Exponential Stability of BAM Fuzzy Cellular Neural Networks with Time-Varying Delays in Leakage Terms and Impulses

Abstract

BAM fuzzy cellular neural networks with time-varying delays in leakage terms and impulses are considered. Some sufficient conditions for the exponential stability of the networks are established by using differential inequality techniques. The results of this paper are completely new and complementary to the previously known results. Finally, an example is given to demonstrate the effectiveness and conservativeness of our theoretical results.

1. Introduction

The bidirectional associative memory (BAM) neural networks were first introduced by Kosko [1–3]. It is a special class of recurrent neural networks that can store bipolar rector pairs. The BAM neural networks are composed of neurons arranged in two layers, the X-layer and Y-layer. Recently, many researchers have studied the dynamics of BAM neural networks with or without delays [4–15]. However, in mathematical modeling of real world problems, uncertainty or vagueness is unavoidable. In order to take vagueness into consideration, fuzzy theory is considered as a suitable method. In [16, 17], the authors first combined those operations with cellular neural networks (FCNNs). Some results have been reported on stability and periodicity of FCNNs. More recently, state estimation problem for the fuzzy BAM neural networks has been obtained in the paper [18, 19] and passivity criteria for the fuzzy BAM neural networks have been studied in the papers [20, 21].

Very recently, a leakage delay, which is the time delay in leakage term of the systems and a considerable factor affecting dynamics for the worse in the systems, is being put to use in the problem of stability for neural networks [22, 23]. However, so far, very little attention has been paid to neural networks with time delay in the leakage (or “forgetting”) term [24–30]. Such time delays in leakage terms are difficult to handle but have great impact on the dynamical behavior of neural networks.

In [31], the authors studied the following BAM fuzzy cellular network with time delay in leakage terms and discrete and unbounded distributed delays: 𝑥^′_𝑖(𝑡)=−𝑑_𝑖𝑥_𝑖𝑡−𝜎₁+^𝑚_𝑗=1𝑎_𝑖𝑗𝑓_𝑗𝑦_𝑗(𝑡+_𝑚𝑗=1)𝑏_𝑖𝑗𝑓_𝑗𝑦_𝑗(𝑡−𝜏(𝑡))+^𝑚_𝑗=1𝑐_𝑖𝑗𝜔_𝑗+^𝑚_𝑗=1𝛼_𝑖𝑗^𝑡_−∞𝑘_𝑗(𝑡−𝑠)𝑓_𝑗𝑦_𝑗(𝑠)+𝑑𝑠_𝑚𝑗=1𝛽_𝑖𝑗^𝑡_−∞𝑘_𝑗(𝑡−𝑠)𝑓_𝑗𝑦_𝑗(𝑠)+𝑑𝑠_𝑚𝑗=1𝑇_𝑖𝑗𝜔_𝑗+^𝑚_𝑗=1𝐻_𝑖𝑗𝜔_𝑗+𝐼_𝑖,𝑦𝑡≥0,𝑖=1,2,…,𝑛,_′𝑗(𝑡)=−𝑑_𝑗𝑦_𝑗𝑡−𝜎₂+^𝑛_𝑖=1𝑎_𝑗𝑖𝑔_𝑖𝑥_𝑖(𝑡)+^𝑛_𝑖=1𝑏̃_𝑗𝑖𝑔_𝑖𝑥_𝑖𝑡−𝜌(𝑡)+^𝑛_𝑖=1𝑐̃_𝑗𝑖𝜔_𝑖+^𝑛_𝑖=1𝛼_𝑗𝑖^𝑡_−∞𝑘_𝑖(𝑡−𝑠)𝑔_𝑖𝑥_𝑖(𝑠)+𝑑𝑠_{𝑚𝑖=𝑛}𝛽_𝑗𝑖^𝑡_−∞𝑘_𝑖(𝑡−𝑠)𝑔_𝑖𝑥_𝑖(𝑠)+𝑑𝑠_𝑛𝑖=1𝑇_𝑗𝑖𝜔_𝑖+^𝑛_𝑖=1𝐻_𝑗𝑖𝜔_𝑖+𝐽_𝑗,𝑡≥0,𝑗=1,2,…,𝑚. (1) However, time-varying delays in the leakage terms inevitably occur in electronic neural networks owing to the unavoidable finite switching speed of amplifiers. It is desirable to study the fuzzy BAM neural networks with time-varying delays in leakage terms. In [32], by using a fixed point theorem and differential inequality techniques, the authors studied the existence and exponential stability of equilibrium point for the following BAM neural network with time-varying delays in leakage terms on time scales: 𝑥^Δ_𝑖(𝑡)=−𝑎_𝑖𝑥_𝑖𝑡−𝛿_𝑖(𝑡)+^𝑚_𝑗=1𝑐_𝑗𝑖𝑓_𝑗𝑦_𝑗𝑡−𝜏_𝑗𝑖(𝑡+_𝑚𝑗=1)𝛼_𝑗𝑖𝑓_𝑗𝑦_𝑗𝑡−𝜏_𝑗𝑖(𝑡)+^𝑚_𝑗=1𝑇_𝑗𝑖𝜇_𝑗+^𝑚_𝑗=1𝛽_𝑗𝑖𝑓_𝑗𝑦_𝑗𝑡−𝜏_𝑗𝑖(𝑡+_𝑚𝑗=1)𝐻_𝑗𝑖𝜇_𝑗+𝐼_𝑖𝑦,𝑡∈𝕋,𝑖=1,2,…,𝑛,_Δ𝑗(𝑡)=−𝑏_𝑗𝑦_𝑗𝑡−𝜂_𝑗(𝑡)+^𝑛_𝑖=1𝑑_𝑖𝑗𝑔_𝑖𝑥_𝑖𝑡−𝜎_𝑖𝑗(𝑡+_𝑛𝑖=1)𝑝_𝑖𝑗𝑔_𝑖𝑥_𝑖𝑡−𝜎_𝑖𝑗(𝑡)+^𝑛_𝑖=1𝐹_𝑖𝑗𝜈_𝑖+^𝑛_𝑖=1𝑞_𝑖𝑗𝑔_𝑖𝑥_𝑖𝑡−𝜎_𝑖𝑗(𝑡)+^𝑛_𝑖=1𝐺_𝑖𝑗𝜈_𝑖+𝐽_𝑗,𝑡∈𝕋,𝑗=1,2,…,𝑚, (2) where 𝕋 is a time scale. Though the nonimpulsive systems have been well studied in theory and in practice, the theory of impulsive differential equations is now being recognized to be richer than the corresponding theory of differential equations without impulses (see [33–35]). What is more, very few results are available on exponential stability of equilibrium point for fuzzy BAM neural networks with time-varying delays in leakage terms and impulses.

