A critical point theorem for a class of non-differentiable functionals with applications

1.   Introduction

Lattice dynamical systems arise from a variety of applications in electrical engineering, biology, chemical reaction, pattern formation and so on, see, e.g., ^{[4,7,14,19,33]}. Many researchers have discussed broadly the deterministic models in ^[6,12,34,39], etc. Stochastic lattice equations, driven by additive independent white noise, was discussed for the first time in ^[2], followed by extensions in ^{[8,13,15,16,21,23,27,32,35,36,37,38,40]}.

In this paper, we will study the long term behavior of the following second order non-autonomous stochastic lattice system driven by additive white noise: for given $\tau\in\mathbb{R}$, $t > \tau$ and $i\in\mathbb{Z}$,

where $u = (u_{i})_{i\in\mathbb{Z}}$ is a sequence in $l^{2}$, $\nu$ and $\lambda$ are positive constants, $\dot{u} = (\dot{u}_{i})_{i\in\mathbb{Z}}$ and $\ddot{u} = (\ddot{u}_{i})_{i\in\mathbb{Z}}$ denote the fist and the second order time derivatives respectively, $Au = ((Au)_{i})_{i\in\mathbb{Z}}$, $A\dot{u} = ((A\dot{u})_{i})_{i\in\mathbb{Z}}$, $A$ is a linear operators defined in (2.2), $a = (a_{i})_{i\in\mathbb{ Z}}\in l^{2}$, $f(u) = \big(f_{i}(u_{i})\big)_{i\in\mathbb{Z}}$ and $h(\dot{u}) = \big(h_{i}(\dot{u}_{i})\big)_{i\in\mathbb{Z}}$ satisfy certain conditions, $g(t) = \big(g_{i}(t)\big)_{i\in\mathbb{Z}}\in L^{2}_{loc}(\mathbb{R}, l^{2})$ is a given time dependent sequence, and $\omega(t) = W(t, \omega)$ is a two-sided real-valued Wiener process on a probability space.

The approximation we use in the paper was first proposed in ^[18,22] where the authors investigated the chaotic behavior of random equations driven by $\mathcal{G}_{\delta}(\theta_{t}\omega)$. Since then, their work was extended by many scholars. To the best of my knowledge, there are three forms of Wong-Zakai approximations $\mathcal{G}_{\delta}(\theta_{t}\omega)$ used recenly, Euler approximation of Brownian ^{[3,10,17,20,25,28,29,30]}, Colored noise ^[5,11,26,31] and Smoothed approximation of Brownian motion by mollifiers ^[9]. In this paper, we will focus on Euler approximation of Brownian and compare the long term behavior of system (1.1) with pathwise deterministic system given by

for $\delta\in\mathbb{R}$ with $\delta\neq0$, $\tau\in\mathbb{R}$, $t > \tau$ and $i\in\mathbb{Z}$, where $\mathcal{G}_{\delta}(\theta_{t}\omega)$ is defined in (3.2). Note that the solution of system (1.2) is written as $u^{\delta}$ to show its dependence on $\delta$.

This paper is organized as follows. In Section 2, we prove the existence and uniqueness of random attractors of system (1.1). Section 3 is devoted to consider the the Wong-Zakai approximations associated with system (1.1). In Section 4, we establish the convergence of solutions and attractors for approximate system (1.2) when $\delta\rightarrow0$.

Throughout this paper, the letter $c$ and $c_{i}(i = 1, 2, \ldots)$ are generic positive constants which may change their values from line to line.

2.   Stochastic lattice system with additive white noise

In this section, we will define a continuous cocycle for second order non-autonomous stochastic lattice system (1.1), and then prove the existence and uniqueness of pullback attractors.

A standard Brownian motion or Wiener process $(W_{t})_{t\in\mathbb{R}}$ (i.e., with two-sided time) in $\mathbb{R}$ is a process with $W_{0} = 0$ and stationary independent increments satisfying $W_{t}-W_{s}\sim \mathcal{N}(0, |t-s|I)$. $\mathcal{F}$ is the Borel $\sigma$-algebra induced by the compact-open topology of $\Omega$, and $P$ is the corresponding Wiener measure on $(\Omega, \mathcal{F})$, where

the probability space $(\Omega, \mathcal{F}, P)$ is called Wiener space. Define the time shift by

Then $(\Omega, \mathcal{F}, P, \{\theta_{t}\}_{t\in\mathbb{R}})$ is a metric dynamical system (see ^[1]) and there exists a $\{\theta_{t}\}_{t\in\mathbb{R}}$-invariant subset $\tilde{\Omega}\subseteq\Omega$ of full measure such that for each $\omega\in\Omega$,

For the sake of convenience, we will abuse the notation slightly and write the space $\tilde{\Omega}$ as $\Omega$.

