Anti-periodic motion and mean-square exponential convergence of nonlocal discrete-time stochastic competitive lattice neural networks with fuzzy logic

Abstract

This paper firstly establishes the discrete-time lattice networks for nonlocal stochastic competitive neural networks with reaction diffusions and fuzzy logic by employing a mix techniques of finite difference to space variables and Mittag-Leffler time Euler difference to time variable. The proposed networks consider both the effects of spatial diffusion and fuzzy logic, whereas most of the existing literatures focus only on discrete-time networks without spatial diffusion. Firstly, the existence of a unique ω-anti-periodic in distribution to the networks is addressed by employing Banach contractive mapping principle and the theory of stochastic calculus. Secondly, global exponential convergence in mean-square sense to the networks is discussed on the basis of constant variation formulas for sequences. Finally, an illustrative example is used to show the feasible of the works in the current paper with the help of MATLAB Toolbox. The work in this paper is pioneering in this regard and it has created a certain research foundations for future studies in this area.

Keywords

Lattice reaction diffusion stochastic finite difference method

1 Introduction

Dynamic neural networks with feedforward and feedback connections between neural layers have potential applications in areas such as visual processing and pattern recognition. To embody the competitive and cooperative properties of neurons, Lemmon et al. [1] discussed a kind of lateral inhibition networks amended by external inputs. Such neural networks characterize the behaviors of dynamics at two activity levels, namely, short-term memory (STM) and long-term memory (LTM). Meyer-Bäse et al. [2 –4] had extended former patterns and have presented a form of competitive neural networks (CNNs) with different time scales. According to the different time scales, short-time memory (STM) is quicker than long-time memory (LTM). Thus short-time memory (STM) means fast neural activity, while long-time memory (LTM) represents the slow activity of unsupervised synaptic modifications induced by inputs. Numerous investigators have researched the issues of synchronization and stability of CNNs, as well as other dynamics behaviors in recent years, see [5 –10]. On the other hand, Yang and Yang [11] in 1996 proposed a novel fuzzy cellular neural network, which incorporates fuzzy logic into the architecture of a cellular neural network. Fuzzy neural networks possess fuzzy logic for template inputs and/or outputs, in addition to summation of product operations. For more details about fuzzy CNNs, please see papers [5, 6].

In biologically based neural systems, the concentration of constituents is not uniform, resulting in diffusion of cytoplasm from higher to lower concentrations. It is said to be diffusion. Since neural systems can be build by scaling down and modeling biological neural networks, diffusion ought to be inserted to neural network patterns. Accordingly, reaction diffusion neural networks were formulated and have shown significant prospects for spatio-temporal pattern storage and matching. Recently, stochastic models have been broadly investigated since they are commonly found in people’s daily lives. Stochastic perturbations in neural networks not only separate neural networks from deterministic neural networks, but also can bring about substantial modifications in dynamic actions of neural networks. In general, the behavior of stochastic systems is highly reliant on time and spatial dependence. Consequently, reaction diffusion is necessary to be taken into account, and this induces the investigations of (stochastic) reaction diffusion CNNs [5 , 12], concretely see the researching topics on the intermittent stabilization [5], exponential synchronization [12], etc.

During the last ten years, a number of CNNs were portrayed in terms of fractional-order systems of dynamics [8 , 13–16], for example, synchronization [13, 14] and multi-stability and stabilization [15, 16], etc. Further, discrete-time neural networks [17 –19] are better fitted for real-time implementations. Firstly, appropriate technology can be used to implement digital controllers rather than analog ones. Secondly, 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 [20 –24. 27]. Thirdly, many processes have a certain regularity, so the study of periodic 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. In particular, anti-periodic motions play an important role in these fields as a special case of periodic phenomena. Up to now, there are few papers focusing on the study of anti-periodic sequences to CNNs in papers [28 –30]. However, almost no paper discusses anti-periodic oscillations of discrete-time CNNs with stochastic perturbations or reaction diffusions. This case triggers one of the main motivations for the discussions of this article.

Recently, there are many researchers pay attention to the discrete-time neural networks. Adhira et al. [17] learned the robust extended dissipative synchronization for a class of delayed discrete-time neural networks through non-fragile state-feedback control by applying a new summation inequality based on extended reciprocally convex matrix inequality. In [31], authors dealt with the global exponential synchronization of a class of discrete-time high-order switched neural networks with time-varying delays by designing the state feedback controllers and used it in audio encryption. By employing the Lyapunov-Krasovskii functional method, the Cauchy matrix inequality and the Schur complement lemma, Huong [32] considered the event-triggeredH_∞ control problem for uncertain neural networks involving time-varying delays and disturbances. Based on the above analysis, we can observe that the exists findings are about neural networks with discrete-time. For all we know, there is no paper concentrating on the study of discrete-time and discrete-space CNNs. Therefore, this paper considers a class of nonlocal CNNs involving time variable and space variable, which extends and implements some results in literatures [17–19 , 31–34].

Recently, exponential Euler differences have become very significant researching topics in literatures [20 –23]. From the viewpoint of numerical computations, lots of numerical methods had been proposed, e.g., exponential Euler integrators [20 –22], exponential Runge-Kutta methods [35], exponential Rosenbrock-type methods [36], etc. From the viewpoint of the theory in difference equations, numerous behaviours of the difference equations governed by exponential Euler differences of integer order were considered by more and more scholars, e.g., stochastic dynamics [20], almost periodicity [21], etc. Furthermore, the works of the predecessors [24 –26] tell us exponential Euler differences can be applied into the discussions of numerical fractional-order differential equations [37 –39]. Correspondingly, it is called Mittag-Leffler Euler differences. On the other hand, finite difference method is also an interesting method for solving partial differential equations [40 –42]. For example, finite difference method is used to study the transverse vibrations of axially moving strings [40]. Ali and Hawwa [41] used the finite difference method to study the dynamics of axially moving beams. The literature [42] applied a meshless method based on the generalized finite difference method to the study of three-dimensional elliptic interface problems. In this paper, we will use exponential Euler difference and finite difference methods to discretize the time and space variables of a nonlocal stochastic competing neural network, respectively.

By employing a mix techniques of the finite difference method and Mittag-Leffler Euler time difference, the objective of the current paper is to achieve the discrete-time and discrete-space schemes corresponding to a continuous-time and continuous-space stochastic fuzzy CNNs with reaction diffusions. And on this basis, the existence of a unique bounded anti-periodic sequence solution in distribution and global exponential stability in the mean-square sense are 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 models is newly introduced.2) The existence of a unique bounded anti-periodic 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 [20–22 , 28–30].

The organization of the rest is as follows. In Section 2, a stochastic lattice CNNs is achieved by using the finite difference methods and Mittag-Leffler Euler difference techniques. The existence of a unique bounded anti-periodic sequence solution in distribution and global exponential convergence in the mean-square sense are discussed in Sections 3-4. In Section 5, an 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 Section 6.

Symbols: $ℝ^{n}$ denotes the space ofn-dimensional real vectors; $ℤ$ is the field of integral numbers; $ℕ_{0} = {0, 1, 2, \dots}$ ; $ℕ = ℕ_{0} \ {0}$ ; $ℕ_{a}^{b} = {a, a + 1, \dots, b}$ for any $a, b \in ℤ$ ;I_J = I ∩ J, $\forall I, J \subseteq ℝ$ . Let $A_{1}, A_{2}, \dots, A_{N}$ be some sets,x₁ ∈ A₁,x₂ ∈ A₂, …, $x_{N} \in A_{N} \Leftrightarrow (x_{1}, x_{2}, \dots, x_{N}) \in (A_{1}, A_{2}, \dots, A_{N})$ .

2 Discrete-time stochastic lattice CNNs with fuzzy logic

Let us introduce the relative conception of fractional calculus in literature [43].

The α-order Caputo fractional derivative of $x \in C^{n} ([a, b], ℝ^{n})$ is defined by $c D_{a}^{α} x (t) = \frac{1}{Γ (n - α)} \int_{a}^{t} \frac{x^{(n)} (s)}{(t - s)^{α - n + 1}} d s, \forall t \in [a, b],$ where $0 < n - 1 < α < n, n \in ℤ_{0}$ . Let α > 0 and $x \in C^{1} ([a, b], ℝ)$ . Then

$lim_{α \to 0^{+}} c D_{a}^{α} x (t) = x (t) - x (a)$ .

$lim_{α \to n} c D_{a}^{α} x (t) = x^{(n)} (t)$ , $n \in ℕ$ .

The Riemann-Liouville fractional integral of $x \in C ([a, b], ℝ)$ is given by $I_{a}^{α} x (t) = \frac{1}{Γ (α)} \int_{a}^{t} (t - s)^{α - 1} x (s) d s, \forall t \in [a, b],$ where α > 0. Let α ∈ (n - 1,n], $n \in ℕ$ . Then

If $x \in C ([a, b], ℝ)$ , then $c D_{a}^{α} I_{a}^{α} x (t) = x (t)$ .

If $x \in C^{n} ([a, b], ℝ)$ , then $\begin{matrix} lim_{α \to n} I_{a}^{α} c D_{a}^{α} x (t) = & x (t) - \sum_{k = 0}^{n - 1} \frac{x^{(k)} (a)}{k!} (t - a)^{k}, \\ \forall t \in [a, b] . \end{matrix}$

The Mittag-Leffler functions are described as $E_{α} (z) : = \sum_{k = 0}^{\infty} \frac{z^{k}}{Γ (α k + 1)},$ $E_{α, β} (z) : = \sum_{k = 0}^{\infty} \frac{z^{k}}{Γ (α k + β)},$ where $z, β \in ℝ$ and α > 0.

Lemma 2.1 ([43]). $\frac{d}{dz} [z^{α} E_{α, α + 1} (λ z^{α})] = z^{α - 1} E_{α, α} (λ z^{α})$ , where $α, λ, z \in ℝ$ .

Lemma 2.2 ([44]). If α ∈ (0, 1] and $λ \in ℝ$ , then

E_α (0) =1, $E_{α, α} (0) = \frac{1}{Γ (α)}$ .

E_α (λt^α) ∈ (0, 1) for λ < 0 andE_α (λt^α) ∈ (1, + ∞) for λ > 0, ∀t > 0.

$E_{α} (λ t_{1}^{α}) > E_{α} (λ t_{2}^{α})$ and $E_{α, α} (λ t_{1}^{α}) > E_{α, α} (λ t_{2}^{α})$ for λ < 0 andt₁ < t₂.

$E_{α} (λ t_{1}^{α}) < E_{α} (λ t_{2}^{α})$ and $E_{α, α} (λ t_{1}^{α}) < E_{α, α} (λ t_{2}^{α})$ for λ > 0 andt₁ < t₂.

