Open Access

Existence and upper semicontinuity of attractors for non-autonomous stochastic lattice FitzHugh-Nagumo systems in weighted spaces

Advances in Difference Equations20162016:310

https://doi.org/10.1186/s13662-016-1009-x

Received: 28 July 2016

Accepted: 21 October 2016

Published: 29 November 2016

Abstract

In this paper, we first consider the existence of random attractors for the non-autonomous stochastic lattice FitzHugh-Nagumo system with random coupled coefficients and additive white noise in a weighted space \(l_{\rho }^{2} \times l_{\rho }^{2} \); then we establish the upper semicontinuity of random attractors as the intensity of noise approaches zero.

Keywords

stochastic lattice dynamical systemrandom attractorweighted spaceswhite noise

MSC

37L5560H15

1 Introduction

Stochastic lattice dynamical systems (SLDSs) arise in a variety of applications where the spatial structure has a discrete character, and uncertainties or random influences are taken into account. In recent years, some work has been done regarding the existence of random attractors for SLDSs (see e.g. [112]). In this work, Bates, Lu and Wang [2] considered the existence of random attractors for first-order non-autonomous SLDS driven by multiplicative white noise. Han, Shen and Zhou [5] considered the existence of random attractors for first-order SLDSs with random coupled coefficients and multiplicative/additive white noise. Wang and Zhou [11] considered non-autonomous stochastic lattice FitzHugh-Nagumo system with random coupled coefficients and multiplicative white noise.

Motivated by [2, 5, 11], we will study the asymptotic behavior of solutions of the following non-autonomous stochastic lattice FitzHugh-Nagumo system with random coupled coefficients and additive white noise: for every \(\tau \in \mathbb{R}\) and \(t>\tau \),
$$ \textstyle\begin{cases} du_{i}= ( \sum_{j=-q}^{q}\eta _{i,j}(\theta _{t}\omega )u_{i+j} -v_{i}+f_{i}( u_{i})+g_{i}(t) )\,dt+c a _{i}\,dw_{i} (t),\quad i\in \mathbb{Z}, \\ dv_{i}= ( \sigma u_{i} - \delta v_{i} + h_{i}(t) )\,dt+c b_{i}\,dw_{i} (t),\quad i\in \mathbb{Z}, \end{cases} $$
(1.1)
with the initial data
$$ u_{i}(\tau )=u_{i,\tau }, \qquad v_{i}(\tau )=v_{i,\tau }, \quad i\in \mathbb{Z}, $$
where c, \(\sigma >0 \) and \(\delta >0\) are constants; \(u_{i}, v_{i}\in \mathbb{R}\); \(f_{i}(u_{i})\), \(g_{i}(t)\), \(h_{i}(t)\in \mathbb{R}\); \(\eta _{i,j}(\omega )\) \((j\in \{-q,\ldots ,0,\ldots ,q\}, q\in \mathbb{N})\) are random variables; \(a =(a _{i})_{i\in \mathbb{Z}},b =(b _{i})_{i\in \mathbb{Z}}\in l^{2}\); \((\theta _{t})_{t\in \mathbb{R}}\) is a metric dynamical system defined on proper probability space \((\Omega ,\mathcal{F},\mathcal{P})\); \(\{w _{i}(t):i\in \mathbb{Z}\}\) are independent two-sided real-valued Wiener processes on \((\Omega ,\mathcal{F} ,\mathcal{P} )\). If \(g_{i}\) and \(h_{i}\) do not depend on t for all \(i\in Z\), then we say (1.1) is an autonomous stochastic lattice FitzHugh-Nagumo system.

The lattice FitzHugh-Nagumo system is used to stimulate the propagation of action potentials in myelinated nerve axons (see [13]). The attractor of stochastic lattice FitzHugh-Nagumo system has been investigated in [4, 14, 15] in the autonomous case. In practice, the coupled mode between two nodes (say, adjacent nodes) is usually random. It is then of great importance to investigate SLDSs with random coupled coefficients. To the best of our knowledge, there are no results on non-autonomous stochastic lattice FitzHugh-Nagumo system with random coupled coefficients and additive white noise in a weighted space.

In this paper, we shall transform the stochastic lattice FitzHugh-Nagumo system (1.1) into a deterministic one with random parameters through two Ornstein-Uhlenbeck processes, and prove the existence of a tempered random attractor in a weighted space \(l_{\rho }^{2}\times l_{\rho }^{2}\) (see (2.2)) for the continuous cocycle (see Definition 3.1) generated by system (1.1), which attracts the random tempered bounded sets in pullback sense. Then we consider the dependence of attractors on the parameters c of the system (1.1) and establish the upper semicontinuity of the random attractor as the intensity c of noise approaches zero.

The rest of this paper is organized as follows. In the next section, we present some mathematical setting for system (1.1). In Section 3, we mainly consider the existence of a tempered random attractor in a weighted space of infinite sequences for system (1.1). Then in Section 4, we consider the upper semicontinuity of the tempered random attractor for system (1.1).

2 Mathematical settings

Throughout this paper, a positive weight function \(\rho :\mathbb{Z}\rightarrow \mathbb{R}^{+}\) is chosen to satisfy
$$ 0< \rho (i)\leqslant M_{0},\qquad \rho (i)\leqslant c_{0}\rho (i\pm 1),\quad \forall i\in \mathbb{Z}, $$
(2.1)
where \(M_{0}\) and \(c_{0}\) are positive constants. For example, for \(i\in \mathbb{Z}\), \(\rho (i)=\frac{1}{(1+\gamma ^{2}i^{2})^{q}}\) (\(q>\frac{1}{ 2}\)) [16] and \(\rho (i)=e^{-\gamma \vert i \vert }\) satisfy condition (2.1), where \(\gamma >0\). Define \(\rho _{i}=\rho (i), \forall i\in \mathbb{Z}\),
$$ l_{\rho }^{2}= \biggl\{ u=(u_{i})_{i\in \mathbb{Z}}:\sum _{i\in \mathbb{Z}}\rho _{i}\vert u_{i} \vert ^{2}< \infty , u_{i}\in \mathbb{R} \biggr\} $$
(2.2)
with norm \(\Vert u \Vert _{\rho , 2}= ( \sum_{i\in \mathbb{Z}}\rho _{i}\vert u_{i} \vert ^{2} ) ^{\frac{1}{2}}\) and inner product \((u,v)_{\rho , 2} = \sum_{i\in \mathbb{Z}} \rho _{i} u_{i}v_{i}\) for \(u=(u_{i})_{i\in \mathbb{Z}}, v=(v_{i})_{i\in \mathbb{Z}} \in l_{\rho }^{2}\). We write \(\Vert \cdot \Vert _{\rho ,2}\) as \(\Vert \cdot \Vert _{\rho }\), \((\cdot ,\cdot )_{\rho , 2} \) as \((\cdot ,\cdot )_{\rho } \), and \(\Vert \cdot \Vert _{\rho }\) as \(\Vert \cdot \Vert \) if \(\rho (i)\equiv 1\). Then \(l_{\rho }^{2}\) is a separable Hilbert space with the norm \(\Vert \cdot \Vert _{\rho }\).

Let \(\Omega =\{\omega \in C(\mathbb{R},l^{2}):\omega (0)=0\}\), \(\mathcal{F} \) is the Borel σ-algebra induced by the compact open topology of Ω, and \(\mathcal{P} \) is the Wiener measure on \((\Omega ,\mathcal{F} )\) (see [17]). The infinite sequence \(\mathbb {e}^{i}\) \((i\in \mathbb{Z})\) denote the element having 1 at position i and 0 for all other components.

Consider the following non-autonomous stochastic lattice FitzHugh-Nagumo system with random coupled coefficients and additive white noise: for every \(\tau \in \mathbb{R}\) and \(t>\tau \),
$$ \left \{ \textstyle\begin{array}{l} du_{i}= ( \sum_{j=-q}^{q}\eta _{i,j}(\theta _{t}\omega )u_{i+j} -v_{i}+f_{i}( u_{i})+g_{i}(t) )\,dt+c a _{i}\,dw_{i} (t), \quad i\in \mathbb{Z}, \\ dv_{i}= ( \sigma u_{i} - \delta v_{i} + h_{i}(t) )\,dt+c b_{i}\,dw_{i}(t), \quad i\in \mathbb{Z}, \\ u_{i}(\tau )=u_{i,\tau },\qquad v_{i}(\tau )=v_{i,\tau }, \quad i\in \mathbb{Z}, \end{array}\displaystyle \right . $$
(2.3)
where c, \(\sigma >0 \) and \(\delta >0\) are constants, \(u_{i},v_{i}\in \mathbb{R}\); \(f_{i}(u_{i}), g_{i}(t), h_{i}(t)\in \mathbb{R}\); \(\eta _{i,j}(\omega )\) \(( j\in \{-q,\ldots ,0,\ldots ,q\}, q\in \mathbb{N} )\) are random variables on probability space \((\Omega ,\mathcal{F},\mathcal{P})\); \(a =(a _{i})_{i\in \mathbb{Z}}\), \(b =(b _{i})_{i\in\mathbb{Z}}\in l^{2}\); \(\{w _{i}(t):i\in \mathbb{Z}\}\) are independent two-sided real-valued Wiener processes on \((\Omega ,\mathcal{F} ,\mathcal{P} )\); \(\theta _{t}\omega (\cdot )=\omega (\cdot +t)-\omega (t)\), for all \(\omega \in \Omega ,t\in \mathbb{R}\). Then \((\Omega ,\mathcal{F},\mathcal{P},\{\theta _{t}\}_{t\in \mathbb{R}})\) is an ergodic metric dynamical system.
System (2.3) can be rewritten as abstract ODEs: for every \(\tau \in \mathbb{R}\) and \(t>\tau \),
$$ \left \{ \textstyle\begin{array}{l} du= (\mathbb{B}(\theta _{t}\omega )u -v+f(u )+g(t ) )\,dt+c \,dW_{1}(t), \\ dv= ( \sigma u-\delta v+h(t ) )\,dt+c \,dW_{2}(t),\\ u (\tau )=u_{ \tau }, \qquad v (\tau )=v_{ \tau }, \end{array}\displaystyle \right . $$
(2.4)
where \(u=(u_{i})_{i\in \mathbb{Z}}\), \(v=(v_{i})_{i\in \mathbb{Z}}\), \(f( u )=(f_{i}( u_{i} ))_{i\in \mathbb{Z}}\) is a nonlinearity satisfying certain conditions, \(g(t)=(g_{i}(t))_{i\in \mathbb{Z}}\) and \(h(t)=(h_{i}(t))_{i\in \mathbb{Z}}\) are given time dependent sequences, \(W_{1}(t)=W_{1}(t,\omega )=\sum_{i\in \mathbb{Z}}a _{i}w _{i}(t)\mathbb {e}^{i}\) and \(W_{2}(t)=W_{2}(t,\omega )=\sum_{i\in \mathbb{Z}}b _{i}w _{i}(t)\mathbb {e}^{i}\) are Brownian motions on \((\Omega ,\mathcal{F} ,\mathcal{P} )\), \(\mathbb{B}(\omega )\) is a linear operator defined by
$$ \bigl(\mathbb{B}(\omega )u\bigr)_{i}=\sum _{j=-q}^{q}\eta _{i,j}(\omega )u_{i+j}. $$
(2.5)

To convert the problem (2.4) into a random differential equation, let \(z(\theta _{t}\omega ):=-\lambda \int _{-\infty }^{0}e^{\lambda s}(\theta _{t}\omega )(s)\,ds\) and \(y(\theta _{t}\omega ):=-\mu \int _{-\infty }^{0}e^{\mu s}(\theta _{t}\omega )(s)\,ds\), \(t\in \mathbb{R}\), \(\omega \in \Omega \), which are Ornstein-Uhlenbeck processes on \((\Omega ,\mathcal{F} ,\mathcal{P} )\) and solve the Ornstein-Uhlenbeck equations \(dz+\lambda z\,dt=dW_{1}(t) \) and \(dy+\mu y\,dt=dW_{2}(t) \), respectively, where \(z(\theta _{t}\omega )=(z_{i}(\theta _{t}\omega ))_{i\in \mathbb{Z}}\) and \(y(\theta _{t}\omega )=(y_{i}(\theta _{t}\omega ))_{i\in \mathbb{Z}}\). From [1719], it is known that the random variables \(\Vert z(\omega ) \Vert \) and \(\Vert y(\omega ) \Vert \) are tempered, and there is a \(\theta _{t}\)-invariant set \(\widetilde{\Omega } \subset \Omega \) of full \(\mathcal{P} \) measure such that \(t\mapsto z(\theta _{t}\omega )\) and \(t\mapsto y(\theta _{t}\omega )\) are continuous in t for every \(\omega \in \widetilde{\Omega } \).

Let \(\widetilde{u}(t,\omega )= u(t,\omega ) -cz(\theta _{t}\omega ) \), \(\widetilde{v}(t,\omega )= v(t,\omega ) -c y(\theta _{t}\omega ) \), \(\omega \in \Omega \), \(t\in \mathbb{R}\), then (2.4) can be written as the following equivalent random system with random coefficients: for every \(\tau \in \mathbb{R}\) and \(t>\tau \),
$$ \left \{ \textstyle\begin{array}{l} \frac{d\widetilde{u}}{dt} = \mathbb{B}(\theta _{t}\omega )\widetilde{u} - \widetilde{v}+ f( \widetilde{u}+ cz(\theta _{t}\omega ) )+c\mathbb{B}(\theta _{t}\omega ) z(\theta _{t}\omega )+ g(t) -c y(\theta _{t}\omega )- c\lambda z(\theta _{t}\omega ), \\ \frac{d\widetilde{v}}{dt}=\sigma \widetilde{u}-\delta \widetilde{v} + h(t) +\sigma c z(\theta _{t}\omega ) + c (\mu -\delta ) y(\theta _{t}\omega ), \\ \widetilde{u}(\tau )=\widetilde{u}_{\tau },\qquad \widetilde{v}(\tau )=\widetilde{v}_{\tau }. \end{array}\displaystyle \right . $$
(2.6)
We will consider (2.6) for \(\omega \in \widetilde{\Omega }\) and write Ω̃ as Ω from now on. In order to obtain the existence and uniqueness of solutions to problem (2.6), we make the following assumptions on \(g_{i}\), \(h_{i}\), \(f_{i}\) and the coefficients \(\eta _{i,j}(\omega )\), \(j\in -q,\ldots ,0,\ldots ,q\), for \(i\in \mathbb{Z}\):
(A1): 
Letting \(\boldsymbol{\beta}(\omega )=\sup \{ \vert \eta _{i,j}(\omega) \vert :j\in\{-q,\ldots ,0,\ldots ,q\},q\in \mathbb{N}\text{ and } i\in \mathbb{Z} \} \geqslant 0\), \(\boldsymbol{\beta}(\theta _{t}\omega )\) belongs to \(L_{\operatorname{loc}}^{1}(\mathbb{R})\) with respect to \(t\in \mathbb{R}\) for each \(\omega \in \Omega \),
$$ \lim_{t\rightarrow \pm \infty } \frac{1}{t} \int _{0}^{t} \boldsymbol{\beta}(\theta _{s}\omega )\,ds=0 ; $$
(2.7)
and \(\boldsymbol{\beta}(\omega )\) is tempered.
(A2): 
For some positive constants \(\alpha ,\beta \) and κ,
$$f_{i}( 0)=0,\qquad f_{i}(u)u\leqslant -\alpha u^{2}+\beta ,\qquad f_{i}^{\prime }(u)\leqslant\kappa , \quad \forall i\in \mathbb{Z}, u\in \mathbb{R}. $$
(A3): 

\(g=(g_{i} )_{i\in \mathbb{Z}} \in L_{\operatorname{loc}}^{2}(\mathbb{R}, l_{\rho }^{2})\), \(h=(h_{i} )_{i\in \mathbb{Z}} \in L_{\operatorname{loc}}^{2}(\mathbb{R}, l_{\rho }^{2})\).

(A4): 
Let \(\boldsymbol{\lambda}=\min \{\frac{\alpha }{2},\delta \}\). There exists a positive constant \(\mathbb {a}\in (0,\boldsymbol{\lambda}) \) such that
$$ \int _{-\infty }^{0}e^{\mathbb {a} s} \bigl(\bigl\Vert g (s+\tau ) \bigr\Vert _{\rho }^{2} + \bigl\Vert h (s+\tau ) \bigr\Vert _{\rho }^{2} \bigr)\,ds< \infty ,\quad \forall \tau \in \mathbb{R}. $$
(2.8)
We call \(v:[\tau ,\tau +T)\rightarrow l_{\rho }^{2}\) a mild solution of the following random lattice differential equations:
$$ \frac{dv}{dt}=G(v,t,\theta _{t}\omega ),\quad v=(v_{i})_{i\in \mathbb{Z}}, G=(G_{i})_{i\in \mathbb{Z}}, t\in [\tau ,\tau +T) ,\tau \in \mathbb{R}, $$
(2.9)
where \(\omega \in \Omega \), if \(v\in C([\tau ,\tau +T),l_{\rho }^{2})\) and
$$ v_{i}(t)=v_{i}(\tau )+ \int _{\tau }^{t}G_{i}\bigl(v(s),s,\theta _{s}\omega \bigr)\,ds ,\quad i\in \mathbb{Z}, t\in [\tau ,\tau+T), \tau \in \mathbb{R}. $$
(2.10)
By Theorem 6.1.7 in [20] and Definition 3.1, we have the following theorem.

