Asymptotic Stability of Impulsive Reaction-Diffusion Cellular Neural Networks with Time-Varying Delays

Abstract

This work addresses the asymptotic stability for a class of impulsive cellular neural networks with time-varying delays and reaction-diffusion. By using the impulsive integral inequality of Gronwall-Bellman type and Hardy-Sobolev inequality as well as piecewise continuous Lyapunov functions, we summarize some new and concise sufficient conditions ensuring the global exponential asymptotic stability of the equilibrium point. The provided stability criteria are applicable to Dirichlet boundary condition and showed to be dependent on all of the reaction-diffusion coefficients, the dimension of the space, the delay, and the boundary of the spatial variables. Two examples are finally illustrated to demonstrate the effectiveness of our obtained results.

1. Introduction

Cellular neural networks (CNNs), proposed by Chua and Yang in 1988 [1, 2], have been the focus of a number of investigations due to their potential applications in various fields such as optimization, linear and nonlinear programming, associative memory, pattern recognition, and computer vision [3–7]. Moreover, on the ground that time delays are unavoidably encountered for the finite switching speed of neurons and amplifiers in implementation of neural networks, it was followed by the introduction of the delayed cellular neural networks (DCNNs) so as to solve some dynamic image processing and pattern recognition problems. Such applications concerning CNNs and DCNNs depend heavily on the dynamical behaviors such as stability, convergence, and oscillatory [8, 9]. Particularly, stability analysis has been a major concern in the designs and applications of the CNNs and DCNNs. The stability of CNNs and DCNNs is a subject of current interest, and considerable theoretical efforts have been put into this topic with many good results reported (see, e.g., [10–13]).

With reference to neural networks, however, it is noteworthy that the state of electronic networks is actually subject to instantaneous perturbations more often than not. On this account, the networks experience abrupt change at certain instants which may be caused by a switching phenomenon, frequency change, or other sudden noise; that is, the networks often exhibit impulsive effects [14, 15]. For instance, according to Arbib [16] and Haykin [17], when a stimulus from the body or the external environment is received by receptors, the electrical impulses will be conveyed to the neural net and impulsive effects arise naturally in the net. As a consequence, in the past few years, scientists have become increasingly interested in the influence that impulses may have on the CNNs and DCNNs and a large number of stability criteria have been derived (see, e.g. [18–22]).

In reality, besides impulsive effects, diffusion effects are also nonignorable since diffusion is unavoidable when electrons are moving in asymmetric electromagnetic fields. As such, the model of neural networks with both impulses and reaction-diffusion should be more accurate to describe the evolutionary process of the systems in question, and it is necessary to consider the effects of both diffusion and impulses on the stability of CNNs and DCNNs.

In the past years, there have been a few theoretical contributions to the stability of CNNs and DCNNs with impulses and diffusion. For instance, Qiu [23] formulated a mathematical model of impulsive neural networks with time-varying delays and reaction-diffusion terms described by impulsive partial differential equations and studied, via delay impulsive differential inequality, the problem of global exponential stability with some stability criteria presented. Remarkably, all of the obtained stability criteria in [23] are independent of the diffusion. In 2008, Li and song [24] investigated a class of impulsive Cohen-Grossberg networks with time-varying delays and reaction-diffusion terms. By establishing a delay inequality with impulsive initial conditions and M-matrix theory, some sufficient conditions ensuring global exponential stability of the equilibrium points are given. Analogous to [23], the proposed stability criteria in [24] are also independent of the diffusion. More recently Pan et al. [25] investigated a class of impulsive Cohen-Grossberg neural networks with time-varying delays and reaction-diffusion in 2010. By the aid of the delay impulsive differential inequality quoted in [23], several sufficient conditions are exploited ensuring global exponential stability of the equilibrium points. Especially, different from [23, 24], the estimate of the exponential convergence rate depends on reaction-diffusion in [25].

In this paper, unlike the methods of impulsive differential inequalities and Poincaré inequality used in [25], we attempt to adopt the new techniques of the impulsive integral inequality of Gronwall-Bellman type and Hardy-Sobolev inequality to investigate the problem of global exponential asymptotic stability for impulsive cellular neural networks with time-varying delays and reaction-diffusion terms. Different from the existing research, we find, besides the reaction-diffusion coefficients, the dimension of the space and the boundary of the spatial variables do influence the stability.

The rest of the paper is organized as follows. In Section 2, the model of impulsive delayed cellular neural networks with reaction-diffusion terms and Dirichlet boundary condition is outlined, and some facts and lemmas are introduced for later reference. By the new agency of the impulsive integral inequality of Gronwall-Bellman type as well as Hardy-Sobolev inequality, we discuss the global exponential asymptotic stability and develop some new criteria in Section 3. To conclude, two illustrative examples are given to verify the effectiveness of our results in Section 4.

2. Preliminaries

Let 𝑅^𝑛 denote the n-dimensional Euclidean space, and Ω⊂𝑅^𝑚 is a bounded open set containing the origin. The boundary of Ω is smooth and mesΩ>0. Let 𝑅₊=[0,∞) and 𝑡₀∈𝑅₊.

We consider the following impulsive neural networks with time delays and reaction-diffusion terms: 𝜕𝑢_𝑖(𝑡,𝑥)𝜕𝑡=^𝑚_𝑠=1𝜕𝜕𝑥_𝑠𝐷^𝑖𝑠𝜕𝑢_𝑖(𝑡,𝑥)𝜕𝑥_𝑠−𝑎^𝑖𝑢_𝑖(𝑡,𝑥)+^𝑛_𝑗=1𝑏_𝑖𝑗𝑓_𝑗𝑢_𝑗(𝑡,𝑥)+^𝑛_𝑗=1𝑡≥𝑡₀𝑐_𝑖𝑗𝑓_𝑗𝑢_𝑗𝑡−𝜏_𝑗(𝑡),𝑥𝑢,𝑡≠𝑡_𝑘,𝑥∈Ω,𝑖=1,2,…,𝑛,𝑘=1,2,…,(2.1)_𝑖𝑡_𝑘+0,𝑥=𝑢_𝑖𝑡_𝑘,𝑥+𝑃_𝑖𝑘𝑢_𝑖𝑡_𝑘,𝑥𝑥∈Ω,𝑘=1,2,…,𝑖=1,2,…,𝑛,(2.2) where 𝑛 corresponds to the numbers of units in a neural network; 𝑥=(𝑥₁,…,𝑥_𝑚)^T∈Ω, 𝑢_𝑖(𝑡,𝑥), denotes the state of the ith neuron at time 𝑡 and in space 𝑥; smooth functions 𝐷_𝑖𝑠=𝐷_𝑖𝑠(𝑡,𝑥,𝑢)≥0 represent transmission diffusion operators of the ith unit; activation functions 𝑓_𝑗(𝑢_𝑗(𝑡,𝑥)) stand for the output of the jth unit at time 𝑡 and in space 𝑥; 𝑏_𝑖𝑗, 𝑐_𝑖𝑗, 𝑎_𝑖 are constants: 𝑏_𝑖𝑗 indicates the strength of the jth unit on the ith unit at time 𝑡 and in space 𝑥, 𝑐_𝑖𝑗 denotes the strength of the jth unit on the ith unit at time 𝑡−𝜏_𝑗(𝑡) and in space 𝑥, where 𝜏_𝑗(𝑡) corresponds to the transmission delay along the axon of the jth unit and satisfies 0≤𝜏_𝑗(𝑡)≤𝜏 (𝜏=const) as well as 𝜏^•_𝑗(𝑡)<1−1/ℎ (ℎ>0), and 𝑎_𝑖>0 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 at time 𝑡 and in space 𝑥. The fixed moments 𝑡_𝑘 (𝑘=1,2,…) are called impulsive moments satisfying 0≤𝑡₀<𝑡₁<𝑡₂<⋯ and lim_𝑘→∞𝑡_𝑘=∞; 𝑢_𝑖(𝑡_𝑘+0,𝑥) and 𝑢_𝑖(𝑡_𝑘−0,𝑥) represent the right-hand and left-hand limit of 𝑢_𝑖(𝑡,𝑥) at time 𝑡_𝑘 and in space 𝑥, respectively. 𝑃_𝑖𝑘(𝑢_𝑖(𝑡_𝑘,𝑥)) stands for the abrupt change of 𝑢_𝑖(𝑡,𝑥) at impulsive moment 𝑡_𝑘 and in space 𝑥.

