Almost Automorphic Solutions to Cellular Neural Networks With Neutral Type Delays and Leakage Delays on Time Scales

Changjin Xu; Maoxin Liao; Peiluan Li; Zixin Liu

doi:10.2991/ijcis.d.200107.001

Download article (PDF)

Next Article In Issue>

Volume 13, Issue 1, 2020, Pages 1 - 11

Almost Automorphic Solutions to Cellular Neural Networks With Neutral Type Delays and Leakage Delays on Time Scales

Authors

Changjin Xu¹^{, *}, Maoxin Liao², Peiluan Li³, Zixin Liu⁴

¹Guizhou Key Laboratory of Economics System Simulation, Guizhou University of Finance and Economics, Guiyang 550004, PR China

²School of Mathematics and Physics, University of South China, Hengyang 421001, PR China

³School of Mathematics and Statistics, Henan University of Science and Technology, Luoyang 471023, PR China

⁴School of Mathematics and Statistics, Guizhou University of Finance and Economics, Guiyang 550004, PR China

^*Corresponding author. Email: xcj403@126.com

Corresponding Author

Changjin Xu

Received 4 August 2019, Accepted 2 January 2020, Available Online 15 January 2020.

DOI: 10.2991/ijcis.d.200107.001 How to use a DOI?
Keywords: Cellular neural networks; Almost automorphic solution; Exponential stability; Leakage delay; Neutral type delay
Abstract: In this paper, cellular neural networks (CNNs) with neutral type delays and time-varying leakage delays are investigated. By applying the existence of the exponential dichotomy of linear dynamic equations on time scales, a fixed point theorem and the theory of calculus on time scales, a set of sufficient conditions which ensure the existence and exponential stability of almost automorphic solutions of the model are obtained. An example with its numerical simulations is given to support the theoretical findings.
Copyright: © 2020 The Authors. Published by Atlantis Press SARL.
Open Access: This is an open access article distributed under the CC BY-NC 4.0 license (http://creativecommons.org/licenses/by-nc/4.0/).

1. INTRODUCTION

Since the cellular neural networks with delay were first introduced and investigated by Roska and Chua [1], they have been extensively applied in various different fields such as classification of pattern and processing of moving images. In recent years, extensive results on the existence and stability of equilibrium points, periodic solutions, almost periodic solutions and anti-periodic solutions for cellular neural networks have been reported. For example, Fan and Shao [2] investigated the positive almost periodic solutions for shunting inhibitory cellular neural networks with time-varying and continuously distributed delays, Li and Wang [3] analyzed the existence and exponential stability of the almost periodic solutions of shunting inhibitory cellular neural networks on time scales, Xia et al. [4] established the sufficient conditions for the existence and exponential stability of almost periodic solution for shunting inhibitory cellular neural networks with impulses, Peng and Wang [5] addressed the existence and exponential stability of anti-periodic solutions to shunting inhibitory cellular neural networks with time-varying delays in leakage terms. For more related work on shunting inhibitory cellular neural networks, one can see [4, 6–17].

Many scholars [18–21] argue that neural networks usually contain some information about the derivative of the past state to further describe and model the dynamics for the complex neural reactions. Then some authors focused on the dynamical behaviors of neutral type neural networks. For example, Rakkiyappan et al. [22] considered the global exponential stability for neutral-type impulsive neural networks, Li et al. [23] discussed the existence of periodic solutions for neutral type cellular neural networks with delays, Bai [24] investigated the global stability of almost periodic solutions of Hopfield neural networks with neutral time-varying delays. In details, we refer the reader to [25–30].

Very recently, a typical time delay called Leakage (or “forgetting”) delay may exist in the negative feedback terms of the neural network system, and these terms are variously known as forgetting or leakage terms [31,32]. Since time delays in the leakage term are difficult to handle but have great impact on the dynamical behavior of neural networks. Therefore, it is meaningful to consider neural networks with time delays in leakage terms [34].

It is well known that both continuous time and discrete time neural networks play an equal roles in various applications [34]. But it is troublesome to study the dynamical properties for continuous and discrete time systems, respectively. In 1990, Hilger [35] proposed the theory of time scales which can deal with both difference and differential calculus in a consistent way. Thus it is significant to investigate the dynamical behaviors of neural networks on time scales. For instance, some authors [3,36–40] investigated periodic solutions, almost periodic solutions and anti-periodic solutions of some neural networks on time scales.

In addition, we shall point out that in real word, almost periodicity is universal than periodicity. Moreover, almost automorphic functions, which were introduced by Bochner, are much more general than almost periodic functions. In addition, the almost automorphic solutions of neural networks can be applied in many areas such as automatic control, image processing, psychophysics, robotics and so on [41–45]. Almost automorphic solutions in the context of differential equations were studied by several authors. We refer the reader to [46–53]. However, to the best of our knowledge, there is no paper published on the almost automorphic solutions of cellular neural networks with neutral type delays and time-varying leakage delays on time scales.

Inspired by the discussion above, in this paper, we consider the following cellular neural networks with neutral type delays and time-varying leakage delays on time scales

xiΔ(t)=−bi(t)xi(t−ηi(t))+∑j=1naij(t)fj(xj(t))+ ∑j=1nbij(t)fj(xj(t−τij(t)))+ ∑j=1ncij(t)fj(xiΔ(t−σij(t)))+Ii(t),(1)

where T is an almost periodic time scale, i=1,2,⋯,n, n corresponds to the number of units in a neural network, xi corresponds to the state vector of the ith unit at time t, fj(xj(t)) denotes the output of the jth unit on ith unit at time t, bij denotes the strength of the jth unit on the ith unit at time t−τij, Ii denotes the external bias on the ith unit at time t, τij corresponds to the transmission delay along the axon of the jth unit, bi represents the rate with which the ith unit will reset its potential to the resting state in isolation when disconnected from the network and external inputs, ηi>0 with t−ηi(t)∈T for all t∈T denote s time delay in leakage term, τij(t)≥0,σij(t)≥0 correspond the transmission delays and satisfy that for t∈T,t−τij∈T,t−σij∈T.

The main aim of this article is to establish some sufficient conditions for the existence and global exponential stability of almost automorphic periodic solutions of (1). By applying the existence of the exponential dichotomy of linear dynamic equations on time scales, a fixed point theorem and the theory of calculus on time scales, we obtain a set of sufficient conditions for the existence and exponential stability of almost automorphic solutions for model (1).

For convenience, we denote by [a,b]T={t|t∈[a,b]∩T}. For a almost automorphic function f:T→R, f+=supt∈R|f(t)|,f−=inft∈R|f(t)|. We denote by R the set of real numbers, by R+ the set of positive real numbers, by X a real Banach space with the norm ||.||. The initial conditions associated with system (1) are of the form:

xi(s)=φi(s),xiΔ(s)=φiΔ(s),s∈(−τ,0]T,(2)

where τ=max{max1≤i≤nηi+,max(i,j){τij+,σij+}},φi∈C1([−τ,0]T,R) and i,j=1,2,⋯,n.

The remainder of the paper is organized as follows. In Section 2, we introduce some lemmas and definitions, which can be used to check the existence of almost automorphic solutions of system (1). In Section 3, we present some sufficient conditions for the existence of almost automorphic solutions of (1). Some sufficient conditions on the global exponential stability of almost automorphic solutions of (1) are established in Section 4. An example is given to illustrate the effectiveness of the obtained results in Section 5. A brief conclusion is drawn in Section 6.