Motivated by the above discussion, in this paper, we consider the following model: 𝑥^′_𝑖(𝑡)=−𝑎_𝑖(𝑡)𝑥_𝑖𝑡−𝛼_𝑖(𝑡)+^𝑚_𝑗=1𝑎_𝑖𝑗(𝑡)𝑓_𝑗𝑦_𝑗(𝑡+_𝑚𝑗=1)𝑏_𝑖𝑗(𝑡)𝑓_𝑗𝑦_𝑗(𝑡−𝜏(𝑡))+^𝑚_𝑗=1𝑐_𝑖𝑗(𝑡)𝜔_𝑗+^𝑚_𝑗=1𝛼_𝑖𝑗(𝑡)^𝑡_−∞𝑘_𝑗(𝑡−𝑠)𝑓_𝑗𝑦_𝑗(𝑠)+𝑑𝑠_𝑚𝑗=1𝛽_𝑖𝑗(𝑡)^𝑡_−∞𝑘_𝑗(𝑡−𝑠)𝑓_𝑗𝑦_𝑗(𝑠)+𝑑𝑠_𝑚𝑗=1𝑇_𝑖𝑗𝜔_𝑗+^𝑚_𝑗=1𝐻_𝑖𝑗𝜔_𝑗+𝐴_𝑖(𝑡),𝑡≥0,𝑡≠𝑡_𝑘,𝑖=1,2,…,𝑛,Δ𝑥_𝑖𝑡_𝑘=𝐼_𝑘𝑥_𝑖𝑡_𝑘𝑦,𝑖=1,2,…,𝑛,𝑘=1,2,…,_′𝑗(𝑡)=−𝑏_𝑗(𝑡)𝑦_𝑗𝑡−𝛽_𝑗(𝑡)+^𝑛_𝑖=1𝑑_𝑗𝑖(𝑡)𝑔_𝑖𝑥_𝑖(𝑡)+^𝑛_𝑖=1𝑝_𝑗𝑖(𝑡)𝑔_𝑖𝑥_𝑖𝑡−𝜌(𝑡)+^𝑛_𝑖=1𝑞_𝑗𝑖(𝑡)𝜇_𝑖+^𝑛_𝑖=1𝛾_𝑗𝑖(𝑡)^𝑡_−∞𝑘_𝑖(𝑡−𝑠)𝑔_𝑖𝑥_𝑖(𝑠)+𝑑𝑠_𝑛𝑖=1𝜂_𝑗𝑖(𝑡)^𝑡_−∞𝑘_𝑖(𝑡−𝑠)𝑔_𝑖𝑥_𝑖(𝑠)+𝑑𝑠_𝑛𝑖=1𝑅_𝑗𝑖𝜇_𝑖+^𝑛_𝑖=1𝑆_𝑗𝑖𝜇_𝑖+𝐵_𝑗(𝑡),𝑡≥0,𝑡≠𝑡_𝑘,𝑗=1,2,…,𝑚,Δ𝑦_𝑗𝑡_𝑘=𝐽_𝑘𝑦_𝑗𝑡_𝑘,𝑗=1,2,…,𝑚,𝑘=1,2,…, (3) where 𝑥_𝑖(𝑡) and 𝑦_𝑗(𝑡) are the states of the 𝑖th neuron and the 𝑗th neuron at time 𝑡, 𝑔_𝑖(𝑡) and 𝑓_𝑗(𝑡) denote the activation functions of the 𝑖th neuron and the 𝑗th neuron at time 𝑡, 𝜇_𝑖 and 𝜔_𝑗 denote the inputs of the 𝑖th neuron and the 𝑗th neuron, 𝐴_𝑖(𝑡) and 𝐵_𝑗(𝑡) denote the bias of the 𝑖th neuron and the 𝑗th neuron at time 𝑡, 𝑎_𝑖(𝑡) and 𝑏_𝑗(𝑡) represent the rates with which the 𝑖th neuron and the 𝑗th neuron at time 𝑡 will reset their potential to the resting state in isolation when disconnected from the networks and external inputs, 𝑎_𝑖𝑗(𝑡), 𝑏_𝑖𝑗(𝑡), 𝑑_𝑗𝑖(𝑡), and 𝑝_𝑗𝑖(𝑡) denote the connection weights of the feedback template at time 𝑡 and 𝑐_𝑖𝑗(𝑡), 𝑞_𝑗𝑖(𝑡) denote the connection weights of the feedforward template at time 𝑡, 𝛼_𝑖𝑗(𝑡), 𝛾_𝑗𝑖(𝑡) and 𝛽_𝑖𝑗(𝑡), 𝜂_𝑗𝑖(𝑡) denote the connection weights of the delays fuzzy feedback MIN template at time 𝑡 and the delays fuzzy feedback MAX template at time 𝑡, 𝑇_𝑖𝑗, 𝑅_𝑗𝑖 and 𝐻_𝑖𝑗, 𝑆_𝑗𝑖 are the elements of the fuzzy feedforward MIN template and fuzzy feedforward MAX template, ⋀ and ⋁ denote the fuzzy AND and fuzzy OR operators, 0≤𝜏(𝑡)≤𝜏 and 0≤𝜌(𝑡)≤𝜌 denote the transmission delays at time 𝑡, Δ𝑥_𝑖(𝑡_𝑘)=𝑥_𝑖(𝑡⁺_𝑘)−𝑥_𝑖(𝑡_𝑘), Δ𝑦_𝑗(𝑡_𝑘)=𝑦_𝑗(𝑡⁺_𝑘)−𝑦_𝑗(𝑡_𝑘) are the impulses at moments 𝑡_𝑘, and 𝑡₁<𝑡₂<⋯ is a strictly increasing sequence such that lim_𝑘→∞𝑡_𝑘=+∞, 𝑘_𝑗(𝑠)≥0, and 𝑘_𝑖(𝑠)≥0 are the feedback kernels and satisfy ∫₀^+∞𝑘_𝑗(𝑠)𝑑𝑠=1, ∫₀^+∞𝑘_𝑖(𝑠)𝑑𝑠=1, 𝑖=1,2,…,𝑛, 𝑗=1,2,…,𝑚.

The initial conditions are given by 𝑥_𝑖(𝑠)=𝜑_𝑖(𝑠)(],𝑦,𝑠∈−∞,0_𝑗(𝑠)=𝜓_𝑗(𝑠),𝑠∈(−∞,0], (4) where 𝜑_𝑖,𝜓_𝑗∈𝐶((−∞,0],𝑅), 𝑖=1,2,…,𝑛, 𝑗=1,2,…,𝑚.

For convenience, for a continuous function 𝑓∶𝑅→𝑅, we denote 𝑓⁺=sup_𝑡∈𝑅|𝑓(𝑡)|,𝑓⁻=inf_𝑡∈𝑅|𝑓(𝑡)|.

Throughout this paper, we make the following assumptions.(H₁)The neuron activation functions 𝑓_𝑗(𝑡) and 𝑔_𝑖(𝑡), 𝑖=1,2,…,𝑛, 𝑗=1,2,…,𝑚, are continuous on 𝑅 and there exist some real constants 𝐿_𝑗,𝑀_𝑖 such that 𝑓_𝑗(𝑢)−𝑓_𝑗(𝑣)≤𝐿_𝑗||,𝑔𝑢−𝑣_𝑖(𝑢)−𝑔_𝑖(𝑣)≤𝑀_𝑖|𝑢−𝑣| (5) for all 𝑢,𝑣∈𝑅, 𝑖=1,2,…,𝑛, 𝑗=1,2,…,𝑚.(H₂)The leakage delays satisfy 𝛼_𝑖(𝑡)≥0, 𝛽_𝑗(𝑡)≥0 and 𝑎_𝑖(𝑡)>0, 𝑏_𝑗(𝑡)>0, 𝑖=1,2,…,𝑛, 𝑗=1,2,…,𝑚.

Definition 1. Let 𝑧^∗(𝑡)=(𝑥^∗₁(𝑡),𝑥^∗₂(𝑡),…,𝑥^∗_𝑛(𝑡),𝑦^∗₁(𝑡),𝑦^∗₂(𝑡),…,𝑦^∗_𝑚(𝑡))^𝑇 be a solution of system (3) with the initial condition 𝜙^∗(𝑡)=(𝜑^∗₁(𝑡),𝜑^∗₂(𝑡),…,𝜑^∗_𝑛(𝑡),𝜓^∗₁(𝑡),𝜓^∗₂(𝑡),…,𝜓^∗_𝑚(𝑡))^𝑇 and let 𝑧(𝑡)=(𝑥₁(𝑡),𝑥₂(𝑡),…,𝑥_𝑛(𝑡),𝑦₁(𝑡),𝑦₂(𝑡),…,𝑦_𝑚(𝑡))^𝑇 be any solution of system (3) with the initial condition 𝜙(𝑡)=(𝜑₁(𝑡),𝜑₂(𝑡),…,𝜑_𝑛(𝑡),𝜓₁(𝑡),𝜓₂(𝑡),…,𝜓_𝑚(𝑡))^𝑇; if these two solutions satisfy 𝑥_𝑖(𝑡)−𝑥^∗_𝑖(𝑡)=𝑂𝑒^−𝜆𝑡,𝑦_𝑗(𝑡)−𝑦^∗_𝑗(𝑡)=𝑂𝑒^−𝜆𝑡,𝑖=1,2,…,𝑛,𝑗=1,2,…,𝑚, (6) then, we say that system (3) is exponentially stable.

Lemma 2 (see [36]). Let 𝑧, 𝑧^′ be two states of system (3); for 𝑖=1,2,…,𝑛, 𝑗=1,2,…,𝑚, one has ^𝑚_𝑗=1𝛼_𝑖𝑗(𝑡)𝑓_𝑗(𝑧)−^𝑚_𝑗=1𝛼_𝑖𝑗(𝑡)𝑓_𝑗𝑧^′≤^𝑚_𝑗=1𝛼_𝑖𝑗(𝑡)𝑓_𝑗(𝑧)−𝑓_𝑗𝑧^′,_𝑚𝑗=1𝛽_𝑖𝑗(𝑡)𝑓_𝑗(𝑧)−^𝑚_𝑗=1𝛽_𝑖𝑗(𝑡)𝑓_𝑗𝑧^′≤^𝑚_𝑗=1𝛽_𝑖𝑗(𝑡)𝑓_𝑗(𝑧)−𝑓_𝑗𝑧^′,_𝑛𝑖=1𝛾_𝑗𝑖(𝑡)𝑔_𝑖(𝑧)−^𝑛_𝑖=1𝛾_𝑗𝑖(𝑡)𝑔_𝑖𝑧^′≤^𝑛_𝑖=1𝛾_𝑗𝑖(𝑡)𝑔_𝑖(𝑧)−𝑔_𝑖𝑧^′,_𝑛𝑗=1𝜂_𝑗𝑖(𝑡)𝑔_𝑖(𝑧)−^𝑛_𝑖=1𝜂_𝑗𝑖(𝑡)𝑔_𝑖𝑧^′≤^𝑛_𝑖=1𝜂_𝑗𝑖(𝑡)𝑔_𝑖(𝑧)−𝑔_𝑖𝑧^′. (7)

Our main purpose of this paper is by using differential inequality techniques to study the exponential stability of (3). The results of this paper are completely new and complementary to the previously known results and the methods used in this paper are different from those used in [31, 32].

