Exponential convergence in the mean-square sense of nonlocal stochastic almost automorphic genetic regulatory lattice networks

Abstract

This paper first establishes the lattice model for nonlocal stochastic genetic regulatory networks with reaction diffusions by employing a mix of the finite difference and Mittag–Leffler time Euler difference techniques. Second, the existence of a unique bounded almost automorphic sequence in distribution and global mean-square exponential convergence to the achieved difference model are investigated. An illustrative example is used to show the feasible of the works of the current paper.

Introduction

The vast number of DNAs, RNAs, proteins, and small molecules in an organism together with the mechanisms regulating gene expression forms gene regulatory networks. It is a network of intracellular genes and their interactions. In the past few years, gene regulatory networks have attracted more and more attention of scholars due to their easy-to-understand properties. Gene regulatory networks are built on the foundation of life sciences and control theory. They are extremely practical and powerful approaches in characterizing highly dynamic and sophisticated interactions of transcription. Therefore, gene regulatory networks are not only the basis for studying various phenomena in living organisms, but also offer a wide range of prospects for applications in systematic biology. In the previous decades, a number of literatures have been reported for gene regulatory networks, please refer Pasquini and Angeli (2021), Augier and Yabo (2022), Kim et al. (2022), Hillerton et al. (2022), Stamova and Stamov (2021), Padmaja and Balasubramaniam (2022), Stamov and Stamova (2021) and Qiao et al. (2020). On the contrary, fractional calculus (Kilbas et al., 2006; Zhang and Xiong, 2020) is more than 300 years. It allows us to characterize a more accurate representation of the problem than the traditional “integer order” approach. Fractional-order gene regulatory networks have played an exceedingly significant position in the evolution of systematic biology, as they provide an efficacious way to describe memory and genetic properties, refer Stamova and Stamov (2021), Padmaja and Balasubramaniam (2022), Stamov and Stamova (2021) and Qiao et al. (2020). Padmaja and Balasubramaniam (2022) considered the $H_{\infty }$ /passivity performance of fractional-order gene regulatory networks in the form of

$$\{\begin{array}{c}{{}^{c}D}_0^{\alpha }{ℳ}_i(t)=-a_i{M}_i(t)+\sum _{j=1}^m{b}_{ij}{f}_j({\mathcal{P}}_j(t))+I_i,\\ {{}^{c}D}_0^{\alpha }{\mathcal{P}}_i(t)=-c_i{\mathcal{P}}_i(t)+d_i{ℳ}_i(t),t>0,i=1,2,\dots ,m,\end{array}$$

(1)

where ${{}^{c}D}_0^{\alpha }$ denotes Caputo fractional-order derivative from initial point $0$ , $\alpha \in (0,1]$ , ${\mathcal{M}}_i$ and ${\mathcal{P}}_i$ represent the concentrations of the $i\mathrm{th}$ mRNA and $i\mathrm{th}$ protein, respectively; $a_i>0$ and $c_i>0$ are the decay rates of mRNA and protein, respectively; $d_i>0$ is the translation rate; $f_j(x)=(x/{\beta }_j{)}^{H_j}/(1+(x/{\beta }_j{)}^{H_j})$ , $H_j$ is the Hill coefficient and ${\beta }_j$ is a positive constant; $I_i=\sum _{j\in W_i}{w}_{\mathrm{ij}}$ , $w_{\mathrm{ij}}\ge 0$ is bounded and $W_i$ is the set of all the j which is a repressor of gene i, $B=(b_{\mathrm{ij}})\in {\mathbb{R}}^{m\times m}$ and

$$b_{\mathrm{ij}}=\{\begin{array}{cc}{w}_{\mathrm{ij}}\hfill & \mathrm{if}\mathrm{transcription}\mathrm{factor}j\mathrm{is}\mathrm{an}\mathrm{activator}\mathrm{of}\mathrm{gene}i,\hfill \\ \multicolumn{1}{c}{0}& \mathrm{if}\mathrm{there}\mathrm{is}\mathrm{no}\mathrm{link}\mathrm{from}\mathrm{node}j\mathrm{to}i,\hfill \\ \multicolumn{1}{c}{-w_{\mathrm{ij}}}& \mathrm{if}\mathrm{transcription}\mathrm{factor}j\mathrm{is}\mathrm{a}\mathrm{repressor}\hfill \\ \mathrm{of}\mathrm{gene}i,i,j=1,2,\dots ,m.\hfill \end{array}$$

In biological systems, the concentration of the constituents is not uniform, resulting in diffusion of cytoplasm from higher to lower concentrations, which is known as diffusion. The gene regulatory networks with reaction diffusions were formulated and have shown significant prospects for spatio-temporal pattern storage and matching, refer Liang et al. (2020) and Li et al. (2017). Recently, stochastic models have been broadly investigated as they are commonly found in people’s daily lives. Stochastic perturbations not only separate the models from deterministic cases, but also can bring about substantial modifications in dynamic actions of gene regulatory networks, refer Coulier et al. (2021) and Xu et al. (2020b). In general, the behaviors of stochastic systems are highly reliant on time and spatial. Consequently, reaction diffusion is necessary to be taken into account, and this induces the investigations of stochastic reaction diffusion systems, refer the researching topics on the exponentially stable in the mean-square sense (Wei et al., 2019) and chaos synchronization (Tai et al., 2019), etc.

Discrete-time neural networks are better fitted for real-time implementations. There are two advantages to working in a discrete-time framework. First, appropriate technology can be used to implement digital controllers rather than analog ones. Second, the synthesized controller is directly implemented in a digital processor. Therefore, control methodologies developed for discrete-time nonlinear systems can be implemented in real systems more effectively (Garrappa and Popolizio, 2011; Shao et al., 2022; Yang et al., 2022; Zhang and Xu, 2019; Zhang et al., 2020; Znidi et al., 2022). A number of processes have a certain regularity; however, they are not completely periodic. So, the study of almost periodic or automorphic sequence has been a significant and interesting topic in the field of difference equations owing to the intensive evolution of the theories of difference equations and the applications in the areas of science and engineering, please refer Li and Shen (2021), Lizama and Mesquita (2013), Aouiti et al. (2021), Xu et al. (2020a) and Feng and Zong (2018) for some relevant works in this field.

For all the authors know, up to now, there are few papers focusing on the study of almost periodicity or automorphy to genetic regulatory networks in literatures (Aouiti and Dridi, 2021; Ayachi, 2021, 2022; Duan et al., 2020; Stamova and Stamov, 2021). Furthermore, almost no paper discusses almost automorphic oscillations of fractional-order stochastic gene regulatory networks with reaction diffusions, and we can observe that almost all of the existing findings are about the time variable. Motivated by the above analysis, by introducing the space variable, this paper discusses the nonlocal stochastic genetic regulatory networks with reaction diffusions in the formula of

$$\{\begin{array}{c}{{}^{c}D}_{t_p}^{{\alpha }_i}{\mathcal{M}}_i(t,x)=\sum _{q=1}^{\mathcal{N}}\frac{\partial }{\partial x_q}\left[D_{\mathrm{iq}}\frac{\partial {\mathcal{M}}_i(t,x)}{\partial x_q}\right]-a_i(x){\mathcal{M}}_i(t,x)\hfill \\ \multicolumn{1}{c}{+\sum _{j=1}^m{b}_{\mathrm{ij}}(t,x)}\\ \multicolumn{1}{c}{\times f_j({\mathcal{P}}_j(t,x))+\sum _{j=1}^m{\omega }_{\mathrm{ij}}(t,x){\sigma }_{\mathrm{ij}}({\mathcal{P}}_j(t,x))\frac{\mathrm{d}{\mathbb{W}}_{1j}(t)}{\mathrm{d}t}}\\ \multicolumn{1}{c}{+I_i(t,x)}\\ \multicolumn{1}{c}{{{}^{c}D}_{t_p}^{{\beta }_i}{\mathcal{P}}_i(t,x)=\sum _{q=1}^{\mathcal{N}}\frac{\partial }{\partial x_q}\left[K_{\mathrm{iq}}\frac{\partial {\mathcal{P}}_i(t,x)}{\partial x_q}\right]-c_i(x){\mathcal{P}}_i(t,x)}\\ \multicolumn{1}{c}{+d_i(t,x){\mathcal{M}}_i(t,x)}\\ \multicolumn{1}{c}{+\sum _{j=1}^m{\varpi }_{\mathrm{ij}}(t,x){\eta }_{\mathrm{ij}}({\mathcal{M}}_j(t,x))\frac{\mathrm{d}{\mathbb{W}}_{2j}(t)}{\mathrm{d}t}}\end{array}$$

(2)

where $(t,x)\in ({\mathcal{J}}_p,\mathrm{\Omega })$ , ${\mathcal{J}}_p=(t_p,t_{p+1}]$ , $t_0=0$ , $\underset{p\to \pm \infty }{lim}{t}_p=\pm \infty$ , $t_p\le t_{p+1}$ , $p\in \mathbb{Z}$ ; $\mathrm{\Omega }=\{x=(x_1,x_2,\dots ,x_{\mathcal{N}}{)}^T\in {\mathbb{R}}^{\ell }:r_{1q}<x_q<r_{2q},r_{1q},r_{2q}\in \mathbb{R},q=1,2,\dots ,\mathcal{N}\}$ ; ${{}^{c}D}_{*}^{{\alpha }_i}$ and ${{}^{c}D}_{*}^{{\beta }_i}$ denote Caputo fractional-order derivatives from initial point $0$ , ${\alpha }_i,{\beta }_i\in (0,1]$ ; $D:=(D_{\mathrm{iq}}{)}_{m\times \mathcal{N}}\ge 0$ and $K:=(K_{\mathrm{iq}}{)}_{m\times \mathcal{N}}\ge 0$ stand for the transmission diffusion matrixes; ${\mathbb{W}}_{1j}$ and ${\mathbb{W}}_{2j}$ denote the Brownian motion on a complete probability space, $i,j=1,2,\dots ,m$ .

The corresponding initial boundary conditions of model (2) are depicted as

$$\{\begin{array}{c}{{\mathcal{M}}_i(0,x)={\varphi }_i(x),x\in \mathrm{\Omega };{\mathcal{M}}_i(t,x)|}_{x\in \partial \mathrm{\Omega }}={\phi }_i(t,x)\hfill \\ \multicolumn{1}{c}{{{\mathcal{P}}_i(0,x)={\tilde{\varphi }}_i(x),x\in \mathrm{\Omega };{\mathcal{P}}_i(t,x)|}_{x\in \partial \mathrm{\Omega }}={\tilde{\phi }}_i(t,x)}\end{array}$$

(3)

where $t\in \mathbb{R},i=1,2,\dots ,m.$

By employing a mix of the finite difference methods and Mittag–Leffler Euler time difference techniques, the objective of the current paper is to achieve the discrete-time and discrete-space schemes corresponding to model (2). In addition, on this basis, the existence and uniqueness of bounded almost automorphic sequence solution in distribution for equation (7) are achieved. Then, the global exponential stability in the mean-square sense is investigated. Compared with the previous literatures, the distinct characteristics of this article are narrated as follows:

1. Based on the finite difference and Mittag–Leffler Euler difference techniques, a novel stochastic lattice model for model (2) is introduced.

2. The existence of a unique bounded almost automorphic sequence solution in distribution is discussed.

3. Global exponential convergence in the mean-square sense is considered.

4. The research findings in this article extend and complement the works in literatures (Aouiti and Dridi, 2021; Ayachi, 2021, 2022; Duan et al., 2020; Stamova and Stamov, 2021; Zhang et al., 2020; Zhang and Xu, 2019).

The organization of the rest is as follows. In “Stochastic lattice networks” section, a stochastic lattice model corresponding model (2) is achieved by using the finite difference methods and Mittag–Leffler Euler difference techniques. The existence of a unique bounded almost automorphic sequence solution in distribution and global exponential convergence in the mean-square sense are discussed in “Almost automorphic motions” and “Global mean-square $\lambda$ -exponential convergence” sections. In “Illustrative example” section, an illustrative example and some numerical simulations are employed to visually expound the current research findings. The conclusions and future works of this paper are presented in “Conclusion and perspectives” section.

Symbols: ${\mathbb{R}}^m$ denotes the space of m-dimensional real vectors; $\mathbb{Z}$ is the field of integral numbers; ${\mathbb{N}}_0=\{0,1,2,\dots \}$ ; $\mathbb{N}={\mathbb{N}}_0\backslash \{0\}$ ; ${\mathbb{N}}_a^b=\{a,a+1,\dots ,b\}$ for any $a,b\in \mathbb{Z}$ ; $I_J=I\cap J$ , $\forall I,J\subseteq \mathbb{R}$ . Let $A_1,A_2,\dots ,A_{\mathcal{N}}$ be some sets, $x_1\in A_1,x_2\in A_2,\dots ,x_{\mathcal{N}}\in A_{\mathcal{N}}\iff (x_1,x_2,\dots ,x_{\mathcal{N}})\in (A_1,A_2,\dots ,A_{\mathcal{N}})$ .

Stochastic lattice networks

Let us introduce the relative conception of fractional calculus as below. The $\alpha$ -order Caputo fractional derivative of $x\in C^n([a,b],{\mathbb{R}}^n)$ is defined by

$${{}^{c}D}_0^{\alpha }x(t)=\frac{1}{\mathrm{\Gamma }(n-\alpha )}\underset{a}{\overset{t}{\int }}\frac{x^{(n)}(s)}{{(t-s)}^{-}n+1}\mathrm{ds},t[a,b]$$

where $0<n-1<\alpha <n,n\in {\mathbb{Z}}_0$ . The Riemann–Liouville fractional integral of $x\in C([a,b],\mathbb{R})$ is given by

$$I_a^{\alpha }x(t)=\frac{1}{\mathrm{\Gamma }(\alpha )}\underset{a}{\overset{t}{\int }}{(t-s)}^{\alpha -1}x(s)\mathrm{d}s,\forall t\in [a,b]$$

where $\alpha >0$ .

The Mittag–Leffler functions play a vital role in this paper, we now present the definitions of one-parameter and two-parameter Mittag–Leffler functions and some properties which will be used in the sequel.

The Mittag–Leffler functions are described as

$${\mathbf{E}}_{\alpha }(z):=\sum _{k=0}^{\infty }\frac{z^k}{\mathrm{\Gamma }(\alpha k+1)},{\mathbf{E}}_{\alpha ,\beta }(z):=\sum _{k=0}^{\infty }\frac{z^k}{\mathrm{\Gamma }(\alpha k+\beta )}$$

where $z,\beta \in \mathbb{R}$ and $\alpha >0$ .

Lemma 1. (Zhang et al., 2021) If $\alpha \in (0,1]$ and $\lambda \in \mathbb{R}$ , then

1. ${\mathbf{E}}_{\alpha }(0)=1$ , ${\mathbf{E}}_{\alpha ,\alpha }(0)=1/\mathrm{\Gamma }(\alpha )$ .

2. ${\mathbf{E}}_{\alpha }(\lambda t^{\alpha })\in (0,1)$ for $\lambda <0$ and ${\mathbf{E}}_{\alpha }(\lambda t^{\alpha })\in (1,+\infty )$ for $\lambda >0$ , $\forall t>0$ .

3. ${\mathbf{E}}_{\alpha }(\lambda t_1^{\alpha })>{\mathbf{E}}_{\alpha }(\lambda t_2^{\alpha })$ and ${\mathbf{E}}_{\alpha ,\alpha }(\lambda t_1^{\alpha })>{\mathbf{E}}_{\alpha ,\alpha }(\lambda t_2^{\alpha })$ for $\lambda <0$ and $t_1<t_2$ .

4. ${\mathbf{E}}_{\alpha }(\lambda t_1^{\alpha })<{\mathbf{E}}_{\alpha }(\lambda t_2^{\alpha })$ and ${\mathbf{E}}_{\alpha ,\alpha }(\lambda t_1^{\alpha })<{\mathbf{E}}_{\alpha ,\alpha }(\lambda t_2^{\alpha })$ for $\lambda >0$ and $t_1<t_2$ .