Theorem 2.1

Let \(T>0\) and (A1)-(A3) hold. Then, for every \(\tau \in \mathbb{R}\) and \(\omega \in \Omega \) and any initial data \((\widetilde{u}_{\tau },\widetilde{v}_{\tau }) \in l_{\rho }^{2} \times l_{\rho }^{2} \), problem (2.6) admits a unique mild solution \((\widetilde{u}(\cdot ,\tau ,\omega ,\widetilde{u}_{\tau }, \widetilde{v}_{\tau }), \widetilde{v}(\cdot ,\tau ,\omega , \widetilde{u}_{\tau },\widetilde{v}_{\tau})) \in C([\tau ,\tau +T),l_{\rho }^{2}\times l_{\rho }^{2})\) with \((\widetilde{u}(\tau ,\tau ,\omega , \widetilde{u}_{\tau },\widetilde{v}_{\tau }), \widetilde{v}(\tau ,\tau ,\omega , \widetilde{u}_{\tau },\widetilde{v}_{\tau })) =(\widetilde{u}_{\tau },\widetilde{v}_{\tau }) \), \((\widetilde{u}( t,\tau ,\omega , \widetilde{u}_{\tau },\widetilde{v}_{\tau }), \widetilde{v}( t,\tau ,\omega , \widetilde{u}_{\tau },\widetilde{v}_{\tau })) \) being continuous in \((\widetilde{u}_{\tau },\widetilde{v}_{\tau }) \) \(\in l_{\rho }^{2}\times l_{\rho }^{2} \); \((\widetilde{u}( t,\tau ,\omega , \widetilde{u}_{\tau },\widetilde{v}_{\tau }), \widetilde{v}( t,\tau ,\omega , \widetilde{u}_{\tau },\widetilde{v}_{\tau }))\) \(\in l^{2} \times l ^{2} \) if \((\widetilde{u}_{\tau },\widetilde{v}_{\tau }) \) \(\in l^{2}\times l ^{2} \). Moreover, (2.6) generates a continuous cocycle \(\Psi _{c}\) over \((\Omega ,\mathcal{F},\mathcal{P},(\theta _{t})_{t\in \mathbb{R}})\) with state space \(l^{2}_{\rho }\times l^{2}_{\rho }\): for \((\widetilde{u}_{\tau },\widetilde{v}_{\tau }) \in l_{\rho }^{2} \times l_{\rho }^{2} , t\in \mathbb{R}^{+}, \tau \in \mathbb{R} \), and \(\omega \in \Omega \),
$$ \Psi _{c} (t,\tau ,\omega ,\widetilde{u}_{\tau }, \widetilde{v}_{\tau }):= \bigl(\widetilde{u}(t+\tau , \tau ,\theta _{-\tau } \omega , \widetilde{u}_{\tau },\widetilde{v}_{\tau }), \widetilde{v}(t+\tau ,\tau , \theta _{-\tau } \omega , \widetilde{u}_{\tau },\widetilde{v}_{\tau })\bigr). $$
(2.11)

3 Existence of random attractors

We first provide some sufficient conditions for the existence of random attractors for a continuous cocycle (or non-autonomous random dynamical system) in weighted spaces of infinite sequences in [2]. The theory of random attractors for autonomous random dynamical system can be found in [2126].

In the following, let \((X,\Vert \cdot \Vert _{X} ) \) be a separable Banach space, and \(\mathcal{D}(X)\) be the collection of all tempered families of nonempty bounded subsets of X.

Definition 3.1

A mapping \(\Phi :\mathbb{R}^{+}\times \mathbb{R}\times \Omega \times X\rightarrow X \) is called a continuous cocycle on X over \(\mathbb{R}\) and \((\Omega ,\mathcal{F},\mathcal{P},\{\theta _{t}\}_{t\in \mathbb{R}})\) if for all \(\tau \in \mathbb{R},\omega \in \Omega \) and \(t,s\in \mathbb{R}^{+}\), the following conditions (1)-(4) are satisfied:
  1. (1)

    \(\Phi (\cdot ,\tau ,\cdot ,\cdot ):\mathbb{R}^{+}\times \Omega \times X\rightarrow X \) is \((\mathcal{B}(\mathbb{R}^{+})\times \mathcal{F}\times \mathcal{B}(X),\mathcal{B}(X))\)-measurable;

     
  2. (2)

    \(\Phi (0,\tau ,\omega ,\cdot )\) is the identity on X;

     
  3. (3)

    \(\Phi (t+s,\tau ,\omega ,\cdot )= \Phi (t,\tau +s,\theta _{s}\omega , \Phi (s,\tau ,\omega ,\cdot ))\);

     
  4. (4)

    \(\Phi (t,\tau ,\omega ,\cdot ):X \rightarrow X\) is continuous.

     

Definition 3.2

Let Φ be a continuous cocycle on X over \(\mathbb{R}\) and \((\Omega ,\mathcal{F},\mathcal{P},\{\theta _{t}\}_{t\in \mathbb{R}})\).
  1. (1)
    A family \(K=\{K(\tau ,\omega ):\tau \in \mathbb{R},\omega \in \Omega \}\in \mathcal{D}(X)\) is called a random absorbing set for Φ if for all \(\tau \in \mathbb{R}\) and \(\omega \in \Omega \) and for every \(D\in \mathcal{D}(X)\), there exists \(T=T(D,\tau ,\omega )>0\) such that
    $$ \Phi \bigl(t,\tau -t,\theta _{-t}\omega ,D(\tau -t,\theta_{-t}\omega )\bigr)\subseteq K(\tau ,\omega )\quad \textit{for all }t\geq T . $$
    (3.1)
     
  2. (2)
    A family \(\mathcal{A}=\{\mathcal{A}(\tau ,\omega ):\tau \in \mathbb{R},\omega \in \Omega \}\in \mathcal{D}(X)\) is called a random attractor for Φ if for all \(t\in \mathbb{R}^{+}\), \(\tau \in \mathbb{R}\) and \(\omega \in \Omega \), (i) \(\mathcal{A}(\tau ,\omega )\) is compact in X and is measurable in ω with respect to \(\mathcal{F}\); (ii) \(\mathcal{A}\) is invariant, that is, \(\Phi (t,\tau ,\omega ,\mathcal{A}(\tau ,\omega ))=\mathcal{A}(\tau +t,\theta _{t}\omega )\); (iii) For every \(D=\{D(\tau ,\omega ):\tau \in \mathbb{R},\omega \in \Omega \}\in \mathcal{D}(X)\),
    $$ \lim_{t\rightarrow \infty }d_{H}\bigl(\Phi \bigl(t,\tau -t,\theta _{-t}\omega ,D(\tau -t,\theta _{-t}\omega )\bigr), \mathcal{A}(\tau ,\omega )\bigr)=0, $$
    (3.2)
    where \(d_{H}\) is the Hausdorff semi-distance given by \(d_{H}(F,G)=\sup_{u\in F}\inf_{v\in G} \Vert u-v \Vert _{X} \), for any \(F,G\subset X\).
     

Theorem 3.3

Let Φ be a continuous cocycle on \(l_{\rho }^{2}\times l_{\rho }^{2}\) over \(\mathbb{R}\) and \((\Omega ,\mathcal{F},\mathcal{P},\{\theta _{t}\}_{t\in \mathbb{R}})\). Suppose that
  1. (a)

    there exists a bounded closed random absorbing set \(B_{0}=\{B_{0}(\tau ,\omega ):\tau \in \mathbb{R},\omega \in \Omega \}\in \mathcal{D}(l_{\rho }^{2}\times l_{\rho }^{2})\) such that, for any \(\tau \in \mathbb{R}\), \(\omega \in \Omega \) and \(B=\{B(\tau ,\omega ):\tau \in \mathbb{R},\omega \in \Omega \}\in \mathcal{D}(l_{\rho }^{2}\times l_{\rho }^{2})\), there exists \(T_{1}=T_{1}(\tau ,\omega ,B)>0\) yielding \(\Phi (t,\tau -t,\theta _{-t}\omega ,B(\tau -t,\theta _{-t}\omega ))\subset B_{0}(\tau ,\omega )\), \(\forall t\geqslant T_{1} \);

     
  2. (b)
    for each \(\tau \in \mathbb{R} \), \(\omega \in \Omega \) and for any \(\varepsilon >0\), there exist \(T_{2}=T_{2}(\tau ,\varepsilon ,\omega ,B_{0})>0\) and \(I_{0}=I_{0}(\tau ,\varepsilon ,\omega ,B_{0})\in \mathbb{N}\) such that
    $$ \sum_{\vert i \vert >I_{0} }\rho _{i}\bigl\vert \Phi _{i}(t,\tau -t,\theta _{-t}\omega,u_{\tau -t}) \bigr\vert ^{2}\leqslant \varepsilon ,\quad \forall t\geqslant T_{2}, u_{\tau -t}\in B_{0}(\tau -t,\theta _{-t}\omega ) . $$
    (3.3)
     
Then Φ possesses a unique random attractor \(\mathcal{A}\) in \(\mathcal{D}(l_{\rho }^{2}\times l_{\rho }^{2})\) given, for every \(\tau \in \mathbb{R}\) and \(\omega \in \Omega \), by
$$ \mathcal{A}(\tau ,\omega )=\bigcap_{\tau \geqslant T_{1}}\overline{ \bigcup_{t\geqslant \tau }\Phi \bigl(t,\tau -t,\theta _{-t}\omega ,B_{0}(\tau -t,\theta _{-t}\omega ) \bigr)}. $$
(3.4)

Next, we will use Theorem 3.3 to prove the existence of a random attractor for the continuous cocycle \(\Psi _{c}\) in \(l_{\rho }^{2} \times l_{\rho }^{2}\) under conditions (A1)-(A4).

Theorem 3.4

If (A1)-(A4) hold, then, for every \(c > 0 \), \(\tau \in \mathbb{R}\), \(\omega \in \Omega \) and for any \(B=\{B(\tau ,\omega ):\tau \in \mathbb{R},\omega \in \Omega \}\in \mathcal{D}(l_{\rho }^{2} \times l_{\rho }^{2})\), there exists \(T=T (\tau ,\omega ,B,c)>0\) such that, for all \(t\geqslant T \) and \((\widetilde{u}_{\tau -t}, \widetilde{v}_{\tau -t} ) \in B(\tau -t, \theta _{-t}\omega ) \), the solution \((\widetilde{u} ,\widetilde{v} ) \) of (2.6) satisfies
$$ \begin{aligned}[b] &\bigl\Vert \widetilde{u} (\tau ,\tau - t, \theta _{-\tau }\omega, \widetilde{u} _{\tau -t} ,\widetilde{v} _{\tau -t}) \bigr\Vert _{ \rho }^{2} +\bigl\Vert \widetilde{v} (\tau ,\tau - t, \theta_{-\tau }\omega ,\widetilde{u} _{\tau -t} ,\widetilde{v}_{\tau -t}) \bigr\Vert _{\rho }^{2} \\ &\qquad {} + \int ^{0}_{ -t} e ^{\int ^{0}_{s} (-\boldsymbol{\lambda}+2(1+q+\widetilde{q})\boldsymbol{\beta}(\theta _{r} \omega ) )\,dr} \bigl\Vert \widetilde{u} (s+\tau ,\tau - t, \theta _{-\tau }\omega,\widetilde{u} _{\tau -t},\widetilde{v} _{\tau -t} ) \bigr\Vert _{ \rho }^{2} \,ds \\ & \quad \leqslant I (c,\tau ,\omega ), \end{aligned} $$
(3.5)
where \(I (c,\tau ,\omega )>0\) is given by
$$ \begin{aligned}[b] &I (c,\tau ,\omega ) \\ &\quad =\mathbb {c}_{0} + \int _{-t}^{0} e^{\int ^{0}_{s} (-\boldsymbol{\lambda}+2 (1+q+\widetilde{q})\boldsymbol{\beta}(\theta _{r } \omega ) )\,dr} \biggl( \frac{8 }{\alpha }\bigl\Vert g(s+\tau ) \bigr\Vert _{\rho }^{2} +\frac{2}{\delta \sigma } \bigl\Vert h(s+\tau ) \bigr\Vert _{\rho }^{2} \biggr)\,ds \\ &\qquad {} + \int ^{0}_{ -t} e ^{\int ^{0}_{s} (-\boldsymbol{\lambda}+2 (1+q+\widetilde{q})\boldsymbol{\beta}(\theta _{r } \omega ) )\,dr} \bigl( \bigl( \mathbb {c}_{1}+ \mathbb {c}_{2}\boldsymbol{\beta}^{2}(\theta _{s}\omega ) \bigr) \bigl\Vert z (\theta _{s } \omega ) \bigr\Vert _{\rho }^{2} + \mathbb {c}_{3}\bigl\Vert y(\theta _{s}\omega ) \bigr\Vert _{\rho }^{2} \bigr)\,ds \\ &\qquad {} +2\beta \sum_{i\in \mathbb{Z}} \rho _{i} \int ^{0}_{ -t} e ^{\int ^{0}_{s} (-\boldsymbol{\lambda}+2 (1+q+\widetilde{q})\boldsymbol{\beta}(\theta _{r } \omega ) )\,dr}\,ds, \end{aligned} $$
(3.6)
where \(\mathbb {c}_{0},\mathbb {c}_{1},\mathbb {c}_{2}\), and \(\mathbb {c}_{3}\) are positive constants independent of \(\tau ,\omega \), and B.

