# Stability Analysis of Impulsive Stochastic Reaction-Diffusion Cellular Neural Network with Distributed Delay via Fixed Point Theory

Authors: Ruofeng Rao; Shouming Zhong
Journal: Complexity
 (2017)

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## Abstract

This paper investigates the stochastically exponential stability of reaction-diffusion impulsive stochastic cellular neural networks (CNN). The reaction-diffusion pulse stochastic system model characterizes the complexity of practical engineering and brings about mathematical difficulties, too. However, the difficulties have been overcome by constructing a new contraction mapping and an appropriate distance on a product space which is guaranteed to be a complete space. This is the first time to employ the fixed point theorem to derive the stability criterion of reaction-diffusion impulsive stochastic CNN with distributed time delays. Finally, an example is provided to illustrate the effectiveness of the proposed methods.


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## Body

## 1. Introduction

In 1988, cellular neural networks (CNN) were originally introduced in [1, 2]. Since then, dynamic neural networks have received extensive attention due to their classification, associative memory, and parallel computing tasks and the ability to solve complex optimization problems. It is generally known that almost all neural networks have similar applications ([3–12]), but the key to the success of these applications lies in the stability of the system. In fact, there are a number of literatures involved in the stability analysis of CNN ([5, 7, 12–14]). In practical engineering, time delay and pulse are unavoidable. Since neural networks usually have spatial properties, due to the existence of parallel paths of various axonal sizes and lengths, it is necessary to introduce continuous distributed delays to simulate them over a given time horizon. Besides, many evolutionary processes, especially the biological neural network in biological systems and bursting rhythm models in pathology, frequency-modulated signal processing systems, are characterized by abrupt changes of states at certain time instants. In addition, electrons have diffusion behavior in inhomogeneous media. Noise disturbance is unavoidable in real nervous systems, which is a major source of instability and poor performance in neural networks. A neural network can be stabilized or destabilized by certain stochastic inputs. The synaptic transmission in real neural networks can be viewed as a noisy process introduced by random fluctuations from the release of neurotransmitters and other probabilistic causes. Hence, the above influent factors should be also taken into consideration in stability analysis of neural networks. So, in this paper, we consider a class of impulsive stochastic reaction-diffusion cellular neural networks with distributed delay. Lyapunov function method is one of the common techniques to solve the stability of neural networks in recent decades. However, every method has its limit. Different methods lead to different criteria for stability criteria which may imply innovations. Fixed point theory and method is one of the alternative methods ([15–22]). Unlike the known literature, we try to employ Banach fixed point theory in this paper to derive the stability of impulsive stochastic reaction-diffusion cellular neural networks with distributed delay. In the next sections, we shall give some model descriptions and preliminaries and employ Banach fixed point theorem, Ho¨lder inequality, Burkholder-Davis-Gundy inequality, and the continuous semigroup of Laplace operators to derive the stochastically exponential stability criterion of the complex system. Of course, to overcome the difficulty of the complex mathematical model, we need to formulate a new contraction mapping on a product space. Moreover, in order to guarantee the completeness of product space, we need to give a reasonable definition of distance. Finally, an example is provided to illustrate the effectiveness of the proposed result.