This paper considers nonlocal stochastic fuzzy CNNs with reaction diffusions in the shape of

${\begin{matrix} STM : ɛ {c D}_{t_{p}}^{α_{i}} u_{i} (t, x) & = & \sum_{q = 1}^{N} \frac{\partial}{\partial x_{q}} [ξ_{iq} \frac{\partial u_{i} (t, x)}{\partial x_{q}}] - c_{i} (x) u_{i} (t, x) + \sum_{j = 1}^{n} a_{ij} (t, x) g_{j} (u_{j} (t, x)) \\ + b_{i} (t, x) \sum_{l = 1}^{m} χ_{l} M_{il} (t, x) + ⋁_{j = 1}^{n} ϱ_{ij} (t, x) g_{j} (u_{j} (t, x)) + ⋀_{j = 1}^{n} ς_{ij} (t, x) \\ \times g_{j} (u_{j} (t, x)) + I_{i} (t, x) + \sum_{j = 1}^{n} μ_{ij} (t, x) σ_{ij} (u_{j} (t, x)) \frac{d B_{j} (t)}{d t}, \\ LTM : {c D}_{t_{p}}^{β_{i}} M_{il} (t, x) & = & - d_{i} (x) M_{il} (t, x) + χ_{l} g_{i} (u_{i} (t, x)), \end{matrix}$ (2.1) where $(t, x) \in J_{p} \times Ω$ , $J_{p} = (t_{p}, t_{p + 1}]$ ,t₀ = 0, $lim_{p \to \pm \infty} t_{p} = \pm \infty$ ,t_p ≤ t_p+1, $p \in ℤ$ ; $Ω = {x = (x_{1}, x_{2}, \dots, x_{N})^{T} \in ℝ^{ℓ} : r_{1 q} \leq x_{q} \leq r_{2 q}, r_{1 q}, r_{2 q} \in ℝ, q = 1, 2, \dots, N}$ ; and denote Caputo frctional-order derivatives from initial point 0, α_i, β_i ∈ (0, 1];n andm stand for the number of STM states and the constant external stimulus, respectively;u_i denotes the neuron of current activity level andM_il is synaptic efficiency;g_i represents the output of neurons;c_i > 0 andd_i > 0 show the time constant of neuron and the disposable scaling constant, respectively;b_i and χ_l describe the strength and the constant of external stimulus, respectively;a_ij and ${\tilde{a}}_{ij}$ show the connection weight and synaptic weight of delayed feedback between theith neuron andjth neuron, respectively; ɛ > 0 stands for the time scale of STM state; $Θ : = (ξ_{iq})_{n \times N} \geq 0$ stands for the transmission diffusion matrixes;I_i is the constant external input; ϱ_ij and ς_ij are the elements of fuzzy feedback MIN template and fuzzy feedback MAX template, respectively; ⋀ and ⋁ denote the fuzzy AND and fuzzy OR operations, respectively;B_j is the Brownian motion on a complete probability space;i,j = 1, 2, …,n,l = 1, 2, …,m.

Let $S_{i} = \sum_{l = 1}^{m} M_{il} χ_{l} = M_{i}^{T} χ$ fori = 1, 2, …,n, whereM_i = (M_i1,M_i2, …,M_im) ^T and χ = (χ₁, χ₂, …, χ_m) ^T. No loss of generality, the input stimulus vector χ can be normalized by unit magnitude |χ|² = 1. ThenEquation (2.1) is converted into

where $(t, x, p, i) \in (J_{p}, Ω, ℤ, ℕ_{1}^{n})$ . The corresponding initial boundary conditions are depicted by ${\begin{matrix} u_{i} (0, x) = φ_{i} (x), x \in Ω; u_{i} (t, x) |_{x \in \partial Ω} = 0, \\ S_{i} (0, x) = ψ_{i} (x), x \in Ω; S_{i} (t, x) |_{x \in \partial Ω} = 0, \end{matrix}$ (2.3) where $(t, i) \in (ℝ, ℕ_{1}^{n})$ .

Let $h_{q} = \frac{r_{2 q} - r_{1 q}}{N_{q}}$ for some $N_{q} \in ℕ$ , $q = 1, 2, \dots, N$ . Define $x_{q}^{ℓ_{q}} = r_{1 q} + ℓ_{q} h_{q}$ for all $(ℓ_{q}, q) \in (ℕ_{0}^{N_{q}}, ℕ_{1}^{N})$ . Set $\partial Ω_{d} = {\bar{Ω}}_{d} ∖ Ω_{d}$ , where $\begin{matrix} {\bar{Ω}}_{d} = & {x^{ℓ} = (x_{1}^{ℓ_{1}}, \dots, x_{N}^{ℓ_{N}})^{T} \in ℝ^{N} : x_{q}^{ℓ_{q}} = r_{1 q} \\ + ℓ_{q} h_{q}, (ℓ_{q}, q) \in (ℕ_{0}^{N_{q}}, ℕ_{1}^{N})}, \\ Ω_{d} = & {x^{ℓ} = (x_{1}^{ℓ_{1}}, \dots, x_{N}^{ℓ_{N}})^{T} \in ℝ^{N} : x_{q}^{ℓ_{q}} = r_{1 q} \\ + ℓ_{q} h_{q}, (ℓ_{q}, q) \in (ℕ_{1}^{N_{q} - 1}, ℕ_{1}^{N})} . \end{matrix}$

By employing the finite difference methods in literature it gets $\begin{matrix} \frac{\partial^{2} u_{i} (t, x)}{\partial x_{q}^{2}} |_{x = x^{ℓ}} \approx \\ \frac{u_{i} (t, x^{ℓ} + h_{q} e_{q}) - 2 u_{i} (t, x^{ℓ}) + u_{i} (t, x^{ℓ} - h_{q} e_{q})}{h_{q}^{2}} \end{matrix}$ where $(t, x^{ℓ}) \in (ℝ, Ω_{d})$ and $(i, q) \in (ℕ_{1}^{n}, ℕ_{1}^{N})$ , ${e_{q} : q = 1, 2, \dots, N}$ is an orthonormal basis of $ℝ^{N}$ denoted by $\begin{matrix} e_{1} = (1, 0, \dots, 0)^{T}, e_{2} = (0, 1, \dots, 0)^{T}, \\ e_{N} = (0, 0, \dots, 1)^{T} . \end{matrix}$

As a consequence,Equation (2.2) is calculated approximately by

${\begin{matrix} c D_{t_{p}}^{α_{i}} u_{i} (t, x^{ℓ}) & \approx & \frac{1}{ɛ} [- c_{i}^{*} (x^{ℓ}) u_{i} (t, x^{ℓ}) + ξ_{i}^{*} (t, u_{i}) + b_{i} (t, x^{ℓ}) S_{i} (t, x^{ℓ}) \\ + \sum_{j = 1}^{n} a_{ij} (t, x^{ℓ}) g_{j} (u_{j} (t, x^{ℓ})) + ⋁_{j = 1}^{n} ϱ_{ij} (t, x^{ℓ}) g_{j} (u_{j} (t, x^{ℓ})) \\ + ⋀_{j = 1}^{n} ς_{ij} (t, x^{ℓ}) g_{j} (u_{j} (t, x^{ℓ})) + I_{i} (t, x^{ℓ}) \\ + \sum_{j = 1}^{n} μ_{ij} (t, x^{ℓ}) σ_{ij} (u_{j} (t, x^{ℓ})) \frac{d B_{j} (t)}{d t}], \\ c D_{t_{p}}^{β_{i}} S_{i} (t, x^{ℓ}) & \approx & - d_{i} (x^{ℓ}) S_{i} (t, x^{ℓ}) + g_{i} (u_{i} (t, x^{ℓ})), \end{matrix}$ (2.4)

where

$\begin{matrix} c_{i}^{*} (x^{ℓ}) = c_{i} (x^{ℓ}) + 2 \sum_{q = 1}^{N} \frac{ξ_{iq}}{h_{q}^{2}}, \\ ξ_{i}^{*} (t, u_{i}) = \sum_{q = 1}^{N} \frac{ξ_{iq}}{h_{q}^{2}} [u_{i} (t, x^{ℓ} + h_{q} e_{q}) + u_{i} (t, x^{ℓ} - h_{q} e_{q})] \end{matrix}$

for all $(t, x^{ℓ}, p, i) \in (J_{p}, Ω_{d}, ℤ, ℕ_{1}^{n})$ .

The initial boundary conditions (2.3) are approximately computed by

${\begin{matrix} u_{i} (0, x^{ℓ}) = φ_{i} (x^{ℓ}), x^{ℓ} \in Ω_{d}; u_{i} (t, x^{ℓ}) |_{x^{ℓ} \in \partial Ω_{d}} = 0, \\ S_{i} (0, x^{ℓ}) = ψ_{i} (x^{ℓ}), x^{ℓ} \in Ω_{d}; S_{i} (t, x^{ℓ}) |_{x^{ℓ} \in \partial Ω_{d}} = 0, \end{matrix}$ (2.5)

where $(t, i) \in (ℝ, ℕ_{1}^{n})$ .

In the whole paper, supposing that it exists a set $Q = {Q_{p} \in Z : Q_{0} = 0, Q_{p} < Q_{p + 1}, p \in Z}$ ensuring $\frac{t_{p + 1} - t_{p}}{Q_{p + 1} - Q_{p}} \equiv h$ for all $p \in ℤ$ . Based on (2.4)–(2.5) and by using the exponential time difference techniques in Ref. [24 –26], it obtains the lattice equations ofEquation (2.2) in the shape of

${\begin{matrix} u_{i}^{x^{ℓ}} (k + 1) & = & λ_{α_{i}}^{x^{ℓ}} (k) u_{i}^{x^{ℓ}} (k) + \sum_{l = μ_{k}}^{k} θ_{α_{i}}^{x^{ℓ}} (k - l, k) [ξ_{i}^{*} (u_{i}^{x^{ℓ}} (l)) + b_{i}^{x^{ℓ}} (l) S_{i}^{x^{ℓ}} (l) \\ + \sum_{j = 1}^{n} a_{ij}^{x^{ℓ}} (l) g_{j} (u_{j}^{x^{ℓ}} (l)) + ⋁_{j = 1}^{n} ϱ_{ij}^{x^{ℓ}} (l) g_{j} (u_{j}^{x^{ℓ}} (l)) + ⋀_{j = 1}^{n} ς_{ij}^{x^{ℓ}} (l) \\ \times g_{j} (u_{j}^{x^{ℓ}} (l)) + I_{i}^{x^{ℓ}} (l) + \sum_{j = 1}^{n} μ_{ij}^{x^{ℓ}} (l) σ_{ij} (u_{j}^{x^{ℓ}} (l)) h^{- 1} Δ_{h} B_{j} (l)], \\ S_{i}^{x^{ℓ}} (k + 1) & = & λ_{β_{i}}^{x^{ℓ}} (k) S_{i}^{x^{ℓ}} (k) + \sum_{l = μ_{k}}^{k} θ_{β_{i}}^{x^{ℓ}} (k - l, k) g_{i} (u_{i}^{x^{ℓ}} (l)) \end{matrix}$ (2.6) with boundary conditions $u_{i}^{x^{ℓ}} (k) |_{x^{ℓ} \in \partial Ω_{d}} = 0 = S_{i}^{x^{ℓ}} (k) |_{x^{ℓ} \in \partial Ω_{d}},$ where $μ_{k} = max_{p \in ℤ} {Q_{p} : Q_{p} \leq k}, ν_{k} = k - μ_{k},$ $\begin{matrix} u_{i}^{x^{ℓ}} (k) = u_{i} (kh, x^{ℓ}), S_{i}^{x^{ℓ}} (k) = S_{i} (kh, x^{ℓ}), \\ ξ_{i}^{*} (u_{i}^{x^{ℓ}} (k)) = ξ_{i}^{*} (kh, u_{i}^{x^{ℓ}} (k)), \end{matrix}$ $b_{i}^{x^{ℓ}} (k) = b_{i} (kh, x^{ℓ}), a_{ij}^{x^{ℓ}} (k) = a_{ij} (kh, x^{ℓ}),$ $\begin{matrix} ϱ_{ij}^{x^{ℓ}} (k) = ϱ_{ij} (kh, x^{ℓ}), ς_{ij}^{x^{ℓ}} (k) = ς_{ij} (kh, x^{ℓ}), \end{matrix}$ $\begin{matrix} μ_{ij}^{x^{ℓ}} (k) = μ_{ij} (kh, x^{ℓ}), I_{i}^{x^{ℓ}} (k) = I_{i} (kh, x^{ℓ}), \\ B_{j} (k) = B_{j} (kh), \end{matrix}$ $\begin{matrix} λ_{α_{i}}^{x^{ℓ}} (k) = \frac{E_{α_{i}} [- \frac{c_{i}^{*} (x^{ℓ})}{ɛ} (ν_{k} h + h)^{α_{i}}]}{E_{α_{i}} [- \frac{c_{i}^{*} (x^{ℓ})}{ɛ} (ν_{k} h)^{α_{i}}]}, \\ λ_{β_{i}}^{x^{ℓ}} (k) = \frac{E_{β_{i}} [- d_{i} (x^{ℓ}) (ν_{k} h + h)^{β_{i}}]}{E_{β_{i}} [- d_{i} (x^{ℓ}) (ν_{k} h)^{β_{i}}]}, \end{matrix}$ $\begin{matrix} θ_{α_{i}}^{x^{ℓ}} (l^{+}, k) = \frac{ɛ}{c_{i}^{*} (x^{ℓ})} [w_{α_{i}}^{x^{ℓ}} (l^{+}) - λ_{α_{i}}^{x^{ℓ}} (k) w_{α_{i}}^{x^{ℓ}} (l^{+} - 1)], \end{matrix}$ $\begin{matrix} θ_{β_{i}}^{x^{ℓ}} (l^{+}, k) = \frac{1}{d_{i}^{*} (x^{ℓ})} [w_{β_{i}}^{x^{ℓ}} (l^{+}) - λ_{β_{i}}^{x^{ℓ}} (k) w_{β_{i}}^{x^{ℓ}} (l^{+} - 1)], \end{matrix}$ $\begin{matrix} w_{α_{i}}^{x^{ℓ}} (- 1) = w_{β_{i}}^{x^{ℓ}} (- 1) = 0, \\ w_{α_{i}}^{x^{ℓ}} (l^{+}) = E_{α_{i}} [- \frac{c_{i}^{*} (x^{ℓ})}{ɛ} (l^{+} h)^{α_{i}}] \\ - E_{α_{i}} [- \frac{c_{i}^{*} (x^{ℓ})}{ɛ} (l^{+} h + h)^{α_{i}}], \end{matrix}$ $\begin{matrix} w_{β_{i}}^{x^{ℓ}} (l^{+}) = & E_{β_{i}} [- d_{i}^{*} (x^{ℓ}) (l^{+} h)^{β_{i}}] \\ - E_{β_{i}} [- d_{i}^{*} (x^{ℓ}) (l^{+} h + h)^{β_{i}}], \end{matrix}$ for all $(x^{ℓ}, l^{+}, k, i, j) \in ({\bar{Ω}}_{d}, ℕ_{0}, ℤ, ℕ_{1}^{n}, ℕ_{1}^{n})$ .