Proof

For each \(\omega \in \Omega \), there exists a sequence \(\eta _{i,j}^{(m)}(t,\omega )\) of continuous functions in \(t\in \mathbb{R}\) (see [27]) such that
$$ \begin{aligned}[b] &\lim_{m\rightarrow \infty} \int _{\tau }^{t}\bigl\vert \eta _{i,j}^{(m)}(s,\omega)-\eta _{i,j}(\theta _{s}\omega ) \bigr\vert ds=0, \\ &\quad \forall t>0,\tau \in \mathbb{R} ,j\in \{-q,\ldots ,0,\ldots ,q\},q\in \mathbb{N} \text{ and } i\in \mathbb{Z}, \end{aligned} $$
(3.7)
and \(\vert \eta _{i,j}^{(m)}(t,\omega ) \vert \leqslant \vert \eta _{i,j}(\theta_{t}\omega) \vert \leqslant \boldsymbol{\beta}(\theta _{t}\omega ) \) for \(t\in \mathbb{R}\). Consider the following random differential equations:
$$ \left \{ \textstyle\begin{array}{l} \frac{d\widetilde{u}^{(m)}}{dt} = \mathbb{B}_{m}(t,\omega )\widetilde{u}^{(m)} - \widetilde{v}^{(m)}+ f( \widetilde{u}^{(m)}+ c z(\theta _{t}\omega ) ) \\ \hphantom{\frac{d\widetilde{u}^{(m)}}{dt} = }+c \mathbb{B}_{m}(t,\omega )z(\theta _{t}\omega )+ g(t) - c \lambda z(\theta _{t}\omega )-cy(\theta _{t}\omega ), \\ \frac{d\widetilde{v}^{(m)}}{dt}=\sigma \widetilde{u}^{(m)}-\delta \widetilde{v}^{(m)} + h(t) +\sigma c z(\theta _{t}\omega ) + c(\mu -\delta ) y(\theta _{t}\omega ), \\ \widetilde{u}^{(m)}(\tau )=\widetilde{u}_{\tau },\qquad \widetilde{v}^{(m)}(\tau )=\widetilde{v}_{\tau }, \end{array}\displaystyle \right . $$
(3.8)
where \((\mathbb{B}_{m}(t,\omega )\widetilde{u}^{(m)})_{i}=\sum_{j=-q}^{q}\eta _{i,j}^{(m)}(t,\omega )\widetilde{u}_{i+j}^{(m)}\). It is easy to see that (3.8) has a unique mild solution \(( \widetilde{u}^{(m)}(\cdot ,\tau ,\omega ,\widetilde{u}_{\tau },\widetilde{v}_{\tau }), \widetilde{v}^{(m)}(\cdot ,\tau ,\omega ,\widetilde{u}_{\tau },\widetilde{v}_{\tau })) \) \(\in C([\tau , +\infty ),l^{2} \times l^{2} )\cap C^{1}((\tau , +\infty ),l^{2}\times l^{2})\) satisfying (3.8). Taking the inner product of (3.8) with \(\widetilde{u} ^{(m)}(t)\) and \(\widetilde{v} ^{(m)}(t)\), respectively, in \(l_{\rho }^{2}\), we have
$$\begin{aligned} &\frac{d}{dt} \biggl( \bigl\Vert \widetilde{u}^{(m)} \bigr\Vert _{\rho }^{2} + \frac{1}{\sigma }\bigl\Vert \widetilde{v}^{(m)} \bigr\Vert _{\rho }^{2} \biggr) \\ &\quad = 2 \bigl( \mathbb{B}_{m} ( t,\omega )\widetilde{u}^{(m)} , \widetilde{u}^{(m)} \bigr) _{\rho }+2 \bigl( f\bigl( \widetilde{u}^{(m)}+cz(\theta _{t}\omega ) \bigr), \widetilde{u}^{(m)} \bigr) _{\rho } \\ & \quad \quad {}+ 2c \bigl( \mathbb{B}_{m} ( t,\omega ) z(\theta _{t} \omega ) , \widetilde{u}^{(m)} \bigr) _{\rho }+ 2 \bigl( g(t), \widetilde{u}^{(m)} \bigr)_{\rho }- 2c \lambda \bigl( z(\theta _{t} \omega ), \widetilde{u}^{(m)} \bigr) _{\rho } \\ &\quad \quad {}-2c \bigl( y(\theta _{t}\omega ), \widetilde{u}^{(m)} \bigr) _{\rho }- \frac{2\delta }{\sigma }\bigl\Vert \widetilde{v}^{(m)} \bigr\Vert ^{2} + \frac{2}{\sigma } \bigl( h(t), \widetilde{v}^{(m)} \bigr)_{\rho } \\ & \quad \quad {}+2c \bigl( z(\theta _{t} \omega ), \widetilde{v}^{(m)} \bigr) _{\rho }+\frac{2c(\mu -\delta )}{\sigma } \bigl( y(\theta _{t} \omega ), \widetilde{v}^{(m)} \bigr) _{\rho }. \end{aligned}$$
(3.9)
Note that
$$\begin{aligned}& 2 \bigl( g(t), \widetilde{u}^{(m)} \bigr)_{\rho }\leqslant \frac{8}{\alpha } \bigl\Vert g (t) \bigr\Vert _{\rho }^{2} + \frac{\alpha }{8} \bigl\Vert \widetilde{u}^{(m)} \bigr\Vert _{\rho }^{2}, \end{aligned}$$
(3.10)
$$\begin{aligned}& - 2c \lambda \bigl( z(\theta _{t} \omega ), \widetilde{u}^{(m)} \bigr) _{\rho }\leqslant \frac{4 c^{2} \lambda ^{2} }{\alpha } \bigl\Vert z(\theta _{t} \omega ) \bigr\Vert _{\rho }^{2} + \frac{\alpha }{4} \bigl\Vert \widetilde{ u}^{(m)} \bigr\Vert _{\rho }^{2}, \end{aligned}$$
(3.11)
$$\begin{aligned}& -2c \bigl( y(\theta _{t}\omega ), \widetilde{u}^{(m)} \bigr) _{\rho }\leqslant \frac{4c^{2}}{\alpha } \bigl\Vert y(\theta _{t}\omega ) \bigr\Vert _{\rho }^{2} + \frac{\alpha }{4} \bigl\Vert \widetilde{ u}^{(m)} \bigr\Vert _{\rho }^{2}, \end{aligned}$$
(3.12)
$$\begin{aligned}& \frac{2}{\sigma } \bigl( h(t), \widetilde{v}^{(m)} \bigr)_{\rho }\leqslant \frac{2 }{\delta \sigma } \bigl\Vert h (t) \bigr\Vert _{\rho }^{2} + \frac{\delta }{ 2\sigma } \bigl\Vert \widetilde{ v}^{(m)} \bigr\Vert _{\rho }^{2}, \end{aligned}$$
(3.13)
$$\begin{aligned}& 2c \bigl( z(\theta _{t} \omega ), \widetilde{v}^{(m)} \bigr) _{\rho }\leqslant \frac{4c^{2}\sigma }{\delta } \bigl\Vert z(\theta _{t} \omega ) \bigr\Vert _{\rho }^{2} + \frac{\delta }{4\sigma } \bigl\Vert \widetilde{v}^{(m)} \bigr\Vert _{\rho }^{2}, \end{aligned}$$
(3.14)
$$\begin{aligned}& \frac{2c(\mu -\delta )}{\sigma } \bigl( y(\theta _{t} \omega ), \widetilde{v}^{(m)} \bigr) _{\rho }\leqslant \frac{4c^{2}(\mu -\delta )^{2}}{\delta \sigma } \bigl\Vert y(\theta _{t}\omega ) \bigr\Vert _{\rho }^{2} + \frac{\delta }{ 4\sigma }\bigl\Vert \widetilde{v}^{(m)} \bigr\Vert _{\rho }^{2}. \end{aligned}$$
(3.15)
By (2.1), we get
$$\begin{aligned}& 2 \bigl( \mathbb{B}_{m} ( t,\omega )\widetilde{u}^{(m)} , \widetilde{u}^{(m)} \bigr)_{\rho }= 2 \sum _{i\in \mathbb{Z}} \Biggl( \rho _{i} \widetilde{u}_{i}^{(m)} \cdot \sum_{j=-q}^{q}\eta _{i,j}^{(m)}(t,\omega )\widetilde{u}_{i+j}^{(m)} \Biggr) \\& \hphantom{2 \bigl( \mathbb{B}_{m} ( t,\omega )\widetilde{u}^{(m)} , \widetilde{u}^{(m)} \bigr)_{\rho }}\leqslant 2 \boldsymbol{\beta}(\theta _{t } \omega )\sum _{i\in \mathbb{Z}} \Biggl( \rho _{i} \bigl\vert \widetilde{u}_{i}^{(m)} \bigr\vert \cdot \sum _{j=-q}^{q} \bigl\vert \widetilde{u}_{i+j}^{(m)} \bigr\vert \Biggr) \\& \hphantom{2 \bigl( \mathbb{B}_{m} ( t,\omega )\widetilde{u}^{(m)} , \widetilde{u}^{(m)} \bigr)_{\rho }}\leqslant 2 (1+q+\widetilde{q})\boldsymbol{\beta}(\theta _{t } \omega ) \bigl\Vert \widetilde{u}^{(m)} \bigr\Vert _{\rho }^{2}, \end{aligned}$$
(3.16)
$$\begin{aligned}& 2c \bigl( \mathbb{B}_{m}(t,\omega )z(\theta _{t}\omega ) , \widetilde{u}^{(m)} \bigr)_{\rho }= 2c\sum _{ i \in \mathbb{Z}} \Biggl(\rho _{i} \widetilde{u}_{ i}^{(m)} \sum_{j=-q}^{q} \eta _{i,j}^{(m)} ( t,\omega )z_{ i+j} (\theta _{t}\omega ) \Biggr) \\& \hphantom{2c \bigl( \mathbb{B}_{m}(t,\omega )z(\theta _{t}\omega ) , \widetilde{u}^{(m)} \bigr)_{\rho }}\leqslant \frac{\alpha }{8} \bigl\Vert \widetilde{u}^{(m)} \bigr\Vert ^{2}_{\rho }+ \frac{8c^{2}(1+2q)(1+2\widetilde{q}) \boldsymbol{\beta}^{2}(\theta _{t }\omega )}{\alpha } \bigl\Vert z(\theta _{t} \omega ) \bigr\Vert ^{2}_{\rho }, \end{aligned}$$
(3.17)
where \(\widetilde{q}= \sum_{k=1}^{q} c_{0}^{k}\). By (A2), we have
$$\begin{aligned} &2 \bigl( f \bigl( \widetilde{u}^{(m)}+c z (\theta _{t }\omega ) \bigr), \widetilde{u}^{(m)} \bigr) _{\rho} \\ &\quad = 2 \sum_{i\in \mathbb{Z}} \rho _{i}f_{i} \bigl( \widetilde{u}_{i}^{(m)}+cz_{i}(\theta _{t } \omega ) \bigr)\cdot \widetilde{u}_{i}^{(m)} \\ &\quad = 2 \sum_{i\in \mathbb{Z}} \rho _{i}f_{i} \bigl( \widetilde{u}_{i}^{(m)}+cz_{i} (\theta _{t } \omega ) \bigr) \cdot \bigl(\widetilde{u}_{i}^{(m)}+c z_{i} (\theta _{t } \omega ) \bigr) \\ &\quad \quad {}- 2 \sum_{i\in \mathbb{Z}} \rho_{i} f_{i} \bigl( \widetilde{u}_{i}^{(m)}+cz_{i} (\theta _{t } \omega ) \bigr) \cdot c z_{i} (\theta _{t } \omega ) \\ &\quad \leqslant 2 \sum_{i\in \mathbb{Z}} \rho _{i}\bigl(-\alpha \bigl(\widetilde{u}_{i}^{(m)}+c z_{i} (\theta _{t } \omega ) \bigr)^{2} +\beta \bigr) \\ &\quad \quad {}+ 2\sum_{i\in \mathbb{Z}} \rho _{i} \kappa \bigl\vert \widetilde{u}_{i}^{(m)}+c z_{i} (\theta _{t } \omega ) \bigr\vert \cdot \bigl\vert c z_{i} ( \theta _{t } \omega ) \bigr\vert \\ &\quad \leqslant - 2\alpha \bigl\Vert \widetilde{u} ^{(m)} \bigr\Vert _{\rho }^{2}- 2\alpha c^{2} \bigl\Vert z (\theta _{t } \omega ) \bigr\Vert _{\rho }^{2} +4 \alpha \sum_{i\in \mathbb{Z}} \rho _{i} \bigl\vert \widetilde{u}_{i}^{(m)} \bigr\vert \cdot \bigl\vert c z_{i} (\theta _{t } \omega ) \bigr\vert + 2 \beta \sum _{i\in \mathbb{Z}} \rho _{i} \\ &\quad \quad{} + 2 \kappa \sum_{i\in \mathbb{Z}} \rho _{i} \bigl\vert \widetilde{u}_{i}^{(m)} \bigr\vert \cdot \bigl\vert cz_{i} (\theta _{t } \omega ) \bigr\vert +2 \kappa c^{2} \bigl\Vert z (\theta _{t } \omega ) \bigr\Vert _{\rho }^{2} \end{aligned}$$
(3.18)
$$\begin{aligned} &\quad \leqslant - 2\alpha \bigl\Vert \widetilde{u} ^{(m)} \bigr\Vert _{\rho }^{2}- 2\alpha c^{2} \bigl\Vert z (\theta _{t } \omega ) \bigr\Vert _{\rho }^{2} +4 \alpha \sum_{i\in \mathbb{Z}} \rho _{i} \bigl\vert \widetilde{u}_{i}^{(m)} \bigr\vert \cdot \bigl\vert c z_{i} (\theta _{t } \omega ) \bigr\vert + 2 \beta \sum _{i\in \mathbb{Z}} \rho _{i} \\ &\quad \quad {}+ 2 \kappa \sum_{i\in \mathbb{Z}} \rho _{i} \bigl\vert \widetilde{u}_{i}^{(m)} \bigr\vert \cdot \bigl\vert c z_{i} (\theta _{t } \omega ) \bigr\vert +2 \kappa c^{2} \bigl\Vert z (\theta _{t } \omega ) \bigr\Vert _{\rho }^{2} \end{aligned}$$
(3.19)
$$\begin{aligned} &\quad \leqslant - 2\alpha \bigl\Vert \widetilde{u} ^{(m)} \bigr\Vert _{\rho }^{2}- 2\alpha c^{2} \bigl\Vert z (\theta _{t } \omega ) \bigr\Vert _{\rho }^{2} + \frac{\alpha }{8} \bigl\Vert \widetilde{u} ^{(m)} \bigr\Vert _{\rho }^{2}+32\alpha c^{2} \bigl\Vert z (\theta _{t } \omega ) \bigr\Vert _{\rho }^{2} + 2 \beta \sum_{i\in \mathbb{Z}} \rho _{i} \\ &\quad\quad {} + \frac{\alpha }{8} \bigl\Vert \widetilde{u} ^{(m)} \bigr\Vert _{\rho }^{2} + \frac{8\kappa c^{2}}{\alpha } \bigl\Vert z (\theta _{t } \omega ) \bigr\Vert _{\rho }^{2} +2 \kappa c^{2} \bigl\Vert z (\theta _{t } \omega ) \bigr\Vert _{\rho }^{2} \end{aligned}$$
(3.20)
$$\begin{aligned} &\quad \leqslant - \frac{7\alpha }{4} \bigl\Vert \widetilde{u} ^{(m)} \bigr\Vert _{\rho }^{2} +c^{2}\biggl(30\alpha + \frac{8\kappa }{\alpha }+2 \kappa \biggr) \bigl\Vert z (\theta _{t } \omega ) \bigr\Vert _{\rho }^{2} + 2 \beta \sum _{i\in \mathbb{Z}} \rho _{i}. \end{aligned}$$
(3.21)
From (3.9)-(3.21), we obtain, for \(t>0\),
$$\begin{aligned} & \frac{d}{dt} \biggl( \bigl\Vert \widetilde{u}^{(m)} \bigr\Vert _{\rho }^{2} + \frac{1}{\sigma }\bigl\Vert \widetilde{v}^{(m)} \bigr\Vert _{\rho }^{2} \biggr)+ \frac{\alpha }{2}\bigl\Vert \widetilde{u}^{(m)} \bigr\Vert _{\rho }^{2} \\ &\quad \leqslant \biggl( 2 (1+q+\widetilde{q})\boldsymbol{\beta}(\theta _{t } \omega ) -\frac{\alpha }{2} \biggr) \bigl\Vert \widetilde{u}^{(m)} \bigr\Vert _{\rho }^{2} - \frac{\delta }{\sigma } \bigl\Vert \widetilde{v}^{(m)} \bigr\Vert _{\rho }^{2} + \frac{8}{\alpha } \bigl\Vert g(t) \bigr\Vert _{\rho }^{2} + \frac{2}{\delta \sigma } \bigl\Vert h(t) \bigr\Vert _{\rho }^{2} \\ & \quad \quad {}+ c^{2} \biggl( \frac{4 \lambda ^{2} }{\alpha } + \frac{4 \sigma }{\delta } + 30\alpha + \frac{8\kappa }{\alpha }+2 \kappa +\frac{8 (1+2q)(1+2\widetilde{q}) \boldsymbol{\beta}^{2}(\theta _{t }\omega )}{\alpha } \biggr) \bigl\Vert z (\theta _{t } \omega ) \bigr\Vert _{\rho }^{2} \\ &\quad \quad {} + 4c^{2} \biggl(\frac{1}{\alpha } +\frac{ (\mu -\delta )^{2}}{\delta \sigma } \biggr) \bigl\Vert y(\theta _{t}\omega ) \bigr\Vert _{\rho }^{2} + 2 \beta \sum_{i\in \mathbb{Z}} \rho _{i}. \end{aligned}$$
(3.22)
Recalling that \(\boldsymbol{\lambda}=\min \{\frac{\alpha }{2},\delta \}\), then we have
$$\begin{aligned} & \frac{d}{dt} \biggl( \bigl\Vert \widetilde{u}^{(m)} \bigr\Vert _{\rho }^{2} + \frac{1}{\sigma }\bigl\Vert \widetilde{v}^{(m)} \bigr\Vert _{\rho }^{2} \biggr)+ \frac{\alpha }{2}\bigl\Vert \widetilde{u}^{(m)} \bigr\Vert _{\rho }^{2} \\ &\quad \leqslant \bigl(-\boldsymbol{\lambda}+ 2 (1+q+\widetilde{q})\boldsymbol{\beta}(\theta _{t } \omega ) \bigr) \biggl(\bigl\Vert \widetilde{u}^{(m)}\bigr\Vert _{\rho }^{2} + \frac{1}{\sigma } \bigl\Vert \widetilde{ v}^{(m)} \bigr\Vert _{\rho }^{2} \biggr) \\ &\quad \quad {} + \frac{8}{\alpha } \bigl\Vert g(t) \bigr\Vert _{\rho }^{2} + \frac{2}{\delta \sigma } \bigl\Vert h(t) \bigr\Vert _{\rho }^{2} \\ &\quad \quad {}+ \bigl(\mathbb {c}_{1}+ \mathbb {c}_{2}\boldsymbol{\beta}^{2}(\theta _{t} \omega ) \bigr) \bigl\Vert z (\theta _{t } \omega ) \bigr\Vert _{\rho }^{2} + \mathbb {c}_{3}\bigl\Vert y(\theta _{t}\omega ) \bigr\Vert _{\rho }^{2}+ 2 \beta \sum_{i\in \mathbb{Z}} \rho _{i}, \end{aligned}$$
(3.23)
where \(\mathbb {c}_{1}= c^{2}( \frac{4 \lambda ^{2} }{\alpha } + \frac{4 \sigma }{\delta } + 30\alpha + \frac{8\kappa }{\alpha }+2 \kappa ) \), \(\mathbb {c}_{2}=\frac{8c^{2}(1+2q)(1+2\widetilde{q}) }{\alpha }\) and \(\mathbb {c}_{3}=4c^{2} (\frac{1}{\alpha } +\frac{ (\mu -\delta )^{2}}{\delta \sigma } ) \). Then we obtain, for \(t>0\),
$$\begin{aligned} &\bigl\Vert \widetilde{u}^{(m)} (\tau ,\tau - t, \omega, \widetilde{u} _{\tau -t},\widetilde{v}_{\tau -t} ) \bigr\Vert _{ \rho }^{2} +\frac{1}{\sigma }\bigl\Vert \widetilde{v}^{(m)}(\tau ,\tau - t, \omega ,\widetilde{u} _{\tau -t},\widetilde{v}_{\tau -t} ) \bigr\Vert _{\rho }^{2} \\ &\qquad {} + \frac{\alpha }{2} \int ^{\tau }_{\tau -t} e ^{\int ^{\tau }_{s} (-\boldsymbol{\lambda}+2 (1+q+\widetilde{q})\boldsymbol{\beta}(\theta _{r } \omega ) )\,dr} \bigl\Vert \widetilde{u}^{(m)} (s,\tau - t, \omega,\widetilde{u} _{\tau -t}, \widetilde{v} _{\tau -t} ) \bigr\Vert _{ \rho }^{2}\,ds \\ &\quad \leqslant e^{\int ^{\tau }_{\tau -t} (-\boldsymbol{\lambda}+2(1+q+\widetilde{q})\boldsymbol{\beta}(\theta _{r } \omega ) )\,dr} \biggl(\Vert \widetilde{u}_{ \tau -t} \Vert _{\rho }^{2} +\frac{1}{\sigma }\Vert \widetilde{v}_{ \tau -t} \Vert _{\rho }^{2} \biggr) \\ &\quad \quad {} + \int ^{\tau }_{\tau -t} e ^{\int ^{\tau }_{s} (-\boldsymbol{\lambda}+2 (1+q+\widetilde{q})\boldsymbol{\beta}(\theta _{r } \omega ) )\,dr } \biggl( \frac{8 }{\alpha } \bigl\Vert g(s ) \bigr\Vert _{\rho }^{2} + \frac{2}{\delta \sigma } \bigl\Vert h(s ) \bigr\Vert _{\rho }^{2} \biggr)\,ds \\ &\quad \quad {} + \int ^{\tau }_{\tau -t} e ^{\int ^{\tau }_{s} (-\boldsymbol{\lambda}+2 (1+q+\widetilde{q})\boldsymbol{\beta}(\theta _{r } \omega ) )\,dr} \bigl( \bigl( \mathbb {c}_{1}+ \mathbb {c}_{2}\boldsymbol{\beta}^{2}(\theta _{s} \omega ) \bigr) \bigl\Vert z (\theta _{s } \omega ) \bigr\Vert _{\rho }^{2} + \mathbb {c}_{3}\bigl\Vert y(\theta _{s}\omega ) \bigr\Vert _{\rho }^{2} \bigr)\,ds \\ &\quad \quad {} +2\beta \sum_{i\in \mathbb{Z}} \rho _{i} \int ^{\tau }_{\tau -t} e ^{\int ^{\tau }_{s} (-\boldsymbol{\lambda}+2 (1+q+\widetilde{q})\boldsymbol{\beta}(\theta _{r } \omega ) )\,dr}\,ds. \end{aligned}$$
(3.24)
From (3.24) and by replacing ω by \(\theta _{-\tau }\omega \), we have
$$\begin{aligned}& \bigl\Vert \widetilde{u}^{(m)} (\tau ,\tau - t, \theta_{-\tau }\omega ,\widetilde{u} _{\tau -t},\widetilde{v} _{\tau -t}) \bigr\Vert _{ \rho }^{2} +\frac{1}{\sigma } \bigl\Vert \widetilde{v}^{(m)} (\tau ,\tau - t, \theta_{-\tau } \omega ,\widetilde{u} _{\tau -t},\widetilde{v}_{\tau -t} ) \bigr\Vert _{\rho }^{2} \\& \qquad {} + \frac{\alpha }{2} \int ^{\tau }_{\tau -t} e ^{\int ^{\tau }_{s} (-\lambda +2 (1+q+\widetilde{q})\boldsymbol{\beta}(\theta _{r-\tau } \omega ) )\,dr} \bigl\Vert \widetilde{u}^{(m)} (s,\tau - t, \theta _{-\tau }\omega, \widetilde{u}_{\tau -t},\widetilde{v} _{\tau -t} ) \bigr\Vert _{ \rho }^{2}\,ds \\& \quad \leqslant e^{\int ^{0}_{-t} (-\boldsymbol{\lambda}+2(1+q+\widetilde{q})\boldsymbol{\beta}(\theta _{r } \omega ) )\,dr} \biggl(\Vert \widetilde{u}_{ \tau -t} \Vert _{\rho }^{2} +\frac{1}{\sigma }\Vert \widetilde{v}_{ \tau -t} \Vert _{\rho }^{2} \biggr) \\& \qquad {} + \int _{-t}^{0} e^{\int ^{0}_{s} (-\boldsymbol{\lambda}+2 (1+q+\widetilde{q})\boldsymbol{\beta}(\theta _{r } \omega ) )\,dr } \biggl( \frac{8 }{\alpha } \bigl\Vert g(s+\tau ) \bigr\Vert _{\rho }^{2} +\frac{2}{\delta \sigma } \bigl\Vert h(s+\tau ) \bigr\Vert _{\rho }^{2} \biggr)\,ds \\& \qquad {} + \int ^{0}_{ -t} e ^{\int ^{0}_{s} (-\boldsymbol{\lambda}+2 (1+q+\widetilde{q})\boldsymbol{\beta}(\theta _{r } \omega ) )\,dr} \bigl( \bigl( \mathbb {c}_{1}+ \mathbb {c}_{2}\boldsymbol{\beta}^{2}(\theta _{s}\omega ) \bigr) \bigl\Vert z (\theta _{s } \omega ) \bigr\Vert _{\rho }^{2} + \mathbb {c}_{3}\bigl\Vert y(\theta _{s}\omega ) \bigr\Vert _{\rho }^{2} \bigr)\,ds \\& \qquad {} +2\beta \sum_{i\in \mathbb{Z}} \rho _{i} \int ^{0}_{ -t} e ^{\int ^{0}_{s} (-\boldsymbol{\lambda}+2 (1+q+\widetilde{q})\boldsymbol{\beta}(\theta _{r } \omega ) )\,dr }\,ds. \end{aligned}$$
(3.25)
Note that (3.25) holds with \(\widetilde{u}^{(m)}(\tau ,\tau - t, \omega ,\widetilde{u}_{\tau -t},\widetilde{v} _{\tau -t} )\) and \(\widetilde{v}^{(m)}(\tau ,\tau - t, \omega ,\widetilde{u} _{\tau -t},\widetilde{v}_{\tau -t} )\) being replaced by \(\widetilde{u}(\tau ,\tau - t, \omega ,\widetilde{u}_{\tau -t},\widetilde{v} _{\tau -t})\) and \(\widetilde{v}(\tau ,\tau - t, \omega ,\widetilde{u} _{\tau -t},\widetilde{v}_{\tau -t})\), then we have
$$\begin{aligned}& \bigl\Vert \widetilde{u} (\tau ,\tau - t, \theta _{-\tau }\omega, \widetilde{u} _{\tau -t} ,\widetilde{v} _{\tau -t}) \bigr\Vert _{ \rho }^{2} +\frac{1}{\sigma }\bigl\Vert \widetilde{v} (\tau ,\tau - t, \theta_{-\tau }\omega ,\widetilde{u} _{\tau -t} , \widetilde{v}_{\tau -t} ) \bigr\Vert _{\rho }^{2} \\& \quad {}+ \frac{\alpha }{2} \int ^{\tau }_{\tau -t} e ^{\int ^{\tau }_{s} (-\boldsymbol{\lambda}+2 (1+q+\widetilde{q})\boldsymbol{\beta}(\theta _{r-\tau } \omega ) )\,dr} \bigl\Vert \widetilde{u} (s,\tau - t, \theta _{-\tau }\omega,\widetilde{u}_{\tau -t}, \widetilde{v} _{\tau -t} ) \bigr\Vert _{ \rho }^{2}\,ds \\& \quad \leqslant e^{\int ^{0}_{-t} (-\boldsymbol{\lambda}+2(1+q+\widetilde{q})\boldsymbol{\beta}(\theta _{r } \omega ) )\,dr} \biggl(\Vert \widetilde{u}_{ \tau -t} \Vert _{\rho }^{2} +\frac{1}{\sigma }\Vert \widetilde{v}_{ \tau -t} \Vert _{\rho }^{2} \biggr) \\& \quad \quad {}+ \int _{-t}^{0} e^{\int ^{0}_{s} (-\boldsymbol{\lambda}+2 (1+q+\widetilde{q})\boldsymbol{\beta}(\theta _{r } \omega ) )\,dr} \biggl( \frac{8 }{\alpha }\bigl\Vert g(s+\tau ) \bigr\Vert _{\rho }^{2} +\frac{2}{\delta \sigma } \bigl\Vert h(s+\tau ) \bigr\Vert _{\rho }^{2} \biggr)\,ds \\& \quad \quad {}+ \int ^{0}_{ -t} e ^{\int ^{0}_{s} (-\boldsymbol{\lambda}+2 (1+q+\widetilde{q})\boldsymbol{\beta}(\theta _{r } \omega ) )\,dr} \bigl( \bigl( \mathbb {c}_{1}+ \mathbb {c}_{2}\boldsymbol{\beta}^{2}(\theta _{s}\omega ) \bigr) \bigl\Vert z (\theta _{s } \omega ) \bigr\Vert _{\rho }^{2} + \mathbb {c}_{3}\bigl\Vert y(\theta _{s}\omega ) \bigr\Vert _{\rho }^{2} \bigr)\,ds \\& \quad\quad {}+2\beta \sum_{i\in \mathbb{Z}} \rho _{i} \int ^{0}_{ -t} e ^{\int ^{0}_{s} (-\boldsymbol{\lambda}+2 (1+q+\widetilde{q})\boldsymbol{\beta}(\theta _{r } \omega ) )\,dr}\,ds. \end{aligned}$$
(3.26)
By (2.7), we find that there exists \(T_{1}=T_{1}(\omega )>0\) such that, for \(t> T_{1}\),
$$\begin{aligned} \int _{-t}^{0} \boldsymbol{\beta}(\theta _{s}\omega )\,ds\leqslant \frac{\boldsymbol{\lambda}}{ 4(1+q+\widetilde{q})} t . \end{aligned}$$
By (A4) and \((\widetilde{u}_{\tau -t} , \widetilde{v}_{\tau -t})\in B(\tau -t, \theta _{-t}\omega ) \in \mathcal{D}( l_{\rho }^{2} \times l_{\rho }^{2}) \), we have
$$\begin{aligned}& \lim_{t\rightarrow +\infty } e^{\int ^{0}_{-t} (-\boldsymbol{\lambda}+2(1+q+\widetilde{q})\boldsymbol{\beta}(\theta _{r } \omega ) )\,dr} \biggl(\Vert \widetilde{u}_{ \tau -t} \Vert _{\rho }^{2} + \frac{1}{\sigma }\Vert \widetilde{v}_{ \tau -t} \Vert _{\rho }^{2} \biggr) \\& \quad \leqslant \limsup_{t\rightarrow +\infty } e^{ \frac{-\boldsymbol{\lambda}t}{2} } \bigl\Vert B( \tau -t,\theta _{-t}\omega ) \bigr\Vert _{\rho }^{2} \\& \quad \leqslant 0. \end{aligned}$$
(3.27)
Therefore, there exists \(T_{2}=T_{2} (\tau ,\omega ,B,c)>0\) such that, for all \(t \geqslant T_{2} \),
$$\begin{aligned} & e^{\int ^{0}_{-t} (-\boldsymbol{\lambda}+2(1+q+\widetilde{q})\boldsymbol{\beta}(\theta _{r } \omega ) )\,dr} \biggl(\Vert \widetilde{u}_{ \tau -t} \Vert _{\rho }^{2} +\frac{1}{\sigma }\Vert \widetilde{v}_{ \tau -t} \Vert _{\rho }^{2} \biggr) \leqslant 1. \end{aligned}$$
(3.28)
Note that \(z(\theta _{t}\omega ) \), \(y(\theta _{t}\omega ) \) and \(\boldsymbol{\beta}(\theta _{t} \omega ) \) are tempered. Then by (A4), we can verify the following integrals are convergent:
$$\begin{aligned}& \int _{-t}^{0} e^{\int ^{0}_{s} (-\boldsymbol{\lambda}+2 (1+q+\widetilde{q})\boldsymbol{\beta}(\theta _{r } \omega ) )\,dr} \biggl( \frac{8 }{\alpha }\bigl\Vert g(s+\tau ) \bigr\Vert _{\rho }^{2} +\frac{2}{\delta \sigma } \bigl\Vert h(s+\tau ) \bigr\Vert _{\rho }^{2} \biggr)\,ds \\& \qquad {} + \int ^{0}_{ -t} e ^{\int ^{0}_{s} (-\boldsymbol{\lambda}+2 (1+q+\widetilde{q})\boldsymbol{\beta}(\theta _{r } \omega ) )\,dr} \bigl( \bigl( \mathbb {c}_{1}+ \mathbb {c}_{2}\boldsymbol{\beta}^{2}(\theta _{s}\omega ) \bigr) \bigl\Vert z (\theta _{s } \omega ) \bigr\Vert _{\rho }^{2} + \mathbb {c}_{3}\bigl\Vert y(\theta _{s}\omega ) \bigr\Vert _{\rho }^{2} \bigr)\,ds \\& \qquad {} +2\beta \sum_{i\in \mathbb{Z}} \rho _{i} \int ^{0}_{ -t} e ^{\int ^{0}_{s} (-\boldsymbol{\lambda}+2 (1+q+\widetilde{q})\boldsymbol{\beta}(\theta _{r } \omega ) )\,dr}\,ds \\& \quad < \infty . \end{aligned}$$
(3.29)
Thus the theorem follows from (3.26), (3.28), and (3.29). □

