Globally Exponential Stability of Periodic Solutions to Impulsive Neural Networks with Time-Varying Delays

Abstract

By using Schaeffer's theorem and Lyapunov functional, sufficient conditions of the existence and globally exponential stability of positive periodic solution to an impulsive neural network with time-varying delays are established. Applications, examples, and numerical analysis are given to illustrate the effectiveness of the main results.

1. Introduction

It is well known that in implementation of neural networks, time delays are inevitably encountered because of the finite switching speed of amplifiers. Specially in electronic neural networks, delays are usually time-varying and often become sources of instability. So it is important to investigate the dynamics of neural networks with delays [1–7]. Recently, the study of the existence of periodic solutions of neural networks has received much attention. The common approaches are based on using Mawhin continuation theorem [1, 2, 8–10], Banach’s fixed point theorem [11–13], fixed point theorem in a cone [14], Schaeffer’s theorem [15, 16], and so on. On the other hand, studies on neural dynamical systems not only involve the existence of periodic solutions, but also involve other dynamical behaviors such as stability of periodic solutions, bifurcations, and chaos. In recent years, the stability of solutions of neural networks has attracted attention of many researchers and many nice results have been obtained [1–3, 5–13, 16–25]. For example, M. Tan and Y. Tan [1] considered the following neural network with variable coefficients and time-varying delays:𝑥^′_𝑖(𝑡)=−𝑐_𝑖(𝑡)𝑥_𝑖(𝑡)+^𝑛_𝑗=1𝑎_𝑖𝑗(𝑡)𝑓_𝑗𝑥_𝑗(𝑡)+^𝑛_𝑗=1𝑏_𝑖𝑗(𝑡)𝑓_𝑗𝑥_𝑗𝑡−𝜏_𝑖𝑗(𝑡)+𝐽_𝑖(𝑡),𝑖=1,2,…,𝑛. (1.1) By using the Mawhin continuation theorem, they discussed the existence and globally exponential stability of periodic solutions.

However, in real world, many physical systems often undergo abrupt changes at certain moments due to instantaneous perturbations, which lead to impulsive effects. In fact, impulsive differential equation represents a more natural framework for mathematical modelling of many real world phenomena such as population dynamic and neural networks. The theory of impulsive differential equations is now being recognized to be richer than the corresponding theory of differential equations without impulse, and various kinds of impulsive differential equations have been extensively studied, see [8–16, 18, 19, 21–25] and references therein. Then, considering impulsive effects, it is necessary and interesting for us to study further the dynamics of system (1.1). Furthermore, as pointed by Gopalsamy and Sariyasa [4], it would be of great interest to study neural networks in periodic environment. On the other hand, to the best of our knowledge, few authors considered the existence of periodic solutions by using Schaeffer’s theorem. Hence, in this paper, by using Schaeffer’s theorem and Lyapunov functional, we aim to discuss the existence and exponential stability of periodic solutions to a class of impulsive neural networks with periodic coefficients and time-varying delays. The model is as follows:𝑥^′_𝑖(𝑡)=−𝑐_𝑖(𝑡)𝑥_𝑖(𝑡)+^𝑛_𝑗=1𝑎_𝑖𝑗(𝑡)𝑓_𝑗𝑥_𝑗(𝑡)+^𝑛_𝑗=1𝑏_𝑖𝑗(𝑡)𝑓_𝑗𝑥_𝑗𝑡−𝜏_𝑖𝑗(𝑡)+𝐽_𝑖(𝑡),𝑡≠𝑡_𝑘,𝑥_𝑖𝑡⁺=1+𝑞^𝑘_𝑖𝑥_𝑖(𝑡),𝑡=𝑡_𝑘, (1.2) with initial conditions𝑥_𝑖(𝑠)=𝜙_𝑖(𝑠),𝜙_𝑖(𝑠)∈𝐶[−𝜏,0],𝑅^𝑛,𝑖=1,2,…,𝑛, (1.3) where 𝑥_𝑖(𝑡) corresponds to the state of the 𝑖th unit, 𝑐_𝑖(𝑡) 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, 𝑓_𝑗(𝑥_𝑗(𝑡)) denotes the output of the 𝑗th unit, 𝑎_𝑖𝑗(𝑡) and 𝑏_𝑖𝑗(𝑡) denote the strength of the 𝑗th unit on the 𝑖th unit, respectively, 𝐽_𝑖(𝑡) is the external bias on the 𝑖th unit, 𝜏_𝑖𝑗(𝑡) corresponds to the transmission delay along the axon of the 𝑗th unit, 𝑡_𝑘 denotes the impulsive moment, and 𝑡₁<𝑡₂<⋯ is a strictly increasing sequence such that lim_𝑘→∞𝑡_𝑘=∞, 𝐶([−𝜏,0],𝑅^𝑛) denotes the Banach space of continuous mapping from [−𝜏,0] to 𝑅^𝑛 equipped with the norm ‖𝜙‖=max_{1≤𝑖≤𝑛}sup_{𝑡∈[0,𝜔]}|𝜙_𝑖(𝑡)| for all 𝜙=(𝜙₁(𝑡),𝜙₂(𝑡),…,𝜙_𝑛(𝑡))^𝑇∈𝐶([−𝜏,0],𝑅^𝑛), where 𝜏=max_{1≤𝑖≤𝑛}sup_{𝑡∈[0,𝜔]}𝜏_𝑖𝑗(𝑡).