We denote by

with the norm as

In particular, $l^{2}$ is a Hilbert space with the inner product $(\cdot, \cdot)$ and norm $\|\cdot\|$ given by

for any $u = (u_{i})_{i\in \mathbb{Z}}, \ v = (v_{i})_{i\in \mathbb{Z}}\in l^{2}$.

Define linear operators $B$, $B^{*}$, and $A$ acting on $l^{2}$ in the following way: for any $u = (u_{i})_{i\in\mathbb{Z}}\in l^{2}$,

and

Then we find that $A = BB^{*} = B^{*}B$ and $(B^{*}u, v) = (u, Bv)$ for all $u, v\in l^{2}$.

Also, we let $F_{i}(s) = \int_{0}^{s}f_{i}(r)dr$, $h(\dot{u}) = (h_{i}(\dot{u}_{i}))_{i\in\mathbb{Z}}$, $f(u) = (f_{i}(u_{i}))_{i\in\mathbb{Z}}$ with $f_{i}, h_{i}\in C^{1}(\mathbb{R}, \mathbb{R})$ satisfy the following assumptions:

and

where $p > 1$, $\alpha_{i}$ and $h_{j}$ are positive constants for $i = 1, 2, 3$ and $j = 1, 2$.

In addition, we let

and

For any $u, v\in l^{2}$, we define a new inner product and norm on $l^{2}$ by

Denote

Then the norms $\|\cdot\|$ and $\|\cdot\|_{\lambda}$ are equivalent to each other.

Let $E = l^{2}_{\lambda}\times l^{2}$ endowed with the inner product and norm

for $\psi_{j} = (u^{(j)}, v^{(j)})^{T} = \big((u^{(j)}_{i}), (v^{(j)}_{i})\big)^{T}_{i\in \mathbb{Z}}\in E, \ j = 1, 2, \quad \psi = (u, v)^{T} = \big((u_{i}), (v_{i})\big)^{T}_{i\in \mathbb{Z}}\in E.$

A family $D = \{D(\tau, \omega):\tau\in\mathbb{R}, \omega\in\Omega\}$ of bounded nonempty subsets of $E$ is called tempered (or subexponentially growing) if for every $\epsilon > 0$, the following holds:

where $\|D\| = \mathop{\sup}\limits_{x\in D}\|x\|_{E}$. In the sequel, we denote by $\mathcal{D}$ the collection of all families of tempered nonempty subsets of $E$, i.e.,

The following conditions will be needed for $g$ when deriving uniform estimates of solutions, for every $\tau\in\mathbb{R}$,

and for any $\varsigma > 0$

where $\gamma = \min\{\frac{\sigma}{2}, \frac{\alpha_{2}\beta}{p+1}\}$.

Let $\bar{v} = \dot{u}+\beta u$ and $\bar{\varphi} = (u, \bar{v})^{T}$, then system (1.1) can be rewritten as

with initial conditions

where

Denote

By (2.10) we have

with initial conditions

where

Note that system (2.11) is a deterministic functional equation and the nonlinearity in (2.11) is locally Lipschitz continuous from $E$ to $E$. Therefore, by the standard theory of functional differential equations, system (2.11) is well-posed. Thus, we can define a continuous cocycle $\Phi_{0}:\mathbb{R}^{+}\times\mathbb{R}\times \Omega\times E\rightarrow E$ associated with system (2.10), where for $\tau\in\mathbb{R}$, $t\in\mathbb{R}^{+}$ and $\omega\in\Omega$

where $v_{\tau} = \bar{v}_{\tau}+a\omega(-\tau)$.

