$ with $\operatorname {Tr}Q=\sum_{n=1}^{\infty}\lambda_{n}<+\infty$, where $\{\lambda_{n}, n\in N\}$ are non-negative real numbers. A V-valued infinite-dimensional fBm with covariance Q can be defined by

$$B^{H}(t)=\sum_{n=1}^{\infty}\sqrt{ \lambda_{n}}e_{n}\beta_{n}^{H}(t), $$

where $\beta_{n}^{H}$ are real, independent fBms with the same Hurst parameter $H\in(\frac{1}{2},1)$.

Let $L_{2}^{0}(V,\mathcal{H})$ denote the space of all $\Phi\in\mathcal {L}(V,\mathcal{H})$ such that $\Phi Q^{\frac{1}{2}}$ is a Hilbert-Schmidt operator. The norm is defined by

$$\Vert \Phi \Vert _{L_{2}^{0}}^{2}=\sum _{n=1}^{\infty} \Vert \sqrt{\lambda_{n}}\Phi e_{n}\Vert ^{2}. $$

Generally, Φ is called a Q-Hilbert-Schmidt operator from V to $\mathcal{H}$.

Definition 2.1

Let $\phi:[0,T]\rightarrow L_{2}^{0}(V,\mathcal{H})$ such that

$$\sum_{n=1}^{\infty} \bigl\Vert K_{H}^{\star} \bigl(\phi Q^{\frac{1}{2}}e_{n} \bigr) \bigr\Vert _{L_{2}^{0}}^{2}< \infty. $$

Then the stochastic integral of ϕ with respect to the infinite-dimensional fBm $B^{H}$ is defined by

$$\begin{aligned} \int_{0}^{t}\phi(s)\,dB^{H}(s):=\sum _{n=1}^{\infty} \int_{0}^{t}\phi(s)Q^{\frac {1}{2}}e_{n} \,d\beta_{n}^{H}(s) =\sum_{n=1}^{\infty} \int_{0}^{t} \bigl(K_{\mathcal{H}}^{\star} \bigl(\phi Q^{\frac {1}{2}}e_{n} \bigr) \bigr) (s)\,dB(s). \end{aligned}$$

Lemma 2.1

[11]

If $\phi:[0,T]\rightarrow L_{2}^{0}(V,\mathcal{H})$ satisfying

$$\sum_{n=1}^{\infty} \bigl\Vert \phi Q^{\frac{1}{2}}e_{n} \bigr\Vert _{L^{\frac {1}{H}}([0,T],\mathcal{H})}< \infty, $$

and for $\tau,\sigma\in[0,T]$ with $\sigma>\tau$, then

$$E \biggl\Vert \int_{\tau}^{\sigma}\phi(s)\,dB^{H}(s) \biggr\Vert ^{2}\leq C_{H}(\sigma-\tau)^{2H-1} \sum _{n=1}^{\infty} \int_{\tau}^{\sigma} \bigl\Vert \phi(s)Q^{\frac{1}{2}}e_{n} \bigr\Vert ^{2}\,ds, $$

where $C_{H}$ is a constant depending on H. If, in addition, $\sum_{n=1}^{\infty} \Vert \phi(t) Q^{\frac{1}{2}}e_{n}\Vert$ is uniformly convergent for $t\in[0,T]$, then

$$E \biggl\Vert \int_{\tau}^{\sigma}\phi(s)\,dB^{H}(s) \biggr\Vert ^{2}\leq C_{H}(\sigma-\tau)^{2H-1} \int_{\tau}^{\sigma} \bigl\Vert \phi(s) \bigr\Vert _{L_{2}^{0}}^{2}\,ds. $$

Remark 1

If $\{\lambda_{n}\}_{n\in\mathbb{N}}$ is a bounded sequence of non-negative real numbers such that the unclear operator Q satisfies $Qe_{n}=\lambda e_{n}$, assuming that there exists a positive constant $k_{\Phi}$ such that

$$\bigl\Vert \Phi(t) \bigr\Vert _{L_{2}^{0}}\leq k_{\Phi},\quad \text{uniformly in }[0,T], $$

then it is obvious that $\sum_{n=1}^{\infty} \Vert \Phi(t) Q^{\frac {1}{2}}e_{n}\Vert$ is uniformly convergent for $t\in[0,T]$.

2.3 Deterministic integro-differential equations in Banach spaces

In this subsection, we recall some basic notations and properties needed in the sequel. For more details on this subsection, we refer to [26].

Let $(X,\Vert \cdot \Vert )$ be a Banach space, $C(I;X)$ denotes the space of all continuous functions from I into X. We consider the following integral-differential equation with nonlocal condition:

$$\begin{aligned} \textstyle\begin{cases} u'(t)=A(t) [u(t)+\int_{0}^{t}B(t,s)u(s)\,ds ]+h(t,u(t)),\\ u(0)+v(x)=u_{0}, \quad t\in[0,T], u_{0}\in X. \end{cases}\displaystyle \end{aligned}$$

(2.1)

Definition 2.2

[26]

A resolvent operator for problem (2.1) is a bounded operator-valued function $R(t,s)\in\mathcal{L}(X,X)$, $0\leq s\leq t\leq T$, such that the properties

• (a) $R(s,s)=I$, $0\leq s\leq T$, $\Vert R(t,s)\Vert \leq Me^{\gamma(t-s)}$ for some constants M and γ;

• (b) for each $x\in X$, $R(t,s)x$ is strongly continuous in s and t;

• (c) for each $x\in X$, $R(t,s)x$ is continuous differentiable in $s\in[0,T]$, $t\in[s,T]$

and

$$\begin{aligned}& \frac{\partial R}{\partial s}(t,s)x=-R(t,s)A(s)x- \int_{s}^{t}R(t,\tau)B(\tau ,s)A(s)x\,d\tau, \\& \frac{\partial R}{\partial t}(t,s)x=A(t)R(t,s)x+ \int_{s}^{t} B(t,\tau)A(\tau )R(\tau,s)x\,d\tau, \end{aligned}$$

with $\frac{\partial R}{\partial t}(t,s)x$, $\frac{\partial R}{\partial s}(t,s)x$ are strongly continuous on $0\leq s\leq t\leq T$.