The initial condition ofEquation (2.6) can be rewritten as $\begin{matrix} u_{i}^{x^{ℓ}} (0) = φ_{i} (x^{ℓ}) : = φ_{i}^{x^{ℓ}}, S_{i}^{x^{ℓ}} (0) = ψ_{i} (x^{ℓ}) : = ψ_{i}^{x^{ℓ}}, \\ \forall (x^{ℓ}, i) \in (Ω_{d}, ℕ_{1}^{n}) . \end{matrix}$

Example 2.1. If the space variable vanishes, thenEquation (2.6) is changed into the classic fractional-order CNNs depicted by

${\begin{matrix} u_{i} (k + 1) & = & λ_{α_{i}} (k) u_{i} (k) + \sum_{l = μ_{k}}^{k} θ_{α_{i}} (k - l, k) [b_{i} (l) S_{i} (l) + \sum_{j = 1}^{n} a_{ij} (l) g_{j} (u_{j} (l)) + ⋁_{j = 1}^{n} ϱ_{ij} (l) \\ \times g_{j} (u_{j} (l)) + ⋀_{j = 1}^{n} ς_{ij} (l) g_{j} (u_{j} (l)) + I_{i} (l) + \sum_{j = 1}^{n} μ_{ij} (l) σ_{ij} (u_{j} (l)) h^{- 1} Δ_{h} B_{j} (l)], \\ S_{i} (k + 1) & = & λ_{β_{i}} (k) S_{i} (k) + \sum_{l = μ_{k}}^{k} θ_{β_{i}} (k - l, k) g_{i} (u_{i} (l)) \end{matrix}$ (2.7) where $\begin{matrix} λ_{α_{i}} (k) = \frac{E_{α_{i}} [- \frac{c_{i}}{ɛ} (ν_{k} h + h)^{α_{i}}]}{E_{α_{i}} [- \frac{c_{i}}{ɛ} (ν_{k} h)^{α_{i}}]}, \\ λ_{β_{i}} (k) = \frac{E_{β_{i}} [- d_{i} (ν_{k} h + h)^{β_{i}}]}{E_{β_{i}} [- d_{i} (ν_{k} h)^{β_{i}}]}, \end{matrix}$ $\begin{matrix} θ_{α_{i}} (l^{+}, k) = \frac{ɛ}{c_{i}} [w_{α_{i}} (l^{+}) - λ_{α_{i}} (k) w_{α_{i}} (l^{+} - 1)], \\ θ_{β_{i}} (l^{+}, k) = \frac{1}{d_{i}} [w_{β_{i}} (l^{+}) - λ_{β_{i}} (k) w_{β_{i}} (l^{+} - 1)], \end{matrix}$ $\begin{matrix} w_{α_{i}} (- 1) = w_{β_{i}} (- 1) = 0, \\ w_{α_{i}} (l^{+}) = E_{α_{i}} [- \frac{c_{i}}{ɛ} (l^{+} h)^{α_{i}}] \\ - E_{α_{i}} [- \frac{c_{i}}{ɛ} (l^{+} h + h)^{α_{i}}], \end{matrix}$ $\begin{matrix} w_{β_{i}} (l^{+}) = E_{β_{i}} [- d_{i} (l^{+} h)^{β_{i}}] \\ - E_{β_{i}} [- d_{i} (l^{+} h + h)^{β_{i}}], \forall (k, i) \in (ℤ, ℕ_{1}^{n}) . \end{matrix}$

Example 2.2. Let α_i = β_i = 1 inEquation (2.7), thenEquation (2.7) is transformed into

${\begin{matrix} u_{i} (k + 1) & = & e^{- \frac{c_{i}}{ɛ} h} u_{i} (k) + \frac{ɛ (1 - e^{- \frac{c_{i}}{ɛ} kh})}{c_{i}} [b_{i} (k) S_{i} (k) + \sum_{j = 1}^{n} a_{ij} (k) g_{j} (u_{j} (k)) \\ + ⋁_{j = 1}^{n} ϱ_{ij} (k) g_{j} (u_{j} (k)) + ⋀_{j = 1}^{n} ς_{ij} (k) g_{j} (u_{j} (k)) + I_{i} (k) \\ + \sum_{j = 1}^{n} μ_{ij} (k) σ_{ij} (u_{j} (k)) h^{- 1} Δ_{h} B_{j} (k)], \\ S_{i} (k + 1) & = & e^{- d_{i} h} S_{i} (k) + \frac{1 - e^{- d_{i} kh}}{d_{i}} g_{i} (u_{i} (k)) \end{matrix}$ (2.8) for all $(k, i) \in (ℤ, ℕ_{1}^{n})$ .

From Lemma 2.3 in literature [22], the lemma below is acquired.

Lemma 2.3 (Constant variation formula). The solution ofEquation (2.6) has the form of

${\begin{matrix} u_{i}^{x^{ℓ}} (k) & = & \prod_{s = k_{0}}^{k - 1} λ_{α_{i}}^{x^{ℓ}} (s) u_{i}^{x^{ℓ}} (k_{0}) + \sum_{q = k_{0}}^{k - 1} \prod_{s = q + 1}^{k - 1} λ_{α_{i}}^{x^{ℓ}} (s) \sum_{l = μ_{q}}^{q} θ_{α_{i}}^{x^{ℓ}} (q - l, q) \\ \times [ξ_{i}^{*} (u_{i}^{x^{ℓ}} (l)) + b_{i}^{x^{ℓ}} (l) S_{i}^{x^{ℓ}} (l) + \sum_{j = 1}^{n} a_{ij}^{x^{ℓ}} (l) g_{j} (u_{j}^{x^{ℓ}} (l)) + ⋁_{j = 1}^{n} ϱ_{ij}^{x^{ℓ}} (l) g_{j} (u_{j}^{x^{ℓ}} (l)) \\ + ⋀_{j = 1}^{n} ς_{ij}^{x^{ℓ}} (l) g_{j} (u_{j}^{x^{ℓ}} (l)) + I_{i}^{x^{ℓ}} (l) + \sum_{j = 1}^{n} μ_{ij}^{x^{ℓ}} (l) σ_{ij} (u_{j}^{x^{ℓ}} (l)) h^{- 1} Δ_{h} B_{j} (l)], \\ S_{i}^{x^{ℓ}} (k) & = & \prod_{s = k_{0}}^{k - 1} λ_{β_{i}}^{x^{ℓ}} (s) S_{i}^{x^{ℓ}} (k_{0}) + \sum_{q = k_{0}}^{k - 1} \prod_{s = q + 1}^{k - 1} λ_{β_{i}}^{x^{ℓ}} (s) \sum_{l = μ_{q}}^{q} θ_{β_{i}}^{x^{ℓ}} (q - l, q) g_{i} (u_{i}^{x^{ℓ}} (l)), x^{ℓ} \in Ω_{d}, \end{matrix}$ (2.9) where $k_{0} \in ℤ$ ; $u_{i}^{x^{ℓ}} (k) |_{x^{ℓ} \in \partial Ω_{d}} = 0 = S_{i}^{x^{ℓ}} (k) |_{x^{ℓ} \in \partial Ω_{d}}$ , $\forall (k, i) \in ([k_{0}, + \infty)_{ℤ}, ℕ_{1}^{n}) .$

According to Lemma 2.3, we obtain

Remark 2.1. The solution ofEquation (2.7) is of the form

${\begin{matrix} u_{i} (k) & = & \prod_{s = k_{0}}^{k - 1} λ_{α_{i}} (s) u_{i} (k_{0}) + \sum_{q = k_{0}}^{k - 1} \prod_{s = q + 1}^{k - 1} λ_{α_{i}} (s) \sum_{l = μ_{q}}^{q} θ_{α_{i}} (q - l, q) \\ \times [b_{i} (l) S_{i} (l) + \sum_{j = 1}^{n} a_{ij} (l) g_{j} (u_{j} (l)) + ⋁_{j = 1}^{n} ϱ_{ij} (l) g_{j} (u_{j} (l)) \\ + ⋀_{j = 1}^{n} ς_{ij} (l) g_{j} (u_{j} (l)) + I_{i} (l) + \sum_{j = 1}^{n} μ_{ij} (l) σ_{ij} (u_{j} (l)) h^{- 1} Δ_{h} B_{j} (l)], \\ S_{i} (k) & = & \prod_{s = k_{0}}^{k - 1} λ_{β_{i}} (s) S_{i} (k_{0}) + \sum_{q = k_{0}}^{k - 1} \prod_{s = q + 1}^{k - 1} λ_{β_{i}} (s) \sum_{l = μ_{q}}^{q} θ_{β_{i}} (q - l, q) g_{i} (u_{i} (l)), \end{matrix}$ where $(k, i) \in ([k_{0}, + \infty)_{ℤ}, ℕ_{1}^{n})$ , $k_{0} \in ℤ$ . The solution ofEquation (2.8) is expressed by

${\begin{matrix} u_{i} (k) & = & e^{- \frac{c_{i}}{ɛ} (k - k_{0}) h} u_{i} (k_{0}) + \frac{ɛ}{c_{i}} \sum_{q = k_{0}}^{k - 1} e^{- \frac{c_{i}}{ɛ} (k - q - 1) h} (1 - e^{- \frac{c_{i}}{ɛ} qh}) \\ \times [b_{i} (q) S_{i} (q) + \sum_{j = 1}^{n} a_{ij} (q) g_{j} (u_{j} (q)) + ⋁_{j = 1}^{n} ϱ_{ij} (l) g_{j} (u_{j} (q)) \\ + ⋀_{j = 1}^{n} ς_{ij} (q) g_{j} (u_{j} (q)) + I_{i} (q) + \sum_{j = 1}^{n} μ_{ij} (l) σ_{ij} (u_{j} (q)) h^{- 1} Δ_{h} B_{j} (q)], \\ S_{i} (k) & = & e^{- d_{i} (k - k_{0}) h} S_{i} (k_{0}) + \frac{1}{d_{i}} \sum_{q = k_{0}}^{k - 1} e^{- d_{i} (k - q - 1) h} (1 - e^{- d_{i} qh}) g_{i} (u_{i} (q)), \end{matrix}$ where $(k, i) \in ([k_{0}, + \infty)_{ℤ}, ℕ_{1}^{n})$ , $k_{0} \in ℤ$ .

