# Antiperiodic Solutions for Quaternion-Valued Shunting Inhibitory Cellular Neural Networks with Distributed Delays and Impulses

Authors: Nina Huo; Yongkun Li
Journal: Complexity
 (2018)

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## Abstract

This paper is concerned with quaternion-valued shunting inhibitory cellular neural networks (QVSICNNs) with distributed delays and impulses. By using a new continuation theorem of the coincidence degree theory, the existence of antiperiodic solutions for QVSICNNs is obtained. By constructing a suitable Lyapunov function, some sufficient conditions are derived to guarantee the global exponential stability of antiperiodic solutions for QVSICNNs. Finally, an example is given to show the feasibility of obtained results.


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## Body

## 1. Introduction

The shunting inhibitory cellular neural networks (SICNNs) [1, 2] have found many applications in psychophysics, speech, perception, robotics, adaptive pattern recognition, vision, and image processing. Since all of these applications heavily rely on the dynamics of SICNNs and time delays are unavoidable in a realistic system [3–13], there have been extensive results about the dynamical behaviors of SICNNs with time delays [4–10]. Besides, a wide variety of evolutionary processes which exist universally in nature and many signal transmission processes in neural networks are often subject to abrupt changes at certain moments due to instantaneous perturbations which lead to impulsive effects. Also, the existence of impulses is frequently a source of instability, bifurcation, and chaos for neural networks [11–13]. Therefore, many researchers have investigated various dynamical behaviors of SICNNs with time delays and impulses [12, 13].On the other hand, quaternion-valued neural networks, which can be seen as a generic extension of complex-valued neural networks (CVNNs) or real-valued neural networks (RVNNs), are much more complicated than CVNNs for their quaternion-valued states, quaternion-valued connection weights, and quaternion-valued activation functions. In the past decades, QVNNs have found many practical applications in aerospace and satellite tracking, processing of polarized waves, image processing, 3D geometrical affine transformation, spatial rotation [14, 15], color night vision [16], and so on. Due to so many practical applications, it is necessary to study the dynamics of QVNNs. At present, only a few of the dynamical behaviors of QVNNs have been studied [17–24]. For example, in [21, 22], the global μ-stability criteria for QVNNs were studied, respectively; in [23], based on Mawhin’s continuation theorem of coincidence degree theory, the existence of periodic solutions for QVNNs was established; in [24], the multiplicity of periodic solutions for QVNNs was discussed by employing Brouwer’s and Leray-Schauder’s fixed point theorems. However, to the best of our knowledge, the antiperiodic oscillation of QVVNs with time-varying delays and impulses has not been reported. Since the existence and stability of antiperiodic solutions are an important topic in nonlinear differential equations and the signal transmission process of neural networks can often be described as an antiperiodic process, the antiperiodic oscillation of neural networks have been