Global attracting set and exponential decay of second-order neutral stochastic functional differential equations driven by fBm

Abstract

In this paper, we are concerned with a class of second-order neutral stochastic functional differential equations driven by a fractional Brownian motion with Hurst parameter $1/2<\hbar <1$ on the Hilbert space. By combining some stochastic analysis theory and new integral inequality techniques, we identify the global attracting sets of the equations under investigation. Some sufficient conditions ensuring the exponential decay of mild solutions in the pth moment to the stochastic systems are obtained. Last, an example is presented to illustrate our theory in the work.

1 Introduction

This paper is devoted to the study of the global attracting set and exponential decay of a class of second-order neutral stochastic functional differential equations driven by a fractional Brownian motion with Hurst parameter $1/2<\hbar <1$. Second-order stochastic systems can capture the dynamic behavior of many natural phenomena. In many real-world scenarios, it seems advantageous to reflect a more complex situation than first-order stochastic differential equations (SDEs). Recently, there has been increasing interest in the study of second-order stochastic differential equations due to their important applications in many areas such as medicine and biology, mathematical physics, electronics and telecommunications. For instance, Ren and Sakthivel [1] investigated the existence, uniqueness and stability of second-order neutral stochastic evolution equations with infinite delay; Liang and Guo [2] probed the behavior for second-order stochastic evolution equations with memory; Arthi et al. [3] discussed the exponential stability for second-order neutral stochastic differential equations with impulses.

One solution for many SDEs is a semimartingale as well as a Markov process. However, in the real world, many objects are not always such processes and they may have long-range aftereffects. Since the work of Mandelbrot and Van Ness [4], the researchers have focused the increasing attention on stochastic models based on the fractional Brownian motion. A fractional Brownian motion (fBm) of Hurst parameter $\hbar \in (0,1)$ is a centered Gaussian process $\beta^{\hbar }=\{ \beta^{\hbar }(t),t\geq 0\}$ with the covariance function

$$ R_{\hbar }(t,s)=\mathbb{E}\bigl(\beta^{\hbar }(t)\beta^{\hbar }(s) \bigr)= \frac{1}{2}\bigl(t^{2\hbar }+s^{2\hbar }-\vert t-s \vert ^{2\hbar }\bigr). $$

When $\hbar =1/2$, the fBm becomes the standard Brownian motion, and the fBm $\beta^{\hbar }$ is neither a semimartingale nor a Markov process if $\hbar \neq 1/2$. However, the fBm $\beta^{\hbar }$, $\hbar >1/2$, is a long-memory process and presents an aggregation behavior. The long-memory property makes the fBm a potential candidate to model noise in mathematical finance (see [5]), in biology (see [6, 7]), in communication networks (see, for instance, [8]), the analysis of global temperature anomaly [9] and electricity markets [10] etc. Stochastic differential equations driven by a fractional Brownian motion (fBm) have attracted the interest of many researchers. For more details on this topic, one can refer to the literature [11–17].

On the other hand, analysis found that the research mainly analyzed the stability of second-order equations to prove exponential stability through some strong conditions, which may be difficult to meet in practice. To find some weaker constraints to ensure the stability of a system, by using the inequality technique, Xu [18] investigated the attracting sets of a class of Volterra differential equations, and the relative stability conditions were further weakened. Attracting sets require the solutions enter some sets at a time and not exit, no matter what the solutions begin with. Attracting sets of first-order stochastic dynamical systems have been extensively studied over the last few decades. For instance, Li and Xu [19] investigated the global attracting sets of neutral stochastic partial functional differential equations; Wang and Li [20] obtained the global attracting sets of impulsive stochastic partial differential equations with infinite delays by establishing some impulsive-integral inequalities. Li [21] studied the global attracting sets and exponential decay in the pth moment of impulsive neutral stochastic functional differential equations driven by fBm. However, it is worth mentioning that the problem of determining the attracting sets of second-order stochastic partial differential equations driven by fBm is more complicated owing to its complexity and still remains open for a while. Hence, techniques and methods for the attracting sets of second-order neutral stochastic partial differential equations driven by fBm should be developed and explored.

To this end, in this paper, we will consider a class of second-order neutral stochastic functional differential equations driven by fBm with Hurst parameter $\hbar \in (1/2,1)$. We aim to investigate the global attracting sets, exponential decay in the pth moment of the kind of second-order stochastic differential equation driven by fBm. We will first develop some new integral inequalities. Subsequently, by using the stochastic analysis techniques, the properties of operator semigroup and combining those new integral inequalities, we will attempt to give the global attracting sets of the considered system. Last, some sufficient conditions ensuring exponential decay in the pth moment of the system under investigation are obtained.

The rest of this paper is organized as follows. In Section 2, we introduce some necessary notations and preliminaries. We devote Section 3 to the global attracting sets of the mild solutions to the equations under investigation. In Section 4, we investigate the exponential decay of equations under investigation. In Section 5, we give an example to illustrate the efficiency of the obtained result. In Section 6, we present our conclusion.

2 Preliminary

In this section we collect some notions, conceptions and lemmas which will be used throughout the paper.

Let $(\Omega ,\mathscr{F},\mathbb{P})$ be a complete probability space. Suppose that there is a one-dimensional fractional Brownian motion $\beta^{\hbar }=\{\beta^{\hbar }(t), t \ge 0\}$ with Hurst parameter $\hbar \in (0,1)$ on $(\Omega ,\mathscr{F},\mathbb{P})$. Now we state the stochastic integral of a deterministic H-valued function with respect to the scalar fractional Brownian motion $\beta^{\hbar }$, $\hbar \in (0, 1)$. To this end, let $T>0$ and denote by $\mathscr{E} _{0}$ the linear space of all H-valued step functions on $[0,T]$, that is, $f\in \mathscr{E}_{0}$ if and only if

$$ f(t)=\sum_{i=1}^{n-1}x_{i}{ \mathbf{1}}_{[t_{i},t_{i+1})}(t), \quad t\in [0, T], $$

where $\mathbf{1}_{[t_{i},t_{i+1})}$ is the indicator function on the set $[t_{i},t_{i+1})$, $x_{i}\in H$, for $i\in \{1,\ldots , n-1\}$ and $0=t_{1}< t_{2}<\cdots <t_{n}=T$. For $f\in \mathscr{E}_{0}$, we define its stochastic integral with respect to $\beta^{\hbar }$ as

$$ \int_{0}^{T} f(s)\,d\beta^{\hbar }(s)=\sum _{i=1}^{n}x_{i}\bigl( \beta^{\hbar }(t_{i+1})-\beta^{\hbar }(t_{i})\bigr). $$

(2.1)

Let $L_{\hbar }: \mathscr{E}_{0}\to L^{2}([0, T]; H)$ be the linear map given by

$$ (L_{\hbar }f) (t) = f(t)k_{\hbar }(T, t) + \int^{T}_{t} \bigl(f(s)-f(t)\bigr) \frac{ \partial k_{\hbar }}{\partial s}(s, t)\,ds \quad \forall f\in {\mathscr{E}}_{0}, $$

(2.2)

where $k_{\hbar }(\cdot , \cdot )$ is the kernel function given in

$$ k_{\hbar }(t, s) =\textstyle\begin{cases} \frac{\tilde{c}_{\hbar }(t-s)^{\hbar -\frac{1}{2}}}{\Gamma (\hbar + \frac{1}{2})} + \frac{\tilde{c}_{\hbar }(\frac{1}{2}-\hbar )}{\Gamma (\hbar +\frac{1}{2})}\int^{t}_{s} (u-s)^{\hbar -\frac{3}{2}} (1- (\frac{s}{u} )^{\frac{1}{2}- \hbar } )\,du &\mbox{if } \hbar \in (0, 1/2), \\ \frac{\hat{c}_{\hbar }}{\Gamma (\hbar -\frac{1}{2})}s^{\frac{1}{2}- \hbar }\int^{t}_{s} (u-s)^{\hbar -\frac{3}{2}}u^{\hbar -\frac{1}{2}}\,du& \mbox{if } \hbar \in (1/2, 1), \end{cases} $$

(2.3)

where $\tilde{c}_{\hbar }$ and $\hat{c}_{\hbar }$ are positive constants depending only on ħ.