2. Exponential Stability

In this section, we will give some sufficient conditions to guarantee the exponential stability of system (3).

Theorem 3. Suppose that (𝐻₁) and (𝐻₂) hold. Let 𝑧^∗(𝑡)=(𝑥^∗₁(𝑡),𝑥^∗₂(𝑡),…,𝑥^∗_𝑛(𝑡),𝑦^∗₁(𝑡),𝑦^∗₂(𝑡),…,𝑦^∗_𝑚(𝑡))^𝑇 be a solution of system (3) with the initial condition 𝜙^∗(𝑡)=(𝜑^∗₁(𝑡),𝜑^∗₂(𝑡),…,𝜑^∗_𝑛(𝑡),𝜓^∗₁(𝑡),𝜓^∗₂(𝑡),…,𝜓^∗_𝑚(𝑡))^𝑇. Furthermore, assume that(H₃)−𝑎_𝑖(𝑡)−𝑎_𝑖(𝑡)𝛼_𝑖(𝑡)𝑎⁺_𝑖+1+𝑎_𝑖(𝑡)𝛼_𝑖(𝑡×_𝑚𝑗=1)𝑎⁺_𝑖𝑗+𝑏⁺_𝑖𝑗+𝛼⁺_𝑖𝑗+𝛽⁺_𝑖𝑗𝐿_𝑗−𝑏<0,𝑖=1,2,…,𝑛,_𝑗(𝑡)−𝑏_𝑗(𝑡)𝛽_𝑗(𝑡)𝑏⁺_𝑗+1+𝑏_𝑗(𝑡)𝛽_𝑗(𝑡)×^𝑛_𝑖=1𝑑⁺_𝑗𝑖+𝑝⁺_𝑗𝑖+𝛾⁺_𝑗𝑖+𝜂⁺_𝑗𝑖𝑀_𝑖<0,𝑗=1,2,…,𝑚. (8)(H₄)There exist constants 𝜃_𝑖𝑘, 𝜃_𝑗𝑘 such that 𝐼_𝑘𝑥_𝑖𝑡_𝑘=−𝜃_𝑖𝑘𝑥_𝑖𝑡_𝑘,0≤𝜃_𝑖𝑘𝐽≤2,𝑖=1,2,…,𝑛,𝑘=1,2,…,_𝑘𝑦_𝑗𝑡_𝑘=−𝜃_𝑗𝑘𝑦_𝑗𝑡_𝑘,0≤𝜃_𝑗𝑘≤2,𝑗=1,2,…,𝑚,𝑘=1,2,…. (9) Then system (3) is exponentially stable.