Lemma 2.1. Suppose that (2.3)–(2.8) hold. Then for every $\tau\in\mathbb{R}$, $\omega\in\Omega$, and $T > 0$, there exists $c = c(\tau, \omega, T) > 0$ such that for all$t\in[\tau, \tau+T]$, the solution $\varphi$ of system (2.11) satisfies

Proof. Taking the inner product $(\cdot, \cdot)_{E}$ on both side of the system (2.11) with $\varphi$, it follows that

For the second term on the left-hand side of (2.12), we have

By the mean value theorem and (2.5), there exists $\xi_{i}\in(0, 1)$ such that

Then

which along with (2.6) and (2.7) implies that

As to the first term on the right-hand side of (2.12), by (2.3) and (2.4) we get

The last term of (2.12) is bounded by

It follows from (2.12)–(2.15) that

where $\gamma = \min\{\frac{\sigma}{2}, \frac{\alpha_{2}\beta}{p+1}\}$. Multiplying (2.16) by $e^{\gamma t}$ and then integrating over $(\tau, t)$ with $t\geq\tau$, we get for every $\omega\in\Omega$

which implies desired result.

Lemma 2.2. Suppose that (2.3)–(2.9) hold. Then the continuous cocycle $\Phi_{0}$ associated with system (2.10) has a closed measurable $\mathcal{D}$-pullback absorbing set $K_{0} = \{K_{0}(\tau, \omega):\tau\in\mathbb{R}, \omega\in\Omega\}\in\mathcal{D}$, where for every $\tau\in\mathbb{R}$ and $\omega\in\Omega$

where $\bar{\varphi}_{\tau-t}\in D(\tau-t, \theta_{-t}\omega)$ and $R_{0}(\tau, \omega)$ is given by

where $c$ is a positive constant independent of $\tau$, $\omega$ and $\mathcal{D}$.

Proof. By (2.17), we get for every $\tau\in\mathbb{R}$, $t\in\mathbb{R}^{+}$ and $\omega\in\Omega$

By (2.1) and (2.8), the last integral on the right-hand side of (2.20) is well defined. For any $s\geq \tau-t$,

which along with (2.20) implies that

By (2.3) and (2.4) we have

Using $\bar{\varphi}_{\tau-t}\in D(\tau-t, \theta_{-t}\omega)$, (2.1) and (2.22), we find

which along with (2.21) implies that there exists $T = T(\tau, \omega, D) > 0$ such that for all $t\geq T$,

where $c$ is a positive constant independent of $\tau$, $\omega$ and $D$. Note that $K_{0}$ given by (2.18) is closed measurable random set in $E$. Given $\tau\in\mathbb{R}$, $\omega\in\Omega$, and $D\in\mathcal{D}$, it follows from (2.24) that for all $t\geq T$,

which implies that $K_{0}$ pullback attracts all elements in $\mathcal{D}$. By (2.1) and (2.9), one can easily check that $K_{0}$ is tempered, which along with (2.25) completes the proof.

Next, we will get uniform estimates on the tails of solutions of system (2.10).

Lemma 2.3. Suppose that (2.3)–(2.9) hold. Then for every $\tau\in\mathbb{R}$, $\omega\in\Omega$, $D = \{D(\tau, \omega):\tau\in\mathbb{R}, \omega\in\Omega\}\in\mathcal{D}$ and $\varepsilon > 0$, there exist $T = T(\tau, \omega, D, \varepsilon) > 0$ and $N = N(\tau, \omega, \varepsilon) > 0$ such that for all$t\geq T$, the solution $\bar{\varphi}$ of system (2.10) satisfies

where $\bar{\varphi}_{\tau-t}\in D(\tau-t, \theta_{-t}\omega)$ and $|\bar{\varphi}_{i}|_{E}^{2} = (1-\nu\beta)|Bu|_{i}^{2}+\lambda |u_{i}|^{2}+|\bar{v}_{i}|^{2}$.

Proof. Let $\eta$ be a smooth function defined on $\mathbb{R}^{+}$ such that $0\leq\eta(s)\leq1$ for all $s\in\mathbb{R}^{+}$, and

Then there exists a constant $C_{0}$ such that $|\eta'(s)|\leq C_{0}$ for $s\in\mathbb{R}^{+}$. Let $k$ be a fixed positive integer which will be specified later, and set $x = (x_{i})_{i\in\mathbb{Z}}$, $y = (y_{i})_{i\in\mathbb{Z}}$ with $x_{i} = \eta(\frac{|i|}{k})u_{i}$, $y_{i} = \eta(\frac{|i|}{k})v_{i}$. Note $\psi = (x, y)^{T} = ((x_{i}), (y_{i}))^{T}_{i\in\mathbb{Z}}$. Taking the inner product of system (2.11) with $\psi$, we have