We mention here that the resolvent operator $R(t,s)$ can be reduced from the evolution operator of the generator $A(t)$ under some suitable conditions (see [26] for the details).

Definition 2.3

[26]

A continuous function $u(\cdot):J\rightarrow X$ is said to be a mild solution to problem (2.1) if, for all $u_{0}\in X$, it satisfies the following integral equation:

$$u(t)=R(t,0) \bigl[u_{0}-v(x) \bigr]+ \int_{0}^{t}R(t,s)h \bigl(s,u(s) \bigr)\,ds. $$

2.4 Hypotheses on stochastic system (1.1)

Let $C(J,\mathcal{H}):= \{x:J\rightarrow\mathcal{H}, x \text{ is continuous almost surely}\}$. The collection of all strongly measurable, square integrable, $\mathcal{H}$-valued random variables, denoted by $L^{2}(\Omega,\mathcal{H})$, is a Banach space with the norm $\Vert x\Vert _{2}=(E\Vert x\Vert ^{2})^{\frac{1}{2}}$. Let $C(J,L^{2}(\Omega,H))$ be the Banach space of all continuous mappings from J into $L^{2}(\Omega,H)$ such that $\sup_{t\in J}E\vert x(t)\vert _{H}^{2}<\infty$. We denote by $\mathcal{C}$ the space of all $\mathcal{F}_{t}$-adapted stochastic processes $x\in C(J,L^{2}(\Omega,H))$ equipped with the norm $\Vert x\Vert _{\mathcal{C}}=(\sup_{s\in J}E\vert x(s)\vert _{H}^{2})^{\frac{1}{2}}$, it is clear that $(\mathcal{C},\Vert \cdot \Vert _{\mathcal{C}})$ is a Banach space.

In this work, we need the following assumptions:

$(A_{1})$ :

The resolvent operator $R(t,s)$, $0\leq s\leq t$, is compact and there exist some positive constants M, γ such that $\Vert R(t,s)\Vert \leq Me^{\gamma( t-s)}$.

$(A_{2})$ :

There exists a constant $K_{f}>0$ such that for all $x,y\in\mathcal{H}$, $t\in[0,T]$,

$$\bigl\Vert f(t,x)-f(t,y) \bigr\Vert ^{2}\leq K_{f} \Vert x-y\Vert ^{2}. $$

$(A_{3})$ :

$\sigma\in L_{2}^{0}(V,\mathcal{H})$ with $\sum_{n=1}^{\infty} \Vert \sigma Q^{\frac{1}{2}}e_{n}\Vert _{L_{2}^{0}}<\infty$.

$(A_{4})$ :

$g:C(J,\mathcal{H})\rightarrow\mathcal{H}$ is square integrable and satisfies:

• (4a) there exists a constant $\alpha\in(0,T)$ such that $g(\xi)=g(\eta)$ for any $\xi,\eta\in C(J,\mathcal{H})$ with $\xi=\eta$ on $[\alpha, T]$;

• (4b) there exists a nondecreasing function $\Gamma:[0,+\infty)\rightarrow(0,+\infty)$ such that

  $$E \bigl\vert g(x) \bigr\vert _{H}^{2}\leq\Gamma \bigl( \Vert x\Vert _{\mathcal{C}}^{2} \bigr). $$

For simplicity, we also assume that $f(t,0)=0$ for all $t\in J$.

At the end of this section, we recall the well-known Leray-Schauder nonlinear alternative, which is employed to prove our results.

Lemma 2.2

[27]

Let X be a closed and convex subset of a Banach space B. Assume that $\mathcal{U}$ is a relatively open subset of X with $0\in\mathcal {U}$ and $\phi: \bar{\mathcal{U}}\rightarrow X$ is a compact map, then either

• (a) ϕ has a fixed point in $\bar{\mathcal{U}}$, or

• (b) there is a point $u\in\partial\mathcal{U}$ and $\lambda\in(0,1)$ with $u\in\lambda\phi(u)$.

3 Main results

In this section, we state and establish our main result. We first present the definition of the mild solution for system (1.1).