## 2. Model Description and Preliminaries

Consider the following reaction-diffusion impulsive stochastic cellular neural networks under Dirichlet boundary value:(1)duit,x=-qidiv∇uit,xdt-aiuit,x-∑j=1nbijfjujt,x-∑j=1ncijfjujt-τt,x-∑j=1nhij∫t-ρttfjujs,xdsdt+σiuit,xdwit,t≠tk,x∈Υ,i∈Nutk+,x=utk-,x+gutk,x,x∈Υ,k=1,2,…uit,x=ζit,x,∀s,x∈-τ,0×Υut,x=0,∀t,x∈0,+∞×∂Υ,where Υ⊂Rm is a bounded domain with the smooth boundary ∂Υ. ui(t,x) is the state variable of the ith neuron at time t and in space variable x for i∈N with N≜{1,2,…,n}. fi denotes the active function of neuron. ai is the rate with which the ith neuron will reset its potential to the resting state in isolation when disconnected from the networks and the external inputs. bij, cij, and hij are elements of feedback template. Let {wi(t),t⩾0} be a real-valued Brownian motion defined on the complete probability space {Ω,F,P} which has natural filtration {Ft}t⩾0. Denote by L2(Υ) the space of all real-valued square integrable functions with the inner product 〈ξ,η〉=∫Υξ(x)η(x)dx, for ξ,η∈L2(Υ) which derives the norm ξ=(∫Υξ2(x)dx)1/2 for ξ∈L2(Υ).  σi(·) is a Borel measurable function. Denote by Δ=∑j=1m∂2/∂xj2 the Laplace operator, with domain DΔ=W01,2Υ∩W02,2(Υ), which generates a strongly continuous semigroup e-qitΔ, where W01,2(Υ) and W02,2(Υ) are the Sobolev spaces with compactly supported sets. div∇ui(t,x) denotes the divergence of ∇ui(t,x) (see, e.g., [25, 26]). qi is the diffusion coefficient, and time delays τ(t),ρ(t)∈[0,τ]. Besides, initial value ζi(t,x) is continuous for s,x∈-τ,0×Υ. The fixed impulsive moments tk(k=1,2,…) satisfy 0<t1<t2<⋯ with limk→+∞tk=+∞.  u(tk+,x) and u(tk-,x) stand for the right-hand and left-hand limit of u(t,x) at time tk, respectively. Further, suppose that u(tk-,x)=limt→tk-u(t,x)=u(tk,x), k=1,2,….In this paper, we assume that(H1)e-qitΔ⩽Me-γt, where M>0 and γ>0 are constants;(H2)fi,gi, and σi are Lipschitz continuous with Lipschitz constants Li>0, Gi, and Ti>0 for i∈N, respectively. In addition, fi(0)=gi(0)=0=σi(0), ∀i∈N.Definition 1.
For anyT>0 and x∈Υ, a stochastic process u={(u1(t,x),u2(t,x),…,un(t,x))}0,T is called a mild solution of impulsive system (1) if, for any i∈N, ui(t,x)∈C([0,T]; L2(Υ)) and, for any t∈[0,T], ui(t,x) is adapted to Ft with (2)Pω:∫0t∫Υuis2dxds<∞=1,and the following stochastic integral equations hold for all i∈N, a.s. for any t∈[0,T] and x∈Υ, (3)uit,x=e-qitΔζ0,x-∫0te-qit-θΔaiuiθ,x-∑j=1nbijfjujθ,x-∑j=1ncijfjujθ-τθ,x-∑j=1nhij∫θ-ρθθfjujs,xdsdθ+∫0te-qit-θΔσiuiθ,xdwiθ+e-qitΔ∑0<tk<teqitkΔgiuitk,x,t⩾0uit,x=ζit,x,∀s,x∈-τ,0×Υ,ut,x=0,∀t,x∈0,+∞×∂Υ.Remark 2.
In Definition1, the mild solution of impulsive system (1) is well defined due to [24, Lemma 3.1].Lemma 3 (Ho¨lder inequality).
Assume that1/p+1/q=1 with p>1, and φ(x)∈Lp(Υ), ϕ∈Lq(Υ); then, (4)∫Υφxϕxdx⩽∫Υφxpdx1/p∫Υϕxqdx1/q.Lemma 4 (Banach contraction mapping principle).
LetΘ be a contraction operator on a complete metric space Γ; then there exists a unique point u∈Γ for which Θ(u)=u.