It is known in this case that

$$ {\mathbb{E}} \biggl\Vert \int^{T}_{0} f(t)\,d\beta^{\hbar }(t) \biggr\Vert ^{2}_{H} = \Vert L_{\hbar }f \Vert ^{2}_{L^{2}([0, T]; H)}. $$

(2.4)

Let $(\mathscr{E}, \Vert \cdot \Vert _{{\mathscr{E}}})$ be the Hilbert space obtained as the completion of the pre-Hilbert space ${\mathscr{E}} _{0}$ under the inner product

$$\langle f, g\rangle_{\mathscr{E}} := \langle L_{\hbar }f,L_{\hbar }g \rangle_{L^{2}([0, T]; H)} \quad \mbox{for any } f,g\in {\mathscr{E}}_{0}. $$

Then stochastic integral (2.1) is extendable to an arbitrary $f\in {\mathscr{E}}$ by isometry (2.4). If $H={\mathbb{R}}^{1}$, it is known (see [22]) that in terms of $L_{\hbar }$ the process $\{\beta (t)=\beta^{\hbar }((L_{\hbar })^{-1}( \mathbf{1}_{[0,t]})), t\ge 0\}$ is a standard real Brownian motion, and $\beta^{\hbar }$ has the following integral representation:

$$ \beta^{\hbar }(t)= \int_{0}^{t}k_{\hbar }(t,s)\,d\beta (s), \quad t\ge 0. $$

Let K be an alternative separable Hilbert space with the inner product $\langle \cdot ,\cdot \rangle_{K}$ and the norm $\Vert \cdot \Vert _{K}$, respectively. We denote by $\mathscr{L}(K,H)$ the space of all linear bounded operators from K into H, equipped with the usual operator norm $\Vert \cdot \Vert$ topology. If $K=H$, we write $\mathscr{L}(K)$ simply for $\mathscr{L}(K,K)$.

Definition 2.1

Let $Q\in \mathscr{L}(K)$ be a positive self-adjoint operator. A K-valued Gaussian process $(B^{\hbar }_{Q}(t), t\ge 0)$ on $(\Omega , {\mathscr{F}}, {\mathbb{P}})$ is called a fractional Q-Brownian motion with Hurst parameter ${\hbar }\in (0, 1)$ if it satisfies:

• (i) for arbitrary $t\ge 0$, the mapping $B^{\hbar }_{Q}(t): K \to L^{2}(\Omega ; {\mathbb{R}}^{1})$ is linear;

• (ii) for arbitrary $k\in K$, $B^{\hbar }_{Q}(t)k$ is a real Gaussian process with mean ${\mathbb{E}}B^{\hbar }_{Q}(t)k=0$ for any $t\ge 0$;

• (iii) ${\mathbb{E}}(B^{\hbar }_{Q}(t)k_{1}\cdot B^{\hbar }_{Q}(s)k _{2})= (1/2)(t^{2{\hbar }} + s^{2{\hbar }} - \vert t-s \vert ^{2{\hbar }})\langle Qk_{1}, k_{2}\rangle_{K}$ for all $s,t\in {\mathbb{R}}_{+}=[0, \infty )$ and $k_{1},k_{2}\in K$.

In particular, if $\operatorname{Tr}Q< \infty$, then $B^{\hbar }_{Q}(t)$ is called a (genuine) fractional Brownian motion with Hurst parameter ${\hbar }\in (0, 1)$ ; and if $Q=I$, this process is called a cylindrical fractional Brownian motion with Hurst parameter ${\hbar }\in (0, 1)$.

In general, it is not necessarily true that there exists a K-valued random process $\hat{B}^{\hbar }_{Q}(\cdot )$ such that $B^{\hbar } _{Q}(t)(k) = \langle \hat{B}^{\hbar }_{Q}(t), k\rangle_{K}$ for each $t\ge 0$, although this is the case when $B^{\hbar }_{Q}$ is a genuine fBm. Suppose that there is a complete orthonormal basis $(e_{n},n \in {\mathbb{N}})$ of K diagonalizing operator Q, i.e., there exists a sequence of $(\lambda_{n}> 0,n\in {\mathbb{N}})$ such that $Qe_{n}=\lambda_{n} e_{n}$ for each $n\in {\mathbb{N}}$, letting $\beta^{\hbar }_{n}(t) = \frac{1}{\sqrt{\lambda_{n}}}B^{\hbar }_{Q}(t)(e _{n})$ for $n\in {\mathbb{N}}$, $t\ge 0$, the scalar processes $(\beta^{\hbar }_{n},n\in {\mathbb{N}})$ are a sequence of fractional Brownian motions which are mutually independent and $B^{\hbar }_{Q}$ can be uniformly represented as

$$B^{\hbar }_{Q}(t)= \sum^{\infty }_{n=1} \sqrt{\lambda_{n}}\beta^{\hbar }_{n}(t)e_{n}, \quad t\ge 0, $$

with its increment covariance operator Q. This series does not necessarily converge almost surely in K. But if Q is a trace class, that is, $\sum^{\infty }_{n=1}\lambda_{n}<\infty$, then the series defines a K-valued stochastic process. For a fixed $\hbar \in (0,1)$ and $T>0$, let $\{{\mathscr{F}}^{B^{\hbar }_{Q}}_{t}\}_{0\leq t\leq T}$ be the natural filtration of the fractional Q-Brownian motion $B^{\hbar }_{Q}(t), 0\leq t\leq T$. For the operator Q, let $B_{Q}(t)$, $t\ge 0$, be a K-valued Q-Wiener process.

We introduce the subspace $K_{Q}=\operatorname{Ran}Q^{1/2}\subset K$, the range of $Q^{1/2}$, which is a Hilbert space endowed with the inner product

$$\langle k_{1}, k_{2}\rangle_{K_{Q}} =\bigl\langle Q^{-1/2}k_{1}, Q^{-1/2}k _{2}\bigr\rangle _{K}\quad\mbox{for any } k_{1},k_{2}\in K_{Q}. $$

Let ${\mathscr{L}}_{2}(K_{Q}, H)$ denote the space of all Hilbert-Schmidt operators from $K_{Q}$ into H. Then ${\mathscr{L}} _{2}(K_{Q}, H)$ turns out to be a separable Hilbert space, equipped with the norm

$$\Vert \Psi \Vert ^{2}_{{\mathscr{L}}_{2}(K_{Q}, H)} =Tr \bigl[\Psi Q \Psi^{*}\bigr]\quad \mbox{for any }\Psi \in {\mathscr{L}}_{2}(K_{Q}, H). $$

For $T\ge 0$, we denote by $\mathcal{U}_{\hbar }^{2}([0, T]; {\mathscr{L}} _{2}(K_{Q}, H))$ the space of all Borel measurable mappings $f: [0, T]\to {\mathscr{L}}_{2}(K_{Q}, H)$ such that $f(\cdot )x \in {\mathscr{E}}$ for each $x\in K$ and $L_{\hbar }f\in L^{2}([0, T]; {\mathscr{L}}_{2}(K_{Q}; H))$. Let $(e_{n},n\in {\mathbb{N}})$ be a complete orthonormal basis diagonalizing operator Q. For any $f\in {\mathcal{U}}_{\hbar }^{2}([0, T]; {\mathscr{L}}_{2}(K_{Q}, H))$, the stochastic integral $\int^{T}_{0} f(t)\,dB^{\hbar }_{Q}(t)$ is defined as

$$ \int^{T}_{0} f(t)\,dB^{\hbar }_{Q}(t) := \sum^{\infty }_{n=1}\sqrt{ \lambda_{n}} \int^{T}_{0} f(t)e_{n}\,d \beta^{\hbar }_{n}(t), $$

(2.5)

provided the infinite series converges in $L^{2}(\Omega ; H)$. It may be verified that this stochastic integral does not depend on the choice of the complete orthonormal basis $\{e_{n}\}_{n=1}^{\infty }$.

Let $\mathbb{R}^{+}=[0,+\infty )$ and $C(X,Y)$ denote the space of continuous mappings from the topological space X to the topological space Y. Especially, for the given constant $r>0$, $C:=C([-r,0]; \mathbb{R})$ denotes the family of all continuous $\mathbb{R}$-valued functions ϕ defined on $[-r, 0]$ with the norm $\Vert \phi \Vert _{C}= \sup_{-r\leq \theta \leq 0}\vert \phi (\theta ) \vert$. Denote $\mathcal{C}=C([-r,0];H)$ equipped with the norm $\Vert \phi \Vert _{ \mathcal{C}}=\sup_{-r\leq \theta \leq 0}\Vert \phi (\theta ) \Vert _{H}$. Define $BC^{b}_{\mathcal{F}_{0}}$ as the family of all bounded $\mathcal{F} _{0}$-measurable, $\mathcal{C}$-valued random variables ϕ, satisfying $\Vert \phi \Vert ^{p}_{Lp}=\sup_{-r\leq \theta \leq 0}\mathbb{E} \Vert \phi (\theta ) \Vert ^{p}_{H}<\infty$ for $p>0$.