Denote by 𝑢(𝑡,𝑥)=𝑢(𝑡,𝑥;𝑡₀,𝜑), 𝑢∈𝑅^𝑛 the solution of system (2.1)-(2.2), satisfying the initial condition 𝑢𝑠,𝑥;𝑡₀,𝜑=𝜑(𝑠,𝑥),𝑡₀−𝜏≤𝑠≤𝑡₀,𝑥∈Ω(2.3) and Dirichlet boundary condition 𝑢𝑡,𝑥;𝑡₀,𝜑=0,𝑡≥𝑡₀,𝑥∈𝜕Ω,(2.4) where the vector-valued function 𝜑(𝑠,𝑥)=(𝜑₁(𝑠,𝑥),…,𝜑_𝑛(𝑠,𝑥))^𝑇 is such that ∫_Ω∑^𝑛𝑖=1𝜑^2𝑖(𝑠,𝑥)𝑑𝑥 is bounded on [𝑡₀−𝜏,𝑡₀] and 𝜑_𝑖(𝑠,𝑥) (𝑖=1,2,…,𝑛) is first-order continuous differentiable as to 𝑠 on [𝑡₀−𝜏,𝑡₀].

The solution 𝑢(𝑡,𝑥)=𝑢(𝑡,𝑥;𝑡₀,𝜑)=(𝑢₁(𝑡,𝑥;𝑡₀,𝜑),…,𝑢_𝑛(𝑡,𝑥;𝑡₀,𝜑))^𝑇 of problems ((2.5)–(2.8)) is, for the time variable 𝑡, a piecewise continuous function with the first kind discontinuity at the points 𝑡_𝑘 (𝑘=1,2,…), where it is continuous from the left, that is the following relations are true:𝑢_𝑖𝑡_𝑘−0,𝑥=𝑢_𝑖𝑡_𝑘,𝑥,𝑢_𝑖𝑡_𝑘+0,𝑥=𝑢_𝑖𝑡_𝑘,𝑥+𝑃_𝑖𝑘𝑢_𝑖𝑡_𝑘,𝑥.(2.5)

Throughout this paper, the norm of 𝑢(𝑡,𝑥;𝑡₀,𝜑) is governed by ‖‖𝑢𝑡,𝑥;𝑡₀,𝜑‖‖_Ω=⎛⎜⎜⎝^𝑛_𝑖=1_Ω𝑢^2𝑖𝑡,𝑥;𝑡₀,𝜑d𝑥⎞⎟⎟⎠^1/2.(2.6)

Before moving on, we introduce two hypotheses as follows.(H1) Activation function 𝑓_𝑗(𝑢_𝑗(𝑡,𝑥)) satisfies 𝑓_𝑖(0)=0, and there exists constant 𝑙_𝑖>0 such that |𝑓_𝑖(𝑦₁)−𝑓_𝑖(𝑦₂)|≤𝑙_𝑖|𝑦₁−𝑦₂| holds for all 𝑦₁,𝑦₂∈𝑅 and 𝑖=1,2,…,𝑛.(H2) The functions 𝑃_𝑖𝑘(𝑢_𝑖(𝑡_𝑘,𝑥)) are continuous on 𝑅 and 𝑃_𝑖𝑘(0)=0, 𝑖=1,2,…,𝑛, 𝑘=1,2,….

According to (H1) and (H2), it is easy to see that problems ((2.5)–(2.8)) admits an equilibrium point 𝑢=0.

Definition 2.1. The equilibrium point 𝑢=0 of problems ((2.5)–(2.8)) is said to be globally exponentially stable if there exist constants 𝜅>0 and 𝑀≥1 such that ‖‖𝑢𝑡,𝑥;𝑡₀,𝜑‖‖_Ω≤𝑀‖𝜑‖_Ω𝑒^{−𝜅(𝑡−𝑡0)},𝑡≥𝑡₀,(2.7) where ‖𝜑‖^2Ω=sup_{𝑡0−𝜏≤𝑠≤𝑡0}∑^𝑛𝑖=1∫_Ω𝜑^2𝑖(𝑠,𝑥)d𝑥.

Lemma 2.2 (Gronwall-Bellman-type impulsive integral inequality [26]). Assume that
(A1) the sequence {𝑡_𝑘} satisfies 0≤𝑡₀<𝑡₁<𝑡₂<⋯, with lim_𝑘→∞𝑡_𝑘=∞,
(A2) 𝑞∈𝑃𝐶¹[𝑅₊,𝑅] and 𝑞(𝑡) is left-continuous at 𝑡_𝑘, 𝑘=1,2,…,
(A3) 𝑝∈𝐶[𝑅₊,𝑅₊] and for 𝑘=1,2,…𝑞(𝑡)≤𝑐+^𝑡𝑡₀𝑝(𝑠)𝑞(𝑠)d𝑠+_{𝑡0<𝑡𝑘<𝑡}𝜂_𝑘𝑞𝑡_𝑘,𝑡≥𝑡₀,(2.8) where 𝜂_𝑘≥0 and 𝑐=const. Then, 𝑞(𝑡)≤𝑐_{𝑡0<𝑡𝑘<𝑡}1+𝜂_𝑘exp^𝑡𝑡₀𝑝(𝑠)d𝑠,𝑡≥𝑡₀.(2.9)

Lemma 2.3 (Hardy-Sobolev inequality [27]). Let Ω⊂𝑅^𝑚(𝑚≥3) be a bounded open set containing the origin and 𝑢∈𝐻¹(Ω)={𝜔∣𝜔∈𝐿²(Ω),𝐷_𝑖𝜔=𝜕𝜔/𝜕𝑥_𝑖∈𝐿²(Ω),1≤𝑖≤𝑚}. Then there exists a positive constant 𝐶_𝑚=𝐶_𝑚(Ω) such that (𝑚−2)²4_Ω𝑢²|𝑥|²𝑑𝑥≤_Ω||∇𝑢||²𝑑𝑥+𝐶_𝑚_𝜕Ω𝑢²𝑑𝜎.(2.10)

Lemma 2.4. If 𝑎>0 and 𝑏>0, then 𝑎𝑏≤(1/𝜀)𝑎²+𝜀𝑏² holds for any 𝜀>0.

3. Main Results

Theorem 3.1. Provided that(1)for 𝑥=(𝑥₁,…,𝑥_𝑚)^𝑇∈Ω(𝑚≥3), there exists a constant 𝛽 such that |𝑥|²=∑^𝑚𝑠=1𝑥^2𝑠<𝛽. In addition, there exists a constant 𝐷>0 such that 𝐷_𝑖𝑠=𝐷_𝑖𝑠(𝑡,𝑥,𝑢)≥𝐷>0. Denote 𝐷(𝑚−2)²/2𝛽=𝜒,(2)𝑃_𝑖𝑘(𝑢_𝑖(𝑡_𝑘,𝑥))=−𝜃_𝑖𝑘𝑢_𝑖(𝑡_𝑘,𝑥), 0≤𝜃_𝑖𝑘≤2,(3)there exists a constant 𝛾 satisfying 𝛾+𝜆+ℎ𝜌𝑒^𝛾𝜏>0 as well as 𝜆+ℎ𝜌𝑒^𝛾𝜏<0, where 𝜆=max_{𝑖=1,…,𝑛}(−𝜒−2𝑎_𝑖+∑^𝑛𝑗=1(𝑏^2𝑖𝑗+𝑐^2𝑖𝑗))+𝜌, 𝜌=𝑛max_{𝑖=1,…,𝑛}(𝑙_𝑖2),then, the equilibrium point 𝑢=0 of problems ((2.5)–(2.8)) is globally exponentially stable with convergence rate – (𝜆+ℎ𝜌𝑒^𝛾𝜏)/2.