considered by many authors, see [13, 25–33]. So, it is necessary to study the antiperiodic solutions for QVNNs.Motivated by the above, in this paper, we are concerned with the following QVSICNN with distributed delays and impulsive effects:(1)ẋpqt=−apqtxpqt−∑Ckl∈Nrp,qCpqklt∫0+∞Kpquxpqtfxklt−udu+Tpqt,t≥0,t≠th,h∈ℕ,Δxpqth=xpqth+−xpqth−=Ipqhxpqth,t=th,p=1,…,m,q=1,…,n,where pq∈11,12,…,1n,…,m1,m2,…,mn≔ℬ; Cpq denotes the cell at the p,q position of the lattice. The r neighborhood Nrp,q of Cpq is given by
(2)Nrp,q=Cpq:maxk−p,l−q≤r,pq∈ℬ.xpq∈ℚ is the activity of the cell Cpq; Tpq:ℚ→ℚ is the external input to Cpq, apqt>0 represents the passive decay rate of the cell activity, Cpqklt≥0 is the connection or coupling strength of postsynaptic activity of the cell transmitted to the cell Cpq, and the activity function f:ℚ→ℚ is a continuous function representing the output or firing rate of the cell Cpq; Kpqt corresponds to the transmission delay kernel; and th,h∈ℕ is a sequence of real numbers such that 0<t1<t2<⋯<th→∞ as h→∞, there exists a μ∈ℕ such that th+μ=th+ω/2,Ipqh+μxpqth+μ=−Ipqh−xpqth,h∈ℕ. Without loss of generality, we also assume that 0,ω/2∩th:h∈ℕ=t1,t2,…,tμ. For convenience, we denote g¯=supt∈ℝgt and g¯=inft∈ℝgt, where g is a bounded function.The main purpose of this paper is to establish the existence of antiperiodic solutions to (1) by using a new continuation theorem of coincidence degree theory and by constructing a suitable Lyapunov function to obtain the global exponential stability of the antiperiodic solution. The results of this paper are completely new and supplement the previously known results.Throughout this paper, we assume Hypotheses 1 to 4.(H1)
Letxpq=xpqR+ixpqI+jxpqJ+kxpqK∈ℚ, assume that the activity function f:ℚ→ℚ of (1) can be expressed as
(3)fxpq=fRxpqR,xpqI,xpqJ,xpqK+ifIxpqR,xpqI,xpqJ,xpqK+jfJxpqR,xpqI,xpqJ,xpqK+kfKxpqR,xpqI,xpqJ,xpqK,where fν∈Cℝ4,ℝ, ν∈R,I,J,K≔E; the impulsive operators Ipqh:ℚ→ℚ can be expressed as
(4)Ipqhxpq=IpqhRxpqR,xpqI,xpqJ,xpqK+iIpqhIxpqR,xpqI,xpqJ,xpqK+jIpqhJxpqR,xpqI,xpqJ,xpqK+kIpqhKxpqR,xpqI,xpqJ,xpqK,where Ipqhν∈Cℝ4,ℝ,ν∈E, pq∈ℬ, and h∈ℕ, and the external inputs Tpq:ℝ→ℚ can be expressed as
(5)Tpqt=TpqRt+iTpqIt+jTpqJt+kTpqKt,where Tpqν∈Cℝ,ℝ,ν∈E, and pq∈ℬ.
(H2)
Functionsapq∈Cℝ,0,+∞,Cpqkl∈Cℝ,ℝ,andTpq∈Cℝ,ℚ, where pq∈ℬ, and there exists a constant ω>0 such that
(6)apqt+ω2=apqt,Cpqklt+ω2=Cpqklt,Tpqt+ω2=−Tpqt,t∈ℝ,and functions f,Ipqh∈Cℚ,ℚ, where pq∈ℬ and h∈ℕ, satisfying
(7)f−x=fx,Ipqh−x=−Ipqhx,x∈ℚ.
(H3)
Forpq∈ℬ, ∫0+∞Kpqudu<+∞.
(H4)
Forpq∈ℬ, uν,andvν∈ℝ, there exist positive constants Lfν, Mfν, and I such that
(8)fνuR,uI,uJ,uK−fνvR,vI,vJ,vK≤LfRuR−vR+LfIuI−vI+LfJuJ−vJ+LfKuK−vK,and
(9)fνuR,uI,uJ,uK≤Mfν,IpqhνuR,uI,uJ,uK≤I,where fν and Ipqhνuandν∈E are mentioned in H1.We will adopt the following notations:(10)T=maxpq∈ℬsupt∈0,ωTpqνt,ν∈E,a−=minpq∈ℬinft∈0,ωapqt,a+=maxpq∈ℬsupt∈0,ωapqt.The initial value of (1) is given by
(11)xpqs=φpqs∈ℚ,s∈−∞,0,where
(12)φpqs=φpqRs+iφpqIs+jφpqJs+kφpqKs,φpqν∈BC−∞,0,ℝ,pq∈ℬ.The remaining part of this paper is organized as follows. In Section3, some definitions are given. In Section 4, we obtain sufficient conditions for the existence of antiperiodic solutions of (1). In Section 5, the global exponential stability of the antiperiodic solution is studied. In Section 6, we give an example to illustrate the feasibility of the obtained results.