Proof. Let 𝑧(𝑡)=(𝑥₁(𝑡),𝑥₂(𝑡),…,𝑥_𝑛(𝑡),𝑦₁(𝑡), 𝑦₂(𝑡),…,𝑦_𝑚(𝑡))^𝑇 be an arbitrary solution of system (3) with the initial condition 𝜙(𝑡)=(𝜑₁(𝑡),𝜑₂(𝑡),…,𝜑_𝑛(𝑡),𝜓₁(𝑡),𝜓₂(𝑡),…,𝜓_𝑚(𝑡))^𝑇. Set 𝑥_𝑖(𝑡)=𝑥_𝑖(𝑡)−𝑥^∗_𝑖(𝑡),𝑖=1,2,…,𝑛,𝑦_𝑗(𝑡)=𝑦_𝑗(𝑡)−𝑦^∗_𝑗(𝑡)𝑓,𝑗=1,2,…,𝑚,_𝑗𝑦_𝑗(𝑡)=𝑓_𝑗𝑦_𝑗(𝑡)+𝑦^∗_𝑗(𝑡)−𝑓_𝑗𝑦^∗_𝑗(𝑡),𝑔𝑗=1,2,…,𝑚,_𝑖𝑥_𝑖(𝑡)=𝑔_𝑖𝑥_𝑖(𝑡)+𝑥^∗_𝑖(𝑡)−𝑔_𝑖𝑥^∗_𝑖(𝑡),𝑖=1,2,…,𝑛. (10)(11) From (3) and (11), for 𝑡>0, 𝑡≠𝑡_𝑘, 𝑘=1,2,…, we have 𝑥^′_𝑖(𝑡)=−𝑎_𝑖(𝑡)𝑥_𝑖𝑡−𝛼_𝑖(𝑡)+^𝑚_𝑗=1𝑎_𝑖𝑗(𝑡)𝑓_𝑗𝑦_𝑗(𝑡+_𝑚𝑗=1)𝑏_𝑖𝑗(𝑡)𝑓_𝑗𝑦_𝑗((𝑡+𝑡−𝜏))_𝑚𝑗=1𝛼_𝑖𝑗(𝑡)^𝑡_−∞𝑘_𝑗(𝑡−𝑠)𝑓_𝑗𝑦_𝑗(𝑠)+𝑑𝑠_𝑚𝑗=1𝛽_𝑖𝑗(𝑡)^𝑡_−∞𝑘_𝑗(𝑡−𝑠)𝑓_𝑗𝑦_𝑗(𝑠)𝑑𝑠,𝑖=1,2,…,𝑛,𝑦^′_𝑗(𝑡)=−𝑏_𝑗(𝑡)𝑦_𝑗𝑡−𝛽_𝑗(𝑡)+^𝑛_𝑖=1𝑑_𝑗𝑖(𝑡)𝑔_𝑖𝑥_𝑖(𝑡+_𝑛𝑖=1)𝑝_𝑗𝑖(𝑡)𝑔_𝑖𝑥_𝑖(𝑡)+𝑡−𝜌_𝑛𝑖=1𝛾_𝑗𝑖(𝑡)^𝑡_−∞𝑘_𝑖(𝑡−𝑠)𝑔_𝑖𝑥_𝑖(𝑠)+𝑑𝑠_𝑛𝑖=1𝜂_𝑗𝑖(𝑡)^𝑡_−∞𝑘_𝑖(𝑡−𝑠)𝑔_𝑖𝑥_𝑖(𝑠)𝑑𝑠,𝑗=1,2,…,𝑚. (12)(13)
According to (H₄), we get 𝑥_𝑖𝑡⁺_𝑘−𝑥^∗_𝑖𝑡⁺_𝑘=𝑥_𝑖𝑡_𝑘+𝐼_𝑘𝑥_𝑖𝑡_𝑘−𝑥^∗_𝑖𝑡_𝑘−𝐼_𝑘𝑥^∗_𝑖𝑡_𝑘=1−𝜃_𝑖𝑘𝑥_𝑖𝑡_𝑘−𝑥^∗_𝑖𝑡_𝑘,𝑦𝑖=1,2,…,𝑛,𝑘=1,2,…,_𝑗𝑡⁺_𝑘−𝑦^∗_𝑗𝑡⁺_𝑘=𝑦_𝑗𝑡_𝑘+𝐽_𝑘𝑦_𝑗𝑡_𝑘−𝑦^∗_𝑗𝑡_𝑘−𝐽_𝑘𝑦^∗_𝑗𝑡_𝑘=1−𝜃_𝑗𝑘𝑦_𝑗𝑡_𝑘−𝑦^∗_𝑗𝑡_𝑘,𝑗=1,2,…,𝑚,𝑘=1,2,…. (14) So, 𝑥_𝑖𝑡⁺_𝑘−𝑥^∗_𝑖𝑡⁺_𝑘=1−𝜃_𝑖𝑘𝑥_𝑖𝑡_𝑘−𝑥^∗_𝑖𝑡_𝑘≤𝑥_𝑖𝑡_𝑘−𝑥^∗_𝑖𝑡_𝑘,𝑦𝑖=1,2,…,𝑛,𝑘=1,2,…,_𝑗𝑡⁺_𝑘−𝑦^∗_𝑗𝑡⁺_𝑘=1−𝜃_𝑗𝑘𝑦_𝑗𝑡_𝑘−𝑦^∗_𝑗𝑡_𝑘≤𝑦_𝑗𝑡_𝑘−𝑦^∗_𝑗𝑡_𝑘,𝑗=1,2,…,𝑚,𝑘=1,2,…, (15) which implies that 𝑥_𝑖𝑡⁺_𝑘=𝑥_𝑖𝑡⁺_𝑘−𝑥^∗_𝑖𝑡⁺_𝑘≤𝑥_𝑖𝑡_𝑘−𝑥^∗_𝑖𝑡_𝑘=𝑥_𝑖𝑡⁻_𝑘,𝑖=1,2,…,𝑛,𝑘=1,2,…,𝑦_𝑗𝑡⁺_𝑘=𝑦_𝑗𝑡⁺_𝑘−𝑦^∗_𝑗𝑡⁺_𝑘≤𝑦_𝑗𝑡_𝑘−𝑦^∗_𝑗𝑡_𝑘=𝑦_𝑗𝑡⁻_𝑘,𝑗=1,2,…,𝑚,𝑘=1,2,…. (16) Define continuous functions Γ_𝑖(𝜔) and Γ_𝑗(𝜔) by setting Γ_𝑖(𝜔)=−𝑎_𝑖(𝑡)𝑒^{𝜔𝛼𝑖(𝑡)}−𝜔−𝑎_𝑖(𝑡)𝑒^{𝜔𝛼𝑖(𝑡)}𝛼_𝑖(𝑡)×𝜔+𝑎⁺_𝑖𝑒^{𝜔𝛼+𝑖}+1+𝑎_𝑖(𝑡)𝑒^{𝜔𝛼𝑖(𝑡)}𝛼_𝑖(𝑡×_𝑚𝑗=1)𝑎⁺_𝑖𝑗+𝑏⁺_𝑖𝑗𝑒^𝜔𝜏+𝛼⁺_𝑖𝑗+𝛽⁺_𝑖𝑗0^+∞𝑘_𝑗(𝑢)𝑒^𝜔𝑢𝑑𝑢𝐿_𝑗,Γ𝑖=1,2,…,𝑛,_𝑗(𝜔)=−𝑏_𝑗(𝑡)𝑒^{𝜔𝛽𝑗(𝑡)}−𝜔−𝑏_𝑗(𝑡)𝑒^{𝜔𝛽𝑗(𝑡)}𝛽_𝑗(𝑡)×𝜔+𝑏⁺_𝑗𝑒^{𝜔𝛽+𝑗}+1+𝑏_𝑗(𝑡)𝑒^{𝜔𝛽𝑗(𝑡)}𝛽_𝑗(𝑡×_𝑛𝑖=1)𝑑⁺_𝑗𝑖+𝑝⁺_𝑗𝑖𝑒^𝜔𝜌+𝛾⁺_𝑗𝑖+𝜂⁺_𝑗𝑖0^+∞𝑘_𝑖(𝑢)𝑒^𝜔𝑢𝑑𝑢𝑀_𝑖,𝑗=1,2,…,𝑚. (17) Then, for 𝑖=1,2,…,𝑛, 𝑗=1,2,…,𝑚, we have Γ_𝑖(0)=−𝑎_𝑖(𝑡)−𝑎_𝑖(𝑡)𝛼_𝑖(𝑡)𝑎⁺_𝑖+1+𝑎_𝑖(𝑡)𝛼_𝑖(𝑡×_𝑚𝑗=1)𝑎⁺_𝑖𝑗+𝑏⁺_𝑖𝑗+𝛼⁺_𝑖𝑗+𝛽⁺_𝑖𝑗𝐿_𝑗Γ<0,_𝑗(0)=−𝑏_𝑗(𝑡)−𝑏_𝑗(𝑡)𝛽_𝑗(𝑡)𝑏⁺_𝑗+1+𝑏_𝑗(𝑡)𝛽_𝑗(𝑡)×^𝑛_𝑖=1𝑑⁺_𝑗𝑖+𝑝⁺_𝑗𝑖+𝛾⁺_𝑗𝑖+𝜂⁺_𝑗𝑖𝑀_𝑖<0. (18) The continuity of Γ_𝑖(𝜔) and Γ_𝑗(𝜔) implies that there exists 𝜆>0 such