Consider the following second-order neutral stochastic functional differential equation driven by fBm with Hurst parameter $1/2<\hbar <1$:

$$ \textstyle\begin{cases} d[x'(t)+G(t,x_{t})] \\ \quad =[Ax(t)+f(t,x_{t})]\,dt+g(t,x_{t})\,d\omega (t)+\sigma (t)\,dB_{Q}^{\hbar }(t), &t\geq 0, \\ x_{0}(t)=\varphi (t)\in BC_{\mathcal{F}_{0}}^{b}([-r,0];H),&t\in [-r,0],x'(0)=\phi_{1}, \end{cases} $$

(2.6)

where $A:D(A)\subset H\rightarrow H$ is the infinitesimal generator of a strongly continuous cosine family on H, $B^{\hbar }_{Q}$ is a fractional Brownian motion with Hurst parameter $\hbar \in (1/2, 1)$ and ω is a standard Wiener process on a real and separable Hilbert space K. The delay term $x_{t}:[-r,0]\rightarrow H$ defined by $x_{t}(\theta )$ for $t\geq 0$ belongs to the space $\mathcal{C}$. $G,f:[0,+\infty )\times \mathcal{C}\rightarrow H$, $g:[0,+\infty ) \times \mathcal{C}\rightarrow {\mathscr{L}}_{2}(K_{Q}, H)$, $\sigma :[0,+\infty )\rightarrow {\mathscr{L}}_{2}(K_{Q}, H)$ are some appropriate mappings specified later.

Now, let us recall some basic concepts and facts on cosine families of operators (see [23]).

Definition 2.2

One parameter family $(T(t))_{t\geq 0}$ is called a strongly continuous cosine family if the following conditions hold:

• (i) $T(0)=I$;

• (ii) $T(t)x$ is continuous in t on $\mathbb{R}$ for all $x\in H$;

• (iii) $T(t+s)+T(t-s)=2T(t)T(s)$ for all $t,s \in \mathbb{R}$.

We also need consider the corresponding strongly continuous sine family $(S(t))_{t\geq 0}$, which is defined as $S(t)x=\int_{0}^{t}T(s)\,ds$, $t\in \mathbb{R}$, $x\in H$. As for the infinitesimal generator $A:D(A)\subset H\rightarrow H$ of a cosine family of operators $(T(t))_{t\geq 0}$, define $Ax=\frac{d^{2}}{dt^{2}}T(t)x | _{t=0}$. A is also a closed and densely defined operator on H.

Throughout this paper, we impose the following assumptions:

• (H1) The cosine family of operators $(T(t))_{t\geq 0}$ and its corresponding sine family $(S(t))_{t\geq 0}$ satisfy the following conditions for all $t\geq 0$:

  $$ \bigl\Vert T(t) \bigr\Vert _{H}\leq Me^{-\lambda_{1} t}, \quad \quad \bigl\Vert S(t) \bigr\Vert _{H}\leq Me^{-\lambda _{2} t}, \quad M\geq 1, \lambda_{1},\lambda_{2}>0. $$

• (H2) The function $\sigma :[0,+\infty )\rightarrow \mathcal{L} ^{0}_{2}(K,H)$ satisfies the following conditions: for the complete orthonormal basis $\{e_{n}\}_{n\in \mathbb{N}}$ in K, we have

  ($\sigma_{1}$):

  $\sum_{n=1}^{\infty }\Vert \sigma (t) Q^{1/2}e_{n} \Vert _{L ^{2}([0,T],H)}<\infty$;

  ($\sigma_{2}$):

  $\sum_{n=1}^{\infty }\Vert \sigma (t)Q^{1/2}e_{n} \Vert _{H}$ is uniformly convergent for $t\in [0,T]$.

• (H3) There exist constants $L_{f}>0$, $L_{g}>0$, $L_{G}>0$, $b_{f}>0$, $b_{g}>0$ and $b_{G}>0$ such that for any $x,y\in \mathcal{C}$ and $t\geq 0$,

  $$\begin{aligned}& \bigl\Vert f(t,x_{t})-f(t,y_{t}) \bigr\Vert _{H}\leq L_{f}\Vert x-y \Vert _{\mathcal{C}}, \quad \quad \bigl\Vert f(t,0) \bigr\Vert \leq b_{f}, \\& \bigl\Vert g(t,x_{t})-g(t,y_{t}) \bigr\Vert _{\mathscr{L}_{2}(K_{Q},H)}\leq L_{g}\Vert x-y \Vert _{\mathcal{C}},\quad \quad \bigl\Vert g(t,0) \bigr\Vert \leq b_{g}, \\& \bigl\Vert G(t,x_{t})-G(t,y_{t}) \bigr\Vert _{H}\leq L_{G}\Vert x-y \Vert _{\mathcal{C}},\quad \quad \bigl\Vert G(t,0) \bigr\Vert \leq b_{G}. \end{aligned}$$

3 Global attracting set and exponential p-stability

In this section, we shall get the global attracting sets of stochastic differential equation (2.6). First, we give the following definition of mild solutions to equation (2.6).

Definition 3.1

An H-valued continuous stochastic process $\{x(t), t\in [-r,T]\}$, $0< T<\infty$ is called a mild solution of (2.6) if $x_{0}(t)=\varphi (t)\in BC_{\mathcal{F}_{0}}^{b}([-r,0];H)$, $t\in [-r,0]$, $x'(0)=\phi _{1}$, and the following conditions hold:

• (i) $x(t)$ is a measurable, $\mathcal{F}_{t}$-adapted process with $\mathbb{E}\int_{0}^{T}\Vert x(t) \Vert ^{2}_{H}\,dt<\infty$;

• (ii) $x(t)$ satisfies the following integral equation:

  $$\begin{aligned} x(t)={}& T(t)\varphi (0)+S(t) \bigl(\phi_{1}-G(0,x_{0}) \bigr) \\ & {}+\int_{0}^{t}T(t-s)G(s,x _{s})\,ds+\int_{0}^{t}S(t-s)f(s,x_{s})\,ds \\ & {}+\int_{0}^{t}S(t-s)g(s,x_{s})\,d\omega (s)+ \int_{0}^{t}S(t-s)\sigma (s)\,dB ^{H}_{Q}(s). \end{aligned}$$

  (3.1)

Remark 3.1

Under assumptions (H1)-(H3), the existence and uniqueness of mild solution to system (2.6) are easily shown by using Picard’s iterative method (see Revathi et al. [24]).

Definition 3.2

(see [18]) A set $S\subset H$ is called the global attracting set of (2.6) if for any initial value $\psi \in BC_{ \mathcal{F}_{0}}^{b}([-r,0]; H)$, the solution process $x(t,\psi )$ of (2.6) converges to S as $t\rightarrow \infty$, i.e.,

$$ \operatorname{dist}\bigl(x(t,\psi ),S\bigr)\rightarrow 0\quad \mbox{as } t\rightarrow \infty , $$

where $\operatorname{dist}(x,S)=\inf_{y\in S}\mathbb{E}\Vert x-y \Vert _{H}$.

In order to get the global attractivity of equation (2.6), we need the following important integral inequalities.

