# Anti-Periodic Dynamics of Quaternion-Valued Fuzzy Cellular Neural Networks with Time-Varying Delays on Time Scales

Authors: Shiping Shen; Bing Li; Yongkun Li
Journal: Discrete Dynamics in Nature and Society
 (2018)

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## Abstract

A class of quaternion-valued fuzzy cellular neural networks with time-varying delays on time scales is proposed. Based on inequality analysis techniques on time scales, a fixed point theorem and the theory of calculus on time scales, the existence, and global exponential stability of anti-periodic solutions for this class of neural networks are established. The obtained results are completely new and supplement to the known results. Finally, a numerical example is given to illustrate the feasibility of our results.


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## Body

## 1. Introduction

Since Yang and Yang [1] first introduced fuzzy cellular neural networks (FCNNs) combining fuzzy operations (fuzzy AND and fuzzy OR) with cellular neural networks, FCNNs have been successfully applied in many fields such as physics, chemistry, biology, economics, sociology, medicine, and meteorology [2]. Because all of their applications heavily rely on their dynamics, in recent years, a lot of meaningful results regarding the dynamics of them are obtained by many researchers (see [3–11] and reference therein). For instance, authors in [8] studied the global stability of equilibria of FCNNs, authors in [9] obtained some sufficient conditions for the existence and stability of a unique periodic solution of FCNNs, authors in [10] investigated the existence and global exponential stability of anti-periodic solutions for neutral type FCNNs with time-varying delays and D operator on time scales, the author in [11] studied the almost periodicity of FCNNs with multi-proportional delays, and in other accounts authors investigated other behaviors of FCNNs with time delays.On the one hand, because quaternion-valued neural networks (QVNNs) as an extension of the real-valued neural networks and complex-valued neural networks can be extensively applied to the fields of robotics, attitude control of satellites, computer graphics, ensemble control, color night vision, and image compression ([12–14]) and one of the benefits by using quaternion is the three-dimensional geometrical affine transformation that can be represented efficiently and compactly, the study of dynamical behaviors for QVNNs has received much attention of many scholars and some good results have been obtained for the stability [15–19], dissipativity [20], periodicity [21], pseduo almost periodicity [22], and synchronization of QVNNs [23, 24].On the other hand, as we know, a special case of the quasi-periodicity of functions is the anti-periodicity andω-anti-periodic functions are 2ω-periodic functions, but not all periodic functions are anti-periodic ones. Since the signal transmission process of neural networks can often be described as an anti-periodic process, the problem of anti-periodic solutions for various types of neural networks has been investigated by many authors ([10, 25–34]).Moreover, continuous time and discrete time systems are very important in implementation and applications. In addition, in a realistic system, the interaction among agents can happen at any time, and maybe some continuous time intervals accompany some discrete moments. So it is necessary and significant to consider both continuous time and discrete time cases at the same time in networked systems. Fortunately, the time scale theory, which was introduced by Hilger [35], can unify the study of continuous and discrete analysis, and the study of dynamic equations on time scales can contain, link, and extend the classical theory of differential and difference equations [36]. Recently, the theory of time scale calculus has been applied in real-valued neural networks [37–43] and complex-valued networks [44]. However, to the best of our knowledge, the existence and global stability of anti-periodic solutions of quaternion-valued fuzzy cellular neural networks (QVFCNNs) on time scales have not been considered yet. Besides, it is well known that time delays are unavoidable in real neural network systems and they may cause the changes of the dynamical behaviors of neural networks [11, 25, 40, 41].Motivated by the above discussion, our main aim of this paper is to study the existence and global stability of anti-periodic solutions of QVFCNNs with time-varying delays. The innovation points of this paper are summarized as follows:(1)
We propose a class of QVFCNNs with time-varying delays on time scales which can unify the continuous time and discrete time cases of QVFCNNs and, what is more, which can contain the QVFCNNs that their time argument may vary in some continuous time intervals accompanying some discrete moments.(2)
The QVFCNNs proposed in this paper contain real-valued FCNNs and complex-valued FCNNs as their special cases.(3)
Our methods of establishing the existence of anti-periodic solutions of QVFCNNs with time-varying delays on time scales are different from those used in [10, 25–34] and can be used to study other types of neural networks.(4)
Our results are completely new and supplement to the known results, and our results show that if the coefficients of leakage terms of QVFCNNs with time-varying delays on time scales are positive regressive, then both the continuous time and discrete time QVFCNNs with time-varying delays have the same dynamics for the anti-periodicity.This paper is organized as follows. In Section2, we give the model description and introduce some definitions and preliminary lemmas and transform the quaternion-valued system (10) into an equivalent real-valued system. In Section 3, we establish the existence of anti-periodic solutions of the considered network based on a fixed point theorem. In Section 4, by using some inequality techniques, we derive some sufficient conditions for the global exponential stability of anti-periodic solutions of the considered network. In Section 5, we give an example to show the feasibility and effectiveness of our main results. This paper ends with a brief conclusion in Section 6.