Theorem 3.5

Assume that (A1)-(A4) hold. Then the continuous cocycle \(\Psi _{c}\) associated with (2.6) has a unique random attractor \(\mathscr{A}_{c}=\{\mathscr{A}_{c}(\tau ,\omega ):\tau \in \mathbb{R},\omega \in \Omega \}\in \mathcal{D}(l_{\rho }^{2}\times l_{\rho }^{2})\).

Proof

By Theorem 3.3, it suffices to prove that, for every \(\varepsilon > 0 \), \(c > 0 \), \(\tau \in \mathbb{R}\), \(\omega \in \Omega \) and for any \(B=\{B(\tau ,\omega ):\tau \in \mathbb{R},\omega \in \Omega \}\in \mathcal{D}(l_{\rho }^{2}\times l_{\rho }^{2})\), there exist \(T=T (\tau ,\omega ,B,c,\varepsilon )>0\) and \(R=R (\tau ,\omega ,c,\varepsilon )>1\) such that, for all \(t\geqslant T \) and \((\widetilde{u}_{\tau -t}, \widetilde{v}_{\tau -t} ) \in B(\tau -t, \theta _{-t}\omega ) \), the solution \((\widetilde{u} ,\widetilde{v} ) \) of (2.6) satisfies
$$ \sum_{\vert i \vert >R}\rho _{i} \bigl(\bigl\vert \widetilde{u}_{i} ( \tau,\tau -t,\theta _{-\tau }\omega , \widetilde{u} _{\tau -t},\widetilde{v} _{\tau -t} ) \bigl\vert ^{2}+ \bigr\vert \widetilde{v}_{i} ( \tau ,\tau -t,\theta _{-\tau }\omega ,\widetilde{u} _{\tau -t},\widetilde{v} _{\tau -t} ) \bigr\vert ^{2} \bigr)\leqslant \varepsilon . $$
(3.30)
Choose a smooth increasing function χ: \(\mathbb{R}^{+}\rightarrow [0,1] \) such that
$$ \chi (s)= \textstyle\begin{cases} 0, & 0\leqslant s\leqslant 1, \\ 1, & s\geqslant 2, \end{cases} $$
(3.31)
and there exists a positive constant \(\mathbb {c}_{\chi}\) such that \(\vert \chi ^{\prime }(s) \vert \leqslant \mathbb {c}_{\chi}\) for \(s\in \mathbb{R}^{+}\).

Let \((\widetilde{u} (\tau ,\tau -t,\omega , \widetilde{u} _{\tau -t}, \widetilde{v} _{\tau -t}), \widetilde{v}(\tau ,\tau -t,\omega , \widetilde{u} _{\tau -t}, \widetilde{v} _{\tau -t}))\) be a mild solution of (2.6) with \(( \widetilde{u}_{\tau -t} , \widetilde{v}_{\tau -t})\in l_{\rho }^{2}\times l_{\rho }^{2}\). For any given \(N>0\) define \(\mathcal{Q}_{N}:l_{\rho }^{2} \times l_{\rho }^{2} \rightarrow l^{2}\times l^{2}\), \(( \widetilde{u} , \widetilde{v} ) = ( \widetilde{u}_{i}, \widetilde{v}_{i} ) _{i\in\mathbb{Z}}\mapsto \) \(\mathcal{Q}_{N}(\widetilde{u},\widetilde{v})=\) \(((\mathcal{Q}_{N}\widetilde{u} )_{i}, (\mathcal{Q}_{N}\widetilde{v} )_{i})_{i\in \mathbb{Z}}\) by \(((\mathcal{Q}_{N}\widetilde{u})_{j}, (\mathcal{Q}_{N}\widetilde{v})_{j}) =(\widetilde{u}_{j},\widetilde{v}_{j})\) if \(\vert j \vert \leqslant N\) and \(((\mathcal{Q}_{N}\widetilde{u})_{j}, (\mathcal{Q}_{N}\widetilde{v})_{j})=(0,0)\) otherwise.