Let $h_q=(r_{2q}-r_{1q})/N_q$ for some $N_q\in \mathbb{N}$ , $q=1,2,\dots ,\mathcal{N}$ . Define $x_q^{{\ell }_q}=r_{1q}+{\ell }_q{h}_q$ for all $({\ell }_q,q)\in ({\mathbb{N}}_0^{N_q},{\mathbb{N}}_1^{\mathcal{N}}).$ Set $\partial {\mathrm{\Omega }}_{\mathrm{d}}={\overline{\mathrm{\Omega }}}_{\mathrm{d}}\backslash {\mathrm{\Omega }}_{\mathrm{d}}$ , where

$$\begin{array}{c}{\overline{\mathrm{\Omega }}}_{\mathrm{d}}=\{x^{\ell }=(x_1^{{\ell }_1},\dots ,x_{\mathcal{N}}^{{\ell }_{\mathcal{N}}}{)}^T\in {\mathbb{R}}^{\mathcal{N}}:x_q^{{\ell }_q}=r_{1q}+{\ell }_q{h}_q,({\ell }_q,q)\in ({\mathbb{N}}_0^{N_q},{\mathbb{N}}_1^{\mathcal{N}})\}\hfill \\ {\mathrm{\Omega }}_{\mathrm{d}}=\{x^{\ell }=(x_1^{{\ell }_1},\dots ,x_{\mathcal{N}}^{{\ell }_{\mathcal{N}}}{)}^T\in {\mathbb{R}}^{\mathcal{N}}:x_q^{{\ell }_q}=r_{1q}+{\ell }_q{h}_q,({\ell }_q,q)\in ({\mathbb{N}}_1^{N_q-1},{\mathbb{N}}_1^{\mathcal{N}})\}\hfill \end{array}$$

By employing the finite difference methods, it gets

$$\{\begin{array}{c}\multicolumn{1}{c}{{\frac{{\partial }^2{\mathcal{M}}_i(t,x)}{\partial x_q^2}|}_{x=x^{\ell }}\approx \frac{{\mathcal{M}}_i(t,x^{\ell }+h_q{\mathbf{e}}_q)-2{\mathcal{M}}_i(t,x^{\ell })+{\mathcal{M}}_i(t,x^{\ell }-h_q{\mathbf{e}}_q)}{h_q^2}}\\ \multicolumn{1}{c}{{\frac{{\partial }^2{\mathcal{P}}_i(t,x)}{\partial x_q^2}|}_{x=x^{\ell }}\approx \frac{{\mathcal{P}}_i(t,x^{\ell }+h_q{\mathbf{e}}_q)-2{\mathcal{P}}_i(t,x^{\ell })+{\mathcal{P}}_i(t,x^{\ell }-h_q{\mathbf{e}}_q)}{h_q^2}}\end{array}$$

(4)

where $(t,x^{\ell })\in (\mathbb{R},{\mathrm{\Omega }}_{\mathrm{d}})$ and $(i,q)\in ({\mathbb{N}}_1^m,{\mathbb{N}}_1^{\mathcal{N}})$ , $\{{\mathbf{e}}_q:q=1,2,\dots ,\mathcal{N}\}$ is an orthonormal basis of ${\mathbb{R}}^{\mathcal{N}}$ denoted by

$${\mathbf{e}}_1=(1,0,\dots ,0{)}^T,{\mathbf{e}}_2=(0,1,\dots ,0{)}^T,{\mathbf{e}}_{\mathcal{N}}=(0,0,\dots ,1{)}^T$$

Making use of $(1)$ , equation (5) is calculated approximately by

$$\{\begin{array}{c}{{}^{c}D}_{t_p}^{{\alpha }_i}{\mathcal{M}}_i(t,x^{\ell })\approx -a_i^{*}(x^{\ell }){\mathcal{M}}_i(t,x^{\ell })+{\mathrm{\Upsilon }}_i(t,{\mathcal{M}}_i)\hfill \\ \multicolumn{1}{c}{+\sum _{j=1}^m{b}_{\mathrm{ij}}(t,x^{\ell })f_j({\mathcal{P}}_j(t,x^{\ell }))}\\ \multicolumn{1}{c}{+\sum _{j=1}^m{\omega }_{\mathrm{ij}}(t,x^{\ell }){\sigma }_{\mathrm{ij}}({\mathcal{P}}_j(t,x^{\ell }))\frac{\mathrm{d}{\mathbb{W}}_{1j}(t)}{\mathrm{d}t}+I_i(t,x^{\ell }),}\\ \multicolumn{1}{c}{{{}^{c}D}_{t_p}^{{\beta }_i}{\mathcal{P}}_i(t,x^{\ell })\approx -c_i^{*}(x^{\ell }){\mathcal{P}}_i(t,x^{\ell })+{\tilde{\mathrm{\Upsilon }}}_i(t,{\mathcal{P}}_i)+d_i(t,x^{\ell }){\mathcal{M}}_i(t,x^{\ell })}\\ \multicolumn{1}{c}{+\sum _{j=1}^m{\varpi }_{\mathrm{ij}}(t,x^{\ell }){\eta }_{\mathrm{ij}}({\mathcal{M}}_j(t,x^{\ell }))\frac{\mathrm{d}{\mathbb{W}}_{2j}(t)}{\mathrm{d}t},}\end{array}$$

(5)

where $a_i^{*}(x^{\ell })=a_i(x^{\ell })+2\sum _{q=1}^{\mathcal{N}}{D}_{\mathrm{iq}}/h_q^2,$ $c_i^{*}(x^{\ell })=c_i(x^{\ell })+2\sum _{q=1}^{\mathcal{N}}2K_{\mathrm{iq}}/h_q^2$

$$\begin{array}{c}{\mathrm{\Upsilon }}_i(t,{\mathcal{M}}_i)=\sum _{q=1}^{\mathcal{N}}\frac{D_{\mathrm{iq}}}{h_q^2}[{\mathcal{M}}_i(t,x^{\ell }+h_q{\mathbf{e}}_q)+{\mathcal{M}}_i(t,x^{\ell }-h_q{\mathbf{e}}_q)]\hfill \\ {\tilde{\mathrm{\Upsilon }}}_i(t,{\mathcal{P}}_i)=\sum _{q=1}^{\mathcal{N}}\frac{K_{\mathrm{iq}}}{h_q^2}[{\mathcal{P}}_i(t,x^{\ell }+h_q{\mathbf{e}}_q)+{\mathcal{P}}_i(t,x^{\ell }-h_q{\mathbf{e}}_q)]\hfill \end{array}$$

$\forall (t,x^{\ell },p,i)\in ({\mathcal{J}}_p,{\mathrm{\Omega }}_{\mathrm{d}},\mathbb{Z},{\mathbb{N}}_1^m).$ Besides, equation (6) is estimated approximately by

$$\{\begin{array}{c}{{\mathcal{M}}_i(0,x^{\ell })={\varphi }_i(x^{\ell }),x^{\ell }\in {\mathrm{\Omega }}_{\mathrm{d}};{\mathcal{M}}_i(t,x^{\ell })|}_{x\in \partial {\mathrm{\Omega }}_{\mathrm{d}}}={\phi }_i(t,x^{\ell })\hfill \\ \multicolumn{1}{c}{{{\mathcal{P}}_i(0,x^{\ell })={\tilde{\varphi }}_i(x^{\ell }),x^{\ell }\in {\mathrm{\Omega }}_{\mathrm{d}};{\mathcal{P}}_i(t,x^{\ell })|}_{x\in \partial {\mathrm{\Omega }}_{\mathrm{d}}}={\tilde{\phi }}_i(t,x^{\ell })}\end{array}$$

(6)

where $t\in \mathbb{R},i\in {\mathbb{N}}_1^m.$

Assume that it exists a set $\mathcal{Q}=\{Q_p\in \mathbb{Z}:Q_0=0,Q_p<Q_{p+1},p\in \mathbb{Z}\}$ ensuring $(t_{p+1}-t_p)/(Q_{p+1}-Q_p)\equiv h$ for all $p\in \mathbb{Z}$ . Based on equations (5) and (6) and by using the exponential time difference techniques in Garrappa and Popolizio (2011), it obtains the lattice model of equation (5) below

$$\{\begin{array}{c}{\mathcal{M}}_i^{x^{\ell }}(k+1)={\lambda }_{{\alpha }_i}^{x^{\ell }}(k){\mathcal{M}}_i^{x^{\ell }}(k)+\sum _{l={\mu }_k}^k{\theta }_{{\alpha }_i}^{x^{\ell }}(k-l,k)[{\mathrm{\Upsilon }}_i({\mathcal{M}}_i^{x^{\ell }}(l))\hfill \\ \multicolumn{1}{c}{+\sum _{j=1}^m{b}_{\mathrm{ij}}^{x^{\ell }}(l)}\\ \multicolumn{1}{c}{\times f_j({\mathcal{P}}_j^{x^{\ell }}(l))+\sum _{j=1}^m{\omega }_{\mathrm{ij}}^{x^{\ell }}(l){\sigma }_{\mathrm{ij}}}\\ \multicolumn{1}{c}{({\mathcal{P}}_j^{x^{\ell }}(l))h^{-1}{\mathrm{\Delta }}_h{\mathbb{W}}_{1j}(l)+I_i^{x^{\ell }}(l)],}\\ \multicolumn{1}{c}{{\mathcal{P}}_i^{x^{\ell }}(k+1)={\lambda }_{{\beta }_i}^{x^{\ell }}(k){\mathcal{P}}_i^{x^{\ell }}(k)+\sum _{l={\mu }_k}^k{\theta }_{{\beta }_i}^{x^{\ell }}(k-l,k)}\\ \multicolumn{1}{c}{[{\tilde{\mathrm{\Upsilon }}}_i({\mathcal{P}}_i^{x^{\ell }}(l))+d_i^{x^{\ell }}(l)}\\ \multicolumn{1}{c}{\times {\mathcal{M}}_i^{x^{\ell }}(l)+\sum _{j=1}^m{\varpi }_{\mathrm{ij}}^{x^{\ell }}(l){\eta }_{\mathrm{ij}}({\mathcal{M}}_j^{x^{\ell }}(l))h^{-1}{\mathrm{\Delta }}_h{\mathbb{W}}_{2j}(l)],}\\ \multicolumn{1}{c}{x^{\ell }\in {\mathrm{\Omega }}_{\mathrm{d}}}\end{array}$$

(7)

with boundary conditions

$$\begin{array}{c}\multicolumn{1}{c}{{{\mathcal{M}}_i^{x^{\ell }}(k)|}_{x^{\ell }\in \partial {\mathrm{\Omega }}_{\mathrm{d}}}={\phi }_{i,k}^{x^{\ell }}:={\phi }_i(\mathrm{kh},x^{\ell }),{{\mathcal{P}}_i^{x^{\ell }}(k)|}_{x^{\ell }\in \partial {\mathrm{\Omega }}_{\mathrm{d}}}}\\ \multicolumn{1}{c}{={\tilde{\phi }}_{i,k}^{x^{\ell }}:={\tilde{\phi }}_i(\mathrm{kh},x^{\ell })}\end{array}$$

where ${\mu }_k=\underset{p\in \mathbb{Z}}{max}\{Q_p:Q_p\le k\},{\nu }_k=k-{\mu }_k,$

$$\begin{array}{c}\multicolumn{1}{c}{{\mathcal{M}}_i^{x^{\ell }}(k)={\mathcal{M}}_i(\mathrm{kh},x^{\ell }),{\mathcal{P}}_i^{x^{\ell }}(k)={\mathcal{P}}_i(\mathrm{kh},x^{\ell }),}\\ \multicolumn{1}{c}{{\mathrm{\Upsilon }}_i({\mathcal{M}}_i^{x^{\ell }}(k))={\mathrm{\Upsilon }}_i(\mathrm{kh},{\mathcal{M}}_i^{x^{\ell }}(k)),}\end{array}$$

$$\begin{array}{c}\multicolumn{1}{c}{{\tilde{\mathrm{\Upsilon }}}_i({\mathcal{P}}_i^{x^{\ell }}(k))={\tilde{\mathrm{\Upsilon }}}_i(\mathrm{kh},{\mathcal{P}}_i^{x^{\ell }}(k)),b_{\mathrm{ij}}^{x^{\ell }}(k)=b_{\mathrm{ij}}(\mathrm{kh},x^{\ell }),}\\ \multicolumn{1}{c}{{\omega }_{\mathrm{ij}}^{x^{\ell }}(k)={\omega }_{\mathrm{ij}}(\mathrm{kh},x^{\ell }),}\end{array}$$

$$I_i^{x^{\ell }}(k)=I_i(\mathrm{kh},x^{\ell }),d_i^{x^{\ell }}(k)=d_i(\mathrm{kh},x^{\ell }),{\varpi }_{\mathrm{ij}}^{x^{\ell }}(k)={\varpi }_{\mathrm{ij}}(\mathrm{kh},x^{\ell }),$$

$$\begin{array}{c}\multicolumn{1}{c}{{\mathrm{\Delta }}_h{\mathbb{W}}_{1j}(k)={\mathbb{W}}_{1j}(\mathrm{kh}+h)-{\mathbb{W}}_{1j}(\mathrm{kh}),{\mathrm{\Delta }}_h{\mathbb{W}}_{2j}(k)}\\ \multicolumn{1}{c}{={\mathbb{W}}_{2j}(\mathrm{kh}+h)-{\mathbb{W}}_{2j}(\mathrm{kh}),}\end{array}$$

$$\begin{array}{c}\multicolumn{1}{c}{{\lambda }_{{\alpha }_i}^{x^{\ell }}(k)=\frac{{\mathbf{E}}_{{\alpha }_i}[-a_i^{*}(x^{\ell })({\nu }_{k}h+h{)}^{{\alpha }_i}]}{{\mathbf{E}}_{{\alpha }_i}[-a_i^{*}(x^{\ell })({\nu }_{k}h{)}^{{\alpha }_i}]},}\\ \multicolumn{1}{c}{{\lambda }_{{\beta }_i}^{x^{\ell }}(k)=\frac{{\mathbf{E}}_{{\beta }_i}[-c_i^{*}(x^{\ell })({\nu }_{k}h+h{)}^{{\beta }_i}]}{{\mathbf{E}}_{{\beta }_i}[-c_i^{*}(x^{\ell })({\nu }_{k}h{)}^{{\beta }_i}]},}\end{array}$$

$${\theta }_{{\alpha }_i}^{x^{\ell }}(l^{+},k)=\frac{1}{a_i^{*}(x^{\ell })}[w_{{\alpha }_i}^{x^{\ell }}(l^{+})-{\lambda }_{{\alpha }_i}^{x^{\ell }}(k)w_{{\alpha }_i}^{x^{\ell }}(l^{+}-1)],$$

$${\theta }_{{\beta }_i}^{x^{\ell }}(l^{+},k)=\frac{1}{c_i^{*}(x^{\ell })}[w_{{\beta }_i}^{x^{\ell }}(l^{+})-{\lambda }_{{\beta }_i}^{x^{\ell }}(k)w_{{\beta }_i}^{x^{\ell }}(l^{+}-1)],$$

$$\begin{array}{c}\multicolumn{1}{c}{w_{{\alpha }_i}^{x^{\ell }}(-1)=w_{{\beta }_i}^{x^{\ell }}(-1)=0,w_{{\alpha }_i}^{x^{\ell }}(l^{+})={\mathbf{E}}_{{\alpha }_i}[-a_i^{*}(x^{\ell })(l^{+}h{)}^{{\alpha }_i}]}\\ \multicolumn{1}{c}{-{\mathbf{E}}_{{\alpha }_i}[-a_i^{*}(x^{\ell })(l^{+}h+h{)}^{{\alpha }_i}],}\end{array}$$

$$w_{{\beta }_i}^{x^{\ell }}(l^{+})={\mathbf{E}}_{{\beta }_i}[-c_i^{*}(x^{\ell })(l^{+}h{)}^{{\beta }_i}]-{\mathbf{E}}_{{\beta }_i}[-c_i^{*}(x^{\ell })(l^{+}h+h{)}^{{\beta }_i}]$$

for all $(x^{\ell },l^{+},k,i,j)\in ({\overline{\mathrm{\Omega }}}_{\mathrm{d}},{\mathbb{N}}_0,\mathbb{Z},{\mathbb{N}}_1^m,{\mathbb{N}}_1^m).$ The initial conditions of equation (7) can be rewritten as

$$\begin{array}{c}\multicolumn{1}{c}{{\mathcal{M}}_i^{x^{\ell }}(0)={\varphi }_i(x^{\ell }):={\varphi }_i^{x^{\ell }},{\mathcal{P}}_i^{x^{\ell }}(0)={\tilde{\varphi }}_i(x^{\ell }):={\tilde{\varphi }}_i^{x^{\ell }},}\\ \multicolumn{1}{c}{\forall (x^{\ell },i)\in ({\mathrm{\Omega }}_{\mathrm{d}},{\mathbb{N}}_1^n)}\end{array}$$

From Lemma 2.3 in literature (Zhang and Xu, 2019), the lemma below holds.

• Lemma 2. Equation (7) can be transformed into

$$\{\begin{array}{c}{\mathcal{M}}_i^{x^{\ell }}(k)=\underset{s=k_0}{\overset{k-1}{\Pi }}{\lambda }_{{\alpha }_i}^{x^{\ell }}(s){\mathcal{M}}_i^{x^{\ell }}(k_0)+\sum _{q=k_0}^{k-1}\underset{s=q+1}{\overset{k-1}{\Pi }}{\lambda }_{{\alpha }_i}^{x^{\ell }}(s)\sum _{l={\mu }_q}^q{\theta }_{{\alpha }_i}^{x^{\ell }}(q-l,q)[{\mathrm{\Upsilon }}_i({\mathcal{M}}_i^{x^{\ell }}(l))\hfill \\ \multicolumn{1}{c}{+\sum _{j=1}^m{b}_{\mathrm{ij}}^{x^{\ell }}(l)f_j({\mathcal{P}}_j^{x^{\ell }}(l))+\sum _{j=1}^m{\omega }_{\mathrm{ij}}^{x^{\ell }}(l){\sigma }_{\mathrm{ij}}({\mathcal{P}}_j^{x^{\ell }}(l))h^{-1}{\mathrm{\Delta }}_h{\mathbb{W}}_{1j}(l)+I_i^{x^{\ell }}(l)],}\\ \multicolumn{1}{c}{{\mathcal{P}}_i^{x^{\ell }}(k)=\underset{s=k_0}{\overset{k-1}{\Pi }}{\lambda }_{{\beta }_i}^{x^{\ell }}(s){\mathcal{P}}_i^{x^{\ell }}(k_0)+\sum _{q=k_0}^{k-1}\underset{s=q+1}{\overset{k-1}{\Pi }}{\lambda }_{{\beta }_i}^{x^{\ell }}(s)\sum _{l={\mu }_q}^q{\theta }_{{\beta }_i}^{x^{\ell }}(q-l,q)[{\tilde{\mathrm{\Upsilon }}}_i({\mathcal{P}}_i^{x^{\ell }}(l))}\\ \multicolumn{1}{c}{+d_i^{x^{\ell }}(l){\mathcal{M}}_i^{x^{\ell }}(l)+\sum _{j=1}^m{\varpi }_{\mathrm{ij}}^{x^{\ell }}(l){\eta }_{\mathrm{ij}}({\mathcal{M}}_j^{x^{\ell }}(l))h^{-1}{\mathrm{\Delta }}_h{\mathbb{W}}_{2j}(l)],}\end{array}(8)$$

(8)

with boundary conditions ${{\mathcal{M}}_i^{x^{\ell }}(k)|}_{x^{\ell }\in \partial {\mathrm{\Omega }}_{\mathrm{d}}}={\phi }_{i,k}^{x^{\ell }},$ ${{\mathcal{P}}_i^{x^{\ell }}(k)|}_{x^{\ell }\in \partial {\mathrm{\Omega }}_{\mathrm{d}}}={\tilde{\phi }}_{i,k}^{x^{\ell }},$ where $\forall (k,i)\in ([k_0,+\infty {)}_{\mathbb{Z}},{\mathbb{N}}_1^m)$ , $k_0\in \mathbb{Z}$ , $x^{\ell }\in {\mathrm{\Omega }}_{\mathrm{d}}$ .