2. PRELIMINARY RESULTS

In this section, we would like to recall some basic definitions and lemmas which are used in what follows.

Definition 2.1.

[54] Let T be a nonempty closed subset (time scale) of R. The forward and backward jump operators σ,ρ:T→T and the graininess μ:T→R are defined, respectively, by

σ(t)=inf{s∈T:s>t},ρ(t) =sup{s∈T:s<t} and μ(t)=σ(t)−t.

Lemma 2.1.

[54] Assume that p,q:T→R are two regressive functions, then

e0(t,s)≡1 and ep(t,t)≡1;
ep(t,s)=1ep(s,t)=e⊖p(s,t);
ep(t,s)ep(s,r)=ep(t,r);
(ep(t,s))Δ=p(t)ep(t,s).

Lemma 2.2.

[54] Let f,g be Δ-differentiable functions on T, then

(ν1f+ν2g)Δ=ν1fΔ+ν2gΔ, for any constants ν1,ν2;
(fg)Δ(t)=fΔ(t)g(t)+f(σ(t))gΔ(t)=f(t)gΔ(t)+fΔ(t)g(σ(t)).

Lemma 2.3.

[54] Assume that p(t)≥0 for t≥s, then ep(t,s)≥1.

Definition 2.2.

[54] A function p:T→R is called regressive provided 1+μ(t)p(t)≠0 for all t∈Tk; p:T→R is called positively regressive provided 1+μ(t)p(t)>0 for all t∈Tk. The set of all regressive and rd-continuous functions p:T→R will be denoted by R=R(T,R) and the set of all positively regressive functions and rd-continuous functions will be denoted by R+=R+(T,R).

Lemma 2.4.

[54] Suppose that p∈R+, then

ep(t,s)>0, for all t,s∈T;
if p(t)≤q(t) for all t≥s, then ep(t,s)≤eq(t,s) for all t≥s.

Lemma 2.5.

[54] If p∈R and a,b,c∈T, then

[ep(c,.)]Δ=−p[ep(c,.)]σ

and

∫abp(t)ep(c,σ(t))Δt=ep(c,a)−ep(c,b).

Lemma 2.6.

[54] Let a∈Tk,b∈T and assume that f:T×Tk→R is continuous at (t,t) where t∈Tk with t>a. Also assume that fΔ(t) is rd-continuous on [a,σ(t)]. Suppose that for each ε>0, there exists a neighborhood U of ϵ∈[a,σ(t)] such that

|f(σ(t),ϵ)−f(s,ϵ)−fΔ(t,ϵ)(σ(t)−s)|≤ε|σ(t)−s|,for alls∈U,

where fΔ denotes the derivative of f with respect to the first variable. Then

g(t):=∫atf(t,ϵ)Δϵ implies gΔ(t):=∫atfΔ(t,ϵ)Δϵ+f(σ(t),t);
h(t):=∫tbf(t,ϵ)Δϵ implies hΔ(t):=∫tbfΔ(t,ϵ)Δϵ−f(σ(t),t).

Next, we recall some definitions of almost automorphic functions on time scales.

Definition 2.3.

[55] A time scale T is called an almost periodic time scale if

Π:={ϵ∈R:t±ϵ∈T,∀t∈T≠{0}.

Definition 2.4.

[54] Let T be an almost periodic time scale.

A function f(t):T→X is said to be almost automorphic, if for any sequence {sn}n=1∞⊂Π, there is a subsequence{ϵn}n=1∞⊂{sn}n=1∞ such that g(t)=limn→∞f(t+ϵn) is well defined for each t∈T and limn→∞g(t−ϵn)=f(t) for each t∈T. Denote by AA(T,X) the set of all such functions;
A continuous function f:T×X→X is said to be almost automorphic, if f(t,x) is almost automorphic in t∈T uniformly in x∈B, where B is any bounded subset of X. Denote by AA(T×X,X) the set of all such functions.

Lemma 2.7.

[53] Let f,g∈AA(T,X). Then we have the following

f+g∈AA(T,X);
α∈AA(T,X) for any constant α∈R;
if φ:X→Y is a continuous function, then the composite function φ∘f:T→Y is almost automorphic.

Lemma 2.8.

[34] Let f∈AA(T×X,X) and f satisfies the Lipschitz condition in x∈X uniformly in t∈T. If φ∈AA(T,X), then f(t,φ(t)) is almost automorphic.

Definition 2.5.

[55] Let x∈Rn and A(t) be a n×n matrix-valued function on T, the linear system

xΔ(t)=A(t)x(t),t∈T(3)

is said to admit an exponential dichotomy on T if there exist positive constants ki,αi,i=1,2, projection P and the fundamental solution matrix X(t) of (3) satisfying

|X(t)PX−1(s)|≤k1e⊖α1(t,s),s,t∈T,t≥s

and

|X(t)(I−P)X−1(s)|≤k2e⊖α2(t,s),s,t∈T,t≤s,

where |.| is a matrix norm on T, that is, if A=(aij)n×n, then we can take |A|=(∑i=1n∑j=1n|aij|2)12.

Lemma 2.9.

[53] Suppose that A(t)∈AA(T,Rn×n) such that {A−1(t)}t∈T and {((I+μ(t))A(t))−1}t∈T are bounded. Moreover, suppose that g∈AA(T,Rn) and (3) admits an exponential dichotomy, then the following system

xΔ(t)=A(t)x(t)+g(t)(4)

has a solution x(t)∈AA(T,Rn) and x(t) is expressed as follows

x(t)=∫−∞tX(t)PX−1(σ(s))g(s)Δs −∫t+∞X(t)(I−P)X−1(σ(s))g(s)Δs,

where X(t) is the fundamental solution matrix of (3), I denotes the n×n-identity matrix.

Lemma 2.10.

[55] Let ci>0 and −ci(t)∈R+,∀t∈T. If min1≤i≤n{inft∈Tci(t)}=m>0, then the linear system

xΔ(t)=diag(−c1(t),−c2(t),⋯,−cn(t))x(t)(5)

admits an exponential dichotomy on T.

Definition 2.6.

[34] Let x∗(t)=(x1∗(t),x2∗(t),⋯,xn∗(t))T be an almost automorphic solution of (1) with initial value φ∗(t)=(φ1∗(t),φ2∗(t), ⋯,φn∗(t))T. If there exist positive constants λ with ⊖λ∈R+ and M>1 such that for an arbitrary solution x(t)=(x1(t),x2(t),⋯,xn(t))T of (1) with initial value φ(t)=(φ1(t),φ2(t),⋯,φn(t))T satisfies

||x−x∗||≤M||φ−φ∗||e⊖λ(t,t0),t0∈[−τ,∞)T,t≥t0.

Then the solution x∗(t) is said to be globally exponentially stable.

3. EXISTENCE OF ALMOST AUTOMORPHIC SOLUTIONS

In this section, we will establish sufficient conditions on the existence of pseudo almost periodic solutions of (1). Let X∗={f∈C1(T,R|f,fΔ∈AA(T,Rn)} with the norm ||f||X∗=max{|f|1,|fΔ|1}, where |f|1=max1≤i≤nfi+,|fΔ|1=max1≤i≤n(fiΔ)+. Then X∗ is a Banach space. Let φ0(t)=(φ10(t),φ20(t),⋯,φn0(t))T , where φi0(t)=∫−∞te−bi(t,σ(s)Ii(s)Δs),i=1,2,⋯,n and L be a constant satisfying L≥max{||φ0||X∗,max1≤i≤n{fj(0)}}. Throughout this article, we assume that

(H1) bi∈C(T,R+) with −bi∈R+ and inft∈T{1−μ(t)bi(t)}=b̄>0,aij,bij,cij,Ii∈C(T,R), τij,σij∈C(T,R+) are almost automorphic, where i,j=1,2,⋯,n.