## 2. Preliminaries

The quaternion was invented in 1843 by Hamilton [34]. The skew field of the quaternion is denoted by
(13)ℚ≔q=q0+iq1+jq2+kq3,where q0,q1,q2,andq3 are real numbers and the elements i,j, and k obey Hamilton’s multiplication rules:
(14)ij=−ji=k,jk=−kj=i,ki=−ik=j,i2=j2=k2=ijk=−1.In order to avoid the difficulty resulting from the noncommutativity of the quaternion multiplication, by Hamilton’s rules and H1, we decompose (1) into the following systems:
(15)ẋpqRt=−apqtxpqRt−∑Ckl∈Nrp,qCpqklt∫0+∞KpqufRt,u,xduxpqRt−∫0+∞KpqufIt,u,xduxpqIt−∫0+∞KpqufJt,u,xduxpqJt−∫0+∞KpqufKt,u,xduxpqKt+TpqRt≜FpqRt,x,t≠th,h∈ℕ,ΔxpqRth=IpqhRxpqth,pq∈ℬ,h∈ℕ,ẋpqIt=−apqtxpqIt−∑Ckl∈Nrp,qCpqklt∫0+∞KpqufRt,u,xduxpqIt+∫0+∞KpqufIt,u,xduxpqRt+∫0+∞KpqufJt,u,xduxpqKt−∫0+∞KpqufKt,u,xduxpqJt+TpqIt≜FpqIt,x,t≠th,h∈ℕ,ΔxpqIth=IpqhIxpqth,pq∈ℬ,h∈ℕ,ẋpqJt=−apqtxpqJt−∑Ckl∈Nrp,qCpqklt∫0+∞KpqufRt,u,xduxpqJt−∫0+∞KpqufIt,u,xduxpqKu+∫0+∞KpqufJt,u,xduxpqRt+∫0+∞KpqufKt,u,xduxpqIt+TpqJt≜FpqJt,x,t≠th,h∈ℕ,ΔxpqJth=IpqhJxpqth,pq∈ℬ,h∈ℕ,ẋpqKt=−apqtxpqKt−∑Ckl∈Nrp,qCpqklt∫0+∞KpqufRt,u,xduxpqKt+∫0+∞KpqufIt,u,xduxpqJt−∫0+∞KpqufJt,u,xduxpqIt+∫0+∞KpqufKt,u,xduxpqRt+TpqKt≜FpqKt,x,t≠th,h∈ℕ,ΔxpqKth=IpqhKxpqth,pq∈ℬ,h∈ℕ,where xpqR+ixpqI+jxpqJ+kxpqK=xpq, fνt,u,x≜fνxklRt−u,xklIt−u,xklJt−u,xklKt−u, and Ipqhνxpqth≜IpqhνxpqRth,xpqIth,xpqJth,xpqKth, pq∈ℬ, h∈ℕ, and ν∈E. That is, (1) can be decomposed as the following real-valued system:
(16)ẋpqνt=Fpqνt,x,t≠th,h∈ℕ,Δxpqνth=Ipqhνxpqth,pq∈ℬ,h∈ℕ,ν∈E,with the initial values
(17)xpqνs=φpqνs,s∈−∞,0,h∈ℕ,pq∈ℬ,ν∈E,where φpqν:−∞,0→ℝ is continuous and bounded.Definition 1.
A piecewise continuous functionx=x11R,x11I,x11J,x11K,…,xmnR,xmnI,xmnJ,xmnKT:ℝ→ℝ4nm is said to be a solution of (16), if
(i)
xs=φs for s∈−∞,0, where φ=φ11R,φ11I,φ11J,φ11K,…,φmnR,φmnI,φmnJ,φmnKT, φpqν∈C−∞,0,ℝ, ν∈E, and pq∈ℬ;
(i)
xt satisfies (16) for t≥0;
(ii)
xt is continuous everywhere except for some th and left continuous at t=th, and the right limit xth+ exists for h∈ℕ.Definition 2.
A solutionx of system (16) is said to be the ω/2-antiperiodic solution of (16), if
(18)xt+ω2=−xt,t≠th,xth+ω2+=−xth+,h∈ℕ.Definition 3.
Letx=x11R,x11I,x11J,x11K,…,xmnR,xmnI,xmnJ,xmnKT be a solution of (16) with the initial value φ=φ11R,φ11I,φ11J,φ11K,…,φmnR,φmnI,φmnJ,φmnKT and y=y11R,y11I,y11J,y11K,…,ymnR,ymnI,ymnJ,xmnKT be an arbitrary solution of (16) with the initial value ψ=ψ11R,ψ11I,ψ11J,ψ11K,…,ψmnR,φmnI,φmnJ,φmnKT. If there exist constants λ>0 and M>0 such that
(19)xt−yt0≤Mφ−ψe−λt,∀t>0,then the solution x of (16) is said to be globally exponentially stable, where
(20)yt−xt0=maxpq∈ℬmaxν∈Eypqνt−xpqνt,φ−ψ=∑pq∈ℬsups∈−∞,0φpqRs−ψpqRs+φpqIs−ψpqIs+φpqJs−ψpqJs+φpqKs−ψpqKs.Remark 1.
Ifx=x11R,x11I,x11J,x11K,…,xmnR,xmnI,xmnJ,xmnKT is an ω/2-antiperiodic solution to (16), then u=x11,…,xmnT, where xpq=xpqR+ixpqI+jxpqJ+kxpqK,pq∈E must be an ω/2-antiperiodic solution to (1). Thus, the problem of finding an ω/2-antiperiodic solution for (1) reduces to finding one for the system of (16). For considering the stability of the solution of (1), we just need to consider the stability of the solutions of (16).