Lemma 3.1

Let $y(t)\in C(\mathbb{R}, \mathbb{R}^{+})$ be a solution of the delay integral inequality

$$ y(t)\leq \textstyle\begin{cases} b_{1}e^{-\lambda_{1}t}+b_{2}e^{-\lambda_{2}t} \\ \quad {}+b_{3}\int_{0}^{t}e^{-\lambda_{1}(t-s)}\Vert y_{s} \Vert _{{C}}\,ds+b_{4}\int_{0}^{t}e^{-\lambda_{2}(t-s)}\Vert y_{s} \Vert _{{C}}\,ds+b_{5},& t\geq 0, \\ \varphi (t),&t\in [-r,0], \end{cases} $$

(3.2)

where $\varphi (t)\in C([-r,0];\mathbb{R}^{+})$, $b_{1},\ldots , b_{5}$ are nonnegative constants and $\lambda_{1}$, $\lambda_{2}>0$. If

$$ \rho :=\frac{b_{3}}{\lambda_{1}}+\frac{b_{4}}{\lambda_{2}}< 1, $$

(3.3)

then there are constants $\lambda \in (0,\lambda_{1}\wedge \lambda _{2})$ and $N\geq 0$ such that

$$ y(t)\leq Ne^{-\lambda t}+\frac{\lambda_{5}}{1-\rho },\quad \forall t\geq 0, $$

(3.4)

where λ and N satisfy that

$$ \bigl\Vert \varphi (0) \bigr\Vert _{C}< N, \quad \textit{and}\quad \frac{b_{1}+b_{2}}{N}+e^{\lambda r}\frac{b_{3}}{\lambda_{1}-\lambda }+e ^{\lambda r} \frac{b_{4}}{\lambda_{2}-\lambda }< 1. $$

(3.5)

Proof

To prove (3.4), we first prove that for any $h>1$,

$$ y(t)< hNe^{-\lambda t}+\frac{\lambda_{5}}{1-\rho },\quad \forall t\geq 0. $$

(3.6)

If (3.6) is not true, by virtue of the continuity of $y(t)$, then there exists $t_{1}>0$ such that

$$ y(t_{1})=hNe^{-\lambda t_{1}}+\frac{\lambda_{5}}{1-\rho }, $$

(3.7)

and

$$ y(t)\leq hNe^{-\lambda t}+\frac{\lambda_{5}}{1-\rho },\quad t\in [-r,t_{1}]. $$

(3.8)

On the other hand, from condition (3.3) and $\varphi (t)\in C([-r,0]; \mathbb{R}^{+})$, we can verify that there exist positive constants λ and N such that (3.5) holds. Thus, from (3.2), (3.5) and (3.8), we can obtain

$$ \begin{aligned} y(t_{1})&\leq b_{1}e^{-\lambda_{1}t_{1}}+b_{2}e^{-\lambda_{2}t_{1}}+b _{3} \int_{0}^{t_{1}}e^{-\lambda_{1}(t_{1}-s)}\Vert y_{s} \Vert _{{C}}\,ds+b_{4} \int_{0}^{t_{1}}e^{-\lambda_{2}(t_{1}-s)}\Vert y_{s} \Vert _{{C}}\,ds+b_{5} \\ &\leq b_{1}e^{-\lambda_{1}t_{1}}+b_{2}e^{-\lambda_{2}t_{1}}+b_{3} \int _{0}^{t_{1}}e^{-\lambda_{1}(t_{1}-s)}\bigl[hNe^{\lambda r}e^{-\lambda s}+(1- \rho )^{-1}b_{5}\bigr]\,ds \\ &\quad {}+b_{4} \int_{0}^{t_{1}}e^{-\lambda_{2}(t_{1}-s)}\bigl[hNe^{\lambda r}e ^{-\lambda s}+(1-\rho )^{-1}b_{5}\bigr] \,ds+b_{5} \\ &\leq b_{1}e^{-\lambda_{1}t_{1}}+b_{2}e^{-\lambda_{2}t_{1}}+b_{3} \int _{0}^{t_{1}}hNe^{-\lambda_{1}(t_{1}-s)}e^{\lambda r}e^{-\lambda s} \,ds \\ &\quad {}+b_{4} \int_{0}^{t_{1}}hNe^{-\lambda_{2}(t_{1}-s)}e^{\lambda r}e^{- \lambda s} \,ds+\biggl(\frac{b_{3}}{\lambda_{1}}+\frac{b_{4}}{\lambda_{2}}\biggr) (1- \rho )^{-1}b_{5}+b_{5} \\ &\leq hNe^{-\lambda t_{1}}\frac{b_{1}+b_{2}}{N}+hNe^{-\lambda t_{1}}e ^{\lambda r}b_{3} \int_{0}^{t_{1}}e^{-\lambda_{1}(t_{1}-s)}e^{-\lambda (t_{1}-s)}\,ds \\ &\quad {}+hNe^{-\lambda t_{1}}e^{\lambda r}b_{4} \int_{0}^{t_{1}}e^{-\lambda _{2}(t_{1}-s)}e^{-\lambda (t_{1}-s)} \,ds+(1-\rho )^{-1}b_{5} \\ &\leq hNe^{-\lambda t_{1}} \biggl(\frac{b_{1}+b_{2}}{N}+e^{\lambda r} \frac{b _{3}}{\lambda_{1}-\lambda }+e^{\lambda r}\frac{b_{4}}{\lambda_{2}- \lambda } \biggr)+(1-\rho )^{-1}b_{5} \\ &< hNe^{-\lambda t_{1}}+(1-\rho )^{-1}b_{5}, \end{aligned} $$

which contradicts equality (3.7). So, (3.6) holds for all $t\geq 0$. Letting $h\rightarrow 1$ in (3.6), we have (3.4). The proof is complete. □

We also need the following lemma.

Lemma 3.2

[21]

For any $\sigma :[0,+\infty )\rightarrow \mathcal{L} ^{0}_{2}(K,H)$ such that (H2) holds and $\sup_{t\geq 0}\Vert \sigma (t) \Vert _{\mathcal{L}^{0}_{2}}<\infty$, and for any $p\geq 2$, $t>0$,

$$ \mathbb{E} \biggl\Vert \int_{0}^{t}S(t-s)\sigma (s)\,dB^{\hbar }_{Q}(s) \biggr\Vert _{H}^{p}\leq C\sup_{t\geq 0} \bigl\Vert \sigma (t) \bigr\Vert _{\mathcal{L}^{0}_{2}} ^{p}, $$

where $C>0$ is a constant depending only on ħ, M, p and r.

Theorem 3.1

Assume that (H1)-(H3) and $\sup_{t\geq 0}\Vert \sigma (t) \Vert _{\mathcal{L} ^{0}_{2}}<\infty$ hold, then $S=\{\phi \in H| \Vert \phi \Vert _{H}^{p}\leq (1- {\rho })^{-1}{J}\}$ is a global attracting set of system (2.6) if the following inequality

$$\begin{aligned} \rho &:= 12^{p-1}M^{p} \lambda_{1}^{-p}L_{G}^{p}+12^{p-1}M^{p} \lambda _{2}^{-p}L_{f}^{p} \\ &\quad {}+ 12^{p-1}M^{p}L_{g}^{p} \lambda_{2}^{p/2-1} \biggl( \frac{p(p-1)}{2} \biggr)^{\frac{p}{2}} \biggl(\frac{p-2}{2(p-1)} \biggr)^{ \frac{p}{2}-1}< 1 \end{aligned}$$

(3.9)

holds for $p>2$, and

$$\begin{aligned} {J}&:=12^{p-1}M^{p}\lambda_{1}^{-p}b_{G}^{p}+12^{p-1}M^{p} \lambda_{2} ^{-p}b_{f}^{p} \\ &\quad {}+12^{p-1}M^{p} \biggl(\frac{p(p-1)}{2} \biggr)^{\frac{p}{2}} \biggl(\frac{p-2}{2(p-1)} \biggr)^{\frac{p}{2}-1}\lambda_{2}^{-\frac{p}{2}}b _{g}^{p}+6^{p-1}C \sup_{t\geq 0}\bigl\Vert \sigma (s) \bigr\Vert ^{p}_{\mathcal{L}^{0} _{2}}. \end{aligned}$$

Proof

From (3.1), we have

$$\begin{aligned} \mathbb{E}\bigl\Vert x(t) \bigr\Vert ^{p}&= \mathbb{E} \biggl\Vert T(t)\varphi (0)+S(t) \bigl( \phi_{1}-G(0,x_{0})\bigr) \\ &\quad {}+ \int_{0}^{t}T(t-s)G(s,x_{s})\,ds+ \int_{0}^{t}S(t-s)f(s,x _{s})\,ds \\ &\quad {}+ \int_{0}^{t}S(t-s)g(s,x_{s})\,d\omega (s)+ \int_{0}^{t}S(t-s)\sigma (s)\,dB ^{H}_{Q}(s) \biggr\Vert _{H}^{p} \\ &\leq 6^{p-1}\mathbb{E}\bigl\Vert T(t)\varphi (0) \bigr\Vert ^{p}_{H}+6^{p-1}\mathbb{E} \bigl\Vert S(t) \bigl(\phi_{1}-G(0,x_{0})\bigr) \bigr\Vert ^{p}_{H} \\ &\quad {}+6^{p-1}\mathbb{E} \biggl\Vert \int_{0}^{t}T(t-s)G(s,x_{s})\,ds \biggr\Vert _{H} ^{p}+6^{p-1}\mathbb{E} \biggl\Vert \int_{0}^{t}S(t-s)f(s,x_{s})\,ds \biggr\Vert ^{p}_{H} \\ &\quad {}+6^{p-1}\mathbb{E} \bigl\Vert S(t-s)g(s,x_{s})\,d\omega (s) \bigr\Vert ^{p}_{H}+6^{p-1} \mathbb{E} \biggl\Vert \int_{0}^{t}S(t-s)\sigma (s)\,dB^{H}_{Q}(s) \biggr\Vert ^{p} _{H} \\ &:= 6^{p-1}\bigl(J_{1}(t)+J_{2}(t)+J_{3}(t)+J_{4}(t)+J_{5}(t)+J_{6}(t) \bigr). \end{aligned}$$