Let $Q^{*} = sup_{p \in ℤ} (Q_{p + 1} - Q_{p}),$ $\begin{matrix} {\bar{λ}}_{α_{i}} = sup_{k \in ℤ, x^{ℓ} \in {\bar{Ω}}_{d}} λ_{α_{i}}^{x^{ℓ}} (k), {\bar{λ}}_{β_{i}} = sup_{k \in ℤ, x^{ℓ} \in {\bar{Ω}}_{d}} λ_{β_{i}}^{x^{ℓ}} (k), \\ {\underline{c}}_{i}^{*} = inf_{x^{ℓ} \in {\bar{Ω}}_{d}} c_{i}^{*} (x^{ℓ}), {\underline{d}}_{i} = inf_{x^{ℓ} \in {\bar{Ω}}_{d}} d_{i} (x^{ℓ}) \end{matrix}$ for all $i \in ℕ_{1}^{n} .$

It easily gets the lemma below.

Lemma 2.4. IfQ^*< + ∞, α_i, β_i ∈ (0, 1] and ${\underline{c}}_{i}^{*}, {\underline{d}}_{i} > 0$ , one has

$\begin{matrix} 0 < λ_{α_{i}}^{x^{ℓ}} (k) \leq {\bar{λ}}_{α_{i}} < 1, 0 < λ_{β_{i}}^{x^{ℓ}} (k) \leq {\bar{λ}}_{β_{i}} < 1, \\ \sum_{l = 0}^{Q^{*}} | θ_{α_{i}}^{x^{ℓ}} (l, k) | \leq \frac{2 ɛ}{{\underline{c}}_{i}^{*}}, \sum_{l = 0}^{Q^{*}} | θ_{β_{i}}^{x^{ℓ}} (l, k) | \leq \frac{2}{{\underline{d}}_{i}} \end{matrix}$ for all $(x^{ℓ}, k, i) \in ({\bar{Ω}}_{d}, ℤ, ℕ_{1}^{n}) .$

Remark 2.2. Papers [17–21 , 27] discussed the dynamical behaviours of various types of discrete-time neural networks without spatial diffusions, e.g., dissipative synchronization, anti-synchronization, event-triggered synchronization, almost periodicity, stability, etc. In literature [27], Zhang et al. considered a kind of discrete-time stochastic CNNs without spatial diffusions by utilizing the method of exponential Euler time difference, and 2p-th mean almost periodic sequence and moment global exponential stability to CNNs had been addressed. Notably, the constant variation formula similarly described in Lemma 2.3 is crucial for the studies of literatures [20 , 27]. The proposed networks and the constant variation formula described in Lemma 2.3 consider the effect of spatial diffusions, so it is superior to the networks in literatures [17–21 , 27].

Remark 2.3. The authors in articles [5 , 12] focused on the researches of continuous-time CNNs with reaction diffusions. However, almost no article is concerned with discrete-time and discrete-space CNNs and then the work of the current paper fills this blank.

3 Anti-periodic sequences in distribution

Let $E (\cdot)$ be the expectation under a complete probability space $(Λ, F, ℙ)$ and $\begin{matrix} B = (B_{11}, \dots, B_{1 n}, B_{21}, \dots, B_{2 n})^{T} \end{matrix}$ be a two-sided standard 2n-dimensional Brownian motion defined on $(Λ, F, ℙ)$ . Set $F_{t} = σ {B (s) : s \leq t}$ for $t \in ℝ$ . Further, $L^{2} (Λ, ℝ^{2 n})$ denotes the family of all square integrable $ℝ^{2 n}$ -valued random variables and $X = B (ℤ \times {\bar{Ω}}_{d}, L^{2} (Λ, ℝ^{2 n}))$ stands for the set of all functions from $ℤ \times {\bar{Ω}}_{d}$ to $L^{2} (Λ, ℝ^{2 n})$ endowed with the norm $\begin{matrix} ∥ S ∥_{X} = sup_{k \in ℤ} max_{1 \leq i \leq n} max_{x^{ℓ} \in {\bar{Ω}}_{d}} \\ {{[E {| u_{i}^{x^{ℓ}} (k) |}^{2}]}^{1 / 2}, {[E {| S_{i}^{x^{ℓ}} (k) |}^{2}]}^{1 / 2}}, \end{matrix}$ where $S = (u_{1}, \dots, u_{n}, S_{1}, \dots, S_{n})^{T} \in X .$ Obviously, $(X, ∥ \cdot ∥_{X})$ becomes a Banach space. In the whole paper, let $φ_{i}^{x^{ℓ}}$ and $ψ_{i}^{x^{ℓ}}$ are $F_{0}$ -adapted, $\forall (x^{ℓ}, i) \in ({\bar{Ω}}_{d}, ℕ_{1}^{n}) .$

Definition 3.1. A discrete-time stochastic process $S = (u_{1}, \dots, u_{n}, S_{1}, \dots, S_{n})^{T} \in X$ is called the solution ofEquation (2.6) if it is $F_{t}$ -adapted and meetsEquation (2.9).

Definition 3.2. Let $ω \in ℝ$ . A discrete-time stochastic process $S \in X$ is called ω-anti-periodic in distribution if the law of $S (\cdot + ω)$ is the same as that of $- S (\cdot)$ .

Lemma 3.1. ([45]) (Minkowski inequality) If $f, g \in L^{2} (Λ, ℝ)$ , then

$\begin{matrix} {(E (f + g)^{2})}^{\frac{1}{2}} \leq {(E f^{2})}^{\frac{1}{2}} + {(E g^{2})}^{\frac{1}{2}} . \end{matrix}$

Lemma 3.2. ([45]) (Hölder inequality) Letp > 1 and $a_{k}, b_{k} : ℤ \to ℝ$ . Then

$\begin{matrix} \sum_{k} | a_{k} b_{k} | \leq [\sum_{k} | a_{k} |]^{1 - 1 / p} [\sum_{k} | a_{k} | | b_{k} |^{p}]^{1 / p} . \end{matrix}$

Lemma 3.3. ([21]) Let ${f (k)}_{k \in ℤ} \subseteq L^{2} (Λ, ℝ)$ and ${B (t)}_{t \in ℝ}$ be a two-sided standard one dimensional Brownian motion. Then

$\begin{matrix} E {| f (k) Δ_{h} B (k) |}^{2} \leq 4 h E | f (k) |^{2}, \end{matrix}$ where Δ_hB (k) = B (kh + h) - B (kh), $\forall k \in ℤ .$

Define $\bar{g} = sup_{k \in ℤ, s \in Ω_{d}} | g (k, s) |,$ where $g : = ℤ \times Ω_{d} \to ℝ$ is a sequence. Let $\begin{matrix} ξ_{i}^{*} = 2 \sum_{q = 1}^{N} \frac{ξ_{iq}}{h_{q}^{2}} for i \in ℕ_{1}^{n}; ρ_{0} = \frac{ρ_{2}}{1 - ρ_{1}}, \end{matrix}$ where $\begin{matrix} ρ_{1} = max_{1 \leq i \leq n} {\frac{2 ɛ}{{\underline{c}}_{i}^{*} (1 - {\bar{λ}}_{α_{i}})} [ξ_{i}^{*} + {\bar{b}}_{i} + \sum_{j = 1}^{n} ({\bar{a}}_{ij} + {\bar{ϱ}}_{ij} \\ + {\bar{ς}}_{ij}) L_{j}^{g} + 2 h^{- \frac{1}{2}} \sum_{j = 1}^{n} {\bar{μ}}_{ij} L_{ij}^{σ}], \frac{2}{{\underline{d}}_{i} (1 - {\bar{λ}}_{β_{i}})} L_{i}^{g}}, \end{matrix}$ $\begin{matrix} ρ_{2} = max_{1 \leq i \leq n, x^{ℓ} \in \partial Ω_{d}} {\frac{2 ɛ}{{\underline{c}}_{i}^{*} (1 - {\bar{λ}}_{α_{i}})} \\ [(\sum_{j = 1}^{n} {\bar{a}}_{ij} + ⋁_{j = 1}^{n} {\bar{ϱ}}_{ij} + ⋀_{j = 1}^{n} {\bar{ς}}_{ij}) | g_{j} (0) | \end{matrix}$ $\begin{matrix} + {\bar{I}}_{i} + 2 h^{- \frac{1}{2}} \sum_{j = 1}^{n} {\bar{μ}}_{ij} | σ_{ij} (0) |], \frac{2}{{\underline{d}}_{i} (1 - {\bar{λ}}_{β_{i}})} | g_{i} (0) |} . \end{matrix}$

Let $S_{0} = {S \in X : ∥ S ∥_{X} \leq ρ_{0}} .$ In accordance withEquation (2.9), define a mapping $P : S_{0} \to X$ as

$\begin{matrix} P S = (P u)_{1}, (P u)_{2}, \dots, (P u)_{n}, \\ (P S)_{1}, (P S)_{2}, \dots, (P S)_{n})^{T}, \end{matrix}$ where

${\begin{matrix} {(P u)}_{i}^{x^{ℓ}} (k) = \sum_{q = - \infty}^{k - 1} \prod_{s = q + 1}^{k - 1} λ_{α_{i}}^{x^{ℓ}} (s) \sum_{l = μ_{q}}^{q} θ_{α_{i}}^{x^{ℓ}} (q - l, q) [ξ_{i}^{*} (u_{i}^{x^{ℓ}} (l)) + b_{i}^{x^{ℓ}} (l) S_{i}^{x^{ℓ}} (l) \\ + \sum_{j = 1}^{n} a_{i j}^{x^{ℓ}} (l) g_{j} (u_{j}^{x^{ℓ}} (l)) + \lor_{j = 1}^{n} ϱ_{i j}^{x^{ℓ}} (l) g_{j} (u_{j}^{x^{ℓ}} (l)) + \land_{j = 1}^{n} ς_{i j}^{x^{ℓ}} (l) g_{j} (u_{j}^{x^{ℓ}} (l)) \\ + I_{i}^{x^{ℓ}} (l) + \sum_{j = 1}^{n} μ_{i j}^{x^{ℓ}} (l) σ_{i j} (u_{j}^{x^{ℓ}} (l)) h^{- 1} Δ_{h} B_{j} (l)], \\ {(P S)}_{i}^{x^{ℓ}} (k) = \sum_{q = - \infty}^{k - 1} \prod_{s = q + 1}^{k - 1} λ_{β_{i}}^{x^{ℓ}} (s) \sum_{l = μ_{q}}^{q} θ_{β_{i}}^{x^{ℓ}} (q - l, q) g_{i} (u_{i}^{x^{ℓ}} (l)), x^{ℓ} \in Ω_{d} \end{matrix}$ (3.1)

with $(P u)_{i}^{x^{ℓ}} (k) |_{x^{ℓ} \in \partial Ω_{d}} = 0 = (P S)_{i}^{x^{ℓ}} (k) |_{x^{ℓ} \in \partial Ω_{d}},$ $\forall (k, i) \in (ℤ, ℕ_{1}^{n}) .$

Here, we needs the following assumptions.