## 2. Model Description and Preliminaries

The quaternion was invented in 1843 by Hamilton [45]. The skew field of quaternion is denoted by (1)Q≔q=qR+qIi+qJj+qKk,where qR, qI, qJ, qK are real numbers and the elements i, j, and k obey the Hamilton’s multiplication rules: (2)ij=-ji=k,jk=-kj=i,ki=-ik=j,i2=j2=k2=ijk=-1.A time scaleT is an arbitrary nonempty closed subset of the real set R with the topology and ordering inherited from R. The forward and the backward jump operators σ,ρ:T→T are defined by (3)σt=inf⁡s∈T,s>t,ρt=sup⁡s∈T,s<t,∀t∈T.The graininess function μ:T→[0,∞) is defined by μ(t)=σ(t)-t. The point t∈T is called left-dense, left-scattered, right-dense, or right-scattered if ρ(t)=t, ρ(t)<t, σ(t)=t or σ(t)>t, respectively. Points that are right-dense and left-dense at the same time are called dense. If T has a left-scattered maximum m, define Tκ=T-{m}; otherwise, set Tκ=T. If T has a right-scattered maximum m, define Tκ=T-{m}; otherwise, set Tκ=T.Assume thatf:T→R is a function and let t∈Tk. Then we define fΔ(t) to be the number (provided it exists) with the property that given any ε>0, there is a neighborhood U of t (i.e., U=(t-δ,t+δ)∩T for some δ>0) such that(4)fσt-fs-fΔtσt-s≤εσt-sfor all s∈U. We call fΔ(t) the delta derivative of f at t.Lemma 1 (see [36]).
Leta∈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 is a neighborhood U of t, independent of τ∈[a,σ(t)], such that(5)fσt,τ-fs,τ-fΔt,τσt-s≤εσt-sfor all s∈U, where fΔ denotes the derivative of f with respect to the first variable. Then (i)
g(t)≔∫atf(t,τ)Δτ implies gΔ(t)≔∫atfΔ(t,τ)Δτ+f(σ(t),t);(ii)
g(t)≔∫tbf(t,τ)Δτ implies gΔ(t)≔∫tbfΔ(t,τ)Δτ-f(σ(t),t).A functionp:T→R is said to be regressive provided 1+μ(t)p(t)≠0,∀t∈Tκ and is said to be positive regressive provided 1+μ(t)p(t)>0,∀t∈Tκ. The set of all regressive and rd-continuous functions p:T→R are denoted by R=R(T)=R(T,R) and the set of all positive regressive and rd-continuous functions p:T→R are denoted by R+=R+(T,R)={p∈R:1+μ(t)p(t)>0,∀t∈T}.Ifp∈R+, then the exponential function is defined by (6)ept,s=exp⁡∫stξμτpτΔτ,∀t,s∈T,with the cylinder transformation(7)ξhz=log⁡1+hzh,ifh≠0,z,ifh=0.Let p,q:T→R be two regressive functions; define(8)p⊕q=p+q+μpq,⊖p=-p1+μp,p⊖q=p⊕⊖q.Lemma 2 (see [36]).
Ifp∈R+, then (i)
ep(t,s)>0, for all t,s∈T;(ii)
ifp(t)≤q(t), then ep(t,s)≤eq(t,s) for all t≥s,t,s∈T.Definition 3 (see [46]).
We say that a time scaleT is periodic if there exists p>0 such that if t∈T, then t±p∈T. For T≠R, the smallest positive p is called the period of the time scale.Definition 4 (see [46]).
LetT≠R be a periodic time scale with period p. We say that the function f:T→R is periodic with period ω if there exists a natural number n such that ω=np, f(t+ω)=f(t) for all t∈T and ω is the smallest positive number such that f(t+ω)=f(t).Remark 5.
From [47], we know that if T is ω-periodic, then the graininess function μ is ω-periodic.Lemma 6 (see [47]).
LetT be ω-periodic and suppose f:T×T→R satisfies the assumptions of Lemma 1. Define g(t)=∫tt+ωf(t,s)Δs. If fΔ(t,s) denotes the derivative of f with respect to t, then (9)gΔt=∫tt+ωfΔt,sΔs+fσt,t+ω-fσt,t.Definition 7.
LetT be an ω-periodic time scale. A function u:T→R is said to be ω-anti-periodic on T if u(t+ω)=-u(t) for all t∈T, ω>0 is a constant.Definition 8.
LetT be an ω-periodic time scale. A function f=fR+fIi+fJj+fKk:T→Qn is called an ω-anti-periodic, where fm:T→Rn,m∈{R,I,J,K}≔M if for every m∈M, fm is ω-anti-periodic.In this paper, we consider the following QVFCNN with time-varying delays on time scales:(10)xlΔt=-altxlt+∑h=1nblhtfhxht-τlht+∑h=1ndlhtμht+⋀h=1nαlht∫t-ηlhttghxhsΔs+⋁h=1nβlht∫t-ξlhttghxhsΔs+⋀h=1nTlhtμht+⋁h=1nSlhtμht+Ilt,where l∈{1,2,…,n}≔Λ, t∈T, T is a periodic time scale; n is the number of neurons in layers; xl(t)∈Q and μh(t)∈Q are the state of the lth neuron at time t and the deviations of the hth neuron at time t, respectively; al(t)>0 represents the rate with which the lth neuron will reset its potential