Existence, Uniqueness, and Stability Analysis of Impulsive Neural Networks with Mixed Time Delays

Abstract

We study a class of impulsive neural networks with mixed time delays and generalized activation functions. The mixed delays include time-varying transmission delay, bounded time-varying distributed delay, and discrete constant delay in the leakage term. By using the contraction mapping theorem, we obtain a sufficient condition to guarantee the global existence and uniqueness of the solution for the addressed neural networks. In addition, a delay-independent sufficient condition for existence of an equilibrium point and some delay-dependent sufficient conditions for stability are derived, respectively, by using topological degree theory and Lyapunov-Krasovskii functional method. The presented results require neither the boundedness, monotonicity, and differentiability of the activation functions nor the differentiability (even differential boundedness) of time-varying delays. Moreover, the proposed stability criteria are given in terms of linear matrix inequalities (LMI), which can be conveniently checked by the MATLAB toolbox. Finally, an example is given to show the effectiveness and less conservativeness of the obtained results.

1. Introduction

As we know, time delay in a system is a common phenomenon that describes the fact that the future state of the system depends not only on the present state but also on the past state and is always unavoidably encountered in many fields such as automatic control, biological chemistry, physical engineer, and neural networks [1–5]. Moreover, time delays can affect the stability of a neural network and create oscillatory and bad dynamic performance [3–5]. Hence, it is significant and necessary to take into account the delay effects on dynamics of neural networks, for example, existence, uniqueness and stability, and so on. To date, neural network models with two categories of time delays, namely, discrete and continuously distributed time delays, have been extensively investigated by many researchers, using some effective approaches; see [6–22] and references therein. For instance, in [6], Kharitonov and Zhabko studied the robust stability of time-delay systems via Lyapunov-Krasovskii functional approach. Wu et al. [9] introduced free-weighting matrix approach and investigated the robust stability problem for time-varying delay systems. Gu introduced the delay decomposition method in [10].

Recently, a special type of time delay, namely, leakage delay (or forgetting delay), is identified and investigated due to its existence in many real systems. In 2007, Gopalsamy [11] proposed the bidirectional associative memory neural networks with constant delay in the leakage term and derived sufficient conditions for existence and stability of equilibrium. Based on this work, Li and Huang [12] investigated the stability of general nonlinear systems with leakage delay, by model transformation, contraction mapping theorem, and degenerate Lyapunov-Krasovskii functional. However, dynamical analysis of neural networks with time delay in leakage term has been little considered due to some theoretical and technical difficulties [23–30]. In fact, time delay in the stabilizing negative feedback term has a tendency to destabilize a system [11] and has great impact on the dynamics of neural networks.

On the other hand, besides delay, impulses are also likely to exist in neural networks. In implementation of electronic networks, the state is subject to instantaneous perturbations and experiences abrupt change at certain moments, which may be caused by switching phenomenon, frequency change, or other sudden noises; that is, it does exhibit impulsive effects; see [13–21, 31–35]. Therefore, impulsive perturbations should be taken into account when studying the dynamics of neural networks. Since the existence of delays and impulses is frequently a source of instability, bifurcation, and chaos for dynamical systems, it is significant to study both delay and impulsive effects on dynamical systems [13–21, 25, 26, 32]. In [25], Li et al. investigated the existence, uniqueness, and stability problems of recurrent neural networks with discrete time delay and time delay in the leakage term under impulsive perturbations, while being without distributed delay. Since neural networks usually have a spatial extent, there is a distribution of propagation delays over a period of time. In these circumstances the signal propagation is not instantaneous and cannot be modelled only with discrete delays and a more appropriate way is to incorporate continuously distributed delays in neural network models. To the best of our knowledge, so far, there has been very little existing work on impulsive neural networks with time delay in the leakage term and discrete and time-varying distributed delays via LMI approach [27].