For the first term of (2.26), we have

where $|\varphi_{i}|_{E}^{2} = (1-\nu\beta)|Bu|_{i}^{2}+\lambda |u_{i}|^{2}+|v_{i}|^{2}$. As to the second term on the left-hand side of (2.26), we get

It is easy to check that

and

By the mean value theorem and (2.5), there exists $\xi_{i}\in(0, 1)$ such that

Then

which along with (2.6) and (2.7) implies that

As to the first term on the right-hand side of (2.26), by (2.3) and (2.4)we get

For the last term of (2.26), we have

As to the first two terms on the right-hand side of (2.30), we get

and

The last two terms of (2.30) is bounded by

It follows from (2.26)–(2.33) that

where $\gamma = \min\{\frac{\sigma}{2}, \frac{\alpha_{2}\beta}{p+1}\}$. Multiplying (2.34) by $e^{\gamma t}$, replacing $\omega$ by $\theta_{-\tau}\omega$ and integrating on $(\tau-t, \tau)$ with $t\in\mathbb{R}^{+}$, we get for every $\omega\in\Omega$

For any $s\geq \tau-t$,

which along with (2.35) implies that

By (2.1) and (2.8), the last four integrals in (2.36) are well defined. By (2.3) and (2.4), we obtain

which along with $\bar{\varphi}_{\tau-t}\in D(\tau-t, \theta_{-t}\omega)$ and (2.1) implies that

Then there exists $T_{1} = T_{1}(\tau, \omega, D, \varepsilon) > 0$ such that for all $t\geq T_{1}$,

By (2.1) and (2.24), there exist $T_{2} = T_{2}(\tau, \omega, D, \varepsilon) > T_{1}$ and $N_{1} = N_{1}(\tau, \omega, \varepsilon) > 0$ such that for all $t\geq T_{2}$ and $k\geq N_{1}$

By (2.8), there exists $N_{2} = N_{2}(\tau, \omega, \varepsilon) > N_{1}$ such that for all $k\geq N_{2}$,

By (2.1) again, we find that there exists $N_{3} = N_{3}(\tau, \omega, \varepsilon) > N_{2}$ such that for all $k\geq N_{3}$,

Then it follows from (2.36)–(2.40) that for all $t\geq T_{2}$ and $k\geq N_{3}$

This concludes the proof.

As a consequence of Lemma 2.2 and Lemma 2.3, we get the existence of $\mathcal{D}$-pullback attractors for $\Phi_{0}$ immediately.

Theorem 2.1. Suppose that (2.3)–(2.9) hold. Then the continuous cocycle $\Phi_{0}$ associated with system (2.10) has a unique $\mathcal{D}$-pullback attractors $\mathcal{A}_{0} = \{\mathcal{A}_{0}(\tau, \omega):\tau\in\mathbb{R}$, $\omega\in\Omega\}\in \mathcal{D}$ in $E$.

3.   Wong-Zakai approximation of second order lattice system

In this section, we will approximate the solutions of system (1.1) by the pathwise Wong-Zakai approximated system (1.2). Given $\delta\neq0$, define a random variable $\mathcal{G}_{\delta}$ by

From (3.1) we find

By (3.2) and the continuity of $\omega$ we get for all $t\in\mathbb{R}$,

Note that this convergence is uniform on a finite interval as stated below.

Lemma 3.1. (^[17]). Let $\tau\in\mathbb{R}$, $\omega\in\Omega$ and $T > 0$. Then for every $\varepsilon > 0$, there exists $\delta_{0} = \delta_{0}(\varepsilon, \tau, \omega, T) > 0$ such that for all $0 < |\delta| < \delta_{0}$ and $t\in[\tau, \tau+T]$,

By Lemma 3.1, we find that there exist $c = c(\tau, \omega, T) > 0$ and $\tilde{\delta}_{0} = \tilde{\delta}_{0}(\tau, \omega, T) > 0$ such that for all $0 < |\delta| < \tilde{\delta}_{0}$ and $t\in[\tau, \tau+T]$,

By (3.3) we find that $\mathcal{G}_{\delta}(\theta_{t}\omega)$ is an approximation of the white noise in a sense. This leads us to consider system (1.2) as an approximation of system (1.1).