For any \(n\geqslant 1\), let \((\widetilde{u} ^{(m)}, \widetilde{v} ^{(m)})=(\widetilde{u}^{(m)} (\tau ,\tau -t,\omega , \mathcal{Q}_{n}\widetilde{u} _{\tau -t}, \mathcal{Q}_{n}\widetilde{v} _{\tau -t}), \widetilde{v}^{(m)}(\tau ,\tau -t,\omega , \mathcal{Q}_{n}\widetilde{u} _{\tau -t}, \mathcal{Q}_{n}\widetilde{v} _{\tau -t})) =(\widetilde{u}_{ i}^{(m)}, \widetilde{v}_{ i}^{(m)})_{i\in \mathbb{Z}}\) be the solution of (3.8). Then taking the inner product \((\widehat{\chi }(r) \widetilde{u}^{(m)} , \widehat{\chi } (r) \widetilde{v}^{(m)} ) = (\chi (\frac{\vert i \vert }{r} )\widetilde{u}_{ i}^{(m)}, \chi (\frac{\vert i \vert }{r} )\widetilde{v}_{ i}^{(m)} ) _{i\in \mathbb{Z}}\) of (3.8) in \(l_{\rho }^{2}\times l_{\rho }^{2}\), we obtain
$$\begin{aligned}& \frac{d }{dt}\sum_{ i \in \mathbb{Z}} \rho _{i} \chi \biggl(\frac{\vert i \vert }{r} \biggr) \biggl( \bigl\vert \widetilde{u}_{ i}^{(m)} \bigr\vert ^{2} + \frac{1 }{\sigma } \bigl\vert \widetilde{v}_{ i}^{(m)} \bigr\vert ^{2} \biggr) \\& \quad = 2 \bigl( \mathbb{B}_{m} ( t,\omega )\widetilde{u}^{(m)} , \widehat{\chi } (r) \widetilde{u}^{(m)} \bigr) _{\rho }+2 \bigl( f\bigl(\widetilde{u}^{(m)} +cz(\theta _{t}\omega ) \bigr), \widehat{\chi } (r) \widetilde{u}^{(m)} \bigr) _{\rho } \\& \quad \quad {}+ 2 \bigl(c \mathbb{B}_{m}(t,\omega )z(\theta _{t}\omega ) , \widehat{\chi } (r) \widetilde{v}^{(m)} \bigr)_{\rho } \\& \quad \quad {} +2 \bigl( g(t), \widehat{\chi } (r) \widetilde{u}^{(m)} \bigr)_{\rho }- 2 \bigl( c\lambda z(\theta _{t}\omega ) , \widehat{\chi } (r) \widetilde{u}^{(m)} \bigr)_{\rho } \\& \quad \quad {}- 2 \bigl( c y(\theta _{t}\omega ), \widehat{\chi } (r) \widetilde{u}^{(m)} \bigr)_{\rho }- \frac{2\delta }{\sigma } \sum_{ i \in \mathbb{Z}} \rho _{i} \chi \biggl(\frac{\vert i \vert }{r} \biggr) \bigl( \widetilde{v}_{ i}^{(m)}\bigr)^{2} \\& \quad \quad {} + \frac{2}{\sigma } \bigl( h(t), \widehat{\chi } (r) \widetilde{v}^{(m)} \bigr)_{\rho }+ 2 \bigl( c z(\theta _{t}\omega ) , \widehat{ \chi } (r) \widetilde{v}^{(m)} \bigr)_{\rho } \\& \quad \quad {}+ \frac{2}{\sigma } \bigl( c (\mu -\delta ) y(\theta _{t}\omega ) , \widehat{\chi } (r) \widetilde{v}^{(m)} \bigr)_{\rho }. \end{aligned}$$
(3.32)
For each term of (3.32), it has been checked that
$$\begin{aligned}& 2 \bigl( \mathbb{B}_{m} ( t,\omega )\widetilde{u}^{(m)}, \widehat{\chi } (r) \widetilde{u}^{(m)} \bigr)_{\rho} \\& \quad = 2 \sum_{ i \in \mathbb{Z}} \Biggl(\rho _{i} \chi \biggl(\frac{\vert i \vert }{r} \biggr)\widetilde{u}_{ i}^{(m)} \sum _{j=-q}^{q} \eta _{i,j}^{(m)} ( t,\omega )\widetilde{u}^{(m)}_{ i+j} \Biggr) \\& \quad \leqslant \boldsymbol{\beta}(\theta _{t} \omega ) \sum _{ i \in \mathbb{Z}} \rho _{i} \chi \biggl(\frac{\vert i \vert }{r} \biggr) \Biggl( (1+2q) \bigl\vert \widetilde{u}_{ i}^{(m)} \bigr\vert ^{2} + \sum_{j=-q}^{q} \bigl\vert \widetilde{u}^{(m)}_{i+j} \bigr\vert ^{2} \Biggr) \\& \quad \leqslant \boldsymbol{\beta}(\theta _{t} \omega ) \sum _{ i \in \mathbb{Z}} \Biggl[\rho _{i} \sum _{j=-q}^{q} \biggl( \biggl( \chi \biggl( \frac{\vert i \vert }{r} \biggr)- \chi \biggl(\frac{\vert i+j \vert }{r} \biggr) \biggr) \bigl\vert \widetilde{u}^{(m)}_{ i+j} \bigr\vert ^{2} + \chi \biggl(\frac{\vert i+j \vert }{r} \biggr) \bigl\vert \widetilde{u}^{(m)}_{ i+j} \bigr\vert ^{2} \biggr) \Biggr] \\& \quad \quad {} +(1+2q)\boldsymbol{\beta}(\theta _{t} \omega ) \sum _{ i \in \mathbb{Z}} \rho _{i} \chi \biggl(\frac{\vert i \vert }{r} \biggr)\bigl\vert \widetilde{u}_{ i}^{(m)} \bigr\vert ^{2} \\& \quad \leqslant \frac{\mathbb {c}_{\chi}q(1+ 2\widetilde{q}) \boldsymbol{\beta}(\theta _{t} \omega )}{r} \bigl\Vert \widetilde{u}^{(m)} \bigr\Vert ^{2}_{\rho }+ 2 (1+q+\widetilde{q}) \boldsymbol{\beta}(\theta _{t} \omega )\sum_{ i \in \mathbb{Z}} \rho _{i} \chi \biggl(\frac{\vert i \vert }{r} \biggr) \bigl\vert \widetilde{u}^{(m)}_{ i } \bigr\vert ^{2}, \end{aligned}$$
(3.33)
$$\begin{aligned}& 2 \bigl( f\bigl(\widetilde{u}^{(m)}+ c z(\theta _{t}\omega ) \bigr), \widehat{\chi } (r) \widetilde{u}^{(m)}\bigr)_{\rho} \\& \quad = 2 \sum_{ i \in \mathbb{Z}}\rho _{i} \chi \biggl( \frac{\vert i \vert }{r} \biggr) f_{i}\bigl( \widetilde{u}_{i}^{(m)} +c z_{i}(\theta _{t}\omega )\bigr) \cdot \bigl( \widetilde{u}_{ i}^{(m)}+c z_{i}(\theta _{t}\omega ) \bigr) \\& \quad \quad {} -2 \sum_{ i \in \mathbb{Z}}\rho _{i} \chi \biggl( \frac{\vert i \vert }{r} \biggr) f_{i}\bigl( \widetilde{u}_{i}^{(m)} +c z_{i}(\theta _{t}\omega )\bigr) \cdot c z_{i}( \theta _{t}\omega ) \\& \quad \leqslant 2 \sum_{i\in \mathbb{Z}} \rho _{i}\chi \biggl(\frac{\vert i \vert }{r} \biggr) \bigl(-\alpha \bigl( \widetilde{u}_{i}^{(m)} +c z_{i}(\theta _{t}\omega )\bigr)^{2}+\beta \bigr) \\& \quad \quad {} +2 \sum_{ i \in \mathbb{Z}}\rho _{i} \chi \biggl( \frac{\vert i \vert }{r} \biggr) \kappa \bigl\vert \widetilde{u}_{i}^{(m)} +c z_{i}(\theta _{t}\omega ) \bigr\vert \cdot \bigl\vert cz_{i}(\theta _{t}\omega ) \bigr\vert \\& \quad \leqslant - 2\alpha \sum_{ i \in \mathbb{Z}} \rho _{i} \chi \biggl(\frac{\vert i \vert }{r} \biggr)\bigl\vert \widetilde{u}_{ i}^{(m)} \bigr\vert ^{2} +2\beta \sum_{i\in \mathbb{Z}} \rho _{i}\chi \biggl( \frac{\vert i \vert }{r} \biggr)+ 4\alpha \sum_{i\in \mathbb{Z}} \rho _{i}\chi \biggl( \frac{\vert i \vert }{r} \biggr) \bigl\vert \widetilde{u}_{i}^{(m)} \bigr\vert \cdot \bigl\vert c z_{i}(\theta _{t}\omega ) \bigr\vert \\& \quad \quad {} + 2\alpha \sum_{i\in \mathbb{Z}} \rho _{i}\chi \biggl( \frac{\vert i \vert }{r} \biggr) \bigl\vert c z_{i}(\theta _{t}\omega ) \bigr\vert ^{2} \\& \quad \quad {}+2 \sum _{ i \in \mathbb{Z}}\rho _{i} \chi \biggl(\frac{\vert i \vert }{r} \biggr) \kappa \bigl\vert \widetilde{u}_{i}^{(m)} \bigr\vert \cdot \bigl\vert cz_{i}(\theta _{t}\omega ) \bigr\vert +2 \sum_{ i \in \mathbb{Z}}\rho _{i} \chi \biggl( \frac{\vert i \vert }{r} \biggr)\kappa \bigl\vert c z_{i}(\theta _{t}\omega ) \bigr\vert ^{2} \\& \quad \leqslant - 2\alpha \sum_{ i \in \mathbb{Z}} \rho _{i} \chi \biggl(\frac{\vert i \vert }{r} \biggr)\bigl\vert \widetilde{u}_{ i}^{(m)} \bigr\vert ^{2} +2\beta \sum_{i\in \mathbb{Z}} \rho _{i}\chi \biggl( \frac{\vert i \vert }{r} \biggr) \\& \quad \quad {}+ 2(2\alpha +\kappa )\sum_{i\in \mathbb{Z}} \rho _{i}\chi \biggl( \frac{\vert i \vert }{r} \biggr) \bigl\vert \widetilde{u}_{i}^{(m)} \bigr\vert \cdot \bigl\vert c z_{i}(\theta _{t}\omega ) \bigr\vert \\& \quad \quad {} + 2(\alpha +\kappa ) \sum_{i\in \mathbb{Z}} \rho _{i}\chi \biggl( \frac{\vert i \vert }{r} \biggr) \bigl\vert cz_{i}(\theta _{t}\omega ) \bigr\vert ^{2} \\& \quad \leqslant - \frac{3\alpha }{2} \sum_{ i \in \mathbb{Z}} \rho _{i} \chi \biggl(\frac{\vert i \vert }{r} \biggr)\bigl\vert \widetilde{u}_{ i}^{(m)} \bigr\vert ^{2} +2\beta \sum_{i\in \mathbb{Z}} \rho _{i}\chi \biggl( \frac{\vert i \vert }{r} \biggr) \\& \quad \quad {} + \biggl( \frac{(2\alpha +\kappa )^{2} }{\alpha } + 2(\alpha +\kappa ) \biggr)c^{2}\sum _{i\in \mathbb{Z}} \rho _{i}\chi \biggl( \frac{\vert i \vert }{r} \biggr) \bigl\vert z_{i}(\theta _{t} \omega ) \bigr\vert ^{2} , \end{aligned}$$
(3.34)
$$\begin{aligned}& 2 \bigl(c\mathbb{B}_{m}(t,\omega )z(\theta _{t}\omega ), \widehat{\chi } (r) \widetilde{u}^{(m)} \bigr)_{\rho} \\& \quad = 2c \sum_{ i \in \mathbb{Z}} \Biggl(\rho _{i} \chi \biggl(\frac{\vert i \vert }{r} \biggr)\widetilde{u}_{ i}^{(m)} \sum _{j=-q}^{q} \eta _{i,j}^{(m)} ( t,\omega ) z _{ i+j}(\theta _{t}\omega ) \Biggr) \\& \quad \leqslant \sum_{ i \in \mathbb{Z}} \rho _{i} \chi \biggl(\frac{\vert i \vert }{r} \biggr) \Biggl( \frac{\alpha }{8}\bigl\vert \widetilde{u}_{ i}^{(m)} \bigr\vert ^{2} + \frac{8c^{2}}{\alpha }(1+2q)\boldsymbol{\beta}^{2} (\theta _{t} \omega ) \sum_{j=-q}^{q} \bigl\vert z_{ i+j}(\theta _{t}\omega ) \bigr\vert ^{2}\Biggr) \\& \quad \leqslant \frac{8c^{2}}{\alpha }(1+2q)\boldsymbol{\beta}^{2} (\theta_{t} \omega ) \sum_{i \in \mathbb{Z}} \rho_{i} \sum_{j=-q}^{q} \biggl( \biggl( \chi \biggl(\frac{\vert i \vert }{r} \biggr)- \chi \biggl(\frac{\vert i+j \vert }{r} \biggr) \biggr) \bigl\vert z_{ i+j}(\theta _{t}\omega )\bigl\vert ^{2} \\& \quad \quad {}+ \chi \biggl(\frac{ \vert i+j\vert }{r} \biggr) \bigr\vert z_{ i+j}(\theta _{t}\omega ) \bigr\vert ^{2} \biggr) \\& \quad \quad {} + \frac{\alpha }{8} \sum_{ i \in \mathbb{Z}} \rho _{i} \chi \biggl(\frac{\vert i \vert }{r} \biggr)\bigl\vert \widetilde{u}_{ i}^{(m)} \bigr\vert ^{2} \\& \quad \leqslant \frac{8c^{2}}{\alpha }(1+2q) (1+2\widetilde{q} )\boldsymbol{\beta}^{2} (\theta _{t} \omega ) \biggl(\frac{\mathbb {c}_{\chi}q }{r} \bigl\Vert z(\theta _{t} \omega ) \bigr\Vert ^{2}_{\rho }+ \sum _{ i \in \mathbb{Z}} \rho _{i} \chi \biggl( \frac{\vert i \vert }{r} \biggr) \bigl\vert z_{ i }(\theta _{t} \omega ) \bigr\vert ^{2} \biggr) \\& \quad \quad {} + \frac{\alpha }{8} \sum_{ i \in \mathbb{Z}} \rho _{i} \chi \biggl(\frac{\vert i \vert }{r} \biggr) \bigl\vert \widetilde{u}^{(m)}_{ i } \bigr\vert ^{2} , \end{aligned}$$
(3.35)
$$\begin{aligned}& 2 \bigl( g(t), \widehat{\chi } (r)\widetilde{u}^{(m)} \bigr)_{\rho }\leqslant \frac{ 2}{\alpha } \sum _{ i \in \mathbb{Z}} \rho _{i} \chi \biggl(\frac{\vert i \vert }{r} \biggr) \bigl\vert g_{ i}(t) \bigr\vert ^{2} + \frac{\alpha }{2} \sum_{i\in \mathbb{Z}} \rho _{i}\chi \biggl(\frac{\vert i \vert }{r} \biggr)\bigl\vert \widetilde{u}_{ i}^{(m)} \bigr\vert ^{2}, \end{aligned}$$
(3.36)
$$\begin{aligned}& - 2 \bigl( c\lambda z(\theta _{t}\omega ) , \widehat{\chi } (r)\widetilde{u}^{(m)} \bigr)_{\rho} \\& \quad \leqslant \frac{ 4c^{2}\lambda ^{2} }{\alpha } \sum_{ i \in \mathbb{Z}} \rho _{i} \chi \biggl(\frac{\vert i \vert }{r} \biggr) \bigl\vert z_{ i}(\theta _{t} \omega ) \bigr\vert ^{2} +\frac{\alpha }{4} \sum_{i\in \mathbb{Z}} \rho _{i}\chi \biggl(\frac{\vert i \vert }{r} \biggr)\bigl\vert \widetilde{u}_{ i}^{(m)} \bigr\vert ^{2}, \end{aligned}$$
(3.37)
$$\begin{aligned}& - 2 \bigl( cy(\theta _{t}\omega ), \widehat{\chi } (r) \widetilde{u}^{(m)} \bigr)_{\rho }\leqslant \frac{ 4c^{2}}{\alpha } \sum_{ i \in \mathbb{Z}} \rho _{i} \chi \biggl( \frac{\vert i \vert }{r} \biggr) \bigl\vert y_{ i}(\theta _{t} \omega ) \bigr\vert ^{2} +\frac{\alpha }{4} \sum _{i\in \mathbb{Z}} \rho _{i}\chi \biggl(\frac{\vert i \vert }{r} \biggr)\bigl\vert \widetilde{u}_{ i}^{(m)} \bigr\vert ^{2}, \end{aligned}$$
(3.38)
$$\begin{aligned}& \frac{2}{\sigma } \bigl( h(t), \widehat{\chi } (r)\widetilde{v}^{(m)} \bigr)_{\rho }\leqslant \frac{ 2}{\delta \sigma } \sum _{ i \in \mathbb{Z}} \rho _{i} \chi \biggl(\frac{\vert i \vert }{r} \biggr) \bigl\vert h_{ i}(t) \bigr\vert ^{2} + \frac{\delta }{2\sigma } \sum_{i\in \mathbb{Z}} \rho _{i}\chi \biggl(\frac{\vert i \vert }{r} \biggr)\bigl\vert \widetilde{v}_{ i}^{(m)} \bigr\vert ^{2}, \end{aligned}$$
(3.39)
$$\begin{aligned}& 2 \bigl(c z(\theta _{t}\omega ) , \widehat{\chi } (r) \widetilde{v}^{(m)} \bigr)_{\rho } \\& \quad \leqslant \frac{ 4c^{2}\sigma }{\delta } \sum_{ i \in \mathbb{Z}} \rho _{i} \chi \biggl( \frac{\vert i \vert }{r} \biggr) \bigl\vert z_{ i}(\theta _{t} \omega ) \bigr\vert ^{2} +\frac{\delta }{4\sigma } \sum _{i\in \mathbb{Z}} \rho _{i}\chi \biggl(\frac{\vert i \vert }{r} \biggr)\bigl\vert \widetilde{v}_{ i}^{(m)} \bigr\vert ^{2}, \end{aligned}$$
(3.40)
$$\begin{aligned}& \frac{2}{\sigma } \bigl( c (\mu -\delta ) y(\theta _{t}\omega ) , \widehat{\chi } (r) \widetilde{v}^{(m)} \bigr)_{\rho } \\& \quad \leqslant \frac{ 4c^{2}(\mu -\delta )^{2}}{\delta \sigma } \sum_{ i \in \mathbb{Z}} \rho _{i} \chi \biggl(\frac{\vert i \vert }{r} \biggr) \bigl\vert y_{ i}(\theta _{t}\omega ) \bigr\vert ^{2} +\frac{\delta }{4\sigma }\sum _{i\in \mathbb{Z}} \rho _{i}\chi \biggl( \frac{\vert i \vert }{r} \biggr)\bigl\vert \widetilde{v}_{ i}^{(m)} \bigr\vert ^{2}. \end{aligned}$$