Throughout this paper, we always assume the following.(𝐻₁)𝑐_𝑖(𝑡)>0,𝑎_𝑖𝑗(𝑡),𝑏_𝑖𝑗(𝑡),𝐽_𝑖(𝑡),𝜏_𝑖𝑗(𝑡) are all continuous 𝜔-periodic functions for 𝑖,𝑗=1,2,…,𝑛.(𝐻₂)𝑓_𝑗∶𝑅→𝑅 is continuous and there exists positive constant 𝑘_𝑗 such that 𝑓_𝑗(𝑢)−𝑓_𝑗(𝑣)≤𝑘_𝑗|𝑢−𝑣| for any 𝑢,𝑣∈𝑅 and 𝑗=1,2,…,𝑛.(𝐻₃)There exists positive integer 𝑝 such that 𝑡_𝑘+𝑝=𝑡_𝑘+𝜔,𝑞^𝑘_𝑖=𝑞_𝑖^𝑘+𝑝. Then [0,𝜔]∩𝑡_𝑘,𝑘=1,2,…=𝑡₁,𝑡₂,…,𝑡_𝑝. (1.4) For convenience, we use the following notations: 1𝑓=𝜔^𝜔₀𝑓(𝑡)𝑑𝑡,𝑓^𝑀=max_{𝑡∈[0,𝜔]}𝑓(𝑡),𝑓^𝑚=min_{𝑡∈[0,𝜔]}𝑓(𝑡), (1.5) where 𝑓(𝑡) is continuous and 𝜔-periodic function.

The rest of this paper is organized as follows. In Section 2, by using Schaeffer’s theorem, sufficient conditions of the existence of 𝜔-periodic solution to system (1.2) with initial conditions (1.3) are established. In Section 3, by using Lyapunov functional, we derive the conditions under which the periodic solution is globally exponentially stable. In Section 4, applications, illustrative examples, and simulations are given to show the effectiveness of the main results. Finally, some conclusions are drawn in Section 5.

2. Existence of Periodic Solution

First we make some preparations. As usual in the theory of impulsive differential equation, by a solution of model (1.2), it means the following.(i)𝑥(𝑡)=(𝑥₁(𝑡),𝑥₂(𝑡),…𝑥_𝑛(𝑡))^𝑇∈𝑅^𝑛,𝑥_𝑖(𝑡) is piecewise continuous such that 𝑥_𝑖(𝑡⁻_𝑘)=𝑥_𝑖(𝑡_𝑘),𝑥_𝑖(𝑡⁺_𝑘) exists, and 𝑥_𝑖(𝑡) is differentiable on (𝑡_𝑘−1,𝑡_𝑘) for 𝑖=1,2,…,𝑛,𝑘=1,2,….(ii)𝑥_𝑖(𝑡) satisfies (1.2) for 𝑖=1,2,…,𝑛.

Definition 2.1. The set 𝐴 is said to be quasi-equicontinuous in [0,𝜔] if for any 𝜖>0, there exists 𝛿>0 such that, if 𝑥∈𝐴,𝑡∈𝑍,𝑡′,𝑡^′′∈(𝑡_𝑘−1,𝑡_𝑘)∩[0,𝜔] and |𝑡^′−𝑡^′′|<𝛿, then |𝑥(𝑡^′)−𝑥(𝑡^′′)|<𝜖.

Lemma 2.2 (see [26, Compactness criterion]). The set 𝐴⊂𝑋 is relatively compact if and only if(i)𝐴 is bounded, that is, ‖𝑥‖≤𝑀 for each 𝑥∈𝐴 and some 𝑀>0,(ii)𝐴 is quasi-equicontinuous in [0,𝜔].

The following lemma is fundamental to our discussion. The method is similar to that of [13, 16], so the proof is omitted here.

Lemma 2.3. 𝑥(𝑡)=(𝑥₁(𝑡),𝑥₂(𝑡),…,𝑥_𝑛(𝑡))^𝑇 is an 𝜔-periodic solution of system (1.2) which is equivalent to 𝑥(𝑡)=(𝑥₁(𝑡),𝑥₂(𝑡),…,𝑥_𝑛(𝑡))^𝑇 is an 𝜔-periodic solution of the following equation: 𝑥_𝑖(𝑡)=^𝜔₀𝐺_𝑖(𝑡,𝑠)^𝑛_𝑗=1𝑎_𝑖𝑗(𝑠)𝑓_𝑗𝑥_𝑗(𝑠)+^𝑛_𝑗=1𝑏_𝑖𝑗(𝑠)𝑓_𝑗𝑥_𝑗𝑠−𝜏_𝑖𝑗(𝑠)+𝐽_𝑖(𝑠)+𝑑𝑠_𝑝𝑘=1𝐺_𝑖𝑡,𝑡_𝑘𝑞^𝑘_𝑖𝑥_𝑖𝑡_𝑘, (2.1) where 𝐺(𝑡,𝑠)=(𝐺₁(𝑡,𝑠),𝐺₂(𝑡,𝑠),…,𝐺_𝑛(𝑡,𝑠))^𝑇, and 𝐺_𝑖()=𝑒𝑡,𝑠^{∫𝜔0𝑐𝑖(𝑢)𝑑𝑢−∫𝑡𝑠𝑐𝑖(𝑢)𝑑𝑢}𝑒^{∫𝜔0𝑐𝑖(𝑢)𝑑𝑢}𝑒−1,0≤𝑠≤𝑡≤𝜔,_{∫𝑠𝑡𝑐𝑖(𝑢)𝑑𝑢}𝑒^{∫𝜔0𝑐𝑖(𝑢)𝑑𝑢}−1,0≤𝑡≤𝑠≤𝜔. (2.2)
It is easy to show that 𝐺_𝑖(𝑡+𝜔,𝑠+𝜔)=𝐺_𝑖(𝑡,𝑠),𝐺_𝑖(𝑡,𝑡+𝜔)−𝐺_𝑖(𝑡,𝑡)=1 and1𝜎_𝑖−1≤𝐺_𝑖()≤𝜎𝑡,𝑠^𝑖𝜎_𝑖,−1 (2.3) where 𝜎_𝑖=𝑒^{∫𝜔0𝑐𝑖(𝑢)𝑑𝑢} and 𝑖=1,2,…,𝑛.