Motivated by aforementioned discussion, in this paper, we consider a class of impulsive neural networks with mixed time delays and generalized activation functions which could be different from each other. The mixed time delays include time-varying transmission delay, bounded time-varying distributed delay, and constant delay in the leakage term. Firstly, by using the contraction mapping theorem, we obtain a delay-independent sufficient condition to guarantee the global existence and uniqueness of the solution for the addressed neural networks. Secondly, we present a delay-independent sufficient condition to guarantee the existence of an equilibrium point by using topological degree theory. Thirdly, some sufficient conditions which are dependent on the leakage delay, time-varying transmission delay, and distributed delay have been derived to guarantee the global asymptotic stability of the equilibrium point by using a new Lyapunov-Krasovskii functional and some analysis technique. The presented results require neither the boundedness, monotonicity, and differentiability of the activation functions nor the differentiability (even differential boundedness) of time-varying delays, which are more effective and less conservative than other existing literatures [27]. In the absence of leakage delay, the obtained results are also new ones. Moreover, the proposed stability criteria are given in terms of linear matrix inequalities (LMI) [36] and can be conveniently checked by the LMI toolbox in MATLAB. Finally, an example is given to show the effectiveness and less conservativeness of the obtained results.

Notations. Let ℝ  (ℝ⁺) denote the set of (positive) real numbers, ℤ₊ denote the set of positive integers, and ℝ^𝑛 denote the 𝑛-dimensional real spaces equipped with the Euclidean norm ‖⋅‖. 𝒜>0 or 𝒜<0 denotes that the matrix 𝒜 is a symmetric and positive definite or negative definite matrix. The notations 𝒜^𝑇 and 𝒜⁻¹ mean the transpose of 𝒜 and the inverse of a square matrix. 𝜆_max(𝒜) or 𝜆_min(𝒜) denotes the maximum eigenvalue or the minimum eigenvalue of matrix 𝒜. 𝐼 denotes the identity matrix with appropriate dimensions and Λ={1,2,…,𝑛}. [⋅]^∗ denotes the integer function. For any interval 𝐽⊆ℝ, set 𝑉⊆ℝ^𝑘(1≤𝑘≤𝑛), 𝐶(𝐽,𝑉)={𝜑∶𝐽→𝑉iscontinuous}, and 𝑃𝐶¹(𝐽,𝑉)={𝜑∶𝐽→𝑉 is continuously differentiable everywhere except at finite number of points 𝑡, at which 𝜑(𝑡⁺), 𝜑(𝑡⁻), 𝜑̇(𝑡⁺), and 𝜑̇(𝑡⁻) exist and 𝜑(𝑡⁺)=𝜑(𝑡), 𝜑̇(𝑡⁺)=𝜑̇(𝑡), where 𝜑̇ denotes the derivative of 𝜑}. For any 𝑥=(𝑥₁,𝑥₂,…,𝑥_𝑛)∈ℝ^𝑛, [𝑥]⁺=(|𝑥₁|,|𝑥₂|,…,|𝑥_𝑛|)^𝑇 and, for any 𝑄=(𝑞_𝑖𝑗)_𝑛×𝑛∈ℤ^𝑛×𝑛, [𝑄]⁺=(|𝑞_𝑖𝑗|)_𝑛×𝑛. For any 𝑡∈ℝ⁺, 𝑥_𝑡 is defined by 𝑥_𝑡=𝑥(𝑡+𝑠), 𝑥_𝑡−=𝑥(𝑡⁻+𝑠), 𝑠∈[−𝜎,0]. In addition, the notation ⋆ always denotes the symmetric block in one symmetric matrix.

2. Preliminaries

Consider the following impulsive neural networks model: 𝑥̇(𝑡)=−𝐷𝑥(𝑡−𝜎)+𝐴𝑓(𝑥(𝑡))+𝐵𝑔(𝑥(𝑡−𝜏(𝑡)))+𝑊^𝑡_{(𝑡)𝑡−𝜌}𝐾(𝑡−𝑠)ℎ(𝑥(𝑠))𝑑𝑠+𝐼,𝑡>0,𝑡≠𝑡_𝑘,Δ𝑥𝑡_𝑘=𝑥𝑡_𝑘−𝑥𝑡⁻_𝑘=𝐼_𝑘𝑥𝑡⁻_𝑘,𝑥_𝑡⁻_𝑘,𝑘∈ℤ₊, (1) where 𝑥(𝑡)=(𝑥₁(𝑡),…,𝑥_𝑛(𝑡))^𝑇 is the neuron state vector of the neural network; 𝐷=diag(𝑑₁,…,𝑑_𝑛) is a diagonal matrix with 𝑑_𝑖>0, 𝑖∈Λ, Λ={1,2,…,𝑛};  𝐴, 𝐵, and 𝑊 are the connection weight matrix, the delayed weight matrix, and the distributively delayed connection weight matrix, respectively; 𝐼 is an external input; 𝜎≥0 is a constant which denotes the leakage delay; 𝜏(𝑡) is a time-varying transmission delay of the neural network; 𝜌(𝑡) is a time-varying distributed delay of the neural network; 𝑓(𝑥(⋅))=(𝑓₁(𝑥₁(⋅)),…,𝑓_𝑛(𝑥_𝑛(⋅)))^𝑇, 𝑔(𝑥(⋅))=(𝑔₁(𝑥₁(⋅)),…,𝑔_𝑛(𝑥_𝑛(⋅)))^𝑇, and ℎ(𝑥(⋅))=(ℎ₁(𝑥₁(⋅)),…,ℎ_𝑛(𝑥_𝑛(⋅)))^𝑇 represent the neuron activation functions; 𝐾(⋅)=diag(𝑘₁(⋅),…,𝑘_𝑛(⋅)) is the delay kernel function.