that Γ_𝑖(𝜆)=−𝑎_𝑖(𝑡)𝑒^{𝜆𝛼𝑖(𝑡)}−𝜆−𝑎_𝑖(𝑡)𝑒^{𝜆𝛼𝑖(𝑡)}𝛼_𝑖(𝑡)×𝜆+𝑎⁺_𝑖𝑒^{𝜆𝛼+𝑖}+1+𝑎_𝑖(𝑡)𝑒^{𝜆𝛼𝑖(𝑡)}𝛼_𝑖(𝑡×_𝑚𝑗=1)𝑎⁺_𝑖𝑗+𝑏⁺_𝑖𝑗𝑒^𝜆𝜏+𝛼⁺_𝑖𝑗+𝛽⁺_𝑖𝑗0^+∞𝑘_𝑗(𝑢)𝑒^𝜆𝑢𝑑𝑢×𝐿_𝑗Γ<0,𝑖=1,2,…,𝑛,_𝑗(𝜆)=−𝑏_𝑗(𝑡)𝑒^{𝜆𝛽𝑗(𝑡)}−𝜆−𝑏_𝑗(𝑡)𝑒^{𝜆𝛽𝑗(𝑡)}𝛽_𝑗(𝑡)×𝜆+𝑏⁺_𝑗𝑒^{𝜆𝛽+𝑗}+1+𝑏_𝑗(𝑡)𝑒^{𝜆𝛽𝑗(𝑡)}𝛽_𝑗(𝑡×_𝑛𝑖=1)𝑑⁺_𝑗𝑖+𝑝⁺_𝑗𝑖𝑒^𝜆𝜌+𝛾⁺_𝑗𝑖+𝜂⁺_𝑗𝑖0^+∞𝑘_𝑖(𝑢)𝑒^𝜆𝑢𝑑𝑢×𝑀_𝑖<0,𝑗=1,2,…,𝑚. (19)
Let 𝑋_𝑖(𝑡)=𝑥_𝑖(𝑡)𝑒^𝜆𝑡,𝑌_𝑗(𝑡)=𝑦_𝑗(𝑡)𝑒^𝜆𝑡,𝑖=1,2,…,𝑛,𝑗=1,2,…,𝑚. (20) From (12) and (20), we obtain 𝑋^′_𝑖(𝑡)=𝜆𝑒^𝜆𝑡𝑥_𝑖(𝑡)+𝑒^𝜆𝑡𝑥^′_𝑖(𝑡)=𝜆𝑋_𝑖(𝑡)+𝑒^𝜆𝑡×−𝑎_𝑖(𝑡)𝑥_𝑖𝑡−𝛼_𝑖(𝑡)+^𝑚_𝑗=1𝑎_𝑖𝑗(𝑡)𝑓_𝑗𝑦_𝑗(𝑡)+^𝑚_𝑗=1𝑏_𝑖𝑗(𝑡)𝑓_𝑗𝑦_𝑗((𝑡+𝑡−𝜏))_𝑚𝑗=1𝛼_𝑖𝑗(𝑡)^𝑡_−∞𝑘_𝑗(𝑡−𝑠)𝑓_𝑗𝑦_𝑗(𝑠)+𝑑𝑠_𝑚𝑗=1𝛽_𝑖𝑗(𝑡)^𝑡_−∞𝑘_𝑗(𝑡−𝑠)𝑓_𝑗𝑦_𝑗(𝑠)𝑑𝑠=𝜆𝑋_𝑖(𝑡)−𝑎_𝑖(𝑡)𝑒^{𝜆𝛼𝑖(𝑡)}𝑋_𝑖𝑡−𝛼_𝑖(𝑡)+^𝑚_𝑗=1𝑎_𝑖𝑗(𝑡)𝑒^𝜆𝑡𝑓_𝑗𝑦_𝑗(𝑡)+^𝑚_𝑗=1𝑏_𝑖𝑗(𝑡)𝑒^𝜆𝑡𝑓_𝑗𝑦_𝑗((𝑡+𝑡−𝜏))_𝑚𝑗=1𝛼_𝑖𝑗(𝑡)𝑒^𝜆𝑡𝑡_−∞𝑘_𝑗(𝑡−𝑠)𝑓_𝑗𝑦_𝑗(𝑠)+𝑑𝑠_𝑚𝑗=1𝛽_𝑖𝑗(𝑡)𝑒^𝜆𝑡𝑡_−∞𝑘_𝑗(𝑡−𝑠)𝑓_𝑗𝑦_𝑗(𝑠)𝑑𝑠,𝑖=1,2,…,𝑛. (21) Similarly, 𝑌^′_𝑗(𝑡)=𝜆𝑌_𝑗(𝑡)−𝑏_𝑗(𝑡)𝑒^{𝜆𝛽𝑗(𝑡)}𝑌_𝑗𝑡−𝛽_𝑗(𝑡+_𝑛𝑖=1)𝑑_𝑗𝑖(𝑡)𝑒^𝜆𝑡𝑔_𝑖𝑥_𝑖(𝑡)+^𝑛_𝑖=1𝑝_𝑗𝑖(𝑡)𝑒^𝜆𝑡𝑔_𝑖𝑥_𝑖(𝑡)+𝑡−𝜌_𝑛𝑖=1𝛾_𝑗𝑖(𝑡)𝑒^𝜆𝑡𝑡_−∞𝑘_𝑖(𝑡−𝑠)𝑔_𝑖𝑥_𝑖(𝑠)+𝑑𝑠_𝑛𝑖=1𝜂_𝑗𝑖(𝑡)𝑒^𝜆𝑡𝑡_−∞𝑘_𝑖(𝑡−𝑠)𝑔_𝑖𝑥_𝑖(𝑠)𝑑𝑠,𝑗=1,2,…,𝑚. (22) We rewrite (21) and (22) as follows: 𝑋^′_𝑖(𝑡)=𝜆𝑋_𝑖(𝑡)−𝑎_𝑖(𝑡)𝑒^{𝜆𝛼𝑖(𝑡)}𝑋_𝑖(𝑡)+𝑎_𝑖(𝑡)𝑒^{𝜆𝛼𝑖(𝑡)}×^𝑡_{𝑡−𝛼𝑖(𝑡)}𝜆𝑋_i(𝑠)−𝑎_𝑖(𝑠)𝑒^{𝜆𝛼𝑖(𝑠)}×𝑋_𝑖𝑠−𝛼_𝑖(𝑠)+^𝑚_𝑗=1𝑎_𝑖𝑗(𝑠)𝑒^𝜆𝑠𝑓_𝑗𝑦_𝑗(𝑠)+^𝑚_𝑗=1𝑏_𝑖𝑗(𝑠)𝑒^𝜆𝑠𝑓_𝑗𝑦_𝑗((𝑠+𝑠−𝜏))_𝑚𝑗=1𝛼_𝑖𝑗(𝑠)𝑒^𝜆𝑠𝑠_−∞𝑘_𝑗(𝑠−𝑢)𝑓_𝑗𝑦_𝑗(𝑢)+𝑑𝑢_𝑚𝑗=1𝛽_𝑖𝑗(𝑠)𝑒^𝜆𝑠𝑠_−∞𝑘_𝑗(𝑠−𝑢)𝑓_𝑗𝑦_𝑗(𝑢)+𝑑𝑢𝑑𝑠_𝑚𝑗=1𝑎_𝑖𝑗(𝑡)𝑒^𝜆𝑡𝑓_𝑗𝑦_𝑗(𝑡)+^𝑚_𝑗=1𝑏_𝑖𝑗(𝑡)𝑒^𝜆𝑡𝑓_𝑗𝑦_𝑗((𝑡+𝑡−𝜏))_𝑚𝑗=1𝛼_𝑖𝑗(𝑡)𝑒^𝜆𝑡𝑡_−∞𝑘_𝑗(𝑡−𝑠)𝑓_𝑗𝑦_𝑗(𝑠)+𝑑𝑠_𝑚𝑗=1𝛽_𝑖𝑗(𝑡)𝑒^𝜆𝑡𝑡_−∞𝑘_𝑗(𝑡−𝑠)𝑓_𝑗𝑦_𝑗(𝑠)𝑌𝑑𝑠,𝑖=1,2,…,𝑛,_′𝑗(𝑡)=𝜆𝑌_𝑗(𝑡)−𝑏_𝑗(𝑡)𝑒^{𝜆𝛽𝑗(𝑡)}𝑌_𝑗(𝑡)+𝑏_𝑗(𝑡)𝑒^{𝜆𝛽𝑗(𝑡)}×^𝑡_{𝑡−𝛽𝑗(𝑡)}𝜆𝑌_𝑗(𝑠)−𝑏_𝑗(𝑠)𝑒^{𝜆𝛽𝑗(𝑠)}×𝑌_𝑗𝑠−𝛽_𝑗(𝑠)+^𝑛_𝑖=1𝑑_𝑗𝑖(𝑠)𝑒^𝜆𝑠𝑔_𝑖𝑥_𝑖(𝑠)+^𝑛_𝑖=1𝑝_𝑗𝑖(𝑠)𝑒^𝜆𝑠𝑔_𝑖𝑥_𝑖(𝑠)+𝑠−𝜌_𝑛𝑖=1𝛾_𝑗𝑖(𝑠)𝑒^𝜆𝑠𝑠_−∞𝑘_𝑖(𝑠−𝑢)𝑔_𝑖𝑥_𝑖(𝑢)+𝑑𝑢_𝑛𝑖=1𝜂_𝑗𝑖(𝑠)𝑒^𝜆𝑠𝑠_−∞𝑘_𝑖(𝑠−𝑢)𝑔_𝑖𝑥_𝑖(𝑢)+𝑑𝑢𝑑𝑠_𝑛𝑖=1𝑑_𝑗𝑖(𝑡)𝑒^𝜆𝑡𝑔_𝑖𝑥_𝑖(𝑡)+^𝑛_𝑖=1𝑝_𝑗𝑖(𝑡)𝑒^𝜆𝑡𝑔_𝑖𝑥_𝑖(𝑡)+𝑡−𝜌_𝑛𝑖=1𝛾_𝑗𝑖(𝑡)𝑒^𝜆𝑡𝑡_−∞𝑘_𝑖(𝑡−𝑠)𝑔_𝑖𝑥_𝑖(𝑠+)𝑑𝑠_𝑛𝑖=1𝜂_𝑗𝑖(𝑡)𝑒^𝜆𝑡𝑡_−∞𝑘_𝑖(𝑡−𝑠)𝑔_𝑖𝑥_𝑖(𝑠)𝑑𝑠,𝑗=1,2,…,𝑚. (23)(24) We define a positive number such that 𝑀=max_{1≤𝑖≤𝑛,1≤𝑗≤𝑚}sup_{[]𝑠∈−𝜏,0}𝑋_𝑖(𝑠),sup_{𝑠∈−𝜌,0}𝑌_𝑗(𝑠). (25) It