## 3. The Existence of Antiperiodic Solutions

In this section, based on a new continuation theorem of coincidence degree theory, we shall study the existence of antiperiodic solutions of (1).Lemma 1.
[35] Let X and Y be Banach spaces, and let L:DomL⊂X→Y be linear and N:X→Y be continuous. Assume that L is one-to-one and K≔L−1N is compact. Furthermore, assume that there exists a bounded and open subset Ω⊂X with 0∈Ω such that equation Lu=λNu has no solutions in ∂Ω∩DomL for any λ∈0,1. Then the problem Lu=Nu has at least one solution in Ω¯.Theorem 1.
Assume that H1–H4 holds. Furthermore, suppose thatA1.
If one has
(21)Γ=a−ω1−a+ω1+a−ω−MfR+MfI+MfJ+MfK∑Ckl∈Nrp,qC¯pqkl∫0+∞Kpqudu>0,then (1) has at least one ω/2-antiperiodic solution.Proof 1.
One has(22)PCℝ,ℝ4mn=x=x11R,x11I,x11J,x11K,…,xmnR,xmnI,xmnJ,xmnKT:ℝ→ℝ4mn,xth,th+1∈Cth,th+1,ℝ4n,xth+,xthexistandxth−=xth,h∈ℕ.Set(23)X=x:x=x11R,x11I,x11J,x11K,…,xmnR,xmnI,xmnJ,xmnKT∈PCℝ,ℝ4mn,xt+ω2=−xt,and
(24)Y=X×ℝ4mn×μ.ThenX is a Banach space with the norm xX=maxpq∈Bmaxν∈Esupt∈ℝxpqνt, and Y is also a Banach space with the norm yY=xX+u,x∈X,u∈ℝ4mn×μ, where · is any norm of ℝ4mn×μ.Define a linear operatorL:DomL⊂X→Y by
(25)Lx=ẋ,Δxt1,Δxt2,…,Δxtμ,where DomL=x:x∈X,ẋ∈X and a continuous operator N:X→Y by
(26)Nx=F11Rt,xF11It,xF11Jt,xF11Kt,x⋮FmnRt,xFmnIt,xFmnJt,xFmnKt,x,I111Rx11t1I111Ix11t1I111Jx11t1I111Kx11t1⋮Imn1Rxmnt1Imn1Ixmnt1Imn1Jxmnt1Imn1Kxmnt1,I112Rx11t2I112Ix11t2I112Jx11t2I112Kx11t2⋮Imn2Rxmnt2Imn2Ixmnt2Imn2Jxmnt2Imn2Kxmnt2,…,I11μRx11tμI11μIx11tμI11μJx11tμI11μKx11tμ⋮ImnμRxmntμImnμIxmntμImnμJxmntμImnμKxmntμ.It is easy to see that(27)kerL=0,ImL=y=g,d1,,…,d11T∈Y:∫0ωgsds=0=Y.Hence,L is reversible. Denote by L−1:ImL→DomL the inverse of L, one has