(3.10)

First, it is easy to see from (H1) that for any $t\ge 0$,

$$ J_{1}(t)= \mathbb{E}\bigl\Vert T(t) \varphi (0) \bigr\Vert _{H}^{p}\leq M^{p}e^{-\lambda _{1}t} \mathbb{E}\bigl\Vert \varphi (0) \bigr\Vert _{H}^{p} $$

(3.11)

and

$$\begin{aligned} J_{2}(t)&= \mathbb{E}\bigl\Vert S(t) \bigl( \phi_{1}-G(0,x_{0})\bigr) \bigr\Vert ^{p}_{H} \\ &\leq M^{p}e^{-p\lambda_{2}t}\mathbb{E}\bigl\Vert \phi_{1}-G(0,x_{0}) \bigr\Vert _{H} ^{p} \\ &\leq M^{p}e^{-p\lambda_{2}t}3^{p-1}\mathbb{E}\bigl(\Vert \phi_{1} \Vert _{H}^{p}+L _{G}\bigl\Vert \varphi (0) \bigr\Vert _{\mathcal{C}}^{p}+b_{G}^{p} \bigr). \end{aligned}$$

(3.12)

By virtue of (H1), (H3) and Hölder’s inequality, we can obtain for any $t\ge 0$ that

$$\begin{aligned} J_{3}(t)={} &\mathbb{E} \biggl\Vert \int_{0}^{t}T(t-s)G(s,x_{s})\,ds \biggr\Vert _{H} ^{p} \\ \leq {}& \mathbb{E} \biggl( \int_{0}^{t}Me^{-\lambda_{1}(t-s)}\bigl\Vert G(s,x_{s}) \bigr\Vert _{H}\,ds \biggr)^{p} \\ \leq {}& \mathbb{E} \biggl( \int_{0}^{t}Me^{-\lambda_{1}(t-s)}\bigl(L_{G} \Vert x_{s}\Vert _{\mathcal{C}}+b_{G}\bigr)\,ds \biggr)^{p} \\ \leq {}&2^{p-1}M^{p}L_{G}^{p} \biggl( \int_{0}^{t}e^{-\lambda_{1}(t-s)} \biggr)^{p-1} \cdot \int_{0}^{t}e^{-\lambda_{1}(t-s)}\mathbb{E}\Vert x_{s} \Vert _{\mathcal{C}}^{p}\,ds+2^{p-1}M^{p} \lambda_{1}^{-p}b_{G}^{p} \\ \leq {}&2^{p-1}M^{p}\lambda_{1}^{1-p}L_{G}^{p} \int_{0}^{t}e^{-\lambda _{1}(t-s)}\mathbb{E}\Vert x_{s} \Vert _{\mathcal{C}}^{p}\,ds+2^{p-1}M^{p} \lambda _{1}^{-p}b_{G}^{p}. \end{aligned}$$

(3.13)

For term $J_{4}(t)$, similarly to $J_{3}(t)$, by virtue of (H1), (H3) and Hölder’s inequality, we also have for any $t\ge 0$ that

$$\begin{aligned} J_{4}(t)= {}&\mathbb{E} \biggl\Vert \int_{0}^{t}S(t-s)f(s,x_{s})\,ds \biggr\Vert _{H} ^{p} \\ \leq {}& \mathbb{E} \biggl( \int_{0}^{t}Me^{-\lambda_{2}(t-s)}\bigl(L_{f} \Vert x_{s} \Vert _{\mathcal{C}}+b_{f}\bigr)\,ds \biggr)^{p} \\ \leq {}&2^{p-1}M^{p}\lambda_{2}^{1-p}L_{f}^{p} \int_{0}^{t}e^{-\lambda _{2}(t-s)}\mathbb{E}\Vert x_{s} \Vert _{\mathcal{C}}^{p}\,ds+2^{p-1}M^{p} \lambda _{2}^{-p}b_{f}^{p}. \end{aligned}$$

(3.14)

Now, let us estimate the terms $J_{5}(t)$. By using Burkhölder-Davis-Gundy’s inequality, assumptions (H1) and (H3) and Hölder’s inequality, we obtain for $p>2$

$$\begin{aligned} J_{5}(t)={} &\mathbb{E} \biggl\Vert \int_{0}^{t}S(t-s)g(s,x_{s})\,d\omega (s) \biggr\Vert ^{p} \\ \leq {}&M^{p} \biggl(\frac{p(p-1)}{2} \biggr)^{\frac{p}{2}} \biggl( \int_{0}^{t}\bigl(e ^{-2\lambda_{2}(t-s)}\mathbb{E}\bigl\Vert g(s,x_{s}) \bigr\Vert ^{2}_{\mathcal{L}^{0} _{2}}\bigr) \,ds \biggr)^{\frac{p}{2}} \\ \leq {}&M^{p} \biggl(\frac{p(p-1)}{2} \biggr)^{\frac{p}{2}} \biggl( \int_{0}^{t}e ^{-\frac{2\lambda_{2}(p-1)}{p-2}(t-s)}\,ds \biggr)^{\frac{p}{2}-1} \int_{0} ^{t}e^{-\lambda_{2}(t-s)}E\bigl\Vert g(s,x_{s}) \bigr\Vert _{\mathcal{L}^{0}_{2}}^{p}\,ds \\ \leq {}&2^{p-1}M^{p}L_{g}^{p} \biggl( \frac{p(p-1)}{2} \biggr)^{\frac{p}{2}} \biggl(\frac{p-2}{2\lambda_{2}(p-1)} \biggr)^{\frac{p}{2}-1} \int_{0}^{t}e ^{-\lambda_{2}(t-s)}E\Vert x_{s} \Vert _{\mathcal{C}}^{p}\,ds \\ &{}+2^{p-1}M^{p} \biggl(\frac{p(p-1)}{2} \biggr)^{\frac{p}{2}} \biggl( \frac{p-2}{2(p-1)} \biggr)^{\frac{p}{2}-1} \lambda_{2}^{-\frac{p}{2}}b _{g}^{p}. \end{aligned}$$

(3.15)

Last, by Lemma 3.2, we have

$$ J_{6}(t)=\mathbb{E} \biggl\Vert \int_{0}^{t}S(t-s)\sigma (s)\,dB^{\hbar }_{Q}(s) \biggr\Vert ^{p}\leq C\sup_{t\geq 0} \bigl\Vert \sigma (s)\bigr\Vert ^{p}_{\mathcal{L}^{0} _{2}}. $$

(3.16)