Let $\bar{v}^{\delta} = \dot{u}^{\delta}+\beta u^{\delta}$ and $\bar{\varphi}_{\delta} = (u^{\delta}, \bar{v}^{\delta})$, the system (1.2) can be rewritten as

with initial conditions

where

Denote

By (3.5) we have

with initial conditions

where

Note that system (3.6) is a deterministic functional equation and the nonlinearity in (3.6) is locally Lipschitz continuous from $E$ to $E$. Therefore, by the standard theory of functional differential equations, system (3.6) is well-posed. Thus, we can define a continuous cocycle $\Phi_{\delta}:\mathbb{R}^{+}\times\mathbb{R}\times \Omega\times E\rightarrow E$ associated with system (3.5), where for $\tau\in\mathbb{R}$, $t\in\mathbb{R}^{+}$ and $\omega\in\Omega$

where $v_{\tau}^{\delta} = \bar{v}_{\tau}^{\delta}-a\int_{-\tau}^{0}\mathcal{G}_{\delta}(\theta_{s}\omega)ds$.

For later purpose, we now show the estimates on the solutions of system (3.6) on a finite time interval.

Lemma 3.2. Suppose that (2.3)–(2.8) hold. Then for every $\tau\in\mathbb{R}$, $\omega\in\Omega$, and $T > 0$, there exist $\delta_{0} = \delta_{0}(\tau, \omega, T) > 0$ and $c = c(\tau, \omega, T) > 0$ such that for all $0 < |\delta| < \delta_{0}$ and$t\in[\tau, \tau+T]$, the solution $\varphi_{\delta}$ of system (3.6) satisfies

Proof. Taking the inner product $(\cdot, \cdot)_{E}$ on both side of the system (3.6) with $\varphi_{\delta}$, it follows that

By the similar calculations in (2.13)–(2.15), we get

and

It follows from (3.7)–(3.10) that

where $\gamma = \min\{\frac{\sigma}{2}, \frac{\alpha_{2}\beta}{p+1}\}$. Multiplying (3.11) by $e^{\gamma t}$ and integrating on $(\tau, t)$ with $t\geq\tau$, we get for every $\omega\in\Omega$

which implies the desired result.

In what follows, we derive uniform estimates on the solutions of system (3.5) when $t$ is sufficiently large.

Lemma 3.3. Suppose that (2.3)–(2.8) hold. Then for every $\delta\neq0$, $\tau\in\mathbb{R}$, $\omega\in\Omega$, and $D = \{D(\tau, \omega):\tau\in\mathbb{R}, \omega\in\Omega\}\in\mathcal{D}$, there exists $T = T(\tau, \omega, D, \delta) > 0$ such that for all$t\geq T$, the solution $\bar{\varphi}_{\delta}$ of system (3.5) satisfies

where $\bar{\varphi}_{\delta, \tau-t}\in D(\tau-t, \theta_{-t}\omega)$ and $R_{\delta}(\tau, \omega)$ is given by

where $c$ is a positive constant independent of $\tau$, $\omega$ and $\delta$.

Proof. Multiplying (3.11) by $e^{\gamma t}$, replacing $\omega$ by $\theta_{-\tau}\omega$ and integrating on $(\tau-t, \tau)$ with $t\in\mathbb{R}^{+}$, we get for every $\omega\in\Omega$

By (2.1), (2.8) and (3.2), the last integral on the right-hand side of (3.13) is well defined. For any $s\geq \tau-t$,

which along with (3.13) shows that

Note that (2.3) and (2.4) implies that

which along with $\bar{\varphi}_{\delta, \tau-t}\in D(\tau-t, \theta_{-t}\omega)$, (2.1) and (3.2) implies that

Then (3.14) and (3.15) can imply the desired estimates.

Next, we show that system (3.5) has a $\mathcal{D}$-pullback absorbing set.

Lemma 3.4. Suppose that (2.3)–(2.9) hold. Then the continuous cocycle $\Phi_{\delta}$ associated with system (3.5) has a closed measurable $\mathcal{D}$-pullback absorbing set $K_{\delta} = \{K_{\delta}(\tau, \omega):\tau\in\mathbb{R}, \omega\in\Omega\}\in\mathcal{D}$, where for every $\tau\in\mathbb{R}$ and $\omega\in\Omega$

where $R_{\delta}(\tau, \omega)$ is given by (3.12).In addition, we have for every $\tau\in\mathbb{R}$ and $\omega\in\Omega$

where $R_{0}(\tau, \omega)$ is defined in (2.19).