(28)L−1yt=∫0tgsds−12∫0ω/2gsds+∑th<tdh−12∑h=1udh,t∈ℝ,where dμ+h=dh for all 1≤h≤μ. Let K≔L−1N, by applying the Arzela-Ascoli theorem, we know that K is compact. Corresponding to the operator equation Lx=λNx,λ∈0,1, we have
(29)ẋpqνt=λFpqνt,x,t≥0,t≠th,Δxpqνth=λIpqhνxpqth,pq∈ℬ,h∈ℕ,ν∈E.Suppose thatx=x11R,x11I,x11J,x11K,…,xmnR,xmnI,xmnJ,xmnKT∈X is a solution of (29) for a certain λ∈0,1, set t0=t0+=0,t2q+1=ω, then we have
(30)∫0ωẋpqRtdt=∑k=12q+1∫th−1+thẋpqRtdt+∑h=12qIpqhRxpqth≤∫0ωapqtxpqRtdt+∫0ω∑Ckl∈Nrp,qCpqklt∫0+∞KpquxpqRtfRt,u,xdu−∫0+∞KpquxpqItfIt,u,xdu−∫0+∞KpquxpqJtfJt,u,xdu−∫0+∞KpquxpqKtfKt,u,xdudt+∫0+∞TpqRtdt+∑h=12qIpqhRxpqth≤∫0ωapqtxpqRtdt+∫0ω∑Ckl∈Nrp,qC¯pqkl∫0+∞KpquxpqRtfRt,u,xdu+∫0+∞KpquxpqItfIt,u,xdu+∫0+∞KpquxpqJtfJt,u,xdu+∫0+∞KpquxpqKtfKt,u,xdudt+∫0ωTpqRtdt+∑h=12qIpqhRxpqth≤a+ωxpqR∞+∫0ω∑Ckl∈Nrp,qC¯pqkl∫0+∞KpquduxXMfR+∫0+∞KpquduxXMfI+∫0+∞KpquduxXMfJ+∫0+∞KpquduxXMfKdt+∫0ωTpqRtdt+∑h=12qIpqhRxpqth≤a+ωxpqR∞+ωMfR+MfI+MfJ+MfKxX∑Ckl∈Nrp,qC¯pqkl∫0+∞Kpqudu+ωT+2qI,pq∈ℬ.Repeating the above procession, forν=I,J,K, we can obtain that
(31)∫0ωẋpqνtdt≤a+ωxpqν∞+ωMfR+MfI+MfJ+MfKxX∑Ckl∈Nrp,qC¯pqkl∫0+∞Kpqudu+ωT+2qI,pq∈ℬ.Integrating both sides of (29) over the interval 0,ω, we can obtain
(32)∫0ωFpqνt,xdt+∑h=12qIpqhxpqhth=0,pq∈ℬ,ν∈E.Hence,(33)∫0ωapqtxpqRtdt=∫0ω∑Ckl∈Nrp,qCpqklt∫0+∞KpqufRt,u,xduxpqRt−∫0+∞KpqufIt,u,xduxpqIt−∫0+∞KpqufJt,u,xduxpqJt−∫0+∞KpqufKt,u,xduxpqKt+TpqRtdt+∑h=12qIpqhRxpqth≤∫0ω∑Ckl∈Nrp,qCpqklt∫0+∞KpqufRt,u,xduxpqRt−∫0+∞KpqufIt,u,xduxpqIt−∫0+∞KpqufJt,u,xduxpqJt−∫0+∞KpqufKt,u,xduxpqKtdt+∫0ωTpqRtdt+∑h=12qIpqhRxpqth≤∫0ω∑Ckl∈Nrp,qC¯pqkl∫0+∞KpquduMfR+MfI+MfJ+MfKxXdt+∫0ωTpqRtdt+∑h=12qIpqhRxpqth≤ωMfR+MfI+MfJ+MfKxX∑Ckl∈Nrp,qC¯pqkl∫0+∞Kpqudu+ωT+2qI,pq∈ℬ.Repeating the above procession, forν=I,J,K, we have