to the resting state in isolation when they are disconnected from the network and the external inputs at time t, αlh(t)∈Q, βlh(t)∈Q, Tlh(t)∈Q, and Slh(t)∈Q are the elements of fuzzy feedback MIN template, fuzzy feedback MAX template, fuzzy feed forward MIN template, and fuzzy feed forward MAX template, respectively; blh(t)∈Q and dlh(t)∈Q are the elements of feedback template and feed forward template, ⋀, ⋁ denote the fuzzy AND and fuzzy OR operations, respectively; fh(·) and gh(·):Q→Q are the activation functions; τlh(t)≥0, ηlh(t)≥0 and ξlh(t)≥0 correspond to transmission delays at time t and satisfy t-τlh(t)∈T, t-ηlh(t)∈T, t-ξlh(t)∈T for t∈T; Il(t)∈Q denotes the input of the lth neuron at time t, l∈Λ.Throughout the rest of the paper, we denote[a,b]T=t∣t∈a,b∩T. For convenience, for a bounded and continuous function f:T→R, we denote f_=inft∈T⁡ft and f¯=supt∈T⁡ft.The initial conditions of system (10) are(11)xls=φls,s∈t0-ν,t0T,where ν=maxl,h∈I⁡{τ¯lh,η¯lh,ξ¯lh}, φl∈Crd([t0-ν,t0]T,Q), l∈Λ.Lemma 9 (see [8]).
Supposex and y are two states of system (15). Then we have (12)⋀j=1nαijtfjx-⋀j=1nαijtfjy≤∑j=1nαijtfjx-fjy,i∈Λ,⋁j=1nβijtfjx-⋁j=1nβijtfjy≤∑j=1nβijtfjx-fjy,i∈Λ.In order to overcome the inconvenience of the noncommutativity of quaternion multiplication, in the following, we will first decompose system (10) into the vector form of the four real-valued systems. To do so, for xh=xhR+xhIi+xhJj+xhKk∈Q, where xhR,xhI,xhJ,xhK∈R, we assume that the activation functions fh(xh) and gh(xh) of (10) can be expressed as (13)fhxh=fhRxhR,xhI,xhJ,xhK+ifhIxhR,xhI,xhJ,xhK+jfhJxhR,xhI,xhJ,xhK+kfhKxhR,xhI,xhJ,xhK,ghxh=ghRxhR,xhI,xhJ,xhK+ighIxhR,xhI,xhJ,xhK+jghJxhR,xhI,xhJ,xhK+kghKxhR,xhI,xhJ,xhK,where fhm,ghm:R4→R, h∈Λ, m∈M.Remark 10.
In system (10), if xh=xhR+xhIi∈C, where xhR,xhI∈R and the activation functions fh,gh:C→C are complex variable functions, that is, (14)fhxh=fhRxhR,xhI+ifhIxhR,xhI,ghxh=ghRxhR,xhI+ighIxhR,xhI,where fhm,ghm:R2→R, h∈Λ, m=R,I, and all the quaternion-valued coefficients of (10) are transformed to the complex-valued coefficients, then system (10) degenerates to a complex-valued system; if all of the activation functions and coefficients of (10) are real variable functions, then system (10) degenerates to a real-valued system.It follows from Hamilton’s multiplication rules that the equivalent real-valued system of QVFCNN (10) can be written as follows:(15)XlΔt=-altXlt+∑h=1nBlhtF~hxht-τlht+∑h=1nDlhtUht+⋀h=1nAlht∫t-ηlhttG~hxhsΔs+⋁h=1nClht∫t-ξlhttG~hxhsΔs+⋀h=1nTlh∗tUht+⋁h=1nSlh∗tUht+Il∗t,l∈Λ,t∈T,where(16)f~hmxh=fhmxhR,xhI,xhJ,xhK,g~hmxh=glmxhR,xhI,xhJ,xhK,Xlt=xlRtxlItxlJtxlKt,Il∗t=IlRtIlItIlJtIlKt,G~hxht=g~hRxhtg~hIxhtg~hJxhtg~hKxht,Uht=μhRtμhItμhJtμhKt,Blht=blhRt-blhIt-blhJt-blhKtblhItblhRt-blhKtblhJtblhJtblhKtblhRt-blhItblhKt-blhJtblhItblhRt,Dlht=dlhRt-dlhIt-dlhJt-dlhKtdlhItdlhRt-dlhKtdlhJtdlhJtdlhKtdlhRt-dlhItdlhKt-dlhJtdlhItdlhRt,Alht=αlhRt-αlhIt-αlhJt-αlhKtαlhItαlhRt-αlhKtαlhJtαlhJtαlhKtαlhRt-αlhItαlhKt-αlhJtαlhItαlhRt,Clht=βlhRt-βlhIt-βlhJt-βlhKtβlhItβlhRt-βlhKtβlhJtβlhJtβlhKtβlhRt-βlhItβlhKt-βlhJtβlhItβlhRt,Slh∗t=SlhRt-SlhIt-SlhJt-SlhKtSlhItSlhRt-SlhKtSlhJtSlhJtSlhKtSlhRt-SlhItSlhKt-SlhJtSlhItSlhRt,Tlh∗t=TlhRt-TlhIt-TlhJt-TlhKtTlhItTlhRt-TlhKtTlhJtTlhJtTlhKtTlhRt-TlhItTlhKt-TlhJtTlhItTlhRt,F~hxht-τlht=f~hRxht-τlhtf~hIxht-τlhtf~hJxht-τlhtf~hKxht-τlht,with the initial conditions(17)Xls=Φls,s∈t0-ν,t0T,where Φl(s)=(φlR(s),φlI(s),φlJ(s),φlK(s))T, φlm(s)∈Crd([t0-ν,t0]T,R), l∈Λ,m∈M.Remark 11.
It is easy to see that ifX(t)=(X1(t),X2(t),…,Xn(t))T is a solution of system (15), where Xl(t)=(xlR(t),xlI(t),xlJ(t),xlK(t)),l∈Λ, then x(t)=(x1(t),x2(t),…,xn(t))T, where xl(t)=xlR(t)+ixlI(t)+jxlJ(t)+kxlK(t),l∈Λ is a solution of (10). Thus, the problem of finding an anti-periodic solution for (10) is reduced to finding one for system (15). For considering the stability of anti-periodic solutions of (10), we just need to consider the stability of anti-periodic solutions of system (15).