Lemma 2.4 (see [27, Schaeffer’s theorem]). Let 𝑋 be a normed space and 𝜙∶𝑋→𝑋 be a compact operator. Define 𝐻𝜙=𝑥∣𝑥∈𝑋,𝑥=𝜆𝜙𝑥,0<𝜆<1. (2.4) Then either(i)set 𝐻(𝜙) is unbounded, or(ii)operator 𝜙 has a fixed point in 𝑋.
In order to use Lemma 2.4, letPC[0,𝜔],R^𝑛=𝑥∶[0,𝜔]⟶𝑅^𝑛∣lim_𝑠→𝑡𝑥(𝑠)=𝑥(𝑡),𝑡≠𝑡_𝑘,lim_{𝑡→𝑡−𝑘}𝑥(𝑡)=𝑥𝑡_𝑘,lim_{𝑡→𝑡+𝑘}𝑥(𝑡)exists,𝑘=1,2,…,𝑝, (2.5) with the norm ‖𝑥‖=max_{1≤𝑖≤𝑛}sup_{𝑡∈[0,𝜔]}|𝑥_𝑖(𝑡)|, then PC([0,𝜔],𝑅^𝑛) is a Banach space.
Define a mapping 𝜙∶PC([0,𝜔],𝑅^𝑛)→PC([0,𝜔],𝑅^𝑛) by (𝜙𝑥)(𝑡)=𝑥(𝑡), where (𝜙𝑥)(𝑡)=((𝜙𝑥)₁(𝑡),(𝜙𝑥)₂(𝑡),…,(𝜙𝑥)_𝑛(𝑡))^𝑇 and𝜙𝑥_𝑖(𝑡)=_𝑡^𝑡+𝜔𝐺_𝑖(𝑡,𝑠)^𝑛_𝑗=1𝑎_𝑖𝑗(𝑠)𝑓_𝑗𝑥_𝑗(𝑠)+^𝑛_𝑗=1𝑏_𝑖𝑗(𝑠)𝑓_𝑗𝑥_𝑗𝑠−𝜏_𝑖𝑗(𝑠)+𝐽_𝑖(𝑠)+𝑑𝑠_𝑝𝑘=1𝐺_𝑖𝑡,𝑡_𝑘𝑞^𝑘_𝑖𝑥_𝑖𝑡_𝑘. (2.6) By Lemma 2.3, it is easy to see that the existence of 𝜔-periodic solution of (1.2) is equivalent to the existence of fixed point of the mapping 𝜙 in PC([0,𝜔],𝑅^𝑛).

Theorem 2.5. Suppose that (𝐻₁)–(𝐻₃) hold. Further,(𝐻₄)max_{1≤𝑖≤𝑛}(𝜎_𝑖/𝜎_𝑖−1)(∑^𝑛_𝑗=1(𝑎_𝑖𝑗+𝑏_𝑖𝑗)𝑘_𝑗+∑^𝑝_𝑘=1|𝑞^𝑘_𝑖|)∶=𝜃<1. Then system (1.2) admits an 𝜔-periodic solution.