follows that 𝑋_𝑖(𝑡[]𝑌)<𝑀,∀𝑡∈−𝜏,0,𝑖=1,2,…,𝑛,_𝑗(𝑡)<𝑀,∀𝑡∈−𝜌,0,𝑗=1,2,…,𝑚. (26) We claim that 𝑋_𝑖(𝑡)<𝑀,𝑌_𝑗(𝑡)<𝑀,∀𝑡>0,𝑡≠𝑡_𝑘,𝑖=1,2,…,𝑛,𝑗=1,2,…,𝑚. (27) If (27) is not valid, then there exist some 𝑖∈{1,2,…,𝑛}, some 𝑗∈{1,2,…,𝑚}, and a first 𝑇₁>0 such that one of the following four cases must occur:(1)𝑋_𝑖(𝑇₁)=𝑀, 𝑋^′_𝑖(𝑇₁)≥0, |𝑋_𝑖(𝑡)|<𝑀, |𝑌_𝑗(𝑡)|<𝑀, for 𝑡<𝑇₁;(2)𝑋_𝑖(𝑇₁)=−𝑀, 𝑋^′_𝑖(𝑇₁)≤0, |𝑋_𝑖(𝑡)|<𝑀, |𝑌_𝑗(𝑡)|<𝑀, for 𝑡<𝑇₁;(3)𝑌_𝑗(𝑇₁)=𝑀, 𝑌^′_𝑗(𝑇₁)≥0, |𝑋_𝑖(𝑡)|<𝑀, |𝑌_𝑗(𝑡)|<𝑀, for 𝑡<𝑇₁;(4)𝑌_𝑗(𝑇₁)=−𝑀, 𝑌^′_𝑗(𝑇₁)≤0, |𝑋_𝑖(𝑡)|<𝑀, |𝑌_𝑗(𝑡)|<𝑀, for 𝑡<𝑇₁. If (1) holds from (19), (23), and (H₃), we have 𝑋^′_𝑖𝑇₁=𝜆𝑋_𝑖𝑇₁−𝑎_𝑖𝑇₁𝑒^{𝜆𝛼𝑖(𝑇1)}𝑋_𝑖𝑇₁+𝑎_𝑖𝑇₁𝑒^{𝜆𝛼𝑖(𝑇1)}×^𝑇1_{𝑇1−𝛼𝑖(𝑇1)}𝜆𝑋_𝑖(𝑠)−𝑎_𝑖(𝑠)𝑒^{𝜆𝛼𝑖(𝑠)}𝑋_𝑖𝑠−𝛼_𝑖(𝑠)+^𝑚_𝑗=1𝑎_𝑖𝑗(𝑠)𝑒^𝜆𝑠𝑓_𝑗𝑦_𝑗(𝑠)+^𝑚_𝑗=1𝑏_𝑖𝑗(𝑠)𝑒^𝜆𝑠𝑓_𝑗𝑦_𝑗((𝑠+𝑠−𝜏))_𝑚𝑗=1𝛼_𝑖𝑗(𝑠)𝑒^𝜆𝑠𝑠_−∞𝑘_𝑗(𝑠−𝑢)𝑓_𝑗𝑦_𝑗(𝑢)+𝑑𝑢_𝑚𝑗=1𝛽_𝑖𝑗(𝑠)𝑒^𝜆𝑠×^𝑠_−∞𝑘_𝑗(𝑠−𝑢)𝑓_𝑗𝑦_𝑗(𝑢+)𝑑𝑢𝑑𝑠_𝑚𝑗=1𝑎_𝑖𝑗𝑇₁𝑒^𝜆𝑇1𝑓_𝑗𝑦_𝑗𝑇₁+^𝑚_𝑗=1𝑏_𝑖𝑗𝑇₁𝑒^𝜆𝑇1𝑓_𝑗𝑦_𝑗((𝑡+𝑡−𝜏))_𝑚𝑗=1𝛼_𝑖𝑗𝑇₁𝑒^{𝜆𝑡𝑇1}_−∞𝑘_𝑗𝑇₁−𝑠𝑓_𝑗𝑦_𝑗(𝑠+)𝑑𝑠_𝑚𝑗=1𝛽_𝑖𝑗𝑇₁𝑒^{𝜆𝑇1𝑇1}_−∞𝑘_𝑗𝑇₁−𝑠𝑓_𝑗𝑦_𝑗(𝑠)𝑑𝑠≤𝜆𝑋_𝑖𝑇₁−𝑎_𝑖𝑇₁𝑒^{𝜆𝛼𝑖(𝑇1)}𝑋_𝑖𝑇₁+𝑎_𝑖𝑇₁𝑒^{𝜆𝛼𝑖(𝑇1)}×^𝑇1_{𝑇1−𝛼𝑖𝑇1}𝜆𝑋_𝑖𝑇₁+𝑎⁺_𝑖𝑒^{𝜆𝛼+𝑖}𝑋_𝑖𝑇₁+^𝑚_𝑗=1𝑎⁺_𝑖𝑗𝐿_𝑗𝑌_𝑗(𝑠+_𝑚𝑗=1)𝑏⁺_𝑖𝑗𝑒^𝜆𝜏𝐿_𝑗𝑌_𝑗((𝑠+𝑠−𝜏))_𝑚𝑗=1𝛼⁺_𝑖𝑗^𝑠_−∞𝑘_𝑗(𝑠−𝑢)𝑒^{𝜆(𝑠−𝑢)}𝐿_𝑗𝑌_𝑗(𝑢+)𝑑𝑢_𝑚𝑗=1𝛽⁺_𝑖𝑗^𝑠_−∞𝑘_𝑗(𝑠−𝑢)𝑒^{𝜆(𝑠−𝑢)}×𝐿_𝑗𝑌_𝑗(𝑢+)𝑑𝑢𝑑𝑠_𝑚𝑗=1𝑎⁺_𝑖𝑗𝐿_𝑗𝑌_𝑗𝑇₁+^𝑚_𝑗=1𝑏⁺_𝑖𝑗𝐿_𝑗𝑒^𝜆𝜏𝑌_𝑗𝑇₁−𝜏𝑇₁+^𝑚_𝑗=1𝛼⁺_𝑖𝑗^𝑇1_−∞𝑘_𝑗𝑇₁−𝑠𝑒^{𝜆(𝑇1−𝑠)}𝐿_𝑗𝑌_𝑗(𝑠+)𝑑𝑠_𝑚𝑗=1𝛽⁺_𝑖𝑗^𝑇1_−∞𝑘_𝑗𝑇₁−𝑠𝑒^{𝜆(𝑇1−𝑠)}𝐿_𝑗𝑌_𝑗(𝑠𝑎)𝑑𝑠≤−_𝑖𝑇₁𝑒^{𝜆𝛼𝑖(𝑇1)}−𝜆−𝑎_𝑖𝑇₁𝑒^{𝜆𝛼𝑖(𝑇1)}×𝛼_𝑖𝑇₁𝜆+𝑎⁺_𝑖𝑒^{𝜆𝛼+𝑖}𝑋_𝑖𝑇₁+𝑎_𝑖𝑇₁𝑒^{𝜆𝛼𝑖(𝑇1)}×^𝑇1_{𝑇1−𝛼𝑖(𝑇1)}^𝑚_𝑗=1𝑎⁺_𝑖𝑗𝐿_𝑗𝑀+^𝑚_𝑗=1𝑏⁺_𝑖𝑗𝑒^𝜆𝜏𝐿_𝑗𝑀+^𝑚_𝑗=1𝛼⁺_𝑖𝑗𝐿_𝑗𝑀×^𝑠_−∞𝑘_𝑗(𝑠−𝑢)𝑒^{𝜆(𝑠−𝑢)}+𝑑𝑢_𝑚𝑗=1𝛽⁺_𝑖𝑗𝐿_𝑗𝑀^𝑠_−∞𝑘_𝑗(𝑠−𝑢)𝑒^{𝜆(𝑠−𝑢)}+𝑑𝑢𝑑𝑠_𝑚𝑗=1𝑎⁺_𝑖𝑗𝐿_𝑗𝑀+^𝑚_𝑗=1𝑏⁺_𝑖𝑗𝑒^𝜆𝜏𝐿_𝑗𝑀+^𝑚_𝑗=1𝛼⁺_𝑖𝑗𝐿_𝑗𝑀×^𝑇1_−∞𝑘_𝑗𝑇₁−𝑠𝑒^{𝜆(𝑇1−𝑠)}+𝑑𝑢_𝑚𝑗=1𝛽⁺_𝑖𝑗𝐿_𝑗𝑀^𝑇1_−∞𝑘_𝑗𝑇₁−𝑠𝑒^{𝜆(𝑇1−𝑠)}𝑎𝑑𝑢≤−_𝑖𝑇₁𝑒^{𝜆𝛼𝑖(𝑇1)}−𝜆−𝑎_𝑖𝑇₁𝑒^{𝜆𝛼𝑖(𝑇1)}×𝛼_𝑖𝑇₁𝜆+𝑎⁺_𝑖𝑒^{𝜆𝛼+𝑖}𝑋_𝑖𝑇₁+1+𝑎_𝑖𝑇₁𝑒^{𝜆𝛼𝑖(𝑇1)}𝛼_𝑖𝑇₁×^𝑚_𝑗=1𝑎⁺_𝑖𝑗+𝑏⁺_𝑖𝑗𝑒^𝜆𝜏+𝛼⁺_𝑖𝑗+𝛽⁺_𝑖𝑗×₀^+∞𝑘_𝑗(𝑢)𝑒^𝜆𝑢𝑑𝑢𝐿_𝑗𝑀=−𝑎_𝑖𝑇₁𝑒^{𝜆𝛼𝑖(𝑇1)}−𝜆−𝑎_𝑖𝑇₁𝑒^{𝜆𝛼𝑖(𝑇1)}×𝛼_𝑖𝑇₁𝜆+𝑎⁺_𝑖𝑒^{𝜆𝛼+𝑖}+1+𝑎_𝑖𝑇₁𝑒^{𝜆𝛼𝑖(𝑇1)}𝛼_𝑖𝑇₁×^𝑚_𝑗=1𝑎⁺_𝑖𝑗+𝑏⁺_𝑖𝑗𝑒^𝜆𝜏+𝛼⁺_𝑖𝑗+𝛽⁺_𝑖𝑗×₀^+∞𝑘_𝑗(𝑢)𝑒^𝜆𝑢𝑑𝑢𝐿_𝑗𝑀<0, (28) and this is a contradiction. Hence, (1) does not hold.