(34)∫0ωapqtxpqνtdt≤ωMfR+MfI+MfJ+MfKxX∑Ckl∈Nrp,qC¯pqkl∫0+∞Kpqudu+ωT+2qI,pq∈ℬ.Since for anyt1,t2∈0,ω, and pq∈ℬ, we have
(35)xpqνt≤xpqνt1+∫0ωẋpqνtdt,xpqνt≥xpqνt2−∫0ωẋpqνtdt,then for any ξpqν,ηpqν∈0,ω, pq∈ℬ, and ν∈E, we obtain
(36)∫0ωapqtxpqνtdt≤∫0ωapqtxpqνξpqνdt+∫0ωapqt∫0ωẋpqνtdtdt,(37)∫0ωapqtxpqνtdt≥∫0ωapqtxpqνηpqνdt−∫0ωapqt∫0ωẋpqνtdtdt.Dividing by∫0ωapqtdt on the both sides of (36) and (37), respectively, we have
(38)xpqνξpqν≥1∫0ωapqtdt∫0ωapqtxpqνtdt−∫0ωẋpqνtdt,pq∈ℬ,ν∈E,xpqνηpqν≤1∫0ωapqtdt∫0ωapqtxpqνtdt+∫0ωẋpqνtdt,pq∈ℬ,ν∈E.Lett¯pqν,t¯pqν∈0,ω such that xpqνt¯pqν=supt∈0,ωxpqνt, xpqνt_pqν=inft∈0,ωxpqνt, pq∈ℬ, and ν∈E; by the arbitrariness of ξpqν,ηpqν, we can obtain
(39)xpqνt¯pqν≥1∫0ωapqtdt∫0ωapqtxpqνtdt−∫0ωẋpqνtdt≥−1∫0ωapqtdt∫0ωapqtxpqνtdt−∫0ωẋpqνtdt≥−1a−ωωMfR+MfI+MfJ+MfKxX∑Ckl∈Nrp,qC¯pqkl∫0+∞Kpqudu+ωT+2qI−a+ωxpqν∞+ωMfR+MfI+MfJ+MfKxX∑Ckl∈Nri,jC¯pqkl∫0+∞Kpqudu+ωT+2qI=−a+ωxpqν∞−1a−+ωMfR+MfI+MfJ+MfKxX∑Ckl∈Nrp,qC¯pqkl×∫0+∞Kpqudu+T+2qIω,pq∈ℬ,ν∈E,and
(40)xpqνt¯pqν≤1∫0ωapqtdt∫0ωapqtxpqνtdt+∫0ωẋpqνtdt≤1∫0ωapqtdt∫0ωapqtxpqνtdt+∫0ωẋpqνtdt≤a+ωxpqν∞+1a−+ωMfR+MfI+MfJ+MfKxX∑Ckl∈Nrp,qC¯pqkl×∫0+∞Kpqudu+T+2qIω,pq∈ℬ,ν∈E.Thus, from (39) and (40) we have
(41)xpqν∞=supt∈0,ωxpqνt≤a+ωxpqν∞+1a−+ωMfR+MfI+MfJ+MfKxX∑Ckl∈Nrp,qC¯pqkl×∫0+∞Kpqudu+T+2qIω,pq∈ℬ,ν∈E,hence
(42)xX≤a+ωxX+1a−+ωMfR+MfI+MfJ+MfKxX∑Ckl∈Nrp,qC¯pqkl×∫0+∞Kpqudu+T+2qIω,that is
(43)xX≤T+2qI/ωa−ω1−a+ω/1+a−ω−MfR+MfI+MfJ+MfK∑Ckl∈Nrp,qC¯pqkl∫0+∞Kpqudu≜A.Obviously,A is independent of λ. Take Ω=x∈X:xX<A+1, then Ω⊂X with 0∈Ω such that equation Lu=λNu has no solutions in ∂Ω∩DomL for any λ∈0,1. Thus, by Lemma 1 we conclude that Lx=Nx has at least one ω/2-antiperiodic solution in X, that is, (16) has at least one ω/2-antiperiodic solution. By Remark 1, we see that (1) has at least one ω/2-antiperiodic solution.