Proof. By Lemma 2.3, it suffices to prove that the mapping 𝜙 admits a fixed point in PC([0,𝜔],𝑅^𝑛).
For any constant 𝐻>0, let Ω={𝑥∣𝑥∈PC([0,𝜔],𝑅^𝑛),‖𝑥‖<𝐻}. For 𝑥∈Ω, from (2.3) and (𝐻₂), we have𝜙𝑥=max_{1≤𝑖≤𝑛}sup_{𝑡∈[0,𝜔]𝑡}^𝑡+𝜔𝐺_𝑖(𝑡,𝑠)^𝑛_𝑗=1𝑎_𝑖𝑗(𝑠)𝑓_𝑗𝑥_𝑗(𝑠)+^𝑛_𝑗=1𝑏_𝑖𝑗(𝑠)𝑓_𝑗𝑥_𝑗𝑠−𝜏_𝑖𝑗(𝑠)+𝐽_𝑖(𝑠)+𝑑𝑠_𝑝𝑘=1𝐺_𝑖𝑡,𝑡_𝑘𝑞^𝑘_𝑖𝑥_𝑖𝑡_𝑘≤max_{1≤𝑖≤𝑛}sup_{𝑡∈[0,𝜔]}𝜎_𝑖𝜎_𝑖−1^𝜔₀^𝑛_𝑗=1𝑎_𝑖𝑗(𝑡)𝑓_𝑗𝑥_𝑗(𝑡)+^𝑛_𝑗=1𝑏_𝑖𝑗(𝑡)𝑓_𝑗𝑥_𝑗𝑡−𝜏_𝑖𝑗(𝑡)+𝐽_𝑖(𝑡+𝜎)𝑑𝑡_𝑖𝜎_𝑖−1^𝑝_𝑘=1𝑞^𝑘_𝑖𝑥_𝑖𝑡_𝑘≤max_{1≤𝑖≤𝑛}sup_{𝑡∈[0,𝜔]}𝜎_𝑖𝜎_𝑖−1^𝜔₀^𝑛_𝑗=1𝑎_𝑖𝑗(𝑡)𝑘_𝑗𝑥_𝑗(𝑡)+𝑓_𝑗(0+_𝑛𝑗=1)𝑏_𝑖𝑗(𝑡)𝑘_𝑗𝑥_𝑗𝑡−𝜏_𝑖𝑗(𝑡)+𝑓_𝑗(0+)𝑑𝑡_𝜔0𝐽_𝑖(𝑡𝜎)𝑑𝑡+^𝑖𝜎_𝑖−1^𝑝_𝑘=1𝑞^𝑘_𝑖𝑥_𝑖𝑡_𝑘≤max_{1≤𝑖≤𝑛}𝜎_𝑖𝜎_𝑖−1^𝑛_𝑗=1𝑎_𝑖𝑗+𝑏_𝑖𝑗𝑘_𝑗+^𝑝_𝑘=1𝑞^𝑘_𝑖‖𝑥‖+𝜎_𝑖𝜎_𝑖−1𝐽_𝑖+^𝑛_𝑗=1𝑓_𝑗(0)𝑎_𝑖𝑗+𝑏_𝑖𝑗≤max_{1≤𝑖≤𝑛}𝜎_𝑖𝜎_𝑖−1^𝑛_𝑗=1𝑎_𝑖𝑗+𝑏_𝑖𝑗𝑘_𝑗+^𝑝_𝑘=1𝑞^𝑘_𝑖𝜎𝐻+^𝑖𝜎_𝑖−1𝐽_𝑖+^𝑛_𝑗=1𝑓_𝑗(0)𝑎_𝑖𝑗+𝑏_𝑖𝑗∶=𝑅. (2.7) It implies that 𝜙(Ω) is uniformly bounded.
For any 𝑡∈[0,𝜔],𝑥∈Ω, we have𝜙𝑥^′_𝑖(𝑡)=𝑑𝑑𝑡_𝑡^𝑡+𝜔𝐺_𝑖(𝑡,𝑠)^𝑛_𝑗=1𝑎_𝑖𝑗(𝑠)𝑓_𝑗𝑥_𝑗(𝑠)+^𝑛_𝑗=1𝑏_𝑖𝑗(𝑡)𝑓_𝑗𝑥_𝑗𝑠−𝜏_𝑖𝑗(𝑠)+𝐽_𝑖(𝑠)+𝑑𝑠_𝑝𝑘=1𝐺_𝑖𝑡,𝑡_𝑘𝑞_𝑖𝑥_𝑖𝑡_𝑘=−𝑐_𝑖(𝑡)𝜙𝑥_𝑖(𝑡)+^𝑛_𝑗=1𝑎_𝑖𝑗(𝑡)𝑓_𝑗𝑥_𝑗(𝑡)+^𝑛_𝑗=1𝑏_𝑖𝑗(𝑡)𝑓_𝑗𝑥_𝑗𝑡−𝜏_𝑖𝑗(𝑡)+𝐽_𝑖(𝑡). (2.8) If 𝑡=𝑡_𝑘, it is obvious that (𝜙𝑥)^′_𝑖(𝑡)=lim_𝑡→𝑡⁻_𝑘(𝜙𝑥)^′_𝑖(𝑡). Hence, from (2.7) and (2.8), we have 𝜙𝑥^′_𝑖(𝑡)≤𝑐^𝑀_𝑖𝑅+^𝑛_𝑗=1𝑎^𝑀_𝑖𝑗+𝑏^𝑀_𝑖𝑗𝑘_𝑗𝐻+𝐽^𝑀_𝑖+^𝑛_𝑗=1𝑓_𝑗(0)𝑎^𝑀_𝑖𝑗+𝑏^𝑀_𝑖𝑗∶=𝛽_𝑖. (2.9) Therefore, 𝜙(Ω)⊂PC([0,𝜔],𝑅^𝑛) is a family of uniformly bounded and equicontinuous subset. By Lemma 2.2, the mapping 𝜙 is compact.
Let 𝑥∈PC([0,𝜔],𝑅^𝑛), and considering the following operator equation:𝑥=𝜆𝜙𝑥,𝜆∈(0,1). (2.10) If 𝑥 is a solution of (2.10), then ‖𝑥‖≤𝜙𝑥≤𝜃‖𝑥‖+max_{1≤𝑖≤𝑛}𝜎_𝑖𝜎_𝑖−1𝐽_𝑖+^𝑛_𝑗=1𝑓_𝑗(0)𝑎_𝑖𝑗+𝑏_𝑖𝑗. (2.11) According to (𝐻₄), we deduce that ‖𝑥‖≤max_{1≤𝑖≤𝑛}𝜎_𝑖/𝜎_𝑖−1𝐽_𝑖+∑^𝑛_𝑗=1𝑓_𝑗(0)𝑎_𝑖𝑗+𝑏_𝑖𝑗1−𝜃∶=𝑀. (2.12) It implies that ‖𝑥‖ is bounded, which is independent of 𝜆∈(0,1). By Lemma 2.4, we obtain that the mapping 𝜙 admits a fixed point in PC([0,𝜔],𝑅^𝑛). Hence system (1.2) admits an 𝜔-periodic solution such that ‖𝑥‖≤𝑀. This completes the proof.

3. Globally Exponentially Stable

In this section, the sufficient conditions ensuring that (1.2) admits a unique 𝜔-periodic solution and all solutions of (1.2) exponentially converge to the unique 𝜔-periodic solution are to be established.