## 3. The Existence of Anti-Periodic Solutions

In this section, we will state and prove the sufficient conditions for the existence of anti-periodic solutions of (15).LetX=φ∣φ=φ1R,φ1I,φ1J,φ1K,φ2R,φ2I,φ2J,φ2K,…,φnR,φnI,φnJ,φnKT≔φ1,φ2,…,φnT∈CT,R4n:φt+ω=-φt,∀t∈T with the norm φX=max⁡maxl∈Λ,m∈M⁡supt∈T⁡φlmt, then X is a Banach space. Let φ0=((φ10)R,(φ10)I,(φ10)J,(φ10)K,(φ20)R,(φ20)I,(φ20)J, (φ20)K,…,(φn0)R,(φn0)I,(φn0)J,(φn0)K)T≔(φ10,φ20,…,φn0)T where φl0(t)=∫tt+ωG-alt,σs(∑h=1nDlh(s)×Uh(s)+⋀h=1nTlh∗(s)Uh(s)+⋁h=1nSlh∗(s)Uh(s)+Il∗(s))Δs, l∈Λ and r be a constant satisfying r≥φ0X.Throughout this paper, we assume that the following conditions hold:H1
Functionsal,τlh,ηlh,ξlh∈C(T,R+) with -al∈R+, blhm,dlhm, αlhm, βlhm, Tlhm, Slhm∈C(T,R) are ω-periodic, μhm,Ilm∈C(T,R) are ω-anti-periodic, and functions fhm, ghm∈C(R4,R) satisfy (18)fhmxhR,xhI,xhJ,xhK=-fhm-xhR,-xhI,-xhJ,-xhK,glmxhR,xhI,xhJ,xhK=-glm-xhR,-xhI,-xhJ,-xhK,wherexlm∈R, l,h∈Λ, m∈M.(H2)
Functionsfhm, ghm∈C(R4,R) and there exist positive constants Lhm, Hhm such that for all xhm, yhm∈R, (19)fhmxhR,xhI,xhJ,xhK-fhmyhR,yhI,yhJ,yhK≤LhRxR-yR+LhIxI-yI+LhJxJ-yJ+LhKxK-yK,ghmxhR,xhI,xhJ,xhK-ghmyhR,yhI,yhJ,yhK≤HhRxR-yR+HhIxI-yI+HhJxJ-yJ+HhKxK-yKandfhm(0,0,0,0)=0, ghm(0,0,0,0)=0, h∈Λ, m∈M.(H3)
(20)maxl∈Λ⁡ωκlPl1+κl≔ξ<1,where κl=e-al(0,ω), l∈Λ, (21)Pl=∑h=1nLhb¯lhR+b¯lhI+b¯lhJ+b¯lhK+∑h=1nη¯lhHhα¯lhR+α¯lhI+α¯lhJ+α¯lhK+∑h=1nξ¯lhHhβ¯lhR+β¯lhI+β¯lhJ+β¯lhK,l∈Λ,Lh=LhR+LhI+LhJ+LhK,Hh=HhR+HhI+HhJ+HhK,h∈Λ.Lemma 12.
Suppose that(H1) holds. Then X(t)=(X1(t),X2(t),…,Xn(t))T is an ω-anti-periodic solution of system (15) if and only if Xl(t) satisfies(22)Xlt=∫tt+ωGlt,σsKls,xΔs,l∈Λ,t∈T,where Gl(t,σ(s))=-e-alt,σs/1+e-al0,ω, (23)Kls,x=∑h=1nBlhsF~hxhs-τlhs+∑h=1nDlhsUhs+⋀h=1nAlhs∫s-ηlhssG~hxhuΔu+⋁h=1nClhs∫s-ξlhssG~hxhuΔu+⋀h=1nTlh∗sUhs+⋁h=1nSlh∗sUhs+Il∗s.Proof.
LetX(t) be an ω-anti-periodic solution of system (15). Set Yl(s)=e-al(t,s)Xl(s), then(24)YlΔs=alse-alt,σsXls+e-alt,σsXlΔs=e-alt,σsalsXls+XlΔs=e-alt,σsKls,x.Integrating (24) from t to t+ω, we have (25)Yt+ω-Yt=∫tt+ωe-alt,σsKls,xΔs.Note that Yl(t)=Xl(t), thus (26)e-alt,t+ωXlt+ω=Xlt+∫tt+ωe-alt,σsKls,xΔs.Since Xl(t+ω)=-Xl(t), we have (27)Xlt=∫tt+ωGlt,σsKls,xΔs,l∈Λ,t∈T,that is, X(t) is a solution of (22).
On the other hand, letX(t) be a solution of (22). By Lemma 6, we obtain (28)XlΔt=-∫tt+ωe-alt,σs1+e-al0,ωKls,xΔsΔ=∫tt+ωalte-alt,σs1+e-al0,ωKls,xΔs-e-alσt,σt+ω1+e-al0,ωKlt+ω,x+e-alσt,σt1+e-al0,ωKlt,x=∫tt+ωalte-alt,σs1+e-al0,ωXlΔs+alsXlsΔs-e-alσt,σt+ω1+e-al0,ωKlt+ω,x+e-alσt,σt1+e-al0,ωKlt,x=alt1+e-al0,ω∫tt+ωe-alt,σsXlΔs+alsXlsΔs+Klt,x=alt1+e-al0,ω∫tt+ωe-alt,sXlsΔΔs+Klt,x=-altXlt+Klt,x,l∈Λ,t∈T.Therefore, (X1(t),X2(t),…,Xn(t))T is an ω-anti-periodic solution of system (15). The proof of Lemma 12 is completed.Remark 13.