Let $Q^{*}=\underset{p\in \mathbb{Z}}{sup}(Q_{p+1}-Q_p),$ ${\overline{\lambda }}_{{\alpha }_i}=\underset{k\in \mathbb{Z},x^{\ell }\in {\mathrm{\Omega }}_{\mathrm{d}}}{sup}{\lambda }_{{\alpha }_i}^{x^{\ell }}(k),$ ${\overline{\lambda }}_{{\beta }_i}=\underset{k\in \mathbb{Z},x^{\ell }\in {\mathrm{\Omega }}_{\mathrm{d}}}{sup}{\lambda }_{{\beta }_i}^{x^{\ell }}(k),$ ${\underset{¯}{a}}_i^{*}=\underset{x^{\ell }\in {\mathrm{\Omega }}_{\mathrm{d}}}{inf}{a}_i^{*}(x^{\ell }),$ ${\underset{¯}{c}}_i^{*}=\underset{x^{\ell }\in {\mathrm{\Omega }}_{\mathrm{d}}}{inf}{c}_i^{*}(x^{\ell }),$ and $i\in {\mathbb{N}}_1^m.$

By Lemma 1 and a direct calculation, it easily gets the lemma below.

Lemma 3. If $Q^{*}<+\infty$ , ${\alpha }_i,{\beta }_i\in (0,1]$ and ${\underset{¯}{a}}_i^{*},{\underset{¯}{c}}_i^{*}>0$ , one has

$$\begin{array}{c}0<{\lambda }_{{\alpha }_i}^{x^{\ell }}(k)\le {\overline{\lambda }}_{{\alpha }_i}<1,0<{\lambda }_{{\beta }_i}^{x^{\ell }}(k)\le {\overline{\lambda }}_{{\beta }_i}<1,\hfill \\ \sum _{l=0}^{Q^{*}}|{\theta }_{{\alpha }_i}^{x^{\ell }}(l,k)|\le \frac{2}{{\underset{¯}{a}}_i^{*}},\sum _{l=0}^{Q^{*}}|{\theta }_{{\beta }_i}^{x^{\ell }}(l,k)|\le \frac{2}{{\underset{¯}{c}}_i^{*}}\hfill \end{array}$$

for all $(x^{\ell },k,i)\in ({\overline{\mathrm{\Omega }}}_{\mathrm{d}},\mathbb{Z},{\mathbb{N}}_1^m)$

Almost automorphic motions

This section introduces some usual notations and concepts concerning the almost automorphic sequence, as well as several crucial inequalities.

Let $\mathbb{E}(·)$ be the expectation under a complete probability space $(\Lambda ,\mathcal{F},\mathbb{P})$ and $\mathbb{W}=({\mathbb{W}}_{11},\dots ,{\mathbb{W}}_{1m},{\mathbb{W}}_{21},\dots ,{\mathbb{W}}_{2m}{)}^T$ be a two-sided standard $2m$ -dimensional Brownian motion defined on $(\Lambda ,\mathcal{F},\mathbb{P})$ . Set ${\mathcal{F}}_t=\sigma \{\mathbb{W}(s):s\le t\}$ for $t\in \mathbb{R}$ . Furthermore, $L^2(\Lambda ,{\mathbb{R}}^{2m})$ denotes the family of all square integrable ${\mathbb{R}}^{2m}$ -valued random variables and $\mathbb{X}=\mathbb{B}(\mathbb{Z}\times {\overline{\mathrm{\Omega }}}_{\mathrm{d}},L^2(\Lambda ,{\mathbb{R}}^{2m}))$ stands for the set of all functions from $\mathbb{Z}\times {\overline{\mathrm{\Omega }}}_{\mathrm{d}}$ to $L^2(\Lambda ,{\mathbb{R}}^{2m})$ endowed with the norm

$$\begin{array}{c}\multicolumn{1}{c}{\parallel \mathcal{Z}{\parallel }_{\mathbb{X}}=\underset{k\in \mathbb{Z}}{sup}\underset{1\le i\le m,x^{\ell }\in {\overline{\mathrm{\Omega }}}_{\mathrm{d}}}{max}max}\\ \multicolumn{1}{c}{\{{\left[\mathbb{E}{|{\mathcal{M}}_i^{x^{\ell }}(k)|}^2\right]}^{1/2},{\left[\mathbb{E}{|{\mathcal{P}}_i^{x^{\ell }}(k)|}^2\right]}^{1/2}\},}\end{array}$$

where $\mathcal{Z}=({\mathcal{M}}_1,\dots ,{\mathcal{M}}_m,{\mathcal{P}}_1,\dots ,{\mathcal{P}}_m{)}^T\in \mathbb{X}.$ Obviously, $(\mathbb{X},\parallel ·{\parallel }_{\mathbb{X}})$ becomes a Banach space. In the whole paper, ${\varphi }_i^{x^{\ell }},{\tilde{\varphi }}_i^{x^{\ell }}$ are ${\mathcal{F}}_0$ -adapted and ${\phi }_{i,k}^{x^{\ell }},{\tilde{\phi }}_{i,k}^{x^{\ell }}$ are ${\mathcal{F}}_k$ -adapted, $\forall (x^{\ell },k,i)\in ({\overline{\mathrm{\Omega }}}_{\mathrm{d}},\mathbb{Z},{\mathbb{N}}_1^m).$

Definition 1. A discrete-time stochastic process $\mathcal{Z}=({\mathcal{M}}_1,\dots ,{\mathcal{M}}_m,{\mathcal{P}}_1,\dots ,{\mathcal{P}}_m{)}^T\in \mathbb{X}$ is called the solution of equation (7) if it is ${\mathcal{F}}_t$ -adapted and meets equation (8).

Definition 2. (Diagana, 2013) $f:\mathbb{Z}\to \mathbb{R}$ is called Bohr almost automorphic sequence if for each sequence $\{k{\prime }_n{\}}_{n\in \mathbb{Z}}$ , it has a sequence $\{k_n{\}}_{n\in \mathbb{Z}}\subseteq \{k{\prime }_n{\}}_{n\in \mathbb{Z}}$ and some function $\tilde{f}:\mathbb{Z}\to \mathbb{R}$ meeting

$$\tilde{f}(k)=\underset{n\to \infty }{lim}f(k+k_n),\underset{n\to \infty }{lim}\tilde{f}(k-k_n)=f(k)$$

uniformly for $k\in \mathbb{Z}$ on any compact subset from $\mathbb{Z}$ .

The random sequence $\{f_n\}$ converges to f in distribution if the corresponding laws of $\{f_n\}$ weakly converge to the law of f. The definition of weak convergence can be found in literature (Cheban and Liu, 2020).

• Definition 3. A mapping $f:\mathbb{Z}\to {\mathbb{R}}^{2m}$ is called Bohr almost automorphic sequence in distribution in case for each sequence $\{k{\prime }_n\}$ , if there exists a subsequence $\{k_n\}$ and some function $\tilde{f}$ such that $\{f(·+k_n)\}$ converges to some function $\tilde{f}(·)$ in distribution and $\{\tilde{f}(·-k_n)\}$ converges to $f(·)$ in distribution uniformly on any compact subset from $\mathbb{Z}$ .

Next, we collect some important inequalities and more details please refer to literatures (Kuang, 2012; Zhang and Xu, 2019).

Lemma 4. (Kuang, 2012) (Minkowski inequality) If $f,g\in L^2(\Lambda ,\mathbb{R})$ , then

$${\left(\mathbb{E}{(f+g)}^2\right)}^{\frac{1}{2}}\le {\left(\mathbb{E}{f}^2\right)}^{\frac{1}{2}}+{\left(\mathbb{E}{g}^2\right)}^{\frac{1}{2}}$$

Lemma 5. (Kuang, 2012) (Hölder inequality) Let $p>1$ and $a_k,b_k:\mathbb{Z}\to \mathbb{R}$ . Then

$$\sum _k|a_k{b}_k|\le {[\sum _k|a_k|]}^{1-1/p}{[\sum _k|a_k||b_k{|}^p]}^{1/p}$$

Lemma 6. (Zhang and Xu, 2019) Let $\{f(k){\}}_{k\in \mathbb{Z}}\subseteq L^2(\Lambda ,\mathbb{R})$ and $\{\mathbb{W}(t){\}}_{t\in \mathbb{R}}$ be a two-sided standard one-dimensional Brownian motion. Then

$$\mathbb{E}{|f(k){\mathrm{\Delta }}_h\mathbb{W}(k)|}^2\le 4h\mathbb{E}|f(k){|}^2$$

where ${\mathrm{\Delta }}_h\mathbb{W}(k)=\mathbb{W}(\mathrm{kh}+h)-\mathbb{W}(\mathrm{kh})$ , $\forall k\in \mathbb{Z}$

Here, we need the following assumptions.

• (H₁) $\{{\nu }_k\}$ is a periodic sequence, that is there exists $k_{*}\in \mathbb{Z}$ such that ${\nu }_{k+k_{*}}={\nu }_k$ , $\forall k\in \mathbb{Z}$ .

• (H₂) $b_{\mathrm{ij}}^{x^{\ell }}(k)$ , ${\omega }_{\mathrm{ij}}^{x^{\ell }}(k)$ , ${\varpi }_{\mathrm{ij}}^{x^{\ell }}(k)$ , $d_i^{x^{\ell }}(k)$ , and $I_i^{x^{\ell }}(k)$ are Bohr almost automorphic sequences with respect to variable $k\in \mathbb{Z}$ , $x^{\ell }\in {\mathrm{\Omega }}_{\mathrm{d}}$ , $i,j\in {\mathbb{N}}_1^m$ .

• (H₃) It exists positive numbers $L_j^f$ , $L_{\mathrm{ij}}^{\sigma }$ , and $L_{\mathrm{ij}}^{\eta }$ such that

  $$\begin{array}{c}\multicolumn{1}{c}{|f_j(x)-f_j(y)|\le L_j^f|x-y|,|{\sigma }_{\mathrm{ij}}(x)-{\sigma }_{\mathrm{ij}}(y)|}\\ \multicolumn{1}{c}{\le L_{\mathrm{ij}}^{\sigma }|x-y|,|{\eta }_{\mathrm{ij}}(x)-{\eta }_{\mathrm{ij}}(y)|\le L_{\mathrm{ij}}^{\eta }|x-y|}\end{array}$$

  for any $x,y\in \mathbb{R}$ , $i,j\in {\mathbb{N}}_1^m$

Define $\overline{f}=\underset{k\in \mathbb{Z},s\in {\mathrm{\Omega }}_{\mathrm{d}}}{sup}|f(k,s)|,$ which $f:=\mathbb{Z}\times {\mathrm{\Omega }}_{\mathrm{d}}\to \mathbb{R}$ is a sequence. Let $D_i^{*}=2\sum _{q=1}^{\mathcal{N}}|D_{\mathrm{iq}}|/h_q^2,$ $K_i^{*}=2\sum _{q=1}^{\mathcal{N}}|K_{\mathrm{iq}}|/h_q^2,$ $i\in {\mathbb{N}}_1^m;$ ${\mathcal{Z}}_0={\rho }_2/(1-{\rho }_1)$ ,

$$\begin{array}{c}{\rho }_1=\underset{1\le i\le m}{max}\{\frac{2}{{\underset{¯}{a}}_i^{*}(1-{\overline{\lambda }}_{{\alpha }_i})}[D_i^{*}+\sum _{j=1}^m{\overline{b}}_{\mathrm{ij}}{L}_j^f+2h^{-\frac{1}{2}}\sum _{j=1}^m{\overline{\omega }}_{\mathrm{ij}}{L}_{\mathrm{ij}}^{\sigma }],\hfill \\ \frac{2}{{\underset{¯}{c}}_i^{*}(1-{\overline{\lambda }}_{{\beta }_i})}[K_i^{*}+{\overline{d}}_i+2h^{-\frac{1}{2}}\sum _{j=1}^m{\overline{\varpi }}_{\mathrm{ij}}{L}_{\mathrm{ij}}^{\eta }]\},\hfill \end{array}$$

$$\begin{array}{l}{\rho }_2=\underset{1\le i\le m}{\max }\underset{k\in \mathbb{Z},x^{\ell }\in {\overline{\mathrm{\Omega }}}_{\mathrm{d}}}{\sup }\{{\left[\mathbb{E}|{\phi }_{i,k}^{x^{\ell }}{|}^2\right]}^{\frac{1}{2}},{\left[\mathbb{E}|{\tilde{\phi }}_{i,k}^{x^{\ell }}{|}^2\right]}^{\frac{1}{2}},\frac{2}{{\underset{\_}{a}}_i^{*}(1-{\overline{\lambda }}_{{\alpha }_i})}[\sum _{j=1}^m{\overline{b}}_{ij}|f_j(0)|\\ +2h^{-\frac{1}{2}}\sum _{j=1}^m{\overline{\omega }}_{ij}|{\sigma }_{ij}(0)|+{\overline{I}}_i],\frac{4h^{-\frac{1}{2}}}{{\underset{\_}{c}}_i^{*}(1-{\overline{\lambda }}_{{\beta }_i})}\sum _{j=1}^m{\overline{\varpi }}_{ij}|{\eta }_{ij}(0)|\}.\end{array}$$

Let ${\mathcal{X}}_0=\{\mathcal{Z}\in \mathbb{X}:\parallel \mathcal{Z}{\parallel }_{\mathbb{X}}\le {\mathcal{Z}}_0\}.$ In accordance with equation (8), define a mapping $\mathrm{\Psi }:{\mathcal{X}}_0\to \mathbb{X}$ as

$$\mathrm{\Psi }\mathcal{Z}=((\mathrm{\Psi }\mathcal{M}{)}_1,(\mathrm{\Psi }\mathcal{M}{)}_2,\dots ,(\mathrm{\Psi }\mathcal{M}{)}_m,(\mathrm{\Psi }\mathcal{P}{)}_1,(\mathrm{\Psi }\mathcal{P}{)}_2,\dots ,(\mathrm{\Psi }\mathcal{P}{)}_m{)}^T$$

where

$$\{\begin{array}{c}(\mathrm{\Psi }\mathcal{M}{)}_{i,k}^{x^{\ell }}=\sum _{q=-\infty }^{k-1}\underset{s=q+1}{\overset{k-1}{\Pi }}{\lambda }_{{\alpha }_i}^{x^{\ell }}(s)\sum _{l={\mu }_q}^q{\theta }_{{\alpha }_i}^{x^{\ell }}(q-l,q)[{\mathrm{\Upsilon }}_i({\mathcal{M}}_i^{x^{\ell }}(l))+\sum _{j=1}^m{b}_{\mathrm{ij}}^{x^{\ell }}(l)\hfill \\ \multicolumn{1}{c}{\times f_j({\mathcal{P}}_j^{x^{\ell }}(l))+\sum _{j=1}^m{\omega }_{\mathrm{ij}}^{x^{\ell }}(l){\sigma }_{\mathrm{ij}}({\mathcal{P}}_j^{x^{\ell }}(l))h^{-1}{\mathrm{\Delta }}_h{\mathbb{W}}_{1j}(l)+I_i^{x^{\ell }}(l)],}\\ \multicolumn{1}{c}{(\mathrm{\Psi }\mathcal{P}{)}_{i,k}^{x^{\ell }}=\sum _{q=-\infty }^{k-1}\underset{s=q+1}{\overset{k-1}{\Pi }}{\lambda }_{{\beta }_i}^{x^{\ell }}(s)\sum _{l={\mu }_q}^q{\theta }_{{\beta }_i}^{x^{\ell }}(q-l,q)[{\tilde{\mathrm{\Upsilon }}}_i({\mathcal{P}}_i^{x^{\ell }}(l))+d_i^{x^{\ell }}(l){\mathcal{M}}_i^{x^{\ell }}(l)}\\ \multicolumn{1}{c}{+\sum _{j=1}^m{\varpi }_{\mathrm{ij}}^{x^{\ell }}(l){\eta }_{\mathrm{ij}}({\mathcal{M}}_j^{x^{\ell }}(l))h^{-1}{\mathrm{\Delta }}_h{\mathbb{W}}_{2j}(l)],x^{\ell }\in {\mathrm{\Omega }}_{\mathrm{d}};n}\end{array}$$