Definition 3.1

A $\mathcal{H}$-valued, $\mathcal{F}_{t}$-adapted stochastic process $x(t)$ is called a mild solution of (1.1), if, for all $t\in[0,T]$, $x(t)$ satisfies

$$\begin{aligned} x(t)=R(t,0) \bigl[x_{0}-g(x) \bigr]+ \int_{0}^{t}R(t,s)f \bigl(s,x(s) \bigr)\,ds + \int_{0}^{t}R(t,s)\sigma \,dB^{H}(s). \end{aligned}$$

Theorem 3.1

Let assumptions $(A_{1})$-$(A_{4})$ be satisfied. Then the system (1.1) has at least one mild solution provided that there exists a constant $N_{0}>0$ such that

$$ \frac{N_{0}}{[6M^{2}(E\vert x_{0}\vert _{H}^{2}+\Gamma(N_{0}))+3M^{2}C_{H}T^{2H-1} \Vert \sigma \Vert _{\mathcal{L}_{2}^{0}}^{2}(2\gamma)^{-1}]e^{(3M^{2}K_{f}T+2\gamma) T}}>1. $$

(3.1)

Proof

We first introduce an equivalent norm $\Vert \cdot \Vert _{\tilde{\mathcal{C}}}$ in the space $\mathcal{C}$ defined by

$$\Vert x\Vert _{\tilde{\mathcal{C}}}^{2}:=\sup_{t\in J}e^{-Lt}E \bigl\vert x(t) \bigr\vert ^{2}, \quad \text{where } L=2M^{2}TK_{f}+2\gamma. $$

It is routine to check that $\tilde{\mathcal{C}}:=(\mathcal{C},\Vert \cdot \Vert _{\tilde{\mathcal{C}}})$ is a Banach space. Let $u\in\mathcal{C}$ be fixed. For $t\in J$, $x\in\tilde{\mathcal{C}}$, define a mapping $P_{u}$ on $\tilde{\mathcal{C}}$ by

$$ \begin{aligned} (P_{u}x) (t)=R(t,0) \bigl[x_{0}-g(u) \bigr]+ \int_{0}^{t}R(t,s)f \bigl(s,x(s) \bigr)\,ds + \int_{0}^{t}R(t,s)\sigma \,dB^{H}(s). \end{aligned} $$

(3.2)

Since $R(t,s)$ is strongly continuous, by the assumptions on f, g, and σ, it is easy to check that $P_{u}$ maps $\tilde{\mathcal{C}}$ into itself. For $x,y\in\tilde{\mathcal{C}}$, by $(A_{1})$ and Hölder’s inequality, we have

$$\begin{aligned} e^{-Lt}E \bigl\vert (P_{u}x) (t)-(P_{u}y) (t) \bigr\vert ^{2} &=e^{-Lt}E \biggl\Vert \int_{0}^{t}R(t,s) \bigl[f \bigl(s,x(s) \bigr)-f \bigl(s,y(s) \bigr) \bigr]\,ds \biggr\Vert ^{2} \\ &\leq e^{-Lt}E \biggl[ \int_{0}^{t} \bigl\Vert R(t,s) \bigl[f \bigl(s,x(s) \bigr)-f \bigl(s,y(s) \bigr) \bigr] \bigr\Vert \,ds \biggr]^{2} \\ &\leq e^{-Lt}M^{2}T \int_{0}^{t}e^{2\gamma(t-s)}E \bigl\Vert f \bigl(s,x(s) \bigr)-f \bigl(s,y(s) \bigr) \bigr\Vert ^{2}\,ds \\ &\leq e^{-Lt}M^{2}TK_{f} \int_{0}^{t}e^{2\gamma(t-s)}E \bigl\Vert x(s)-y(s) \bigr\Vert ^{2}\,ds \\ &\leq M^{2}TK_{f} \int_{0}^{t}e^{(2\gamma-L)(t-s)}e^{-Ls}E \bigl\Vert x(s)-y(s) \bigr\Vert ^{2}\,ds \\ &\leq\frac{M^{2}TK_{f}}{L-2\gamma}\sup_{s\in J}e^{-Ls}E \bigl\Vert x(s)-y(s) \bigr\Vert ^{2}, \end{aligned}$$

recalling the choice of L, we have

$$\begin{aligned} e^{-Lt}E \bigl\vert (P_{u}x) (t)-(P_{u}y) (t) \bigr\vert ^{2}\leq\frac{1}{2}\sup_{s\in J}e^{-Ls}E \bigl\Vert x(s)-y(s) \bigr\Vert ^{2}, \end{aligned}$$

which implies that $\Vert P_{u}x-P_{u}y\Vert _{\tilde{\mathcal{C}}}^{2}\leq\frac {1}{2}\Vert x-y\Vert _{\tilde{\mathcal{C}}}^{2}$. By the Banach contraction mapping principle, it follows that $P_{u}$ has a unique fixed point $x_{u}\in \tilde{\mathcal{C}}$ such that

$$\begin{aligned} (x_{u}) (t)=R(t,0) \bigl[x_{0}-g(u) \bigr]+ \int_{0}^{t}R(t,s)f \bigl(s,x_{u}(s) \bigr)\,ds + \int_{0}^{t}R(t,s)\sigma \,dB^{H}(s). \end{aligned}$$

Based on this fact, for some $\alpha\in(0,T)$, we set

$$\begin{aligned} \bar{u}(t):= \textstyle\begin{cases} u(t),&t\in(\alpha,T]; \\ u(\alpha), &t\in[0,\alpha]. \end{cases}\displaystyle \end{aligned}$$

From (3.2), it follows that

$$\begin{aligned} x_{\bar{u}}(t)&=R(t,0) \bigl[x_{0}-g(\bar{u}) \bigr]+ \int_{0}^{t}R(t,s)f \bigl(s,x_{\bar{u}}(s) \bigr)\,ds + \int_{0}^{t}R(t,s)\sigma \,dB^{H}(s). \end{aligned}$$

Let $\mathcal{C}_{\alpha}:=C([\alpha,T],L^{2}(\Omega,H))$. Define a map $\Phi:\mathcal{C}_{\alpha}\rightarrow\mathcal{C}_{\alpha}$ by

$$\begin{aligned} (\Phi u) (t)=x_{\bar{u}}(t),\quad t\in[\alpha,T]. \end{aligned}$$

We will show that the operator Φ satisfies all the conditions of Lemma 2.2. The proof will be divided into the following steps.