Proof. Multiplying both sides of (2.1) by 𝑢_𝑖(𝑡,𝑥) and integrating with respect to spatial variable 𝑥 on Ω, we get d∫_Ω𝑢_𝑖2(𝑡,𝑥)d𝑥d𝑡=2^𝑚_𝑠=1_Ω𝑢_𝑖(𝑡,𝑥)𝜕𝜕𝑥_𝑠𝐷^𝑖𝑠𝜕𝑢_𝑖(𝑡,𝑥)𝜕𝑥_𝑠d𝑥−2𝑎^𝑖_Ω+2_𝑛𝑢^2𝑖(𝑡,𝑥)d𝑥_𝑗=1𝑏_𝑖𝑗_Ω+2_𝑛𝑢_𝑖(𝑡,𝑥)𝑓_𝑗𝑢_𝑗(𝑡,𝑥)d𝑥_𝑗=1𝑐_𝑖𝑗_Ω𝑢_𝑖(𝑡,𝑥)𝑓_𝑗𝑢_𝑗𝑡−𝜏_𝑗(𝑡),𝑥d𝑥𝑡≥𝑡₀,𝑡≠𝑡_𝑘,𝑘=1,2,….(3.1)
Regarding the right-hand part of (3.1), the first term becomes by using Green formula, Dirichlet boundary condition, Lemma 2.3, and condition 1 of Theorem 3.12^𝑚_𝑠=1_Ω𝑢_𝑖(𝑡,𝑥)𝜕𝜕𝑥_𝑠𝐷^𝑖𝑠𝜕𝑢_𝑖(𝑡,𝑥)𝜕𝑥_𝑠d𝑥=−2^𝑚_𝑠=1_Ω𝐷_𝑖𝑠𝜕𝑢^𝑖(𝑡,𝑥)𝜕𝑥_𝑠²≤−𝐷d𝑥(𝑚−2)²2_Ω𝑢^2𝑖(𝑡,𝑥)|𝑥|²𝑑𝑥≤−𝐷(𝑚−2)²2𝛽_Ω𝑢^2𝑖(𝑡,𝑥)𝑑𝑥≜−𝜒_Ω𝑢^2𝑖(𝑡,𝑥)d𝑥.(3.2)
Moreover, we derive from (H1) that 2^𝑛_𝑗=1𝑏_𝑖𝑗_Ω𝑢_𝑖(𝑡,𝑥)𝑓_𝑗𝑢_𝑗(𝑡,𝑥)d𝑥≤2^𝑛_𝑗=1≤2_𝑛||𝑏_𝑖𝑗||_Ω||𝑢_𝑖(𝑡,𝑥)||||𝑓_𝑗𝑢_𝑗(𝑡,𝑥)||d𝑥_𝑗=1_Ω≤𝑙_𝑗||𝑏_𝑖𝑗||||𝑢_𝑖(𝑡,𝑥)||||𝑢_𝑗(𝑡,𝑥)||d𝑥_𝑛_𝑗=1_Ω𝑏^2𝑖𝑗2𝑢^2𝑖(𝑡,𝑥)+𝑙^2𝑗𝑢^2𝑖(𝑡,𝑥)d𝑥,_𝑛_𝑗=1𝑐_𝑖𝑗_Ω𝑢_𝑖(𝑡,𝑥)𝑓_𝑗𝑢_𝑗𝑡−𝜏_𝑗(𝑡),𝑥𝑑𝑥≤2^𝑛_𝑗=1≤2_𝑛||𝑐_𝑖𝑗||_Ω||𝑢_𝑖(𝑡,𝑥)||||𝑓_𝑗𝑢_𝑗𝑡−𝜏_𝑗(𝑡),𝑥||𝑑𝑥_𝑗=1_Ω≤𝑙_𝑗||𝑐_𝑖𝑗||||𝑢_𝑖(𝑡,𝑥)||||𝑢_𝑗𝑡−𝜏_𝑗(𝑡),𝑥||𝑑𝑥_𝑛_𝑗=1_Ω𝑐^2𝑖𝑗𝑢^2𝑖(𝑡,𝑥)+𝑙^2𝑗𝑢^2𝑖𝑡−𝜏_𝑗(𝑡),𝑥𝑑𝑥.(3.3) Consequently, substituting ((2.10)–(3.14)) into (3.1) produces d∫_Ω𝑢^2𝑖(𝑡,𝑥)d𝑥d𝑡≤−𝜒_Ω𝑢^2𝑖(𝑡,𝑥)d𝑥−2𝑎_𝑖_Ω+𝑢^2𝑖(𝑡,𝑥)d𝑥_𝑛_𝑗=1_Ω𝑏^2𝑖𝑗+𝑢^2𝑖(𝑡,𝑥)+𝑙^2𝑗𝑢^2𝑖(𝑡,𝑥)d𝑥_𝑛_𝑗=1_Ω𝑐^2𝑖𝑗𝑢^2𝑖(𝑡,𝑥)+𝑙^2𝑗𝑢^2𝑖𝑡−𝜏_𝑗(𝑡),𝑥d𝑥(3.4)for𝑡≥𝑡₀,𝑡≠𝑡_𝑘,𝑘=1,2,….
We define a Lyapunov function 𝑉_𝑖(𝑡) as 𝑉_𝑖(𝑡)=∫_Ω𝑢^2𝑖(𝑡,𝑥)d𝑥. It is easy to find that 𝑉_𝑖(𝑡) is a piecewise continuous function with points of discontinuity of the first kind 𝑡_𝑘 (𝑘=1,2,…), where it is continuous from the left, that is, 𝑉_𝑖(𝑡_𝑘−0)=𝑉_𝑖(𝑡_𝑘) (𝑘=1,2,…). In addition, due to 𝑉_𝑖(𝑡₀+0)≤𝑉_𝑖(𝑡₀) and the following estimate derived from condition 2 of Theorem 3.1𝑢^2𝑖𝑡_𝑘+0,𝑥=−𝜃_𝑖𝑘𝑢_𝑖𝑡_𝑘,𝑥+𝑢_𝑖𝑡_𝑘,𝑥²=1−𝜃_𝑖𝑘²𝑢^2𝑖𝑡_𝑘,𝑥≤𝑢^2𝑖𝑡_𝑘,𝑥(𝑘=1,2,…),(3.5) we have𝑉_𝑖𝑡_𝑘+0≤𝑉_𝑖𝑡_𝑘,𝑘=0,1,2,….(3.6) holds for 𝑡=𝑡_𝑘 (𝑘=0,1,2,…). Put 𝑡∈(𝑡_𝑘,𝑡_𝑘+1), 𝑘=0,1,2,…. Then for the derivative d𝑉_𝑖(𝑡)/d𝑡 of 𝑉_𝑖 with respect to problems ((2.5)–(2.8)), it results from (3.4) that d𝑉_𝑖(𝑡)d𝑡≤−𝜒_Ω𝑢^2𝑖(𝑡,𝑥)d𝑥−2𝑎_𝑖_Ω𝑢^2𝑖(𝑡,𝑥)d𝑥+^𝑛_𝑗=1_Ω𝑏^2𝑖𝑗+𝑢^2𝑖(𝑡,𝑥)+𝑙^2𝑗𝑢^2𝑖(𝑡,𝑥)d𝑥_𝑛_𝑗=1_Ω𝑐^2𝑖𝑗𝑢^2𝑖(𝑡,𝑥)+𝑙^2𝑗𝑢^2𝑗𝑡−𝜏_𝑗(𝑡),𝑥d𝑥≤−𝜒−2𝑎_𝑖+^𝑛_𝑗=1𝑏^2𝑖𝑗+^𝑛_𝑗=1𝑐^2𝑖𝑗𝑉^𝑖+max_{𝑖=1,…,𝑛}(𝑡)𝑙^2𝑖^𝑛_𝑗=1𝑉_𝑗(𝑡)+max_{𝑖=1,…,𝑛}𝑙_𝑖2^𝑛_𝑗=1𝑉_𝑗𝑡−𝜏_𝑗(𝑡)𝑡∈𝑡_𝑘,𝑡_𝑘+1,𝑘=0,1,2,….(3.7) Choose 𝑉(𝑡) of the form 𝑉(𝑡)=∑^𝑛𝑖=1𝑉_𝑖(𝑡). From (3.7), one then reads d𝑉(𝑡)d𝑡≤max^{𝑖=1,…,𝑛}−𝜒−2𝑎^𝑖+^𝑛_𝑗=1𝑏^2𝑖𝑗+𝑛max_{𝑖=1,…,𝑛}+𝑐^2𝑖𝑗+𝑛max_{𝑖=1,…,𝑛}𝑙_𝑖2𝑉(𝑡)=𝜆𝑉(𝑡)+𝜌𝑙_𝑖2𝑛^𝑛_𝑗=1𝑉_𝑗𝑡−𝜏_𝑗_𝑗=1𝑉_𝑗𝑡−𝜏_𝑗(𝑡)𝑡∈𝑡_𝑘,𝑡_𝑘+1,𝑘=0,1,2,….(3.8)
Construct 𝑉^∗(𝑡)=e^{𝛾(𝑡−𝑡0)}𝑉(𝑡), where 𝛾 satisfies 𝛾+𝜆+ℎ𝜌𝑒^𝛾𝜏>0 and 𝜆+ℎ𝜌𝑒^𝛾𝜏<0. Evidently, 𝑉^∗(𝑡) is also a piecewise continuous function with points of discontinuity of the first kind 𝑡_𝑘 (𝑘=1,2,…), in which it is continuous from the left, that is 𝑉^∗(𝑡_𝑘−0)=𝑉^∗(𝑡_𝑘) (𝑘=1,2,…). Moreover, at 𝑡=𝑡_𝑘 (𝑘=0,1,2,…), we find by use of (3.6) 𝑉^∗𝑡_𝑘+0≤𝑉^∗𝑡_𝑘,𝑘=0,1,2,….(3.9)