(9)

$${(\mathrm{\Psi }\mathcal{M}{)}_i^{x^{\ell }}(k)|}_{x^{\ell }\in \partial {\mathrm{\Omega }}_{\mathrm{d}}}={{\phi }_{i,k}^{x^{\ell }},(\mathrm{\Psi }\mathcal{P}{)}_i^{x^{\ell }}(k)|}_{x^{\ell }\in \partial {\mathrm{\Omega }}_{\mathrm{d}}}={\tilde{\phi }}_{i,k}^{x^{\ell }},\forall (k,i)\in (\mathbb{Z},{\mathbb{N}}_1^m)$$

Proposition 1. $\mathrm{\Psi }$ is well defined and maps ${\mathcal{X}}_0$ to ${\mathcal{X}}_0$ if $(H_3)$ and $(H_4)$ below hold.

$$\left(H_4\right){\rho }_1<1.$$

Proof. Suppose that $\mathcal{Z}=({\mathcal{M}}_1,\dots ,{\mathcal{M}}_m,{\mathcal{P}}_1,\dots ,{\mathcal{P}}_m{)}^T\in {\mathcal{X}}_0$ . Based on equation (9), Lemmas 4, 5, and 6, it gets

$$\begin{array}{c}{\left[\mathbb{E}{|(\mathrm{\Psi }\mathcal{M}{)}_i^{x^{\ell }}(k)|}^2\right]}^{\frac{1}{2}}=\{\mathbb{E}[\sum _{q=-\infty }^{k-1}\underset{s=q+1}{\overset{k-1}{\Pi }}{\lambda }_{{\alpha }_i}^{x^{\ell }}(s)\sum _{l={\mu }_q}^q{\theta }_{{\alpha }_i}^{x^{\ell }}(q-l,q)\hfill \\ \times ({\mathrm{\Upsilon }}_i({\mathcal{M}}_i^{x^{\ell }}(l))+\sum _{j=1}^m{b}_{\mathrm{ij}}^{x^{\ell }}(l)f_j({\mathcal{P}}_j^{x^{\ell }}(l))\hfill \\ +\sum _{j=1}^m{\omega }_{\mathrm{ij}}^{x^{\ell }}(l){\sigma }_{\mathrm{ij}}({\mathcal{P}}_j^{x^{\ell }}(l))h^{-1}{\mathrm{\Delta }}_h{\mathbb{W}}_{1j}(l)+I_i^{x^{\ell }}(l))\left]2\right\}\frac{1}{2}\hfill \\ \le \{\mathbb{E}\{{[\sum _{q=-\infty }^{k-1}\underset{s=q+1}{\overset{k-1}{\Pi }}{\lambda }_{{\alpha }_i}^{x^{\ell }}(s)]}^{\frac{1}{2}}\{\sum _{q=-\infty }^{k-1}\underset{s=q+1}{\overset{k-1}{\Pi }}{\lambda }_{{\alpha }_i}^{x^{\ell }}(s)\hfill \\ \times [\sum _{l={\mu }_q}^q{\theta }_{{\alpha }_i}^{x^{\ell }}(q-l,q)({\mathrm{\Upsilon }}_i({\mathcal{M}}_i^{x^{\ell }}(l))+\sum _{j=1}^m{b}_{\mathrm{ij}}^{x^{\ell }}(l)f_j({\mathcal{P}}_j^{x^{\ell }}(l))\hfill \\ +\sum _{j=1}^m{\omega }_{\mathrm{ij}}^{x^{\ell }}(l){\sigma }_{\mathrm{ij}}({\mathcal{P}}_j^{x^{\ell }}(l))h^{-1}{\mathrm{\Delta }}_h{\mathbb{W}}_{1j}(l)+I_i^{x^{\ell }}(l))]2\left\}\frac{1}{2}\right\}2\}\frac{1}{2}\hfill \\ \le \sum _{q=-\infty }^{k-1}\underset{s=q+1}{\overset{k-1}{\Pi }}{\lambda }_{{\alpha }_i}^{x^{\ell }}(s)\{\mathbb{E}[\sum _{l={\mu }_q}^q{\theta }_{{\alpha }_i}^{x^{\ell }}(q-l,q)\hfill \end{array}$$

$$\begin{array}{c}({\mathrm{\Upsilon }}_i({\mathcal{M}}_i^{x^{\ell }}(l))+\sum _{j=1}^m{b}_{\mathrm{ij}}^{x^{\ell }}(l)f_j({\mathcal{P}}_j^{x^{\ell }}(l))\hfill \\ {{+\sum _{j=1}^m{\omega }_{\mathrm{ij}}^{x^{\ell }}(l){\sigma }_{\mathrm{ij}}({\mathcal{P}}_j^{x^{\ell }}(l))h^{-1}{\mathrm{\Delta }}_h{\mathbb{W}}_{1j}(l)+I_i^{x^{\ell }}(l))]}^2\}}^{\frac{1}{2}}\hfill \\ \le \frac{2}{-a_i^{*}(1-{\overline{\lambda }}_{{\alpha }_i})}\{[D_i^{*}+\sum _{j=1}^m{\overline{b}}_{\mathrm{ij}}{L}_j^f]\parallel \mathcal{Z}{\parallel }_{\mathbb{X}}\hfill \\ +\sum _{j=1}^m{\overline{b}}_{\mathrm{ij}}|f_j(0)|+2h^{-\frac{1}{2}}\sum _{j=1}^m{\overline{\omega }}_{\mathrm{ij}}|{\sigma }_{\mathrm{ij}}(0)|+{\overline{I}}_i\hfill \\ +\sum _{j=1}^m{\overline{\omega }}_{\mathrm{ij}}{L}_{\mathrm{ij}}^{\sigma }{\left[\mathbb{E}{|{\mathcal{P}}_j^{x^{\ell }}(l)h^{-1}{\mathrm{\Delta }}_h{\mathbb{W}}_{1j}(l)|}^2\right]}^{\frac{1}{2}}\}\hfill \\ \le \frac{2}{\mathrm{-}{a}_i^{*}(1-{\overline{\lambda }}_{{\alpha }_i})}[D_i^{*}+\sum _{j=1}^m{\overline{b}}_{\mathrm{ij}}{L}_j^f+2h^{-\frac{1}{2}}\sum _{j=1}^m{\overline{\omega }}_{\mathrm{ij}}{L}_{\mathrm{ij}}^{\sigma }]\parallel \mathcal{Z}{\parallel }_{\mathbb{X}}\hfill \\ +\frac{2}{\mathrm{-}{a}_i^{*}(1-{\overline{\lambda }}_{{\alpha }_i})}{[\sum _{j=1}^m{\overline{b}}_{\mathrm{ij}}|f_j(0)|+2h^{-\frac{1}{2}}\sum _{j=1}^m{\overline{\omega }}_{\mathrm{ij}}|{\sigma }_{\mathrm{ij}}(0)|+\overline{I}]}_i\hfill \\ \le {\rho }_1{\mathcal{X}}_0+{\rho }_2,\forall (\mathcal{Z},x^{\ell },k,i)\in ({\mathcal{X}}_0,{\mathrm{\Omega }}_{\mathrm{d}},\mathbb{Z},{\mathbb{N}}_1^m)\hfill \end{array}$$

and by a similar derivation as the above, it acquires

$$\begin{array}{c}{\left[\mathbb{E}{|(\mathrm{\Psi }\mathcal{P}{)}_i^{x^{\ell }}(k)|}^2\right]}^{\frac{1}{2}}=\{\mathbb{E}[\sum _{q=-\infty }^{k-1}\underset{s=q+1}{\overset{k-1}{\Pi }}{\lambda }_{{\beta }_i}^{x^{\ell }}(s)\sum _{l={\mu }_q}^q{\theta }_{{\beta }_i}^{x^{\ell }}(q-l,q)({\tilde{\mathrm{\Upsilon }}}_i({\mathcal{P}}_i^{x^{\ell }}(l))\hfill \\ \multicolumn{1}{c}{{{+d_i^{x^{\ell }}(l){\mathcal{M}}_i^{x^{\ell }}(l)+\sum _{j=1}^m{\varpi }_{\mathrm{ij}}^{x^{\ell }}(l){\eta }_{\mathrm{ij}}({\mathcal{M}}_j^{x^{\ell }}(l))h^{-1}{\mathrm{\Delta }}_h{\mathbb{W}}_{2j}(l))]}^2\}}^{\frac{1}{2}}}\\ \multicolumn{1}{c}{\le \frac{2}{\mathrm{-}{c}_i^{*}(1-{\overline{\lambda }}_{{\beta }_i})}[K_i^{*}+{\overline{d}}_i+2h^{-\frac{1}{2}}\sum _{j=1}^m{\overline{\varpi }}_{\mathrm{ij}}]L_{\mathrm{ij}}^{\eta }\parallel \mathcal{Z}{\parallel }_{\mathbb{X}}}\\ \multicolumn{1}{c}{+\frac{2}{\mathrm{-}{c}_i^{*}(1-{\overline{\lambda }}_{{\beta }_i})}\times 2h^{-\frac{1}{2}}\sum _{j=1}^m{\overline{\varpi }}_{\mathrm{ij}}|{\eta }_{\mathrm{ij}}(0)|}\\ \multicolumn{1}{c}{\le {\rho }_1{\mathcal{X}}_0+{\rho }_2,\forall (\mathcal{Z},x^{\ell },k,i)\in ({\mathcal{X}}_0,{\mathrm{\Omega }}_{\mathrm{d}},\mathbb{Z},{\mathbb{N}}_1^m).}\end{array}$$

(10)

Summarizing the above analyses, it leads to $\parallel \mathrm{\Psi }\mathcal{Z}{\parallel }_{\mathbb{X}}\le {\rho }_1{\mathcal{Z}}_0+{\rho }_2={\mathcal{Z}}_0,\forall \mathcal{Z}\in {\mathcal{X}}_0.$ Therefore, $\mathrm{\Psi }$ is well defined and maps ${\mathcal{X}}_0$ to itself. The proof is achieved.

• Proposition 2. Equation (7) possesses a unique bounded solution in ${\mathcal{X}}_0$ if $(H_3)$ - $(H_4)$ hold.

Proof. By Proposition 1, $\mathrm{\Psi }:{\mathcal{X}}_0\to {\mathcal{X}}_0$ . Assume that

$$\begin{array}{c}\multicolumn{1}{c}{\mathcal{Z}=({\mathcal{M}}_1,\dots ,{\mathcal{M}}_m,{\mathcal{P}}_1,\dots ,{\mathcal{P}}_m{)}^T,}\\ \multicolumn{1}{c}{\tilde{\mathcal{Z}}=({\tilde{\mathcal{M}}}_1,\dots ,{\tilde{\mathcal{M}}}_m,{\tilde{\mathcal{P}}}_1,\dots ,{\tilde{\mathcal{P}}}_m{)}^T\in {\mathcal{X}}_0,}\end{array}$$

it derives from equation (9) that

$$\begin{array}{c}{\left[\mathbb{E}{|(\mathrm{\Psi }\mathcal{M}{)}_i^{x^{\ell }}(k)-(\mathrm{\Psi }\tilde{\mathcal{M}}{)}_i^{x^{\ell }}(k)|}^2\right]}^{\frac{1}{2}}\hfill \\ \multicolumn{1}{c}{\le \{\mathbb{E}|\sum _{q=-\infty }^{k-1}\underset{s=q+1}{\overset{k-1}{\Pi }}{\lambda }_{{\alpha }_i}^{x^{\ell }}(s)\sum _{l={\mu }_q}^q{\theta }_{{\alpha }_i}^{x^{\ell }}(q-l,q)}\\ \multicolumn{1}{c}{\times [|{\mathrm{\Upsilon }}_i({\mathcal{M}}_i^{x^{\ell }}(l))-{\mathrm{\Upsilon }}_i({\tilde{\mathcal{M}}}_i^{x^{\ell }}(l))|+\sum _{j=1}^m{\overline{b}}_{\mathrm{ij}}|f_j({\mathcal{P}}_j^{x^{\ell }}(l))-f_j({\tilde{\mathcal{P}}}_j^{x^{\ell }}(l))|}\\ \multicolumn{1}{c}{{{+\sum _{j=1}^m{h}^{-1}|{\mathrm{\Delta }}_h{\mathbb{W}}_{1j}(l)|{\overline{\omega }}_{\mathrm{ij}}|{\sigma }_{\mathrm{ij}}({\mathcal{P}}_j^{x^{\ell }}(l))-{\sigma }_{\mathrm{ij}}({\tilde{\mathcal{P}}}_j^{x^{\ell }}(l))|]|}^2\}}^{\frac{1}{2}}}\\ \multicolumn{1}{c}{\le \sum _{q=-\infty }^{k-1}\underset{s=q+1}{\overset{k-1}{\Pi }}{\lambda }_{{\alpha }_i}^{x^{\ell }}(s)\{\mathbb{E}[\sum _{l={\mu }_q}^q|{\theta }_{{\alpha }_i}^{x^{\ell }}(l,q)|}\\ \multicolumn{1}{c}{\times (|{\mathrm{\Upsilon }}_i({\mathcal{M}}_i^{x^{\ell }}(l))-{\mathrm{\Upsilon }}_i({\tilde{\mathcal{M}}}_i^{x^{\ell }}(l))|+\sum _{j=1}^m{\overline{b}}_{\mathrm{ij}}|f_j({\mathcal{P}}_j^{x^{\ell }}(l))-f_j({\tilde{\mathcal{P}}}_j^{x^{\ell }}(l))|}\\ \multicolumn{1}{c}{{{+\sum _{j=1}^m{h}^{-1}|{\mathrm{\Delta }}_h{\mathbb{W}}_{1j}(l)|{\overline{\omega }}_{\mathrm{ij}}|{\sigma }_{\mathrm{ij}}({\mathcal{P}}_j^{x^{\ell }}(l))-{\sigma }_{\mathrm{ij}}({\tilde{\mathcal{P}}}_j^{x^{\ell }}(l))|)]}^2\}}^{\frac{1}{2}}}\\ \multicolumn{1}{c}{\le \frac{2}{\mathrm{-}{a}_i^{*}(1-{\overline{\lambda }}_{{\alpha }_i})}[D_i^{*}+\sum _{j=1}^m{\overline{b}}_{\mathrm{ij}}{L}_j^f+2h^{-\frac{1}{2}}\sum _{j=1}^m{\overline{\omega }}_{\mathrm{ij}}{L}_{\mathrm{ij}}^{\sigma }]\parallel \mathcal{Z}-\tilde{\mathcal{Z}}{\parallel }_{\mathbb{X}}}\\ \multicolumn{1}{c}{\le {\rho }_1\parallel \mathcal{Z}-\tilde{\mathcal{Z}}{\parallel }_{\mathbb{X}},\forall (x^{\ell },k,i)\in ({\overline{\mathrm{\Omega }}}_{\mathrm{d}},\mathbb{Z},{\mathbb{N}}_1^m).}\end{array}$$

Similarly,

$$\begin{array}{c}\multicolumn{1}{c}{{\left[\mathbb{E}{|(\mathrm{\Psi }\mathcal{P}{)}_i^{x^{\ell }}(k)-(\mathrm{\Psi }\tilde{\mathcal{P}}{)}_i^{x^{\ell }}(k)|}^2\right]}^{\frac{1}{2}}\le {\rho }_1\parallel \mathcal{Z}-\tilde{\mathcal{Z}}{\parallel }_{\mathbb{X}},}\\ \multicolumn{1}{c}{\forall (x^{\ell },k,i)\in ({\overline{\mathrm{\Omega }}}_{\mathrm{d}},\mathbb{Z},{\mathbb{N}}_1^m).}\end{array}$$

So $\parallel \mathrm{\Psi }\mathcal{Z}-\mathrm{\Psi }\tilde{\mathcal{Z}}{\parallel }_{\mathbb{X}}\le {\rho }_1\parallel \mathcal{Z}-\tilde{\mathcal{Z}}{\parallel }_{\mathbb{X}},$ $\forall \mathcal{Z},\tilde{\mathcal{Z}}\in {\mathcal{X}}_0$ . From $(H_4)$ , $\mathrm{\Psi }$ is a contractive mapping and $\mathrm{\Psi }$ has a unique fixed point $\mathsf{Z}=\mathrm{\Psi }\mathsf{Z}\in {\mathcal{X}}_0\subseteq \mathbb{X}$ solving equation (7). This finishes the proof.

Let

$$\begin{array}{l}{\widehat{\rho }}_1=\underset{1\le i\le m}{\max }\frac{2}{{\underset{\_}{a}}_i^{*}(1-{\overline{\lambda }}_{{\alpha }_i})}\left[D_i^{*}+\sum _{j=1}^m{\overline{b}}_{ij}{L}_j^f+2h^{-\frac{1}{2}}\sum _{j=1}^m{\overline{\omega }}_{ij}{L}_{ij}^{\sigma }\right],\\ {\stackrel{⌣}{\rho }}_1=\underset{1\le i\le m}{\max }\frac{2}{{\underset{\_}{c}}_i^{*}(1-{\overline{\lambda }}_{{\beta }_i})}\left[K_i^{*}+{\overline{d}}_i+2h^{-\frac{1}{2}}\sum _{j=1}^m{\overline{\varpi }}_{ij}{L}_{ij}^{\eta }\right].\end{array}$$

• Theorem 1. A unique Bohr almost automorphic sequence in distribution solves equation (7) if $(H_1)$ - $(H_4)$ hold.