Step 1. Φ maps bounded sets into bounded sets in $C_{\alpha}$. Let $r>0$ and

$$B_{r}(\alpha):= \Bigl\{ x\in\mathcal{C}_{\alpha}: \sup _{t\in[\alpha,T]}E \bigl\vert x(t) \bigr\vert ^{2}\leq r \Bigr\} . $$

It is obvious that $B_{r}(\alpha)$ is a bounded closed convex subset in $\mathcal{C}_{\alpha}$. We now show that there exists a constant $L_{0}>0$ such that $\Vert \Phi v\Vert _{\mathcal{C}}^{2}\leq L_{0}$ for each $v\in B_{r}(\alpha)$.

Let $v\in B_{r}(\alpha)$ and $t\in J$. Recalling the definition of Φ, by the assumptions $(A_{1})$-$(A_{4})$ together with Hölder’s inequality we have

$$\begin{aligned} e^{-2\gamma t}E \bigl\Vert x_{\bar{v}}(t) \bigr\Vert ^{2}& \leq3e^{-2\gamma t}E \bigl\Vert R(t,0) \bigl[x_{0}-g(\bar{v}) \bigr] \bigr\Vert ^{2} \\ &\quad {}+ 3e^{-2\gamma t}E \biggl\Vert \int_{0}^{t}R(t,s)f \bigl(s,x_{\bar{v}}(s) \bigr)\,ds \biggr\Vert ^{2} +3e^{-2\gamma t}E \biggl\Vert \int_{0}^{t}R(t,s)\sigma \,dB^{H}(s) \biggr\Vert ^{2} \\ &\leq6M^{2} \bigl[E\Vert x_{0}\Vert ^{2}+E \bigl\Vert g(\bar{v}) \bigr\Vert ^{2} \bigr]+3e^{-2\gamma t}M^{2}T \int _{0}^{t}e^{2\gamma(t-s)}E \bigl\Vert f \bigl(s,x_{\bar{v}}(s) \bigr) \bigr\Vert ^{2}\,ds \\ &\quad{} +3e^{-2\gamma t}M^{2}C_{H}T^{2H-1} \int_{0}^{t}e^{2\gamma(t-s)}\Vert \sigma \Vert _{\mathcal{L}_{2}^{0}}^{2}\,ds \\ &\leq L_{1}+3M^{2}T \int_{0}^{t}e^{-2\gamma s}E \bigl\Vert x_{\bar{v}}(s) \bigr\Vert ^{2}\,ds, \end{aligned}$$

where $L_{1}=6M^{2}[E\Vert x_{0}\Vert ^{2}+\Gamma(r)]+\frac{3M^{2}C_{H}T^{2H-1}\Vert \sigma \Vert _{\mathcal{L}_{2}^{0}}^{2}}{2\gamma}$. This gives, by Gronwall’s inequality,

$$\Vert \Pi v\Vert _{\mathcal{C}}^{2}= \Vert x_{\bar{v}} \Vert _{\mathcal{C}}^{2}\leq L_{1}e^{3M^{2}T+2\gamma T}:=L_{0}. $$

Step 2. Φ maps bounded sets into equicontinuous sets of $\mathcal{C}_{\alpha}$. Let $v\in B_{r}(\alpha)$ and $\alpha\leq t_{1}< t_{2}\leq T$, we have

$$ \begin{aligned}[b] &E \bigl\Vert (\Phi v) (t_{2})-(\Phi v) (t_{1}) \bigr\Vert ^{2} \\ & \quad \leq3 \bigl\Vert R(t_{2},0)-R(t_{1},0) \bigr\Vert ^{2}E \bigl\Vert x_{0}-g(\bar{v})) \bigr\Vert ^{2} \\ &\quad \quad{} +3E \biggl\Vert \int_{0}^{t_{1}} \bigl[R(t_{2},s)-R(t_{1},s) \bigr]f \bigl(s,x_{\bar{v}}(s) \bigr)\,ds + \int _{t_{1}}^{t_{2}}R(t_{2},s)f \bigl(s,x_{\bar{v}}(s) \bigr)\,ds \biggr\Vert ^{2} \\ &\quad \quad{} +3E \biggl\Vert \int_{0}^{t_{1}} \bigl[R(t_{2},s)-R(t_{1},s) \bigr]\sigma \,dB^{H}(s) + \int _{t_{1}}^{t_{2}}R(t_{2},s)\sigma \,dB^{H}(s) \biggr\Vert ^{2} \\ &\quad =I_{1}+I_{2}+I_{3}. \end{aligned} $$

(3.3)

By the strong continuity of $R(t,s)$ and the assumption of g, we have

$$\begin{aligned} I_{1}\leq6 \bigl\Vert R(t_{2},0)-R(t_{1},0) \bigr\Vert ^{2} \bigl[E\Vert x_{0}\Vert ^{2}+ \Gamma(r) \bigr]\rightarrow 0, \quad \text{as }t_{1}\rightarrow t_{2}. \end{aligned}$$

For the term $I_{2}$, by Hölder’s inequality and $(A_{1})$ we have

$$ \begin{aligned}[b] I_{2}&\leq6t_{1} \int_{0}^{t_{1}} \bigl\Vert R(t_{2},s)-R(t_{1},s) \bigr\Vert ^{2}E \bigl\Vert f \bigl(s,x_{\bar{v}}(s) \bigr) \bigr\Vert ^{2}\,ds \\ &\quad{} +6M^{2}(t_{2}-t_{1}) \int_{t_{1}}^{t_{2}}e^{2\gamma(t_{2}-s)}E \bigl\Vert f \bigl(s,x_{\bar {v}}(s) \bigr) \bigr\Vert ^{2}\,ds, \end{aligned} $$