It is easy to see that(29)11+κl≤Glt,σs≤κl1+κl,σs∈t,t+ωT,l∈Λ,t∈T.Theorem 14.
Let(H1)-(H3) hold. Then, system (15) has a unique ω-anti-periodic solution in the region X∗={φ∈X:φ-φ0X≤ξr/1-ξ}.Proof.
Define an operatorΦ:X∗→X by(30)Φφ=Φφ1,Φφ2,…,ΦφnT⟶X1φ,X2φ,…,XnφT,where Xlφ(t)=∫tt+ωGl(t,σ(s))Kl(s,φ)Δs,t∈T,l∈Λ and Gl,Kl are defined the same as those in Lemma 12.
Forφ∈X∗, then φX≤φ-φ0X+φ0X≤ξr/1-ξ+r=r/1-ξ. First, we show that for any φ∈X∗, Φφ∈X∗. From (H2), we have (31)Φφlt+ω=∫t+ωt+ω+ωGlt+ω,σs∑h=1nBlhsF~hφhs-τlhs+∑h=1nDlhsUhs+⋀h=1nAlhs∫s-ηlhssG~hφhuΔu+⋁h=1nClhs∫s-ξlhssG~hφhuΔu+⋀h=1nTlh∗sUhs+⋁h=1nSlh∗sUhs+Il∗sΔs=∫tt+ωGlt+ω,σs+ω∑h=1nBlhs+ωF~hφhs+ω-τlhs+ω+∑h=1nDlhs+ωUhs+ω+⋀h=1nAlhs+ω∫s+ω-ηlhs+ωs+ωG~hφhuΔu+⋁h=1nClhs+ω∫s+ω-ξlhs+ωs+ωG~hφhuΔu+⋀h=1nTlh∗s+ωUhs+ω+⋁h=1nSlh∗s+ωUhs+ω+Il∗s+ωΔs=-∫tt+ωGlt,σs∑h=1nBlhsF~hφhs-τlhs+∑h=1nDlhsUhs+⋀h=1nAlhs∫s-ηlhssG~hφhuΔu+⋁h=1nClhs∫s-ξlhssG~hφhuΔu+⋀h=1nTlh∗sUhs+⋁h=1nSlh∗sUhs+Il∗sΔs=-Φφlt,l∈Λ,t∈T.From (H1)-(H3), we have (32)ΦφlRt-φ0lRt=∫tt+ωGlt,σs∑j=1nblhRsf~hRφhs-τlhs-blhIsf~hIφhs-τlhs-blhJsf~hJφhs-τlhs-blhKsf~hKφhs-τlhs+⋀h=1nαlhRs∫s-ηlhssg~hRφhuΔu-⋀h=1nαlhIs∫s-ηlhssg~hIφhuΔu-⋀h=1nαlhJs∫s-ηlhssg~hJφhuΔu-⋀h=1nαlhKs∫s-ηlhssg~hKφhuΔu+⋁h=1nβlhRs∫s-ξlhssg~hRφhuΔu-⋁h=1nβlhIs∫s-ξlhssg~hIφhuΔu-⋁h=1nβlhJs∫s-ξlhssg~hJφhuΔu-⋁h=1nβlhKs∫s-ξlhssg~hKφhuΔuΔs≤∫tt+ωGlt,σs∑h=1nLhb¯lhR+b¯lhI+b¯lhJ+b¯lhK+∑h=1nη¯lhHhα¯lhR+α¯lhI+α¯lhJ+α¯lhK+∑h=1nξ¯lhHhβ¯lhR+β¯lhI+β¯lhJ+β¯lhKΔsφX≤ωκl1+κl∑h=1nLhb¯lhR+b¯lhI+b¯lhJ+b¯lhK+∑h=1nη¯lhHhα¯lhR+α¯lhI+α¯lhJ+α¯lhK+∑h=1nξ¯lhHhβ¯lhR+β¯lhI+β¯lhJ+β¯lhKr1-ξ=ωκlPl1+κlr1-ξ≤ξr1-ξ,l∈Λ.In a similar way, one can obtain (33)Φφlmt-φ0lmt≤ξr1-ξ,m=I,J,K,l∈Λ.Therefore, (34)Φφ-φ0≤ξr1-ξ,which implies that Φφ∈X∗, so the mapping Φ is a self-mapping from X∗ to X∗. Next, we shall prove that Φ is a contraction mapping. In fact, for any φ,ψ∈X∗, we can get (35)Φφ-ΦψlRt=∫tt+ωGlt,σs∑j=1nblhRsf~hRφhs-τlhs-f~hRψhs-τlhs-blhIsf~hIφhs-τlhs-f~hIψhs-τlhs-blhJsf~hJφhs-τlhs-f~hJψhs-τlhs-blhKsf~hkφhs-τlhs-f~hKψhs-τlhs+⋀h=1nαlhRs∫s-ηlhssg~hRφhu-g~hRψhuΔu-⋀h=1nαlhIs∫s-ηlhssg~hIφhu-g~hIψhuΔu-⋀h=1nαlhJs∫s-ηlhssg~hJφhu-g~hJψhuΔu-⋀h=1nαlhKs∫s-ηlhssg~hKφhu-g~hKψhuΔu+⋁h=1nβlhRs∫s-ξlhssg~hRφhu-g~hRψhuΔu-⋁h=1nβlhIs∫s-ξlhssg~hIφhu-g~hIψhuΔu-⋁h=1nβlhJs∫s-ξlhssg~hJφhu-g~hJψhuΔu-⋁h=1nβlhKs∫s-ξlhssg~hKφhu-g~hKψhuΔuΔs≤∫tt+ωGlt,σs∑h=1nLhb¯lhR+b¯lhI+b¯lhJ+b¯lhk+∑h=1nη¯lhHhα¯lhR+α¯lhI+α¯lhJ+α¯lhK+∑h=1nξ¯lhHhβ¯lhR+β¯lhI+β¯lhJ+β¯lhKΔsφ-ψX≤ωκl1+κl∑h=1nLhb¯lhR+b¯lhI+b¯lhJ+b¯lhK+∑h=1nη¯lhHhα¯lhR+α¯lhI+α¯lhJ+α¯lhK+∑h=1nξ¯lhHhβ¯lhR+β¯lhI+β¯lhJ+β¯lhKφ-ψX≤ξφ-ψX,l∈Λ.In a similar way, we can obtain (36)Φφ-Φψlmt≤ξφ-ψX,m=I,J,K,l∈Λ.Hence, (37)Φφ-ΦψX≤ξφ-ψX.Therefore, by (H3), Φ is a contraction mapping. Thus, system (15) has a unique ω-anti-periodic solution in the region X∗=φ∈X:φ-φ0X≤ξr/1-ξ. This completes the proof of Theorem 14.