Substituting (3.11)-(3.16) into (3.10), we obtain

$$\begin{aligned} \mathbb{E}\bigl\Vert x(t) \bigr\Vert ^{p}\leq {}&6^{p-1} M^{p}e^{-\lambda_{1}t}\mathbb{E} \bigl\Vert \varphi (0) \bigr\Vert _{H}^{p} \\ &{}+18^{p-1}M^{p}e^{-\lambda_{2}t} \mathbb{E}\bigl(\Vert \phi_{1} \Vert _{H}^{p}+L_{G} \bigl\Vert \varphi (0) \bigr\Vert _{\mathcal{C}}^{p}+b_{G}^{p} \bigr) \\ &{}+12^{p-1}M^{p}\lambda_{1}^{1-p}L_{G}^{p} \int_{0}^{t}e^{-\lambda_{1}(t-s)} \mathbb{E}\Vert x_{s} \Vert _{\mathcal{C}}^{p}\,ds+12^{p-1}M^{p} \lambda_{1}^{-p}b _{G}^{p} \\ &{}+12^{p-1}M^{p}\lambda_{2}^{1-p}L_{f}^{p} \int_{0}^{t}e^{-\lambda_{2}(t-s)} \mathbb{E}\Vert x_{s} \Vert _{\mathcal{C}}^{p}\,ds+12^{p-1}M^{p} \lambda_{2}^{-p}b _{f}^{p} \\ &{}+12^{p-1}M^{p}L_{g}^{p} \biggl( \frac{p(p-1)}{2} \biggr)^{\frac{p}{2}} \biggl(\frac{p-2}{2\lambda_{2}(p-1)} \biggr)^{\frac{p}{2}-1} \int_{0}^{t}e ^{-\lambda_{2}(t-s)}E\Vert x_{s} \Vert _{\mathcal{C}}^{p}\,ds \\ &{}+12^{p-1}M^{p} \biggl(\frac{p(p-1)}{2} \biggr)^{\frac{p}{2}} \biggl( \frac{p-2}{2(p-1)} \biggr)^{\frac{p}{2}-1} \lambda_{2}^{-\frac{p}{2}}b _{g}^{p} \\ &{}+6^{p-1}C \sup_{t\geq 0}\bigl\Vert \sigma (s) \bigr\Vert ^{p}_{\mathcal{L}^{0} _{2}}. \end{aligned}$$

(3.17)

Let $b_{1}:=6^{p-1} M^{p}\mathbb{E}\Vert \varphi (0) \Vert _{H}^{p}$, $b_{2}:=18^{p-1}M^{p}\mathbb{E}(\Vert \phi_{1} \Vert _{H}^{p}+L_{G}\Vert \varphi (0) \Vert _{\mathcal{C}}^{p}+b_{G}^{p})$, $b_{3}:=12^{p-1}M^{p}\lambda_{1} ^{1-p}L_{G}^{p}$, and

$$ b_{4}:=12^{p-1}M^{p}\lambda_{2}^{1-p}L_{f}^{p}+12^{p-1}M^{p}L_{g}^{p} \biggl(\frac{p(p-1)}{2} \biggr)^{\frac{p}{2}} \biggl( \frac{p-2}{2\lambda_{2}(p-1)} \biggr)^{\frac{p}{2}-1}. $$

From (3.9), we know $\rho :=\frac{b_{3}}{\lambda_{1}}+\frac{b_{4}}{ \lambda_{2}}<1$. Since $\varphi \in PC^{B}_{\mathcal{F}_{0}}([-r,0];H)$, so there exist ${N}>0$, $\lambda \in (0,\lambda_{1}\wedge \lambda_{2})$ such that

$$ \bigl\Vert \varphi (0) \bigr\Vert _{C}< N, \quad \mbox{and}\quad \frac{b_{1}+b_{2}}{N}+e^{\lambda r}\frac{b_{3}}{\lambda_{1}-\lambda }+e^{\lambda r} \frac{b_{4}}{\lambda _{2}-\lambda }< 1. $$

It follows from Lemma 3.1 that

$$ \mathbb{E}\bigl\Vert x(t) \bigr\Vert _{H}^{p}\leq {N}e^{-\lambda t}+ \frac{b_{5}}{1-{\rho }}. $$

So, by Definition 3.2 we know that S is a global attracting set of the mild solution to (2.6). □

Remark 3.2

Notice that (3.15) has no meaning if $p=2$. But we can re-estimate $J_{5}(t)$ if $p=2$. Thus, we have also the following corollary.

Corollary 3.1

Assume that (H1)-(H3) and $\sup_{t\geq 0}\Vert \sigma (t) \Vert _{\mathcal{L} ^{0}_{2}}<\infty$ hold. If the following inequality

$$ \begin{aligned} {\rho }:=12M^{2}\lambda_{1}^{-2}L_{G}^{2}+12M^{2} \lambda_{2}^{-2}L _{f}^{2}+ 12M^{2}\lambda_{2}^{-1}L_{g}^{2}< 1 \end{aligned} $$

(3.18)

and

$$ {J}:=12M^{2}\lambda_{1}^{-2}b_{G}^{2}+12M^{2} \lambda_{2}^{-2}b_{f} ^{2}+12M^{2}L_{g}^{2} \lambda_{2}^{-1}b_{g}^{2}+6C\sup _{t\geq 0}\bigl\Vert \sigma (s) \bigr\Vert ^{2}_{\mathcal{L}^{0}_{2}} $$

hold, then $S=\{\phi \in H| \Vert \phi \Vert _{H}^{2}\leq (1-{\rho })^{-1} {J}\}$ is a global attracting set of system (2.6).

Proof

We only need re-estimate $J_{5}(t)$. By

$$\begin{aligned} J_{5}(t)= {}&\mathbb{E} \biggl\Vert \int_{0}^{t}S(t-s)g(s,x_{s})\,d\omega (s) \biggr\Vert ^{2} \\ \leq {}&M^{2} \int_{0}^{t}e^{-2\lambda_{2}(t-s)}\mathbb{E}\bigl\Vert g(s,x_{s}) \bigr\Vert ^{2}_{\mathcal{L}^{0}_{2}}\,ds \\ \leq {}&2M^{2}L_{g}^{2} \int_{0}^{t}e^{-\lambda_{2}(t-s)}E\Vert x_{s} \Vert _{ \mathcal{C}}^{2}\,ds+2M^{2}L_{g}^{2} \lambda_{2}^{-1}b_{g}^{2}. \end{aligned}$$

(3.19)

Then, by letting $p=2$ in (3.10)-(3.14) and (3.16) and using (3.19), we can obtain

$$\begin{aligned} \mathbb{E}\bigl\Vert x(t) \bigr\Vert ^{2} \leq {}&6 M^{2}e^{-\lambda_{1}t}\mathbb{E}\bigl\Vert \varphi (0) \bigr\Vert _{H}^{2}+18M^{2}e^{-\lambda_{2}t}\mathbb{E} \bigl(\Vert \phi_{1} \Vert _{H}^{2}+L_{G} \bigl\Vert \varphi (0) \bigr\Vert _{\mathcal{C}}^{2}+b_{G}^{2} \bigr) \\ &+12M^{2}\lambda_{1}^{-1}L_{G}^{2} \int_{0}^{t}e^{-\lambda_{1}(t-s)} \mathbb{E}\Vert x_{s} \Vert _{\mathcal{C}}^{2}\,ds+12M^{2} \lambda_{1}^{-2}b_{G} ^{2} \\ &{}+12M^{2}\lambda_{2}^{-1}L_{f}^{2} \int_{0}^{t}e^{-\lambda_{2}(t-s)} \mathbb{E}\Vert x_{s} \Vert _{\mathcal{C}}^{2}\,ds+12M^{2} \lambda_{2}^{-2}b_{f} ^{2} \\ &{}+12M^{2}L_{g}^{2} \int_{0}^{t}e^{-\lambda_{2}(t-s)}E\Vert x_{s} \Vert _{ \mathcal{C}}^{2}\,ds+12M^{2}L_{g}^{2} \lambda_{2}^{-1}b_{g}^{2}+6C \sup _{t\geq 0}\bigl\Vert \sigma (s) \bigr\Vert ^{2}_{\mathcal{L}^{0}_{2}}. \end{aligned}$$

(3.20)

Similar to the proof of Theorem 3.1, by using Lemma 3.1, we can deduce that the desired results are true. The proof is complete. □

4 Exponential decay

In this section, we shall focus on the exponential decay in the pth moment of the mild solution of equation (2.6).

Definition 4.1

The mild solution of (2.6) with initial $\phi \in C([-r, 0]; H)$ is said to have exponential decay in the pth moment if there exists a pair of positive constants $\lambda >0$ and $M = M(\varphi ) \geq 0$ such that

$$ \mathbb{E}\bigl\Vert x(t,\varphi ) \bigr\Vert ^{p}_{H} \leq Me^{-\lambda t},\quad t\geq 0. $$

In particular, system (2.6) is said to have exponential decay in the mean square when $p=2$.