Set 𝑡∈(𝑡_𝑘,𝑡_𝑘+1), 𝑘=0,1,2,…. By virtue of (3.8), one has d𝑉^∗(𝑡)d𝑡=𝛾e^{𝛾(𝑡−𝑡0)}𝑉(𝑡)+e^{𝛾(𝑡−𝑡0)}d𝑉(𝑡)d𝑡≤𝛾e^{𝛾(𝑡−𝑡0)}𝑉(𝑡)+𝜆𝑉(𝑡)+𝜌^𝑛_𝑗=1𝑉_𝑗𝑡−𝜏_𝑗(𝑡)e^{𝛾(𝑡−𝑡0)}=(𝛾+𝜆)𝑉^∗(𝑡)+𝜌e^{𝛾(𝑡−𝑡0)𝑛}_𝑗=1𝑉_𝑗𝑡−𝜏_𝑗(𝑡)𝑡∈𝑡_𝑘,𝑡_𝑘+1,𝑘=0,1,2,….(3.10)
Choose small enough 𝜀>0. Integrating (3.10) from 𝑡_𝑘+𝜀 to 𝑡 gives 𝑉^∗(𝑡)≤+𝑉^∗𝑡_𝑘+𝜀+(𝛾+𝜆)^𝑡𝑡_𝑘+𝜀𝑉^∗(𝑠)𝑑𝑠_{𝑡𝑡𝑘+𝜀}𝜌e^{𝛾(𝑠−𝑡0)𝑛}_𝑗=1𝑉_𝑗𝑠−𝜏_𝑗(𝑠)𝑑𝑠,𝑡∈𝑡_𝑘,𝑡_𝑘+1,𝑘=0,1,2,…(3.11)
which yields after letting 𝜀→0 in (3.11) 𝑉^∗(𝑡)≤+𝑉^∗𝑡_𝑘+0+(𝛾+𝜆)^𝑡𝑡_𝑘𝑉^∗(𝑠)𝑑𝑠_𝑡𝑡𝑘𝜌e^{𝛾(𝑠−𝑡0)𝑛}_𝑗=1𝑉_𝑗𝑠−𝜏_𝑗(𝑠)𝑑𝑠,𝑡∈𝑡_𝑘,𝑡_𝑘+1,𝑘=0,1,2,….(3.12)
We now proceed to estimate the value of 𝑉^∗(𝑡) at 𝑡=𝑡_𝑘+1, 𝑘=0,1,2,…. For small enough 𝜀>0, we put 𝑡=𝑡_𝑘+1−𝜀. Now an application of (3.12) leads to, for 𝑘=0,1,2,…,𝑉^∗𝑡_𝑘+1−𝜀≤𝑉^∗𝑡_𝑘+0+(𝛾+𝜆)^{𝑡𝑘+1𝑡−𝜀}_𝑘𝑉^∗(𝑠)𝑑𝑠+^{𝑡𝑘+1𝑡−𝜀}_𝑘𝜌e^{𝛾(𝑠−𝑡0)𝑛}_𝑗=1𝑉_𝑗𝑠−𝜏_𝑗(𝑠)𝑑𝑠.(3.13)
If we let 𝜀→0 in (3.13), there results 𝑉^∗𝑡_𝑘+1−0≤+𝑉^∗𝑡_𝑘+0+(𝛾+𝜆)^𝑡𝑘+1_𝑡𝑘𝑉^∗(𝑠)𝑑𝑠_{𝑡𝑘+1𝑡𝑘}𝜌e^{𝛾(𝑠−𝑡0)𝑛}_𝑗=1𝑉_𝑗𝑠−𝜏_𝑗(𝑠)𝑑𝑠,𝑘=0,1,2,….(3.14)
Note that 𝑉^∗(𝑡_𝑘+1−0)=𝑉^∗(𝑡_𝑘+1) is applicable for 𝑘=0,1,2,…. Thus, 𝑉^∗𝑡_𝑘+1≤𝑉^∗𝑡_𝑘+0+(𝛾+𝜆)^𝑡𝑘+1_𝑡𝑘𝑉^∗(𝑠)𝑑𝑠+^𝑡𝑘+1_𝑡𝑘𝜌e^{𝛾(𝑠−𝑡0)𝑛}_𝑗=1𝑉_𝑗𝑠−𝜏_𝑗(𝑠)𝑑𝑠(3.15)
holds for 𝑘=0,1,2,…. By synthesizing (3.12) and (3.15), we then arrive at 𝑉^∗(𝑡)≤+𝑉^∗𝑡_𝑘+0+(𝛾+𝜆)^𝑡𝑡_𝑘𝑉^∗(𝑠)𝑑𝑠_𝑡𝑡𝑘𝜌e^{𝛾(𝑠−𝑡0)𝑛}_𝑗=1𝑉_𝑗𝑠−𝜏_𝑗(𝑠)𝑑𝑠𝑡∈𝑡_𝑘,𝑡_𝑘+1,𝑘=0,1,2,….(3.16)
This, together with (3.9), results in 𝑉^∗(𝑡)≤𝑉^∗𝑡_𝑘+(𝛾+𝜆)^𝑡𝑡_𝑘𝑉^∗(𝑠)𝑑𝑠+^𝑡𝑡_𝑘𝜌e^{𝛾(𝑠−𝑡0)𝑛}_𝑗=1𝑉_𝑗𝑠−𝜏_𝑗(𝑠)𝑑𝑠(3.17)for𝑡∈(𝑡_𝑘,𝑡_𝑘+1],𝑘=0,1,2,….
Recalling assumptions that 0≤𝜏_𝑗(𝑡)≤𝜏 and 𝜏^•_𝑗(𝑡)<1−1/ℎ (ℎ>0), we have ^𝑡𝑡_𝑘𝜌e^{𝛾(𝑠−𝑡0)𝑛}_𝑗=1𝑉_𝑗𝑠−𝜏_𝑗((𝑠))𝑑𝑠=^𝑛_𝑗=1^{𝑡−𝜏𝑗𝑡(𝑡)}_{𝑘−𝜏𝑗𝑡𝑘}𝜌e^{𝛾(𝜃+𝜏𝑗(𝑠)−𝑡0)}𝑉_𝑗(𝜃)1≤ℎ𝜌𝑒_𝑛𝛾𝜏1−𝜏^•_𝑗(𝑠)𝑑𝜃_𝑗=1^{𝑡−𝜏𝑗𝑡(𝑡)}_{𝑘−𝜏𝑗𝑡𝑘}e^{𝛾(𝜃−𝑡0)}𝑉_𝑗(𝜃)𝑑𝜃.(3.18) Hence, 𝑉^∗(𝑡)≤𝑉^∗𝑡_𝑘+(𝛾+𝜆)^𝑡𝑡_𝑘𝑉^∗(𝑠)𝑑𝑠+ℎ𝜌𝑒^𝑛𝛾𝜏_𝑗=1^{𝑡−𝜏𝑗𝑡(𝑡)}_{𝑘−𝜏𝑗𝑡𝑘}𝑡∈𝑡_𝑘e^{𝛾(𝑠−𝑡0)}𝑉_𝑗(𝑠)𝑑𝑠,𝑡_𝑘+1,𝑘=0,1,2,….(3.19)
By induction argument, we reach 𝑉^∗𝑡_𝑘≤𝑉^∗𝑡_𝑘−1+(𝛾+𝜆)^𝑡𝑘_𝑡𝑘−1𝑉^∗(𝑠)𝑑𝑠+ℎ𝜌𝑒^𝑛𝛾𝜏_𝑗=1^{𝑡𝑘−𝜏𝑗(𝑡𝑘)𝑡}_{𝑘−1−𝜏𝑗𝑡𝑘−1}𝑉⋮e^{𝛾(𝑠−𝑡0)}𝑉_𝑗(𝑠)𝑑𝑠,_∗𝑉𝑡₂≤𝑉^∗𝑡₁+(𝛾+𝜆)^𝑡2_𝑡1𝑉^∗(𝑠)𝑑𝑠+ℎ𝜌𝑒^𝑛𝛾𝜏_𝑗=1^{𝑡2−𝜏𝑗(𝑡2)𝑡}_{1−𝜏𝑗𝑡1}e^{𝛾(𝑠−𝑡0)}𝑉_𝑗(𝑠)𝑑𝑠,_∗𝑡₁≤𝑉^∗𝑡₀+(𝛾+𝜆)^𝑡1_𝑡0𝑉^∗(𝑠)𝑑𝑠+ℎ𝜌𝑒^𝑛𝛾𝜏_𝑗=1^{𝑡1−𝜏𝑗(𝑡1)𝑡}_{0−𝜏𝑗𝑡0}e^{𝛾(𝑠−𝑡0)}𝑉_𝑗(𝑠)𝑑𝑠.(3.20)
Therefore, 𝑉^∗(𝑡)≤𝑉^∗𝑡₀+(𝛾+𝜆)^𝑡𝑡₀𝑉^∗(𝑠)𝑑𝑠+ℎ𝜌𝑒^𝑛𝛾𝜏_𝑗=1^{𝑡−𝜏𝑗𝑡(𝑡)}_{0−𝜏𝑗𝑡0}≤𝑉_∗e^{𝛾(𝑠−𝑡0)}𝑉_𝑗(𝑠)𝑑𝑠=𝑉_∗𝑡₀+(𝛾+𝜆)^𝑡𝑡₀𝑉^∗(𝑠)𝑑𝑠+ℎ𝜌𝑒^𝑛𝛾𝜏_𝑗=1^𝑡𝑡_{0−𝜏𝑗𝑡0}e^{𝛾(𝑠−𝑡0)}𝑉_𝑗(𝑠)𝑑𝑠+ℎ𝜌𝑒𝑡₀_𝑛𝛾𝜏+(𝛾+𝜆+ℎ𝜌𝑒^𝛾𝜏)^𝑡𝑡₀𝑉^∗(𝑠)𝑑𝑠_𝑗=1^𝑡0_{𝑡0−𝜏𝑗𝑡0}e^{𝛾(𝑠−𝑡0)}𝑉_𝑗(𝑠)𝑑𝑠𝑡∈𝑡_𝑘,𝑡_𝑘+1,𝑘=0,1,2,….(3.21) Since ℎ𝜌𝑒^𝑛𝛾𝜏_𝑗=1^𝑡0_{𝑡0−𝜏𝑗𝑡0}e^{𝛾(𝑠−𝑡0)}𝑉_𝑗(𝑠)𝑑𝑠≤ℎ𝜌𝑒^𝑛𝛾𝜏_𝑗=1^𝑡0_𝑡0−𝜏=ℎ𝜌𝑒_𝛾𝜏𝑉_𝑗(𝑠)𝑑𝑠^𝑡0_𝑡0−𝜏^𝑛_𝑗=1_Ω𝜑^2𝑗(𝑠,𝑥)𝑑𝑥𝑑𝑠≤𝜏ℎ𝜌𝑒^𝛾𝜏‖𝜑‖^2Ω,(3.22) we claim 𝑉^∗(𝑡)≤𝑉^∗𝑡₀+𝜏ℎ𝜌𝑒^𝛾𝜏‖𝜑‖^2Ω+(𝛾+𝜆+ℎ𝜌𝑒^𝛾𝜏)^𝑡𝑡₀𝑉^∗(𝑠)𝑑𝑠𝑡∈𝑡_𝑘,𝑡_𝑘+1,𝑘=0,1,2….(3.23)
According to Lemma 2.2, we claim 𝑉^∗(𝑡)≤𝑉^∗𝑡₀+𝜏ℎ𝜌𝑒^𝛾𝜏‖𝜑‖^2Ωexp(𝛾+𝜆+ℎ𝜌𝑒^𝛾𝜏)𝑡−𝑡₀,𝑡≥𝑡₀(3.24) which reduces to‖‖𝑢𝑡,𝑥;𝑡₀,𝜑‖‖_Ω≤√1+𝜏ℎ𝜌𝑒^𝛾𝜏‖𝜑‖_Ωexp𝜆+ℎ𝜌𝑒^𝛾𝜏2𝑡−𝑡⁰,𝑡≥𝑡₀.(3.25) This completes the proof.