If (2) holds, then, from (H₃), (19), and (24), we have 𝑋^′_𝑖𝑇₁=𝜆𝑋_𝑖𝑇₁−𝑎_𝑖𝑇₁𝑒^{𝜆𝛼𝑖(𝑇1)}𝑋_𝑖𝑇₁+𝑎_𝑖𝑇₁𝑒^{𝜆𝛼𝑖(𝑇1)}×^𝑇1_{𝑇1−𝛼𝑖(𝑇1)}𝜆𝑋_𝑖(𝑠)−𝑎_𝑖(𝑠)×𝑒^{𝜆𝛼𝑖(𝑠)}𝑋_𝑖𝑠−𝛼_𝑖(𝑠)+^𝑚_𝑗=1𝑎_𝑖𝑗(𝑠)𝑒^𝜆𝑠𝑓_𝑗𝑦_𝑗(𝑠)+^𝑚_𝑗=1𝑏_𝑖𝑗(𝑠)𝑒^𝜆𝑠𝑓_𝑗𝑦_𝑗((𝑠+𝑠−𝜏))_𝑚𝑗=1𝛼_𝑖𝑗(𝑠)𝑒^𝜆𝑠𝑠_−∞𝑘_𝑗(𝑠−𝑢)𝑓_𝑗𝑦_𝑗(𝑢)+𝑑𝑢_𝑚𝑗=1𝛽_𝑖𝑗(𝑠)𝑒^𝜆𝑠×^𝑠_−∞𝑘_𝑗(𝑠−𝑢)𝑓_𝑗𝑦_𝑗(𝑢)+𝑑𝑢𝑑𝑠_𝑚𝑗=1𝑎_𝑖𝑗𝑇₁𝑒^𝜆𝑇1𝑓_𝑗𝑦_𝑗𝑇₁+^𝑚_𝑗=1𝑏_𝑖𝑗𝑇₁𝑒^𝜆𝑇1𝑓_𝑗𝑦_𝑗((𝑡+𝑡−𝜏))_𝑚𝑗=1𝛼_𝑖𝑗𝑇₁𝑒^{𝜆𝑡𝑇1}_−∞𝑘_𝑗𝑇₁−𝑠𝑓_𝑗𝑦_𝑗(𝑠)+𝑑𝑠_𝑚𝑗=1𝛽_𝑖𝑗𝑇₁𝑒^{𝜆𝑇1𝑇1}_−∞𝑘_𝑗𝑇₁−𝑠𝑓_𝑗𝑦_𝑗(𝑠)𝑑𝑠≥𝜆𝑋_𝑖𝑇₁−𝑎_𝑖𝑇₁𝑒^{𝜆𝛼𝑖(𝑇1)}𝑋_𝑖𝑇₁+𝑎_𝑖𝑇₁𝑒^{𝜆𝛼𝑖(𝑇1)}×^𝑇1_{𝑇1−𝛼𝑖(𝑇1)}𝜆𝑋_𝑖𝑇₁+𝑎⁺_𝑖𝑒^{𝜆𝛼+𝑖}𝑋_𝑖𝑇₁−^𝑚_𝑗=1𝑎⁺_𝑖𝑗𝐿_𝑗𝑌_𝑗(𝑠−_𝑚𝑗=1)𝑏⁺_𝑖𝑗𝑒^𝜆𝜏𝐿_𝑗𝑌_𝑗((𝑠−𝑠−𝜏))_𝑚𝑗=1𝛼⁺_𝑖𝑗^𝑠_−∞𝑘_𝑗(𝑠−𝑢)𝑒^{𝜆(𝑠−𝑢)}𝐿_𝑗𝑌_𝑗(𝑢−)𝑑𝑢_𝑚𝑗=1𝛽⁺_𝑖𝑗^𝑠_−∞𝑘_𝑗(𝑠−𝑢)𝑒^{𝜆(𝑠−𝑢)}×𝐿_𝑗𝑌_𝑗(𝑢−)𝑑𝑢𝑑𝑠_𝑚𝑗=1𝑎⁺_𝑖𝑗𝐿_𝑗𝑌_𝑗𝑇₁−^𝑚_𝑗=1𝑏⁺_𝑖𝑗𝐿_𝑗𝑒^𝜆𝜏𝑌_𝑗𝑇₁−𝜏𝑇₁−^𝑚_𝑗=1𝛼⁺_𝑖𝑗^𝑇1_−∞𝑘_𝑗𝑇₁−𝑠𝑒^{𝜆(𝑇1−𝑠)}𝐿_𝑗𝑌_𝑗(𝑠−)𝑑𝑠_𝑚𝑗=1𝛽⁺_𝑖𝑗^𝑇1_−∞𝑘_𝑗𝑇₁−𝑠𝑒^{𝜆(𝑇1−𝑠)}𝐿_𝑗𝑌_𝑗(𝑠𝑎)𝑑𝑠≥−_𝑖𝑇₁𝑒^{𝜆𝛼𝑖(𝑇1)}−𝜆−𝑎_𝑖𝑇₁𝑒^{𝜆𝛼𝑖(𝑇1)}×𝛼_𝑖𝑇₁𝜆+𝑎⁺_𝑖𝑒^{𝜆𝛼+𝑖}𝑋_𝑖𝑇₁+𝑎_𝑖𝑇₁𝑒^{𝜆𝛼𝑖(𝑇1)}×^𝑇1_{𝑇1−𝛼𝑖(𝑇1)}−^𝑚_𝑗=1𝑎⁺_𝑖𝑗𝐿_𝑗𝑀−^𝑚_𝑗=1𝑏⁺_𝑖𝑗𝑒^𝜆𝜏𝐿_𝑗𝑀−^𝑚_𝑗=1𝛼⁺_𝑖𝑗𝐿_𝑗𝑀×^𝑠_−∞𝑘_𝑗(𝑠−𝑢)𝑒^{𝜆(𝑠−𝑢)}−𝑑𝑢_𝑚𝑗=1𝛽⁺_𝑖𝑗𝐿_𝑗𝑀^𝑠_−∞𝑘_𝑗(𝑠−𝑢)𝑒^{𝜆(𝑠−𝑢)}−𝑑𝑢𝑑𝑠_𝑚𝑗=1𝑎⁺_𝑖𝑗𝐿_𝑗𝑀−^𝑚_𝑗=1𝑏⁺_𝑖𝑗𝑒^𝜆𝜏𝐿_𝑗𝑀−^𝑚_𝑗=1𝛼⁺_𝑖𝑗𝐿_𝑗𝑀×^𝑇1_−∞𝑘_𝑗𝑇₁−𝑠𝑒^{𝜆(𝑇1−𝑠)}−𝑑𝑢_𝑚𝑗=1𝛽⁺_𝑖𝑗𝐿_𝑗𝑀^𝑇1_−∞𝑘_𝑗𝑇₁−𝑠𝑒^{𝜆(𝑇1−𝑠)}≥−𝑎𝑑𝑢_𝑖𝑇₁𝑒^{𝜆𝛼𝑖(𝑇1)}−𝜆−𝑎_𝑖𝑇₁𝑒^{𝜆𝛼𝑖(𝑇1)}×𝛼_𝑖𝑇₁𝜆+𝑎⁺_𝑖𝑒^{𝜆𝛼+𝑖}𝑋_𝑖𝑇₁+1+𝑎_𝑖𝑇₁𝑒^{𝜆𝛼𝑖(𝑇1)}𝛼_𝑖𝑇₁×^𝑚_𝑗=1𝑎⁺_𝑖𝑗+𝑏⁺_𝑖𝑗𝑒^𝜆𝜏+𝛼⁺_𝑖𝑗+𝛽⁺_𝑖𝑗×₀^+∞𝑘_𝑗(𝑢)𝑒^𝜆𝑢𝑑𝑢𝐿_𝑗(−𝑀)>0. (29) This is also a contradiction.
Similarly, if (3) (or (4)) holds, we can derive a contradiction. Therefore, (27) holds.
Furthermore, together with (16) and (17), we have 𝑥_𝑖𝑡⁺_𝑘=𝑥_𝑖𝑡⁺_𝑘−𝑥^∗_𝑖𝑡⁺_𝑘≤𝑥_𝑖𝑡_𝑘−𝑥^∗_𝑖𝑡_𝑘=𝑥_𝑖𝑡⁻_𝑘=𝑋_𝑖𝑡⁻_𝑘𝑒^{−𝜆𝑡𝑘}≤𝑀𝑒^{−𝜆𝑡𝑘},𝑦_𝑗𝑡⁺_𝑘=𝑦_𝑗𝑡⁺_𝑘−𝑦^∗_𝑗𝑡⁺_𝑘≤𝑦_𝑗𝑡_𝑘−𝑦^∗_𝑗𝑡_𝑘=𝑦_𝑗𝑡⁻_𝑘=𝑌_𝑗𝑡⁻_𝑘𝑒^{−𝜆𝑡𝑘}≤𝑀𝑒^{−𝜆𝑡𝑘}, (30) where 𝑖=1,2,…,𝑛, 𝑗=1,2,…,𝑚, and 𝑘=1,2,….
From (27) and (30), we get 𝑥_𝑖(𝑡)−𝑥^∗_𝑖(𝑡)=𝑂𝑒^−𝜆𝑡,𝑦_𝑗(𝑡)−𝑦^∗_𝑗(𝑡)=𝑂𝑒^−𝜆𝑡 (31) for all 𝑡>0, 𝑖=1,2,…,𝑛, and 𝑗=1,2,…,𝑚. Therefore, system (3) is exponentially stable. This completes the proof.