## 4. Global Exponential Stability of Anti-Periodic Solutions

In this section, we will study the global exponential stability of anti-periodic solutions of (15).Theorem 15.
Assume that(H1)-(H3) hold. Suppose further that (H4)
a_l>max⁡Pl,l∈Λ,
then system (15) has an ω-anti-periodic solution that is globally exponentially stable.Proof.
From Theorem14, we see that system (15) has an ω-anti-periodic solution Z∗(t)=(X1∗(t),X2∗(t),…,Xn∗(t))T with initial value φ∗(s)=(φ1∗(t),φ2∗(t),…,φn∗(t))T. Suppose that Z(t)=(X1(t),X2(t),…,Xn(t))T is an arbitrary solution of system (15) with initial value φ(s)=(φ1(t),φ2(t),…,φn(t))T. Then(38)Zt-Z∗tlΔt=-altXlt-Xl∗t+∑h=1nBlhtF~hxht-τlht-F~hxh∗t-τlht+⋀h=1nAlht∫t-ηlhttG~hxhs-G~hxh∗sΔs+⋁h=1nClht∫t-ξlhttG~hxhs-GhxhsΔs,l∈Λ,t∈T.Let Θl be defined by (39)Θlθ=a_l-θ-exp⁡θsups∈T⁡μs∑h=1nexp⁡θτ¯lhLhb¯lhR+b¯lhI+b¯lhJ+b¯lhK+∑h=1nη¯lhHhα¯lhR+α¯lhI+α¯lhJ+α¯lhK+∑h=1nξ¯lhHhβ¯lhR+β¯lhI+β¯lhJ+β¯lhK,l∈Λ. By (H5), for l∈Λ, we find (40)Θl0=a_l-Pl>0.Since Θl is continuous on [0,+∞) and Θl(θ)→-∞, as θ→+∞, so there exists θl such that Θl(θl)=0 and Θl(θ)>0 for θ∈(0,θl), l∈Λ. By choosing c=minl∈Λ⁡{θl}, we have Θl(c)≥0, l∈Λ. So, we can choose a positive constant 0<λ<min⁡{c,minl∈Λ⁡{a_l}} such that (41)Θlλ>0,l∈Λ,which imply that (42)exp⁡λsups∈T⁡μsa_l-λ∑h=1nexp⁡λτ¯lhLhb¯lhR+b¯lhI+b¯lhJ+b¯lhK+∑h=1nη¯lhHhα¯lhR+α¯lhI+α¯lhJ+α¯lhK+∑h=1nξ¯lhHhβ¯lhR+β¯lhI+β¯lhJ+β¯lhK<1,where l∈Λ. Let(43)M∗=min⁡al_Pl,l∈Λ,by (H4), we have M∗>1. Thus, (44)1M∗<exp⁡λsups∈T⁡μsa_l-λ∑h=1nexp⁡λτ¯lhLhb¯lhR+b¯lhI+b¯lhJ+b¯lhK+∑h=1nη¯lhHhα¯lhR+α¯lhI+α¯lhJ+α¯lhK+∑h=1nξ¯lhHhβ¯lhR+β¯lhI+β¯lhJ+β¯lhK.
LetZ(t)-Z∗(t)=maxl∈Λ⁡maxm∈M⁡xlmt-xl∗mt, ϖ0=maxl∈Λ⁡{maxm∈M⁡{sups∈[t0-ν,t0]T⁡|φlm(s)-(φl∗)m(s)|}}. For any ε>0, it is obvious that(45)Zt0-Z∗t0<ϖ0+εand(46)Zt-Z∗t<ϖ0+εe⊖λt,t0<M∗ϖ0+εe⊖λt,t0,∀t∈t0-ν,t0T.We claim that(47)Zt-Z∗t<M∗ϖ0+εe⊖λt,t0,∀t∈t0,+∞T.If (47) is not true, then there must be a first time t1∈(t0,+∞)T such that(48)Zt1-Z∗t1≥M∗ϖ0+εe⊖λt1,t0,Zt-Z∗t<M∗ϖ0+εe⊖λt,t0,t∈t0-ν,t1T.Therefore, there must exist a constant C≥1 such