Stability of Impulsive Neural Networks with Time-Varying and Distributed Delays

Abstract

This work is devoted to investigating the stability of impulsive cellular neural networks with time-varying and distributed delays. We use the new method of fixed point theory to obtain some new and concise sufficient conditions to ensure the existence and uniqueness of solution and the global exponential stability of trivial equilibrium. The presented algebraic criteria are easily checked and do not require the differentiability of delays.

1. Introduction

Since cellular neural networks (CNNs) were proposed by Chua and Yang in 1988 [1, 2], many researchers have put great effort into this subject due to their numerous successful applications in various fields such as optimization, linear and nonlinear programming, associative memory, pattern recognition, and computer vision.

Owing to the finite switching speed of amplifiers, there is no doubt that time delays exist in the communication and response of neurons. Moreover, as neural networks usually have a spatial extent due to the presences of a multitude of parallel pathways with a variety of axon sizes and lengths, there is a distribution of conduction velocities along these pathways and a distribution of propagation designed with discrete delays. Therefore, a more appropriate and ideal way is to incorporate continuously distributed delays with a result that a more effective model of cellular neural networks with time-varying and distributed delays proposed.

In fact, beside delay effects, stochastic and impulsive as well as diffusing effects are also likely to exist in neural networks. So far, there have been many results [3–11] on the study of dynamic behaviors of complex CNNs such as impulsive delayed reaction-diffusion CNNs and stochastic delayed reaction-diffusion CNNs. Summing up the existing researches on the stability of complex CNNs, we see that the primary method is Lyapunov theory. However, there are also lots of difficulties in the applications of corresponding theories to specific problems. It is therefore necessary to seek some new methods to deal with the stability in order to overcome those difficulties.

Recently, it is inspiring that Burton and other authors have applied the fixed point theory to investigate the stability of deterministic systems and obtained some more applicable conclusions, for example, see the monograph [12] and the work in [13–24]. In addition, more recently, there have been a few papers where the fixed point theory is employed to investigate the stability of stochastic (delayed) differential equations, for instance, see [25–31]. Precisely, in [26–28], Luo used the fixed point theory to study the exponential stability of mild solutions for stochastic partial differential equations with bounded delays and with infinite delays. In [29, 30], Sakthivel used the fixed point theory to discuss the asymptotic stability in pth moment of mild solutions to nonlinear impulsive stochastic partial differential equations with bounded delays and with infinite delays. In [31], Luo used the fixed point theory to study the exponential stability of stochastic Volterra-Levin equations. We wonder if we can obtain some new and more applicable stability criteria of complex CNNs by applying the fixed point theory.

With this motivation, in this paper, we aim to discuss the global exponential stability of impulsive CNNs with time-varying and distributed delays. It is worth noting that our research technique is based on the contraction mapping principle rather than the usual method of Lyapunov theory. We deal with, by employing the fixed point theorem, the existence and uniqueness of solution and the global exponential stability of trivial equilibrium at the same time, for which Lyapunov method feels helpless. The obtained stability criteria are easily checked and do not require the differentiability of delays.

2. Preliminaries

Let 𝑅^𝑛 denote the n-dimensional Euclidean space and ‖⋅‖ represent the Euclidean norm 𝒩≜{1,2,…,𝑛} and 𝑅₊=[0,∞). 𝐶[𝑋,𝑌] corresponds to the space of continuous mappings from the topological space 𝑋 to the topological space 𝑌.

In this paper, we consider the following impulsive cellular neural networks with time-varying and distributed delays: d𝑥_𝑖(𝑡)d𝑡=−𝑎_𝑖𝑥_𝑖(𝑡)+^𝑛_𝑗=1𝑏_𝑖𝑗𝑓_𝑗𝑥_𝑗(𝑡+_𝑛𝑗=1)𝑐_𝑖𝑗𝑔_𝑗𝑥_𝑗𝑡−𝜏_𝑖𝑗(𝑡)+^𝑛_𝑗=1𝑑_𝑖𝑗0^𝜌(𝑡)𝜎_𝑗𝑥_𝑗(𝑡−𝜃)d𝜃𝑡≥0,𝑡≠𝑡_𝑘,Δ𝑥_𝑖𝑡_𝑘=𝑥_𝑖𝑡_𝑘+0−𝑥_𝑖𝑡_𝑘=𝐼_𝑖𝑘𝑥_𝑖𝑡_𝑘,𝑘=1,2,…, (1)(2) where 𝑖∈𝒩 and 𝑛 is the number of neurons in the neural network. 𝑥_𝑖(𝑡) corresponds to the state of the 𝑖th neuron at time 𝑡. 𝑓_𝑗, 𝑔_𝑗, and 𝜎_𝑗 denote the activation functions, respectively. The constant 𝑎_𝑖>0 represents the rate with which the 𝑖th neuron will reset its potential to the resting state when disconnected from the network and external inputs. The constants 𝑏_𝑖𝑗, 𝑐_𝑖𝑗, and 𝑑_𝑖𝑗 represent the connection weights of the jth neuron to the 𝑖th neuron, respectively. 𝜏_𝑖𝑗(𝑡) and 𝜌(𝑡) correspond to the transmission delays meeting 0≤𝜏_𝑖𝑗(𝑡)≤𝜏 (𝜏=constant) and 0≤𝜌(𝑡)≤𝜌 (𝜌=constant). The fixed impulsive moments 𝑡_𝑘 (𝑘=1,2,…) satisfy 0=𝑡₀<𝑡₁<𝑡₂<⋯ and lim_𝑘→∞𝑡_𝑘=∞. 𝑥_𝑖(𝑡_𝑘+0) and 𝑥_𝑖(𝑡_𝑘−0) stand for the right-hand and left-hand limits of 𝑥_𝑖(𝑡) at time 𝑡_𝑘, respectively. 𝐼_𝑖𝑘(𝑥_𝑖(𝑡_𝑘)) shows the impulsive perturbation of the 𝑖th neuron at the impulsive moment 𝑡_𝑘.

Throughout this paper, we always assume that 𝑓_𝑖(0)=𝑔_𝑖(0)=𝜎_𝑖(0)=𝐼_𝑖𝑘(0)=0 for 𝑖∈𝒩 and 𝑘=1,2,…. Thereby, problems (1) and (2) admit a trivial equilibrium 𝐱=0.

Denote by 𝐱(𝑡)≜𝐱(𝑡;𝑠,𝜑)=(𝑥₁(𝑡;𝑠,𝜑₁),…,𝑥_𝑛(𝑡;𝑠,𝜑_𝑛))^𝑇∈𝑅^𝑛 the solution to (1) and (2) with the initial condition 𝑥_𝑖(𝑠)=𝜑_𝑖(𝑠),−𝑚^∗≤𝑠≤0,𝑖∈𝒩, (3) where 𝑚^∗=max{𝜏,𝜌}, 𝜑_𝑖(𝑠)∈𝐶[[−𝑚^∗,0],𝑅] and 𝜑(𝑠)=(𝜑₁(𝑠),…,𝜑_𝑛(𝑠))^𝑇∈𝑅^𝑛.

The solution 𝐱(𝑡)≜𝐱(𝑡;𝑠,𝜑)∈𝑅^𝑛 to (1)–(3) is, for the time variable 𝑡, a piecewise continuous vector-valued function with the first-kind discontinuous points 𝑡_𝑘 (𝑘=1,2,…), where it is left-continuous; that is, the following relations are true: 𝑥_𝑖𝑡_𝑘−0=𝑥_𝑖𝑡_𝑘,𝑥_𝑖𝑡_𝑘+0=𝑥_𝑖𝑡_𝑘+𝐼_𝑖𝑘𝑥_𝑖𝑡_𝑘,𝑖∈𝒩,𝑘=1,2,…. (4)

Definition 1. The trivial equilibrium 𝐱=0 is said to be globally exponentially stable if for any initial condition 𝜑(𝑠)∈𝐶[[−𝑚^∗,0],𝑅^𝑛], there exists a pair of positive constants 𝜆 and 𝑀 such that 𝐱𝑡;𝑠,𝜑≤𝑀e^−𝜆𝑡,∀𝑡≥0. (5)
The consideration of this paper is based on the following fixed point theorem.