Proof. Equation (7) possesses a unique solution $\mathsf{Z}$ in ${\mathcal{X}}_0$ on the basis of Proposition 2. From $(H_1)$ - $(H_2)$ , for any $\{k{\prime }_n\}$ , it has a subsequence $\{k_n\}$ and some functions ${\tilde{b}}_{\mathrm{ij}}$ , ${\tilde{\omega }}_{\mathrm{ij}}$ , ${\tilde{\varpi }}_{\mathrm{ij}}$ , ${\tilde{d}}_i$ , and ${\tilde{I}}_i$ such that $\underset{n\to \infty }{lim}{\nu }_{k+k_n}={\nu }_{k+k\prime }$ uniformly for $k\in \mathbb{Z}$ and

$$\underset{n\to \infty }{lim}{b}_{\mathrm{ij}}^{x^{\ell }}(k+k_n)={\tilde{b}}_{\mathrm{ij}}^{x^{\ell }}(k),\underset{n\to \infty }{lim}{\tilde{b}}_{\mathrm{ij}}^{x^{\ell }}(k-k_n)=b_{\mathrm{ij}}^{x^{\ell }}(k),$$

$$\underset{n\to \infty }{lim}{\omega }_{\mathrm{ij}}^{x^{\ell }}(k+k_n)={\tilde{\omega }}_{\mathrm{ij}}^{x^{\ell }}(k),\underset{n\to \infty }{lim}{\tilde{\omega }}_{\mathrm{ij}}^{x^{\ell }}(k-k_n)={\omega }_{\mathrm{ij}}^{x^{\ell }}(k),$$

$$\underset{n\to \infty }{lim}{\varpi }_{\mathrm{ij}}^{x^{\ell }}(k+k_n)={\tilde{\varpi }}_{\mathrm{ij}}^{x^{\ell }}(k),\underset{n\to \infty }{lim}{\tilde{\varpi }}_{\mathrm{ij}}^{x^{\ell }}(k-k_n)={\varpi }_{\mathrm{ij}}^{x^{\ell }}(k),$$

$$\underset{n\to \infty }{lim}{d}_i^{x^{\ell }}(k+k_n)={\tilde{d}}_i^{x^{\ell }}(k),\underset{n\to \infty }{lim}{\tilde{d}}_i^{x^{\ell }}(k-k_n)=d_i^{x^{\ell }}(k),$$

$$\underset{n\to \infty }{lim}{I}_i^{x^{\ell }}(k+k_n)={\tilde{I}}_i^{x^{\ell }}(k),\underset{n\to \infty }{lim}{\tilde{I}}_i^{x^{\ell }}(k-k_n)=I_i^{x^{\ell }}(k)$$

uniformly in k on any compact subset from $\mathbb{Z}$ , $k\prime \in [0,k_{*}{]}_{\mathbb{Z}}$ , $x^{\ell }\in {\mathrm{\Omega }}_{\mathrm{d}}$ , $i=1,2,\dots ,m$ .

Let $\tilde{\mathcal{Z}}=({\tilde{\mathcal{M}}}_1,\dots ,{\tilde{\mathcal{M}}}_m,{\tilde{\mathcal{P}}}_1,\dots ,{\tilde{\mathcal{P}}}_m{)}^T$ be the solution of

$$\{\begin{array}{c}{\tilde{\mathcal{M}}}_i^{x^{\ell }}(k)=\sum _{q=-\infty }^{k-1}\underset{s=q+1}{\overset{k-1}{\Pi }}{\lambda }_{{\alpha }_i}^{x^{\ell }}(s+k\prime )\sum _{l=0}^{{\nu }_{q+k\prime }}{\theta }_{{\alpha }_i}^{x^{\ell }}(l,q+k\prime )\hfill \\ \multicolumn{1}{c}{[{\mathrm{\Upsilon }}_i({\tilde{\mathcal{M}}}_i^{x^{\ell }}(q-l))}\\ \multicolumn{1}{c}{+\sum _{j=1}^m{\tilde{b}}_{\mathrm{ij}}^{x^{\ell }}(q-l)f_j({\tilde{\mathcal{P}}}_j^{x^{\ell }}(q-l))}\\ \multicolumn{1}{c}{+\sum _{j=1}^m{\tilde{\omega }}_{\mathrm{ij}}^{x^{\ell }}(q-l){\sigma }_{\mathrm{ij}}({\tilde{\mathcal{P}}}_j^{x^{\ell }}(q-l))}\\ \multicolumn{1}{c}{\times h^{-1}{\mathrm{\Delta }}_h{\mathbb{W}}_{1j}(q-l)+{\tilde{I}}_i^{x^{\ell }}(q-l)],}\\ \multicolumn{1}{c}{{\tilde{\mathcal{P}}}_i^{x^{\ell }}(k)=\sum _{q=-\infty }^{k-1}\underset{s=q+1}{\overset{k-1}{\Pi }}{\lambda }_{{\beta }_i}^{x^{\ell }}(s+k\prime )\sum _{l=0}^{{\nu }_{q+k\prime }}{\theta }_{{\beta }_i}^{x^{\ell }}(l,q+k\prime )}\\ \multicolumn{1}{c}{[{\tilde{\mathrm{\Upsilon }}}_i({\tilde{\mathcal{P}}}_i^{x^{\ell }}(q-l))}\\ \multicolumn{1}{c}{+{\tilde{d}}_i^{x^{\ell }}(q-l){\tilde{\mathcal{M}}}_i^{x^{\ell }}(q-l)+\sum _{j=1}^m{\tilde{\varpi }}_{\mathrm{ij}}^{x^{\ell }}(q-l)}\\ \multicolumn{1}{c}{{\eta }_{\mathrm{ij}}({\tilde{\mathcal{M}}}_j^{x^{\ell }}(q-l))}\\ \multicolumn{1}{c}{\times h^{-1}{\mathrm{\Delta }}_h{\mathbb{W}}_{2j}(q-l)],x^{\ell }\in {\mathrm{\Omega }}_{\mathrm{d}};}\end{array}$$

(11)

$${\tilde{\mathcal{M}}}_i^{x^{\ell }}(k){|}_{x^{\ell }\in \partial {\mathrm{\Omega }}_{\mathrm{d}}}={\phi }_{i,k}^{x^{\ell }},{\tilde{\mathcal{P}}}_i^{x^{\ell }}(k){|}_{x^{\ell }\in \partial {\mathrm{\Omega }}_{\mathrm{d}}}={\tilde{\phi }}_{i,k}^{x^{\ell }},\forall (k,i)\in (\mathbb{Z},{\mathbb{N}}_1^m).$$

According to Proposition 2, the unique solution $\mathsf{Z}=({\mathsf{M}}_1,\dots ,{\mathsf{M}}_m,{\mathsf{P}}_1,\dots ,{\mathsf{P}}_m{)}^T\in {\mathcal{X}}_0$ of equation (7) satisfies

$$\{\begin{array}{c}{\mathsf{M}}_i^{x^{\ell }}(k+k_n)=\sum _{q=-\infty }^{k-1}\underset{s=q+1}{\overset{k-1}{\Pi }}{\lambda }_{{\alpha }_i}^{x^{\ell }}(s+k_n)\sum _{l=0}^{{\nu }_{q+k_n}}{\theta }_{{\alpha }_i}^{x^{\ell }}(l,q+k_n)[{\mathrm{\Upsilon }}_i({\mathsf{M}}_i^{x^{\ell }}(q-l+k_n))\hfill \\ \multicolumn{1}{c}{+\sum _{j=1}^m{b}_{\mathrm{ij}}^{x^{\ell }}(q-l+k_n)f_j({\mathsf{P}}_j^{x^{\ell }}(q-l+k_n))+\sum _{j=1}^m{\omega }_{\mathrm{ij}}^{x^{\ell }}(q-l+k_n)}\\ \multicolumn{1}{c}{\times {\sigma }_{\mathrm{ij}}({\mathsf{P}}_j^{x^{\ell }}(q-l+k_n))h^{-1}{\mathrm{\Delta }}_h{\mathbb{W}}_{1j}(q-l+k_n)+I_i^{x^{\ell }}(q-l+k_n)],}\\ \multicolumn{1}{c}{{\mathsf{P}}_i^{x^{\ell }}(k+k_n)=\sum _{q=-\infty }^{k-1}\underset{s=q+1}{\overset{k-1}{\Pi }}{\lambda }_{{\beta }_i}^{x^{\ell }}(s+k_n)\sum _{l=0}^{{\nu }_{q+k_n}}{\theta }_{{\beta }_i}^{x^{\ell }}(l,q+k_n)[{\tilde{\mathrm{\Upsilon }}}_i({\mathsf{P}}_i^{x^{\ell }}(q-l+k_n))}\\ \multicolumn{1}{c}{+d_i^{x^{\ell }}(q-l+k_n){\mathsf{M}}_i^{x^{\ell }}(q-l+k_n)+\sum _{j=1}^m{\varpi }_{\mathrm{ij}}^{x^{\ell }}(q-l+k_n)}\\ \multicolumn{1}{c}{\times {\eta }_{\mathrm{ij}}({\mathsf{M}}_j^{x^{\ell }}(q-l+k_n))h^{-1}{\mathrm{\Delta }}_h{\mathbb{W}}_{2j}(q-l+k_n)],x^{\ell }\in {\mathrm{\Omega }}_{\mathrm{d}};}\end{array}$$

(12)

$${{\mathsf{M}}_i^{x^{\ell }}(k)|}_{x^{\ell }\in \partial {\mathrm{\Omega }}_{\mathrm{d}}}={\phi }_{i,k}^{x^{\ell }},{{\mathsf{P}}_i^{x^{\ell }}(k)|}_{x^{\ell }\in \partial {\mathrm{\Omega }}_{\mathrm{d}}}={\tilde{\phi }}_{i,k}^{x^{\ell }},\forall (k,i)\in (\mathbb{Z},{\mathbb{N}}_1^m).$$

It should be remarked that ${\mathrm{\Delta }}_h{\mathbb{W}}_{\upsilon j}(k+k_n)$ has the same law as ${\mathrm{\Delta }}_h{\mathbb{W}}_{\upsilon j}(k)$ , $\forall (n,k,j)\in (\mathbb{N},\mathbb{Z},{\mathbb{N}}_1^m)$ , $\upsilon =1,2$ . Let us discuss the stochastic process ${\mathsf{Z}}_n=({\mathsf{M}}_{1,n},\dots ,{\mathsf{M}}_{m,n},{\mathsf{P}}_{1,n},\dots ,{\mathsf{P}}_{m,n}{)}^T$ below

$$\{\begin{array}{c}{\mathsf{M}}_{i,n}^{x^{\ell }}(k)=\sum _{q=-\infty }^{k-1}\underset{s=q+1}{\overset{k-1}{\Pi }}{\lambda }_{{\alpha }_i}^{x^{\ell }}(s+k_n)\sum _{l=0}^{{\nu }_{q+k_n}}{\theta }_{{\alpha }_i}^{x^{\ell }}(l,q+k_n)[{\mathrm{\Upsilon }}_i({\mathsf{M}}_{i,n}^{x^{\ell }}(q-l))\hfill \\ \multicolumn{1}{c}{+\sum _{j=1}^m{b}_{\mathrm{ij}}^{x^{\ell }}(q-l+k_n)f_j({\mathsf{P}}_{j,n}^{x^{\ell }}(q-l))+\sum _{j=1}^m{\omega }_{\mathrm{ij}}^{x^{\ell }}(q-l+k_n)}\\ \multicolumn{1}{c}{\times {\sigma }_{\mathrm{ij}}({\mathsf{P}}_{j,n}^{x^{\ell }}(q-l))h^{-1}{\mathrm{\Delta }}_h{\mathbb{W}}_{1j}(q-l)+I_i^{x^{\ell }}(q-l+k_n)],}\\ \multicolumn{1}{c}{{\mathsf{P}}_{i,n}^{x^{\ell }}(k)=\sum _{q=-\infty }^{k-1}\underset{s=q+1}{\overset{k-1}{\Pi }}{\lambda }_{{\beta }_i}^{x^{\ell }}(s+k_n)\sum _{l=0}^{{\nu }_{q+k_n}}{\theta }_{{\beta }_i}^{x^{\ell }}(l,q+k_n)[{\tilde{\mathrm{\Upsilon }}}_i{\mathsf{P}}_{i,n}^{x^{\ell }}(q-l))}\\ \multicolumn{1}{c}{+d_i^{x^{\ell }}(q-l+k_n){\mathsf{M}}_{i,n}^{x^{\ell }}(q-l)+\sum _{j=1}^m{\varpi }_{\mathrm{ij}}^{x^{\ell }}(q-l+k_n)}\\ \multicolumn{1}{c}{\times {\eta }_{\mathrm{ij}}({\mathsf{M}}_{j,n}^{x^{\ell }}(q-l))h^{-1}{\mathrm{\Delta }}_h{\mathbb{W}}_{2j}(q-l)],x^{\ell }\in {\mathrm{\Omega }}_{\mathrm{d}};}\end{array}$$

(13)

${\mathsf{M}}_{i,n}^{x^{\ell }}(k){|}_{x^{\ell }\in \partial {\mathrm{\Omega }}_{\mathrm{d}}}={\phi }_{i,k}^{x^{\ell }},{\mathsf{P}}_{i,n}^{x^{\ell }}(k)|x^{\ell }\in \partial {\mathrm{\Omega }}_{\mathrm{d}}={\tilde{\phi }}_{i,k}^{x^{\ell }},$ $\forall (k,i)\in (\mathbb{Z},{\mathbb{N}}_1^m).$ Clearly, $\left({\mathsf{M}}_{i,n}^{x^{\ell }}(k),{\mathsf{P}}_{i,n}^{x^{\ell }}(k)\right)$ has the same distribution as $\left({\mathsf{M}}_i^{x^{\ell }}(k+k_n),{\mathsf{P}}_i^{x^{\ell }}(k+k_n)\right)$ , $\forall (x^{\ell },n,k,i)\in ({\overline{\mathrm{\Omega }}}_{\mathrm{d}},\mathbb{N},\mathbb{Z},{\mathbb{N}}_1^m)$ . Similar to $\mathsf{Z}$ , $\tilde{\mathcal{Z}}$ and ${\mathsf{Z}}_n$ are unique and bounded in ${\mathcal{X}}_0$ , $\forall n\in \mathbb{N}$ .

For simplify, let ${\mathbf{U}}_{i,n}^{x^{\ell }}(k)={\mathsf{M}}_{i,n}^{x^{\ell }}(k)-{\tilde{\mathcal{M}}}_i^{x^{\ell }}(k)$ , ${\mathbf{V}}_{i,n}^{x^{\ell }}(k)={\mathsf{P}}_{i,n}^{x^{\ell }}(k)-{\tilde{\mathcal{P}}}_i^{x^{\ell }}(k)$ ,

$$\{\begin{array}{c}{\hat{\mathcal{F}}}_i^{x^{\ell }}({\mathsf{Z}}_n(k)):={\mathrm{\Upsilon }}_i({\mathsf{M}}_{i,n}^{x^{\ell }}(k))+\sum _{j=1}^m{b}_{\mathrm{ij}}^{x^{\ell }}(k+k_n)f_j({\mathsf{P}}_{j,n}^{x^{\ell }}(k))\hfill \\ \multicolumn{1}{c}{+\sum _{j=1}^m{\omega }_{\mathrm{ij}}^{x^{\ell }}(k+k_n){\sigma }_{\mathrm{ij}}({\mathsf{P}}_{j,n}^{x^{\ell }}(k))h^{-1}{\mathrm{\Delta }}_h{\mathbb{W}}_{1j}(k)+I_i^{x^{\ell }}(k+k_n),}\\ \multicolumn{1}{c}{{\stackrel{⌣}{\mathcal{F}}}_i^{x^{\ell }}({\mathsf{Z}}_n(k)):={\tilde{\mathrm{\Upsilon }}}_i({\mathsf{P}}_{i,n}^{x^{\ell }}(k))+d_i^{x^{\ell }}(k+k_n){\mathsf{M}}_{i,n}^{x^{\ell }}(k)}\\ \multicolumn{1}{c}{+\sum _{j=1}^m{\varpi }_{\mathrm{ij}}^{x^{\ell }}(k+k_n){\eta }_{\mathrm{ij}}({\mathsf{M}}_{j,n}^{x^{\ell }}(k))h^{-1}{\mathrm{\Delta }}_h{\mathbb{W}}_{2j}(k)}\end{array}$$

and

$$\{\begin{array}{c}{\hat{\mathcal{G}}}_i^{x^{\ell }}(\tilde{\mathcal{Z}}(k)):={\mathrm{\Upsilon }}_i({\tilde{\mathcal{M}}}_i^{x^{\ell }}(k))+\sum _{j=1}^m{\tilde{b}}_{\mathrm{ij}}^{x^{\ell }}(k)f_j({\tilde{\mathcal{P}}}_j^{x^{\ell }}(k))\hfill \\ \multicolumn{1}{c}{+\sum _{j=1}^m{\tilde{\omega }}_{\mathrm{ij}}^{x^{\ell }}(k){\sigma }_{\mathrm{ij}}({\tilde{\mathcal{P}}}_j^{x^{\ell }}(k))h^{-1}{\mathrm{\Delta }}_h{\mathbb{W}}_{1j}(k)+{\tilde{I}}_i^{x^{\ell }}(k),}\\ \multicolumn{1}{c}{{\stackrel{⌣}{\mathcal{G}}}_i^{x^{\ell }}(\tilde{\mathcal{Z}}(k)):={\tilde{\mathrm{\Upsilon }}}_i({\tilde{\mathcal{P}}}_i^{x^{\ell }}(k))+{\tilde{d}}_i^{x^{\ell }}(k){\tilde{\mathcal{M}}}_i^{x^{\ell }}(k)}\\ \multicolumn{1}{c}{+\sum _{j=1}^m{\tilde{\varpi }}_{\mathrm{ij}}^{x^{\ell }}(k){\eta }_{\mathrm{ij}}({\tilde{\mathcal{M}}}_j^{x^{\ell }}(k))h^{-1}{\mathrm{\Delta }}_h{\mathbb{W}}_{2j}(k),}\end{array}$$

$\forall (x^{\ell },n,k,i)\in ({\mathrm{\Omega }}_{\mathrm{d}},\mathbb{N},\mathbb{Z},{\mathbb{N}}_1^m)$ . Because ${\mathsf{Z}}_n\in {\mathcal{X}}_0$ , it must exist a positive constant ${\mathcal{F}}_0$ such that ${\sup }_{k\in \mathbb{Z}}{\max }_{1\le i\le m,x^{\ell }\in {\mathrm{\Omega }}_{\mathrm{d}}}{\left[\mathbb{E}|{\widehat{ℱ}}_i^{x^{\ell }}(\mathrm{Z}(k{))|}^2+|{\stackrel{⌣}{ℱ}}_i^{x^{\ell }}({\mathrm{Z}}_n(k{))|}^2\right]}^{\frac{1}{2}}\le {ℱ}_0$ , $\forall m\in \mathbb{Z}$ .