that(49)Zt1-Z∗t1=CM∗ϖ0+εe⊖λt1,t0,Zt-Z∗t<CM∗ϖ0+εe⊖λt,t0,t∈t0-ν,t1T.Multiplying both sides of (38) by e-dl(t0,σ(t)) and integrating over [t0,t]T, we get (50)Zt-Z∗tl=Zt0-Z∗t0le-alt,t0+∫t0te-alt,σs×∑h=1nBlhsF~hxhs-τlhs-F~hxh∗s-τlhs+⋀h=1nAlhs∫s-ηlhssG~hxhu-G~hxh∗uΔu+⋁h=1nClhs∫s-ξlhssG~hxhu-G~hxh∗uΔuΔs,t∈T,l∈Λ.In view of (45), (46), (49) and M∗>1, we have(51)xlRt1-x∗lRt1=xlRt0-xRl∗t0e-alt1,t0+∫t0t1e-alt1,σs∑j=1nblhRsf~hRxhs-τlhs-f~hRxh∗s-τlhs-blhIsf~hIxhs-τlhs-f~hIxh∗s-τlhs-blhJsf~hJxhs-τlhs-f~hJxh∗s-τlhs-blhKsf~hKxhs-τlhs-f~hKxh∗s-τlhs+⋀h=1nαlhRs∫s-ηlhssg~hRxhu-g~hRxh∗uΔu-⋀h=1nαlhIs∫s-ηlhssg~hIxhu-g~hIxh∗uΔu-⋀h=1nαlhJs∫s-ηlhssg~hJxhu-g~hJxh∗uΔu-⋀h=1nαlhKs∫s-ηlhssg~hKxhu-g~hKxh∗uΔu+⋁h=1nβlhRs∫s-ξlhssg~hRxhu-g~hRxh∗uΔu-⋁h=1nβlhIs∫s-ξlhssg~hIxhu-g~hIxh∗uΔu-⋁h=1nβlhJs∫s-ξlhssg~hJxhu-g~hJxh∗uΔu-⋁h=1nβlhKs∫s-ξlhssg~hKxhu-g~hKxh∗uΔuΔs≤e-al⊕λt1,t0CM∗+∫t0t1e-al⊕λt1,σseλσs,s-τlhs∑h=1nLhb¯lhR+b¯lhI+b¯lhJ+b¯lhK+eλσs,s∑h=1nη¯lhHhα¯lhR+α¯lhI+α¯lhJ+α¯lhK+eλσs,s×∑h=1nξ¯lhHhβ¯lhR+β¯lhI+β¯lhJ+β¯lhKΔsCM∗ϖ0+εe⊖λt1,t0≤1M∗-exp⁡λsups∈T⁡μsa_l-λexp⁡λτ¯lh∑h=1nLhb¯lhR+b¯lhI+b¯lhJ+b¯lhK+∑h=1nη¯lhHhα¯lhR+α¯lhI+α¯lhJ+α¯lhK+∑h=1nξ¯lhHhβ¯lhR+β¯lhI+β¯lhJ+β¯lhKe-a_l-λt1,t0+exp⁡λsups∈T⁡μsa_l-λexp⁡λτ¯lh∑h=1nLhb¯lhR+b¯lhI+b¯lhJ+b¯lhK+∑h=1nη¯lhHhα¯lhR+α¯lhI+α¯lhJ+α¯lhK+∑h=1nξ¯lhHhβ¯lhR+β¯lhI+β¯lhJ+β¯lhKCM∗ϖ0+εe⊖λt1,t0<CM∗ϖ0+εe⊖λt1,t0,l∈Λ.
Similarly, we have(52)xlmt1-x∗lmt1<CM∗ϖ0+εe⊖λt1,t0,m=I,J,K,l∈Λ.Hence (53)Zt1-Z∗t1=max⁡xlmt1-x∗lmt1,m∈M,l∈Λ<CM∗ϖ0+εe⊖λt1,t0,which contradicts the first equation of (48). Therefore, (47) holds. Letting ε→0+ leads to (54)Zt-Z∗t≤M∗ϖ0e⊖λt,t0,∀t∈t0,+∞T.Hence, the ω-anti-periodic solution of system (15) is globally exponentially stable. The proof of Theorem 15 is completed.Remark 16.
In view of Remark11, we have that, under conditions of Theorems 14 and 15, system (10) has a unique anti-periodic solution that is globally exponentially stable.Remark 17.
In view of Theorem14, Theorem 15, and Remark 16, we see that under condition -al∈R+, l∈Λ both the continuous time and the discrete time cases of system (10) have the same dynamics for the anti-periodicity.