(H2) fj∈C(R,R) and there exist constants Lj>0 and Mj>0 such that for any u,v∈R,

|fj(u)−fj(v)|≤Lj|u−v|,|fj(u)|≤Mj,

where j=1,2,⋯,n.

(H3)

max1≤i≤nϱibi−,1+bi+bi−ϱi≤12,max1≤i≤nςibi−,1+bi+bi−ςi≤1

where

ϱi=bi+ηi++∑j=1n(aij++bij++cij+)(Lj+1),ςi=bi+ηi++∑j=1n[(aij++bij++cij+)Lj].

Theorem 3.1.

If (H1)–(H3) are satisfied. Then there exists a unique almost automorphic solution of system (1) in X0={φ∈X∗|||φ−φ0||X∗≤L}.

Proof

For any given φ∈X∗, we consider the following system

xiΔ(t)=−bi(t)xi(t)+Θi(t,φ)+Ii(t),i=1,2,⋯,n,(6)

where

Θi(t,φ)=bi(t)∫t−ηi(t)tφiΔ(s)Δs+∑j=1naij(t)fj(φj(t)) +∑j=1nbij(t)fj(φj(t−τij(t))) +∑j=1ncij(t)fj(φiΔ(t−σij(t))),i=1,2,⋯,n.(7)

It follows from Lemma 2.10 that the linear system

xiΔ(t)=−bi(t)xi(t),i=1,2,⋯,n,(8)

admits an exponential dichotomy on T. Thus, in view of Lemma 2.9, we derive that system (6) has exactly one almost automorphic solution as follows

xiφ(t)=∫−∞te−bi(t,σ(s))[Θi(s,φ)+Ii(s)]Δs,i=1,2,⋯,n,(9)

For φ∈X∗, then

||φ||X∗≤||φ−φ0||X∗+||φ0||X∗≤2L.(10)

Define an operator as follows

Φ:X∗→X∗,(φ1,φ2,⋯,φn)T→(x1φ,x2φ,⋯,xnφ)T.(11)

First we show that for any φ∈X∗, we have Φφ∈X∗. Note that, for i=1,2,⋅,n, we have

|Θi(s,φ)=|bi(s)∫s−ηi(s)sφiΔ(θ)Δθ+∑j=1naij(s)fj(φj(s)) +∑j=1nbij(s)fj(φj(s−τij(s))) +∑j=1ncij(s)fj(φiΔ(s−σij(s)))|

≤bi+ηi+||φ||X∗+∑j=1naij+(|fj(φj(s))−fj(0)|+|fj(0)|) + ∑j=1nbij+(|fj(φj(s−τij(s)))−fj(0)|+|fj(0)|) + ∑j=1ncij+(|fj(φiΔ(s−σij(s)))−fj(0)|+|fj(0)|) ≤bi+ηi+||φ||X∗+∑j=1naij+(Lj||φ||X∗+|fj(0)|) + ∑j=1nbij+(Lj||φ||X∗+|fj(0)|) + ∑j=1ncij+(Lj||φ||X∗+|fj(0)|) =[bi+ηi++∑j=1n(aij++bij++cij+)Lj]||φ||X∗ + ∑j=1n[aij++bij++cij+)|fj(0)|] ≤2L{bi+ηi++∑j=1n[(aij++bij++cij+)(Lj+1)]}.(12)

Thus we get

On the other hand, for i=1,2,⋯,n, we have

|(Φ(φ−φ0))iΔ(t)|=|(∫−∞te−bi(t,σ(s))Θi(s,φ)Δs)tΔ|=|Θi(t,φ)−bi(t)∫−∞te−bi(t,σ(s))Θi(s,φ)Δs|≤|Θi(t,φ)|+|bi(t)|∫−∞te−bi(t,σ(s))|Θi(s,φ)|Δs≤2Lϱi(1+bi+bi−),i=1,2,…,n.(14)

It follows from (H3) that

||Φφ−φ0||X∗≤max1≤i≤nϱibi−,1+bi+bi−ϱi≤L(15)

which implies that Φφ∈X∗. Next, we show that Φ is a contraction. For any φ=(φ1,φ2,⋯,φn)T,ψ=(ψ1,ψ2,⋯,ψn)T∈X∗, for i=1,2,⋯,n, we denote

ϒi(s,φ,ψ)=bi(t)∫s−ηi(s)s(φiΔ(s)−ψiΔ(s))Δs + ∑j=1naij(t)[fj(φj(t))−fj(ψj(t))] + ∑j=1nbij(t)[fj(φj(t−τij(t)))−fj(ψj(t−τij(t)))] + ∑j=1ncij(t)[fj(φiΔ(t−σij(t)))−fj(ψiΔ(t−σij(t)))].(16)

Then

|(Φφ−Φψ)i(t)|=|∫−∞te−bi(t,σ(s))ϒi(s,φ,ψ)Δs|≤∫−∞te−bi(t,σ(s))|ϒi(s,φ,ψ)|Δs≤∫−∞te−bi(t,σ(s)){bi+ηi++∑j=1n[(aij++bij++cij+)Lj]}Δs||φ−ψ||X∗≤ςibi−||φ−ψ||X∗,i=1,2,⋯,n(17)

and

|(Φφ−Φψ)iΔ(t)|=|∫−∞te−bi(t,σ(s))(ϒi(s,φ,ψ))tΔΔs|=|ϒi(s,φ,ψ)−bi(t)∫−∞te−bi(t,σ(s))ϒi(s,φ,ψ)Δs|≤|ϒi(s,φ,ψ)|+|bi(t)|∫−∞te−bi(t,σ(s))|ϒi(s,φ,ψ)|Δs≤{bi+ηi++∑j=1n[(aij++bij++cij+)Lj]}||φ−ψ||X∗+ bi+∫−∞te−bi(t,σ(s))×{bi+ηi++∑j=1n[(aij++bij++cij+)Lj]}Δs||φ−ψ||X∗≤(1+bi+bi−)||φ−ψ||X∗,i=1,2,⋯,n(18)

In view of (H3), we get that ||Φφ−Φφ||<||φ−ψ||. Then Φ is a contraction. Thus Φ has a fixed point in X0, that is, (1) has a unique almost automorphic solution in X0. The proof of Theorem 3.1 is completed.

4. EXPONENTIAL STABILITY OF ALMOST AUTOMORPHIC SOLUTIONS

In this section, we will obtain the exponential stability of the almost automorphic solutions of system (1).

Theorem 4.1.

Suppose that (H1)–(H3) are fulfilled. Then the almost automorphic solution of system (1) is globally exponentially stable.