Proof. Note $K_{\delta}$ given by (3.16) is closed measurable random set in $E$. Given $\tau\in\mathbb{R}$, $\omega\in\Omega$, and $D\in\mathcal{D}$, it follows from Lemma 3.3 that there exists $T_{0} = T_{0}(\tau, \omega, D, \delta)$ such that for all $t\geq T_{0}$,

which implies that $K_{\delta}$ pullback attracts all elements in $\mathcal{D}$. By (2.1), (2.8) and (3.2), we can prove $K_{\delta}(\tau, \omega)$ is tempered. The convergence (3.17) can be obtained by Lebesgue dominated convergence as in ^[17].

We are now in a position to derive uniform estimates on the tail of solutions of system (3.5).

Lemma 3.5. Suppose that (2.3)–(2.8) hold. Then for every $\tau\in\mathbb{R}$, $\omega\in\Omega$ and $\varepsilon > 0$, there exist $\delta_{0} = \delta_{0}(\omega) > 0$, $T = T(\tau, \omega, \varepsilon) > 0$ and $N = N(\tau, \omega, \varepsilon) > 0$ such that for all$t\geq T$ and $0 < |\delta| < \delta_{0}$, the solution $\bar{\varphi}_{\delta}$ of system (3.5) satisfies

where $\bar{\varphi}_{\delta, \tau-t}\in K_{\delta}(\tau-t, \theta_{-t}\omega)$ and $|\bar{\varphi}_{\delta, i}|_{E}^{2} = (1-\nu\beta)|Bu^{\delta}|_{i}^{2}+\lambda |u^{\delta}_{i}|^{2}+|\bar{v}^{\delta}_{i}|^{2}$.

Proof. Let $\eta$ be a smooth function defined in Lemma 2.3, and set $x = (x_{i})_{i\in\mathbb{Z}}$, $y = (y_{i})_{i\in\mathbb{Z}}$ with $x_{i} = \eta(\frac{|i|}{k})u^{\delta}_{i}$, $y_{i} = \eta(\frac{|i|}{k})v^{\delta}_{i}$. Note $\psi = (x, y)^{T} = ((x_{i}), (y_{i}))^{T}_{i\in\mathbb{Z}}$. Taking the inner product of system (3.6) with $\psi$, we have

For the first term of (3.18), we have

where $|\varphi_{\delta, i}|_{E}^{2} = (1-\nu\beta)|Bu^{\delta}|_{i}^{2}+\lambda |u^{\delta}_{i}|^{2}+|v^{\delta}_{i}|^{2}$. By the similar calculations in (2.28)–(2.33), we get

and

It follows from (3.18)–(3.22) that

where $\gamma = \min\{\frac{\sigma}{2}, \frac{\alpha_{2}\beta}{p+1}\}$. Multiplying (3.23) by $e^{\gamma t}$, replacing $\omega$ by $\theta_{-\tau}\omega$ and integrating on $(\tau-t, \tau)$ with $t\in\mathbb{R}^{+}$, we get for every $\omega\in\Omega$

For any $s\geq\tau-t$,

which along with (3.24) shows that

By (2.1) and (2.8), the last four integrals on the right-hand side of (3.24) are well defined. Note that (2.3) and (2.4) implies that

Since $\bar{\varphi}_{\delta, \tau-t}\in K_{\delta}(\tau-t, \theta_{-t}\omega)$, we find

which along with (2.1) and (3.2) shows that there exist $T_{1} = T_{1}(\tau, \omega, \varepsilon) > 0$ and $\delta_{0} > 0$ such that for all $t\geq T_{1}$ and $0 < |\delta| < \delta_{0}$,

By Lemma 3.3, (2.1) and (3.2), there exist $T_{2} = T_{2}(\tau, \omega, \varepsilon) > T_{1}$ and $N_{1} = N_{1}(\tau, \varepsilon) > 0$ such that for all $t\geq T_{2}$, $k\geq N_{1}$ and $0 < |\delta| < \delta_{0}$

By (2.8), there exists $N_{2} = N_{2}(\tau, \varepsilon) > N_{1}$ such that for all $k\geq N_{2}$,

By (2.1) and (3.2) again, we find that there exists $N_{3} = N_{3}(\tau, \varepsilon) > N_{2}$ such that for all $k\geq N_{3}$ and $0 < |\delta| < \delta_{0}$,

Then it follows from (3.25)–(3.29) that for all $t\geq T_{2}$, $k\geq N_{3}$ and $0 < |\delta| < \delta_{0}$,

This concludes the proof.