{ν_k} is a ω-periodic sequence and ${I_{i}^{x^{ℓ}} (k)}$ is anti-ω-periodic sequence for eachx^ℓ ∈ Ω_d, $ω \in ℤ$ ; i.e., ν_k+ω = ν_k and $I_{i}^{x^{ℓ}} (k + ω) = - I_{i}^{x^{ℓ}} (k)$ , $\forall (x^{ℓ}, k, i) \in (Ω_{d}, ℤ, ℕ_{1}^{n})$ .

$a_{ij}^{x^{ℓ}} (k)$ , $b_{i}^{x^{ℓ}} (k)$ , $ϱ_{ij}^{x^{ℓ}} (k)$ , $ς_{ij}^{x^{ℓ}} (k)$ and $μ_{ij}^{x^{ℓ}} (k)$ are ω-periodic sequences with respect to variable $k \in ℤ$ , $\forall (x^{ℓ}, i, j) \in (Ω_{d}, ℕ_{1}^{n}, ℕ_{1}^{n})$ .

g_j and σ_ij are odd mappings, and it exists positive numbers $L_{j}^{g}$ and $L_{ij}^{σ}$ such that $\begin{matrix} | g_{j} (x) - g_{j} (y) | \leq L_{j}^{g} | x - y |, \\ | σ_{ij} (x) - σ_{ij} (y) | \leq L_{ij}^{σ} | x - y | \end{matrix}$ for any $x, y \in ℝ$ , $i, j \in ℕ_{1}^{n}$ .

Proposition 3.1. P is well defined and maps $S_{0}$ to $S_{0}$ if (H₃) and (H₄) below are satisfied.

ρ₁ < 1.

Proof. Let $S = (u_{1}, \dots, u_{n}, S_{1}, \dots, S_{n})^{T} \in S_{0}$ . FromEquation (3.1), Lemmas 3.1, 3.2 and 3.3, it gets $\begin{matrix} {[E {| (P M)_{i}^{x^{ℓ}} (k) |}^{2}]}^{\frac{1}{2}} & \leq & {E {[\sum_{q = - \infty}^{k - 1} \prod_{s = q + 1}^{k - 1} λ_{α_{i}}^{x^{ℓ}} (s)]^{\frac{1}{2}} {\sum_{q = - \infty}^{k - 1} \prod_{s = q + 1}^{k - 1} λ_{α_{i}}^{x^{ℓ}} (s) [\sum_{l = μ_{q}}^{q} θ_{α_{i}}^{x^{ℓ}} (q - l, q) \\ \times (ξ_{i}^{*} (u_{i}^{x^{ℓ}} (l)) + b_{i}^{x^{ℓ}} (l) S_{i}^{x^{ℓ}} (l) + \sum_{j = 1}^{n} a_{ij}^{x^{ℓ}} (l) g_{j} (u_{j}^{x^{ℓ}} (l)) \\ + ⋁_{j = 1}^{n} ϱ_{ij}^{x^{ℓ}} (l) g_{j} (u_{j}^{x^{ℓ}} (l)) + ⋀_{j = 1}^{n} ς_{ij}^{x^{ℓ}} (l) g_{j} (u_{j}^{x^{ℓ}} (l)) + I_{i}^{x^{ℓ}} (l) \\ + \sum_{j = 1}^{n} μ_{ij}^{x^{ℓ}} (l) σ_{ij} (u_{j}^{x^{ℓ}} (l)) h^{- 1} Δ_{h} B_{j} (l))]^{2}}^{\frac{1}{2}}}^{2}}^{\frac{1}{2}} \\ \leq & \frac{2 ɛ}{{\underline{c}}_{i}^{*} (1 - {\bar{λ}}_{α_{i}})} {[ξ_{i}^{*} + {\bar{b}}_{i} + \sum_{j = 1}^{n} ({\bar{a}}_{ij} + {\bar{ϱ}}_{ij} + {\bar{ς}}_{ij}) L_{j}^{g}] ∥ S ∥_{X} \\ + (\sum_{j = 1}^{n} {\bar{a}}_{ij} + ⋁_{j = 1}^{n} {\bar{ϱ}}_{ij} + ⋀_{j = 1}^{n} {\bar{ς}}_{ij}) | g_{j} (0) | + {\bar{I}}_{i} + 2 h^{- \frac{1}{2}} \sum_{j = 1}^{n} {\bar{μ}}_{ij} | σ_{ij} (0) | \\ + \sum_{j = 1}^{n} {\bar{μ}}_{ij} L_{ij}^{σ} {[E {| u_{j}^{x^{ℓ}} (l) h^{- 1} Δ_{h} B_{j} (l) |}^{2}]}^{\frac{1}{2}}} \\ \leq & \frac{2 ɛ}{{\underline{c}}_{i}^{*} (1 - {\bar{λ}}_{α_{i}})} [ξ_{i}^{*} + {\bar{b}}_{i} + \sum_{j = 1}^{n} ({\bar{a}}_{ij} + {\bar{ϱ}}_{ij} + {\bar{ς}}_{ij}) L_{j}^{g} + 2 h^{- \frac{1}{2}} \sum_{j = 1}^{n} {\bar{μ}}_{ij} L_{ij}^{σ}] \\ \times ∥ S ∥_{X} + \frac{2 ɛ}{{\underline{c}}_{i}^{*} (1 - {\bar{λ}}_{α_{i}})} [(\sum_{j = 1}^{n} {\bar{a}}_{ij} + ⋁_{j = 1}^{n} {\bar{ϱ}}_{ij} + ⋀_{j = 1}^{n} {\bar{ς}}_{ij}) | g_{j} (0) | \\ + {\bar{I}}_{i} + 2 h^{- \frac{1}{2}} \sum_{j = 1}^{n} {\bar{μ}}_{ij} | σ_{ij} (0) |] \\ \leq & ρ_{1} ρ_{0} + ρ_{2}, \forall (S, x^{ℓ}, k, i) \in (S_{0}, Ω_{d}, ℤ, ℕ_{1}^{n}), \end{matrix}$ and by a similar derivation as the above, it acquires $\begin{matrix} {[E {| {(P S)}_{i}^{x^{ℓ}} (k) |}^{2}]}^{\frac{1}{2}} = {E {[\sum_{q = - \infty}^{k - 1} \prod_{s = q + 1}^{k - 1} λ_{β_{i}}^{x^{ℓ}} (s) \sum_{l = μ_{q}}^{q} θ_{β_{i}}^{x^{ℓ}} (q - l, q) g_{i} (u_{i}^{x^{ℓ}} (l))]}^{2}}^{\frac{1}{2}} \\ \leq \frac{2}{{\underline{d}}_{i} (1 - {\bar{λ}}_{β_{i}})} L_{i}^{g} ‖ S ‖_{X} + \frac{2}{{\underline{d}}_{i} (1 - {\bar{λ}}_{β_{i}})} | g_{i} (0) | \\ \leq ρ_{1} ρ_{0} + ρ_{2}, \forall (S, x^{ℓ}, k, i) \in (S_{0}, Ω_{d}, ℤ, ℕ_{1}^{n}) \end{matrix}$ Summarizing the above analyses, it leads to $∥ P S ∥_{X} \leq ρ_{1} ρ_{0} + ρ_{2} = ρ_{0}$ , $\forall S \in S_{0} .$ Therefore,P is well defined and maps $S_{0}$ to $S_{0}$ . The proof is complete.□

Remark 3.1. In terms of assumption (H₄), one has $\begin{matrix} 2 ɛ [ξ_{i}^{*} + {\bar{b}}_{i} + \sum_{j = 1}^{n} ({\bar{a}}_{ij} + {\bar{ϱ}}_{ij} + {\bar{ς}}_{ij}) L_{j}^{g} \\ + 2 h^{- \frac{1}{2}} \sum_{j = 1}^{n} {\bar{μ}}_{ij} L_{ij}^{σ}] < {\underline{c}}_{i}^{*}, 2 L_{i}^{g} < {\underline{d}}_{i} \end{matrix}$

for anyi = 1, 2, …,n. Evidently, $2 L_{i}^{g} < {\underline{d}}_{i}$ is easy to realize for the large disposable scaling constantd_i and if the time constant of neuronc_i large enough, then (H₄) is fulfilled for any small ɛ,i = 1, 2, …,n, which shows that condition (H₄) is reasonable.

Proposition 3.2. Equation (2.6) possesses a unique bounded solution in $S_{0}$ if (H₃)-(H₄) hold.

Proof. By Proposition 3.1, $P : S_{0} \to S_{0}$ . Let $\begin{matrix} S = (u_{1}, \dots, u_{n}, S_{1}, \dots, S_{n})^{T}, \tilde{S} = ({\tilde{u}}_{1}, \dots, {\tilde{u}}_{n}, \\ {\tilde{S}}_{1}, \dots, {\tilde{S}}_{n})^{T} \in S_{0} . \end{matrix}$ It derives fromEquation (3.1) that $\begin{matrix} {[E {| {(P u)}_{i}^{x^{ℓ}} (k) - {(P \tilde{u})}_{i}^{x^{ℓ}} (k) |}^{2}]}^{\frac{1}{2}} \\ \leq {E | \sum_{q = - \infty}^{k - 1} \prod_{s = q + 1}^{k - 1} λ_{α_{i}}^{x^{ℓ}} (s) \sum_{l = μ_{q}}^{q} θ_{α_{i}}^{x^{ℓ}} (q - l, q) [| ξ_{i}^{*} (u_{i}^{x^{ℓ}} (l)) - ξ_{i}^{*} ({\tilde{u}}_{i}^{x^{ℓ}} (l)) | \\ + {\bar{b}}_{i} (l) | S_{i}^{x^{ℓ}} (l) - {\tilde{S}}_{i}^{x^{ℓ}} (l) | + \sum_{j = 1}^{n} ({\bar{a}}_{i j} + {\bar{ϱ}}_{i j} + {\bar{ς}}_{i j}) | g_{j} (u_{j}^{x^{ℓ}} (l)) - g_{j} ({\tilde{u}}_{j}^{x^{ℓ}} (l)) | \\ + \sum_{j = 1}^{n} h^{- 1} | Δ_{h} B_{j} (l) | {\bar{μ}}_{i j} | σ_{i j} (u_{j}^{x^{ℓ}} (l)) - σ_{i j} ({\tilde{u}}_{j}^{x^{ℓ}} (l)) |] |^{2}}^{\frac{1}{2}} \\ \leq \sum_{q = - \infty}^{k - 1} \prod_{s = q + 1}^{k - 1} λ_{α_{i}}^{x^{ℓ}} (s) {E [\sum_{l = μ_{q}}^{q} | θ_{α_{i}}^{x^{ℓ}} (q - l, q) | (| ξ_{i}^{*} (u_{i}^{x^{ℓ}} (l)) - ξ_{i}^{*} ({\tilde{u}}_{i}^{x^{ℓ}} (l)) | \end{matrix}$ $\begin{matrix} + {\bar{b}}_{i} | S_{i}^{x^{ℓ}} (l) - {\tilde{S}}_{i}^{x^{ℓ}} (l) | + \sum_{j = 1}^{n} ({\bar{a}}_{i j} + {\bar{ϱ}}_{i j} + {\bar{ς}}_{i j}) | g_{j} (u_{j}^{x^{ℓ}} (l)) - g_{j} ({\tilde{u}}_{j}^{x^{ℓ}} (l)) | \\ + \sum_{j = 1}^{n} h^{- 1} | Δ_{h} B_{j} (l) | {\bar{μ}}_{i j} | σ_{i j} (u_{j}^{x^{ℓ}} (l)) - σ_{i j} ({\tilde{u}}_{j}^{x^{ℓ}} (l)) |)]^{2}}^{\frac{1}{2}} \\ \leq \frac{2 ϵ}{{\underline{c}}_{i}^{*} (1 - {\bar{λ}}_{α_{i}})} [θ_{i}^{*} + {\bar{b}}_{i} + \sum_{j = 1}^{n} ({\bar{a}}_{i j} + {\bar{ϱ}}_{i j} + {\bar{ς}}_{i j}) L_{j}^{g} + 2 h^{- \frac{1}{2}} \sum_{j = 1}^{n} {\bar{μ}}_{i j} L_{i j}^{σ}] ‖ S - \tilde{S} ‖_{X} \\ \leq ρ_{1} ‖ S - \tilde{S} ‖_{X}, \forall (x^{ℓ}, k, i) \in ({\bar{Ω}}_{d}, ℤ, ℕ_{1}^{n}) \end{matrix}$ Similarly, it obtains $\begin{matrix} {[E {| (P S_{i}^{x^{ℓ}} (k) - (P \tilde{S})_{i}^{x^{ℓ}} (k) |}^{2}]}^{\frac{1}{2}} \\ \leq ρ_{1} ∥ S - \tilde{S} ∥_{X}, \forall (x^{ℓ}, k, i) \in ({\bar{Ω}}_{d}, ℤ, ℕ_{1}^{n}) . \end{matrix}$ So $∥ P S - P \tilde{S} ∥_{X} \leq ρ_{1} ∥ S - \tilde{S} ∥_{X},$ $\forall S, \tilde{S} \in S_{0}$ . From (H₄),P is contractive andP has a unique fixed point $\vec{S} = P \vec{S} \in S_{0} \subseteq X$ solvingEquation (2.6). The proof is complete.□