(3.41)
By putting (3.33)-(3.41) into (3.32), we have
$$\begin{aligned}& \frac{d }{dt} \sum_{ i \in \mathbb{Z}} \rho _{i} \chi \biggl(\frac{\vert i \vert }{r} \biggr) \biggl( \bigl\vert \widetilde{u}_{ i}^{(m)} \bigr\vert ^{2} + \frac{1 }{\sigma } \bigl\vert \widetilde{v}_{ i}^{(m)} \bigr\vert ^{2} \biggr) \\& \quad \leqslant \bigl( 2 (1+q+\widetilde{q}) \boldsymbol{\beta}(\theta _{t} \omega )-\alpha \bigr) \sum_{ i \in \mathbb{Z}} \rho _{i} \chi \biggl(\frac{\vert i \vert }{r} \biggr)\bigl\vert \widetilde{u}_{ i}^{(m)} \bigr\vert ^{2}- \frac{\delta }{\sigma } \sum _{ i \in \mathbb{Z}} \rho _{i} \chi \biggl(\frac{\vert i \vert }{r} \biggr)\bigl\vert \widetilde{v}_{ i}^{(m)} \bigr\vert ^{2} \\& \quad \quad {} + \frac{\mathbb {c}_{\chi}q(1+ 2\widetilde{q})\boldsymbol{\beta}(\theta _{t} \omega )}{r} \bigl\Vert \widetilde{u}^{(m)} \bigr\Vert ^{2}_{\rho }+\frac{2}{\alpha } \sum_{ i \in \mathbb{Z}} \rho _{i} \chi \biggl( \frac{\vert i \vert }{r} \biggr) \bigl\vert g_{ i}(t) \bigr\vert ^{2} + \frac{ 2}{\delta \sigma } \sum_{ i \in \mathbb{Z}} \rho _{i} \chi \biggl(\frac{\vert i \vert }{r} \biggr) \bigl\vert h_{ i}(t) \bigr\vert ^{2} \\& \quad \quad {} +2\beta \sum_{i\in \mathbb{Z}} \rho _{i}\chi \biggl(\frac{\vert i \vert }{r} \biggr) +\frac{8c^{2}\mathbb {c}_{\chi }q(1+2q)(1+2\widetilde{q} )}{\alpha r} \boldsymbol{\beta}^{2} ( \theta _{t} \omega ) \bigl\Vert z(\theta _{t} \omega ) \bigr\Vert _{\rho }^{2} \\& \quad \quad {} + c^{2} \biggl( \frac{4\sigma }{\delta } + \frac{4\lambda ^{2}}{\alpha }+ \frac{(2\alpha +\kappa )^{2} }{\alpha } + 2(\alpha +\kappa ) \\& \quad \quad {} +\frac{8(1+2q)(1+2\widetilde{q} )}{\alpha }\boldsymbol{\beta}^{2} (\theta _{t} \omega ) \biggr)\sum_{i\in \mathbb{Z}} \rho _{i}\chi \biggl( \frac{\vert i \vert }{r} \biggr) \bigl\vert z_{i}(\theta _{t}\omega ) \bigr\vert ^{2} \\& \quad \quad {} + \biggl( \frac{4c^{2} (\mu -\delta )^{2} }{\delta \sigma } + \frac{4c^{2}}{\alpha } \biggr) \sum _{ i \in \mathbb{Z}} \rho _{i} \chi \biggl(\frac{\vert i \vert }{r} \biggr) \bigl\vert y_{ i}(\theta _{t}\omega ) \bigr\vert ^{2} . \end{aligned}$$
(3.42)
Recalling \(\boldsymbol{\lambda}=\min \{\frac{\alpha }{2},\delta \}\), multiplying (3.42) by \(e^{\int ^{t}_{0}( 2 (1+q+\widetilde{q})\boldsymbol{\beta}(\theta _{r} \omega ) -\boldsymbol{\lambda})\,dr}\) and then integrating over \([\tau -t,\tau ]\) with \(t>0\), we get
$$\begin{aligned}& \sum_{ i \in \mathbb{Z}} \rho _{i} \chi \biggl( \frac{\vert i \vert }{r} \biggr) \biggl(\bigl\vert \widetilde{u}_{ i}^{(m)}( \tau ,\tau -t, \omega ,\mathcal{Q}_{n} \widetilde{u}_{\tau -t}, \mathcal{Q}_{n}\widetilde{v}_{\tau -t}) \bigl\vert ^{2} \\& \qquad {}+ \frac{1 }{\sigma } \bigr\vert \widetilde{v}_{ i}^{(m)} ( \tau , \tau -t,\omega ,\mathcal{Q}_{n}\widetilde{u}_{\tau -t},\mathcal{Q}_{n}\widetilde{v}_{\tau -t}) \bigr\vert ^{2} \biggr) \\& \quad \leqslant e ^{ \int _{\tau -t}^{\tau }( 2 (1+q+\widetilde{q}) \boldsymbol{\beta}(\theta _{r}\omega )- \boldsymbol{\lambda})\,dr} \biggl( \Vert \mathcal{Q}_{n} \widetilde{u}_{\tau -t } \Vert _{ \rho }^{2} + \frac{1 }{\sigma }\Vert \mathcal{Q}_{n}\widetilde{v}_{\tau -t } \Vert _{ \rho }^{2} \biggr) \\& \quad \quad {} +\frac{ \mathbb {c}_{\chi}q(1+2\widetilde{q})}{r} \int _{\tau -t}^{\tau }e^{ \int _{s}^{\tau } ( 2 (1+q+\widetilde{q}) \boldsymbol{\beta}(\theta _{r}\omega )- \boldsymbol{\lambda})\,dr } \\& \quad \quad {}\times \boldsymbol{\beta}(\theta _{s}\omega )\bigl\Vert \widetilde{u}^{(m)}(s , \tau -t, \omega,\mathcal{Q}_{n}\widetilde{u}_{\tau -t}, \mathcal{Q}_{n}\widetilde{v}_{\tau -t}) \bigr\Vert _{\rho }^{2}\,ds \\& \quad \quad {} +\sum_{i\in \mathbb{Z}}\rho _{i}\chi \biggl( \frac{\vert i \vert }{r} \biggr) \int _{\tau -t}^{\tau }e^{ \int _{s}^{\tau } (2 (1+q+\widetilde{q}) \boldsymbol{\beta}(\theta _{r}\omega )- \boldsymbol{\lambda})\,dr } \biggl( \frac{2}{\alpha }\bigl\vert g _{i}(s ) \bigr\vert ^{2} + \frac{2}{\delta \sigma }\bigl\vert h _{i}(s ) \bigr\vert ^{2} + 2 \beta \biggr)\,ds \\& \quad \quad {} + \mathbb {c}_{4} \int _{\tau -t}^{\tau }e^{ \int _{s}^{\tau } ( 2 (1+q+\widetilde{q}) \boldsymbol{\beta}(\theta _{r}\omega )- \boldsymbol{\lambda})\,dr } \boldsymbol{\beta}^{2}(\theta _{s}\omega )\bigl\Vert z(\theta _{s}\omega ) \bigr\Vert _{\rho }^{2}\,ds \\& \quad \quad {} + \int _{\tau -t}^{\tau }e^{ \int _{s}^{\tau } (2 (1+q+\widetilde{q}) \boldsymbol{\beta}(\theta _{r}\omega )- \boldsymbol{\lambda})\,dr } \sum _{i\in \mathbb{Z}}\rho _{i}\chi \biggl(\frac{\vert i \vert }{r} \biggr) \bigl( \bigl(\mathbb {c}_{5} + \mathbb {c}_{6} \boldsymbol{\beta}^{2}(\theta _{s}\omega ) \bigr) \bigl\vert z_{i}(\theta _{s}\omega ) \bigr\vert \\& \quad \quad {} + \mathbb {c}_{7}\bigl\vert y_{i}(\theta _{s} \omega ) \bigr\vert \bigr)\,ds, \end{aligned}$$
(3.43)
where \(\mathbb {c}_{4}= \frac{8c^{2}c_{\chi }q(1+2q)(1+2\widetilde{q} )}{\alpha r} \), \(\mathbb {c}_{5}= c^{2} (\frac{4\sigma }{\delta } + \frac{4\lambda ^{2}}{\alpha }+ \frac{(2\alpha +\kappa )^{2} }{\alpha } + 2(\alpha +\kappa ) )\), \(\mathbb {c}_{6}= \frac{8c^{2}(1+2q)(1+2\widetilde{q} )}{\alpha } \), and \(\mathbb {c}_{7}= \frac{4c^{2} (\mu -\delta )^{2} }{\delta \sigma } + \frac{4c^{2}}{\alpha } \). Replacing ω and m in (3.43) by \(\theta _{-\tau } \omega \) and \(m_{k}\), respectively, and letting \(k\rightarrow \infty \), then we obtain
$$\begin{aligned}& \sum_{ i \in \mathbb{Z}} \rho _{i} \chi \biggl(\frac{\vert i \vert }{r} \biggr) \biggl(\bigl\vert \widetilde{u}_{ i} (\tau , \tau -t,\theta _{-\tau }\omega ,\mathcal{Q}_{n}\widetilde{u}_{\tau -t},\mathcal{Q}_{n}\widetilde{v}_{\tau -t}) \bigl\vert ^{2} \\& \quad \quad {} + \frac{1 }{\sigma } \bigr\vert \widetilde{v}_{ i} ( \tau , \tau -t, \theta _{-\tau }\omega,\mathcal{Q}_{n}\widetilde{u}_{\tau -t},\mathcal{Q}_{n} \widetilde{v}_{\tau -t}) \bigr\vert ^{2} \biggr) \\& \quad \leqslant e ^{ \int _{ -t}^{0} ( 2 (1+q+\widetilde{q}) \boldsymbol{\beta}(\theta _{r}\omega )- \boldsymbol{\lambda})\,dr} \biggl( \Vert \mathcal{Q}_{n} \widetilde{u}_{\tau -t } \Vert _{ \rho }^{2} + \frac{1 }{\sigma }\Vert \mathcal{Q}_{n}\widetilde{v}_{\tau -t } \Vert _{ \rho }^{2} \biggr) \\& \quad \quad {} +\frac{ \mathbb {c}_{\chi}q(1+2\widetilde{q})}{r} \int _{ -t}^{0}e^{ \int _{s}^{0} ( 2(1+q+\widetilde{q}) \boldsymbol{\beta}(\theta _{r}\omega )- \boldsymbol{\lambda})\,dr } \\& \quad \quad {} \times \boldsymbol{\beta}(\theta _{s}\omega )\bigl\Vert \widetilde{u} (s+\tau , \tau -t, \theta _{-\tau }\omega,\mathcal{Q}_{n}\widetilde{u}_{\tau -t}, \mathcal{Q}_{n}\widetilde{v}_{\tau -t}) \bigr\Vert _{\rho }^{2}\,ds \\& \quad \quad {} +\sum_{i\in \mathbb{Z}}\rho _{i}\chi \biggl( \frac{\vert i \vert }{r} \biggr) \int _{ -t}^{0} e^{ \int _{s}^{0} (2 (1+q+\widetilde{q}) \boldsymbol{\beta}(\theta _{r}\omega )- \boldsymbol{\lambda})\,dr } \biggl( \frac{8}{\alpha }\bigl\vert g _{i}(s+\tau ) \bigr\vert ^{2} + \frac{2}{\delta \sigma }\bigl\vert h_{i}(s+\tau ) \bigr\vert ^{2} + 2\beta \biggr)\,ds \\& \quad \quad {} + \mathbb {c}_{4} \int _{ -t}^{0}e^{ \int _{s}^{0 } ( 2 (1+q+\widetilde{q}) \boldsymbol{\beta}(\theta _{r}\omega )- \boldsymbol{\lambda})\,dr } \boldsymbol{\beta}^{2}(\theta _{s}\omega )\bigl\Vert z(\theta _{s}\omega ) \bigr\Vert _{\rho }^{2}\,ds \\& \quad \quad {} +\int _{ -t}^{0} e^{ \int _{s}^{0 } (2 (1+q+\widetilde{q}) \boldsymbol{\beta}(\theta _{r}\omega )- \boldsymbol{\lambda})\,dr } \\& \quad \quad {} \times \sum_{i\in \mathbb{Z}}\rho _{i}\chi \biggl(\frac{\vert i \vert }{r} \biggr) \bigl( \bigl( \mathbb {c}_{5} + \mathbb {c}_{6} \boldsymbol{\beta}^{2}(\theta _{s}\omega ) \bigr) \bigl\vert z_{i}(\theta _{s}\omega ) \bigr\vert +\mathbb {c}_{7}\bigl\vert y_{i}(\theta _{s} \omega ) \bigr\vert \bigr)\,ds. \end{aligned}$$
(3.44)
We now estimate each term on the right-hand side of (3.44). For the first term on the right-hand side of (3.44), since \(( \mathcal{Q}_{n}\widetilde{u}_{\tau -t }, \mathcal{Q}_{n}\widetilde{v}_{\tau -t } ) \in B (\tau -t, \theta _{-t}\omega )\), and B is tempered, then there exists \(T_{1}=T_{1}(\tau ,\varepsilon ,\omega ,B )> 0\) such that if \(t>T_{1}\), then
$$\begin{aligned} e^{ \int _{-t}^{0} (2 (1+q+\widetilde{q}) \boldsymbol{\beta}(\theta _{r}\omega )- \boldsymbol{\lambda})\,dr} \biggl( \Vert \mathcal{Q}_{n} \widetilde{u}_{\tau -t } \Vert _{ \rho }^{2} + \frac{1 }{\sigma }\Vert \mathcal{Q}_{n}\widetilde{v}_{\tau -t } \Vert _{ \rho }^{2} \biggr) < \varepsilon . \end{aligned}$$
(3.45)
For the second term on the right-hand side of (3.44), by (A1) and Theorem 3.4, there exist \(T_{2}=T_{2}(\tau ,\varepsilon ,\omega ,B)>0\) and \(R_{1}=R_{1}(\varepsilon ,\omega )>0\) such that, for all \(t>T_{2}\) and \(r>R_{1}\),
$$\begin{aligned}& \frac{\mathbb {c}_{\chi}q(1+2\widetilde{q})}{r} \int _{-t}^{0} e^{\int _{s}^{0} (2(1+q+\widetilde{q}) \boldsymbol{\beta}(\theta _{r}\omega )- \boldsymbol{\lambda})\,dr } \\& \quad {} \times \boldsymbol{\beta}(\theta _{s}\omega )\bigl\Vert \widetilde{u} (s+\tau , \tau -t,\theta _{-\tau } \omega,\mathcal{Q}_{n}\widetilde{u}_{\tau -t}, \mathcal{Q}_{n} \widetilde{v}_{\tau -t} ) \bigr\Vert _{\rho }^{2}\,ds < \varepsilon . \end{aligned}$$
(3.46)
For the third term on the right-hand side of (3.44), by (A4), there exist \(R_{2}=R_{2}(\varepsilon ,\omega )>0\) and \(T_{3}=T_{3}(\varepsilon ,\omega )>0\), such that if \(r>R_{2}\) and \(t>T_{3}\), then
$$\begin{aligned}& \sum_{i\in \mathbb{Z}}\rho _{i}\chi \biggl( \frac{\vert i \vert }{r} \biggr) \int _{-t}^{0} e^{ \int _{s}^{0} (2 (1+q+\widetilde{q}) \boldsymbol{\beta}(\theta _{r}\omega )- \boldsymbol{\lambda})\,dr} \\& \quad {} \times \biggl( \frac{2}{\alpha }\bigl\vert g _{i}(s+\tau ) \bigr\vert ^{2} + \frac{2}{\delta \sigma }\bigl\vert h_{i}(s+\tau ) \bigr\vert ^{2} + 2\beta \biggr)\,ds< \varepsilon . \end{aligned}$$
(3.47)
For the fourth and fifth terms on the right-hand side of (3.44), since \(z(\theta _{t}\omega ) \), \(y(\theta _{t}\omega ) \) and \(\boldsymbol{\beta}(\theta _{t} \omega ) \) are tempered, then there exist \(R_{3}=R_{3}(\varepsilon ,\omega )>0\) and \(T_{4}=T_{4}(\varepsilon ,\omega )>0\), such that if \(r>R_{3}\) and \(t>T_{4}\), then
$$\begin{aligned}& \mathbb {c}_{4} \int _{ -t}^{0}e^{ \int _{s}^{0 } ( 2 (1+q+\widetilde{q}) \boldsymbol{\beta}(\theta _{r}\omega )- \boldsymbol{\lambda})\,dr } \boldsymbol{\beta}^{2}(\theta _{s}\omega )\bigl\Vert z(\theta _{s}\omega ) \bigr\Vert _{\rho }^{2}\,ds \\& \quad \quad {} + \int _{ -t}^{0} e^{ \int _{s}^{0 } (2 (1+q+\widetilde{q})\boldsymbol{\beta}(\theta _{r}\omega )- \boldsymbol{\lambda})\,dr } \\& \quad \quad {} \times \sum_{i\in\mathbb{Z}}\rho _{i}\chi \biggl(\frac{\vert i \vert }{r} \biggr) \bigl( \bigl( \mathbb {c}_{5} + \mathbb {c}_{6} \boldsymbol{\beta}^{2}(\theta _{s}\omega ) \bigr) \bigl\vert z_{i}(\theta _{s}\omega ) \bigr\vert + \mathbb {c}_{7}\bigl\vert y_{i}(\theta _{s} \omega ) \bigr\vert \bigr)\,ds \\& \quad < \varepsilon . \end{aligned}$$
(3.48)
Let \(T=\max \{T_{1},T_{2},T_{3},T_{4}\}\) and \(R =2\max \{R_{1},R_{2},R_{3}\}\). By (3.45)-(3.48), we have, for all \(t>T \) and \(r>R \),
$$\begin{aligned}& \sum_{\vert i \vert >R}\rho _{i} \biggl(\bigl\vert \widetilde{u}_{i} ( \tau ,\tau -t,\theta _{-\tau }\omega, \mathcal{Q}_{n}\widetilde{u}_{\tau -t},\mathcal{Q}_{n} \widetilde{v}_{\tau -t} ) \bigl\vert ^{2} \\& \quad {} +\frac{1}{\sigma } \bigl\vert \widetilde{v}_{i} ( \tau ,\tau -t,\theta_{-\tau }\omega ,\mathcal{Q}_{n}\widetilde{u}_{\tau -t}, \mathcal{Q}_{n}\widetilde{v}_{\tau -t} ) \bigr\vert ^{2} \biggr)< 4 \varepsilon , \end{aligned}$$
(3.49)
for any \(n \geqslant 1\). Let \(n \rightarrow \infty \), we see that (3.30) holds. The proof is completed. □