Definition 3.1. Let 𝑥^∗(𝑡) be an 𝜔-periodic solution of system (1.2) with initial value 𝜙^∗. If there exist constants 𝛼>0,𝑃≥1, for every solution 𝑥(𝑡) of (1.2) with initial 𝜙, such that 𝑥_𝑖(𝑡)−𝑥^∗_𝑖(𝑡)≤𝑃𝜙−𝜙^∗𝑒^−𝛼𝑡forany𝑡>0,𝑖=1,2,…,𝑛, (3.1) then 𝑥^∗(𝑡) is said to be globally exponentially stable.

Theorem 3.2. Suppose that (𝐻₁)–(𝐻₄) hold. Further,(𝐻₅)  −𝑐^𝑚_𝑖+∑^𝑛_𝑗=1(𝑎^𝑀_𝑖𝑗+𝑏^𝑀_𝑖𝑗)𝑘_𝑗<0,𝑖=1,2,…,𝑛,(𝐻₆)  lnmax_{1≤𝑖≤𝑛}|1+𝑞^𝑘_𝑖|/𝑡_𝑘−𝑡_𝑘−1≤𝜁<𝛼,where max_{1≤𝑖≤𝑛}|1+𝑞^𝑘_𝑖|≥1,𝜁>0 is a constant, 𝛼 is a constant determined in (3.5).
Then system (1.2) admits a unique 𝜔-periodic solution, which is globally exponentially stable.