Theorem 2 (see [32]). Let Υ be a contraction operator on a complete metric space Θ, then there exists a unique point 𝜁∈Θ for which Υ(𝜁)=𝜁.

3. Main Results

In this section, we will, for (1)–(3), use the contraction mapping principle to prove the existence and uniqueness of the solution and the global exponential stability of trivial equilibrium all at once. Before proceeding, we firstly introduce some assumptions as follows.(A1) There exist nonnegative constants 𝑙_𝑗 such that for any 𝜂,𝜐∈𝑅, 𝑓_𝑗𝜂−𝑓_𝑗(𝜐)≤𝑙_𝑗𝜂−𝜐,𝑗∈𝒩. (6)(A2) There exist nonnegative constants 𝑘_𝑗 such that for any 𝜂,𝜐∈𝑅, 𝑔_𝑗𝜂−𝑔_𝑗(𝜐)≤𝑘_𝑗𝜂−𝜐,𝑗∈𝒩. (7)(A3) There exist nonnegative constants 𝑝_𝑗𝑘 such that for any 𝜂,𝜐∈𝑅, 𝐼_𝑗𝑘𝜂−𝐼_𝑗𝑘(𝜐)≤𝑝_𝑗𝑘𝜂−𝜐,𝑗∈𝒩,𝑘=1,2,…. (8)(A4) There exist nonnegative constants 𝜔_𝑗 such that for any 𝜂,𝜐∈𝑅, 𝜎_𝑗𝜂−𝜎_𝑗(𝜐)≤𝜔_𝑗𝜂−𝜐,𝑗∈𝒩. (9)

Let ℋ=ℋ₁×⋯×ℋ_𝑛, and let ℋ_𝑖 (𝑖∈𝒩) be the space embracing functions 𝜙_𝑖(𝑡)∶[−𝑚^∗,+∞)→𝑅, wherein 𝜙_𝑖(𝑡) satisfies the following: (1)𝜙_𝑖(𝑡) is continuous on 𝑡≠𝑡_𝑘 (𝑘=1,2,…),(2)lim_𝑡→𝑡⁻_𝑘𝜙_𝑖(𝑡) and lim_𝑡→𝑡⁺_𝑘𝜙_𝑖(𝑡) exist; moreover, lim_𝑡→𝑡⁻_𝑘𝜙_𝑖(𝑡)=𝜙_𝑖(𝑡_𝑘) for 𝑘=1,2,…,(3)𝜙_𝑖(𝑠)=𝜑_𝑖(𝑠) on 𝑠∈[−𝑚^∗,0],(4)e^𝛼𝑡𝜙_𝑖(𝑡)→0 as 𝑡→∞, where 𝛼=const and 0<𝛼<min_𝑖∈𝒩{𝑎_𝑖},

where 𝑡_𝑘 and 𝜑_𝑖(𝑠) are defined as shown in Section 2. Also ℋ is a complete metric space when it is equipped with a metric defined by 𝑑𝐪(𝑡),𝐡(𝑡)=^𝑛_𝑖=1sup_{𝑡≥−𝑚∗}𝑞_𝑖(𝑡)−ℎ_𝑖(𝑡), (10) where 𝐪(𝑡)=(𝑞₁(𝑡),…,𝑞_𝑛(𝑡))∈ℋ and 𝐡(𝑡)=(ℎ₁(𝑡),…,ℎ_𝑛(𝑡))∈ℋ.

Theorem 3. Assume that conditions (A1)–(A4) hold provided that (i)there exists a constant 𝜇 such that inf_𝑘=1,2,…{𝑡_𝑘−𝑡_𝑘−1}≥𝜇,(ii)there exist constants 𝑝_𝑖 such that 𝑝_𝑖𝑘≤𝑝_𝑖𝜇 for 𝑖∈𝒩 and 𝑘=1,2,…, (iii)∑^𝑛_𝑖=1{(1/𝑎_𝑖)max_𝑗∈𝒩|𝑏_𝑖𝑗𝑙_𝑗|+(1/𝑎_𝑖)max_𝑗∈𝒩|𝑐_𝑖𝑗𝑘_𝑗|+(𝜌/𝑎_𝑖)max_𝑗∈𝒩|𝜔_𝑗𝑑_𝑖𝑗|}+max_𝑖∈𝒩{𝑝_𝑖(𝜇+(1/𝑎_𝑖))}≜𝜒<1,
and then the trivial equilibrium 𝐱=0 is globally exponentially stable.