## 3. Main Result: Stochastically Exponential Stability

Theorem 5.
Assume that (H1) and (H2) hold. Then, CNN (1) is stochastically exponentially mean square stable if the following condition holds: (5)0<κ<1,where μ=infk=1,2,…(tk+1-tk)>0 and(6)κ≜6M21γ2maxi∈Nai2+n1γ2maxi∈N∑j=1nbij2+cij2Lj2+nτ2γ2+2M21+1γ2μ2maxi∈NGi2+2γmaxi∈NTi2.Proof.
Firstly, we need to formulate a contraction mapping on a product space.
LetΓi be the Banach space of all Ft-adapted mean square continuous processes consisting of functions ui(t,x) at t⩾0 with t≠tk such that E(eαtui(t,x)2)→0 as t→+∞, where α∈(0,γ) is a positive scalar. Now, we construct an operator Θ≜(Θ1,Θ2,…,Θi,…,Θn) with Θi:Γi→Γi as follows:(7)Θiuit,x=e-qitΔζ0,x-∫0te-qit-θΔaiuiθ,x-∑j=1nbijfjujθ,x-∑j=1ncijfjujθ-τθ,x-∑j=1nhij∫θ-ρθθfjujs,xdsdθ+∫0te-qit-θΔσiuiθ,xdwiθ+e-qitΔ∑0<tk<teqitkΔgiuitk,x,t⩾0,(8)Θiuit,x=ζit,x,s,x∈-τ,0×ΥΘiuit,x=0,∀t,x∈0,+∞×∂Υ.Equipped with the following distance:(9)distu,v=Emaxi∈Nsupt⩾-τuit,x-vit,x21/2,∀u,v∈Γ1×Γ2×⋯×Γn,Γ1×Γ2×⋯×Γn becomes a complete metric space, where u=u(t,x)=(u1(t,x),u2(t,x),…,un(t,x))T, v=v(t,x)=(v1(t,x),v2(t,x),…,vn(t,x))T.
Next, we are to apply contractive mapping theory to complete the proof via three steps.
Step 1. From (7), for t∈0,+∞∖{tk}k=1∞, we claim that Θi(ui)(t) is mean square continuous. Indeed, let δ be a small enough scalar: (10)EΘiuit+δ,x-Θiuit,x2⩽4Ee-qit+δΔζ0,x-e-qitΔζ0,x2+4E∫0t+δe-qit+δ-θΔaiuiθ,x-∑j=1nbijfjujθ,x-∑j=1ncijfjujθ-τθ,x-∑j=1nhij∫θ-ρθθfjujs,xdsdθ-∫0te-qit-θΔaiuiθ,x-∑j=1nbijfjujθ,x-∑j=1ncijfjujθ-τθ,x-∑j=1nhij∫θ-ρθθfjujs,xdsdθ2+4E∫0t+δe-qit+δ-θΔσiuiθ,xdwiθ-∫0te-qit-θΔσiuiθ,xdwiθ2+4Ee-qit+δΔ∑0<tk<t+δeqitkΔgiuitk,x-e-qitΔ∑0<tk<teqitkΔgiuitk,x2.Firstly, we estimate (11)Ee-qit+δΔζ0,x-e-qitΔζ0,x2⩽Ee-qiδΔ-1e-qitΔζ0,x2⟶0,ifδ⟶0.Next, we evaluate(12)E∫0t+δe-qit+δ-θΔaiuiθ,x-∑j=1nbijfjujθ,x-∑j=1ncijfjujθ-τθ,x-∑j=1nhij∫θ-ρθθfjujs,xdsdθ-∫0te-qit-θΔaiuiθ,x-∑j=1nbijfjujθ,x-∑j=1ncijfjujθ-τθ,x-∑j=1nhij∫θ-ρθθfjujs,xdsdθ2⩽2E∫tt+δe-qit+δ-θΔaiuiθ,x-∑j=1nbijfjujθ,x-∑j=1ncijfjujθ-τθ,x-∑j=1nhij∫θ-ρθθfjujs,xdsdθ2+2E∫0te-qit-θΔaiuiθ,x-∑j=1nbijfjujθ,x-∑j=1ncijfjujθ-τθ,x-∑j=1nhij∫θ-ρθθfjujs,xdse-qiδΔ-1dθ2⟶0,ifδ⟶0.Via Burkholder-Davis-Gundy inequality, we can conclude that if δ→0, (13)E∫0t+δe-qit+δ-θΔσiuiθ,xdwiθ-∫0te-qit-θΔσiuiθ,xdwiθ2⩽8E∫0tM2e-2γt-θσiuiθ,xe-qiδΔ-12dθ+8E∫tt+δM2e-2γt+δ-θσiuiθ,x2dθ⟶0.
Due tot≠tk, it is obvious that (14)Ee-qit+δΔ∑0<tk<t+δeqitkΔgiuitk,x-e-qitΔ∑0<tk<teqitkΔgiuitk,x2⟶0,ifδ⟶0.