## 5. An Example

In this section, we give an example to illustrate the feasibility and effectiveness of our results obtained in Sections3 and 4.Example 18.
In system (10), let n=2, ω=π. The coefficients are taken as follows: (55)f~hx=g~hx=180hxR+1-xJ-1+ih80sin⁡22xI+xK-jh80sin⁡xJ+h80ktanh⁡xI+xK+xR,a1t=0.8+0.02cos⁡t,a2t=0.95+0.01cos⁡t,Ilt=0.2lcos⁡t+i0.1+0.2sin⁡t+0.2jsin⁡t+0.1kcos⁡t,blht=0.4sin⁡2t+0.1hicos⁡2t+0.1ljcos⁡2t+0.3ksin⁡2t,dlht=0.1lcos⁡2t+0.3isin⁡2t+0.1hjcos⁡2t+0.1ksin⁡2t,αlht=0.1hcos⁡2t+0.2hisin⁡2t+0.2jsin⁡2t+0.1ksin⁡2t,βlht=0.3sin⁡2t+0.1icos⁡2t+0.2hjcos⁡2t+0.2ksin⁡2t,Tlht=0.2+0.1hsin⁡2t+0.1icos⁡2t+0.1ljsin⁡2t+0.1lkcos⁡2t,Slht=0.1cos⁡2t+0.1hisin⁡2t+0.1jcos⁡2t+0.2lksin⁡2t,μht=0.1lsin⁡t+0.2hicos⁡t+0.2jcos⁡t+0.1ksin⁡t,if T=R, then we take(56)τlht=cos2⁡t,ηlht=h2cos⁡t,ξlht=l2sin2⁡t,and if T=Z, then we take(57)τlht=1+cos2⁡π2+πt,ηlht=h2sin⁡πt,ξlht=l2sin2⁡πt,l,h=1,2.By calculating, we have(58)L1m=H1m=0.0125,L2m=H2m=0.025,a¯1=0.82,a¯2=0.96,m∈M,a_1=0.8,a_2=0.95,b¯11R=0.4,b¯11I=b¯11J=b¯12J=0.1,b¯11K=0.3,b¯12R=b¯21R=0.4,b¯12I=0.2,b¯12K=0.3,b¯21I=0.1,b¯21J=0.2,b¯21K=0.3,b¯22R=0.4,b¯22I=b¯22J=0.2,b¯22K=d¯11I=0.3,d¯11R=d¯11J=d¯11K=0.1,d¯12R=d¯12K=0.1,d¯12I=0.3,d¯12J=d¯21R=0.2,d¯21I=0.1,d¯21J=d¯21K=0.1,d¯22R=d¯22J=0.2,d¯22I=0.3,d¯22K=0.1,τ¯11=τ¯12=τ¯21=τ¯22=1,α¯11R=α¯11K=0.1,α¯11J=α¯11I=α¯12R=α¯12J=0.2,α¯12I=0.4,α¯12K=0.1,α¯21R=0.1,α¯21I=α¯21J=α¯22J=α¯22R=0.2,α¯21K=α¯22K=0.1,α¯22I=0.4,β¯11R=β¯21R=β¯12R=β¯22R=0.3,β¯11I=β¯12I=β¯21I=β¯22I=0.1,β¯12J=β¯22J=0.4,β¯11J=β¯21J=0.2,β¯12K=β¯11K=β¯21K=β¯22K=0.2,when T=R,(59)ξ¯11=ξ¯12=0.5,ξ¯21=ξ¯22=1,η¯11=η¯21=0.5,η¯12=η¯22=1,when T=Z,(60)ξ¯11=ξ¯12=ξ¯21=ξ¯22=η¯11=η¯21=η¯12=η¯22=0.Therefore, whether T=R or T=Z, we have -al∈R+ and it is easy to verify that conditions (H1) and (H2) are satisfied. By a simple calculation, we have (61)ξ≤max⁡P1a_1,P2a_2=max⁡0.4,0.5=0.5<1,which implies that conditions (H3) and (H4) are also satisfied. Therefore, according to Theorems 14 and 15, (10) has a unique π-anti-periodic solution, which is globally exponentially stable (Figures 1 and 2).Figure 1
Responses of the four parts ofx with continuous time t.Figure 2
Responses of the four parts ofx with discrete time t.


## 6. Conclusion

In this paper, we have proposed a class of QVFCNNs with time-varying delays on time scales which can unify the continuous time and discrete time cases of QVFCNNs and contain real-valued FCNNs and complex-valued FCNNs as their special cases. Based on a novel method that is different from those used in [10, 25–34], we established the existence and global exponential stability of anti-periodic solutions of the QVFCNNs. To the best of our knowledge, this is the first paper to study the existence of anti-periodic solutions for QVFCNNs with time-varying delays on time scales. The method of this paper may be applied to study other types of the QVNNs on times scales such as quaternion-valued recurrent neural networks, quaternion-valued BAM neural networks, quaternion-valued shunting inhibitory cellular neural networks, quaternion-valued Cohen-Grossberg neural networks, and so on.



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*Source: 5290786-2018-06-28.xml*