(3.4)

applying properties (b) of Definition 2.2 we get

$$\lim_{t_{1}\rightarrow t_{2}} \bigl\Vert R(t_{2},s)-R(t_{1},s) \bigr\Vert ^{2}E \bigl\Vert f \bigl(s,x_{\bar{v}}(s) \bigr) \bigr\Vert ^{2}=0, $$

by the assumption on f, we further derive that

$$\begin{aligned} & \int_{0}^{t_{1}} \bigl\Vert R(t_{2},s)-R(t_{1},s) \bigr\Vert ^{2}E \bigl\Vert f \bigl(s,x_{\bar{v}}(s) \bigr) \bigr\Vert ^{2}\,ds \\ &\quad = \int_{0}^{t_{1}} \bigl\Vert R(t_{2},s)-R(t_{1},s) \bigr\Vert ^{2}E \bigl\Vert f \bigl(s,x_{\bar{v}}(s) \bigr)-f(s,0) \bigr\Vert ^{2}\,ds \\ &\quad \leq2M^{2} \int_{0}^{t_{1}} \bigl[e^{2\gamma(t_{2}-s)}+e^{2\gamma(t_{1}-s)} \bigr]\,ds\sup_{t\in[\alpha,T]} E \bigl\Vert x_{\bar{v}}(t) \bigr\Vert ^{2}< \infty, \end{aligned}$$

we conclude, by the dominated convergence theorem, that the first term of (3.4) tends to 0 independently of v. Similarly, we can deduce that the second term of (3.4) also tends to 0 independently of v.

For $I_{3}$, by Lemma 2.1 and the properties of $R(t,s)$ we have

$$\begin{aligned} I_{3}&\leq6C_{H}t_{1}^{2H-1} \int_{0}^{t_{1}} \bigl\Vert R(t_{2},s)-R(t_{1},s) \bigr\Vert ^{2}\Vert \sigma \Vert _{\mathcal{L}_{2}^{0}}^{2}\,ds \\ &\quad{} +6C_{H}(t_{2}-t_{1})^{2H-1} \int_{t_{1}}^{t_{2}}e^{2\gamma(t_{2}-s)}\Vert \sigma \Vert _{\mathcal{L}_{2}^{0}}^{2}\,ds \\ &\rightarrow0,\quad \text{as }t_{1}\rightarrow t_{2}, \end{aligned}$$

where $C_{H}$ is the constant in Lemma 2.1.

Thus, the right hand side of (3.3) tends to 0 independently of v when $t_{1}\rightarrow t_{2}$. From the assumption $(A_{1})$, the compactness of $R(t,s)$, $t>s>0$ implies the continuity in the uniform operator topology. Therefore, the set $\{\Phi v,v\in B_{r}(\alpha)\}$ is equicontinuous.

Step 3. Φ maps $B_{r}(\alpha)$ into a relatively compact set in $\mathcal{H}$. That is, for every $t\in[\alpha,T]$, the set $\Pi (t):=\{(\Phi v)(t),v\in B_{r}(\alpha)\}$ is relatively compact in H.

Let $t\in[\alpha,T]$ be fixed and $0<\epsilon<t$. For $v\in B_{r}(\alpha )$, we define an operator $\Phi^{\epsilon}$ on $B_{r}(\alpha)$ by

$$\begin{aligned} \bigl(\Phi^{\epsilon}v \bigr) (t)&=R(t,0) \bigl[x_{0}-g(\bar{v}) \bigr]+ \int_{0}^{t-\epsilon }R(t,s)f \bigl(s,x_{\bar{v}}(s) \bigr)\,ds \\ &\quad{} + \int_{0}^{t-\epsilon}R(t,s)\sigma \,dB^{H}(s). \end{aligned}$$

By the compactness of $R(t,s)$, $t,s>0$, one sees that the set $\Phi^{\epsilon}(t)=\{( \Phi^{\epsilon}v)(t),v\in B_{r}(\alpha)\}$ is relatively compact in H for each $0<\epsilon<t$. Moreover, applying Hölder’s inequality together with Lemma 2.1 we have

$$\begin{aligned} &E \bigl\Vert (\Phi v) (t)- \bigl( \Phi^{\epsilon}v \bigr) (t) \bigr\Vert ^{2} \\ &\quad \leq 2 E \biggl\Vert \int_{t-\epsilon}^{t}R(t,s)f \bigl(s,x_{\bar{v}}(s) \bigr)\,ds \biggr\Vert ^{2} +2E \biggl\Vert \int_{t-\epsilon}^{t}R(t,s)\sigma \,dB^{H}(s) \biggr\Vert ^{2} \\ &\quad \leq2M^{2}\epsilon \int_{t-\epsilon}^{t} e^{2\gamma(t-s)}E \bigl\Vert f \bigl(s,x_{\bar {v}}(s) \bigr) \bigr\Vert ^{2}\,ds +2M^{2}C_{H}\epsilon^{2H-1} \int_{t-\epsilon}^{t} e^{2\gamma(t-s)}\Vert \sigma \Vert _{\mathcal{L}_{2}^{0}}^{2}\,ds, \end{aligned}$$

which implies that the relatively compact set $\{(\Phi^{\epsilon }v):v\in B_{r}(\alpha)\}$ arbitrarily close to the set $\{(\Phi v):v\in B_{r}(\alpha)\}$. Thus, the set $\{(\Phi v),v\in B_{r}(\alpha)\}$ is relatively compact in H.