So, we have proved from (10)–(14) that Θi(ui)(t) is mean square continuous at t⩾0 with t≠tk.
Next, we claim that(15)limδ→0+Θiuitk+δ=Θiuitk+guitk,limδ→0-Θiuitk+δ=Θiuitk.
Indeed, obviously, (11)–(13) hold for all t=tk, too. In addition, let δ>0 be small enough: (16)e-qitk+δΔ∑0<tj<tk+δeqitjΔgiuitj,x-e-qitkΔ∑0<tj<tkeqitjΔgiuitj,x=giutk,x,δ⟶0+.On the other hand, let δ<0 be small enough: (17)e-qitk+δΔ∑0<tj<tk+δeqitjΔgiuitj,x-e-qitkΔ∑0<tj<tkeqitjΔgiuitj,x=0,δ⟶0-.
This together with (16) implies that (15) holds.
Step 2. We claim that (18)EeαtΘiuit,x2⟶0,ift⟶+∞.
Indeed, we have the following inequality similar to (10): (19)EeαtΘiuit,x2⩽4Eeαte-qitΔζ0,x2+4Eeαt∫0te-qit-θΔaiuiθ,x-∑j=1nbijfjujθ,x-∑j=1ncijfjujθ-τθ,x-∑j=1nhij∫θ-ρθθfjujs,xdsdθ2+4Eeαt∫0te-qit-θΔσiuiθ,xdwiθ2+4Eeαte-qitΔ∑0<tk<teqitkΔgiuitk,x2,t⩾0.Condition (H1) yields (20)Eeαte-qitΔζ0,x2⩽EM2e-2γ-αtζ0,x2⟶0,ift⟶+∞.
For any givenε>0, the assumption E(eαtui(t,x)2)→0 tells us that there exists t∗>0 such that (21)Eeαtuit,x2<ε,∀t⩾t∗.Moreover, Ho¨lder inequality gives (22)Eeαt∫0te-qit-θΔaiuiθ,xdθ2⩽M2ai2γEeαt∫0te-γt-θuiθ,x2dθ⩽M2ai2γEe-γ-αtt∗eγt∗maxθ∈0,t∗uiθ,x2+ε1γ-α,which together with the arbitrariness of ε derives (23)Eeαt∫0te-qit-θΔaiuiθ,xdθ2⟶0,ift⟶+∞.
Besides,(24)Eeαt∫0te-qit-θΔ∑j=1nbijfjujθ,xdθ2⩽EM∑j=1nbijLje-γ-αt∫0teγθujθ,xdθ2.Using similar methods of (21) and (22), we can deduce from (24) that (25)Eeαt∫0te-qit-θΔ∑j=1nbijfjujθ,xdθ2⟶0,ift→+∞.
Similar to that of (24) and (22), we can also obtain (26)Eeαt∫0te-qit-θΔ∑j=1ncijfjujθ-τθ,xdθ2⩽EM∑j=1ncijLj1γe-γ-αteγτt∗+τmaxs∈-τ,t∗+τeγsujs,x2+εeγτ1γ-α.Now, similar to that of (22), we know from (26) that (27)Eeαt∫0te-qit-θΔ∑j=1ncijfjujθ-τθ,xdθ2⟶0,ift⟶+∞.Similarly, Ho¨lder inequality yields (28)Eeαt∫0te-qit-θΔ∑j=1nhij∫θ-ρθθfjujs,xdsdθ2⩽M2E∑j=1nhijLjτγe-γ-αtτmaxθ∈-τ,t∗+τujs,x21γeγt∗+τ+ετeατ1γ-α.Similar to (22), we can conclude from (28) that (29)Eeαt∫0te-qit-θΔ∑j=1nhij∫θ-ρθθfjujs,xdsdθ2⟶0,ift⟶+∞.
Hence,(30)Eeαt∫0te-qit-θΔaiuiθ,x-∑j=1nbijfjujθ,x-∑j=1ncijfjujθ-τθ,x-∑j=1nhij∫θ-ρθθfjujs,xdsdθ2⩽4Eeαt∫0te-qit-θΔaiuiθ,xdθ2+4Eeαt∫0te-qit-θΔ∑j=1nbijfjujθ,xdθ2+4Eeαt∫0te-qit-θΔ∑j=1ncijfjujθ-τθ,xdθ2+4Eeαt∫0te-qit-θΔ∑j=1nhij∫θ-ρθθfjujs,xdsdθ2⟶0,ift⟶+∞.
Burkholder-Davis-Gundy inequality andHo¨lder inequality derive(31)Eeαt∫0te-qit-θΔσiuiθ,xdwiθ2⩽8EM2Ti2e-2γ-αtt∗maxθ∈0,t∗e2γθuiθ,x2+8εEM2Ti214γγ-α,which together with the arbitrariness of ε implies (32)Eeαt∫0te-qit-θΔσiuiθ,xdwiθ2⟶0,ift⟶+∞.Next, we may assume that tl-1<t∗⩽tl and tj-1<t⩽tj.