Proof

By Theorem 3.1, we know that (1) has an almost automorphic solution x(t)=(x1(t),x2(t),⋯,xn(t))T with initial condition φ(t)=(φ1(t),φ2(t),⋯,φn(t))T. Suppose that y(t)=(y1(t),y2(t),⋯,yn(t))T is an arbitrary solution of (1) with initial condition ψ(t)=(ψ1(t),ψ2(t),⋯,ψn(t))T. Denote u(t)=(u1(t),u2(t),⋯,un(t))T, where ui(t)=yi(t)−xi(t),i=1,2,⋯,n. Then it follows from (1) that

uiΔ(t)=−bi(t)ui(t−ηi(t))+∑j=1naij(t)[fj(yj(t))−fj(xj(t))] + ∑j=1nbij(t)[fj(yj(t−τij(t)))−fj(xj(t−τij(t)))] + ∑j=1ncij(t)[fj(yiΔ(t−σij(t)))−fj(xiΔ(t−σij(t)))], i=1,2,⋅,n.(19)

The initial condition of (19) is

ϕi(s)=φi(s)−ψi(s),ϕiΔ(s) =φiΔ(s)−ψiΔ(s),s∈[−τ,0]T,i=1,2,⋯,n.(20)

Rewrite (19) as the form

uiΔ(t)=−bi(t)ui(t)+bi(t)∫t−ηi(t)tuiΔ(s)Δs +∑j=1naij(t)[fj(yj(t))−fj(xj(t))] +∑j=1nbij(t)[fj(yj(t−τij(t)))−fj(xj(t−τij(t)))] +∑j=1ncij(t)[fj(yiΔ(t−σij(t)))−fj(xiΔ(t−σij(t)))], i=1,2,…,n.(21)

It follows from (21) that for i=1,2,⋯,n and t≥t0,t0∈[−τ,0]T,

ui(t)=ui(t0)e−bi(t,t0)+∫t0te−bi(t,σ(s))bi(s)∫s−ηi(s)suiΔ(ϑ)Δϑ+ ∑j=1naij(s)[fj(yj(s))−fj(xj(s))]+ ∑j=1nbij(s)[fj(yj(s−τij(s)))−fj(xj(s−τij(s)))]+∑j=1ncij(s)[fj(yiΔ(s−σij(s)))−fj(xiΔ(s−σij(s)))]∫s−ηi(s)sΔs,(22)

where i=1,2,⋯,n. For μ∈R, define Πi(ω) and Γi(ω) as follows

Πi(ω)=bi−−ω−eωsups∈Tμ(s)[bi+ηi+eωηi++∑j=1naij+Lj +∑j=1nbij+Ljeωτij++∑j=1ncij+Ljeωσij+],(23)

Γi(ω)=bi−−ω−(bi+eωsups∈Tμ(s)+bi−−ω) [bi+ηi+eωηi++∑j=1naij+Lj+∑j=1nbij+Ljeωτij++∑j=1ncij+Ljeωσij+],(24)

where i=1,2,⋯,n. By (H3), we get

Πi(0)=bi−−bi+ηi++∑j=1n(aij++bij++cij+)Lj>0,(25)

Γi(0)=bi−−bi++bi−bi+ηi++∑j=1n(aij++bij++cij+)Lj>0.(26)

Since Πi(ω) and Γi(ω) are continuous on [0,+∞) and limω→+∞Πi(ω)=−∞,limω→+∞Γi(ω)=−∞, then there exist ωi,ωi∗>0 such that Πi(ωi)=0,Γi(ωi∗)=0 and Πi(ω)>0 for ω∈(0,ωi), Γi(ω)>0 for ω∈(0,ωi∗),i=1,2,⋯,n. By choosing a positive constant ω0=min{ω1,ω2,⋯,ωn,ω1∗,ω2∗,⋯,ωn∗}, we get Πi(ω0)≥0 and Γi(ω0)≥0,i=1,2,⋯,n. Thus we can choose a positive constant 0<ξ<min{ω0,min1≤i≤n{bi−}} such that

Πi(ξ)>0,Γi(ξ)>0,i=1,2,⋯,n,

which implies that

eξsups∈Tμ(s)bi−−ξ[bi+ηi+eξηi++∑j=1naij+Lj+∑j=1nbij+Ljeξτij++∑j=1ncij+Ljeξσij+]<1(27)

and

[1+bi+eξsups∈Tμ(s)bi−−ξ][bi+ηi+eξηi++∑j=1naij+Lj+∑j=1nbij+Ljeξτij++∑j=1ncij+Ljeξσij+]<1,(28)

where i=1,2,⋯,n. Let

M=max1≤i≤nbi−bi+ηi++∑j=1n(aij++bij++cij+)Lj(29)

By (H3), we know that M>1. Then we get

1M<eξsups∈Tμ(s)bi−−ξ[bi+ηi+eξηi++∑j=1naij+Lj+∑j=1nbij+Ljeξτij++∑j=1ncij+Ljeξσij+].(30)

Moreover, we have that e⊖ξ(t,t0)>1, where t∈[−τ,t0]T. Then

||u||X∗≤Me⊖ξ(t,t0)||φ−ψ||X∗, for all t∈[−τ,t0]T.(31)

We claim that

||u||X∗≤Me⊖ξ(t,t0)||φ−ψ||X∗, for all t∈[−t0,+∞]T.(32)

To prove this (32), we show that for any p>1, the following inequality holds

||u||X∗≤pMe⊖ξ(t,t0)||φ−ψ||X∗, for all t∈[−t0,+∞]T.(33)

which implies that

|ui(t)|≤pMe⊖ξ(t,t0)||φ−ψ||X∗, for all t∈[−t0,+∞]T.(34)

and

|uiΔ(t)|≤pMe⊖ξ(t,t0)||φ−ψ||X∗, for all t∈[−t0,+∞]T.(35)

By way of contradiction, assume that (33) does not hold. Now we consider the two cases.

Case 1. (34) is not true and (35) is true. Then there exists t∗∈(t0,+∞)T and i∗∈{1,2,⋯,n} such that

|ui∗(t∗)|≥pMe⊖ξ(t∗,t0)||φ−ψ||X∗,|ui∗(t)|<pMe⊖ξ(t,t0)||φ−ψ||X∗,for allt∈[−t0,t∗]T,

|uk(t)|<pMe⊖ξ(t,t0)||φ−ψ||X∗,fork≠i∗,t∈[−t0,t∗]T,k=1,2,⋯,n.

Therefore, there exists a constant γ1≥1 such that

|ui∗(t∗)|=γ1pMe⊖ξ(t∗,t0)||φ−ψ||X∗,|ui∗(t)|<γ1pMe⊖ξ(t,t0)||φ−ψ||X∗,for allt∈[−t0,t∗]T.

|uk(t)|<γ1pMe⊖ξ(t,t0)||φ−ψ||X∗,fork≠i∗,t∈[−t0,t∗]T,k=1,2,⋯,n.