Step 4. We show that there exists an open set $\mathbf {D}\subseteq\mathcal{C}_{\alpha}$ such that, for all $v\in\partial\mathbf{D}$, $v\notin\{\lambda(\Phi v):\lambda\in (0,1)\}$.

Let $v\in\mathcal{C}_{\alpha}$ be a possible solution of $v=\lambda(\Phi v)$ for some $\lambda\in(0,1)$. Then, for each $t\in J$, we have

$$\begin{aligned} v(t)&=\lambda(\Phi v) (t)=\lambda x_{\bar{v}}(t) \\ &=\lambda R(t,0) \bigl[x_{0}-g(\bar{v}) \bigr] +\lambda \int_{0}^{t}R(t,s)f \bigl(s,x_{\bar{v}}(s) \bigr)\,ds +\lambda \int_{0}^{t}R(t,s)\sigma \,dB^{H}(s). \end{aligned}$$

It is clear that

$$ \begin{aligned} \Vert v\Vert _{\mathcal{C}}^{2} \leq \Vert \Phi v\Vert _{\mathcal{C}}^{2}=\Vert x_{\bar{v}} \Vert _{\mathcal{C}}^{2}. \end{aligned} $$

(3.5)

On the other hand, by the assumptions on $R(t,s)$, f, and g, we can use a similar argument to before, yielding

$$\begin{aligned} e^{-2\gamma t}E \bigl\Vert x_{\bar{v}}(t) \bigr\Vert ^{2} & \leq 6M^{2} \bigl[E\Vert x_{0}\Vert ^{2}+\Gamma \bigl(\Vert \bar{v}\Vert _{\mathcal{C}}^{2} \bigr) \bigr]+3M^{2}T \int _{0}^{t}e^{-2\gamma s} E \bigl\Vert f \bigl(s,x_{\bar{v}}(s) \bigr) \bigr\Vert ^{2}\,ds \\ &\quad{} +3M^{2}C_{H}T^{2H-1} \int_{0}^{t}e^{-2\gamma s}\Vert \sigma \Vert _{\mathcal {L}_{2}^{0}}^{2}\,ds \\ &\leq6M^{2} \bigl[E\Vert x_{0}\Vert ^{2}+ \Gamma \bigl(\Vert \bar{v}\Vert _{\mathcal{C}}^{2} \bigr) \bigr] +3M^{2}HT^{2H-1}(2\gamma)^{-1}\Vert \sigma \Vert _{\mathcal{L}_{2}^{0}}^{2} \\ &\quad{} +3M^{2}TK_{f} \int_{0}^{t}e^{-2\gamma s} E \bigl\Vert x_{\bar{v}}(s)) \bigr\Vert ^{2}\,ds. \end{aligned}$$

Applying Gronwall’s inequality it follows that

$$E \bigl\vert x_{\bar{v}}(t) \bigr\vert _{H}^{2}\leq \bigl[6M^{2} \bigl(E\Vert x_{0}\Vert ^{2}+ \Gamma \bigl(\Vert \bar{v}\Vert _{\mathcal{C}}^{2} \bigr) \bigr) +3M^{2}C_{H}T^{2H-1}(2\gamma)^{-1}\Vert \sigma \Vert _{\mathcal {L}_{2}^{0}}^{2} \bigr]e^{3M^{2}K_{f}T^{2}+2\gamma T}. $$

Recalling (3.5), we have

$$\Vert v\Vert _{\mathcal{C}}^{2}\leq \bigl[6M^{2} \bigl(E \Vert x_{0}\Vert ^{2}+\Gamma \bigl(\Vert \bar{v} \Vert _{\mathcal{C}}^{2} \bigr) \bigr) +3M^{2}C_{H}T^{2H-1}(2 \gamma)^{-1}\Vert \sigma \Vert _{\mathcal {L}_{2}^{0}}^{2} \bigr]e^{3M^{2}K_{f}T^{2}+2\gamma T}. $$

That is,

$$\frac{\Vert v\Vert _{\mathcal{C}}^{2}}{[6M^{2}(E\Vert x_{0}\Vert ^{2}+\Gamma(\Vert \bar{v}\Vert _{\mathcal{C}}^{2})) +3M^{2}C_{H}T^{2H-1}(2\gamma)^{-1}\Vert \sigma \Vert _{\mathcal {L}_{2}^{0}}^{2}]e^{3M^{2}K_{f}T^{2}+2\gamma T}}\leq1. $$

By assumption (3.1), there exists a constant $M^{\star}$ such that $\Vert v\Vert _{\mathcal {C}}^{2}\neq M^{\star}$.

Set

$$\mathbf{D}= \Bigl\{ v\in C_{\alpha}: \sup_{t\in[\alpha,T]}E \bigl\Vert v(s) \bigr\Vert ^{2}< M^{\star} \Bigr\} . $$

It is obvious that D is an open subset of $B_{r}(\alpha)$ for all $r\geq M^{\star}$. We infer that, for all $v\in\partial\mathbf {D}$, $v\notin\{\lambda(\Phi v):\lambda\in(0,1)\}$.