## 4. The Global Exponential Stability of Antiperiodic Solution

In this section, by constructing a suitable Lyapunov function we derive sufficient conditions ensuring the global exponential stability of antiperiodic solutions for (1).Theorem 2.
Let H1–H4 and A1 hold. Furthermore, assume thatA2.
The impulsive operators satisfy
(44)Ipqhxpqth=−γpqhxpqth,0≤γpqh≤2,pq∈ℬ,h∈ℕ.
A3.
There exists a positive constantλ>0 satisfying
(45)δ=maxpq∈ℬδpq<0,where
(46)δpq=λ−a−+∑Ckl∈Nrp,qC¯pqkl∫0+∞KpquMfR+MfI+MfJ+MfK+4LfR+LfI+LfJ+LfKdu,then, the ω/2-antiperiodic solution of (1) is globally exponentially stable.Proof 2.
By Theorem1, (16) has an ω/2-periodic solution. Let xt be the ω/2-periodic solution with the initial value φt and yt be an arbitrary solution with the initial value ψt. Set zpqνt=xpqνt−ypqνt, pq∈ℬ, and ν∈E. By (16), for t>0 and t≠th, we have
(47)D+zpqνt≤−a−zpqνt+∑Ckl∈Nrp,qC¯pqkl∫0+∞KpquMfRzpqRt+MfIzpqIt+MfJzpqJt+MfKzpqKt+4LfRzklRt−u+LfIzklIt−u+LfJzklJt−u+LfKzklKt−udu,pq∈ℬ,ν∈E.Fort=th,h∈ℕ, from A2 we can have the following:
(48)xpqνth+−ypqνth+=1−γpqhxpqνth−ypqνth≤xpqνth−ypqνth,that is
(49)zpqνth+≤zpqνth,where pq∈ℬ and ν∈E.Construct the Lyapunov functionVt as follows:
(50)Vt=VRt+VIt+VJt+VKt,where Vνt=∑pq∈Bzpqνtteλt, pq∈ℬ, and ν∈E.Calculating the upper right derivative ofVRt along the solutions of (16), for t≠th, we obtain
(51)D+VRt=∑pq∈ℬλeλtzpqRt+signzpqRtD+zpqRteλt≤∑pq∈ℬλ−a−eλtzpqR+eλt∑Ckl∈Nrp,qC¯pqkl∫0+∞KpquMfRzpqRt+MfIzpqIt+MfJzpqJt+MfKzpqKt+4LfRzklRt−u+LfIzklIt−u+LfJzklJt−udu≤eλt∑pq∈ℬλ−a−+∑Ckl∈Nrp,qC¯pqkl∫0+∞KpquMfR+MfI+MfJ+MfK+4LfR+LfI+LfJ+LfKduz.Repeat the same calculation and we can get(52)D+Vνt≤eλt∑pq∈ℬλ−a−+∑Ckl∈Nrp,qC¯pqkl∫0+∞KpquMfR+MfI+MfJ+MfK+4LfR+LfI+LfJ+LfKduz,ν∈E,t≠th.It follows from A3, (51), and (52) that for t≠th,
(53)D+Vt≤0.By (49), we also have
(54)Vth+=VRth++VIth++VJth++VKth+=∑pq∈ℬzpqRth+eλth++∑pq∈ℬzpqIth+eλth++∑pq∈ℬzpqJth+eλth++∑pq∈ℬzpqKth+eλth+≤∑pq∈ℬzpqRtheλth+∑pq∈ℬzpqItheλth+∑pq∈ℬzpqJtheλth+∑pq∈ℬzpqKtheλth=Vth,pq∈ℬ,h∈ℕ.Hence,Vt≤V0 for all t≥0.On the other hand, we have(55)V0=VR0+VI0+VJ0+VK0=∑pq∈ℬzpqR0+zpqI0+zpqJ0+zpqK0≤∑pq∈ℬsups∈−∞,0zpqRs+zpqIs+zpqJs+zpqKs≤∑pq∈ℬsups∈−∞,0φpqRs−ψpqRs+φpqIs−ψpqIs+φpqJs−ψpqJs+φpqKs−ψpqKs=φ−ψ.LetM=1, and we can easily obtain the following:
(56)xt−yt0≤Vte−λt≤Vte−λt≤Mφ−ψe−λt,t≥0.Therefore, theω/2-antiperiodic solution of (16) is globally exponentially stable. According to Remark 1, we know that the ω/2-antiperiodic solution of (16) is globally exponentially stable. The proof is complete.