According to equations (11) and (13), it gains

$${\left[\mathbb{E}{|{\mathbf{U}}_{i,n}^{x^{\ell }}(k)|}^2\right]}^{\frac{1}{2}}\le {\xi }_{i,n}^1(k)+{\xi }_{i,n}^2(k)+{\xi }_{i,n}^3(k)+{\xi }_{i,n}^4(k),$$

(14)

where

$$\begin{array}{c}{\xi }_{i,n}^1(k):=\{\mathbb{E}|\sum _{q=-\infty }^{k-2}\underset{s=q+1}{\overset{k-1}{\Pi }}{\lambda }_{{\alpha }_i}^{x^{\ell }}(s+k_n)\hfill \\ \multicolumn{1}{c}{{{\times \sum _{l=0}^{{\nu }_{q+k_n}}[{\theta }_{{\alpha }_i}^{x^{\ell }}(l,q+k_n)-{\theta }_{{\alpha }_i}^{x^{\ell }}(l,q+k\prime )]{\hat{\mathcal{F}}}_i^{x^{\ell }}({\mathsf{Z}}_n(q-l))|}^2\}}^{\frac{1}{2}}}\\ \multicolumn{1}{c}{\le \frac{{\mathcal{F}}_0}{1-{\overline{\lambda }}_{{\alpha }_i}}\underset{q\in \mathbb{Z}}{sup}\underset{x^{\ell }\in {\mathrm{\Omega }}_{\mathrm{d}}}{max}\sum _{l=0}^{Q^{*}}|{\theta }_{{\alpha }_i}^{x^{\ell }}(l,q+k_n)-{\theta }_{{\alpha }_i}^{x^{\ell }}(l,q+k\prime )|:={\zeta }_{i,n}^1,}\end{array}$$

$$\begin{array}{c}{\xi }_{i,n}^2(k):=\{\mathbb{E}|\sum _{q=-\infty }^{k-2}\underset{s=q+1}{\overset{k-1}{\Pi }}{\lambda }_{{\alpha }_i}^{x^{\ell }}(s+k_n)\hfill \\ \multicolumn{1}{c}{{\times [\sum _{l=0}^{{\nu }_{q+k_n}}-\sum _{l=0}^{{\nu }_{q+k\prime }}]{\theta }_{{\alpha }_i}^{x^{\ell }}(l,q+k\prime ){\hat{\mathcal{F}}}_i^{x^{\ell }}({\mathsf{Z}}_n(q-l)){|}^2\}}^{\frac{1}{2}}}\\ \multicolumn{1}{c}{\le \frac{{\mathcal{F}}_0}{1-{\overline{\lambda }}_{{\alpha }_i}}\underset{q\in \mathbb{Z}}{sup}\underset{x^{\ell }\in {\mathrm{\Omega }}_{\mathrm{d}}}{max}\sum _{l=min\{{\nu }_{q+k_n},{\nu }_{q+k\prime }\}}^{max\{{\nu }_{q+k_n},{\nu }_{q+k\prime }\}}|{\theta }_{{\alpha }_i}^{x^{\ell }}(l,q+k\prime )|:={\zeta }_{i,n}^2,}\end{array}$$

$$\begin{array}{c}{\xi }_{i,n}^3(k):=\{\mathbb{E}|\sum _{q=-\infty }^{k-2}\underset{s=q+1}{\overset{k-1}{\Pi }}[{\lambda }_{{\alpha }_i}^{x^{\ell }}(s+k_n)-{\lambda }_{{\alpha }_i}^{x^{\ell }}(s+k\prime )]\hfill \\ \multicolumn{1}{c}{{{\times \sum _{l=0}^{{\nu }_{q+k\prime }}{\theta }_{{\alpha }_i}^{x^{\ell }}(l,q+k\prime ){\hat{\mathcal{F}}}_i^{x^{\ell }}({\mathsf{Z}}_n(q-l))|}^2\}}^{\frac{1}{2}}}\\ \multicolumn{1}{c}{\le \frac{2}{\mathrm{-}{a}_i^{*}}\sum _{q=-\infty }^{k-2}\underset{s=q+1}{\overset{k-1}{\Pi }}|{\lambda }_{{\alpha }_i}^{x^{\ell }}(s+k_n)-{\lambda }_{{\alpha }_i}^{x^{\ell }}(s+k\prime )|{\mathcal{F}}_0}\\ \multicolumn{1}{c}{\le \frac{2{\mathcal{F}}_0}{\mathrm{-}{a}_i^{*}(1-{\overline{\lambda }}_{{\alpha }_i})}\underset{s\in \mathbb{Z}}{sup}\underset{x^{\ell }\in {\mathrm{\Omega }}_{\mathrm{d}}}{max}|{\lambda }_{{\alpha }_i}^{x^{\ell }}(s+k_n)-{\lambda }_{{\alpha }_i}^{x^{\ell }}(s+k\prime )|:={\zeta }_{i,n}^3,}\end{array}$$

$$\begin{array}{c}{\xi }_{i,n}^4(k):=\{\mathbb{E}|\sum _{q=-\infty }^{k-2}\underset{s=q+1}{\overset{k-1}{\Pi }}{\lambda }_{{\alpha }_i}^{x^{\ell }}(s+k\prime )\hfill \\ \multicolumn{1}{c}{{{\times \sum _{l=0}^{{\nu }_{q+k\prime }}{\theta }_{{\alpha }_i}^{x^{\ell }}(l,q+k\prime )[{\hat{\mathcal{F}}}_i^{x^{\ell }}({\mathsf{Z}}_n(q-l))-{\hat{\mathcal{G}}}_i^{x^{\ell }}(\tilde{\mathcal{Z}}(q-l))]|}^2\}}^{\frac{1}{2}}}\\ \multicolumn{1}{c}{+\{\mathbb{E}|\sum _{l=0}^{{\nu }_{k-1+k_n}}{\theta }_{{\alpha }_i}^{x^{\ell }}(l,k-1+k_n){\hat{\mathcal{F}}}_i^{x^{\ell }}({\mathsf{Z}}_n(k-1-l))}\\ \multicolumn{1}{c}{{{-\sum _{l=0}^{{\nu }_{k-1+k\prime }}{\theta }_{{\alpha }_i}^{x^{\ell }}(l,k-1+k\prime ){\hat{\mathcal{G}}}_i^{x^{\ell }}(\tilde{\mathcal{Z}}(k-1-l))|}^2\}}^{\frac{1}{2}}}\\ \multicolumn{1}{c}{\le {\tilde{\xi }}_{i,n}^4(k)+\frac{2}{\mathrm{-}{a}_i^{*}}\sum _{q=-\infty }^{k-2}{\overline{\lambda }}_{{\alpha }_i}^{k-q-1}[D_i^{*}+\sum _{j=1}^m{\overline{b}}_{\mathrm{ij}}{L}_j^f+2h^{-\frac{1}{2}}\sum _{j=1}^m{\overline{\omega }}_{\mathrm{ij}}{L}_{\mathrm{ij}}^{\sigma }]}\\ \multicolumn{1}{c}{\times \underset{1\le i\le m,x^{\ell }\in {\mathrm{\Omega }}_{\mathrm{d}},l\in {[0,Q^{*}]}_{\mathbb{Z}}}{max}\{{[\mathbb{E}|{\mathbf{U}}_{i,n}^{x^{\ell }}(q-l){|}^2]}^{\frac{1}{2}},{[\mathbb{E}|{\mathbf{V}}_{i,n}^{x^{\ell }}(q-l){|}^2]}^{\frac{1}{2}}\}}\\ \multicolumn{1}{c}{+\underset{k\in \mathbb{Z}}{sup}\sum _{l=0}^{Q^{*}}|{\theta }_{{\alpha }_i}^{x^{\ell }}(l,k-1+k_n)-{\theta }_{{\alpha }_i}^{x^{\ell }}(l,k-1+k\prime )|{\mathcal{F}}_0}\\ \multicolumn{1}{c}{+\sum _{l=min\{{\nu }_{k-1+k_n},{\nu }_{k-1+k\prime }\}}^{max\{{\nu }_{k-1+k_n},{\nu }_{k-1+k\prime }\}}|{\theta }_{{\alpha }_i}^{x^{\ell }}(l,k-1+k\prime )|{\mathcal{F}}_0}\\ \multicolumn{1}{c}{+(1-{\overline{\lambda }}_{{\alpha }_i}){\rho }_1\underset{1\le i\le m}{max}\underset{q\in {(-\infty ,\mathcal{L}]}_{\mathbb{Z}},x^{\ell }\in {\mathrm{\Omega }}_{\mathrm{d}}}{sup}\{{[\mathbb{E}|{\mathbf{U}}_{i,n}^{x^{\ell }}(q){|}^2]}^{\frac{1}{2}},{[\mathbb{E}|{\mathbf{V}}_{i,n}^{x^{\ell }}(q){|}^2]}^{\frac{1}{2}}\}}\\ \multicolumn{1}{c}{\le {\tilde{\xi }}_{i,n}^4(k)+{\xi }_{i,n}^1+{\xi }_{i,n}^2}\\ \multicolumn{1}{c}{+{\rho }_1\underset{1\le i\le m,q\in {(-\infty ,\mathcal{L}]}_{\mathbb{Z}}}{max}\underset{x^{\ell }\in {\mathrm{\Omega }}_{\mathrm{d}}}{sup}\{{[\mathbb{E}|{\mathbf{U}}_{i,n}^{x^{\ell }}(q){|}^2]}^{\frac{1}{2}},{[\mathbb{E}|{\mathbf{V}}_{i,n}^{x^{\ell }}(q){|}^2]}^{\frac{1}{2}}\}}\end{array}$$

for any positive integer $\mathcal{L}$ , in which ${\tilde{\xi }}_{i,n}^4(k):=\sum _{q=-\infty }^{k-1}{\overline{\lambda }}_{{\alpha }_i}^{k-q-1}\sum _{l=0}^{Q^{*}}|{\theta }_{{\alpha }_i}^{x^{\ell }}(l,q+k\prime )|{\widetilde{\stackrel{~}{\xi }}}_{i,n}^4(q-l)$ ,

$$\begin{array}{c}{\widetilde{\stackrel{~}{\xi }}}_{i,n}^4(k):=\sum _{j=1}^m|b_{\mathrm{ij}}^{x^{\ell }}(k+k_n)-{\tilde{b}}_{\mathrm{ij}}^{x^{\ell }}(k)|(L_j^f{\mathcal{Z}}_0+|f_j(0)|)\hfill \\ \multicolumn{1}{c}{+\sum _{j=1}^m|{\omega }_{\mathrm{ij}}^{x^{\ell }}(k+k_n)-{\tilde{\omega }}_{\mathrm{ij}}^{x^{\ell }}(k)|(L_{\mathrm{ij}}^{\sigma }{\mathcal{Z}}_0+|{\sigma }_{\mathrm{ij}}(0)|)2h^{-\frac{1}{2}}}\\ \multicolumn{1}{c}{+|I_i^{x^{\ell }}(k+k_n)-{\tilde{I}}_i^{x^{\ell }}(k)|}\end{array}$$

for all $(x^{\ell },n,k,i)\in ({\overline{\mathrm{\Omega }}}_{\mathrm{d}},\mathbb{N},(-\infty ,\mathcal{L}{]}_{\mathbb{Z}},{\mathbb{N}}_1^m)$ .

Taking a real sequence $\{{\mathbf{l}}_s>0{\}}_{s\in \mathbb{N}}$ with $\underset{s\to \infty }{lim}{\mathbf{l}}_s=+\infty$ . For $k\in [-{\mathbf{l}}_s,\mathcal{L}{]}_{\mathbb{Z}}$ , it computes

$${\tilde{\xi }}_{i,n}^4(k)\le \sum _{q=-\infty }^{\mathcal{L}-1}{\overline{\lambda }}_{{\alpha }_i}^{-{\mathbf{l}}_s-q-1}\sum _{l=0}^{Q^{*}}|{\theta }_{{\alpha }_i}^{x^{\ell }}(l,q+k\prime )|{\widetilde{\stackrel{~}{\xi }}}_{i,n}^4(q-l):={\zeta }_{i,n,s}^4$$

for all $(n,s,k,i)\in (\mathbb{N},\mathbb{N},(-{\mathbf{l}}_s,\mathcal{L}{]}_{\mathbb{Z}},{\mathbb{N}}_1^m)$ . For each $s\in \mathbb{N}$ , ${\zeta }_{i,n,s}^4$ is obviously independent of k and $\underset{n\to \infty }{lim}{\zeta }_{i,n,s}^4=0$ by Lebesgue dominated convergence theorem, $\forall (n,k,i)\in (\mathbb{N},(-{\mathbf{l}}_s,\mathcal{L}{]}_{\mathbb{Z}},{\mathbb{N}}_1^m)$ .

Let ${\hat{\zeta }}_{n,s}:=\underset{1\le i\le m,x^{\ell }\in {\mathrm{\Omega }}_{\mathrm{d}}}{max}(2{\zeta }_{i,n}^1+2{\zeta }_{i,n}^2+{\zeta }_{i,n}^3+{\zeta }_{i,n,s}^4)$ . Then ${\hat{\zeta }}_{n,s}<\infty ,$ $\underset{n\to \infty }{lim}{\hat{\zeta }}_{n,s}=0$ for each $s\in \mathbb{N}$ . Consequently, equation (14) leads to

$$\begin{array}{c}\underset{k\in {(-{\mathbf{l}}_s,\mathcal{L}]}_{\mathbb{Z}}}{sup}\underset{1\le i\le m,x^{\ell }\in {\overline{\mathrm{\Omega }}}_{\mathrm{d}}}{max}{\left[\mathbb{E}{|{\mathbf{U}}_{i,n}^{x^{\ell }}(k)|}^2\right]}^{\frac{1}{2}}\hfill \\ \multicolumn{1}{c}{\le {\hat{\rho }}_1\underset{k\in {(-\infty ,\mathcal{L}]}_{\mathbb{Z}}}{sup}\underset{1\le i\le m,x^{\ell }\in {\overline{\mathrm{\Omega }}}_{\mathrm{d}}}{max}\{{[\mathbb{E}|{\mathbf{U}}_{i,n}^{x^{\ell }}(k){|}^2]}^{\frac{1}{2}},{[\mathbb{E}|{\mathbf{V}}_{i,n}^{x^{\ell }}(k){|}^2]}^{\frac{1}{2}}\}}\\ \multicolumn{1}{c}{+{\hat{\zeta }}_{n,s},}\end{array}$$

(15)

for $n,s\in \mathbb{N}$ . Similarly, it also exists ${\stackrel{⌣}{\zeta }}_{n,s}$ satisfying ${lim}_{n\to \infty }{\stackrel{⌣}{\zeta }}_{n,s}=0$ and

$$\begin{array}{l}\underset{k\in {(-{\mathbf{l}}_s,ℒ]}_{\mathbb{Z}}}{\sup }\underset{1\le i\le m,x^{\ell }\in {\overline{\mathrm{\Omega }}}_{\mathrm{d}}}{\max }{\left[\mathbb{E}{\left|{\mathbf{V}}_{i,n}^{x^{\ell }}(k)\right|}^2\right]}^{\frac{1}{2}}\hfill \\ \le {\stackrel{⌣}{\rho }}_1\underset{k\in {(-\infty ,ℒ]}_{\mathbb{Z}}}{\sup }\underset{1\le i\le m,x^{\ell }\in {\overline{\mathrm{\Omega }}}_{\mathrm{d}}}{\max }\left\{{\left[\mathbb{E}|{\mathbf{U}}_{i,n}^{x^{\ell }}(k){|}^2\right]}^{\frac{1}{2}},{\left[\mathbb{E}|{\mathbf{V}}_{i,n}^{x^{\ell }}(k){|}^2\right]}^{\frac{1}{2}}\right\}+{\stackrel{⌣}{\zeta }}_{n,s},\hfill \end{array}.$$