Proof. Multiplying both sides of (1) with e^𝑎𝑖𝑡 gives, for 𝑡>0 and 𝑡≠𝑡_𝑘, de^𝑎𝑖𝑡𝑥_𝑖(𝑡)=e^𝑎𝑖𝑡d𝑥_𝑖(𝑡)+𝑎_𝑖𝑥_𝑖(𝑡)e^𝑎𝑖𝑡d𝑡=e^{𝑎𝑖𝑡𝑛}_𝑗=1𝑏_𝑖𝑗𝑓_𝑗𝑥_𝑗(𝑡)+^𝑛_𝑗=1𝑐_𝑖𝑗𝑔_𝑗𝑥_𝑗𝑡−𝜏_𝑖𝑗(𝑡)+^𝑛_𝑗=1𝑑_𝑖𝑗0^𝜌(𝑡)𝜎_𝑗𝑥_𝑗(𝑡−𝜃)d𝜃d𝑡, (11) which yields after integrating from 𝑡_𝑘−1+𝜀 (𝜀>0) to 𝑡∈(𝑡_𝑘−1,𝑡_𝑘) (𝑘=1,2,…) that 𝑥_𝑖(𝑡)e^𝑎𝑖𝑡=𝑥_𝑖𝑡_𝑘−1+𝜀e^{𝑎𝑖(𝑡𝑘−1+𝜀)}+^𝑡_{𝑡𝑘−1+𝜀}e^{𝑎𝑖𝑠𝑛}_𝑗∑₌₁𝑏_𝑖𝑗𝑓_𝑗𝑥_𝑗(𝑠)+∑^𝑛_𝑗=1𝑐_𝑖𝑗𝑔_𝑗𝑥_𝑗𝑠−𝜏_𝑖𝑗(𝑠)+∑^𝑛_𝑗=1𝑑_𝑖𝑗∫₀^𝜌(𝑠)𝜎_𝑗𝑥_𝑗(𝑠−𝜃)d𝜃d𝑠. (12)
Letting 𝜀→0 in (12), we have, for 𝑡∈(𝑡_𝑘−1,𝑡_𝑘) (𝑘=1,2,…), 𝑥_𝑖(𝑡)e^𝑎𝑖𝑡=𝑥_𝑖𝑡_𝑘−1+0e^{𝑎𝑖𝑡𝑘−1}+^𝑡_𝑡𝑘−1e^{𝑎𝑖𝑠𝑛}_𝑗∑₌₁𝑏_𝑖𝑗𝑓_𝑗𝑥_𝑗(𝑠)+∑^𝑛_𝑗=1𝑐_𝑖𝑗𝑔_𝑗𝑥_𝑗𝑠−𝜏_𝑖𝑗(𝑠)+∑^𝑛_𝑗=1𝑑_𝑖𝑗∫₀^𝜌(𝑠)𝜎_𝑗𝑥_𝑗(𝑠−𝜃)d𝜃d𝑠. (13)
Setting 𝑡=𝑡_𝑘−𝜀 (𝜀>0) in (13), we get 𝑥_𝑖𝑡_𝑘−𝜀e^{𝑎𝑖(𝑡𝑘−𝜀)}=𝑥_𝑖𝑡_𝑘−1+0e^{𝑎𝑖𝑡𝑘−1}+^𝑡𝑘_𝑡^−𝜀_𝑘−1e^{𝑎𝑖𝑠𝑛}_𝑗∑₌₁𝑏_𝑖𝑗𝑓_𝑗𝑥_𝑗(𝑠)+∑^𝑛_𝑗=1𝑐_𝑖𝑗𝑔_𝑗𝑥_𝑗𝑠−𝜏_𝑖𝑗(𝑠)+∑^𝑛_𝑗=1𝑑_𝑖𝑗∫₀^𝜌(𝑠)𝜎_𝑗𝑥_𝑗(𝑠−𝜃)d𝜃d𝑠, (14) which generates by letting 𝜀→0𝑥_𝑖𝑡_𝑘−0e^{𝑎𝑖𝑡𝑘}=𝑥_𝑖𝑡_𝑘−1+0e^{𝑎𝑖𝑡𝑘−1}+^𝑡𝑘_𝑡𝑘−1e^{𝑎𝑖𝑠𝑛}_𝑗∑₌₁𝑏_𝑖𝑗𝑓_𝑗𝑥_𝑗(𝑠)+∑^𝑛_𝑗=1𝑐_𝑖𝑗𝑔_𝑗𝑥_𝑗𝑠−𝜏_𝑖𝑗(𝑠)+∑^𝑛_𝑗=1𝑑_𝑖𝑗∫₀^𝜌(𝑠)𝜎_𝑗𝑥_𝑗(𝑠−𝜃)d𝜃d𝑠. (15)
Noting 𝑥_𝑖(𝑡_𝑘−0)=𝑥_𝑖(𝑡_𝑘), (15) can be rearranged as 𝑥_𝑖𝑡_𝑘e^{𝑎𝑖𝑡𝑘}=𝑥_𝑖𝑡_𝑘−1+0e^{𝑎𝑖𝑡𝑘−1}+^𝑡𝑘_𝑡𝑘−1e^{𝑎𝑖𝑠𝑛}_𝑗∑₌₁𝑏_𝑖𝑗𝑓_𝑗𝑥_𝑗(𝑠)+∑^𝑛_𝑗=1𝑐_𝑖𝑗𝑔_𝑗𝑥_𝑗𝑠−𝜏_𝑖𝑗(𝑠)+∑^𝑛_𝑗=1𝑑_𝑖𝑗∫₀^𝜌(𝑠)𝜎_𝑗𝑥_𝑗(𝑠−𝜃)d𝜃d𝑠. (16)
Combining (13) and (16), we derive that 𝑥_𝑖(𝑡)e^𝑎𝑖𝑡=𝑥_𝑖𝑡_𝑘−1+0e^{𝑎𝑖𝑡𝑘−1}+^𝑡_𝑡𝑘−1e^{𝑎𝑖𝑠𝑛}_𝑗∑₌₁𝑏_𝑖𝑗𝑓_𝑗𝑥_𝑗(𝑠)+∑^𝑛_𝑗=1𝑐_𝑖𝑗𝑔_𝑗𝑥_𝑗𝑠−𝜏_𝑖𝑗(𝑠)+∑^𝑛_𝑗=1𝑑_𝑖𝑗∫₀^𝜌(𝑠)𝜎_𝑗𝑥_𝑗(𝑠−𝜃)d𝜃d𝑠 (17) is true for 𝑡∈(𝑡_𝑘−1,𝑡_𝑘] (𝑘=1,2,…). Hence, we get, for 𝑡∈(𝑡_𝑘−1,𝑡_𝑘] (𝑘=1,2,…), 𝑥_𝑖(𝑡)e^𝑎𝑖𝑡=𝑥_𝑖𝑡_𝑘−1+𝐼_{𝑖(𝑘−1)}𝑥_𝑖𝑡_𝑘−1e^{𝑎𝑖𝑡𝑘−1}+^𝑡_𝑡𝑘−1e^{𝑎𝑖𝑠𝑛}_𝑗∑₌₁𝑏_𝑖𝑗𝑓_𝑗𝑥_𝑗(𝑠)+∑^𝑛_𝑗=1𝑐_𝑖𝑗𝑔_𝑗𝑥_𝑗𝑠−𝜏_𝑖𝑗(𝑠)+∑^𝑛_𝑗=1𝑑_𝑖𝑗∫₀^𝜌(𝑠)𝜎_𝑗𝑥_𝑗(𝑠−𝜃)d𝜃d𝑠=𝑥_𝑖𝑡_𝑘−1e^{𝑎𝑖𝑡𝑘−1}+^𝑡_𝑡𝑘−1e^{𝑎𝑖𝑠𝑛}_𝑗∑₌₁𝑏_𝑖𝑗𝑓_𝑗𝑥_𝑗(𝑠)+∑^𝑛_𝑗=1𝑐_𝑖𝑗𝑔_𝑗𝑥_𝑗𝑠−𝜏_𝑖𝑗(𝑠)+∑^𝑛_𝑗=1𝑑_𝑖𝑗∫₀^𝜌(𝑠)𝜎_𝑗𝑥_𝑗(𝑠−𝜃)d𝜃d𝑠+𝐼_{𝑖(𝑘−1)}𝑥_𝑖𝑡_𝑘−1e^{𝑎𝑖𝑡𝑘−1}, (18) which results in 𝑥_𝑖𝑡_𝑘−1e^{𝑎𝑖𝑡𝑘−1}=𝑥_𝑖𝑡_𝑘−2e^{𝑎𝑖𝑡𝑘−2}+^𝑡𝑘−1_𝑡𝑘−2e^{𝑎𝑖𝑠𝑛}_𝑗∑₌₁𝑏_𝑖𝑗𝑓_𝑗𝑥_𝑗(𝑠)+∑^𝑛_𝑗=1𝑐_𝑖𝑗𝑔_𝑗𝑥_𝑗𝑠−𝜏_𝑖𝑗(𝑠)+∑^𝑛_𝑗=1𝑑_𝑖𝑗∫₀^𝜌(𝑠)𝜎_𝑗𝑥_𝑗(𝑠−𝜃)d𝜃d𝑠+𝐼_{𝑖(𝑘−2)}𝑥_𝑖𝑡_𝑘−2e^{𝑎𝑖𝑡𝑘−2}⋮𝑥_𝑖𝑡₂e^{𝑎𝑖𝑡2}=𝑥_𝑖𝑡₁e^{𝑎𝑖𝑡1}+^𝑡2_𝑡1e^{𝑎𝑖𝑠𝑛}_𝑗∑₌₁𝑏_𝑖𝑗𝑓_𝑗𝑥_𝑗(𝑠)+∑^𝑛_𝑗=1𝑐_𝑖𝑗𝑔_𝑗𝑥_𝑗𝑠−𝜏_𝑖𝑗(𝑠)+∑^𝑛_𝑗=1𝑑_𝑖𝑗∫₀^𝜌(𝑠)𝜎_𝑗𝑥_𝑗(𝑠−𝜃)d𝜃d𝑠+𝐼_𝑖1𝑥_𝑖𝑡₁e^{𝑎𝑖𝑡1},𝑥_𝑖𝑡₁e^{𝑎𝑖𝑡1}=𝜑_𝑖(0)+^𝑡1₀e^{𝑎𝑖𝑠𝑛}_𝑗∑₌₁𝑏_𝑖𝑗𝑓_𝑗𝑥_𝑗(𝑠)+∑^𝑛_𝑗=1𝑐_𝑖𝑗𝑔_𝑗𝑥_𝑗𝑠−𝜏_𝑖𝑗(𝑠)+∑^𝑛_𝑗=1𝑑_𝑖𝑗∫₀^𝜌(𝑠)𝜎_𝑗𝑥_𝑗(𝑠−𝜃)d𝜃d𝑠. (19)
We therefore conclude, for 𝑡>0, 𝑥_𝑖(𝑡)=𝜑_𝑖(0)e^{−𝑎𝑖𝑡}+e^{−𝑎𝑖𝑡𝑡}₀e^{𝑎𝑖𝑠𝑛}_𝑗∑₌₁𝑏_𝑖𝑗𝑓_𝑗𝑥_𝑗(𝑠)+∑^𝑛_𝑗=1𝑐_𝑖𝑗𝑔_𝑗𝑥_𝑗𝑠−𝜏_𝑖𝑗(𝑠+_𝑛)_𝑗∑₌₁𝑑_𝑖𝑗∫₀^𝜌(𝑠)𝜎_𝑗𝑥_𝑗(𝑠−𝜃)d𝜃d𝑠+e^{−𝑎𝑖𝑡}_{0<𝑡𝑘<𝑡}𝐼_𝑖𝑘𝑥_𝑖𝑡_𝑘e^{𝑎𝑖𝑡𝑘}. (20)
Note that 𝑥_𝑖(0)=𝜑_𝑖(0) in (20). We then define the following operator 𝜋 acting on ℋ, for 𝐲(𝑡)=(𝑦₁(𝑡),…,𝑦_𝑛(𝑡))∈ℋ: 𝜋(𝐲)(𝑡)=𝜋𝑦₁(𝑡),…,𝜋𝑦_𝑛(𝑡), (21) where 𝜋(𝑦_𝑖)(𝑡)∶[−𝑚^∗,+∞)→𝑅 (𝑖∈𝒩) obeys the rule as follows: 𝜋𝑦_𝑖(𝑡)=𝜑_𝑖(0)e^{−𝑎𝑖𝑡}+e^{−𝑎𝑖𝑡𝑡}₀e^{𝑎𝑖𝑠𝑛}_𝑗∑₌₁𝑏_𝑖𝑗𝑓_𝑗𝑦_𝑗(𝑠)+∑^𝑛_𝑗=1𝑐_𝑖𝑗𝑔_𝑗𝑦_𝑗𝑠−𝜏_𝑖𝑗(𝑠+_𝑛)_𝑗∑₌₁𝑑_𝑖𝑗∫₀^𝜌(𝑠)𝜎_𝑗𝑦_𝑗(𝑠−𝜃)d𝜃d𝑠+e^{−𝑎𝑖𝑡}_{0<𝑡𝑘<𝑡}𝐼_𝑖𝑘𝑦_𝑖𝑡_𝑘e^{𝑎𝑖𝑡𝑘} (22) on 𝑡≥0 and 𝜋(𝑦_𝑖)(𝑠)=𝜑_𝑖(𝑠) on 𝑠∈[−𝑚^∗,0].