By (22), for i=1,2,⋯,n, we get

|ui∗(t∗)|=|ui∗(t0)e−bi∗(t∗,t0)+∫t0t∗e−bi∗(t∗,σ(s)) {bi∗(s)∫s−ηi∗(s)sui∗Δ(ϑ)Δϑ +∑j=1nai∗j(s)[fj(yj(s))−fj(xj(s))] +∑j=1nbi∗j(s)[fj(yj(s−τi∗j(s)))−fj(xj(s−τi∗j(s)))] +∑j=1nci∗j(s)[fj(yi∗Δ(s−σi∗j(s)))−fj(xi∗Δ(s−σi∗j(s)))]}Δs

≤e−bi∗(t∗,t0)||φ−ψ||X∗+γ1pMe⊖ξ(t∗,t0)||φ−ψ||X∗ × |∫t0t∗e−bi∗(t∗,σ(s))eξ(t∗,σ(s)){bi∗+∫s−ηi∗(s)seξ(σ(s),ϑ)Δϑ + ∑j=1nai∗j+Ljeξ(σ(s),s)+∑j=1nbi∗j+Ljeξ(σ(s),s−τi∗j(s)) ∫s−ηi(s)s+ ∑j=1nci∗j+Ljeξ(σ(s),s−σi∗j(s))}Δs ≤e−bi∗(t∗,t0)||φ−ψ||X∗+γ1pMe⊖ξ(t∗,t0)||φ−ψ||X∗ × |∫t0t∗e−bi∗⊕ξ(t∗,σ(s)){∑j=1nbi∗+ηi∗+eξ(σ(s),s−ηi∗(s)) + ∑j=1nai∗j+Ljeξ(σ(s),s)+∑j=1nbi∗j+Ljeξ(σ(s),s−τi∗j(s)) ∫s−ηi(s)s+ ∑j=1nci∗j+Ljeξ(σ(s),s−σi∗j(s))}Δs ≤e−bi∗(t∗,t0)||φ−ψ||X∗+γ1pMe⊖ξ(t∗,t0)||φ−ψ||X∗ × |∫t0t∗e−bi∗⊕ξ(t∗,σ(s)){∑j=1nbi∗+ηi∗+eξ(ηi∗++supt∈Tμ(s)) + ∑j=1nai∗j+Ljeξsupt∈Tμ(s)+∑j=1nbi∗j+Ljeξ(τi∗j++supt∈Tμ(s)) ∫s−ηi(s)s+ ∑j=1nci∗j+Ljeξ(σi∗j++supt∈Tμ(s))}Δs =γ1pMe⊖ξ(t∗,t0)||φ−ψ||X∗{∫t0t∗1γ1pMe−bi∗⊕ξ(t∗,t0)+eξsupt∈Tμ(s) × [bi∗+ηi∗+eξηi∗++∑j=1nai∗j+Lj+∑j=1nbi∗j+Ljeξτi∗j++∑j=1nci∗j+Ljeξσi∗j+] ×∫t0t∗e−bi∗⊕ξ(t∗,σ(s))Δs}(36)

≤γ1pMe⊖ξ(t∗,t0)||φ−ψ||X∗{∑j=1n1Me−(bi∗−ξ)(t∗,t0)+eξsupt∈Tμ(s) × [(bi∗+ηi∗+eξηi∗++∑j=1nai∗j+Lj+∑j=1nbi∗j+Ljeξτi∗j++∑j=1nci∗j+Ljeξσi∗j+) × 1−(bi∗−ξ)∫t0t∗(−(bi∗−ξ))e−(bi∗−ξ)(t∗,σ(s))Δs]} =γ1pMe⊖ξ(t∗,t0)||φ−ψ||X∗{[1M−eξsupt∈Tμ(s)bi∗−−ξ(∑j=1nbi∗+ηi∗+eξηi∗+ +∑j=1nai∗j+Lj+∑j=1nbi∗j+Ljeξτi∗j++∑j=1nci∗j+Ljeξσi∗j+)]e−(bi∗−ξ)(t∗,t0) + eξsupt∈Tμ(s)bi∗−−ξ(bi∗+ηi∗+eξηi∗++∑j=1nai∗j+Lj+∑j=1nbi∗j+Ljeξτi∗j+ +∑j=1nci∗j+Ljeξσi∗j+)} <γ1pMe⊖ξ(t∗,t0)||φ−ψ||X∗,(37)

which is a contradiction.

Case 2. (35) is not true and (34) is true. Then there exists t∗∗∈(t0,+∞)T and i∗∗∈{1,2,⋯,n} such that

|ui∗∗Δ(t∗∗)|≥pMe⊖ξ(t∗∗,t0)||φ−ψ||X∗,|ui∗∗Δ(t)|<pMe⊖ξ(t,t0)||φ−ψ||X∗,for allt∈[−t0,t∗∗]T,

|ukΔ(t)|<pMe⊖ξ(t,t0)||φ−ψ||X∗,fork≠i∗∗,t∈[−t0,t∗∗]T,k=1,2,⋯,n.

Therefore, there exists a constant γ2≥1 such that

|ui∗∗Δ(t∗∗)|=γ2pMe⊖ξ(t∗∗,t0)||φ−ψ||X∗,|ui∗∗(t)| <γ2pMe⊖ξ(t,t0)||φ−ψ||X∗,for allt∈[−t0,t∗∗]T.

|ukΔ(t)|<γ2pMe⊖ξ(t,t0)||φ−ψ||X∗,fork≠i∗∗,t∈[−t0,t∗∗]T,k=1,2,⋯,n.