(16)

for $n,s\in \mathbb{N}$ . Summarizing equations (15) and (16), it induces

$$\begin{array}{l}\underset{k\in {(-{\mathbf{l}}_s,ℒ]}_{\mathbb{Z}}}{\sup }\underset{1\le i\le m,x^{\ell }\in {\overline{\mathrm{\Omega }}}_{\mathrm{d}}}{\max }\left\{{\left[\mathbb{E}|{\mathbf{U}}_{i,n}^{x^{\ell }}(k){|}^2\right]}^{\frac{1}{2}},{\left[\mathbb{E}|{\mathbf{V}}_{i,n}^{x^{\ell }}(k){|}^2\right]}^{\frac{1}{2}}\right\}\hfill \\ \le {\rho }_1\underset{k\in {(-\infty ,ℒ]}_{\mathbb{Z}}}{\sup }\underset{1\le i\le m,x^{\ell }\in {\overline{\mathrm{\Omega }}}_{\mathrm{d}}}{\max }\left\{{\left[\mathbb{E}|{\mathbf{U}}_{i,n}^{x^{\ell }}(k){|}^2\right]}^{\frac{1}{2}},{\left[\mathbb{E}|{\mathbf{V}}_{i,n}^{x^{\ell }}(k){|}^2\right]}^{\frac{1}{2}}\right\}+{\widehat{\zeta }}_{n,s}+{\stackrel{⌣}{\zeta }}_{n,s}.\hfill \end{array}$$

(17)

for $n,s\in \mathbb{N}$ . In equation (17), letting $n\to \infty$ and $s\to \infty$ in turn, it gains

$$\begin{array}{c}\underset{n\to \infty }{lim}\underset{k\in {[-\mathcal{L},\mathcal{L}]}_{\mathbb{Z}}}{sup}{max}_{1\le i\le m,x^{\ell }\in {\overline{\mathrm{\Omega }}}_{\mathrm{d}}}\{{[\mathbb{E}|{\mathbf{U}}_{i,n}^{x^{\ell }}(k){|}^2]}^{\frac{1}{2}},{[\mathbb{E}|{\mathbf{V}}_{i,n}^{x^{\ell }}(k){|}^2]}^{\frac{1}{2}}\}\hfill \\ \multicolumn{1}{c}{\le \underset{n\to \infty }{lim}\underset{k\in {(-\infty ,\mathcal{L}]}_{\mathbb{Z}}}{sup}{max}_{1\le i\le m,x^{\ell }\in {\overline{\mathrm{\Omega }}}_{\mathrm{d}}}}\\ \multicolumn{1}{c}{\{{[\mathbb{E}|{\mathbf{U}}_{i,n}^{x^{\ell }}(k){|}^2]}^{\frac{1}{2}},{[\mathbb{E}|{\mathbf{V}}_{i,n}^{x^{\ell }}(k){|}^2]}^{\frac{1}{2}}\}=0.}\end{array}$$

Because $({\mathsf{M}}_{i,n}^{x^{\ell }}(k),{\mathsf{P}}_{i,n}^{x^{\ell }}(k))$ has the same distribution as $({\mathsf{M}}_i^{x^{\ell }}(k+k_n),{\mathsf{P}}_i^{x^{\ell }}(k+k_n))$ , $\mathsf{Z}(k+k_n)$ converges to $\tilde{\mathcal{Z}}(k)$ in distribution uniformly on any compact subset $[-\mathcal{L},\mathcal{L}{]}_{\mathbb{Z}}$ , $\forall (x^{\ell },n,k,i,\mathcal{L})\in ({\overline{\mathrm{\Omega }}}_{\mathrm{d}},\mathbb{N},\mathbb{Z},{\mathbb{N}}_1^m,\mathbb{N})$ . Similarly, it easily obtains $\tilde{\mathcal{Z}}(k-k_n)$ converges to $\mathsf{Z}(k)$ in distribution uniformly on any compact subset $[-\mathcal{L},\mathcal{L}{]}_{\mathbb{Z}}$ , $\forall (x^{\ell },n,k,i,\mathcal{L})\in ({\overline{\mathrm{\Omega }}}_{\mathrm{d}},\mathbb{N},\mathbb{Z},{\mathbb{N}}_1^m,\mathbb{N})$ . The proof is achieved.

• Remark 1. In literatures (Aouiti and Dridi, 2021; Ayachi, 2021, 2022; Duan et al., 2020; Stamova and Stamov, 2021), the authors discussed the (weighted) pseudo almost periodicity and Stepanov-like pseudo-weighted automorphy to genetic regulatory networks. However, almost no paper regards almost automorphic sequences of discrete-time stochastic gene regulatory lattice networks. Thus, the current work extends the works in papers (Aouiti and Dridi, 2021; Ayachi, 2021, 2022; Duan et al., 2020; Stamova and Stamov, 2021).

Global mean-square $\lambda$ -exponential convergence

In this section, we will explore the global exponential convergence in the mean-square sense of equation (7).

Let $\mathcal{Z}=({\mathcal{M}}_1,\dots ,{\mathcal{M}}_m,{\mathcal{P}}_1,\dots ,{\mathcal{P}}_m{)}^T$ and $\tilde{\mathcal{Z}}=({\tilde{\mathcal{M}}}_1,\dots ,{\tilde{\mathcal{M}}}_m,{\tilde{\mathcal{P}}}_1,\dots ,{\tilde{\mathcal{P}}}_m{)}^T$ be any two solutions of equation (7) with initial boundary conditions

$$\begin{array}{c}\multicolumn{1}{c}{{{\mathcal{M}}_i^{x^{\ell }}(0)={\varphi }_i^{x^{\ell }},{\tilde{\mathcal{M}}}_i^{x^{\ell }}(0)={\tilde{\varphi }}_i^{x^{\ell }},\forall x^{\ell }\in {\mathrm{\Omega }}_{\mathrm{d}},{\mathcal{M}}_i^{x^{\ell }}(k)|}_{x^{\ell }\in \partial {\mathrm{\Omega }}_{\mathrm{d}}}}\\ \multicolumn{1}{c}{={{\tilde{\mathcal{M}}}_i^{x^{\ell }}(k)|}_{x^{\ell }\in \partial {\mathrm{\Omega }}_{\mathrm{d}}}={\phi }_{i,k}^{x^{\ell }}}\end{array}$$

$$\begin{array}{c}\multicolumn{1}{c}{{{\mathcal{P}}_i^{x^{\ell }}(0)={\psi }_i^{x^{\ell }},{\tilde{\mathcal{P}}}_i^{x^{\ell }}(0)={\tilde{\psi }}_i^{x^{\ell }},\forall x^{\ell }\in {\mathrm{\Omega }}_{\mathrm{d}},{\mathcal{P}}_i^{x^{\ell }}(k)|}_{x^{\ell }\in \partial {\mathrm{\Omega }}_{\mathrm{d}}}}\\ \multicolumn{1}{c}{={{\tilde{\mathcal{P}}}_i^{x^{\ell }}(k)|}_{x^{\ell }\in \partial {\mathrm{\Omega }}_{\mathrm{d}}}={\tilde{\phi }}_{i,k}^{x^{\ell }}}\end{array}$$

for all $(k,i)\in ({\mathbb{N}}_0,{\mathbb{N}}_1^m).$

Set $\mathcal{W}=({\mathcal{U}}_1,\dots ,{\mathcal{U}}_n,{\mathcal{V}}_1,\dots ,{\mathcal{V}}_n{)}^T$ with ${\mathcal{U}}_i^{x^{\ell }}(k)={\mathcal{M}}_i^{x^{\ell }}(k)-{\tilde{\mathcal{M}}}_i^{x^{\ell }}(k)$ and ${\mathcal{V}}_i^{x^{\ell }}(k)={\mathcal{P}}_i^{x^{\ell }}(k)-{\tilde{\mathcal{P}}}_i^{x^{\ell }}(k)$ for all $(x^{\ell },k,i)\in ({\mathrm{\Omega }}_{\mathrm{d}},{\mathbb{N}}_0,{\mathbb{N}}_1^m).$ Equation (7) is said to be globally mean-square $\overline{\lambda }$ -exponential convergent if it exists $M>0$ and $0<\iota <1$ such that

$$\begin{array}{c}\multicolumn{1}{c}{\parallel \mathcal{W}(k){\parallel }_{{\mathrm{\Omega }}_{\mathrm{d}}}\le M\parallel \varphi {\parallel }_{{\mathrm{\Omega }}_{\mathrm{d}}}{\overline{\lambda }}^{\iota k},\forall k\ge 0,}\\ \multicolumn{1}{c}{\overline{\lambda }=\underset{1\le i\le m}{max}\{{\overline{\lambda }}_{{\alpha }_i},{\overline{\lambda }}_{{\beta }_i}\}\in (0,1)}\end{array}$$

where

$$\parallel \mathcal{W}(k){\parallel }_{{\mathrm{\Omega }}_{\mathrm{d}}}=\underset{1\le i\le m,x^{\ell }\in {\mathrm{\Omega }}_{\mathrm{d}}}{max}\{\parallel {\mathcal{U}}_i^{x^{\ell }}(k){\parallel }_{{\mathrm{\Omega }}_{\mathrm{d}}},\parallel {\mathcal{V}}_i^{x^{\ell }}(k){\parallel }_{{\mathrm{\Omega }}_{\mathrm{d}}}\},$$

$$\parallel {\mathcal{U}}_i^{x^{\ell }}(k){\parallel }_{{\mathrm{\Omega }}_{\mathrm{d}}}={[\mathbb{E}|{\mathcal{U}}_i^{x^{\ell }}(k{)|}^2]}^{\frac{1}{2}},\parallel {\mathcal{V}}_i^{x^{\ell }}(k){\parallel }_{{\mathrm{\Omega }}_{\mathrm{d}}}={[\mathbb{E}|{\mathcal{V}}_i^{x^{\ell }}(k{)|}^2]}^{\frac{1}{2}}$$

and

$$\parallel \varphi {\parallel }_{{\mathrm{\Omega }}_{\mathrm{d}}}=\underset{1\le i\le m,x^{\ell }\in {\mathrm{\Omega }}_{\mathrm{d}}}{max}\{{[\mathbb{E}|{\varphi }_i^{x^{\ell }}-{\tilde{\varphi }}_i^{x^{\ell }}{|}^2]}^{\frac{1}{2}},{[\mathbb{E}|{\psi }_i^{x^{\ell }}-{\tilde{\psi }}_i^{x^{\ell }}{|}^2]}^{\frac{1}{2}}\}$$

Here, $\iota$ is called the convergent rate of equation (7)

• Theorem 2. Equation (7) is globally mean-square $\lambda$ -exponentially convergent if $(H_1)$ - $(H_2)$ hold.

Proof. By equation (8), it achieves

$$\begin{array}{c}|{\mathcal{U}}_i^{x^{\ell }}(k)|\le \underset{s=0}{\overset{k-1}{\Pi }}{\lambda }_{{\alpha }_i}^{x^{\ell }}(s)|{\varphi }_i^{x^{\ell }}-{\tilde{\varphi }}_i^{x^{\ell }}|+\sum _{q=0}^{k-1}\underset{s=q+1}{\overset{k-1}{\Pi }}{\lambda }_{{\alpha }_i}^{x^{\ell }}(s)\hfill \\ \multicolumn{1}{c}{\times \sum _{l=0}^{{\nu }_q}{\theta }_{{\alpha }_i}^{x^{\ell }}(l,q)[|{\mathrm{\Upsilon }}_i({\mathcal{M}}_i^{x^{\ell }}(q-l))-{\mathrm{\Upsilon }}_i({\tilde{\mathcal{M}}}_i^{x^{\ell }}(q-l))|}\\ \multicolumn{1}{c}{+\sum _{j=1}^m{\overline{b}}_{\mathrm{ij}}|f_j({\mathcal{P}}_j^{x^{\ell }}(q-l))-f_j({\tilde{\mathcal{P}}}_j^{x^{\ell }}(q-l))|}\\ \multicolumn{1}{c}{+\sum _{j=1}^m{h}^{-1}{\mathrm{\Delta }}_h|{\mathbb{W}}_{1j}(q-l){\overline{\omega }}_{\mathrm{ij}}|}\\ \multicolumn{1}{c}{{{\times |{\sigma }_{\mathrm{ij}}({\mathcal{P}}_j^{x^{\ell }}(q-l))-{\sigma }_{\mathrm{ij}}({\tilde{\mathcal{P}}}_j^{x^{\ell }}(q-l))|]|}^2\}}^{\frac{1}{2}}}\\ \multicolumn{1}{c}{\le {\overline{\lambda }}_{{\alpha }_i}^k\parallel \varphi {\parallel }_{{\mathrm{\Omega }}_{\mathrm{d}}}+\sum _{q=0}^{k-1}{\overline{\lambda }}_{{\alpha }_i}^{k-q-1}\sum _{l=0}^{{\nu }_q}|{\theta }_{{\alpha }_i}^{x^{\ell }}(l,q)|}\\ \multicolumn{1}{c}{[D_i^{*}|{\mathcal{U}}_i^{x^{\ell }}(q-l)|+\sum _{j=1}^m{\overline{b}}_{\mathrm{ij}}{L}_j^f|{\mathcal{V}}_j^{x^{\ell }}(q-l)|}\\ \multicolumn{1}{c}{+\sum _{j=1}^m{h}^{-1}{\overline{\omega }}_{\mathrm{ij}}{L}_{\mathrm{ij}}^{\sigma }|{\mathcal{V}}_j^{x^{\ell }}(q-l)||{\mathrm{\Delta }}_h{\mathbb{W}}_{1j}(q-l)|]}\end{array}$$

for all $(x^{\ell },k,i)\in ({\mathrm{\Omega }}_{\mathrm{d}},{\mathbb{N}}_0,{\mathbb{N}}_1^m).$ Thus, it derives

$$\begin{array}{c}\parallel {\mathcal{U}}_i^{x^{\ell }}(k){\parallel }_{{\mathrm{\Omega }}_{\mathrm{d}}}\le {\overline{\lambda }}^k\parallel \varphi {\parallel }_{{\mathrm{\Omega }}_{\mathrm{d}}}+\sum _{q=0}^{k-1}{\overline{\lambda }}_{{\alpha }_i}^{k-q-1}\sum _{l=0}^{{\nu }_q}|{\theta }_{{\alpha }_i}^{x^{\ell }}(l,q)|\hfill \\ \multicolumn{1}{c}{\times [D_i^{*}+\sum _{j=1}^m{\overline{b}}_{\mathrm{ij}}{L}_j^f+2h^{-\frac{1}{2}}\sum _{j=1}^m{\overline{\omega }}_{\mathrm{ij}}{L}_{\mathrm{ij}}^{\sigma }]\parallel \mathcal{W}(q-l){\parallel }_{{\mathrm{\Omega }}_{\mathrm{d}}},}\end{array}$$

(18)

where $(x^{\ell },k,i)\in ({\mathrm{\Omega }}_{\mathrm{d}},{\mathbb{N}}_0,{\mathbb{N}}_1^m).$

Similarly, it gets

$$\begin{array}{c}\parallel {\mathcal{V}}_i^{x^{\ell }}(k){\parallel }_{{\mathrm{\Omega }}_{\mathrm{d}}}\le {\overline{\lambda }}^k\parallel \varphi {\parallel }_{{\mathrm{\Omega }}_{\mathrm{d}}}+\sum _{q=0}^{k-1}{\overline{\lambda }}_{{\beta }_i}^{k-q-1}\sum _{l=0}^{{\nu }_q}|{\theta }_{{\beta }_i}^{x^{\ell }}(l,q)|\hfill \\ \multicolumn{1}{c}{\times [K_i^{*}+{\overline{d}}_i+2h^{-\frac{1}{2}}\sum _{j=1}^m{\overline{\varpi }}_{\mathrm{ij}}{L}_{\mathrm{ij}}^{\eta }]\parallel \mathcal{W}(q-l){\parallel }_{{\mathrm{\Omega }}_{\mathrm{d}}},}\end{array}$$

(19)

where $(x^{\ell },k,i)\in ({\mathrm{\Omega }}_{\mathrm{d}},{\mathbb{N}}_0,{\mathbb{N}}_1^m).$

Owing to $(H_4)$ , it has $M>1$ and $0<\iota <1$ ensuring

$$\{\begin{array}{c}\frac{1}{M}+{max}_{1\le i\le m}\frac{2{\overline{\lambda }}_{{\alpha }_i}^{-\iota (Q^{*}+1)}}{\mathrm{-}{a}_i^{*}(1-{\overline{\lambda }}_{{\alpha }_i}^{1-\iota })}[D_i^{*}+\sum _{j=1}^m{\overline{b}}_{\mathrm{ij}}{L}_j^f+2h^{-\frac{1}{2}}\sum _{j=1}^m{\overline{\omega }}_{\mathrm{ij}}{L}_{\mathrm{ij}}^{\sigma }]<1,\hfill \\ \multicolumn{1}{c}{\frac{1}{M}+{max}_{1\le i\le m}\frac{2{\overline{\lambda }}_{{\beta }_i}^{-\iota (Q^{*}+1)}}{\mathrm{-}{c}_i^{*}(1-{\overline{\lambda }}_{{\beta }_i}^{1-\iota })}[K_i^{*}+{\overline{d}}_i+2h^{-\frac{1}{2}}\sum _{j=1}^m{\overline{\varpi }}_{\mathrm{ij}}{L}_{\mathrm{ij}}^{\eta }]<1.}\end{array}$$

(20)

Supposing that $\parallel \mathcal{W}(k){\parallel }_{{\mathrm{\Omega }}_{\mathrm{d}}}\le M\parallel \varphi {\parallel }_{{\mathrm{\Omega }}_{\mathrm{d}}}{\overline{\lambda }}^{\iota k}$ for all $k\in {\mathbb{N}}_0.$ If not, there must be one of the following cases holds.