In addition, one can deduce from (H1)(33)Eeαte-qitΔ∑0<tk⩽t∗eqitkΔgiuitk,x2⩽EMeα/2te-γt∑0<tk⩽tlMeγtkGiuitk,x2⟶0,ift⟶+∞.Besides, we can estimate by means of definite integral (34)Eeαte-qitΔ∑t∗<tk<teqitkΔgiuitk,x2⩽εGi2M4Ee-1/22γ-αt∑tl⩽tk⩽tj-1e1/22γ-αtk2⩽εEM2Gi1+2μ2γ-α2.
Moreover, the arbitrariness ofε implies (35)Eeαte-qitΔ∑t∗<tk<teqitkΔgiuitk,x2⟶0,ift⟶+∞.Hence, if t→+∞, (36)Eeαte-qitΔ∑0<tk<teqitkΔgiuitk,x2⩽2Eeαte-qitΔ∑0<tk⩽t∗eqitkΔgiuitk,x2+2Eeαte-qitΔ∑t∗<tk<teqitkΔgiuitk,x2⟶0.
Combining (19), (20), (23), (30), (32), and (36) results in (18).
Step 3. Finally, we claim that Θ is a contractive mapping on Γ1×Γ2×⋯×Γn.
Indeed, from the above two steps, we knowΘi(Γi)⊂Γi, and then ΘΓ1×Γ2×⋯×Γn⊂Γ1×Γ2×⋯×Γn.
On the other hand, for anyi∈N and u,v∈Γ1×Γ2×⋯×Γn, (37)Emaxi∈Nsupt⩾-τΘiuit,x-Θivit,x2⩽6Emaxi∈Nsupt⩾-τ∫0te-qit-θΔaiuiθ,x-viθ,xdθ2+6Emaxi∈Nsupt⩾-τ∫0te-qit-θΔ∑j=1nbijfjujθ,x-fjvjθ,xdθ2+6Emaxi∈Nsupt⩾-τ∫0te-qit-θΔ∑j=1ncijfjujθ-τθ,x-fjvjθ-τθ,xdθ2+6Emaxi∈Nsupt⩾-τ∫0te-qit-θΔ∑j=1nhij∫θ-ρθθfjujs,x-fjvjs,xdsdθ2+6Emaxi∈Nsupt⩾-τ∫0te-qit-θΔσiuiθ,x-σiviθ,xdwiθ2+6Emaxi∈Nsupt⩾-τe-qitΔ∑0<tk<teqitkΔgiuitk,x-givitk,x2.
Besides, it follows by theHo¨lder inequality that (38)Emaxi∈Nsupt⩾-τ∫0te-qit-θΔaiuiθ,x-viθ,xdθ2⩽M21γ2maxi∈Nai2Emaxi∈Nsupt⩾-τuiθ,x-viθ,x2,Emaxi∈Nsupt⩾-τ∫0te-qit-θΔ∑j=1nbijfjujθ,x-fjvjθ,xdθ2⩽nM21γ2maxi∈N∑j=1nbij2Lj2Emaxi∈Nsupt⩾-τuit,x-vit,x2,Emaxi∈Nsupt⩾-τ∫0te-qit-θΔ∑j=1ncijfjujθ-τθ,x-fjvjθ-τθ,xdθ2⩽nM21γ2maxi∈N∑j=1ncij2Lj2Emaxi∈Nsupt⩾-τuit,x-vit,x2,Emaxi∈Nsupt⩾-τ∫0te-qit-θΔ∑j=1nhij∫θ-ρθθfjujs,x-fjvjs,xdsdθ2⩽nM2τ21γ2maxi∈N∑j=1nhij2Lj2Emaxi∈Nsupt⩾-τuit,x-vit,x2,Emaxi∈Nsupt⩾-τe-qitΔ∑0<tk<teqitkΔgiuitk,x-givitk,x2⩽M4maxi∈NGi22Emaxi∈Nsupt⩾-τuit,x-vit,x2+2Emaxi∈Nsupt⩾-τ1μe-γt∫0teγsuis,x-vis,xds2⩽2M41+1γ2μ2maxi∈NGi2Emaxi∈Nsupt⩾-τuit,x-vit,x2,where we assume that tj-1<t⩽tj.
In addition, it follows from Burkholder-Davis-Gundy inequality that(39)Emaxi∈Nsupt⩾-τ∫0te-qit-θΔσiuiθ,x-σiviθ,xdwiθ2⩽2γM2maxi∈NTi2Emaxi∈Nsupt⩾-τuiθ,x-viθ,x2.
Now, combining (37)–(39) gives (40)distΘu,Θv⩽κdistu,v,∀u,v∈Γ1×Γ2×⋯×Γn,where κ is defined as (6), satisfying 0<κ<1. This implies that Θ:Γ1×Γ2×⋯×Γn→Γ1×Γ2×⋯×Γn is a contraction mapping such that there exists the fixed point u≜(u1(t,x),u2(t,x),…,un(t,x)) of Θ in Γ1×Γ2×⋯×Γn, which implies that u is a solution of CNN (1), satisfying E(eαtΘi(ui(t,x))2)→0, t→+∞ so that Emaxi∈Nsupt⩾-τ(eαtΘi(ui(t,x))2)→0, t→+∞. Therefore, CNN (1) is stochastically exponentially mean square stable.