Theorem 3.1. A unique anti-ω-periodic sequence in distribution solvesEquation (2.6) if (H₁)-(H₄) hold.

Proof.Equation (2.6) possesses a unique solution $\vec{S}$ in $S_{0}$ on the basis of Proposition 3.2. According to Proposition 3.2, the unique solution $\vec{S} = ({\vec{u}}_{1}, \dots, {\vec{u}}_{n}, {\vec{S}}_{1}, \dots, {\vec{S}}_{n})^{T} \in S_{0}$ ofEquation (2.6) meets ${\vec{u}}_{i}^{x^{ℓ}} (k + ω) = \sum_{q = - \infty}^{k - 1} \prod_{s = q + 1}^{k - 1} λ_{α_{i}}^{x^{ℓ}} (s) \sum_{l = μ_{q}}^{q} θ_{α_{i}}^{x^{ℓ}} (q - l, q) [ξ_{i}^{*} ({\vec{u}}_{i}^{x^{ℓ}} (l + ω))$ $+ b_{i}^{x^{ℓ}} (l) {\vec{S}}_{i}^{x^{ℓ}} (l + ω) + \sum_{j = 1}^{n} a_{ij}^{x^{ℓ}} (l) g_{j} ({\vec{u}}_{j}^{x^{ℓ}} (l + ω))$ $+ ⋁_{j = 1}^{n} ϱ_{ij}^{x^{ℓ}} (l) g_{j} ({\vec{u}}_{j}^{x^{ℓ}} (l + ω)) + ⋀_{j = 1}^{n} ς_{ij}^{x^{ℓ}} (l) g_{j} ({\vec{u}}_{j}^{x^{ℓ}} (l + ω))$ $- I_{i}^{x^{ℓ}} (l) + \sum_{j = 1}^{n} μ_{ij}^{x^{ℓ}} (l) σ_{ij} ({\vec{u}}_{j}^{x^{ℓ}} (l + ω)) h^{- 1} Δ_{h} B_{j} (l + ω)],$ ${\vec{S}}_{i}^{x^{ℓ}} (k + ω) = \sum_{q = - \infty}^{k - 1} \prod_{s = q + 1}^{k - 1} λ_{β_{i}}^{x^{ℓ}} (s) \sum_{l = μ_{q}}^{q} θ_{β_{i}}^{x^{ℓ}} (q - l, q) g_{i} ({\vec{u}}_{i}^{x^{ℓ}} (l + ω)), \forall x^{ℓ} \in Ω_{d},$ where ${\vec{u}}_{i}^{x^{ℓ}} (k) |_{x^{ℓ} \in \partial Ω_{d}} = 0 = {\vec{S}}_{i}^{x^{ℓ}} (k) |_{x^{ℓ} \in \partial Ω_{d}},$ $\forall (k, i) \in (ℤ, ℕ_{1}^{n}) .$

Let us discuss the stochastic process ${\vec{S}}_{ω} = ({\vec{u}}_{1, ω}, \dots, {\vec{u}}_{n, ω}, {\vec{S}}_{1, ω}, \dots, {\vec{S}}_{n, ω})^{T}$ below ${\vec{u}}_{i, ω}^{x^{ℓ}} (k) = \sum_{q = - \infty}^{k - 1} \prod_{s = q + 1}^{k - 1} λ_{α_{i}}^{x^{ℓ}} (s) \sum_{l = μ_{q}}^{q} θ_{α_{i}}^{x^{ℓ}} (q - l, q) [ξ_{i}^{*} ({\vec{u}}_{i, ω}^{x^{ℓ}} (l))$ $+ b_{i}^{x^{ℓ}} (l) {\vec{S}}_{i, ω}^{x^{ℓ}} (l) + \sum_{j = 1}^{n} a_{ij}^{x^{ℓ}} (l) g_{j} ({\vec{u}}_{j, ω}^{x^{ℓ}} (l))$ $+ ⋁_{j = 1}^{n} ϱ_{ij}^{x^{ℓ}} (l) g_{j} ({\vec{u}}_{j, ω}^{x^{ℓ}} (l)) + ⋀_{j = 1}^{n} ς_{ij}^{x^{ℓ}} (l) g_{j} ({\vec{u}}_{j, ω}^{x^{ℓ}} (l))$ $- I_{i}^{x^{ℓ}} (l) + \sum_{j = 1}^{n} μ_{ij}^{x^{ℓ}} (l) σ_{ij} ({\vec{u}}_{j, ω}^{x^{ℓ}} (l)) h^{- 1} Δ_{h} B_{j} (l)],$ ${\vec{S}}_{i, ω}^{x^{ℓ}} (k) = \sum_{q = - \infty}^{k - 1} \prod_{s = q + 1}^{k - 1} λ_{β_{i}}^{x^{ℓ}} (s) \sum_{l = μ_{q}}^{q} θ_{β_{i}}^{x^{ℓ}} (q - l, q) g_{i} ({\vec{u}}_{i, ω}^{x^{ℓ}} (l)), \forall x^{ℓ} \in Ω_{d};$ where ${\vec{u}}_{i, ω}^{x^{ℓ}} (k) |_{x^{ℓ} \in \partial Ω_{d}} = 0 = {\vec{S}}_{i, ω}^{x^{ℓ}} (k) |_{x^{ℓ} \in \partial Ω_{d}},$ $\forall (k, i) \in (ℤ, ℕ_{1}^{n}) .$ Similar to $\vec{S}$ , ${\vec{S}}_{ω}$ is unique and bounded in $S_{0}$ . Noting that Δ_hB_j (k + ω) has the same law as Δ_hB_j (k), $\forall (k, j) \in (ℤ, ℕ_{1}^{n})$ . Then ${\vec{S}}_{ω} (\cdot)$ has the same distribution as $\vec{S} (\cdot + ω)$ .

Resembling the derivation in Proposition 3.2, it calculates $\begin{matrix} {[E {| {\vec{u}}_{i, ω}^{x^{ℓ}} (k) + {\vec{u}}_{i}^{x^{ℓ}} (k) |}^{2}]}^{\frac{1}{2}} \leq ρ_{1} ‖ {\vec{S}}_{ω} + \vec{S} ‖_{X}, \\ {[E {| {\vec{S}}_{i, ω}^{x^{ℓ}} (k) + {\vec{S}}_{i}^{x^{ℓ}} (k) |}^{2}]}^{\frac{1}{2}} \leq ρ_{1} ‖ {\vec{S}}_{ω} + \vec{S} ‖_{X}, \end{matrix}$ where $(x^{ℓ}, k, i) \in ({\bar{Ω}}_{d}, ℤ, ℕ_{1}^{n})$ . Together with assumption (H₄), it leads to $∥ {\vec{S}}_{ω} + \vec{S} ∥_{X} = 0$ . So the law of ${\vec{S}}_{ω}$ is equal to that of $- \vec{S}$ . Recalling that ${\vec{S}}_{ω} (\cdot)$ has the same distribution as $\vec{S} (\cdot + ω)$ . Then $\vec{S} (\cdot + ω)$ has the same distribution as $- \vec{S}$ . This finishes the proof.□

Remark 3.2. In literatures [28 –30], the authors discussed anti-periodic oscillations to continuous-time CNNs without spatial diffusions. In literature [27], Zhang et al. investigated 2p-th mean almost periodic sequence for a kind of discrete-time stochastic CNNs without spatial diffusions. To date, nevertheless, the anti-periodic sequence of discrete-time CNNs, let alone CNNs affected by both spatial diffusions and stochastic perturbations, has not been deeply addressed. Thus the current work extends the results in papers [27 –30].

4 Mean-square λ-exponential convergence

Let $S = (u_{1}, \dots, u_{n}, S_{1}, \dots, S_{n})^{T}$ and $\tilde{S} = ({\tilde{u}}_{1}, \dots, {\tilde{u}}_{n}, {\tilde{S}}_{1}, \dots, {\tilde{S}}_{n})^{T}$ be any two solutions ofEquation (2.6) with initial boundary conditions $\begin{matrix} u_{i}^{x^{ℓ}} (0) = φ_{i}^{x^{ℓ}}, {\tilde{u}}_{i}^{x^{ℓ}} (0) = {\tilde{φ}}_{i}^{x^{ℓ}}, \\ \forall x^{ℓ} \in Ω_{d}, u_{i}^{x^{ℓ}} (k) |_{x^{ℓ} \in \partial Ω_{d}} = {\tilde{u}}_{i}^{x^{ℓ}} (k) |_{x^{ℓ} \in \partial Ω_{d}} = 0, \end{matrix}$ $\begin{matrix} S_{i}^{x^{ℓ}} (0) = φ_{i}^{x^{ℓ}}, {\tilde{S}}_{i}^{x^{ℓ}} (0) = {\tilde{φ}}_{i}^{x^{ℓ}}, \\ \forall x^{ℓ} \in Ω_{d}, S_{i}^{x^{ℓ}} (k) |_{x^{ℓ} \in \partial Ω_{d}} = {\tilde{S}}_{i}^{x^{ℓ}} (k) |_{x^{ℓ} \in \partial Ω_{d}} = 0 \end{matrix}$ for all $(k, i) \in (ℕ_{0}, ℕ_{1}^{n}) .$

SetV = (U₁, …,U_n,S₁, …,S_n) ^T with $U_{i}^{x^{ℓ}} (k) = u_{i}^{x^{ℓ}} (k) - {\tilde{u}}_{i}^{x^{ℓ}} (k)$ and $S_{i}^{x^{ℓ}} (k) = S_{i}^{x^{ℓ}} (k) - {\tilde{S}}_{i}^{x^{ℓ}} (k)$ for all $(x^{ℓ}, k, i) \in (Ω_{d}, ℕ_{0}, ℕ_{1}^{n}) .$ Equation (2.6) is said to be globally mean-square λ-exponential convergent if it existsM > 0 and 0 < τ < 1 such that $\begin{matrix} ∥ V (k) ∥_{Ω_{d}} \leq M ∥ φ ∥_{Ω_{d}} λ^{τ k}, \forall k \geq 0, \\ λ = max_{1 \leq i \leq n} {{\bar{λ}}_{α_{i}}, {\bar{λ}}_{β_{i}}} \in (0, 1), \end{matrix}$ where $\begin{matrix} ∥ V (k) ∥_{Ω_{d}} \\ = max_{1 \leq i \leq n, x^{ℓ} \in Ω_{d}} {∥ U_{i}^{x^{ℓ}} (k) ∥_{Ω_{d}}, ∥ S_{i}^{x^{ℓ}} (k) ∥_{Ω_{d}}}, \end{matrix}$ $\begin{matrix} ∥ U_{i}^{x^{ℓ}} (k) ∥_{Ω_{d}} = [E | U_{i}^{x^{ℓ}} (k) |^{2}]^{\frac{1}{2}}, \\ ∥ S_{i}^{x^{ℓ}} (k) ∥_{Ω_{d}} = [E | S_{i}^{x^{ℓ}} (k) |^{2}]^{\frac{1}{2}}, \end{matrix}$ and $\begin{matrix} ∥ φ ∥_{Ω_{d}} = max_{1 \leq i \leq n, x^{ℓ} \in Ω_{d}} \\ {[E | φ_{i}^{x^{ℓ}} - {\tilde{φ}}_{i}^{x^{ℓ}} |^{2}]^{\frac{1}{2}}, [E | φ_{i}^{x^{ℓ}} - {\tilde{φ}}_{i}^{x^{ℓ}} |^{2}]^{\frac{1}{2}}} . \end{matrix}$

Theorem 4.1. Equation (2.6) is globally mean-square λ-exponentially convergent if (H₃)-(H₄) hold.