By (22), for i=1,2,⋯,n, we have

|ui∗∗Δ(t∗∗)|=|−bi∗∗(t∗∗)ui∗∗(t0)e−bi∗∗(t∗∗,t0)+bi∗∗(t∗∗) ∫t∗∗−ηi∗∗t∗∗ui∗∗Δ(s)Δs + ∑j=1nai∗∗j(t∗∗)[fj(yj(t∗∗))−fj(xj(t∗∗))] + ∑j=1nbi∗∗j(t∗∗)[fj(yj(t∗∗−τi∗∗j(t∗∗))) − fj(xj(t∗∗−τi∗∗j(t∗∗)))] + ∑j=1nci∗∗j(t∗∗)[fj(yi∗∗Δ(t∗∗−σi∗∗j(t∗∗))) − fj(xi∗∗Δ(t∗∗−σi∗∗j(t∗∗)))] − bi∗∗(t∗∗)∫t0t∗∗e−bi∗∗(t∗∗,σ(s)) × {bi∗∗(s)∫s−ηi∗∗sui∗∗Δ(ϑ)Δϑ + ∑j=1nai∗∗j(s)[fj(yj(s))−fj(xj(s))] + ∑j=1nbi∗∗j(s)[fj(yj(s−τi∗∗j(s))) −fj(xj(s−τi∗∗j(s)))] + ∑j=1nci∗∗j(s)[fj(ysΔ(s−σi∗∗j(s))) − fj(xi∗∗Δ(s−σi∗∗j(s)))]∫s−ηi∗∗s}Δs| ≤e−bi∗∗(t∗∗,t0)e−bi∗∗(t∗∗,t0)||φ−ψ||X∗ + γ2pMe⊖ξ(t∗∗,t0)||φ−ψ||X∗ × (bi∗∗+∫s−ηi∗∗seξ(σ(s),ϑ)Δϑ +∑j=1nai∗∗j+Ljeξ(σ(t∗∗),t∗∗) + ∑j=1nbi∗∗j+Ljeξ(σ(t∗∗),t∗∗−τi∗∗j(t∗∗)) ∫s−ηi(s)s+ ∑j=1nci∗∗j+Ljeξ(σ(t∗∗),t∗∗−σi∗∗j(t∗∗)) + bi∗∗+γ2pMe⊖ξ(t∗∗,t0)||φ−ψ||X∗ {∫t0t∗∗e−bi∗∗(t∗∗,σ(s))eξ(t∗∗,σ(s)) × [bi∗∗+∫s−ηi∗∗seξ(σ(s),ϑ)Δϑ+∑j=1nai∗∗j+Ljeξ(σ(s),s) + ∑j=1nbi∗∗j+Ljeξ(σ(s),s−τi∗∗j(s)) ∫s−ηi(s)s+ ∑j=1nci∗∗j+Ljeξ(σ(s),s−σi∗∗j(s))Δs ≤e−bi∗∗(t∗∗,t0)||φ−ψ||X∗+γ2pMe⊖ξ(t∗∗,t0)||φ−ψ||X∗ × (bi∗∗+ηi∗∗+eξ(t∗∗,t∗∗−ηi∗∗(t∗∗)+∑j=1nai∗∗j+Ljeξ(σ(t∗∗),t∗∗) + ∑j=1nbi∗∗j+Ljeξ(σ(t∗∗),t∗∗−τi∗∗j(t∗∗)) + ∑j=1nci∗∗j+Ljeξ(σ(t∗∗),t∗∗−σi∗∗j(t∗∗))) + bi∗∗+γ2pMe⊖ξ(t∗∗,t0)||φ−ψ||X∗{∫t0t∗∗e−bi∗∗⊕ξ(t∗∗,σ(s)) × [bi∗∗+ηi∗∗+eξ(σ(s),s−ηi∗∗(s))+∑j=1nai∗∗j+Ljeξ(σ(s),s) + ∑j=1nbi∗∗j+Ljeξ(σ(s),s−τi∗∗j(s)) ∫s−ηi(s)s+ ∑j=1nci∗∗j+Ljeξ(σ(s),s−σi∗∗j(s))]Δs ≤e−bi∗∗(t∗∗,t0)||φ−ψ||X∗+γ2pMe⊖ξ(t∗∗,t0)||φ−ψ||X∗ × (bi∗∗+ηi∗∗+eξηi∗∗+∑j=1nai∗∗j+Lj+∑j=1nbi∗∗j+Ljeξτi∗∗j + ∑j=1nci∗∗j+Ljeξσi∗∗j) × (1+bi∗∗+eξsupt∈Tμ(s)∫t0t∗∗e−bi∗∗⊕ξ(t∗∗,σ(s))Δs) ≤γ2pMe⊖ξ(t∗∗,t0)||φ−ψ||X∗{bi∗∗+Me−bi∗∗⊕ξ(t∗∗,t0) + (bi∗∗+ηi∗∗+eξηi∗∗++∑j=1nai∗∗j+Lj +∑j=1nbi∗∗j+Ljeξτi∗∗j++∑j=1nci∗∗j+Ljeξσi∗∗j+) × (1+bi∗∗j+eξsupt∈Tμ(s)∫t0t∗∗e−bi∗∗⊕ξ(t∗∗,σ(s))Δs)} ≤γ2pMe⊖ξ(t∗∗,t0)||φ−ψ||X∗{[1M−eξsupt∈Tμ(s)bi∗∗−−ξ (bi∗∗+ηi∗∗+eξηi∗∗++∑j=1nai∗∗j+Lj+∑j=1nbi∗∗j+Ljeξτi∗∗j++∑j=1nci∗∗j+Ljeξσi∗∗j+)] × bi∗∗+e−(bi∗∗−ξ)(t∗∗,t0)+(1+bi∗∗j+eξsupt∈Tμ(s)bi∗∗j−−ξ) (bi∗∗+ηi∗∗+eξηi∗∗++∑j=1nai∗∗j+Lj+∑j=1nbi∗∗j+Ljeξτi∗∗j++∑j=1nci∗∗j+Ljeξσi∗∗j+)} <γ2pMe⊖ξ(t∗∗,t0)||φ−ψ||X∗,(38)

which is also a contradiction. Based on the two cases above, we can conclude that (33) holds. Let p→1, then (32) holds. We can take ⊖λ=⊖ξ, then λ>0 and ⊖λ∈R+. Then we derive

||u||X∗≤M||φ−ψ||X∗e⊖λ(t,t0),t∈−τ,∞)T,t≥t0,(39)

which means that the almost automorphic solution of (1) is globally exponentially stable. The proof of Theorem 4.1 is completed.

Remark 4.1.

In [2–4,7], the scholars considered the almost periodic solution for different type neural networks. In [5,8,10,11,39], the authors investigated anti-periodic solutions to various neural networks. The research topic of [2–5,7,8,10,11,39] did not involve almost automorphic solution. In this article, we have analyzed the almost automorphic solutions to cellular neural networks with neutral type delays and time-varying leakage delays on time scales. The obtained theoretical results in [2–5,7,8,10,11,39] cannot be applied to system (1) to derive the existence and the exponential stability of almost automorphic solutions for system (1). From this viewpoint, we can say that the main results on the existence and the exponential stability of almost automorphic solutions for system (1) are completely new and complement previous publications.

5. AN EXAMPLE

Considering the following cellular neural networks with neutral type delays and time-varying leakage delays on time scales

{x1Δ(t)=−b1(t)x1(t−η1(t))+∑j=12a1j(t)fj(xj(t)) +∑j=12b1j(t)fj(xj(t−τ1j(t))) +∑j=12c1j(t)fj(x1Δ(t−σ1j(t)))+I1(t),x2Δ(t)=−b2(t)x2(t−η2(t))+∑j=12a2j(t)fj(xj(t)) +∑j=12b2j(t)fj(xj(t−τ2j(t))) +∑j=12c2j(t)fj(x2Δ(t−σ2j(t)))+I2(t),(40)

where f1(u1)=sin0.3u1,f2(u2)=sin0.2u2 and

a11(t)a12(t)a21(t)a22(t)=0.01+0.04cos2t0.02+0.01cos3t0.01+0.03cos5t0.01+0.02cos3t,b11(t)b12(t)b21(t)b22(t)=0.01+0.02cos5t0.02+0.03cos2t0.02+0.03cos3t0.02+0.03cos3t,c11(t)c12(t)c21(t)c22(t)=0.02+0.02cos5t0.02+0.03cos2t0.03+0.01cos3t0.03+0.02cos2t,b1(t)b2(t)η1(t)η2(t)=0.02+0.01cos2t0.03+0.01cos3t0.02+0.01cos5t0.01+0.02cos2t,I1(t)I2(t)=0.03+0.01sin5t0.04+0.01sin3t.

Then we get L1=0.3,L2=0.2,M1=0.3,M2=0.2 and

[a11+a12+a21+a22+]=[0.050.030.040.03],[b11+b12+b21+b22+]=[0.030.050.050.05],[c11+c12+c21+c22+]=[0.040.050.040.05],[b1−b2−η1+η2+]=[0.10.20.30.3].

It is not difficult to verify that all assumptions in Theorems 4.1 and 4.2 are fulfilled. Thus we can conclude that (1) has an almost automorphic solution, which is globally exponentially stable. The results are verified by the numerical simulations in Figures 1 and 2.

6. CONCLUSIONS

In this paper, we investigate a class of cellular neural networks with neutral type delays and time-varying leakage delays. Applying the existence of the exponential dichotomy of linear dynamic equations on time scales, a fixed point theorem and the theory of calculus on time scales, we establish a series of sufficient conditions for the existence and exponential stability of almost automorphic solutions for the cellular neural networks with neutral type delays and time-varying leakage delays on time scales. We show that the existence and global exponential stability of almost automorphic solutions for system (1) only depends on time delays ηi(i=1,2,⋯,n) (the delays in the leakage term) and does not depend on time delays τij(i,j=1,2,⋯,n) and σij(i,j=1,2,⋯,n), which implies that the delays in the leakage term do harm to the existence and global exponential stability of almost automorphic solutions. To the best of our knowledge, it is the first time to deal with the almost automorphic solution for cellular neural networks with neutral type delays and time-varying leakage delays on time scales. The idea of this manuscript can be applied directly to investigate a lot of numerous network systems. The theoretical predictions of this manuscript show that under a suitable parameter condition, the cellular neural networks with neutral type delays and leakage delays will display almost automorphic oscillatory phenomenon. In real life, the almost automorphic oscillatory behavior plays an important role in helping us process visual information successfully. It can be effectively applied in predicting the law of brain cell activity, which is useful to serve the diagnosis of diseases. In addition, we know that the quaternion-valued cellular neural networks can be regarded as a generalization of real-valued and complex-valued cellular neural networks. So far, there are very few publications that consider almost automorphic solutions of quaternion-valued cellular neural networks. In the near future, we will focus on this topic.