Remark 3.2. According to the conditions of Theorem 3.1, we see that the reaction-diffusion term do influence the stability of problem ((2.5)–(2.8)). Moreover, besides the reaction-diffusion coefficients, the dimension of the space and the boundary of spatial variables have also an effect on the stability of the equilibrium point 𝑢=0.

Theorem 3.3 .. Providing that(1) for 𝑥=(𝑥₁,…,𝑥_𝑚)^𝑇∈Ω(𝑚≥3), there exist constants 𝛽 such that |𝑥|²=∑^𝑚𝑠=1𝑥^2𝑠<𝛽, in addition, there exists constant 𝐷>0 such that 𝐷_𝑖𝑠=𝐷_𝑖𝑠(𝑡,𝑥,𝑢)≥𝐷>0; denote 𝐷(𝑚−2)²/2𝛽=𝜒,(2)𝑃_𝑖𝑘(𝑢_𝑖(𝑡_𝑘,𝑥))=−𝜃_𝑖𝑘𝑢_𝑖(𝑡_𝑘,𝑥), 1−√1+𝛼≤𝜃_𝑖𝑘≤1+√1+𝛼, 𝛼≥0,(3)inf_𝑘=1,2…(𝑡_𝑘−𝑡_𝑘−1)>𝜇, (4)there exists constant 𝛾 which satisfies 𝛾+𝜆+ℎ𝜌𝑒^𝛾𝜏>0 and 𝜆+ℎ𝜌𝑒^𝛾𝜏+ln(1+𝛼)/𝜇<0, where 𝜆=max_{𝑖=1,…,𝑛}(−𝜒−2𝑎_𝑖+∑^𝑛𝑗=1(𝑏^2𝑖𝑗+𝑐^2𝑖𝑗))+𝜌 and 𝜌=𝑛max_{𝑖=1,…,𝑛}(𝑙_𝑖2), then, the equilibrium point 𝑢=0 of problem ((2.5)–(2.8)) is globally exponentially stable with convergence rate –(1/2)(𝜆+ℎ𝜌𝑒^𝛾𝜏+ln(1+𝛼)/𝜇).