Proof. ByEquation (2.9), it achieves $\begin{matrix} | U_{i}^{x^{ℓ}} (k) | \leq \prod_{s = 0}^{k - 1} λ_{α_{i}}^{x^{ℓ}} (s) | φ_{i}^{x^{ℓ}} - {\tilde{φ}}_{i}^{x^{ℓ}} | + \sum_{q = 0}^{k - 1} \prod_{s = q + 1}^{k - 1} λ_{α_{i}}^{x^{ℓ}} (s) \sum_{l = 0}^{ν_{q}} | θ_{α_{i}}^{x^{ℓ}} (l, q) | \\ \times [| ξ_{i}^{*} (u_{i}^{x^{ℓ}} (q - l)) - ξ_{i}^{*} ({\tilde{u}}_{i}^{x^{ℓ}} (q - l)) | + {\bar{b}}_{i} | S_{i}^{x^{ℓ}} (q - l) - {\tilde{S}}_{i}^{x^{ℓ}} (q - l) | \\ + \sum_{j = 1}^{n} ({\bar{a}}_{i j} + {\bar{ϱ}}_{i j} + {\bar{ς}}_{i j}) | g_{j} (u_{j}^{x^{ℓ}} (q - l)) - g_{j} ({\tilde{u}}_{j}^{x^{ℓ}} (q - l)) | \\ + \sum_{j = 1}^{n} {\bar{μ}}_{i j} | σ_{i j} (u_{j}^{x^{ℓ}} (q - l)) - σ_{i j} (u_{j}^{x^{ℓ}} (q - l)) | h^{- 1} | Δ_{h} B_{j} (q - l) |] \\ \leq {\bar{λ}}_{α_{i}}^{k} | φ_{i}^{x^{ℓ}} - {\tilde{φ}}_{i}^{x^{ℓ}} | + \sum_{q = 0}^{k - 1} {\bar{λ}}_{α_{i}}^{k - q - 1} \sum_{l = 0}^{ν_{q}} | θ_{α_{i}}^{x^{ℓ}} (l, q) | [ξ_{i}^{*} | U_{i}^{x^{ℓ}} (q - l) | + {\bar{b}}_{i} | S_{i}^{x^{ℓ}} (q - l) | \\ + \sum_{j = 1}^{n} ({\bar{a}}_{i j} + {\bar{ϱ}}_{i j} + {\bar{ς}}_{i j}) L_{j}^{g} | U_{j}^{x^{ℓ}} (q - l) | + \sum_{j = 1}^{n} h^{- 1} {\bar{μ}}_{i j} L_{i j}^{σ} | U_{j}^{x^{ℓ}} (q - l) | | Δ_{h} B_{j} (q - l) |] \end{matrix}$ for all $(x^{ℓ}, k, i) \in (Ω_{d}, ℕ_{0}, ℕ_{1}^{n}) .$ Thus, it derives $\begin{matrix} ‖ U_{i}^{x^{ℓ}} (k) ‖_{Ω_{d}} \leq {\bar{λ}}^{k} ‖ φ ‖_{Ω_{d}} + \sum_{q = 0}^{k - 1} {\bar{λ}}_{α_{i}}^{k - q - 1} \sum_{l = 0}^{ν_{q}} | θ_{α_{i}}^{x^{ℓ}} (l, q) | [ξ_{i}^{*} + {\bar{b}}_{i} \\ + \sum_{j = 1}^{n} ({\bar{a}}_{i j} + {\bar{ϱ}}_{i j} + {\bar{ς}}_{i j}) L_{j}^{g} + 2 h^{- \frac{1}{2}} \sum_{j = 1}^{n} {\bar{μ}}_{i j} L_{i j}^{σ}] ‖ V (q - l) ‖_{Ω_{d}}, \end{matrix}$ (4.1) where $(x^{ℓ}, k, i) \in (Ω_{d}, ℕ_{0}, ℕ_{1}^{n}) .$

Similarly, it gets $‖ S_{i}^{x^{ℓ}} (k) ‖_{Ω_{d}} \leq {\bar{λ}}^{k} ‖ φ ‖_{Ω_{d}} + \sum_{q = 0}^{k - 1} {\bar{λ}}_{β_{i}}^{k - q - 1} \sum_{l = 0}^{ν_{q}} | θ_{β_{i}}^{x^{ℓ}} (l, q) | L_{i}^{g} ‖ V (q - l) ‖_{Ω_{d}},$ (4.2) where $(x^{ℓ}, k, i) \in (Ω_{d}, ℕ_{0}, ℕ_{1}^{n}) .$

Owing to (H₄), it hasM > 1 and 0 < τ < 1 ensuring

${\begin{matrix} \frac{1}{M} + \max_{1 \leq i \leq n} \frac{2 ϵ {\bar{λ}}_{α_{i}}^{- τ (Q^{*} + 1)}}{{\underline{c}}_{i}^{*} (1 - {\bar{λ}}_{α_{i}}^{1 - τ})} [ξ_{i}^{*} + {\bar{b}}_{i} + \sum_{j = 1}^{n} ({\bar{a}}_{i j} + {\bar{ϱ}}_{i j} + {\bar{ς}}_{i j}) L_{j}^{g} + 2 h^{- \frac{1}{2}} \sum_{j = 1}^{n} {\bar{μ}}_{i j} L_{i j}^{σ}] < 1, \\ \frac{1}{M} + \max_{1 \leq i \leq n} \frac{2 {\bar{λ}}_{β_{i}}^{- τ (Q^{*} + 1)}}{{\underline{d}}_{i} (1 - {\bar{λ}}_{β_{i}}^{1 - τ})} L_{i}^{g} < 1. \end{matrix}$

Supposing that $∥ V (k) ∥_{Ω_{d}} \leq M ∥ φ ∥_{Ω_{d}} {\bar{λ}}^{τ k}$ for all $k \in ℕ_{0} .$ If not, there must be one of the following cases holds.

It exists $i_{0} \in ℕ_{1}^{n}$ and $k_{0} \in ℕ$ causing $\begin{matrix} ∥ U_{i_{0}}^{x^{ℓ}} (k_{0}) ∥_{Ω_{d}} > M ∥ φ ∥_{Ω_{d}} {\bar{λ}}^{τ k_{0}}; ∥ V (k) ∥_{Ω_{d}} \leq M ∥ φ ∥_{Ω_{d}} {\bar{λ}}^{τ k}, \forall k \in [0, k_{0})_{ℤ} . \end{matrix}$

It exists $i_{1} \in ℕ_{1}^{n}$ and $k_{1} \in ℕ$ causing $\begin{matrix} ∥ S_{i_{1}}^{x^{ℓ}} (k_{1}) ∥_{Ω_{d}} > M ∥ φ ∥_{Ω_{d}} {\bar{λ}}^{τ k_{1}}; ∥ V (k) ∥_{Ω_{d}} \leq M ∥ φ ∥_{Ω_{d}} {\bar{λ}}^{τ k}, \forall k \in [0, k_{1})_{ℤ} . \end{matrix}$

In the light of (4.1), it results in $\begin{matrix} ‖ U_{i_{0}}^{x^{ℓ}} (k_{0}) ‖_{Ω_{d}} \leq {\bar{λ}}^{k_{0}} ‖ φ ‖_{Ω_{d}} + \sum_{q = 0}^{k_{0} - 1} {\bar{λ}}_{α_{i_{0}}}^{k_{0} - q - 1} \sum_{l = 0}^{ν_{q}} | θ_{α_{i_{0}}}^{x^{ℓ}} (l, q) | [ξ_{i_{0}}^{*} + {\bar{b}}_{i_{0}} \\ + \sum_{j = 1}^{n} ({\bar{a}}_{i_{0} j} + {\bar{ϱ}}_{i_{0} j} + {\bar{ς}}_{i_{0} j}) L_{j}^{g} + 2 h^{- \frac{1}{2}} \sum_{j = 1}^{n} {\bar{μ}}_{i_{0} j} L_{i_{0} j}^{σ}] ‖ V (q - l) ‖_{Ω_{d}} \\ \leq {\bar{λ}}^{k_{0}} ‖ φ ‖_{Ω_{d}} + \sum_{q = 0}^{k_{0} - 1} {\bar{λ}}_{α_{i_{0}}}^{(1 - τ) (k_{0} - q - 1)} \sum_{l = 0}^{ν_{q}} | θ_{α_{i_{0}}}^{x^{ℓ}} (l, q) | [ξ_{i_{0}}^{*} + {\bar{b}}_{i_{0}} \\ + \sum_{j = 1}^{n} ({\bar{a}}_{i_{0} j} + {\bar{ϱ}}_{i_{0} j} + {\bar{ς}}_{i_{0} j}) L_{j}^{g} + 2 h^{- \frac{1}{2}} \sum_{j = 1}^{n} {\bar{μ}}_{i_{0} j} L_{i_{0} j}^{σ}] M ‖ φ ‖_{Ω_{d}} {\bar{λ}}^{τ (q - l)} \end{matrix}$ (4.3) $\begin{matrix} \leq {\bar{λ}}^{k_{0}} ‖ φ ‖_{Ω_{d}} + \frac{2 ϵ {\bar{λ}}_{α_{i_{0}}}^{- τ (Q^{*} + 1)}}{{\underline{c}}_{i_{0}}^{*} (1 - {\bar{λ}}_{α_{i_{0}}}^{1 - τ})} [ξ_{i_{0}}^{*} + {\bar{b}}_{i_{0}} \\ + \sum_{j = 1}^{n} ({\bar{a}}_{i_{0} j} + {\bar{ϱ}}_{i_{0} j} + {\bar{ς}}_{i_{0} j}) L_{j}^{g} + 2 h^{- \frac{1}{2}} \sum_{j = 1}^{n} {\bar{μ}}_{i_{0} j} L_{i_{0} j}^{σ}] M ‖ φ ‖_{Ω_{d}} {\bar{λ}}^{τ k_{0}} \\ \leq {\frac{1}{M} {\bar{λ}}^{(1 - τ) k_{0}} + \frac{2 ϵ {\bar{λ}}_{α_{i_{0}}}^{- τ (Q^{*} + 1)}}{{\underline{c}}_{i_{0}}^{*} (1 - {\bar{λ}}_{α_{i_{0}}}^{1 - τ})} [ξ_{i_{0}}^{*} + {\bar{b}}_{i_{0}} \\ + \sum_{j = 1}^{n} ({\bar{a}}_{i_{0} j} + {\bar{ϱ}}_{i_{0} j} + {\bar{ς}}_{i_{0} j}) L_{j}^{g} + 2 h^{- \frac{1}{2}} \sum_{j = 1}^{n} {\bar{μ}}_{i_{0} j} L_{i_{0} j}^{σ}]} M ‖ φ ‖_{Ω_{d}} {\bar{λ}}^{τ k_{0}} \\ \leq M ‖ φ ‖_{Ω_{d}} {\bar{λ}}^{τ k_{0}} . \end{matrix}$ This induces a confliction with (1). Similarly, by (4.2), it acquires $‖ S_{i_{1}}^{x^{ℓ}} (k_{1}) ‖_{Ω_{d}} \leq {\frac{1}{M} {\bar{λ}}^{(1 - τ) k_{1}} + \frac{2 {\bar{λ}}_{β_{i_{1}}}^{- τ (Q^{*} + 1)}}{{\underline{d}}_{i_{1}} (1 - {\bar{λ}}_{β_{i_{1}}}^{1 - τ})} L_{i}^{g}} M ‖ φ ‖_{Ω_{d}} {\bar{λ}}^{τ k_{1}} \leq M ‖ φ ‖_{Ω_{d}} {\bar{λ}}^{τ k_{1}} .$ It is a confliction with (2). Then $∥ V (k) ∥_{Ω_{d}} \leq M ∥ φ ∥_{Ω_{d}} {\bar{λ}}^{τ k}$ , $\forall k \in ℕ_{0} .$ That is,Equation (2.6) is global mean-square λ-exponentially convergent. The proof is finished.