By Lemma 3.4, $\Phi_{\delta}$ has a closed $\mathcal{D}$-pullback absorbing set, and Lemma 3.5 shows that $\Phi_{\delta}$ is asymptotically null in $E$ with respect to $\mathcal{D}$. Therefore, we get the existence of $\mathcal{D}$-pullback attractors for $\Phi_{\delta}$.

Lemma 3.6. Suppose that (2.3)–(2.9) hold. Then the continuous cocycle $\Phi_{\delta}$ associated with (3.5) has a unique $\mathcal{D}$-pullback attractors $\mathcal{A}_{\delta} = \{\mathcal{A}_{\delta}(\tau, \omega):\tau\in\mathbb{R}$, $\omega\in\Omega\}\in \mathcal{D}$ in $E$.

For the attractor $\mathcal{A}_{\delta}$ of $\Phi_{\delta}$, we have the uniform compactness as showed below.

Lemma 3.7. Suppose that (2.3)–(2.9) hold. Then for every $\tau\in\mathbb{R}$, $\omega\in\Omega$, there exists $\delta_{0} = \delta_{0}(\omega) > 0$ such that$\mathop{\bigcup}\limits_{0 < |\delta| < \delta_{0}}\mathcal{A}_{\delta}(\tau, \omega)$ is precompact in $E$.

Proof. Given $\varepsilon > 0$, we will prove that $\mathop{\bigcup}\limits_{0 < |\delta| < \delta_{0}}\mathcal{A}_{\delta}(\tau, \omega)$ has a finite covering of balls of radius less than $\varepsilon$. By (3.2) we have

By $\lim_{\delta\rightarrow0}\int^{\delta}_{0}\frac{\omega(r)}{\delta}dr = 0$, there exists $\delta_{1} = \delta_{1}(\omega) > 0$ such that for all $0 < |\delta| < \delta_{1}$,

Similarly, there exists $l_{1}$ between $s$ and $s+\delta$ such that $\int^{s+\delta}_{s}\frac{\omega(l)}{\delta}dl = \omega(l_{1})$, which along with (2.1) implies that there exists $T_{1} = T_{1}(\omega) < 0$ such that for all $s\leq T_{1}$ and $|\delta|\leq1$,

Let $\delta_{2} = \min\{\delta_{1}, 1\}$. By (3.30)–(3.32) we get for all $0 < |\delta| < \delta_{2}$ and $s\leq T_{1}$,

By (3.4), there exist $\delta_{0} = \delta_{0}(\omega)\in(0, \delta_{2})$ and $c_{1}(\omega) > 0$ such that for all $0 < |\delta|\leq\delta_{0}$ and $T_{1}\leq s\leq0$,

which along with (3.33) implies that for all $0 < |\delta| < \delta_{0}$ and $s\leq0$,

where $c_{2}(\omega) = 2+c_{1}(\omega)$. Denote by

and

with $c$ and $c_{2}$ being as in (3.12) and (3.34). By (3.12) and (3.35) we find that for all $0 < |\delta| < \delta_{0}$,

By (3.35) and (3.36), we find that $K_{\delta}(\tau, \omega)\subseteq B(\tau, \omega)$ for all $0 < |\delta| < \delta_{0}$, $\tau\in\mathbb{R}$ and $\omega\in\Omega$. Therefore, for every $\tau\in\mathbb{R}$, $\omega\in\Omega$,

By Lemma 3.5, there exist $T = T(\tau, \omega, \varepsilon) > 0$ and $N = N(\tau, \omega, \varepsilon) > 0$ such that for all $t\geq T$ and $0 < |\delta| < \delta_{0}$,

for any $\bar{\varphi}_{\delta, \tau-t}\in K_{\delta}(\tau-t, \theta_{-t}\omega)$. By (3.38) and the invariance of $\mathcal{A}_{\delta}$, we obtain

We find that (3.37) implies the set $\{(\bar{\varphi}_{i})_{|i| < N}:\bar{\varphi}\in \mathop{\bigcup}\limits_{0 < |\delta| < \delta_{0}}\mathcal{A}_{\delta}(\tau, \omega)\}$ is bounded in a finite dimensional space and hence is precompact. This along with (3.39) implies $\mathop{\bigcup}\limits_{0 < |\delta| < \delta_{0}}\mathcal{A}_{\delta}(\tau, \omega)$ has a finite covering of balls of radius less than $\varepsilon$ in $E$. This completes the proof.