Proof. Define a Lyapunov function 𝑉 of the form 𝑉(𝑡)=∑^𝑛𝑖=1𝑉_𝑖(𝑡), where 𝑉_𝑖(𝑡)=∫_Ω𝑢^2𝑖(𝑡,𝑥)d𝑥. Obviously, 𝑉(𝑡) is a piecewise continuous function with points of discontinuity of the first kind 𝑡_𝑘, 𝑘=1,2,…, where it is continuous from the left, that is, 𝑉₁(𝑡_𝑘−0)=𝑉₁(𝑡_𝑘) (𝑘=1,2,…). Furthermore, for 𝑡=𝑡_𝑘 (𝑘=0,1,2,…), it follows from condition 2 of Theorem 3.3 that 𝑢^2𝑖𝑡_𝑘+0,𝑥−𝑢^2𝑖𝑡_𝑘,𝑥=1−𝜃_𝑖𝑘²𝑢^2𝑖𝑡_𝑘,𝑥−𝑢^2𝑖𝑡_𝑘,𝑥≤𝛼𝑢^2𝑖𝑡_𝑘,𝑥.(3.26) Thereby, 𝑉𝑡_𝑘+0≤𝛼𝑉𝑡_𝑘+𝑉𝑡_𝑘,𝑘=0,1,2,….(3.27)
Construct another Lyapunov function defined by 𝑉^∗(𝑡)=e^{𝛾(𝑡−𝑡0)}𝑉(𝑡), where 𝛾 satisfies 𝛾+𝜆+ℎ𝜌𝑒^𝛾𝜏>0 and 𝜆+ℎ𝜌𝑒^𝛾𝜏+(ln(1+𝛼)/𝜇)<0. Then, 𝑉^∗(𝑡) is also a piecewise continuous function with points of discontinuity of the first kind 𝑡_𝑘, 𝑘=1,2,…, where it is continuous from the left, that is 𝑉^∗(𝑡_𝑘−0)=𝑉^∗(𝑡_𝑘) (𝑘=1,2,…). And for 𝑡=𝑡_𝑘 (𝑘=0,1,2,…), it results from (3.27) that 𝑉^∗𝑡_𝑘+0≤𝛼𝑉^∗𝑡_𝑘+𝑉^∗𝑡_𝑘,𝑘=0,1,2,….(3.28)
Set 𝑡∈(𝑡_𝑘,𝑡_𝑘+1], 𝑘=0,1,2,…. Following the same procedure as in Theorem 3.1, we get 𝑉^∗(𝑡)≤𝑉^∗𝑡_𝑘+0+(𝛾+𝜆)^𝑡𝑡_𝑘𝑉^∗(𝑠)𝑑𝑠+ℎ𝜌𝑒^𝛾𝜏×^𝑛_𝑗=1^{𝑡−𝜏𝑗𝑡(𝑡)}_{𝑘−𝜏𝑗𝑡𝑘}e^{𝛾(𝜃−𝑡0)}𝑉_𝑗(𝜃)𝑑𝜃𝑡∈𝑡_𝑘,𝑡_𝑘+1,𝑘=0,1,2,….(3.29)
The relations (3.28) and (3.29) yield 𝑉^∗(𝑡)−𝑉^∗𝑡_𝑘≤𝛼𝑉^∗𝑡_𝑘+(𝛾+𝜆)^𝑡𝑡_𝑘𝑉^∗(𝑠)𝑑𝑠+ℎ𝜌𝑒^𝛾𝜏×^𝑛_𝑗=1^{𝑡−𝜏𝑗𝑡(𝑡)}_{𝑘−𝜏𝑗𝑡𝑘}e^{𝛾(𝜃−𝑡0)}𝑉_𝑗(𝜃)𝑑𝜃𝑡∈𝑡_𝑘,𝑡_𝑘+1,𝑘=0,1,2,….(3.30)
By induction argument, we arrive at 𝑉^∗𝑡_𝑘−𝑉^∗𝑡_𝑘−1≤𝛼𝑉^∗𝑡_𝑘−1+(𝛾+𝜆)^𝑡𝑘_𝑡𝑘−1𝑉^∗(𝑠)𝑑𝑠+ℎ𝜌𝑒^𝑛𝛾𝜏_𝑗=1^{𝑡𝑘−𝜏𝑗(𝑡𝑘)𝑡}_{𝑘−1−𝜏𝑗𝑡𝑘−1}𝑉⋮e^{𝛾(𝜃−𝑡0)}𝑉_𝑗(𝜃)𝑑𝜃,_∗𝑉𝑡₂−𝑉^∗𝑡₁≤𝛼𝑉^∗𝑡₁+(𝛾+𝜆)^𝑡2_𝑡1𝑉^∗(𝑠)𝑑𝑠+ℎ𝜌𝑒^𝑛𝛾𝜏_𝑗=1^{𝑡2−𝜏𝑗(𝑡2)𝑡}_{1−𝜏𝑗𝑡1}e^{𝛾(𝜃−𝑡0)}𝑉_𝑗(𝜃)𝑑𝜃,_∗𝑡₁−𝑉^∗𝑡₀≤𝛼𝑉^∗𝑡₀+(𝛾+𝜆)^𝑡1_𝑡0𝑉^∗(𝑠)𝑑𝑠+ℎ𝜌𝑒^𝑛𝛾𝜏_𝑗=1^{𝑡1−𝜏𝑗(𝑡1)𝑡}_{0−𝜏𝑗𝑡0}e^{𝛾(𝜃−𝑡0)}𝑉_𝑗(𝜃)𝑑𝜃.(3.31) Hence, 𝑉^∗(𝑡)−𝑉^∗𝑡₀≤+𝛼𝑉ℎ𝜌^∗𝑒𝑡_{0𝑛𝛾𝜏}+(𝛾+𝜆)^𝑡𝑡₀𝑉^∗(𝑠)𝑑𝑠_𝑗=1^{𝑡−𝜏𝑗𝑡(𝑡)}_{0−𝜏𝑗𝑡0}≤𝛼𝑉_∗e^{𝛾(𝜃−𝑡0)}𝑉_𝑗(𝜃)𝑑𝜃+𝛼_{𝑡0<𝑡𝑘<𝑡}𝑉𝑡_𝑘+ℎ𝜌𝑒𝑡_{0𝑛𝛾𝜏}+(𝛾+𝜆+ℎ𝜌𝑒^𝛾𝜏)^𝑡𝑡₀𝑉^∗(𝑠)𝑑𝑠_𝑗=1^𝑡0_{𝑡0−𝜏𝑗𝑡0}e^{𝛾(𝜃−𝑡0)}𝑉_𝑗(𝜃)𝑑𝜃+𝛼_{𝑡0<𝑡𝑘<𝑡}𝑉𝑡_𝑘𝑡∈𝑡_𝑘,𝑡_𝑘+1,𝑘=0,1,2,….(3.32)
Introducing ℎ𝜌𝑒^𝛾𝜏∑^𝑛𝑗=1∫^𝑡0_{𝑡0−𝜏𝑗(𝑡0)}e^{𝛾(𝜃−𝑡0)}𝑉_𝑗(𝜃)𝑑𝜃≤𝜏ℎ𝜌𝑒^𝛾𝜏‖𝜑‖^2Ω as shown in the proof of Theorem 3.1 into (3.32), the expression becomes 𝑉^∗(𝑡)−𝑉^∗𝑡₀≤+𝛼𝑉𝛼^∗𝑡₀_{𝑡0<𝑡𝑘<𝑡}+𝜏ℎ𝜌𝑒^𝛾𝜏‖𝜑‖^2Ω+(𝛾+𝜆+ℎ𝜌𝑒^𝛾𝜏)^𝑡𝑡₀𝑉^∗(𝑠)𝑑𝑠𝑉𝑡_𝑘𝑡∈𝑡_𝑘,𝑡_𝑘+1,𝑘=0,1,2….(3.33)
It then results from Lemma 2.2 that 𝑉^∗(𝑡)=≤((𝛼𝛼++11))𝑉𝑉^∗_∗𝑡₀+𝜏ℎ𝜌𝑒^𝛾𝜏‖𝜑‖^2Ω_{𝑡0<𝑡𝑘<𝑡}(1+𝛼)exp(𝛾+𝜆+ℎ𝜌𝑒^𝛾𝜏)𝑡−𝑡₀𝑡₀+𝜏ℎ𝜌𝑒^𝛾𝜏‖𝜑‖^2Ω(1+𝛼)^𝑘exp(𝛾+𝜆+ℎ𝜌𝑒^𝛾𝜏)𝑡−𝑡₀,𝑡≥𝑡₀.(3.34)
On the other hand, since inf_𝑘=1,2,…(𝑡_𝑘−𝑡_𝑘−1)>𝜇, one has 𝑘<(𝑡_𝑘−𝑡₀)/𝜇. Thereby, (1+𝛼)^𝑘<𝑒𝑥𝑝ln(1+𝛼)𝜇𝑡^𝑘−𝑡₀<𝑒𝑥𝑝ln(1+𝛼)𝜇𝑡−𝑡⁰.(3.35)
And (3.34) can be rewritten as 𝑉^∗(𝑡)≤(𝛼+1)𝑉^∗𝑡₀+𝜏ℎ𝜌𝑒^𝛾𝜏‖𝜑‖^2Ωexp𝛾+𝜆+ℎ𝜌𝑒^𝛾𝜏+ln(1+𝛼)𝜇𝑡−𝑡⁰(3.36) which implies ‖‖𝑢𝑡,𝑥;𝑡₀,𝜑‖‖_Ω≤√(𝛼+1+𝜏ℎ𝜌𝑒^𝛾𝜏)‖𝜑‖_Ωexp12𝜆+ℎ𝜌𝑒^𝛾𝜏+ln(1+𝛼)𝜇𝑡−𝑡⁰,𝑡≥𝑡₀.(3.37) The proof is completed.