Proof. By Theorem 2.5, system (1.2) admits an 𝜔-periodic solution 𝑥^∗(𝑡)=(𝑥^∗₁(𝑡),𝑥^∗₂(𝑡),…,𝑥^∗_𝑛(𝑡)) with initial value 𝜙^∗. Let 𝑥(𝑡)=(𝑥₁(𝑡),𝑥₂(𝑡),…,𝑥_𝑛(𝑡)) be an arbitrary solution of (1.2) with initial value 𝜙. Define 𝑧_𝑖(𝑡)=𝑥^∗_𝑖(𝑡)−𝑥_𝑖(𝑡) and 𝑔_𝑗(𝑧_𝑗(𝑡))=𝑓_𝑗(𝑧_𝑗(𝑡)+𝑥_𝑗(𝑡))−𝑓_𝑗(𝑥_𝑗(𝑡)), then we have 𝑧^′_𝑖(𝑡)=−𝑐_𝑖(𝑡)𝑧_𝑖(𝑡)+^𝑛_𝑗=1𝑎_𝑖𝑗(𝑡)𝑔_𝑗𝑧_𝑗(𝑡)+^𝑛_𝑗=1𝑏_𝑖𝑗(𝑡)𝑔_𝑗𝑧_𝑗𝑡−𝜏_𝑖𝑗(𝑡),𝑡≠𝑡_𝑘,𝑧_𝑖𝑡⁺=1+𝑞^𝑘_𝑖𝑧_𝑖(𝑡),𝑡=𝑡_𝑘. (3.2) By (𝐻₅), we have −𝑐^𝑚_𝑖+∑^𝑛_𝑗=1(𝑎^𝑀_𝑖𝑗+𝑏^𝑀_𝑖𝑗)𝑘_𝑗<0 for 𝑖=1,2,…,𝑛. Let ℎ_𝑖(𝜆)=𝜆−𝑐^𝑚_𝑖+𝑒^𝑛𝜆𝜏_𝑗=1𝑎^𝑀_𝑖𝑗+𝑏^𝑀_𝑖𝑗𝑘_𝑗. (3.3) It is clear that ℎ_𝑖(𝜆) is continuous on 𝑅 and ℎ_𝑖(0)<0,𝑖=1,2,…,𝑛. In addition, 𝑑ℎ_𝑖(𝜆)𝑑𝜆=1+𝜏𝑒^𝑛𝜆𝜏_𝑗=1𝑎^𝑀_𝑖𝑗+𝑏^𝑀_𝑖𝑗𝑘_𝑗>0, (3.4) and ℎ_𝑖(+∞)=+∞, then ℎ_𝑖(𝜆) is strictly monotone increasing. Therefore, there exists a unique 𝜆_𝑖>0 such that 𝜆_𝑖−𝑐^𝑚_𝑖+𝑒^𝜆𝑖𝜏∑^𝑛_𝑗=1(𝑎^𝑀_𝑖𝑗+𝑏^𝑀_𝑖𝑗)𝑘_𝑗=0 for 𝑖=1,2,…,𝑛. Let 𝛼=min𝜆₁,𝜆₂,…,𝜆_𝑛, (3.5) then ℎ_𝑖(𝛼)=𝛼−𝑐^𝑚_𝑖+𝑒^𝑛𝛼𝜏_𝑗=1𝑎^𝑀_𝑖𝑗+𝑏^𝑀_𝑖𝑗𝑘_𝑗≤0,𝑖=1,2,…,𝑛. (3.6) Obviously, for 𝑡∈[−𝜏,0] and the above 𝛼, we have 𝑧_𝑖(𝑡)≤𝜙−𝜙^∗≤𝜙−𝜙^∗𝑒^−𝛼𝑡,𝑖=1,2,…,𝑛, (3.7) where ‖𝜙−𝜙^∗‖=max_{1≤𝑖≤𝑛}sup_{−𝜏≤𝑠≤0}|𝜙_𝑖(𝑠)−𝜙^∗_𝑖(𝑠)|.
Define 𝑉(𝑡)=(𝑉₁(𝑡),𝑉₂(𝑡),…,𝑉_𝑛(𝑡))^𝑇 by𝑉_𝑖(𝑡)=𝑒^𝛼𝑡𝑧_𝑖(𝑡),𝑖=1,2,…,𝑛. (3.8) In view of (3.2) and (3.8), for 𝑡≠𝑡_𝑘, we have 𝑑⁺𝑉_𝑖(𝑡)𝑑𝑡=𝛼𝑒^𝛼𝑡𝑧_𝑖(𝑡)+𝑒^𝛼𝑡sgn𝑧_𝑖(𝑡)−𝑐_𝑖(𝑡)𝑧_𝑖(𝑡)+^𝑛_𝑗=1𝑎_𝑖𝑗(𝑡)𝑔_𝑗𝑧_𝑗(𝑡)+^𝑛_𝑗=1𝑏_𝑖𝑗(𝑡)𝑔_𝑗𝑧_𝑗𝑡−𝜏_𝑖𝑗(𝑡)≤𝛼−𝑐_𝑖(𝑡)𝑒^𝛼𝑡𝑧_𝑖(𝑡)+𝑒^𝑛𝛼𝑡_𝑗=1𝑎_𝑖𝑗(𝑡)𝑘_𝑗𝑧_𝑗(𝑡)+𝑒^𝑛𝛼𝑡_𝑗=1𝑏_𝑖𝑗(𝑡)𝑘_𝑗𝑧_𝑗𝑡−𝜏_𝑖𝑗(𝑡)≤𝛼−𝑐^𝑚_𝑖𝑉_𝑖(𝑡)+^𝑛_𝑗=1𝑎^𝑀_𝑖𝑗𝑘_𝑗𝑉_𝑗(𝑡)+𝑒^𝑛𝛼𝜏_𝑗=1𝑏^𝑀_𝑖𝑗𝑘_𝑗𝑉_𝑗𝑡−𝜏_𝑖𝑗(𝑡). (3.9) We claim that 𝑉_𝑖(𝑡)=𝑒^𝛼𝑡𝑧_𝑖(𝑡)≤𝜙−𝜙^∗for𝑡∈0,𝑡₁,𝑖=1,2,…,𝑛. (3.10) If not, then there exist 𝑖₀∈{1,2,…,𝑛} and 0<𝑡<𝑡₁ such that 𝑉_𝑖0𝑡=𝜙−𝜙^∗,𝑑⁺𝑉_𝑖0𝑡𝑑𝑡>0,𝑉_𝑖(𝑡)≤𝜙−𝜙^∗, (3.11) for 𝑡∈(−𝜏,𝑡],𝑖=1,2,…,𝑛. Then, it follows from (3.9) and (3.11) that 𝑑0<⁺𝑉_𝑖0𝑡≤𝑑𝑡𝛼−𝑐^𝑚_𝑖𝑉_𝑖𝑡+^𝑛_𝑗=1𝑎^𝑀_𝑖𝑗𝑘_𝑗𝑉_𝑗𝑡+𝑒^𝑛𝛼𝜏_𝑗=1𝑏^𝑀_𝑖𝑗𝑘_𝑗𝑉_𝑗𝑡−𝜏_𝑖𝑗𝑡≤𝛼−𝑐^𝑚_𝑖0+𝑒^𝑛𝛼𝜏_𝑗=1𝑎^𝑀_𝑖𝑗+𝑏^𝑀_𝑖𝑗𝑘_𝑗𝜙−𝜙^∗. (3.12) Equation (3.12) leads to 𝛼−𝑐^𝑚_𝑖0+𝑒^𝑛𝛼𝜏_𝑗=1𝑎^𝑀_𝑖0𝑗+𝑏^𝑀_𝑖0𝑗𝑘_𝑗>0, (3.13) which contradicts (3.6). Thus (3.10) holds, that is, 𝑧_𝑖(𝑡)≤𝜙−𝜙^∗𝑒^−𝛼𝑡,forany𝑡∈0,𝑡₁,𝑖=1,2,…,𝑛. (3.14) If 𝑡=𝑡₁, we have 𝑧_𝑖𝑡⁺₁=1+𝑞¹_𝑖𝑧_𝑖𝑡₁=1+𝑞¹_𝑖lim_{𝑡→𝑡−1}𝑧_𝑖(𝑡)≤1+𝑞¹_𝑖𝜙−𝜙^∗𝑒^−𝛼𝑡1, (3.15) for 𝑖=1,2,…,𝑛. Similar to the steps of (3.10)–(3.14), we can derive that 𝑧_𝑖(𝑡)≤1+𝑞¹_𝑖𝜙−𝜙^∗𝑒^−𝛼𝑡,for𝑡∈𝑡₁,𝑡₂,𝑖=1,2,…,𝑛. (3.16) If 𝑡=𝑡₂, then 𝑧_𝑖𝑡⁺₂=1+𝑞²_𝑖𝑧_𝑖𝑡₂≤1+𝑞¹_𝑖1+𝑞²_𝑖𝜙−𝜙^∗𝑒^−𝛼𝑡2. (3.17) By repeating the same procedure, then 𝑧_𝑖(𝑡)≤1+𝑞¹_𝑖1+𝑞²_𝑖⋯1+𝑞^𝑝_𝑖⋅𝜙−𝜙^∗𝑒^−𝛼𝑡,𝑡∈𝑡_𝑝,𝑡_𝑝+1,𝑖=1,2,…,𝑛. (3.18) It follows from (𝐻₆) that |1+𝑞^𝑘_𝑖|≤𝑒^{𝜁(𝑡𝑘−𝑡𝑘−1)}, which leads to 1+𝑞¹_𝑖1+𝑞²_𝑖⋯1+𝑞^𝑝_𝑖≤𝑒^{𝜁(𝑡1−𝑡0)}𝑒^{𝜁(𝑡2−𝑡1)}⋯𝑒^{𝜁(𝑡𝑝−𝑡𝑝−1)}≤𝑒^𝜁𝑡𝑒^{𝜁(𝜔−𝑡𝑝)}, (3.19) for any 𝑡∈[𝑡_𝑘,𝑡_𝑘+1),𝑖=1,2,…,𝑛,𝑘=1,2,…. So the combination (3.18) and (3.19) gives 𝑧_𝑖(𝑡)≤𝑒^{𝜁(𝜔−𝑡𝑝)}𝜙−𝜙^∗𝑒^{−(𝛼−𝜁)𝑡},𝑡∈𝑡_𝑘,𝑡_𝑘+1,𝑖=1,2,…,𝑛,𝑘=1,2,…. (3.20) In addition, it is clear that 𝑧_𝑖(𝑡)≤𝑒^{𝜁(𝜔−𝑡𝑝)}𝜙−𝜙^∗𝑒^{−(𝛼−𝜁)𝑡},𝑡∈0,𝑡₁,𝑖=1,2,…,𝑛. (3.21) Therefore, from (3.20) and (3.21), for any 𝑡>0, we have 𝑥_𝑖(𝑡)−𝑥^∗_𝑖(𝑡)=𝑧_𝑖(𝑡)≤𝑒^{𝜁(𝜔−𝑡𝑝)}𝜙−𝜙^∗𝑒^{−(𝛼−𝜁)𝑡},𝑖=1,2,…,𝑛. (3.22) It implies that the 𝜔-periodic solution 𝑥^∗(𝑡) of (1.2) is globally exponentially stable. Hence, (1.2) admits a unique 𝜔-periodic solution, which is globally exponentially stable. This completes the proof.