• 1. It exists $i_0\in {\mathbb{N}}_1^m$ and $k_0\in \mathbb{N}$ causing

$$\begin{array}{c}\multicolumn{1}{c}{\parallel {\mathcal{U}}_{i_0}^{x^{\ell }}(k_0){\parallel }_{{\mathrm{\Omega }}_{\mathrm{d}}}>M\parallel \varphi {\parallel }_{{\mathrm{\Omega }}_{\mathrm{d}}}{\overline{\lambda }}^{\iota k_0};\parallel \mathcal{W}(k){\parallel }_{{\mathrm{\Omega }}_{\mathrm{d}}}\le M\parallel \varphi {\parallel }_{{\mathrm{\Omega }}_{\mathrm{d}}}{\overline{\lambda }}^{\iota k},}\\ \multicolumn{1}{c}{\forall k\in [0,k_0{)}_{\mathbb{Z}}.}\end{array}$$

• 2. It exists $i_1\in {\mathbb{N}}_1^m$ and $k_1\in \mathbb{N}$ causing

$$\begin{array}{c}\multicolumn{1}{c}{\parallel {\mathcal{V}}_{i_1}^{x^{\ell }}(k_1){\parallel }_{{\mathrm{\Omega }}_{\mathrm{d}}}>M\parallel \varphi {\parallel }_{{\mathrm{\Omega }}_{\mathrm{d}}}{\overline{\lambda }}^{\iota k_1};\parallel \mathcal{W}(k){\parallel }_{{\mathrm{\Omega }}_{\mathrm{d}}}\le M\parallel \varphi {\parallel }_{{\mathrm{\Omega }}_{\mathrm{d}}}{\overline{\lambda }}^{\iota k},}\\ \multicolumn{1}{c}{\forall k\in [0,k_1{)}_{\mathbb{Z}}.}\end{array}$$

In the light of equation (18), it results in

$$\begin{array}{c}\parallel {\mathcal{U}}_{i_0}^{x^{\ell }}(k_0){\parallel }_{{\mathrm{\Omega }}_{\mathrm{d}}}\le {\overline{\lambda }}^{k_0}\parallel \varphi {\parallel }_{{\mathrm{\Omega }}_{\mathrm{d}}}+\sum _{q=0}^{k_0-1}{\overline{\lambda }}_{{\alpha }_{i_0}}^{k_0-q-1}\sum _{l=0}^{{\nu }_q}|{\theta }_{{\alpha }_{i_0}}^{x^{\ell }}(l,q)|\hfill \\ \multicolumn{1}{c}{\times [D_{i_0}^{*}+\sum _{j=1}^m{\overline{b}}_{i_{0}j}{L}_j^f+2h^{-\frac{1}{2}}\sum _{j=1}^m{\overline{\omega }}_{i_{0}j}{L}_{i_{0}j}^{\sigma }]}\\ \multicolumn{1}{c}{\parallel \mathcal{W}(q-l){\parallel }_{{\mathrm{\Omega }}_{\mathrm{d}}}}\\ \multicolumn{1}{c}{\le {\overline{\lambda }}^{k_0}\parallel \varphi {\parallel }_{{\mathrm{\Omega }}_{\mathrm{d}}}+\sum _{q=0}^{k_0-1}{\overline{\lambda }}_{{\alpha }_{i_0}}^{(1-\iota )(k_0-q-1)}\sum _{l=0}^{{\nu }_q}|{\theta }_{{\alpha }_{i_0}}^{x^{\ell }}(l,q)|}\\ \multicolumn{1}{c}{\times [D_{i_0}^{*}+\sum _{j=1}^m{\overline{b}}_{i_{0}j}{L}_j^f+2h^{-\frac{1}{2}}\sum _{j=1}^m{\overline{\omega }}_{i_{0}j}{L}_{i_{0}j}^{\sigma }]}\\ \multicolumn{1}{c}{M\parallel \varphi {\parallel }_{{\mathrm{\Omega }}_{\mathrm{d}}}{\overline{\lambda }}^{\iota (q-l)}}\\ \multicolumn{1}{c}{\le {\overline{\lambda }}^{k_0}\parallel \varphi {\parallel }_{{\mathrm{\Omega }}_{\mathrm{d}}}+\frac{2{\overline{\lambda }}_{{\alpha }_{i_0}}^{-\iota (Q^{*}+1)}}{\mathrm{-}{a}_{i_0}^{*}(1-{\overline{\lambda }}_{{\alpha }_{i_0}}^{1-\iota })}[D_{i_0}^{*}}\\ \multicolumn{1}{c}{+\sum _{j=1}^m{\overline{b}}_{i_{0}j}{L}_j^f+2h^{-\frac{1}{2}}\sum _{j=1}^m{\overline{\omega }}_{i_{0}j}{L}_{i_{0}j}^{\sigma }]M\parallel \varphi {\parallel }_{{\mathrm{\Omega }}_{\mathrm{d}}}{\overline{\lambda }}^{\iota k_0}}\\ \multicolumn{1}{c}{\le \{\frac{1}{M}{\overline{\lambda }}^{(1-\iota )k_0}+\frac{2{\overline{\lambda }}_{{\alpha }_{i_0}}^{-\iota (Q^{*}+1)}}{\mathrm{-}{a}_{i_0}^{*}(1-{\overline{\lambda }}_{{\alpha }_{i_0}}^{1-\iota })}[D_{i_0}^{*}}\\ \multicolumn{1}{c}{+\sum _{j=1}^m{\overline{b}}_{i_{0}j}{L}_j^f+2h^{-\frac{1}{2}}\sum _{j=1}^m{\overline{\omega }}_{i_{0}j}{L}_{i_{0}j}^{\sigma }]\}M\parallel \varphi {\parallel }_{{\mathrm{\Omega }}_{\mathrm{d}}}{\overline{\lambda }}^{\iota k_0}}\\ \multicolumn{1}{c}{\le M\parallel \varphi {\parallel }_{{\mathrm{\Omega }}_{\mathrm{d}}}{\overline{\lambda }}^{\iota k_0}.((17)\mathrm{is}\mathrm{used})}\end{array}$$

(21)

This induces a confliction with $(1)$ . Similarly, by $(16)$ , it acquires

$$\begin{array}{c}\parallel {\mathcal{V}}_{i_1}^{x^{\ell }}(k_1){\parallel }_{{\mathrm{\Omega }}_{\mathrm{d}}}\le \{\frac{1}{M}{\overline{\lambda }}^{(1-\iota )k_1}+\frac{2{\overline{\lambda }}_{{\beta }_{i_1}}^{-\iota (Q^{*}+1)}}{\mathrm{-}{c}_{i_1}^{*}(1-{\overline{\lambda }}_{{\beta }_{i_1}}^{1-\iota })}[K_{i_1}^{*}\hfill \\ \multicolumn{1}{c}{+{\overline{d}}_{i_1}+2h^{-\frac{1}{2}}\sum _{j=1}^m{\overline{\varpi }}_{i_{1}j}{L}_{i_{1}j}^{\eta }]\}M\parallel \varphi {\parallel }_{{\mathrm{\Omega }}_{\mathrm{d}}}{\overline{\lambda }}^{\iota k_1}}\\ \multicolumn{1}{c}{\le M\parallel \varphi {\parallel }_{{\mathrm{\Omega }}_{\mathrm{d}}}{\overline{\lambda }}^{\iota k_1}.}\end{array}$$

(22)

This is also a confliction with $(2)$ . Then $\parallel \mathcal{W}(k){\parallel }_{{\mathrm{\Omega }}_{\mathrm{d}}}\le M\parallel \varphi {\parallel }_{{\mathrm{\Omega }}_{\mathrm{d}}}{\overline{\lambda }}^{\iota k}$ for all $k\in {\mathbb{N}}_0.$ That is, equation (7) is globally mean-square $\lambda$ -exponentially convergent. The proof is finished.

• Remark 2. By employing time exponential Euler differences, literatures (Zhang and Xu, 2019; Zhang et al., 2020) discussed exponential convergence of discrete-time stochastic models without diffusions. Apparently, the achieved difference models in literatures (Zhang and Xu, 2019; Zhang et al., 2020) can be viewed as a special case of model (4) to some extent. Thus, the work of this article supplements and extends the corresponding results in papers (Zhang and Xu, 2019; Zhang et al., 2020).

Illustrative example

Considering the following nonlocal stochastic genetic regulatory networks with reaction diffusions in the form of

$$\{\begin{array}{c}{{}^{c}D}_{7p}^{0.7}\mathcal{M}(t,x)=0.15\frac{{\partial }^2\mathcal{M}(t,x)}{\partial x^2}-18\mathcal{M}(t,x)\hfill \\ \multicolumn{1}{c}{+\stackrel{2}{\sin }(\sqrt{3}t+x)f(\mathcal{P}(t,x))}\\ \multicolumn{1}{c}{+2\sigma (\mathcal{P}(t,x))\frac{\mathrm{d}{\mathbb{W}}_1(t)}{\mathrm{d}t}+0.2\stackrel{2}{\sin }\left(\frac{1}{2+\cos (\sqrt{5}t)+\cos (\sqrt{7}t)}\right),}\\ \multicolumn{1}{c}{{{}^{c}D}_{7p}^{0.6}\mathcal{P}(t,x)=0.1\frac{{\partial }^2\mathcal{P}(t,x)}{\partial x^2}-20\mathcal{P}(t,x)+0.2|\cos (\sqrt{17}t+x)|}\\ \multicolumn{1}{c}{\times \mathcal{M}(t,x)+3\eta (\mathcal{M}(t,x))\frac{\mathrm{d}{\mathbb{W}}_2(t)}{\mathrm{d}t},}\end{array}$$

(23)

where $(t,x)\in ((7p,7p+7],(0,10))$ , $p\in {\mathbb{Z}}_0$ ; $f(s)=\frac{0.01s^2}{1+s^2}$ , $\sigma (s)=0.01|\cos s|$ , $\eta (s)=0.01|\sin s|$ for $s\in \mathbb{R}$ . The initial boundary conditions are depicted as

$$\{\begin{array}{c}\mathcal{M}(0,x)=0.2x\sin (0.1(10-x)),x\in (0,10);\hfill \\ \multicolumn{1}{c}{\mathcal{M}(t,0)=\mathcal{M}(t,10)=0,t\in {\mathbb{R}}_0,}\\ \multicolumn{1}{c}{\mathcal{P}(0,x)=0.3x\sin (0.1(10-x)),x\in (0,10);}\\ \multicolumn{1}{c}{\mathcal{P}(t,0)=\mathcal{P}(t,10)=0,t\in {\mathbb{R}}_0.}\end{array}$$

Taking $h=1$ and $h_1=0.5$ . It obtains the lattice equations corresponding to equation (9) in the shape of

$$\{\begin{array}{l}{ℳ}^{\iota }(k+1)={\lambda }_{0.7}^{\iota }(k){ℳ}^{\iota }(k)+\sum _{l={\mu }_k}^k{\theta }_{0.7}^{\iota }(k-l,k)[\mathrm{\Upsilon }({ℳ}^{\iota }(l))+{\sin }^2(\sqrt{3}l+\iota )f(\mathcal{P}(l,\iota ))\hfill \\ +2\sigma (\mathcal{P}(l,\iota ))h^{-1}{\mathrm{\Delta }}_h{\mathbb{W}}_1(l)+0.2{\sin }^2\left(\frac{1}{2+\cos (\sqrt{5}l)+\cos (\sqrt{7}l)}\right)],\hfill \\ {\mathcal{P}}^{\iota }(k+1)={\lambda }_{0.6}^{\iota }(k){\mathcal{P}}^{\iota }(k)+\sum _{l={\mu }_k}^k{\theta }_{0.6}^{\iota }(k-l,k)[\tilde{\mathrm{\Upsilon }}({\mathcal{P}}^{\iota }(l))+0.2|\cos (\sqrt{17}l+\iota )|\hfill \\ \times ℳ(l,\iota )+3\eta (ℳ(l,\iota ))h^{-1}{\mathrm{\Delta }}_h{\mathbb{W}}_2(l)]\hfill \end{array}$$

(24)

with initial condition

$${\mathcal{M}}^{\iota }(0)=0.2\iota \sin (0.1(10-\iota )),{\mathcal{P}}^{\iota }(0)=0.3\iota \sin (0.1(10-\iota ))$$

for $\iota =0.5\ell$ , $\ell =0,1,\dots ,20$ and boundary condition

$${\mathcal{M}}^{\iota }(k){|}_{\iota \in \{0,10\}}={\mathcal{P}}^{\iota }(k){|}_{\iota \in \{0,10\}}=0,$$

${\nu }_k=k-{\mu }_k$ , ${\mu }_k=\underset{p\in {\mathbb{Z}}_0}{max}\{7p:7p\le k\}$ , ${\lambda }_{0.7}^{\iota }$ , ${\theta }_{0.7}^{\iota }$ , ${\lambda }_{0.6}^{\iota }$ and ${\theta }_{0.6}^{\iota }$ are defined as those in (4) with $a^{*}=19.2$ and $c^{*}=20.8$ , respectively;

$$\begin{array}{c}\multicolumn{1}{c}{\mathrm{\Upsilon }({\mathcal{M}}^{\iota }(l))=\frac{0.15}{0.25}[{\mathcal{M}}^{\iota +0.5}(l)+{\mathcal{M}}^{\iota -0.5}(l)],}\\ \multicolumn{1}{c}{\tilde{\mathrm{\Upsilon }}({\mathcal{P}}^{\iota }(l))=\frac{0.1}{0.25}[{\mathcal{P}}^{\iota +0.5}(l)+{\mathcal{P}}^{\iota -0.5}(l)]}\end{array}$$

for all $l,k\in {\mathbb{Z}}_0$ and $\iota =0.5\ell$ , $\ell =1,\dots ,19$ .

By a calculation, it gets ${\rho }_1\le 0.8882,$ ${\rho }_2\le 0.1705,$ ${\mathcal{Z}}_0\le 1.5250.$ Therefore, conditions $(H_1)$ - $(H_4)$ hold and equation (24) admits a unique almost automorphic sequence solution, which is globally mean-square $\lambda$ -exponentially convergent (see Figures 1–6).

Figure 1.

Almost automorphy of ${\mathcal{M}}^{\iota }(·)$ ( $\iota =3.5,7.5$ ) to equation (24).

Figure 2.

Almost automorphy of ${\mathcal{P}}^{\iota }(·)$ ( $\iota =3.5,7.5$ ) to equation (24).

Figure 3.

Almost automorphy of ${\mathcal{M}}^{\iota }(·)$ ( $\iota =0,0.5,\dots ,10$ ) to equation (24).

Figure 4.

Almost automorphy of ${\mathcal{P}}^{\iota }(·)$ ( $\iota =0,0.5,\dots ,10$ ) to equation (24).

Figure 5.

$\lambda$ -exponential convergence of ${\mathcal{M}}^{\iota }(·)$ ( $\varsigma =0,0.5,\dots ,10$ ) to equation (24).

Figure 6.

$\lambda$ -exponential convergence of ${\mathcal{P}}^{\iota }(·)$ ( $\iota =0,0.5,\dots ,10$ ) to equation (24).

Figures 1–6 show the trajectories of the solutions to equation (24) in two- and three-dimensional spaces. Figures 1 and 2 depict the almost automorphic oscillations of states $\mathcal{M}$ and $\mathcal{P}$ of equation (24) with space variable $\iota =3.5,7.5$ . Figures 3 to 6 picture three-dimensional overall views of states $\mathcal{M}$ and $\mathcal{P}$ of equation (24) with space variable $\iota \in [0,10{]}_{\mathbb{Z}}$ and time variable $k\in [0,50{]}_{\mathbb{Z}}$ .

• Remark 3. Exponential Euler difference methods were proposed and had been applied into the study of applied numerical time-varying models, please refer Mohamad and Gopalsamy (2000), Zhang and Li (2022a), Huang et al. (2014) and Zhang and Li (2022b). For the fractional-order multivariable models, few reports are published up to now, and the work of this paper will fill this gap.

Conclusion and perspectives

By employing the finite difference and Mittag–Leffler time Euler difference techniques, a novel lattice model for nonlocal stochastic genetic regulatory networks with reaction diffusions is set up. Furthermore, the existence of a unique global bounded almost automorphic sequence solution in distribution and global exponential convergence in the mean-square sense have been investigated for the achieved stochastic lattice model.

According to the works in this literature, it will be many problems worthy of further discussion.

1. This paper only discusses ${\alpha }_i,{\beta }_i\in (0,1](i=1,2,\dots ,m)$ , and other cases are supposed to be studied.

2. Reimann–Liouville derivatives should be studied in the further.

3. Other dynamics ought to be discussed, for example, bifurcation, chaos, and control.