In what follows, we will apply the contraction mapping principle to prove the existence and uniqueness of solution and the global exponential stability of trivial equilibrium at the same time. The subsequent proof can be divided into two steps.
Step 1. We need to prove that 𝜋(ℋ)⊂ℋ. For 𝑦_𝑖(𝑡)∈ℋ_𝑖 (𝑖∈𝒩), it is necessary to show that 𝜋(𝑦_𝑖)(𝑡)⊂ℋ_𝑖. As defined above, we see that 𝜋(𝑦_𝑖)(𝑠)=𝜑_𝑖(𝑠) on 𝑠∈[−𝑚^∗,0]. Owing to the continuity of 𝜑_𝑖(𝑠) on 𝑠∈[−𝑚^∗,0], we immediately know that 𝜋(𝑦_𝑖)(𝑡) is continuous on 𝑡∈[−𝑚^∗,0].
Choose a fixed time 𝑡>0, and it is then derived from (22) that 𝜋𝑦_𝑖(𝑡+𝑟)−𝜋𝑦_𝑖(𝑡)=𝑄₁+𝑄₂+𝑄₃+𝑄₄+𝑄₅,𝑡>0, (23) where, 𝑄₁=𝜑_𝑖(0)e^{−𝑎𝑖(𝑡+𝑟)}−𝜑_𝑖(0)e^{−𝑎𝑖𝑡},𝑄₂=e^{−𝑎𝑖(𝑡+𝑟)}₀^𝑡+𝑟e^{𝑎𝑖𝑠𝑛}_𝑗=1𝑏_𝑖𝑗𝑓_𝑗𝑦_𝑗(𝑠)d𝑠−e^{−𝑎𝑖𝑡𝑡}₀e^{𝑎𝑖𝑠𝑛}_𝑗=1𝑏_𝑖𝑗𝑓_𝑗𝑦_𝑗(𝑠)𝑄d𝑠,₃=e^{−𝑎𝑖(𝑡+𝑟)}₀^𝑡+𝑟e^{𝑎𝑖𝑠𝑛}_𝑗=1𝑐_𝑖𝑗𝑔_𝑗𝑦_𝑗𝑠−𝜏_𝑖𝑗(𝑠)d𝑠−e^{−𝑎𝑖𝑡𝑡}₀e^{𝑎𝑖𝑠𝑛}_𝑗=1𝑐_𝑖𝑗𝑔_𝑗𝑦_𝑗𝑠−𝜏_𝑖𝑗(𝑠)𝑄d𝑠,₄=e^{−𝑎𝑖(𝑡+𝑟)}₀^𝑡+𝑟e^{𝑎𝑖𝑠𝑛}_𝑗=1𝑑_𝑖𝑗0^𝜌(𝑠)𝜎_𝑗𝑦_𝑗(𝑠−𝜃)d𝜃d𝑠−e^{−𝑎𝑖𝑡𝑡}₀e^{𝑎𝑖𝑠𝑛}_𝑗=1𝑑_𝑖𝑗0^𝜌(𝑠)𝜎_𝑗𝑦_𝑗()𝑄𝑠−𝜃d𝜃d𝑠,₅=e^{−𝑎𝑖(𝑡+𝑟)}_{0<𝑡𝑘<()𝑡+𝑟}𝐼_𝑖𝑘𝑦_𝑖𝑡_𝑘e^{𝑎𝑖𝑡𝑘}−e^{−𝑎𝑖𝑡}_{0<𝑡𝑘<𝑡}𝐼_𝑖𝑘𝑦_𝑖𝑡_𝑘e^{𝑎𝑖𝑡𝑘}. (24)
Since 𝑦_𝑖(𝑡)∈ℋ_𝑖, we know that 𝑦_𝑖(𝑡) is continuous on 𝑡≠𝑡_𝑘 (𝑘=1,2,…); moreover, lim_𝑡→𝑡⁻_𝑘𝑦_𝑖(𝑡) and lim_𝑡→𝑡⁺_𝑘𝑦_𝑖(𝑡) exist, in addition, lim_𝑡→𝑡⁻_𝑘𝑦_𝑖(𝑡)=𝑦_𝑖(𝑡_𝑘).
Letting 𝑡≠𝑡_𝑘 (𝑘=1,2,…) in (23), it is easy to see that 𝑄_𝑖→0 as 𝑟→0 for 𝑖=1,…,5. Thus, 𝜋(𝑦_𝑖)(𝑡+𝑟)−𝜋(𝑦_𝑖)(𝑡)→0 as 𝑟→0 holds on 𝑡>0 and 𝑡≠𝑡_𝑘 (𝑘=1,2,…).
Letting 𝑡=𝑡_𝑘 (𝑘=1,2,…) in (23), it is not difficult to find that 𝑄_𝑖→0 as 𝑟→0 for 𝑖=1,…,4. Letting 𝑟<0 be small enough, we compute 𝑄₅=e^{−𝑎𝑖(𝑡𝑘+𝑟)}_{0<𝑡𝑚<𝑡𝑘+𝑟}𝐼_𝑖𝑚𝑦_𝑖𝑡_𝑚e^{𝑎𝑖𝑡𝑚}−e^{−𝑎𝑖𝑡𝑘}_{0<𝑡𝑚<𝑡𝑘}𝐼_𝑖𝑚𝑦_𝑖𝑡_𝑚e^{𝑎𝑖𝑡𝑚}=e^{−𝑎𝑖(𝑡𝑘+𝑟)}−e^{−𝑎𝑖𝑡𝑘}_{0<𝑡𝑚<𝑡𝑘}𝐼_𝑖𝑚𝑦_𝑖𝑡_𝑚e^{𝑎𝑖𝑡𝑚} (25) which implies lim_𝑟→0⁻𝑄₅=0. Letting 𝑟>0 be small enough, we have 𝑄₅=e^{−𝑎𝑖(𝑡𝑘+𝑟)}_{0<𝑡𝑚<𝑡𝑘+𝑟}𝐼_𝑖𝑚𝑦_𝑖𝑡_𝑚e^{𝑎𝑖𝑡𝑚}−e^{−𝑎𝑖𝑡𝑘}_{0<𝑡𝑚<𝑡𝑘}𝐼_𝑖𝑚𝑦_𝑖𝑡_𝑚e^{𝑎𝑖𝑡𝑚}=e^{−𝑎𝑖(𝑡𝑘+𝑟)}_{0<𝑡𝑚<𝑡𝑘}𝐼_𝑖𝑚𝑦_𝑖𝑡_𝑚e^{𝑎𝑖𝑡𝑚}+𝐼_𝑖𝑘𝑦_𝑖𝑡_𝑘e^{𝑎𝑖𝑡𝑘}−e^{−𝑎𝑖𝑡𝑘}_{0<𝑡𝑚<𝑡𝑘}𝐼_𝑖𝑚𝑦_𝑖𝑡_𝑚e^{𝑎𝑖𝑡𝑚}=e^{−𝑎𝑖(𝑡𝑘+𝑟)}−e^{−𝑎𝑖𝑡𝑘}_{0<𝑡𝑚<𝑡𝑘}𝐼_𝑖𝑚𝑦_𝑖𝑡_𝑚e^{𝑎𝑖𝑡𝑚}+e^{−𝑎𝑖(𝑡𝑘+𝑟)}𝐼_𝑖𝑘𝑦_𝑖𝑡_𝑘e^{𝑎𝑖𝑡𝑘}, (26) which implies lim_𝑟→0⁺𝑄₅=e^{−𝑎𝑖𝑡𝑘}𝐼_𝑖𝑘(𝑦_𝑖(𝑡_𝑘))e^{𝑎𝑖𝑡𝑘}.
According to the above discussion, we see that 𝜋(𝑦_𝑖)(𝑡)∶[−𝑚^∗,+∞)→𝑅 is continuous on ≠𝑡_𝑘(𝑘=1,2,…), while for 𝑡=𝑡_𝑘 (𝑘=1,2,…), lim_𝑡→𝑡⁻_𝑘𝜋(𝑦_𝑖)(𝑡) and lim_𝑡→𝑡⁺_𝑘𝜋(𝑦_𝑖)(𝑡) exist; moreover, lim_𝑡→𝑡⁻_𝑘𝜋(𝑦_𝑖)(𝑡)=𝜋(𝑦_𝑖)(𝑡_𝑘)≠lim_𝑡→𝑡⁺_𝑘𝜋(𝑦_𝑖)(𝑡).
Next, we will prove that e^𝛼𝑡𝜋(𝑦_𝑖)(𝑡)→0 as 𝑡→∞ for 𝑖∈𝒩. To begin with, we give the expression of e^𝛼𝑡𝜋(𝑦_𝑖)(𝑡) as follows: e^𝛼𝑡𝜋𝑦_𝑖(𝑡)=𝑊₁+𝑊₂+𝑊₃+𝑊₄+𝑊₅,𝑡>0, (27) Where?𝑊₁=𝜑_𝑖(0)e^{−(𝑎𝑖−𝛼)𝑡},?𝑊₂=e^𝛼𝑡e^{−𝑎𝑖𝑡}∫^𝑡₀e^𝑎𝑖𝑠∑^𝑛_𝑗=1𝑏_𝑖𝑗𝑓_𝑗(𝑦_𝑗(𝑠))d𝑠,?𝑊₅=e^𝛼𝑡e^{−𝑎𝑖𝑡}∑_{0<𝑡𝑘<𝑡}{𝐼_𝑖𝑘(𝑦_𝑖(𝑡_𝑘))e^{𝑎𝑖𝑡𝑘}},?𝑊₃=e^𝛼𝑡e^{−𝑎𝑖𝑡}∫^𝑡₀e^𝑎𝑖𝑠∑^𝑛_𝑗=1𝑐_𝑖𝑗𝑔_𝑗(𝑦_𝑗(𝑠−𝜏_𝑖𝑗(𝑠)))d𝑠, and?𝑊₄=e^𝛼𝑡e^{−𝑎𝑖𝑡}∫^𝑡₀e^𝑎𝑖𝑠∑^𝑛_𝑗=1𝑑_𝑖𝑗∫₀^𝜌(𝑠)𝜎_𝑗(𝑦_𝑗(𝑠−𝜃))d𝜃d𝑠.