Theorem 4.1

Assume that (H1)-(H3) with $b_{G}=b_{f}=b_{g}=0$ and

$$ \int_{0}^{+\infty }e^{\lambda_{2}s}\bigl\Vert \sigma (s) \bigr\Vert ^{2}_{\mathcal{L} ^{0}_{2}}\,ds< \infty $$

(4.1)

hold, and the following inequality

$$\begin{aligned} \rho &:= 12^{p-1}M^{p} \lambda_{1}^{-p}L_{G}^{p}+12^{p-1}M^{p} \lambda _{2}^{-p}L_{f}^{p}+ 12^{p-1}M^{p}L_{g}^{p} \lambda_{2}^{p/2-1} \biggl( \frac{p(p-1)}{2} \biggr)^{\frac{p}{2}} \biggl(\frac{p-2}{2(p-1)} \biggr)^{ \frac{p}{2}-1} \\ &< 1 \end{aligned}$$

(4.2)

holds for $p>2$. Then the mild solution of system (2.6) has exponential decay in the pth moment.

Proof

From (3.1), we have

$$\begin{aligned} \mathbb{E}\bigl\Vert x(t) \bigr\Vert ^{p}= {}&\mathbb{E} \biggl\Vert T(t)\varphi (0)+S(t) \bigl( \phi_{1}-G(0,x_{0})\bigr) \\ &{}+ \int_{0}^{t}T(t-s)G(s,x_{s})\,ds+ \int_{0}^{t}S(t-s)f(s,x _{s})\,ds \\ &{}+ \int_{0}^{t}S(t-s)g(s,x_{s})\,d\omega (s)+ \int_{0}^{t}S(t-s)\sigma (s)\,dB ^{\hbar }_{Q}(s) \biggr\Vert _{H}^{p} \\ \leq {}&6^{p-1}\mathbb{E}\bigl\Vert T(t)\varphi (0) \bigr\Vert ^{p}_{H}+6^{p-1}\mathbb{E} \bigl\Vert S(t) \bigl(\phi_{1}-G(0,x_{0})\bigr) \bigr\Vert ^{p}_{H} \\ &{}+6^{p-1}\mathbb{E} \biggl\Vert \int_{0}^{t}T(t-s)G(s,x_{s})\,ds \biggr\Vert _{H} ^{p}+6^{p-1}\mathbb{E} \biggl\Vert \int_{0}^{t}S(t-s)f(s,x_{s})\,ds \biggr\Vert ^{p}_{H} \\ &{}+6^{p-1}\mathbb{E} \bigl\Vert S(t-s)g(s,x_{s})\,d\omega (s) \bigr\Vert ^{p}_{H}+6^{p-1} \mathbb{E} \biggl\Vert \int_{0}^{t}S(t-s)\sigma (s)\,dB^{\hbar }_{Q}(s) \biggr\Vert ^{p}_{H} \\ := {}&6^{p-1}\bigl(J_{1}(t)+J_{2}(t)+J_{3}(t)+J_{4}(t)+J_{5}(t)+J_{6}(t) \bigr). \end{aligned}$$

(4.3)

Since for every $t>0$, $\int_{0}^{t}S(t-s)\sigma (s)\,dB^{\hbar }_{Q}(s)$ is a centered Gaussian random variable and by Kahane-Khintchine’s inequality, there exists a constant $C_{p}$ such that

$$ J_{6}(t)=\mathbb{E} \biggl\Vert \int_{0}^{t}S(t-s)\sigma (s)\,dB^{\hbar }_{Q}(s) \biggr\Vert ^{p}\leq C_{p} \biggl(\mathbb{E} \biggl\Vert \int_{0}^{t}S(t-s)\sigma (s)\,dB^{\hbar }_{Q}(s) \biggr\Vert ^{2} \biggr)^{p/2}. $$

Choosing suitable $\varepsilon > 0$ small enough such that $(\lambda _{2}-\varepsilon )p\geq 2\lambda_{2}$ and $\delta :=\lambda_{2}- \varepsilon >0$, by assumption (H2) and Lemma 2 of [13], we derive that

$$\begin{aligned} \mathbb{E} \biggl\Vert \int_{0}^{t}S(t-s)\sigma (s)\,dB^{\hbar }_{Q}(s) \biggr\Vert ^{2}\leq {}&M^{2}c_{\hbar }\hbar (2\hbar -1)t^{2\hbar -1} \int_{0}^{t}e ^{-2\lambda_{2}(t-s)}\bigl\Vert \sigma (s) \bigr\Vert _{\mathcal{L}^{0}_{2}}^{2}\,ds \\ \leq {}&M^{2}c_{\hbar }\hbar (2\hbar -1)t^{2\hbar -1} \int_{0}^{t}e^{- \lambda_{2}(t-s)}\bigl\Vert \sigma (s) \bigr\Vert _{\mathcal{L}^{0}_{2}}^{2}\,ds \\ \leq {}&e^{-\delta t}M^{2}c_{\hbar }\hbar (2\hbar -1)t^{2\hbar -1}e ^{-\varepsilon t} \int_{0}^{t}e^{\lambda_{2}s}\bigl\Vert \sigma (s) \bigr\Vert _{ \mathcal{L}^{0}_{2}}^{2}\,ds. \end{aligned}$$

(4.4)

Therefore, (4.1) ensures the existence of a positive constant K such that

$$ M^{2}c_{\hbar }\hbar (2\hbar -1)t^{2\hbar -1}e^{-\varepsilon t} \int _{0}^{t}e^{\lambda_{2}s}\bigl\Vert \sigma (s) \bigr\Vert _{\mathcal{L}^{0}_{2}}^{2}\,ds \leq K \quad \mbox{for all } t\geq 0. $$

Then

$$ J_{6}(t)\leq C_{p}K^{p/2}e^{-\lambda_{2}t}. $$

(4.5)

Notice that $b_{G}=b_{f}=b_{g}=0$, then substituting (3.11)-(3.15) and (4.5) into (4.3), we obtain

$$\begin{aligned} \mathbb{E}\bigl\Vert x(t) \bigr\Vert ^{p}\leq {}&6^{p-1} M^{p}e^{-\lambda_{1}t}\mathbb{E} \bigl\Vert \varphi (0) \bigr\Vert _{H}^{p} \\ &{}+12^{p-1}M^{p}e^{-\lambda_{2}t} \mathbb{E}\bigl(\Vert \phi_{1} \Vert _{H}^{p}+L_{G} \bigl\Vert \varphi (0) \bigr\Vert _{\mathcal{C}}^{p} \bigr)+6^{p-1}C _{p}K^{p/2}e^{-\lambda_{2}t} \\ &{}+12^{p-1}M^{p}\lambda_{1}^{1-p}L_{G}^{p} \int_{0}^{t}e^{-\lambda_{1}(t-s)} \mathbb{E}\Vert x_{s} \Vert _{\mathcal{C}}^{p}\,ds \\ &{}+12^{p-1}M^{p} \lambda_{2}^{1-p}L _{f}^{p} \int_{0}^{t}e^{-\lambda_{2}(t-s)}\mathbb{E}\Vert x_{s} \Vert _{ \mathcal{C}}^{p}\,ds \\ &{}+12^{p-1}M^{p}L_{g}^{p} \biggl( \frac{p(p-1)}{2} \biggr)^{\frac{p}{2}} \biggl(\frac{p-2}{2\lambda_{2}(p-1)} \biggr)^{\frac{p}{2}-1} \int_{0}^{t}e ^{-\lambda_{2}(t-s)}E\Vert x_{s} \Vert _{\mathcal{C}}^{p}\,ds. \end{aligned}$$

(4.6)

Let $b_{1}:=6^{p-1} M^{p}\mathbb{E}\Vert \varphi (0) \Vert _{H}^{p}$, $b_{2}:=12^{p-1}M^{p}\mathbb{E}(\Vert \phi_{1} \Vert _{H}^{p}+L_{G}\Vert \varphi (0) \Vert _{\mathcal{C}}^{p})+6^{p-1}C_{p}K^{p/2}$, $b_{3}:=12^{p-1}M^{p} \lambda_{1}^{1-p}L_{G}^{p}$, and

$$ b_{4}:=12^{p-1}M^{p}\lambda_{2}^{1-p}L_{f}^{p}+12^{p-1}M^{p}L_{g}^{p} \biggl(\frac{p(p-1)}{2} \biggr)^{\frac{p}{2}} \biggl( \frac{p-2}{2\lambda_{2}(p-1)} \biggr)^{\frac{p}{2}-1}. $$

From (4.2), we know $\rho :=\frac{b_{3}}{\lambda_{1}}+\frac{b_{4}}{ \lambda_{2}}<1$. Since $\varphi \in PC^{B}_{\mathcal{F}_{0}}([-r,0];H)$, so there exist ${N}>0$, $\lambda \in (0,\lambda_{1}\wedge \lambda_{2})$ such that

$$ \bigl\Vert \varphi (0) \bigr\Vert _{C}< N, \quad \mbox{and}\quad \frac{b_{1}+b_{2}}{N}+e^{\lambda r}\frac{b_{3}}{\lambda_{1}-\lambda }+e^{\lambda r} \frac{b_{4}}{\lambda _{2}-\lambda }< 1. $$

It follows from Lemma 3.1 that

$$ \mathbb{E}\bigl\Vert x(t) \bigr\Vert _{H}^{p}\leq {N}e^{-\lambda t}. $$

The proof is complete. □

Similar to Corollary 3.1, we also have the following Corollary 4.1.