4.   Upper semicontinuity of pullback attractors

In this section, we will study the limiting of solutions of (3.5) as $\delta\rightarrow0$. Hereafter, we need an additional condition on $f$: For all $i\in\mathbb{Z}$ and $s\in\mathbb{R}$,

where $\alpha_{4}$ is a positive constant, $\kappa = (\kappa_{i})_{i\in\mathbb{Z}}\in l^{2}$ and $p > 1$.

Lemma 4.1. Suppose that (2.3)–(2.7) and (4.1) hold. Let $\bar{\varphi}$ and $\bar{\varphi}_{\delta}$ are the solutions of (2.10) and (3.5), respectively. Then for every $\tau\in\mathbb{R}$, $\omega\in\Omega$, $T > 0$ and $\varepsilon\in(0, 1)$, there exist $\delta_{0} = \delta_{0}(\tau, \omega, T, \varepsilon) > 0$ and $c = c(\tau, \omega, T) > 0$ such that for all$t\in[\tau, \tau+T]$ and $0 < |\delta| < \delta_{0}$,

Proof. Let $\tilde{\varphi} = \varphi_{\delta}-\varphi$ and $\tilde{\varphi} = (\tilde{u}, \tilde{v})^{T}$, where $\tilde{u} = u^{\delta}-u$, $\tilde{v} = v^{\delta}-v$, $\varphi$ and $\varphi_{\delta}$ are the solutions of (2.11) and (3.6), respectively. By (2.11) and (3.6) we get

where

Taking the inner product of (4.2) with $\tilde{\varphi}$ in $E$, we have

For the second term on the left-hand side of (4.3), using the similar calculations in (2.13) we have

For the first term on the right-hand side of (4.3), by (4.1) we get

As to the last term of (4.3), we have

It follows from (4.3)–(4.6) that

By Lemma 2.1 and Lemma 3.2, there exists $\delta_{1} = \delta_{1}(\tau, \omega, T) > 0$ and $c_{1} = c_{1}(\tau, \omega, T) > 0$ such that for all $0 < |\delta| < \delta_{1}$ and $t\in[\tau, \tau+T]$,

which along with (4.7) shows that for all $0 < |\delta| < \delta_{1}$ and $t\in[\tau, \tau+T]$

Applying Gronwall's inequality and Lemma 3.1 to (4.8), we see that for every $\varepsilon\in(0, 1)$, there exists $\delta_{0} = \delta_{0}(\tau, \omega, T, \varepsilon)\in(0, \delta_{1})$ such that for all $0 < |\delta| < \delta_{0}$ and $t\in[\tau, \tau+T]$

On the other hand, we have

which along with (4.9) implies the desired result.

Finally, we establish the upper semicontinuity of random attractors as $\delta\rightarrow0$.

Theorem 4.1. Suppose that (2.3)–(2.9) and (4.1) hold. Then for every $\tau\in\mathbb{R}$ and $\omega\in\Omega$,

where $d_{E}(\mathcal{A}_{\delta}(\tau, \omega), \mathcal{A}_{0}(\tau, \omega)) = \mathop{\sup}\limits_{x\in\mathcal{A}_{\delta}(\tau, \omega)}\mathop{\inf}\limits_{y\in\mathcal{A}_{0}(\tau, \omega)}\|x-y\|_{E}$.

Proof. Let $\delta_{n}\rightarrow0$ and $\bar{\varphi}_{\delta_{n}, \tau}\rightarrow \bar{\varphi}_{\tau}$ in $E$. Then by Lemma 4.1, we find that for all $\tau\in\mathbb{R}$, $t\geq0$ and $\omega\in\Omega$,

By (3.16)–(3.17) we have, for all $\tau\in\mathbb{R}$ and $\omega\in\Omega$,

Then by (4.11), (4.12) and Lemma 3.7, (4.10) follows from Theorem 3.1 in ^[24] immediately.

5.   Conclusions

In this paper we use similar idea in ^[30] but apply to second order non-autonomous stochastic lattice dynamical systems with additive noise. we establish the convergence of solutions of Wong-zakai approximations and the upper semicontinuity of random attractors of the approximate random system as the step-length of the Wiener shift approaches zero. In addition, as to the second order non-autonomous stochastic lattice dynamical systems with multiplicative noise, we can use the similar method in ^[29] to get the corresponding results.

Acknowledgements

The authors would like to thank anonymous referees and editors for their valuable comments and constructive suggestions.

Conflict of interest

The authors declare no conflict of interest.