First, it is obvious that lim_𝑡→∞𝑊₁=0 as 𝑎_𝑖−𝛼>0. Furthermore, for 𝑦_𝑗(𝑡)∈ℋ_𝑗 (𝑗∈𝒩), we see lim_𝑡→∞e^𝛼𝑡𝑦_𝑗(𝑡)=0. Then, for any 𝜀>0, there exists a 𝑇_𝑗>0 such that 𝑠≥𝑇_𝑗 implies |e^𝛼𝑠𝑦_𝑗(𝑠)|<𝜀. Choose 𝑇^∗=max_𝑗∈𝒩{𝑇_𝑗}. It is derived form (A1) that 𝑊₂≤e^𝛼𝑡e^{−𝑎𝑖𝑡𝑡}₀e^{𝑎𝑖𝑠𝑛}_𝑗=1𝑏_𝑖𝑗𝑙_𝑗𝑦_𝑗(𝑠)d𝑠=e^𝛼𝑡e^{−𝑎𝑖𝑡𝑡}₀e^𝑎𝑖𝑠e^{𝑛−𝛼𝑠}_𝑗=1𝑏_𝑖𝑗𝑙_𝑗e^𝛼𝑠𝑦_𝑗(𝑠)d𝑠=e^{−(𝑎𝑖−𝛼)𝑡𝑇∗}₀e^{(𝑎𝑖𝑛−𝛼)𝑠}_𝑗=1𝑏_𝑖𝑗𝑙_𝑗e^𝛼𝑠𝑦_𝑗(𝑠)d𝑠+e^{−(𝑎𝑖−𝛼)𝑡𝑡}_𝑇∗e^{(𝑎𝑖𝑛−𝛼)𝑠}_𝑗=1𝑏_𝑖𝑗𝑙_𝑗e^𝛼𝑠𝑦_𝑗(𝑠)d𝑠≤e^{−(𝑎𝑖𝑛−𝛼)𝑡}_𝑗=1𝑏_𝑖𝑗𝑙_𝑗sup_{𝑠∈0,𝑇∗}e^𝛼𝑠𝑦_𝑗(𝑠)^𝑇∗₀e^{(𝑎𝑖−𝛼)𝑠}d𝑠+𝜀^𝑛_𝑗=1𝑏_𝑖𝑗𝑙_𝑗e^{−(𝑎𝑖−𝛼)𝑡𝑡}_𝑇∗e^{(𝑎𝑖−𝛼)𝑠}d𝑠≤e^{−(𝑎𝑖𝑛−𝛼)𝑡}_𝑗=1𝑏_𝑖𝑗𝑙_𝑗sup_{𝑠∈0,𝑇∗}e^𝛼𝑠𝑦_𝑗(𝑠)^𝑇∗₀e^{(𝑎𝑖−𝛼)𝑠}+𝜀d𝑠𝑎_𝑖−𝛼^𝑛_𝑗=1𝑏_𝑖𝑗𝑙_𝑗, (28) which leads to 𝑊₂→0 as 𝑡→∞.
Similarly, for the given 𝜀>0 above, there also exists a 𝑇^′_𝑗>0 such that 𝑠≥𝑇^′_𝑗−𝜏 implies |e^𝛼𝑠𝑦_𝑗(𝑠)|<𝜀. Select 𝑇=max_𝑗∈𝒩{𝑇^′_𝑗}. It follows from (A2) that 𝑊₃≤e^𝛼𝑡e^{−𝑎𝑖𝑡𝑡}₀e^{𝑎𝑖𝑠𝑛}_𝑗=1𝑐_𝑖𝑗𝑘_𝑗𝑦_𝑗𝑠−𝜏_𝑖𝑗(𝑠)d𝑠≤e^{−(𝑎𝑖−𝛼)𝑡}×^𝑡₀e^𝑎𝑖𝑠e^{−𝛼{𝑠−𝜏}}×^𝑛_𝑗=1𝑐_𝑖𝑗𝑘_𝑗e^{𝛼[𝑠−𝜏𝑖𝑗(𝑠)]}𝑦_𝑗𝑠−𝜏_𝑖𝑗(𝑠)d𝑠=e^𝛼𝜏e^{−(𝑎𝑖−𝛼)𝑡}×₀𝑇e^{(𝑎𝑖𝑛−𝛼)𝑠}_𝑗=1𝑐_𝑖𝑗𝑘_𝑗e^{𝛼[𝑠−𝜏𝑖𝑗(𝑠)]}𝑦_𝑗𝑠−𝜏_𝑖𝑗(𝑠)d𝑠+e^𝛼𝜏e^{−(𝑎𝑖−𝛼)𝑡}×^𝑡𝑇e^{(𝑎𝑖𝑛−𝛼)𝑠}_𝑗=1𝑐_𝑖𝑗𝑘_𝑗e^{𝛼[𝑠−𝜏𝑖𝑗(𝑠)]}𝑦_𝑗𝑠−𝜏_𝑖𝑗(𝑠)d𝑠≤e^𝑛𝛼𝜏_𝑗=1𝑐_𝑖𝑗𝑘_𝑗sup_{𝑠∈−𝜏,}𝑇e^𝛼𝑠𝑦_𝑗(𝑠)×e^{−(𝑎𝑖−𝛼)𝑡}₀𝑇e^{(𝑎𝑖−𝛼)𝑠}d𝑠+e^𝛼𝜏𝜀^𝑛_𝑗=1𝑐_𝑖𝑗𝑘_𝑗e^{−(𝑎𝑖−𝛼)𝑡𝑡}𝑇e^{(𝑎𝑖−𝛼)𝑠}d𝑠≤e^𝑛𝛼𝜏_𝑗=1𝑐_𝑖𝑗𝑘_𝑗sup_{𝑠∈−𝜏,}𝑇e^𝛼𝑠𝑦_𝑗(𝑠)×e^{−(𝑎𝑖−𝛼)𝑡}₀𝑇e^{(𝑎𝑖−𝛼)𝑠}ed𝑠+^𝛼𝜏𝜀𝑎_𝑖−𝛼^𝑛_𝑗=1𝑐_𝑖𝑗𝑘_𝑗, (29) which results in 𝑊₃→0 as 𝑡→∞. In addition, it is derived from (A4) that 𝑊₄≤e^𝛼𝑡e^{−𝑎𝑖𝑡𝑡}₀e^{𝑎𝑖𝑠𝑛}_𝑗=1𝑑_𝑖𝑗^𝜌₀𝜔_𝑗𝑦_𝑗(𝑠−𝜃)d𝜃d𝑠=e^𝛼𝑡e^{−𝑎𝑖𝑡}×^𝑡₀e^𝑎𝑖𝑠e^{𝑛−𝛼𝑠}_𝑗=1𝑑_𝑖𝑗^𝜌₀e^𝛼𝜃𝜔_𝑗e^{𝛼(𝑠−𝜃)}𝑦_𝑗(𝑠−𝜃)d𝜃d𝑠≤e^𝛼𝑡e^{−𝑎𝑖𝑡}×^𝑡₀e^𝑎𝑖𝑠e^−𝛼𝑠×^𝑛_𝑗=1𝑑_𝑖𝑗sup_{𝜁∈𝑠−𝜌,𝑠}e^𝛼𝜁𝑦_𝑗(𝜁)^𝜌₀e^𝛼𝜃𝜔_𝑗d𝜃d𝑠. (30)
Since e^𝛼𝜁|𝑦_𝑗(𝜁)|→0 as 𝜁→∞, we know that, for any 𝜀>0, there exists a 𝑇_𝑗^′′>0 such that 𝜁>𝑇_𝑗^′′−𝜌 implies e^𝛼𝜁|𝑦_𝑗(𝜁)|<𝜀. Selecting 𝑇=max_𝑗∈𝒩{𝑇_𝑗^′′}, it follows from (30) that 𝑊₄≤e^{(𝛼−𝑎𝑖)𝑡}×^𝑇₀e^{(𝑎𝑖𝑛−𝛼)𝑠}_𝑗=1𝑑_𝑖𝑗sup_{𝜁∈𝑠−𝜌,𝑠}e^𝛼𝜁𝑦_𝑗(𝜁×_𝜌0)e^𝛼𝜃𝜔_𝑗d𝜃d𝑠+e^{(𝛼−𝑎𝑖)𝑡}×^𝑡_𝑇e^{(𝑎𝑖𝑛−𝛼)𝑠}_𝑗=1𝑑_𝑖𝑗sup_{𝜁∈𝑠−𝜌,𝑠}e^𝛼𝜁𝑦_𝑗(𝜁×_𝜌0)e^𝛼𝜃𝜔_𝑗≤ed𝜃d𝑠_𝛼𝜌𝛼^𝑛_𝑗=1𝑑_𝑖𝑗𝜔_𝑗sup_{𝜁∈−𝜌,𝑇}e^𝛼𝜁𝑦_𝑗(𝜁)×e^{(𝛼−𝑎𝑖)𝑡𝑇}₀e^{(𝑎𝑖−𝛼)𝑠}+ed𝑠_𝛼𝜌𝛼^𝑛_𝑗=1𝑑_𝑖𝑗𝜔_𝑗sup_{𝜁∈𝑇−𝜌,𝑡}e^𝛼𝜁𝑦_𝑗(𝜁)×e^{(𝛼−𝑎𝑖)𝑡𝑡}_𝑇e^{(𝑎𝑖−𝛼)𝑠}d𝑠≤e^{(𝛼−𝑎𝑖)𝑡}𝑒^𝛼𝜌𝛼^𝑛_𝑗=1𝑑_𝑖𝑗𝜔_𝑗sup_{𝜁∈−𝜌,𝑇}e^𝛼𝜁𝑦_𝑗(𝜁×_𝑇0)e^{(𝑎𝑖−𝛼)𝑠}d𝑠+𝜀^𝑛_𝑗=1𝑑_𝑖𝑗𝜔_𝑗e^𝛼𝜌𝛼𝑎_𝑖,−𝛼 (31) which yields 𝑊₄→0 as 𝑡→∞.
Furthermore, from (A3), we see that |𝐼_𝑖𝑘(𝑥_𝑖(𝑡_𝑘))|≤𝑝_𝑖𝑘|𝑦_𝑖(𝑡_𝑘)|. So, 𝑊₅≤e^𝛼𝑡e^{−𝑎𝑖𝑡}_{0<𝑡𝑘<𝑡}𝑝_𝑖𝑘𝑦_𝑖𝑡_𝑘e^{𝑎𝑖𝑡𝑘}. (32)