Remark 4.1. Here, λ^τ describes the convergent rate ofEquation (2.6). If the closer λ^τ tends to 0, i.e., the bigger τ > 0 is, the fasterEquation (2.6) reaches global mean-square stability. Also,M has the same feature as λ^τ. So λ^τ andM are measurement metrics of global mean-square stability toEquation (2.6).

Remark 4.2. Literatures [3 , 28–30] discussed the exponential convergence of continuous-time CNNs without diffusions or stochastic disturbances. In literature [27], Zhang et al. studied 2p-th moment global exponential stability to a kind of discrete-time stochastic CNNs without spatial diffusions. By considering the influence of spacial diffusions, the discussed networks in this paper have a more comprehensive field of application than those in papers [3 , 27–30].

Fig. 1

Anti-12-periodic motion of $u_{1}^{ϖ} (\cdot)$ (ϖ = 5, 9) toEquation (5.2).

5 Illustrative example

Considering nonlocal fuzzy stochastic CNNs with reaction diffusions in the form of

${\begin{matrix} c D_{6 p}^{0.5} u_{1} (t, x) = \frac{1}{0.3} {0.1 \frac{\partial^{2} u_{1} (t, x)}{\partial x^{2}} - 15 u_{1} (t, x) + 0.9 \cos (o t + x) S (t, x) \\ + 3 \sin (o t + 5 x) g_{2} (u_{2} (t, x)) + 0.1 \land_{j = 1}^{2} ς_{j} (t, x) g_{j} (u_{j} (t, x)) \\ + 0.32 \sin (\frac{o}{2} t + \sqrt{13} x) + 6 σ_{11} (u_{1} (t, x)) \frac{d B_{1} (t)}{d t}}, \\ c D_{6 p}^{0.5} u_{2} (t, x) = \frac{1}{0.3} {0.13 \frac{\partial^{2} u_{2} (t, x)}{\partial x^{2}} - 12 u_{2} (t, x) + 041 S (t, x) \\ + \cos (o t + 11 x) g_{1} (u_{1} (t, x)) + 0.2 \lor_{j = 1}^{2} ϱ_{j} (t, x) g_{j} (u_{j} (t, x)) \\ + 0.41 \cos (\frac{o}{2} t + 2 x) + σ_{22} (u_{2} (t, x)) \frac{d B_{2} (t)}{d t}}, \\ c D_{6 p}^{0.8} S (t, x) = - 17 S (t, x) + g_{1} (u_{1} (t, x)), \end{matrix}$ (5.1) where (t,x) ∈ ((6p, 6p + 6] , (0, 10)), $p \in ℤ_{0}$ . Let $o = \frac{π}{6}$ , ς₁ (t,x) = ς₂ (t,x) = | sin(ot + 3x) |, ϱ₁ (t,x) = ϱ₂ (t,x) = | cos(ot + x) |,g₁ (s) = g₂ (s) =0.1s, σ₁₁ (s) =0.01 sin s = σ₂₂ (s) for (t,x) ∈ ((6p, 6p + 6] , (0, 10)), $p \in ℤ_{0}$ , $s \in ℝ$ . The corresponding initial boundary conditions are depicted by

${\begin{matrix} u_{1} (0, x) = u_{2} (0, x) = 0.2 (x - 10) sin x, S (0, x) = 0.5 x sin (x - 10), \forall x \in (0, 10), \\ u_{1} (t, 0) = u_{1} (t, 10) = u_{2} (t, 0) = u_{2} (t, 10) = 0, S (t, 0) = S (t, 10) = 0, \forall t \in ℝ_{0} . \end{matrix}$

Fig. 2

Anti-12-periodic motion of $u_{2}^{ϖ} (\cdot)$ (ϖ = 5, 9) toEquation (5.2).

Takingh = 1 andh₁ = 0.5. It obtains the lattice equations ofEquation (5.1) denoted by

${\begin{matrix} u_{1}^{ϖ} (k + 1) & = & λ_{0.5}^{ϖ} (k) u_{1}^{ϖ} (k) + \sum_{l = μ_{k}}^{k} θ_{0.5}^{ϖ} (k - l, k) [ξ_{1}^{*} (u_{1}^{ϖ} (l)) + 0.9 cos (ol + ϖ) S (l, ϖ) \\ + 3 sin (ol + 5 ϖ) g_{2} (u_{2} (l, ϖ)) + 0.1 ⋀_{j = 1}^{2} ς_{j} (l, ϖ) g_{j} (u_{j} (l, ϖ)) \\ + 0.32 sin (\frac{o}{2} l + \sqrt{13} ϖ) + 6 σ_{11} (u_{1} (l, ϖ)) h^{- 1} Δ_{h} B_{1} (l)], \\ u_{2}^{ϖ} (k + 1) & = & λ_{0.5}^{ϖ} (k) u_{2}^{ϖ} (k) + \sum_{l = μ_{k}}^{k} θ_{0.5}^{ϖ} (k - l, k) [ξ_{2}^{*} (u_{2}^{ϖ} (l)) + 0.41 S (l, ϖ) \\ + cos (ol + 11 ϖ) g_{1} (u_{1} (l, ϖ)) + 0.2 ⋁_{j = 1}^{2} ϱ_{j} (l, ϖ) g_{j} (u_{j} (l, ϖ)) \\ + 0.41 cos (\frac{o}{2} l + 2 ϖ) + σ_{22} (u_{2} (l, ϖ)) h^{- 1} Δ_{h} B_{2} (l)], \\ S^{ϖ} (k + 1) & = & λ_{0.8}^{ϖ} (k) S^{ϖ} (k) + \sum_{l = μ_{k}}^{k} θ_{0.8}^{ϖ} (k - l, k) g_{1} (u_{1} (l, ϖ)) \end{matrix}$ (5.2) with initial conditions $\begin{matrix} u_{1} (0, ϖ) = u_{2} (0, ϖ) = 0.2 (ϖ - 10) sin ϖ, \\ S (0, ϖ) = 0.5 ϖ sin (ϖ - 10) \end{matrix}$ for ϖ = 0.5ℓ, ℓ=0, 1, …, 20, and boundary conditions $\begin{matrix} u_{1}^{ϖ} (k) |_{ϖ \in {0, 10}} \\ = u_{2}^{ϖ} (k) |_{ϖ \in {0, 10}} = S^{ϖ} (k) |_{ϖ \in {0, 10}} = 0, \end{matrix}$

ν_k = k - μ_k, $μ_{k} = max_{p \in ℤ_{0}} {10 p : 10 p \leq k}$ , $λ_{0.5}^{ζ}$ , $θ_{0.5}^{ζ}$ , $λ_{0.8}^{ζ}$ and $θ_{0.8}^{ζ}$ are defined as those in (2.6) with $c_{1}^{*} = 15.8$ , $c_{2}^{*} = 13.04$ andd₁ = 17.88, respectively; $\begin{matrix} ξ_{1}^{*} (u_{1}^{ϖ} (k)) = \frac{0.1}{0.25} [u_{1}^{ϖ + 0.5} (k) + u_{1}^{ϖ - 0.5} (k)], \\ ξ_{2}^{*} (u_{2}^{ϖ} (k)) = \frac{0.13}{0.25} [u_{2}^{ϖ + 0.5} (k) + u_{2}^{ϖ - 0.5} (k)] \end{matrix}$ for all $k \in ℤ_{0}$ and ϖ = 0.5ℓ, ℓ=1, …, 19.

By a calculation, it gets $\begin{matrix} ρ_{1} = 0.9646 < 1, ρ_{2} = 0.3012, \\ ρ_{0} = 8.5085 . \end{matrix}$ Thus, (H₁)-(H₄) hold andEquation (5.2) admits a unique anti-12-periodic sequence solution, which is globally mean-square λ-exponentially convergent, seeFigs. 3 –6.

Fig. 3

Anti-12-periodic motion ofS^ϖ (·)(ϖ = 5, 9) toEquation (5.2).

Fig. 4

Anti-12-periodic motion and λ-exponential convergence of $u_{1}^{ϖ} (\cdot)$ (ϖ = 0, 0.5, …, 10).

Fig. 5

Anti-12-periodic motion and λ-exponential convergence of $u_{2}^{ϖ} (\cdot)$ (ϖ = 0, 0.5, …, 10).

Fig. 6

Anti-12-periodic motion and λ-exponential convergence ofS^ϖ (·)(ϖ = 0, 0.5, …, 10).

6 Conclusions and perspectives

By employing the mix techniques of finite difference and Mittag-Leffler time Euler difference, a novel discrete-time lattice model for nonlocal stochastic CNNs with reaction diffusions and fuzzy logic has been built. Furthermore, the existence of a unique global bounded anti-periodic sequence solution in distribution and global exponential stability in mean-square sense for the achieved stochastic discrete-time lattice model have been investigated with the helps of Banach contractive mapping principle, constant variation formulas for sequences and the theory of stochastic calculus. In considerations of fuzzy logic, stochastic perturbations and spatial diffusions in CNNs, it allows the networks addressed in this paper to have a wider field of application. In particular, the addition of spatial diffusions has increased the difficulty of our study. Meanwhile, it also makes the networks and the content of our study more meaningful than the extant literatures. The work in this paper is groundbreaking and it has laid the theoretical and practical foundations for future research in discrete-time and discrete-space stochastic fuzzy CNNs.

According to the current works, it will be many problems worthy of further discussion.

This paper only discusses $α_{i}, β_{i} \in (0, 1], i \in ℕ_{1}^{n}$ and other cases can be studied.

Reimann-Liouville derivatives should be studied in the further.

Other dynamics ought to be discussed, e.g., bifurcation, chaos and control, etc.

Time delays should be considered in the future works.

For the space variables, other more precise differences should be discussed in the future.

Acknowledgments

This work is supported by Scientific Research Fund Project of Education Department of Yunnan Province under Grant No. 2020J1223 and 2022J1097.

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