Remark 3.4. Theorem 3.1 is in fact the special case of Theorem 3.3 by choosing 𝛼=0. Due to Lemma 2.4, we know 2^𝑛_𝑗=1𝑏_𝑖𝑗_Ω𝑢_𝑖(𝑡,𝑥)𝑓𝑢_𝑗(𝑡,𝑥)d𝑥≤^𝑛_𝑗=1_Ω𝜀¹𝑏^2𝑖𝑗𝑢^2𝑖(𝑡,𝑥)+𝑙^2𝑗𝜀₁2𝑢^2𝑖(𝑡,𝑥)d𝑥,_𝑛_𝑗=1𝑐_𝑖𝑗_Ω𝑢_𝑖(𝑡,𝑥)𝑓𝑢_𝑗𝑡−𝜏_𝑗,𝑥d𝑥≤^𝑛_𝑗=1_Ω𝜀²𝑐^2𝑖𝑗𝑢^2𝑖(𝑡,𝑥)+𝑙^2𝑗𝜀₂𝑢^2𝑖𝑡−𝜏_𝑗,𝑥d𝑥(3.38) hold for any 𝜀₁,𝜀₂>0.
In the sequel, we follow the same procedures as in Theorems 3.1 and 3.3 to find the following theorems.

Theorem 3.5. Provided that(1)for 𝑥=(𝑥₁,…,𝑥_𝑚)^𝑇∈Ω(𝑚≥3), there exists a constant 𝛽 such that |𝑥|²=∑^𝑚𝑠=1𝑥^2𝑠<𝛽. in addition, there exists a constant 𝐷>0 such that 𝐷_𝑖𝑠=𝐷_𝑖𝑠(𝑡,𝑥,𝑢)≥𝐷>0; denote 𝐷(𝑚−2)²/2𝛽=𝜒,(2)𝑃_𝑖𝑘(𝑢_𝑖(𝑡_𝑘,𝑥))=−𝜃_𝑖𝑘𝑢_𝑖(𝑡_𝑘,𝑥), 0≤𝜃_𝑖𝑘≤2,(3)there exist constants 𝛾 and 𝜀₁,𝜀₂>0 such that 𝛾+𝜆+ℎ𝜌𝑒^𝛾𝜏>0 and 𝜆+ℎ𝜌𝑒^𝛾𝜏<0, where 𝜆=max_{𝑖=1,…,𝑛}(−𝜒−2𝑎_𝑖+∑^𝑛𝑗=1(𝜀₁𝑏^2𝑖𝑗+𝜀₂𝑐^2𝑖𝑗))+(𝑛/𝜀₁)max_{𝑖=1,…,𝑛}(𝑙^2𝑖) and 𝜌=(𝑛/𝜀₂)max_{𝑖=1,…,𝑛}(𝑙^2𝑖), then, the equilibrium point 𝑢=0 of problem ((2.5)–(2.8)) is globally exponentially stable with convergence rate –(𝜆+ℎ𝜌𝑒^𝛾𝜏)/2.

Theorem 3.6. Assume that(1)for 𝑥=(𝑥₁,…,𝑥_𝑚)^𝑇∈Ω(𝑚≥3), there exists a constant 𝛽 such that |𝑥|²=∑^𝑚𝑠=1𝑥^2𝑠<𝛽; In addition, there exists a constant 𝐷>0 such that 𝐷_𝑖𝑠=𝐷_𝑖𝑠(𝑡,𝑥,𝑢)≥𝐷>0; denote 𝐷(𝑚−2)²/2𝛽=𝜒,(2)𝑃_𝑖𝑘(𝑢_𝑖(𝑡_𝑘,𝑥))=−𝜃_𝑖𝑘𝑢_𝑖(𝑡_𝑘,𝑥), 1−√1+𝛼≤𝜃_𝑖𝑘≤1+√1+𝛼, 𝛼≥0,(3)inf_𝑘=1,2,…(𝑡_𝑘−𝑡_𝑘−1)>𝜇,(4)there exist constants 𝛾 and 𝜀₁,𝜀₂>0 such that 𝛾+𝜆+ℎ𝜌𝑒^𝛾𝜏>0 and 𝜆+ℎ𝜌𝑒^𝛾𝜏+ln(1+𝛼)/𝜇<0, where 𝜆=max_{𝑖=1,…,𝑛}(−𝜒−2𝑎_𝑖+∑^𝑛_𝑗=1(𝜀₁𝑏^2𝑖𝑗+𝜀₂𝑐^2𝑖𝑗))+(𝑛/𝜀₁)max_{𝑖=1,…,𝑛}(𝑙^2𝑖) and 𝜌=(𝑛/𝜀₂)max_{𝑖=1,…,𝑛}(𝑙^2𝑖).
Then, the equilibrium point 𝑢=0 of problem ((2.5)–(2.8)) is globally exponentially stable with convergence rate −(1/2)(𝜆+ℎ𝜌𝑒^𝛾𝜏+ln(1+𝛼)/𝜇).
Further, on the condition that |𝑃_𝑖𝑘(𝑢_𝑖(𝑡_𝑘,𝑥))|≤𝜃_𝑖𝑘|𝑢_𝑖(𝑡_𝑘,𝑥)|, where 𝜃^2𝑖𝑘<(𝛼−1)/2 and 𝛼≥1, we obtain, for 𝑡=𝑡_𝑘 (𝑘=1,2,…), 𝑢^2𝑖𝑡_𝑘+0,𝑥−𝑢^2𝑖𝑡_𝑘,𝑥=≤2𝑃𝑢_𝑖𝑘𝑖𝑢_𝑖𝑡_𝑘,𝑥+𝑢_𝑖𝑡_𝑘,𝑥²−𝑢^2𝑖𝑡_𝑘,𝑥≤2+2𝑡_𝑘𝜃,𝑥_2𝑖𝑘²+2𝑃_𝑖𝑘𝑢_𝑖𝑡_𝑘,𝑥²−𝑢^2𝑖𝑡_𝑘,𝑥𝑢_𝑖𝑡_𝑘,𝑥²−𝑢^2𝑖𝑡_𝑘,𝑥≤𝛼𝑢^2𝑖𝑡_𝑘,𝑥.(3.39)
Identical with the proof of Theorem 3.3, we present the theorem as follows.