As 𝑦_𝑖(𝑡)∈ℋ_𝑖, we have lim_𝑡→∞e^𝛼𝑡𝑦_𝑖(𝑡)=0. Then, for any 𝜀>0, there exists a nonimpulsive point 𝑇_𝑖>0 such that 𝑠≥𝑇_𝑖 implies |e^𝛼𝑠𝑦_𝑖(𝑠)|<𝜀. It then follows from conditions (i) and (ii) that 𝑊₅≤e^𝛼𝑡e^{−𝑎𝑖𝑡}_{0<𝑡𝑘<𝑇𝑖}𝑝_𝑖𝑘𝑦_𝑖𝑡_𝑘e^{𝑎𝑖𝑡𝑘}+_{𝑇𝑖<𝑡𝑘<𝑡}𝑝_𝑖𝑘𝑦_𝑖𝑡_𝑘e^𝛼𝑡𝑘e^{(𝑎𝑖−𝛼)𝑡𝑘}≤e^𝛼𝑡e^{−𝑎𝑖𝑡}_{0<𝑡𝑘<𝑇𝑖}𝑝_𝑖𝑘𝑦_𝑖𝑡_𝑘e^{𝑎𝑖𝑡𝑘}+e^𝛼𝑡e^{−𝑎𝑖𝑡}𝑝_𝑖𝜀_{𝑇𝑖<𝑡𝑘<𝑡}𝜇e^{(𝑎𝑖−𝛼)𝑡𝑘}≤e^{−(𝑎𝑖−𝛼)𝑡}_{0<𝑡𝑘<𝑇𝑖}𝑝_𝑖𝑘𝑦_𝑖𝑡_𝑘e^{𝑎𝑖𝑡𝑘}+e^{−(𝑎𝑖−𝛼)𝑡}𝑝_𝑖𝜀_{𝑇𝑖<𝑡𝑟<𝑡𝑘}e^{(𝑎𝑖−𝛼)𝑡𝑟}𝑡_𝑟+1−𝑡_𝑟+𝜇e^{(𝑎𝑖−𝛼)𝑡𝑘}≤e^{−(𝑎𝑖−𝛼)𝑡}_{0<𝑡𝑘<𝑇𝑖}𝑝_𝑖𝑘𝑦_𝑖𝑡_𝑘e^{𝑎𝑖𝑡𝑘}+e^{−(𝑎𝑖−𝛼)𝑡}𝑝_𝑖𝜀^𝑡_𝑇𝑖e^{(𝑎𝑖−𝛼)𝑠}d𝑠+e^{−(𝑎𝑖−𝛼)𝑡}𝑝_𝑖𝜀𝜇e^{(𝑎𝑖−𝛼)𝑡}≤e^{−(𝑎𝑖−𝛼)𝑡}_{0<𝑡𝑘<𝑇𝑖}𝑝_𝑖𝑘𝑦_𝑖𝑡_𝑘e^{𝑎𝑖𝑡𝑘}+𝑝_𝑖𝜀𝑎_𝑖−𝛼+𝑝_𝑖𝜀𝜇, (33) which means that 𝑊₅→0 as 𝑡→∞.
Now, we can derive from (27) that e^𝛼𝑡𝜋(𝑦_𝑖)(𝑡)→0 as 𝑡→∞ for 𝑖∈𝒩. It is therefore concluded that 𝜋(𝑦_𝑖)(𝑡)⊂ℋ_𝑖 which results in 𝜋(ℋ)⊂ℋ.
Step 2. We need to prove that 𝜋 is contractive. For 𝐳=(𝑧₁(𝑡),…,𝑧_𝑛(𝑡))∈ℋ and 𝐲=(𝑦₁(𝑡),…,𝑦_𝑛(𝑡))∈ℋ, we estimate 𝜋𝑦_𝑖(𝑡)−𝜋𝑧_𝑖(𝑡)≤𝐽₁+𝐽₂+𝐽₃+𝐽₄, (34) where 𝐽₁=e^{−𝑎𝑖𝑡𝑡}₀e^{𝑎𝑖𝑠𝑛}_𝑗=1𝑏_𝑖𝑗𝑓_𝑗𝑦_𝑗(𝑠)−𝑓_𝑗𝑧_𝑗(𝑠𝐽)d𝑠,₂=e^{−𝑎𝑖𝑡𝑡}₀e^{𝑎𝑖𝑠𝑛}_𝑗=1𝑐_𝑖𝑗𝑔_𝑗𝑦_𝑗𝑠−𝜏_𝑖𝑗(𝑠)−𝑔_𝑗𝑧_𝑗𝑠−𝜏_𝑖𝑗(𝑠)𝐽d𝑠,₃=e^{−𝑎𝑖𝑡𝑡}₀e^{𝑎𝑖𝑠𝑛}_𝑗=1𝑑_𝑖𝑗0^𝜌(𝑠)𝜎_𝑗𝑦_𝑗(𝑠−𝜃)−𝜎_𝑗𝑧_𝑗(𝐽𝑠−𝜃)d𝜃d𝑠,₄=e^{−𝑎𝑖𝑡}_{0<𝑡𝑘<𝑡}e^{𝑎𝑖𝑡𝑘}𝐼_𝑖𝑘𝑦_𝑖𝑡_𝑘−𝐼_𝑖𝑘𝑧_𝑖𝑡_𝑘. (35)
Note that 𝐽₁≤e^{−𝑎𝑖𝑡𝑡}₀e^{𝑎𝑖𝑠𝑛}_𝑗=1𝑏_𝑖𝑗𝑙_𝑗𝑦_𝑗(𝑠)−𝑧_𝑗(𝑠)d𝑠≤max_𝑗∈𝒩𝑏_𝑖𝑗𝑙_𝑗^𝑛_𝑗=1sup_{[]𝑠∈0,𝑡}𝑦_𝑗(𝑠)−𝑧_𝑗(𝑠)×e^{−𝑎𝑖𝑡𝑡}₀e^𝑎𝑖𝑠≤1d𝑠𝑎_𝑖max_𝑗∈𝒩𝑏_𝑖𝑗𝑙_𝑗^𝑛_𝑗=1sup_{[]𝑠∈0,𝑡}𝑦_𝑗(𝑠)−𝑧_𝑗(𝑠,𝐽₂)≤e^{−𝑎𝑖𝑡𝑡}₀e^{𝑎𝑖𝑠𝑛}_𝑗=1𝑐_𝑖𝑗𝑘_𝑗𝑦_𝑗𝑠−𝜏_𝑖𝑗(𝑠)−𝑧_𝑗𝑠−𝜏_𝑖𝑗(𝑠)d𝑠≤max_𝑗∈𝒩𝑐_𝑖𝑗𝑘_𝑗^𝑛_𝑗=1sup_{[]𝜉∈−𝜏,𝑡}𝑦_𝑗(𝜉)−𝑧_𝑗(𝜉)×e^{−𝑎𝑖𝑡𝑡}₀e^𝑎𝑖𝑠≤1d𝑠𝑎_𝑖max_𝑗∈𝒩𝑐_𝑖𝑗𝑘_𝑗^𝑛_𝑗=1sup_{[]𝜉∈−𝜏,𝑡}𝑦_𝑗(𝜉)−𝑧_𝑗(𝜉,𝐽₃)≤e^{−𝑎𝑖𝑡𝑡}₀e^{𝑎𝑖𝑠𝑛}_𝑗=1𝑑_𝑖𝑗0^𝜌(𝑠)𝜔_𝑗𝑦_𝑗(𝑠−𝜃)−𝑧_𝑗(𝑠−𝜃)d𝜃d𝑠≤e^{−𝑎𝑖𝑡𝑡}₀e^{𝑎𝑖𝑠𝑛}_𝑗=1𝑑_𝑖𝑗sup_{𝜉∈𝑠−𝜌,𝑠}𝑦_𝑗(𝜉)−𝑧_𝑗(𝜉×_0𝜌(𝑠))𝜔_𝑗d𝜃d𝑠≤e^{−𝑎𝑖𝑡𝑡}₀e^{𝑎𝑖𝑠𝑛}_𝑗=1𝜔_𝑗𝑑_𝑖𝑗sup_{𝜉∈−𝜌,𝑡}𝑦_𝑗(𝜉)−𝑧_𝑗(𝜉)𝜌(𝑠)d𝑠≤max_𝑗∈𝒩𝜔_𝑗𝑑_𝑖𝑗^𝑛_𝑗=1sup_{𝜉∈−𝜌,𝑡}𝑦_𝑗(𝜉)−𝑧_𝑗(𝜉)×e^{−𝑎𝑖𝑡𝑡}₀e^𝑎𝑖𝑠𝜌(𝑠)≤𝜌d𝑠𝑎_𝑖max_𝑗∈𝒩𝜔_𝑗𝑑_𝑖𝑗^𝑛_𝑗=1sup_{𝜉∈−𝜌,𝑡}𝑦_𝑗(𝜉)−𝑧_𝑗(𝜉,𝐽₄)≤e^{−𝑎𝑖𝑡}_{0<𝑡𝑘<𝑡}e^{𝑎𝑖𝑡𝑘}𝑝_𝑖𝑘𝑦_𝑖𝑡_𝑘−𝑧_𝑖𝑡_𝑘≤𝑝_𝑖e^{−𝑎𝑖𝑡}sup_{[]𝑠∈0,𝑡}𝑦_𝑖(𝑠)−𝑧_𝑖(𝑠)_{0<𝑡𝑘<𝑡}e^{𝑎𝑖𝑡𝑘}𝜇≤𝑝_𝑖e^{−𝑎𝑖𝑡}sup_{[]𝑠∈0,𝑡}𝑦_𝑖(𝑠)−𝑧_𝑖(𝑠×_{0<𝑡𝑟<𝑡𝑘})e^{𝑎𝑖𝑡𝑟}𝑡_𝑟+1−𝑡_𝑟+e^{𝑎𝑖𝑡𝑘}𝜇≤𝑝_𝑖sup_{[]𝑠∈0,𝑡}𝑦_𝑖(𝑠)−𝑧_𝑖(𝑠)e^{−𝑎𝑖𝑡𝑡}₀e^𝑎𝑖𝑠d𝑠+e^𝑎𝑖𝑡𝜇≤𝑝_𝑖1𝜇+𝑎_𝑖sup_{[]𝑠∈0,𝑡}𝑦_𝑖(𝑠)−𝑧_𝑖(𝑠). (36)
It is then derived from (36) that sup_{[𝑡∈−𝑚∗],𝑇}𝜋𝑦_𝑖(𝑡)−𝜋𝑧_𝑖(𝑡≤1)𝑎_𝑖max_𝑗∈𝒩𝑏_𝑖𝑗𝑙_𝑗^𝑛_𝑗=1sup_{[𝑠∈−𝑚∗],𝑇}𝑦_𝑗(𝑠)−𝑧_𝑗(𝑠+1)𝑎_𝑖max_𝑗∈𝒩𝑐_𝑖𝑗𝑘_𝑗^𝑛_𝑗=1sup_{[𝜉∈−𝑚∗],𝑇}𝑦_𝑗(𝜉)−𝑧_𝑗(𝜉+𝜌)𝑎_𝑖max_𝑗∈𝒩𝜔_𝑗𝑑_𝑖𝑗^𝑛_𝑗=1sup_{[𝜉∈−𝑚∗],𝑇}𝑦_𝑗(𝜉)−𝑧_𝑗(𝜉)+𝑝_𝑖1𝜇+𝑎_𝑖sup_{[𝑠∈−𝑚∗],𝑇}𝑦_𝑖(𝑠)−𝑧_𝑖(𝑠), (37) which means that ^𝑛_𝑖=1sup_{[𝑡∈−𝑚∗],𝑇}𝜋𝑦_𝑖(𝑡)−𝜋𝑧_𝑖(𝑡)≤𝜒^𝑛_𝑗=1sup_{[𝑠∈−𝑚∗],𝑇}𝑦_𝑗(𝑠)−𝑧_𝑗(𝑠), (38) where 𝜒≜^𝑛_𝑖=11𝑎_𝑖max_𝑗∈𝒩𝑏_𝑖𝑗𝑙_𝑗+1𝑎_𝑖max_𝑗∈𝒩𝑐_𝑖𝑗𝑘_𝑗+𝜌𝑎_𝑖max_𝑗∈𝒩𝜔_𝑗𝑑_𝑖𝑗+max_𝑖∈𝒩𝑝_𝑖1𝜇+𝑎_𝑖. (39)
In view of condition (iii), we know that 𝜋 is a contraction mapping, and hence, there exists a unique fixed point 𝐲(⋅) of 𝜋 in ℋ which means that 𝐲^T(⋅) is the solution to (1)–(3) and e^𝛼𝑡‖𝐲^T(⋅)‖→0 as 𝑡→∞. This completes the proof.