## 5. An Illustrative Example

In this section, we give an example to show the feasibility and effectiveness of the results obtained in this paper.Example 1.
Consider the following QVSICNN:(57)ẋpqt=−apqtxpqt−∑Ckl∈Nrp,qCpqklt∫0+∞Kpquxpqtfxklt−udu+Tpqt,t≥0,t≠th,Δxpqth=xpqth+−xpqth−=Ipqhxpqth,t=th,where ẋpqt=ẋpqRt+iẋpqIt+jẋpqJt+kẋpqKt∈ℚ, Kpqu=e−u,p,q=1,2, and the coefficients are as follows:
(58)fx=0.01sinxR+i0.02sinxI+j0.03sinxJ+k0.04sinxK,a11t=a12t=a21t=a22t=18πsin2t+14π,C11t=C12t=C21t=C22t=0.01cos2t,T11t=T12t=T21t=T22t=0.2sint+i0.1cost+j0.3sint+k0.5cost,and the impulsive functions Ipqhxpqth=−0.001xpqth,p,q=1,2,th=hω/2,andh∈ℕ.
By a simple calculation, we can get(59)a−=14π,a+=18π+14π,ω=2π,γpqh=0.001,I=0.001,∫0+∞Kpqudu=1,MfR=0.01,MfI=0.02,MfJ=0.03,MfK=0.04,LfR=LfI=LfJ=LfK=0.04,∑Ckl∈N11,1C¯11kl=∑Ckl∈N11,2C¯12kl=∑Ckl∈N12,1C¯21kl=∑Ckl∈N12,2C¯22kl=0.04,Γ=a−ω1−a+ω1+a−ω−MfR+MfI+MfJ+MfK∑Ckl∈Nrp,qC¯pqkl∫0+∞Kpqudu≈0.08>0.
Takingλ=0.01, we obtain
(60)max1≤i,j≤2δpq=λ−a−+∑Ckl∈Nrp,qC¯pqkl∫0+∞KpquMfR+MfI+MfJ+MfK+4LfR+LfI+LfJ+LfKdu≈−0.04<0.
Thus conditions H1–H4 and A1–A3 hold. Therefore, according to Theorems1 and 2, (57) has at least one π-antiperiodic solution, which is globally exponentially stable (see Figures 1 and 2).Figure 1
The states of four parts ofx11, x12, x21, and x22.Figure 2
Curves ofx in 3-dimensional space for stable case.


## 6. Conclusion

In this paper, we investigated the existence and global exponential stability of antiperiodic solutions for a class of QVSICNNs with impulsive effects. We introduce a new method different from all other antiperiodic solutions of neural networks in a previous work. By using a new continuation theorem of the coincidence degree theory and constructing a suitable Lyapunov function, we obtain the existence and global exponential stability results for an antiperiodic solution. However, in this paper, we only investigate the antiperiodic solution problem of QVSICNNs with impulsive effects. In future work, periodic solution, almost periodic solution, and pseudo almost periodic solution in the quaternion field can be considered.



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