Theorem 3.7. Assume that(1)for 𝑥=(𝑥₁,…,𝑥_𝑚)^𝑇∈Ω(𝑚≥3), there exists a constant 𝛽 such that |𝑥|²=∑^𝑚𝑠=1𝑥^2𝑠<𝛽. in addition, there exists a constant 𝐷>0 such that 𝐷_𝑖𝑠=𝐷_𝑖𝑠(𝑡,𝑥,𝑢)≥𝐷>0. Denote 𝐷(𝑚−2)²/2𝛽=𝜒,(2)|𝑃_𝑖𝑘(𝑢_𝑖(𝑡_𝑘,𝑥))|≤𝜃_𝑖𝑘|𝑢_𝑖(𝑡_𝑘,𝑥)|, where 𝜃^2𝑖𝑘≤(𝛼−1)/2 and 𝛼≥1,(3)inf_𝑘=1,2,…(𝑡_𝑘−𝑡_𝑘−1)>𝜇,(4)there exist constants𝛾and𝜀₁,𝜀₂>0such that𝛾+𝜆+ℎ𝜌𝑒^𝛾𝜏>0and𝜆+ℎ𝜌𝑒^𝛾𝜏+ln(1+𝛼)/𝜇<0,where𝜆=max_{𝑖=1,…,𝑛}(−𝜒−2𝑎_𝑖+∑^𝑛𝑗=1(𝜀₁𝑏^2𝑖𝑗+𝜀₂𝑐^2𝑖𝑗))+(𝑛/𝜀₁)max_{𝑖=1,…,𝑛}(𝑙_𝑖2)and𝜌=(𝑛/𝜀₂)max_{𝑖=1,…,𝑛}(𝑙_𝑖2).
Then, the equilibrium point 𝑢=0 of problem ((2.5)–(2.8)) is globally exponentially stable with convergence rate −1/2(𝜆+ℎ𝜌𝑒^𝛾𝜏+ln(1+𝛼)/𝜇).

Remark 3.8. Different from Theorems 3.1–3.6, the impulsive part in Theorem 3.7 could be nonlinear and this will be of more applicability. Actually, Theorems 3.1–3.6 can be regarded as the special cases of Theorem 3.7.

4. Examples

Example 4.1. Consider the following impulsive reaction-diffusion delayed neural network: 𝜕𝑢_𝑖(𝑡,𝑥)𝜕𝑡=^𝑚_𝑠=1𝜕𝜕𝑥_𝑠𝐷^𝑖𝑠𝜕𝑢_𝑖(𝑡,𝑥)𝜕𝑥_𝑠−𝑎^𝑖𝑢_𝑖(𝑡,𝑥)+^𝑛_𝑗=1𝑏_𝑖𝑗𝑓_𝑗𝑢_𝑗(𝑡,𝑥)+^𝑛_𝑗=1𝑡≥0,𝑡≠𝑡_𝑘𝑐_𝑖𝑗𝑓_𝑗𝑢_𝑗𝑡−𝜏_𝑗(𝑡),𝑥,𝑥∈Ω,𝑘=1,2,…,𝑖=1,…,𝑛(4.1) with the impulsive effects characterized by 𝑢₁𝑡_𝑘+0,𝑥=𝑢₁𝑡_𝑘,𝑥+1.343𝑢₁𝑡_𝑘,𝑥,𝑢₂𝑡_𝑘+0,𝑥=𝑢₂𝑡_𝑘𝑘,𝑥=+1,12.,3…43,𝑢₂𝑥𝑡_𝑘∈,𝑥Ω(4.2) and initial condition (2.3) and Dirichlet condition (2.4), where 𝑛=2, 𝑚=4, Ω={(𝑥₁,…,𝑥₄)^𝑇|∑^4𝑖=1𝑥^2𝑖<4}, 𝑎₁=𝑎₂=6.5, (𝐷_𝑖𝑠)_2×4=^{1.22.32.53.11.83.22.73.4}, (𝑏_𝑖𝑗)_2×2=^{−0.231.3−0.13}, (𝑐_𝑖𝑗)_2×2=^{−0.1−0.20.1−0.3}, 𝑓_𝑗(𝑢_𝑗)=(1/4)(|𝑢_𝑗+1|−|𝑢_𝑗−1|), 0≤𝜏_𝑗(𝑡)≤0.5, and 𝜏^•_𝑗(𝑡)<0. For 𝛽=4 and 𝐷=1.2, we compute 𝜒=0.6. This, together with the chosen 𝑙_𝑖=1/2, yields 𝜆=max_{𝑖=1,…,𝑛}−𝜒−2𝑎^𝑖+^𝑛_𝑗=1𝑏^2𝑖𝑗+𝑐^2𝑖𝑗+𝑛max_{𝑖=1,…,𝑛}𝑙^2𝑖=−4,𝜌=𝑛max_{𝑖=1,…,𝑛}𝑙^2𝑖=12.(4.3)
By selecting 𝛾=2.6, 𝜏=0.5 and ℎ=1, we estimate 𝛾+𝜆+ℎ𝜌𝑒^𝛾𝜏=2.6−4+12𝑒^1.3>0,𝜆+ℎ𝜌𝑒^𝛾𝜏=−4+12𝑒^1.3<0.(4.4)
According to Theorem 3.1, we therefore conclude that the system in Example 4.1 is globally exponential stable.

Example 4.2. Consider the following impulsive reaction-diffusion delayed neural network: 𝜕𝑢_𝑖(𝑡,𝑥)𝜕𝑡=^𝑚_𝑠=1𝜕𝜕𝑥_𝑠𝐷^𝑖𝑠𝜕𝑢_𝑖(𝑡,𝑥)𝜕𝑥_𝑠−𝑎^𝑖𝑢_𝑖(𝑡,𝑥)+^𝑛_𝑗=1𝑏_𝑖𝑗𝑓_𝑗𝑢_𝑗(𝑡,𝑥)+^𝑛_𝑗=1𝑡≥0,𝑡≠𝑡_𝑘𝑐_𝑖𝑗𝑓_𝑗𝑢_𝑗𝑡−𝜏_𝑗(𝑡),𝑥,𝑥∈Ω,𝑘=1,2,…,𝑖=1,…,𝑛(4.5) with the impulsive effects featured by 𝑢₁𝑡_𝑘+0,𝑥=𝑢₁𝑡_𝑘,𝑥+arctan0.5𝑢₁𝑡_𝑘,𝑥,𝑢₂𝑡_𝑘+0,𝑥=𝑢₂𝑡_𝑘,𝑥+ar𝑘ct=an1,02.,5…𝑢₂,𝑡𝑥_𝑘,∈𝑥Ω(4.6) and initial condition (2.3) and Dirichlet condition (2.4), where 𝑛=2, 𝑚=4, Ω={(𝑥₁,…,𝑥₄)^𝑇|∑^4𝑖=1𝑥^2𝑖<4}, 𝑎₁=𝑎₂=6.5, (𝐷_𝑖𝑠)_2×4=^{1.22.32.53.11.83.22.73.4}, (𝑏_𝑖𝑗)_2×2=^{−0.231.3−0.13}, (𝑐_𝑖𝑗)_2×2=^{−0.1−0.20.1−0.3}, 𝑓_𝑗(𝑢_𝑗)=1/4(|𝑢_𝑗+1|−|𝑢_𝑗−1|), 0≤𝜏_𝑗(𝑡)≤0.5, 𝜏^•_𝑗(𝑡)<0, and inf_𝑘=1,2,…(𝑡_𝑘−𝑡_𝑘−1)>1. For 𝛽=4 and 𝐷=1.2, we compute 𝜒=0.6. This, together with 𝑙_𝑖=1/2 and 𝜀₁=𝜀₂=1, yields 𝜌=𝑛𝜀₂max_{𝑖=1,…,𝑛}𝑙^2𝑖=12,𝜆=max^{𝑖=1,…,𝑛}−𝜒−2𝑎^𝑖+^𝑛_𝑗=1𝜀₁𝑏^2𝑖𝑗+𝜀₂𝑐^2𝑖𝑗+𝑛𝜀₁max_{𝑖=1,…,𝑛}𝑙^2𝑖=−4.(4.7)
Select 𝛼=1.5 by setting 𝜃_𝑖𝑘=0.5. Hence, we compute by letting 𝜇=1, 𝛾=3, 𝜏=0.5, and ℎ=1 that 𝛾+𝜆+ℎ𝜌𝑒^𝛾𝜏=3−4+12𝑒^1.5>0,𝜆+ℎ𝜌𝑒^𝛾𝜏+ln(1+𝛼)𝜇=−4+21𝑒^1.5+ln2.5<0.(4.8)
It is then concluded from Theorem 3.7 that the system in Example 4.2 is globally exponentially stable.

Acknowledgment

The work is supported by National Natural Science Foundation of China under Grant 60904028.