Remark 3.3. Theorem 3.2 implies that the impulse 𝑞^𝑘_𝑖 affects the existence and exponential stability of the periodic solution of system (1.2). It shows the dynamics of impulsive differential system (1.2) is richer than the corresponding system (1.1) without impulse.

4. Applications and Examples

In (1.2), if 𝑎_𝑖𝑗(𝑡)≡0, then (1.2) reads:𝑥^′_𝑖(𝑡)=−𝑐_𝑖(𝑡)𝑥_𝑖(𝑡)+^𝑛_𝑗=1𝑏_𝑖𝑗(𝑡)𝑓_𝑗𝑥_𝑗𝑡−𝜏_𝑖𝑗(𝑡)+𝐽_𝑖(𝑡),𝑡≠𝑡_𝑘,𝑥_𝑖𝑡⁺=1+𝑞^𝑘_𝑖𝑥_𝑖(𝑡),𝑡=𝑡_𝑘. (4.1) For system (4.1), we have the following result.

Proposition 4.1. Suppose that (𝐻₁)–(𝐻₃) hold. Further,(𝐻₇)max_{1≤𝑖≤𝑛}(𝜎_𝑖/(𝜎_𝑖−1))(∑^𝑛_𝑗=1𝑏_𝑖𝑗𝑘_𝑗+∑^𝑝_𝑘=1|𝑞^𝑘_𝑖|)∶=𝜃<1,(𝐻₈)−𝑐^𝑚_𝑖+∑^𝑛_𝑗=1𝑏^𝑀_𝑖𝑗𝑘_𝑗<0,(𝐻₉)lnmax_{1≤𝑖≤𝑛}|1+𝑞^𝑘_𝑖|/(𝑡_𝑘−𝑡_𝑘−1)≤𝜁<𝛼, where max_{1≤𝑖≤𝑛}|1+𝑞^𝑘_𝑖|≥1,𝜁>0 is a constant, 𝛼 is determined in Theorem 3.2.
Then system (4.1) admits a unique 𝜔-periodic solution, which is globally exponentially stable.

If the impulses are absent in system (1.2), that is, 𝑞^𝑘_𝑖≡0, then (1.2) leads to (1.1). Similarly we have the following.

Proposition 4.2. Suppose that (𝐻₁)–(𝐻₃) hold. Further,(𝐻₁₀)  max_{1≤𝑖≤𝑛}(𝜎_𝑖/(𝜎_𝑖−1))∑^𝑛_𝑗=1(𝑎_𝑖𝑗+𝑏_𝑖𝑗)𝑘_𝑗∶=𝜃<1,(𝐻₁₁)  −𝑐^𝑚_𝑖+∑^𝑛_𝑗=1(𝑎^𝑀_𝑖𝑗+𝑏^𝑀_𝑖𝑗)𝑘_𝑗<0,then system (1.1) admits a unique 𝜔-periodic solution, which is globally exponentially stable.

Remark 4.3. Proposition 4.2 implies that the sufficient conditions of the existence and globally exponential stability of periodic solution to (1.1) are independent of the time-varying delays, while the corresponding results obtained by authors [5] are dependent on delays. Without effect from time-varying delays, our results are better for people to keep the stability of system (1.1). Although the authors [1] also established similar conditions which are independent of delays, their employed tool and analysis techniques are very different so that their main results are different from ours. Particularly, (1.1) is the special case of (1.2) without impulse. Hence, in this sense, results of this paper complement or improve some previously known results [1, 5].

Finally, two examples and numerical analysis are given to show the usefulness of the main results.