Lemma 4. Assume conditions (A1)–(A4) hold. Provided that (i)inf_𝑘=1,2,…{𝑡_𝑘−𝑡_𝑘−1}≥1, (ii)there exist constants 𝑝_𝑖 such that 𝑝_𝑖𝑘≤𝑝_𝑖 for 𝑖∈𝒩 and 𝑘=1,2,…, (iii)∑^𝑛_𝑖=1{(1/𝑎_𝑖)max_𝑗∈𝒩|𝑏_𝑖𝑗𝑙_𝑗|+(1/𝑎_𝑖)max_𝑗∈𝒩|𝑐_𝑖𝑗𝑘_𝑗|+(𝜌/𝑎_𝑖)max_𝑗∈𝒩|𝜔_𝑗𝑑_𝑖𝑗|}+max_𝑖∈𝒩{𝑝_𝑖(1+(1/𝑎_𝑖))}≜𝜒<1,
then the trivial equilibrium 𝐱=0 is globally exponentially stable.

Proof. Lemma 4 is a direct conclusion by letting 𝜇=1 in Theorem 3.

Remark 5. In Theorem 3, we use the fixed point theorem to prove the existence and uniqueness of solution and the global exponential stability of trivial equilibrium all at once, while Lyapunov method fails to do this.

Remark 6. The presented sufficient conditions in Theorem 3 and Lemma 4 do not require even the differentiability of delays, let alone the monotone decreasing behavior of delays which is necessary in some relevant works.