4 Upper semicontinuity of random attractors

In this section, we first present a criterion concerning upper semicontinuity of non-autonomous random attractors with respect to a parameter in [28]. Similar results can be found in [29, 30] for deterministic equations and in [26, 31] for autonomous stochastic equations.

Theorem 4.1

Let \(\Phi _{c}\) be a continuous cocycle on X over \(\mathbb{R}\) and \((\Omega ,\mathcal{F},\mathcal{P},\{\theta _{t}\}_{t\in \mathbb{R}})\). Suppose that
  1. (i)

    \(\Phi _{c}\) has a closed measurable random absorbing set \(K_{c}=\{K_{c}(\tau ,\omega ):\tau \in \mathbb{R},\omega \in \Omega \}\) in \(\mathcal{D}(X)\) and a unique random attractor \(\mathcal{A}_{c}=\{\mathcal{A}_{c}(\tau ,\omega ):\tau \in \mathbb{R},\omega \in \Omega \}\) in \(\mathcal{D}(X)\).

     
  2. (ii)
    For each \(\tau \in \mathbb{R}\) and \(\omega \in \Omega \), \(K_{0}(\tau ,\omega ) =\{u\in X :\Vert u \Vert _{X} \leqslant r_{0}(\tau ,\omega ) \} \) and
    $$ \limsup_{c\rightarrow 0}\bigl\Vert K_{c}(\tau ,\omega ) \bigr\Vert _{X} =\limsup_{c\rightarrow 0}\sup _{x\in K_{c}(\tau ,\omega ) }\Vert x \Vert _{X}\leqslant r_{0}(\tau ,\omega ), $$
    (4.1)
    where \(r_{0}(\tau ,\omega )\) is a positive valued tempered random variable.
     
  3. (iii)

    There exists \(\varepsilon >0\) such that, for every \(\tau \in \mathbb{R}\) and \(\omega \in \Omega \), \(\bigcup_{ \vert c \vert \leqslant \varepsilon }\mathcal{A}_{c}(\tau ,\omega ) \) is precompact in X.

     
  4. (iv)
    For \(t>0\), \(\tau \in \mathbb{R}\), \(\omega \in \Omega \), \(c_{n}\rightarrow 0\) when \(n\rightarrow \infty \), and \(x_{n},x_{0}\in X\) with \(x_{n}\rightarrow x_{0} \) when \(n\rightarrow \infty \), we have
    $$ \lim_{n\rightarrow \infty }\Phi _{c_{n}} (t,\tau ,\omega ,x_{n})=\Phi _{0}(t,\tau ,\omega ,x_{0}). $$
    (4.2)
     
Then, for \(\tau \in \mathbb{R}\) and \(\omega \in \Omega \),
$$ d_{H}\bigl( \mathcal{A}_{c}(\tau ,\omega ), \mathcal{A}_{0}(\tau ,\omega )\bigr)=\sup_{u\in \mathcal{A}_{c}(\tau ,\omega )}\inf _{v\in \mathcal{A}_{0}(\tau ,\omega )}\Vert u-v \Vert _{X}\rightarrow 0 \quad\textit{as } c\rightarrow 0. $$
(4.3)

Next, we use Theorem 4.1 to prove an upper semicontinuity of random attractors \(\mathcal{A}_{c}(\tau ,\omega )\) to \(\mathcal{A}_{0} (\tau ,\omega )\) as \(c\rightarrow 0\).

To indicate the dependence of solutions on c, we will write the solution of (2.6) as \(\widetilde{\varphi }^{(c)}=(\widetilde{u}^{(c)},\widetilde{v}^{(c)} ) \). When \(c=0\), the system (2.4) reduces to the limiting system:
$$ \left \{ \textstyle\begin{array}{l} du= (\mathbb{B}(\theta _{t}\omega )u-v +f(u )+g(t ) )\,dt , \\ dv= ( \sigma u-\delta v+h(t ) )\,dt ,\\ u (\tau )=u_{ \tau }, \qquad v (\tau )=v_{ \tau }. \end{array}\displaystyle \right . $$
(4.4)
Let \(\varphi =(u, v)\) be a mild solution of (4.4) with initial data \((u_{\tau },v_{\tau })\).

Theorem 4.2

Assume that (A1)-(A4) hold. Then, for every \(\tau \in \mathbb{R}\) and \(\omega \in \Omega \), we have
$$\begin{aligned}& d_{H}\bigl( \mathcal{A}_{c}(\tau ,\omega ), \mathcal{A}_{0}(\tau ,\omega )\bigr) \\& \quad =\sup_{(\widetilde{u}^{(c)},\widetilde{v}^{(c)})\in \mathcal{A}_{c}(\tau ,\omega )}\inf_{(u,v)\in \mathcal{A}_{0}(\tau ,\omega )}\bigl(\bigl\Vert \widetilde{u}^{(c)}-u \bigr\Vert ^{2}_{\rho }+ \bigl\Vert \widetilde{v}^{(c)}-v \bigr\Vert _{\rho }^{2} \bigr) ^{\frac{1}{2}}\rightarrow 0 \quad \textit{as } c\rightarrow 0. \end{aligned}$$
(4.5)

Proof

Let \(I (c,\tau ,\omega )\) be as in Theorem 3.4. (i) By Theorems 3.4 and 3.5, \(\Phi _{c}\) has a closed measurable random absorbing set \(B_{c}=\{B_{c}(\tau ,\omega ):\tau \in \mathbb{R},\omega \in \Omega \} \in \mathcal{D}(l_{\rho }^{2}\times l_{\rho }^{2})\), where \(B_{c}(\tau ,\omega ) =\{(\widetilde{u}^{(c)},\widetilde{v}^{(c)})\in l_{\rho }^{2}\times l_{\rho }^{2} :\Vert \widetilde{u}^{(c)} \Vert _{\rho }^{2}+\Vert \widetilde{v}^{(c)} \Vert _{\rho }^{2} \leqslant I (c,\tau ,\omega ) \} \), and a unique random attractor \(\mathcal{A}_{c}=\{\mathcal{A}_{c}(\tau ,\omega ):\tau \in \mathbb{R},\omega \in \Omega \}\) in \(\mathcal{D}(l_{\rho }^{2}\times l_{\rho }^{2})\), for each \(\tau \in \mathbb{R}\) and \(\omega \in \Omega \), \(\mathcal{A}_{c}(\tau ,\omega )\subseteq B_{c}(\tau ,\omega ) \).