Example 4.4. Let 𝑥^′₁(𝑡)=−𝑐₁(𝑡)𝑥₁(𝑡)+𝑏₁₁(𝑡)𝑓𝑥₁𝑡−𝜏₁₁(𝑡)+𝑏₁₂(𝑡)𝑓𝑥₂𝑡−𝜏₁₂(𝑡)+𝐽₁(𝑡),𝑥^′₂(𝑡)=−𝑐₂(𝑡)𝑥₂(𝑡)+𝑏₂₁(𝑡)𝑓𝑥₁t−𝜏₂₁(𝑡)+𝑏₂₂(𝑡)𝑓𝑥₂𝑡−𝜏₂₂(𝑡)+𝐽₂(𝑡),Δ𝑥_𝑖(𝑡)=1+𝑞^𝑘_𝑖𝑥_𝑖(𝑡),𝑖=1,2, (4.2) where 𝑓(𝑥)=𝑥,𝑐₁(𝑡)=1+sin𝜋𝑡/4,𝑐₂(𝑡)=2+cos𝜋𝑡/4,𝑎₁₁(𝑡)=𝑎₁₂(𝑡)=𝑎₂₁(𝑡)=𝑎₂₂(𝑡)=0,𝑏₁₁(𝑡)=1/8+sin𝜋𝑡/16,  𝑏₁₂(𝑡)=1/8+cos𝜋𝑡/16,𝑏₂₁(𝑡)=1/16−cos𝜋𝑡/32,𝑏₂₂(𝑡)=1/16−sin𝜋𝑡/24,𝜏₁₁(𝑡)=𝜏₁₂(𝑡)=cos𝜋𝑡,  𝜏₂₁(𝑡)=𝜏₂₂(𝑡)=1/2−sin𝜋𝑡/3,𝐽₁(𝑡)=5+2cos𝜋𝑡,𝐽₂(𝑡)=7−sin𝜋𝑡,𝑞^𝑘_𝑖=1/8,𝑡_𝑘=𝑘−1/2. Then 𝑘₁=𝑘₂=1,𝜔=2,𝜏=1,{𝑡_𝑘,𝑘=1,2,…}∩[0,2]={𝑡₁,𝑡₂}.
By easy computation, 𝜎₁=𝑒²,𝜎₂=𝑒⁴, and 𝜃≈0.8674<1, which implies (𝐻₄) holds. On the other hand, it is easy to verify that (𝐻₅) holds. By verification, 𝛼>1/4>𝑙𝑛(1+(1/8)), namely, (𝐻₆) holds too. From Theorems 2.5 and 3.2, we obtain that (4.2) has a unique 2-periodic solution, which is globally exponentially stable, see Figure 1.

(a)

(b)

(c)

Figure 1 

Dynamics of (4.2)—(a) time series of 𝑥1, (b) time series of 𝑥2, and (c) portrait of (𝑥1,𝑥2).

Example 4.5. Let 𝑥^′₁(𝑡)1=−2+sin2𝜋𝑡4𝑥₁(𝑡)+14+cos2𝜋𝑡6𝑓𝑥₁((+1𝑡−2+sin2𝜋𝑡))4−sin2𝜋𝑡8𝑓𝑥₂((,𝑥𝑡−3−sin2𝜋𝑡))+cos2𝜋𝑡_′2(𝑡)1=−2+cos2𝜋𝑡4𝑥₂(𝑡)+13+cos2𝜋𝑡4𝑓𝑥₁((+1𝑡−5−sin2𝜋𝑡))6−cos2𝜋𝑡8𝑓𝑥₂(𝑡−(1+cos2𝜋𝑡))+sin2𝜋𝑡, (4.3) where 𝑓(𝑥)=(1/4)𝑥 for 𝑥∈𝑅, 𝑐₁(𝑡)=1/2+sin2𝜋𝑡/4,𝑐₂(𝑡)=1/2+cos2𝜋𝑡/4,𝑎₁₁(𝑡)=𝑎₁₂(𝑡)=𝑎₂₁(𝑡)=𝑎₂₂(𝑡)=0, 𝑏₁₁(𝑡)=1/4+cos2𝜋𝑡/6, 𝑏₁₂(𝑡)=1/4−sin2𝜋𝑡/8, 𝑏₂₁(𝑡)=1/3+cos2𝜋𝑡/4, 𝑏₂₂(𝑡)=1/6−cos2𝜋𝑡/8. Then 𝑘₁=𝑘₂=(1/4),𝜔=1.
By computation, 𝜃≈0.318<1, which implies that (𝐻₁₀) holds. It is easy to verify that (𝐻₁₁) holds too. From Proposition 4.2, system (4.3) has a unique 1-periodic solution, which is globally exponentially stable, see Figure 2. However, by calculation, conditions of the results of [1] fail, then one cannot obtain the existence of periodic solution of system (4.3) by results of reference [1], which further shows that the results complement or improve previously known results.

(a)

(b)

Figure 2 

Dynamics of (4.3)—(a) time series of 𝑥1, (b) time series of 𝑥2.

5. Conclusions

In this paper, the existence and globally exponential stability of the periodic solution of system (1.2) are studied. Model (1.2) is very general, including such models as continuous bidirectional associative memory networks, cellular neural networks, and Hopfield-type neural networks (see, e.g., [6, 7, 28]). The main methods employed here are Schaeffer’ theorem, differential inequality techniques, and Lyapunov functional, which are very different from [1]. The sufficient conditions obtained here are new and complement or improve the previously known results [1, 5–7]. Finally, applications, two illustrative examples and simulations, are given to show the effectiveness of the main results.

Acknowledgments

The authors would like to thank the reviewers for their valuable comments and constructive suggestions, which are very useful for improving the quality of this paper. This paper is supported by National Natural Science Foundation of China (11161015) and Doctoral Foundation of Guilin University of Technology (2010).