4. Example

Consider the following two-dimensional impulsive cellular neural network with time-varying and distributed delays. d𝑥_𝑖(𝑡)d𝑡=−𝑎_𝑖𝑥_𝑖(𝑡)+²_𝑗=1𝑏_𝑖𝑗𝑓_𝑗𝑥_𝑗(𝑡+_2𝑗=1)𝑐_𝑖𝑗𝑔_𝑗𝑥_𝑗𝑡−𝜏_𝑖𝑗(𝑡)+²_𝑗=1𝑑_𝑖𝑗0^𝜌(𝑡)𝜎_𝑗𝑥_𝑗(𝑡−𝜃)d𝜃,𝑡≥0,𝑡≠𝑡_𝑘,Δ𝑥_𝑖𝑡_𝑘=𝑥_𝑖𝑡_𝑘+0−𝑥_𝑖𝑡_𝑘=arctan0.4𝑥_𝑖𝑡_𝑘,𝑘=1,2,…, (40) with the initial conditions 𝑥₁(𝑠)=cos(𝑠), 𝑥₂(𝑠)=sin(𝑠) on −𝑚^∗≤𝑠≤0, where 𝜏_𝑖𝑗(𝑡)=0.8+0.4cos(𝑡), 𝜌(𝑡)=0.5+0.3sin(𝑡), 𝑚^∗ is defined as shown in (3), 𝑎₁=𝑎₂=7, 𝑏_𝑖𝑗=0, 𝑐₁₁=0, 𝑐₁₂=1/7, 𝑐₂₁=−1/7, 𝑐₂₂=−1/7, 𝑑₁₁=3/7, 𝑑₁₂=2/7, 𝑑₂₁=0, 𝑑₂₂=1/7, 𝑓_𝑗(𝑠)=𝑔_𝑗(𝑠)=𝜎_𝑗(𝑠)=(|𝑠+1|−|𝑠−1|)/2, and 𝑡_𝑘=𝑡_𝑘−1+0.5𝑘.

It is easily to find that 𝜇=0.5, 𝑙_𝑗=𝑘_𝑗=𝜔_𝑗=1, and 𝑝_𝑖𝑘=0.4. Let 𝑝_𝑖=0.8 and compute ²_𝑖=11𝑎_𝑖max_𝑗∈𝒩𝑏_𝑖𝑗𝑙_𝑗+1𝑎_𝑖max_𝑗∈𝒩𝑐_𝑖𝑗𝑘_𝑗+𝜌𝑎_𝑖max_𝑗∈𝒩𝜔_𝑗𝑑_𝑖𝑗+max_𝑖∈𝒩𝑝_𝑖1𝜇+𝑎_𝑖<1. (41)

From Theorem 3, we conclude that the trivial equilibrium 𝐱=0 of this two-dimensional impulsive cellular neural network with time-varying and distributed delays is globally exponentially stable.

5. Conclusions

This article is a new attempt of applying the fixed point theory to the stability analysis of impulsive neural networks with time-varying and distributed delays, which is different from the existing relevant publications where Lyapunov theory is the main technique. From what have been discussed above, we see that the contraction mapping principle is effective for not only the investigation of the existence and uniqueness of solution but also for the stability analysis of trivial equilibrium. In the future, we will continue to explore the application of other kinds of fixed point theorems to the stability research of complex neural networks.

Acknowledgment

This work is supported by the National Natural Science Foundation of China under Grant no. 61174077 and Natural Science Foundation of Jiangsu of China under Grant no. BK2012682.