From step 1 to step 4 together with the Arzela-Ascoli theorem, it suffices to show that $\Phi:\bar{\mathbf{D}}\rightarrow\mathcal{C}_{\alpha}$ is a compact operator, where $\bar{\mathbf{D}}$ is the closure of D. As a consequence of Lemma 2.2, it follows that Φ has a fixed point $v_{0}\in\bar{\mathbf{D}}$. Let $y=x_{v_{0}}$. By (3.2), we have

$$\begin{aligned} y(t)&=R(t,0) \bigl[x_{0}-g(v_{0}) \bigr]+ \int_{0}^{t}R(t,s)f \bigl(s,x_{v_{0}}(s) \bigr)\,ds + \int_{0}^{t}R(t,s)\sigma \,dB^{H}(s) \\ &=R(t,0) \bigl[x_{0}-g(v_{0}) \bigr]+ \int_{0}^{t}R(t,s)f \bigl(s,y(s) \bigr)\,ds + \int_{0}^{t}R(t,s)\sigma \,dB^{H}(s). \end{aligned}$$

Noting that for $t\in[\alpha,T]$,

$$y(t)=x_{v_{0}}(t)=x_{\bar{v_{0}}}(t)=(\Phi v_{0}) (t)=v_{0}(t), $$

which indicates, by the assumptions on g, that $g(v_{0})=g(y)$. We conclude that $y(t)$ is a mild solution of system (1.1). The proof is completed. □

We conclude this section with a comment on a special case of the nonlocal Cauchy problem, namely, where $g(x)$ is given by $g(x)=\sum_{i=0}^{k}g_{i}(x(t_{i}))$. More precisely, consider the following stochastic integral-differential equation with nonlocal condition:

$$ \textstyle\begin{cases} { d}x(t)= A(t)[x(t)+\int_{0}^{t}B(t,s)x(s)\,ds+f(t,x(t))]\,{ d}t+\sigma\,{ d}B^{H}(t),\\ \quad t\in J:=[0,T], \\ x(0)+\sum_{i=0}^{k}g_{i}(x(t_{i}))=x_{0}, \quad 0< t_{1}< t_{2}< \cdots< t_{k}< T, k\in \mathbb{N}^{+}. \end{cases} $$

(3.6)

We need the following assumptions.

$(A_{5})$ :

For every i, $i=1,2,\ldots,k$, $g_{i}(\cdot ):C(J,\mathcal{H})\rightarrow\mathcal{H}$ satisfy:

• (5a) there exists a constant $\alpha\in(0,T)$ such that $g_{i}(\xi)=g_{i}(\eta)$ for any $\xi,\eta\in C(J,\mathcal{H})$ with $\xi =\eta$ on $[\alpha, T]$;

• (5b) there exist some positive constants $m_{i}$, $n_{i}$, $i=1,2,\ldots,k$ such that

  $$\bigl\vert g_{i}(x) \bigr\vert ^{2}\leq m_{i} \Vert x\Vert ^{2}+n_{i},\quad\quad 6kM^{2}K^{\star} \sum_{i=1}^{k}m_{i}< 1, $$

  where $K^{\star}=e^{6M^{2}T^{2}K_{f}+2\gamma T}$.

Theorem 3.2

Under the assumptions $(A_{1})$-$(A_{3})$ and $(A_{5})$, the system (3.6) has at least one mild solution.

Proof

Since the proof resembles the arguments of Theorem 3.1, we sketch it only.

We define a map g on $\mathcal{H}$ by

$$g(u)=\sum_{i=0}^{k}g_{i} \bigl(u(t_{i}) \bigr),\quad 0< t_{1}< t_{2}< \cdots< t_{k}< T, k\in \mathbb{N}^{+}. $$

Let L, $\tilde{\mathcal{C}}$, $P_{u}$, ū, Φ, and $B_{r}(\alpha)$ be as in Theorem 3.1. Thus, $P_{u}$ has a unique fixed point $x_{u}$. By (3.2), we have

$$\begin{aligned} (x_{\bar{u}}) (t)&=R(t,0) \Biggl[x_{0}-\sum _{i=0}^{k}g_{i} \bigl( \bar{u}(t_{i}) \bigr) \Biggr] \\ &\quad{} + \int _{0}^{t}R(t,s)f \bigl(s,x_{\bar{u}}(s) \bigr)\,ds + \int_{0}^{t}R(t,s)\sigma \,dB^{H}(s). \end{aligned}$$

By using the assumptions on f, g, and σ, we can deduce that

$$\begin{aligned} &e^{-2\gamma t}E \bigl\Vert x_{\bar{u}}(t) \bigr\Vert ^{2} \\ & \quad \leq6M^{2} \Biggl[E\Vert x_{0}\Vert ^{2}+k\sum _{i=1}^{k}E \bigl\Vert g_{i} \bigl(\bar{u}(t_{i}) \bigr) \bigr\Vert ^{2} \Biggr] \\ & \quad \quad{} +3M^{2}T \int_{0}^{t}e^{-2\gamma s}E \bigl\Vert f \bigl(s,x_{\bar{u}}(s) \bigr) \bigr\Vert ^{2}\,ds +3M^{2}C_{H}T^{2H-1} \int_{0}^{t}e^{-2\gamma s}\Vert \sigma \Vert _{\mathcal{L}_{2}^{0}}^{2}\,ds \\ &\quad \leq6M^{2} \Biggl[E\Vert x_{0}\Vert ^{2}+k \sum_{i=1}^{k} \bigl(m_{i}\Vert \bar{u}\Vert _{\mathcal{C}}^{2}+n_{i} \bigr) \Biggr]+ 3M^{2}C_{H}T^{2H-1}(2\gamma)^{-1}\Vert \sigma \Vert _{\mathcal{L}_{2}^{0}}^{2} \\ &\quad \quad{} +6M^{2}T(2\gamma)^{-1}\sup_{s\in J}E \bigl\Vert f(s,0) \bigr\Vert ^{2}+6M^{2}TK_{f} \int _{0}^{t}e^{-2\gamma s}E \bigl\Vert x_{\bar{u}}(s) \bigr\Vert ^{2}\,ds. \end{aligned}$$