3. An Example

In this section, we present an example to illustrate the feasibility of our results obtained in previous sections.

Example 4. Consider the following fuzzy BAM neural networks with distributed delays and impulses: 𝑥^′_𝑖(𝑡)=−𝑎_𝑖(𝑡)𝑥_𝑖𝑡−𝛼_𝑖(𝑡)+²_𝑗=1𝑎_𝑖𝑗(𝑡)𝑓_𝑗𝑦_𝑗(𝑡+_2𝑗=1)𝑏_𝑖𝑗(𝑡)𝑓_𝑗𝑦_𝑗(𝑡−𝜏(𝑡))+²_𝑗=1𝑐_𝑖𝑗(𝑡)𝜔_𝑗+²_𝑗=1𝛼_𝑖𝑗(𝑡)^𝑡_−∞𝑘_𝑗(𝑡−𝑠)𝑓_𝑗𝑦_𝑗(𝑠)+𝑑𝑠_2𝑗=1𝛽_𝑖𝑗(𝑡)^𝑡_−∞𝑘_𝑗(𝑡−𝑠)𝑓_𝑗𝑦_𝑗(𝑠)+𝑑𝑠_2𝑗=1𝑇_𝑖𝑗𝜔_𝑗+^𝑚_𝑗=1𝐻_𝑖𝑗𝜔_𝑗+𝐴_𝑖(𝑡),𝑡≥0,𝑡≠𝑡_𝑘,Δ𝑥_𝑖𝑡_𝑘=−𝜃_𝑖𝑘𝑥_𝑖𝑡_𝑘𝑦,𝑘=1,2,…,_′𝑗(𝑡)=−𝑏_𝑗(𝑡)𝑦_𝑗𝑡−𝛽_𝑗(𝑡)+²_𝑖=1𝑑_𝑗𝑖(𝑡)𝑔_𝑖𝑥_𝑖(𝑡)+²_𝑖=1𝑝_𝑗𝑖(𝑡)𝑔_𝑖𝑥_𝑖(𝑡+𝑡−𝜌)_2𝑖=1𝑞_𝑗𝑖(𝑡)𝜇_𝑖+^𝑛_𝑖=1𝛾_𝑗𝑖(𝑡)^𝑡_−∞𝑘_𝑖(𝑡−𝑠)𝑔_𝑖𝑥_𝑖(𝑠)+𝑑𝑠_2𝑖=1𝜂_𝑗𝑖(𝑡)^𝑡_−∞𝑘_𝑖(𝑡−𝑠)𝑔_𝑖𝑥_𝑖(𝑠)+𝑑𝑠_2𝑖=1𝑅_𝑗𝑖𝜇_𝑖+^𝑛_𝑖=1𝑆_𝑗𝑖𝜇_𝑖+𝐵_𝑗(𝑡),𝑡≥0,𝑡≠𝑡_𝑘,Δ𝑦_𝑗𝑡_𝑘=−𝜃_𝑗𝑘𝑦_𝑗𝑡_𝑘,𝑘=1,2,…, (32) where 𝑖,𝑗=1,2, 𝑓_𝑗(𝑥)=𝑔_𝑖(𝑥)=(1/8)(|𝑥+1|−|𝑥−1|), and 𝑡₁<𝑡₂<⋯ is strictly increasing sequences such that lim_𝑘→∞𝑡_𝑘=+∞; the coefficients are as follows: 𝑎₁(𝑡)=12(9+sin𝑡),𝑎₂(𝑡)||,𝛼=4+sin𝑡₁(𝑡)=𝛼₂(𝑡)=1(),𝑏201+sin𝑡₁(𝑡)=12(9+cos𝑡),𝑏₂(𝑡)||,𝛽=4+cos𝑡₁(𝑡)=𝛽₂(𝑡)=1(),𝑎201+cos𝑡₁₁(𝑡)=𝑎₁₂(𝑡)=𝑎₂₁(𝑡)=𝑎₂₂(𝑡)=19(),𝑏1+2sin𝑡₁₁(𝑡)=𝑏₁₂(𝑡)=𝑏₂₁(𝑡)=𝑏₂₂(𝑡)=19(),𝑑1+2cos𝑡₁₁(𝑡)=𝑑₁₂(𝑡)=𝑑₂₁(𝑡)=𝑑₂₂(𝑡)=16(),𝑝1+sin𝑡₁₁(𝑡)=𝑝₁₂(𝑡)=𝑝₂₁(𝑡)=𝑝₂₂(𝑡)=16(),𝛼1+cos𝑡₁₁(𝑡)=𝛼₁₂(𝑡)=𝛼₂₁(𝑡)=𝛼₂₂(𝑡)=13𝛽sin𝑡,₁₁(𝑡)=𝛽₁₂(𝑡)=𝛽₂₁(𝑡)=𝛽₂₂(𝑡)=13𝛾cos𝑡,₁₁(𝑡)=𝛾₁₂(𝑡)=𝛾₂₁(𝑡)=𝛾₂₂(𝑡)=1(),𝜂121+3sin𝑡₁₁(𝑡)=𝜂₁₂(𝑡)=𝜂₂₁(𝑡)=𝜂₂₂(𝑡)=1(),𝑇121+3cos𝑡_𝑖𝑗=𝐻_𝑖𝑗=𝑅_𝑖𝑗=𝑆_𝑖𝑗𝜔=1,_𝑗=𝜇_𝑖,𝐴=1𝑖,𝑗=1,2₁(𝑡)=𝐴₂(𝑡)=12cos𝑡,𝐵₁(𝑡)=𝐵₂(𝑡)=12𝜃sin𝑡,_𝑖𝑘1=1−3sin(2+𝑘),𝜃_𝑗𝑘2=1+3cos(3𝑘),𝑖,𝑗=1,2. (33) By calculating, we have 𝑎⁺₁=𝑎⁺₂=𝑏⁺₁=𝑏⁺₂𝑎=5,₋₁=𝑎⁻₂=𝑏⁻₁=𝑏⁻₂𝛼=4,₊₁=𝛼⁺₂=𝛽⁺₁=𝛽⁺₂=1,𝑎10_+𝑖𝑗=𝑏⁺_𝑖𝑗=𝑑⁺_𝑗𝑖=𝑝⁺_𝑗𝑖=𝛼⁺_𝑖𝑗=𝛽⁺_𝑖𝑗=𝛾⁺_𝑗𝑖=𝜂⁺_𝑗𝑖=13,𝐿𝑖,𝑗=1,2_𝑗=𝑀_𝑖=14𝑖,𝑗=1,2. (34) We can see that system (32) satisfies the conditions (H₁), (H₂), and (H₄); for 𝑖=1,2, 𝑗=1,2, we have −𝑎⁻_𝑖1−𝑎⁺_𝑖𝛼⁺_𝑖+1+𝑎⁺_𝑖𝛼⁺_𝑖ײ_𝑗=1𝑎⁺_𝑖𝑗+𝑏⁺_𝑖𝑗+𝛼⁺_𝑖𝑗+𝛽⁺_𝑖𝑗𝐿_𝑗<0,−𝑏⁻_𝑗1−𝑏⁺_𝑗𝛽⁺_𝑗+1+𝑏⁺_𝑗𝛽⁺_𝑗ײ_𝑖=1𝑑⁺_𝑗𝑖+𝑝⁺_𝑗𝑖+𝛾⁺_𝑗𝑖+𝜂⁺_𝑗𝑖𝑀_𝑖<0, (35) which implies that (H₃) holds. Therefore, from Theorem 3, system (32) is exponentially stable (Figure 1 illustrates our plausible results).

(a)

(b)

(c)

(d)

Figure 1 

Phase responses of states 𝑥1, 𝑥2, 𝑦1, and 𝑦2.

4. Conclusion

In this paper, we consider a class of BAM fuzzy cellular neural networks with time-varying delays in leakage terms and impulses. By using differential inequality techniques, we obtain sufficient conditions for the exponential stability of this class of networks. Our results are completely new and complementary to the previously known results. Finally, an example is given to demonstrate the effectiveness and conservativeness of our theoretical results.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

This work is supported by the National Natural Science Foundation of People’s Republic of China under Grant 11361072.