## 4. Numerical Example

Consider the following impulsive stochastic reaction-diffusion CNN with distributed delay:(41)duit,x=-qidiv∇uit,xdt-aiuit,x-∑j=1nbijsinj10ujt,x-∑j=1ncijsinj10ujt-τt,x-∑j=1nhij∫t-ρttsinj10ujs,xdsdt+sin0.05iuit,xdwit,t≠tk,x∈Υ,i∈Nutk+,x=utk-,x+0.1jsin⁡utk,x,x∈Υ,k=1,2,…uit,x=ζit,x,s,x∈-τ,0×Υut,x=0,u∈0,+∞×∂Υ,where we suppose Υ=(0,π), n=2, τ=3, μ=1.5, ai=0.5i, bij=0.01(i+j)=cij=hij, and qi=-1. Then, via computing the eigenfunctions of -Δ, we can obtain that etΔ⩽e-π2t, t⩾0, so that we can take γ=π2, M=1. In addition, differential mean value theorem yields (42)sinj10ujt,x-sinj10vjt,x⩽j10ujt,x-vjt,x, and then we get Lj=j/10, j=1,2. Similarly, we can compute that Gi=0.1i, Ti=0.05i, and i=1,2. Finally, we can compute (6) on a computer running Matlab software, obtaining κ=0.8716∈(0,1). Therefore, Theorem 5 tells us that CNN (41) is stochastically exponentially mean square stable.Table1 is presented to compare the complexity of neural networks investigated in various literatures via fixed point theorems and techniques.Table 1
Comparison of the complexity of system models in the literature related to fixed point theory.
Theorem5
[16]
[15]
[23]
[24]
Using fixed point theory
Yes
Yes
Yes
Yes
Yes
Impulse model
Yes
No
No
Yes
Yes
Distributed delays
Yes
No
No
No
No
Reaction-diffusion model
Yes
No
No
No
No
Itô stochastic model
Yes
No
No
No
No
Equations type
Integrodifferential (partial) eq.
Differential eq.
Differential eq.
Differential eq.
Integrodifferential eq.
Stability type
Stochastically exponential
Exponential
Exponential
Exponential
ExponentialRemark 6.
Impulsive reaction-diffusion Itô-type stochastic model gives a lot of mathematical difficulties in deriving the stability criterion. Motivated by some methods and techniques of the above-mentioned literature  ([3–31]),  this is the first time for us to analyze such a complex model by way of fixed point theorem. Our model is closer to real engineering so that it is more complex than those of the previous literature, and we utilize Banach fixed point theorem, Ho¨lder inequality, Burkholder-Davis-Gundy inequality, and the continuous semigroup of Laplace operators to overcome the difficulties. Besides, the distance defined in this paper satisfies the triangle inequality, which is another point different from those of previous related literatures.


## 5. Conclusions

Since our CNN model involves pulse and Laplacian operators, our model is different from the previous model ([15–22]), which also implies some difficulties in mathematical techniques. Motivated by the previous literature related to fixed point theory ([15–22, 25–31]), the authors employed Banach fixed point theorem, Ho¨lder inequality, Burkholder-Davis-Gundy inequality, and the continuous semigroup of Laplace operators to derive the stochastically exponential stability criterion of impulsive stochastic reaction-diffusion cellular neural networks with distributed delay.



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