CONFLICT OF INTEREST

The authors declare that they have no competing interests.

AUTHORS' CONTRIBUTIONS

The study was conceived and designed by Changjin Xu and Maoxin Liao and experiments performed by Peiluan Li and Zixin Liu. All authors read and approved the manuscript.

ACKNOWLEDGMENTS

The work is supported by National Natural Science Foundation of China (No. 61673008), Project of High-level Innovative Talents of Guizhou Province ([2016]5651), Major Research Project of The Innovation Group of The Education Department of Guizhou Province ([2017]039), Innovative Exploration Project of Guizhou University of Finance and Economics ([2017]5736-015), Project of Key Laboratory of Guizhou Province with Financial and Physical Features ([2017]004), Hunan Provincial Key Laboratory of Mathematical Modeling and Analysis in Engineering (Changsha University of Science & Technology)(2018MMAEZD21), University Science and Technology Top Talents Project of Guizhou Province (KY[2018]047), Guizhou University of Finance and Economics (2018XZD01) and Foundation of Science and Technology of Guizhou Province ([2019]1051). The authors would like to thank the referees and the editor for helpful suggestions incorporated into this paper.

REFERENCES

1.T. Roska and L.O. Chua, Cellular neural networks with nonlinear and delay-type templates, Int. J. Circuit Theory Appl., Vol. 20, 1992, pp. 469-481.

2.Q.Y. Fan and J.Y. Shao, Positive almost periodic solutions for shunting inhibitory cellular neural networks with time-varying and continuously distributed delays, Commun. Nonlinear Sci. Numer. Simul., Vol. 15, 2010, pp. 1655-1663.

3.Y.K. Li and C. Wang, Almost periodic solutions of shunting inhibitory cellular neural networks on time scales, Commun. Nonlinear Sci. Numer. Simul., Vol. 17, 2012, pp. 3258-3266.

4.Y.H. Xia, J.D. Cao, and Z.K. Huang, Existence and exponential stability of almost periodic solution for shunting inhibitory cellular neural networks with impulses, Chaos Soliton. Fract., Vol. 34, 2007, pp. 1599-1607.

5.L.Q. Peng and W.T. Wang, Anti-periodic solutions for shunting inhibitory cellular neural networks with time-varying delays in leakage terms, Neurocomputing, Vol. 111, 2013, pp. 27-33.

6.A. Bouzerdoum and R.B. Pinter, Shunting inhibitory cellular neural networks: derivation and stability analysis, IEEE Trans. Circuits Syst. I Fundam. Theory Appl., Vol. 40, 1993, pp. 215-221.

7.L. Chen and H.Y. Zhao, Global stability of almost periodic solution of shunting inhibitory cellular neural networks with variable coefficients, Chaos Soliton. Fract., Vol. 35, 2008, pp. 351-357.

8.J.Y. Shao, Anti-periodic solutions for shunting inhibitory cellular neural networks with time-varying delays, Phys. Lett. A, Vol. 372, 2008, pp. 5011-5016.

9.Y.Q. Yang and J.D. Cao, Stability and periodicity in delayed cellular neural networks with impulsive effects, Nonlinear Anal. Real World Appl., Vol. 8, 2007, pp. 362-374.

10.A.P. Zhang, Existence and exponential stability of anti-periodic solutions for HCNNs with time-varying leakage delays, Adv. Differ. Equ., Vol. 2013, 2013, pp. 162.

11.Z.D. Huang, L.Q. Peng, and M. Xu, Anti-periodic solutions for high-order cellular neural networks with time-varying delays, Electr. J. Differ. Equ., Vol. 2010, 2010, pp. 1-9.

12.Z.L. Li, M.H. Dong, S.P. Wen, X. Hu, P. Zhou, and Z.G. Zeng, CLU-CNNs: object detection for medical images, Neurocomputing, Vol. 350, 2019, pp. 53-59.

13.M.H. Dong, S.P. Wen, Z.G. Zeng, Z. Yan, and T.W. Huang, Sparse fully convolutional network for face labeling, Neurocomputing, Vol. 331, 2019, pp. 465-472.

14.Z. Yan, W.W. Liu, S.P. Wen, and Y. Yang, Multi-label image classification by feature attention network, IEEE Access, Vol. 7, 2019, pp. 98005-98013.

15.Y.J. Fan, X. Huang, Z. Wang, and Y.X. Li, Nonlinear dynamics and chaos in a simplified memristor-based fractional-order neural network with discontinuous memductance function, Nonlinear Dyn., Vol. 93, 2018, pp. 611-627.

16.Y.J. Fan, X. Huang, H. Shen, and J.D. Cao, Switching event-triggered control for global stabilization of delayed memristive neural networks: an exponential attenuation scheme, Neural Netw., Vol. 117, 2019, pp. 216-224.

17.Z. Wang, X.H. Wang, Y.X. Li, and X. Huang, Stability and Hopf bifurcation of fractional-order complex-valued single neuron model with time delay, Int. J. Bifurc. Chaos., Vol. 27, 2017, pp. 1750209.

18.J.H. Park, C.H. Park, O.M. Kwon, and S.M. Lee, A new stability criterion for bidirectional associative memory neural networks of neutral-type, Appl. Math. Comput., Vol. 199, 2008, pp. 716-722.

19.R. Rakkiyappan and P. Balasubramaniam, New global exponential stability results for neutral type neural networks with distributed time delays, Neurocomputing, Vol. 71, 2008, pp. 1039-1045.

20.J. Liu and G. Zong, New delay-dependent asymptotic stability conditions concerning BAM neural networks of neutral type, Neurocomputing, Vol. 72, 2009, pp. 2549-2555.

21.R. Samidurai, S.M. Anthoni, and K. Balachandran, Global exponential stability of neutral-type impulsive neural networks with discrete and distributed delays, Nonlinear Anal. Hybrid Syst., Vol. 4, 2010, pp. 103-112.

22.R. Rakkiyappan, P. Balasubramaniam, and J. Cao, Global exponential stability results for neutral-type impulsive neural networks, Nonlinear Anal. Real World Appl., Vol. 11, 2010, pp. 122-130.

23.Y.K. Li, L. Zhao, and X. Chen, Existence of periodic solutions for neutral type cellular neural networks with delays, Appl. Math. Model., Vol. 36, 2012, pp. 1173-1183.

24.C. Bai, Global stability of almost periodic solutions of Hopfield neural networks with neutral time-varying delays, Appl. Math. Comput., Vol. 203, 2008, pp. 72-79.

25.B. Xiao, Existence and uniqueness of almost periodic solutions for a class of Hopfield neural networks with neutral delays, Appl. Math. Lett., Vol. 22, 2009, pp. 528-533.

26.K. Wang and Y. Zhu, Stability of almost periodic solution for a generalized neutral-type neural networks with delays, Neurocomputing, Vol. 73, 2010, pp. 3300-3307.