By Gronwall’s inequality it follows that

$$\begin{aligned} E \bigl\Vert x_{\bar{u}}(t) \bigr\Vert ^{2}\leq \Biggl( \bar{L}+6M^{2}k\sum_{i=1}^{k}m_{i} \Vert \bar {u}\Vert _{\mathcal{C}}^{2} \Biggr)K^{\star}, \end{aligned}$$

where

$$\bar{L}=6M^{2} \Biggl[E\Vert x_{0}\Vert ^{2}+k \sum_{i=1}^{k}n_{i}+T(2 \gamma)^{-1}\sup_{s\in J}E \bigl\Vert f(s,0) \bigr\Vert ^{2} \Biggr]+3M^{2}C_{H}T^{2H-1}(2 \gamma)^{-1}\Vert \sigma \Vert _{\mathcal{L}_{2}^{0}}^{2}. $$

Noting that

$$\Vert u\Vert _{\mathcal{C}}^{2}\leq \Vert x_{\bar{u}} \Vert _{\mathcal{C}}^{2}, $$

we further derive that

$$\Vert u\Vert _{\mathcal{C}}^{2}\leq\frac{K^{\star}\bar{L}}{1-6M^{2}kK^{\star} \sum_{i=1}^{k}m_{i}}. $$

Let $\epsilon>0$ be fixed. Set $M^{\star}=\frac{K^{\star}\bar {L}}{1-6M^{2}kK^{\star}\sum_{i=1}^{k}m_{i}}+\epsilon$ and

$$D= \Bigl\{ v\in C \bigl([\alpha,T],\mathcal{H} \bigr):\sup_{t\in[\alpha,T]}E \bigl\Vert v(t) \bigr\Vert ^{2}< M^{\star} \Bigr\} . $$

Then D is an open subset of $B_{r}(\alpha)$ for all $r\geq M^{\star}$. In the sequel, we can employ similar arguments to step 1-step 4 of Theorem 3.1 and prove the existence of mild solutions for system (3.6), we omit it here. This completes the proof. □

4 An example

As an application, we present an example to illustrate our results. Considering the following stochastic integro-differential equation with nonlocal condition:

$$ \textstyle\begin{cases} \frac{\partial x(t,z)}{\partial t}=\frac{\partial^{2}}{\partial z^{2}}a(t)x(t,z)+\int_{0}^{t}B(t,s)x(t,z)\,ds+\sin tx(t,z)+\frac {dB^{H}(t)}{dt},\\ \quad x\in[0,\pi], t\in[0,T], \\ x(t,0)=x(t,\pi)=0, \quad t\in[0,T],\\ x(0,z)=\sum_{i=1}^{k}h_{i}(z)x(t_{i},z), \quad z\in[0,\pi], \end{cases} $$

(4.1)

where $0< t_{1}< t_{2}<\cdots<t_{k}<T$, $B^{H}(t)$ stands for a cylindrical fBm defined on a complete probability space $(\Omega, \mathcal{F},P,\{{\mathscr {F}}_{t}\} )$, $h_{i}\in L^{2}([0,\pi])$, and $a(t)$ are continuous functions.

To rewrite the stochastic differential equation in the abstract form (1.1), let $\mathcal{H}=K=V=L^{2}([0,\pi])$ with the norm $\Vert \cdot \Vert$. The operator $A(t)$ is defined by $A(t)x(z)=a(t)\frac{\partial ^{2}x}{\partial z^{2}}$ with the domain

$$\mathcal{D}(A)= \bigl\{ x\in\mathcal{H}, x, x{''} \text{ are absolutely continuous}, x{''}\in\mathcal{H} \text{ and }x(0)=x(\pi)=0 \bigr\} . $$

Then $A(t)$ generates an evolution operator and $R(t,s)$ can be reduced by this evolution operator such that $(A_{1})$ is satisfied (see [26] for details).

Let

$$\begin{aligned} f \bigl(t,x(t,\cdot) \bigr)=\sin tx(t,\cdot),\quad\quad g_{i}(x) ( \cdot)=h_{i}(\cdot )x(t_{i},\cdot), \quad\quad \sigma=I. \end{aligned}$$

Then system (4.1) has an abstract formulation given by (3.6) and the assumptions $(A_{2})$, $(A_{3})$, and $(A_{5})$ are satisfied with $K_{f}=1$, $m_{i}=\sup_{z\in[0,\pi]}\Vert h_{i}(z)\Vert ^{2}$, $n_{i}=0$. By Theorem 3.2, the system (4.1) has a mild solution on $[0,T]$ provided that $6kM^{2}K^{\star }\sum_{i=1}^{k}m_{i}<1$.

5 Conclusion

In this paper, we have investigated the existence of mild solutions for a class of nonlocal stochastic integro-differential equations driven by fractional Brownian motion in a Hilbert space. By employing the Leray-Schauder nonlinear alternative, the existence of mild solutions is proved, a special case of this result is given. An example is presented to illustrate our theoretical results. In a sequel of this paper we will study the asymptotic behavior of the mild solutions, and we are also interested in studying the existence and asymptotic stability of stochastic evolution equations driven by fractional Brownian motion with the Hurst parameter $H\in(0,\frac{1}{2})$.