Corollary 4.1

Assume that (H1)-(H3) are satisfied with $b_{G}=b_{f}=b_{g}=0$,

$$ \int_{0}^{+\infty }e^{\lambda_{2}s}\bigl\Vert \sigma (s) \bigr\Vert ^{2}_{\mathcal{L} ^{0}_{2}}\,ds< \infty $$

(4.7)

and the following inequality

$$ \begin{aligned} {\rho }:=12M^{2}\lambda_{1}^{-2}L_{G}^{2}+12M^{2} \lambda_{2}^{-2}L _{f}^{2}+ 12M^{2}\lambda_{2}^{-1}L_{g}^{2}< 1 \end{aligned} $$

(4.8)

holds. Then the mild solution of system (2.6) has exponential decay in the mean square.

Proof

We only need re-estimate $J_{5}(t)$. By Burkhölder-Davis-Gundy’s inequality, we have

$$\begin{aligned} J_{5}(t)= {}&\mathbb{E} \biggl\Vert \int_{0}^{t}S(t-s)g(s,x_{s})\,d\omega (s) \biggr\Vert ^{2} \\ \leq {}&M^{2} \int_{0}^{t}e^{-2\lambda_{2}(t-s)}\mathbb{E}\bigl\Vert g(s,x_{s}) \bigr\Vert ^{2}_{\mathcal{L}^{0}_{2}}\,ds \\ \leq {}&2M^{2}L_{g}^{2} \int_{0}^{t}e^{-\lambda_{2}(t-s)}E\Vert x_{s} \Vert _{ \mathcal{C}}^{2}\,ds. \end{aligned}$$

(4.9)

Notice that $b_{G}=b_{f}=b_{g}=0$, then let $p=2$ in (3.10)-(3.14) and (4.5), we can obtain

$$\begin{aligned} \mathbb{E}\bigl\Vert x(t) \bigr\Vert ^{2} \leq{} &6 M^{2}e^{-\lambda_{1}t}\mathbb{E}\bigl\Vert \varphi (0) \bigr\Vert _{H}^{2} \\ &{}+12M^{2}e^{-\lambda_{2}t}\mathbb{E} \bigl(\Vert \phi_{1} \Vert _{H}^{2}+L_{G} \bigl\Vert \varphi (0) \bigr\Vert _{\mathcal{C}}^{2}\bigr) \\ &{}+12M^{2}\lambda_{1}^{-1}L_{G}^{2} \int_{0}^{t}e^{-\lambda_{1}(t-s)} \mathbb{E}\Vert x_{s} \Vert _{\mathcal{C}}^{2}\,ds \\ &{}+12M^{2}\lambda_{2}^{-1}L_{f}^{2} \int_{0}^{t}e^{-\lambda_{2}(t-s)} \mathbb{E}\Vert x_{s} \Vert _{\mathcal{C}}^{2}\,ds \\ &{}+12M^{2}L_{g}^{2} \int_{0}^{t}e^{-\lambda_{2}(t-s)}E\Vert x_{s} \Vert _{ \mathcal{C}}^{2}\,ds+Ke^{-\lambda_{2}t}. \end{aligned}$$

(4.10)

Similar to the proof of Theorem 4.1, by using Lemma 3.1, we can deduce that the desired results are true. The proof is complete. □

5 Example

Example 5.1

We consider the following second-order neutral stochastic partial functional differential equation driven by the fractional Brownian motion:

$$ \textstyle\begin{cases} \partial [\frac{\partial }{\partial t}u(t,x)+v_{1}u(t,x(t-1))] \\ \quad =[\frac{ \partial^{2}}{\partial x^{2}}u(t,x)+v_{2}u(t,x(t-1))]\,dt \\ \quad \quad +v_{3}u(t,x(t-1))\,d\omega (t)+\sigma (t)\,dB^{\hbar }_{Q}(t),&0\leq x\leq \pi , t\geq 0, \\ u(t,x)=\varphi (t,x),& s\in [-1,0],x\in [0,\pi ], \\ & u(t,0)=u(t,\pi )=0, \\ \frac{\partial }{\partial t}u(0,x)=u_{1}(x),&x\in [0,\pi ], \end{cases} $$

(5.1)

where $v_{i}\geq 0$, $i=1,2,3$, are constants, $\varphi (s,\cdot ) \in H$, $\varphi (\cdot ,x)\in BC^{b}_{\mathcal{F}_{0}}([-1,0];H)$, $H=L^{2}[0,\pi ]$, $\omega (t)$ is a Wiener process and $B^{\hbar } _{Q}(t)$ is a fractional Brownian motion.

Define the infinitesimal operator $A:H\rightarrow H$ by $A=\frac{ \partial^{2}}{\partial X^{2}}$, whose domain $D(A)=\{x\in H:x,x' \mbox{ are absolutely continuous, } x(0)=0, x'\in H\}$, then

$$ Ax=-\sum_{n=1}^{\infty }n^{2}\langle x,e_{n}\rangle_{H}e_{n}, \quad x \in D(A), $$

where $e_{n}(x)=\sqrt{\frac{2}{\pi }}\sin nx$, $n=1,2,\ldots$ , is a complete orthonormal set of eigenvectors of A.

Let

$$\begin{aligned}& G\bigl(t,x(t-1)\bigr)=v_{1}u\bigl(t,x(t-1)\bigr), \\& f\bigl(t,x(t-1)\bigr)=v_{2}u\bigl(t,x(t-1)\bigr), \\& g\bigl(t,x(t-1)\bigr)=v_{3}u\bigl(t,x(t-1)\bigr). \end{aligned}$$

Then we can get (H1) is satisfied with $M=1$ and $\lambda_{1}= \lambda_{2}=\pi$, and (H3) is satisfied with $L_{G}=v_{1}$, $L_{f}=v_{2}$, $L_{g}=v_{3}$ and $b_{G}=b_{f}=b_{g}=0$. Assume that $\sigma (t)$ satisfies assumption (H2) such that $\sup_{t\geq 0}\Vert \sigma (s) \Vert ^{2}_{\mathcal{L}^{0}_{2}}<\infty$. Thus, by virtue of Corollary 3.1, we know that

$$ S_{1}=\bigl\{ x\in H: \Vert x \Vert _{H}\leq \sqrt{(1- \hat{\rho })^{-1}\hat{J}} \bigr\} $$

is a global attracting set of system (5.1) with $\hat{J}=6C\sup_{t \geq 0}\Vert \sigma (s) \Vert ^{2}_{\mathcal{L}^{0}_{2}}$ provided that $\hat{\rho }:=12\pi^{-2}v_{1}+12\pi^{-2}v_{2}+12\pi^{-1}v_{3}<1$.

In addition, if $\int_{0}^{+\infty }e^{\lambda_{2}s}\Vert \sigma (s) \Vert ^{2}_{\mathcal{L}^{0}_{2}}\,ds<\infty$ and $\hat{\rho }<1$, then by Corollary 4.1, we know the mild solution of system (5.1) has exponential decay in the mean square.

6 Conclusion

In this paper, by establishing new integral inequalities, we obtain the global attracting sets and exponential decay of second-order neutral stochastic functional differential equations driven by fBm with Hurst parameter $H\in (1/2,1)$. By estimating the pth moment of fractional Brownian noise and using new integral inequalities, we obtain some sufficient conditions ensuring the exponential decay in the pth moment of the mild solution of the considered equations. In our next paper, we will explore the global attracting sets and the exponential decay in the pth moment of second-order neutral stochastic functional differential equations driven by the fractional Brownian motion with Hurst parameter $H\in (0,1/2)$ and the global attracting sets and the exponential decay in the pth moment of second-order neutral stochastic functional differential equations driven by a multifractional Brownian motion. Besides, we also will attempt to explore the global attracting sets and the exponential decay in the pth moment of second-order stochastic differential inclusions with time-delay.