27.P. Balasubramaniam, V.M. Revathi, and J.H. Park, L2 − L∞ Filtering for neutral Markovian switching system with mode-dependent time-varying delays and partially unknown transition probabilities, Appl. Math. Comput., Vol. 219, 2013, pp. 9524-9542.

28.S. Lakshmanan, J.H. Park, H.Y. Jung, O.M. Kwon, and R. Rakkiyappan, A delay partitioning approach to delay-dependent stability analysis for neutral type neural networks with discrete and distributed delays, Neurocomputing, Vol. 111, 2013, pp. 81-89.

29.J.H. Park, Synchronization of cellular neural networks of neutral type via dynamic feedback controller, Chaos Soliton. Fract., Vol. 42, 2009, pp. 1299-1304.

30.S.M. Lee, O.M. Kwon, and J.H. Park, A novel delay-dependent criterion for delayed neural networks of neutral type, Phys. Lett. A, Vol. 374, 2010, pp. 1843-1848.

31.P. Balasubramaniam, V. Vembarasan, and R. Rakkiyappan, leakage delays in T-S fuzzy cellular neural networks, Neural Process. Lett., Vol. 33, 2011, pp. 111-136.

32.H. Zhang and M. Yang, Global exponential stability of almost periodic solutions for SICNNs with continuously distributed leakage delays, Abstr. Appl. Anal., Vol. 2013, 2013, pp. 1-14.

33.B.W. Liu, Pseudo almost periodic solution for CNNs with continuously distributed leakage delays, Neural Process Lett., Vol. 42, 2015, pp. 233-256.

34.Y.K. Li and L. Yang, Almost automorphic solution for neutral type high-order Hopfield neural networks with delays in leakage terms on time scales, Appl. Math. Comput., Vol. 242, 2014, pp. 679-693.

35.S. Hilger, Analysis on measure chains—a unified approach to continuous and discrete calculus, Results Math., Vol. 18, 1990, pp. 18-56.

36.Y.K. Li, X. Chen, and L. Zhao, Stability and existence of periodic solutions to delayed Cohen-Grossberg BAM neural networks with impulses on time scales, Neurocomputing, Vol. 72, 2009, pp. 1621-1630.

37.Z.Q. Zhang and K.Y. Liu, Existence and global exponential stability of a periodic solution to interval general bidirectional associative memory (BAM) neural networks with multiple delays on time scales, Neural Netw., Vol. 24, 2011, pp. 427-439.

38.Y.K. Li and L. Yang, Almost periodic solutions for neutral-type BAM neural networks with delays on time scales, J. Appl. Math., Vol. 2013, 2013, pp. 1-13.

39.Y.K. Li, L. Yang, and W.Q. Wu, Anti-periodic solutions for a class of Cohen-Grossberg neural networks with time-varying delays on time scales, Int. J. Syst. Sci., Vol. 42, 2011, pp. 1127-1132.

40.Y.K. Li and J.Y. Shu, Anti-periodic solutions to impulsive shunting inhibitory cellular neural networks with distributed delays on time scales, Commun. Nonlinear Sci. Numer. Simul., Vol. 16, 2011, pp. 3326-3336.

41.L. Li, Z. Wang, Y.X. Li, H. Shen, and J.W. Lu, Hopf bifurcation analysis of a complex-valued neural network model with discrete and distributed delays, Appl. Math. Comput., Vol. 330, 2018, pp. 152-169.

42.Z. Wang, L. Li, Y.Y. Li, and Z.S. Cheng, Stability and Hopf bifurcation of a three-neuron network with multiple discrete and distributed delays, Neural Process. Lett., Vol. 48, 2018, pp. 1481-1502.

43.Y.J. Fan, X. Huang, Z. Wang, and Y.X. Li, Improved quasi-synchronization criteria for delayed fractional-order memristor-based neural networks via linear feedback control, Neurocomputing, Vol. 306, 2018, pp. 68-79.

44.J. Jia, X. Huang, Y.X. Li, J.D. Cao, and Alsaedi Ahmed, Global stabilization of fractional-order mMemristor-based neural networks with time delay, IEEE Trans. Neural Netw. Learn. Syst., 2019, pp. 1-13.

45.Y. Fan, X. Huang, Y. Li, J. Xia, and G. Chen, Aperiodically intermittent control for quasi-synchronization of delayed memristive neural networks: an interval matrix and matrix measure combined method, IEEE Trans. Syst. Man Cybern. Syst., Vol. 49, 2018, pp. 2254-2265.

46.G.M. N’Guéré, Existence and uniqueness of almost automorphic mild solutions to some semilinear abstract differential equations, a unified approach, Semigroup Forum, Vol. 69, 2004, pp. 80-86.

47.J.A. Goldstein and G.M. N’Guérékata, Almost automorphic solutions of semilinear evolution equations, Proc. Am. Math. Soc., Vol. 133, 2005, pp. 2401-2408.

48.K. Ezzinbi, S. Fatajou, and G.M. N’Guérékata, Almost automorphic solutions for some partial functional differential equations, J. Math. Anal. Appl., Vol. 328, 2007, pp. 344-358.

49.F. Ché and Z.B. Nahia, Global attractivity and existence of weighted pseudo almost automorphic solution for GHNNs with delays and variable coefficients, Gulf J. Math., Vol. 1, 2013, pp. 5-24.

50.F. Chérif, Sufficient conditions for global stability and existence of almost automorphic solution of a class of RNNs, Differ. Equ. Dyn. Syst., Vol. 22, 2014, pp. 191-207.

51.G.M. N’Guéré, Topics in Almost Automorphy, Springer, New York, 2005.

52.C. Wang and Y.K. Li, Weighted pseudo almost automorphic functions with applications to abstract dynamic equations on time scales, Ann. Pol. Math., Vol. 108, 2013, pp. 225-240.

53.C. Lizama and J.G. Mesquita, Almost automorphic solutions of dynamic equations on time scales, J. Funct. Anal., Vol. 265, 2013, pp. 2267-2311.

54.M. Bohner and A. Peterson, Dynamic Equations on Time Scales: An Introduction with Applications, Birkhä, Boston, 2001.

55.Y.K. Li and C. Wang, Uniformly almost periodic functions and almost periodic solutions to dynamic equations on time scales, Abstr. Appl. Anal., Vol. 2011, 2011, pp. 1-22.

Download article (PDF)

Next Article In Issue>

Journal: International Journal of Computational Intelligence Systems
Volume-Issue: 13 - 1
Pages: 1 - 11
Publication Date: 2020/01/15
ISSN (Online): 1875-6883
ISSN (Print): 1875-6891
DOI: 10.2991/ijcis.d.200107.001 How to use a DOI?
Open Access: This is an open access article distributed under the CC BY-NC 4.0 license (http://creativecommons.org/licenses/by-nc/4.0/).

Cite this article

ris enw bib

TY  - JOUR
AU  - Changjin Xu
AU  - Maoxin Liao
AU  - Peiluan Li
AU  - Zixin Liu
PY  - 2020
DA  - 2020/01/15
TI  - Almost Automorphic Solutions to Cellular Neural Networks With Neutral Type Delays and Leakage Delays on Time Scales
JO  - International Journal of Computational Intelligence Systems
SP  - 1
EP  - 11
VL  - 13
IS  - 1
SN  - 1875-6883
UR  - https://doi.org/10.2991/ijcis.d.200107.001
DO  - 10.2991/ijcis.d.200107.001
ID  - Xu2020
ER  -

download .riscopy to clipboard