(ii) Given \(\vert c \vert <1\). By (3.6), we have
$$ I (c,\tau ,\omega )\leqslant I(1,\tau ,\omega ) < \infty $$
(4.6)
and
$$ \limsup_{c\rightarrow 0 } I (c,\tau ,\omega )\leqslant I ( 1,\tau ,\omega) . $$
(4.7)
So, for every \(\tau \in \mathbb{R}\) and \(\omega \in \Omega \),
$$ \limsup_{c\rightarrow 0}\bigl\Vert B_{c}(\tau ,\omega ) \bigr\Vert _{\rho }=\limsup_{c\rightarrow 0}\sup _{x\in B_{c}(\tau ,\omega ) }\Vert x \Vert _{ l_{\rho }^{2}\times l_{\rho }^{2}}\leqslant I^{\frac{1}{2}} ( 1,\tau ,\omega ). $$
(4.8)
Moreover, \(B_{0}(\tau ,\omega ) =\{(u,v)\in l_{\rho }^{2}\times l_{\rho }^{2} :\Vert u \Vert _{\rho }^{2}+\Vert v \Vert _{\rho }^{2} \leqslant I ( 1, \tau ,\omega ) \} \) is a closed tempered random absorbing set for the continuous cocycle \(\Phi _{0}\) associated with the limiting system (4.4), and
$$ \bigcup_{ \vert c \vert \leqslant 1 }\mathcal{A}_{c}(\tau , \omega ) \subseteq \bigcup_{ \vert c \vert \leqslant 1 } B_{c}( \tau ,\omega ) \subseteq B_{0}(\tau ,\omega ). $$
(4.9)
(iii) Given \(\vert c \vert <1\). Let us prove the precompactness of \(\bigcup_{ \vert c \vert \leqslant 1}\mathcal{A}_{c}(\tau ,\omega ) \) for every \(\tau \in \mathbb{R}\) and \(\omega \in \Omega \). For one thing, by Theorem 3.5, for every \(\varepsilon > 0 \), \(c > 0 \), \(\tau \in \mathbb{R}\), \(\omega \in \Omega \), there exist \(T=T (\tau ,\omega ,B_{0},c,\varepsilon )>0\) and \(R=R (\tau ,\omega ,c,\varepsilon )>1\) such that, for all \(t\geqslant T \) and \((\widetilde{u}_{\tau -t}^{(c)}, \widetilde{v}_{\tau -t}^{(c)} ) \in B_{0}(\tau -t, \theta _{-t}\omega ) \), the solution \((\widetilde{u} ^{(c)}, \widetilde{v} ^{(c)} )\) of (2.6) satisfies
$$ \sum_{\vert i \vert >R}\rho _{i} \bigl(\bigl\vert \widetilde{u}_{i}^{(c)} \bigl( \tau,\tau -t,\theta _{-\tau }\omega ,\widetilde{u}_{\tau -t}^{(c)}, \widetilde{v}_{\tau -t}^{(c)} \bigr) \bigr\vert ^{2} + \bigl\vert \widetilde{v}_{i}^{(c)} \bigl( \tau ,\tau -t, \theta _{-\tau }\omega,\widetilde{u}_{\tau -t}^{(c)}, \widetilde{v}_{\tau -t}^{(c)} \bigr) \bigr\vert ^{2} \bigr)\leqslant \varepsilon , $$
(4.10)
which, along with (4.9) and the invariance of \(\mathcal{A}_{c}(\tau ,\omega ) \), we have, for every \(\tau \in \mathbb{R}\), \(\omega \in \Omega \), \(t \geqslant T\),
$$ \sup_{(\widetilde{u}^{(c)},\widetilde{v} ^{(c)})\in \bigcup _{ \vert c \vert \leqslant 1 }\mathcal{A}_{c}(\tau ,\omega )}\sum_{\vert i \vert >R}\rho _{i} \bigl(\bigl\vert \widetilde{u}_{i}^{(c)} \bigr\vert ^{2} +\bigl\vert \widetilde{v}_{i}^{(c)} \bigr\vert ^{2} \bigr)\leqslant \varepsilon . $$
By (4.9) we find that the set \(\{(\widetilde{u}^{(c)}_{i},\widetilde{v}^{(c)}_{i})_{\vert i \vert \leqslant R}: (\widetilde{u}^{(c)},\widetilde{v}^{(c)} )\in \bigcup_{ \vert c \vert \leqslant 1 }\mathcal{A}_{c}(\tau ,\omega ) \} \) is bounded in a finite-dimensional space and hence \(\bigcup_{ \vert c \vert \leqslant 1}\mathcal{A}_{c}(\tau ,\omega ) \) is precompact in \(l_{\rho }^{2}\times l_{\rho }^{2} \).
(iv) Let \(U =\widetilde{u}^{(c)}-u \), \(V= \widetilde{v}^{(c)}-v \), \((U, V)= \widetilde{\varphi }^{(c)}- \varphi \). Let \((\widetilde{ u }^{(c,m)}( t ,\tau , \omega ,\widetilde{u}^{(c)}_{\tau } , \widetilde{v}^{(c)}_{\tau } ), \widetilde{v} ^{(c,m)}( t ,\tau , \omega, \widetilde{u}^{(c)}_{\tau }, \widetilde{v}^{(c)}_{\tau } ) ) \) and \((u^{(m)}( t ,\tau , \omega,u_{\tau },v_{\tau }),v^{(m)}( t ,\tau , \omega,u_{\tau },v_{\tau } )) \) be the solutions of the following random differential equations with initial data:
$$ \left \{ \textstyle\begin{array}{l} \frac{d\widetilde{u}^{(c,m)}}{dt} = \mathbb{B}_{m}(t,\omega )\widetilde{u}^{(c,m)} - \widetilde{v}^{(c,m)}+ f( \widetilde{u}^{(c,m)}+ c z(\theta _{t}\omega ) ) \\ \quad \quad \quad {}+c \mathbb{B}_{m}(t,\omega )z(\theta _{t}\omega ) + g(t)- c\lambda z(\theta _{t}\omega )-c y(\theta _{t}\omega ), \\ \frac{d\widetilde{v}^{(c,m)}}{dt}=\sigma \widetilde{u}^{(c,m)}-\delta \widetilde{v}^{(c,m)} + h(t) +\sigma cz(\theta _{t}\omega ) + c (\mu -\delta ) y(\theta _{t}\omega ), \\ \widetilde{u}^{(c,m)}(\tau )=\widetilde{u}_{\tau },\qquad \widetilde{v}^{(c,m)}(\tau )=\widetilde{v}_{\tau }, \end{array}\displaystyle \right . $$
(4.11)
and
$$ \left \{ \textstyle\begin{array}{l} \frac{du^{( m)}}{dt}=\mathbb{B}_{m}(t,\omega )u^{( m)}- v^{( m)} + f( u^{( m)} )+ g(t) , \\ \frac{dv^{( m)}}{dt}=\sigma u^{( m)}-\delta v^{( m)} +h(t) , \\ u^{( m)}(\tau )=u _{\tau },\qquad v^{( m)}(\tau )=v _{\tau }, \end{array}\displaystyle \right . $$
(4.12)
respectively. Then \(\widetilde{\varphi }^{(c,m)}( \cdot ,\tau , \omega ,\widetilde{u}^{(c)}_{\tau }, \widetilde{v}^{(c)}_{\tau } )\), \(\varphi ^{( m)}( \cdot ,\tau , \omega ,u_{\tau },v_{\tau } ) \in C([\tau , +\infty ), l_{\rho }^{2} \times l_{\rho }^{2} )\) and satisfy the differential equations (4.11) and (4.12), respectively. Moreover, \(\widetilde{\varphi }^{(c )}( t ,\tau , \omega ,\widetilde{u}^{(c)}_{\tau }, \widetilde{v}^{(c)}_{\tau } )\) and \(\varphi( t ,\tau , \omega ,u_{\tau },v_{\tau } ) \) are limit functions of subsequences of \(\{\widetilde{\varphi }^{(c,m)}( t ,\tau , \omega ,\widetilde{u}^{(c)}_{\tau }, \widetilde{v}^{(c)}_{\tau } )\}\) and \(\{\varphi ^{( m)}( t ,\tau , \omega ,u_{\tau },v_{\tau } ) \}\) \(\in l_{\rho }^{2} \times l_{\rho }^{2} \). So \(\widetilde{\varphi } ^{(c )}( t ,\tau , \omega ,\widetilde{u}^{(c)}_{\tau },\widetilde{v}^{(c)}_{\tau } ) -\varphi( t ,\tau , \omega ,u_{\tau },v_{\tau } ) \) is a limit function of a subsequence of \(\{\widetilde{\varphi } ^{(c,m)}( t ,\tau , \omega ,\widetilde{u}^{(c)}_{\tau },\widetilde{v}^{(c)}_{\tau } ) -\varphi ^{( m)}( t ,\tau , \omega ,u_{\tau },v_{\tau } ) \}\) in \(l_{\rho }^{2}\times l_{\rho }^{2}\), and \((U^{(m )}( t ,\tau , \omega, \widetilde{u}^{(c)}_{\tau },\widetilde{v}^{(c)}_{\tau },u_{\tau },v_{\tau } ), V ^{(m )}( t ,\tau , \omega,\widetilde{u}^{(c)}_{\tau }, \widetilde{v}^{(c)}_{\tau },u_{\tau },v_{\tau } ))\) satisfies
$$ \left \{ \textstyle\begin{array}{l} \frac{dU^{( m)}}{dt}=\mathbb{B}_{m}(t,\omega )U^{( m)}- V^{( m)} + f( \widetilde{u}^{(c,m)}+c z(\theta _{t}\omega ))- f( u^{( m)} )\\ \quad \quad \quad {} + c\mathbb{B}_{m}(t,\omega )z(\theta _{t}\omega ) -c \lambda z(\theta _{t}\omega )- c y(\theta _{t}\omega ) , \\ \frac{dV^{( m)}}{dt}=\sigma U^{( m)}-\delta V^{( m)}+ \sigma c z(\theta _{t}\omega )+ c(\mu -\delta ) y(\theta _{t}\omega ), \\ U^{( m)}(\tau )=\widetilde{u}^{(c )} _{\tau }-u_{\tau } , \qquad V^{( m)}(\tau )=\widetilde{v}^{(c)} _{\tau }-v_{\tau } . \end{array}\displaystyle \right . $$
(4.13)
By taking the inner product of (4.13) with \((U^{( m)},V^{( m)})\) in \(l_{\rho }^{2}\times l_{\rho }^{2}\), we get
$$\begin{aligned}& \frac{1}{2} \frac{d}{dt} \biggl( \bigl\Vert U^{( m)} \bigr\Vert ^{2}_{\rho }+ \frac{1}{\sigma }\bigl\Vert V^{( m)} \bigr\Vert ^{2}_{\rho } \biggr) \\& \quad = \bigl(\mathbb{B}_{m}(t,\omega )U^{( m)}, U^{( m)} \bigr)_{\rho }+ \bigl( f\bigl( \widetilde{u}^{(c,m)} + c z(\theta _{t}\omega ) \bigr)- f\bigl( u^{( m)} \bigr), U^{( m)} \bigr)_{\rho } \\& \quad \quad {} + \bigl( c\mathbb{B}_{m}(t,\omega )z(\theta _{t}\omega ) , U^{( m)} \bigr)_{\rho }- c \bigl( \lambda z(\theta _{t}\omega )+ y(\theta _{t}\omega ), U^{( m)} \bigr)_{\rho } \\& \quad \quad {} -\frac{\delta }{\sigma }\bigl\Vert V^{( m)} \bigr\Vert ^{2}_{\rho }+ \biggl( c z(\theta _{t}\omega ) + \frac{c (\mu -\delta )}{\sigma }y(\theta _{t}\omega ) , V^{( m)} \biggr)_{\rho }. \end{aligned}$$
(4.14)
Now let us estimate the terms in (4.14):
$$\begin{aligned}& \bigl(\mathbb{B}_{m}(t,\omega )U^{( m)}, U^{( m)} \bigr)_{\rho }\leqslant (1+q+\widetilde{q})\boldsymbol{\beta}( \theta _{t} \omega ) \bigl\Vert U^{( m)} \bigr\Vert ^{2}_{\rho }, \end{aligned}$$
(4.15)
$$\begin{aligned}& \bigl( f\bigl( \widetilde{u}^{(c,m)} + c z(\theta _{t} \omega ) \bigr)- f\bigl( u^{( m)} \bigr), U^{( m)} \bigr)_{\rho } \\& \quad = \sum_{ i \in \mathbb{Z}}\rho _{i} \bigl( f_{ i}\bigl( \widetilde{u}_{ i}^{(c,m)}+ c z_{ i}(\theta _{t}\omega ) \bigr)- f_{ i}\bigl( u_{ i}^{( m)} + c z_{ i}(\theta _{t} \omega ) \bigr) \bigr) \cdot U_{ i}^{( m)} \\& \quad \quad {} + \sum_{ i \in \mathbb{Z}}\rho _{i} \bigl( f_{ i}\bigl( u_{ i}^{( m)}+ c z_{ i}( \theta _{t}\omega ) \bigr)- f_{ i}\bigl( u_{ i}^{( m)} \bigr) \bigr) \cdot U_{ i}^{( m)} \\& \quad \leqslant \kappa \sum_{ i \in \mathbb{Z}}\rho _{i}\bigl\vert \widetilde{u}_{ i}^{(c,m)} - u_{ i}^{( m)}\bigr\vert \cdot \bigl\vert U_{ i}^{( m)} \bigr\vert +\kappa \sum_{ i \in \mathbb{Z}}\rho _{i} \bigl\vert c z_{ i}(\theta _{t}\omega)\bigr\vert \cdot \bigl\vert U_{ i}^{( m)} \bigr\vert \\& \quad \leqslant \frac{5\kappa }{4} \bigl\Vert U^{( m)} \bigr\Vert ^{2}_{\rho }+\kappa c^{2}\bigl\Vert z(\theta _{t}\omega) \bigr\Vert ^{2}_{\rho }, \end{aligned}$$
(4.16)
$$\begin{aligned}& \bigl( c \mathbb{B}_{m}(t,\omega )z(\theta _{t}\omega ) , U^{( m)} \bigr)_{\rho} \\& \quad \leqslant \frac{\kappa }{4} \bigl\Vert U^{( m)} \bigr\Vert ^{2}_{\rho }+ \frac{c^{2}(1+2q)(1+2\widetilde{q}) \boldsymbol{\beta}^{2}(\theta _{t }\omega )}{\kappa } \bigl\Vert z(\theta _{t} \omega ) \bigr\Vert ^{2}_{\rho }, \end{aligned}$$
(4.17)
$$\begin{aligned}& - c \bigl( \lambda z(\theta _{t}\omega )+ y(\theta _{t} \omega ), U^{( m)} \bigr)_{\rho} \\& \quad \leqslant \frac{c^{2}\lambda ^{2} }{ \kappa } \bigl\Vert z(\theta _{t}\omega ) \bigr\Vert _{\rho }^{2} + \frac{c^{2} }{ \kappa } \bigl\Vert y(\theta _{t}\omega ) \bigr\Vert _{\rho }^{2} + \frac{ \kappa }{2}\bigl\Vert U^{(m)} \bigr\Vert _{\rho }^{2}, \end{aligned}$$
(4.18)
$$\begin{aligned}& \biggl( c z(\theta _{t}\omega ) +\frac{c (\mu -\delta )}{\sigma }y(\theta _{t}\omega ) , V^{( m)} \biggr)_{\rho } \\& \quad \leqslant \frac{\sigma c^{2} }{ 2 \delta } \bigl\Vert z(\theta _{t}\omega ) \bigr\Vert _{\rho }^{2} + \frac{c^{2} (\mu -\delta )^{2}}{2 \delta \sigma }\bigl\Vert y(\theta _{t}\omega ) \bigr\Vert _{\rho }^{2} + \frac{\delta }{ \sigma } \bigl\Vert V^{(m)} \bigr\Vert _{\rho }^{2}. \end{aligned}$$
(4.19)
It follows from (4.15)-(4.19) that
$$\begin{aligned}& \frac{d}{dt} \biggl( \bigl\Vert U^{( m)} \bigr\Vert ^{2}_{\rho }+ \frac{1}{\sigma }\bigl\Vert V^{( m)} \bigr\Vert ^{2}_{\rho } \biggr) \\& \quad \leqslant \bigl( 2(1+q+\widetilde{q})\boldsymbol{\beta}( \theta _{t} \omega ) + 2\kappa \bigr) \bigl\Vert U^{( m)} \bigr\Vert ^{2}_{\rho } \\& \quad \quad {}+ c^{2} \biggl(\frac{(1+2q)(1+2\widetilde{q}) \boldsymbol{\beta}^{2}(\theta _{t }\omega )}{\kappa } + \frac{\sigma }{ 2 \delta } + \kappa \biggr) \bigl\Vert z(\theta _{t} \omega ) \bigr\Vert ^{2}_{\rho } \\& \quad \quad {}+ \frac{c^{2} (\mu -\delta )^{2}}{2 \delta \sigma }\bigl\Vert y(\theta _{t}\omega ) \bigr\Vert _{\rho }^{2}. \end{aligned}$$
(4.20)
Applying the Gronwall inequality to (4.20) from τ to \(t+\tau \), we have
$$\begin{aligned}& \bigl\Vert U ^{ (m)}\bigl( t+\tau ,\tau , \omega ,\widetilde{u}^{(c)}_{\tau },\widetilde{v}^{(c)}_{\tau }, u_{\tau },v_{\tau } \bigr)\bigl\Vert ^{2}_{\rho }+\frac{1}{\sigma } \bigr\Vert V^{( m)}\bigl( t+\tau ,\tau , \omega, \widetilde{u}^{(c)}_{\tau }, \widetilde{v}^{(c)}_{\tau }, u_{\tau},v_{\tau } \bigr) \bigr\Vert ^{2}_{\rho } \\& \quad \leqslant e^{\int ^{t+\tau }_{\tau } (2(1+q+\widetilde{q})\boldsymbol{\beta}( \theta _{r} \omega ) +2\kappa )\,dr } \biggl( \bigl\Vert \widetilde{u}^{(c)}_{\tau }-u_{\tau } \bigr\Vert ^{2}_{\rho }+ \frac{1}{\sigma } \bigl\Vert \widetilde{v}^{(c)}_{\tau }-v_{\tau } \bigr\Vert ^{2}_{\rho } \biggr) \\& \quad \quad {} +\int ^{t+\tau } _{\tau }e^{\int ^{t+\tau }_{s} ( 2(1+q+\widetilde{q})\boldsymbol{\beta}( \theta _{r} \omega ) +2\kappa )\,dr } \bigl(\bigl( \mathbb {c}_{8}+ \mathbb {c}_{9}\boldsymbol{\beta}^{2}( \theta _{s} \omega ) \bigr) \bigl\Vert z( \theta _{s} \omega ) \bigr\Vert _{\rho }^{2} +\mathbb {c}_{10} \bigl\Vert y( \theta _{s} \omega ) \bigr\Vert _{\rho }^{2}\bigr)\,ds , \end{aligned}$$
(4.21)
where \(\mathbb {c}_{8}= \frac{\sigma c^{2} }{ 2 \delta } + \kappa c^{2}\), \(\mathbb {c}_{9}= \frac{c^{2}(1+2q)(1+2\widetilde{q}) }{\kappa } \) and \(\mathbb {c}_{10}= \frac{c^{2} (\mu -\delta )^{2}}{2 \delta \sigma } \). Replacing ω in the above by \(\theta _{-\tau }\omega \), we have
$$\begin{aligned}& \bigl\Vert U ^{ (m)}\bigl( t+\tau ,\tau , \theta _{-\tau }\omega,\widetilde{u}^{(c)}_{\tau },\widetilde{v}^{(c)}_{\tau }, u_{\tau },v_{\tau } \bigr)\bigr\Vert ^{2}_{\rho }+ \frac{1}{\sigma } \bigl\Vert V^{( m)}\bigl( t+\tau ,\tau , \theta_{-\tau }\omega,\widetilde{u}^{(c)}_{\tau },\widetilde{v}^{(c)}_{\tau }, u_{\tau },v_{\tau } \bigr) \bigr\Vert ^{2}_{\rho} \\& \quad \leqslant e^{\int ^{t }_{0} ( 2(1+q+\widetilde{q})\boldsymbol{\beta}( \theta _{r} \omega ) +2\kappa )\,dr } \biggl( \bigl\Vert \widetilde{u}^{(c)}_{\tau }-u_{\tau } \bigr\Vert ^{2}_{\rho }+ \frac{1}{\sigma } \bigl\Vert \widetilde{v}^{(c)}_{\tau }-v_{\tau } \bigr\Vert ^{2}_{\rho } \biggr) \\& \quad \quad {} +\int ^{t } _{0} e^{\int ^{t }_{s} ( 2(1+q+\widetilde{q})\boldsymbol{\beta}( \theta _{r} \omega ) +2\kappa )\,dr } \bigl(\bigl(\mathbb {c}_{8}+ \mathbb {c}_{9}\boldsymbol{\beta}^{2}( \theta _{s} \omega ) \bigr) \bigl\Vert z( \theta _{s} \omega ) \bigr\Vert _{\rho }^{2} +\mathbb {c}_{10} \bigl\Vert y( \theta _{s} \omega ) \bigr\Vert _{\rho}^{2}\bigr)\,ds . \end{aligned}$$
(4.22)
From (4.22), we find that, for \(\tau \in \mathbb{R}\), \(t\in \mathbb{R}^{+}\), \(\omega \in \Omega \), \(c \rightarrow 0\) and \((\widetilde{u}^{(c)}_{\tau }, \widetilde{v}^{(c)}_{\tau } )\), \(( u_{\tau },v_{\tau })\in l_{\rho }^{2}\times l_{\rho }^{2}\) with \((\widetilde{u}^{(c)}_{\tau }, \widetilde{v}^{(c)}_{\tau } )\rightarrow ( u_{\tau },v_{\tau })\),
$$\begin{aligned} \lim_{ c \rightarrow 0} \widetilde{\varphi }^{(c,m)} \bigl( t+\tau ,\tau , \theta _{-\tau }\omega ,\widetilde{u}^{(c)}_{\tau }, \widetilde{v}^{(c)}_{\tau } \bigr) = \varphi ^{( m)} ( t+ \tau ,\tau , \theta _{-\tau }\omega ,u_{\tau },v_{\tau } ) \quad \text{ in } l^{2}_{\rho }\times l_{\rho }^{2} . \end{aligned}$$
(4.23)
Let \(\{c_{n} \} \subset [-1,1] \) be a sequence of numbers with \(c_{n} \rightarrow 0 \) when \(n \rightarrow +\infty \). Then \(\widetilde{\varphi }^{(c )}( t+\tau ,\tau , \theta _{-\tau }\omega ,\widetilde{u}^{(c)}_{\tau }, \widetilde{v}^{(c)}_{\tau } )\) and \(\varphi( t+\tau ,\tau , \theta _{-\tau }\omega ,u_{\tau },v_{\tau } ) \) being limit functions of subsequences of \(\{\widetilde{\varphi }^{(c,m)}( t+\tau ,\tau , \theta _{-\tau }\omega ,\widetilde{u}^{(c)}_{\tau }, \widetilde{v}^{(c)}_{\tau } ) \}\) and \(\{\varphi ^{( m)}( t+\tau ,\tau , \theta _{-\tau }\omega ,u_{\tau },v_{\tau } ) \} \) in \(l_{\rho }^{2} \times l_{\rho }^{2}\) implies that, for \(\tau \in \mathbb{R}\), \(t\in \mathbb{R}^{+}\), \(\omega \in \Omega \), \(c_{n} \rightarrow 0\) and \((\widetilde{u}^{(c_{n} )}_{\tau }, \widetilde{v}^{(c_{n} )}_{\tau } )\), \(( u_{\tau },v_{\tau })\in l_{\rho }^{2}\times l_{\rho }^{2}\) with \((\widetilde{u}^{(c_{n} )}_{\tau }, \widetilde{v}^{(c_{n} )}_{\tau } )\rightarrow ( u_{\tau },v_{\tau })\), the following holds:
$$ \lim_{ n\rightarrow \infty } \widetilde{\varphi }^{(c _{n})}\bigl( t+\tau , \tau , \theta _{-\tau }\omega ,\widetilde{u}^{(c_{n})}_{\tau }, \widetilde{v}^{(c_{n})}_{\tau } \bigr)=\varphi( t+\tau ,\tau , \theta _{-\tau }\omega ,u_{\tau },v_{\tau } ) \quad \text{in } l_{\rho }^{2} \times l_{\rho }^{2}. $$
(4.24)
We completed the proof. □

Declarations

Acknowledgements

The authors would like to express their sincere thanks to the anonymous referees for their time and comments. This work is supported by the National Natural Science Foundation of China under Grant Nos. 11326114, 11401244 and 11471290; Natural Science Research Project of Ordinary Universities in Jiangsu Province under Grant No. 14KJB110003; Zhejiang Normal University Foundation under Grant No. ZC304011068; Zhejiang Natural Science Foundation under Grant No. LY14A010012.

Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors’ Affiliations

(1)
School of Mathematical Science, Huaiyin Normal University, Huaian, P.R. China
(2)
Department of Mathematics, Zhejiang Normal University, Jinhua, P.R. China

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