Throughout this paper, we make the following assumptions.(H₁)𝜏(𝑡) represents the discrete transmission delay with 0≤𝜏(𝑡)≤𝜏; 𝜌(𝑡) represents the time-varying distributed delay with 0≤𝜌(𝑡)≤𝜌, where 𝜏, 𝜌 are two positive constants.(H₂)The delay kernels 𝑘_𝑗(⋅), 𝑗∈Λ, are some real valued continuous functions defined on [0,𝜌] and satisfy 𝑘_𝑗(𝑠)≤|𝑘(𝑠)|,^𝜌₀|𝑘(𝑠)|𝑑𝑠=𝜅, (2) where 𝜅 is a positive constant.(H₃)The neuron activation functions 𝑓_𝑗, 𝑔_𝑗, and ℎ_𝑗, 𝑗∈Λ, are continuous on ℝ and satisfy 𝜎⁻_𝑗≤𝑓_𝑗(𝑢)−𝑓_𝑗(𝑣)𝑢−𝑣≤𝜎⁺_𝑗,𝛿⁻_𝑗≤𝑔_𝑗(𝑢)−𝑔_𝑗(𝑣)𝑢−𝑣≤𝛿⁺_𝑗,𝜁⁻_𝑗≤ℎ_𝑗(𝑢)−ℎ_𝑗(𝑣)𝑢−𝑣≤𝜁⁺_𝑗, (3) for any 𝑢, 𝑣∈ℝ, 𝑢≠𝑣, 𝑗∈Λ, where 𝜎⁻_𝑗, 𝜎⁺_𝑗, 𝛿⁻_𝑗, 𝛿⁺_𝑗, 𝜁⁻_𝑗, and 𝜁⁺_𝑗 are some real constants and they may be positive, zero, or negative.(H₄)𝐼_𝑘(⋅)∶ℝ×ℝ^𝑛→ℝ^𝑛, 𝑘∈ℤ₊, are some continuous functions.(H₅)The impulse times 𝑡_𝑘 satisfy 0=𝑡₀<𝑡₁<⋯<𝑡_𝑘→∞ and inf_𝑘∈ℤ+{𝑡_𝑘−𝑡_𝑘−1}>0.

We will consider model (1) with the initial condition 𝑥(𝑠)=𝜑(𝑠),𝑠∈−𝜂,0, (4) where 𝜂=max{𝜎,𝜏,𝜌}, 𝜑(⋅)=(𝜑₁,…,𝜑_𝑛)^𝑇∈𝑃𝐶¹([−𝜂,0]), ℝ^𝑛, whose norm is defined by 𝜑²_𝜂=maxmax^𝑛_{−𝜂≤𝜃≤0𝑖=1}𝜑²_𝑖(𝜃),max^𝑛_{−𝜏≤𝜃≤0𝑖=1}𝜑̇²_𝑖(𝜃). (5)

Definition 1 (see [37]). Assume that Ω∈ℝ^𝑛 is a bounded and open set and ℱ(𝑢)∶Ω→ℝ^𝑛 is a continuous and differentiable function. If 𝑝∈ℱ(𝜕Ω) and 𝐽_ℱ(𝑢)≠0 for any 𝑢∈ℱ⁻¹(𝑝), where 𝜕Ω denotes the boundary of Ω and 𝐽_ℱ denotes the Jacobian determinant relative to ℱ, then the topological degree relative to Ω and 𝑝 is defined by degℱ,Ω,𝑝=_𝑢∈ℱ⁻¹_(𝑝)sgn𝐽_ℱ(𝑢),ℱ⁻¹𝑝≠∅,0,ℱ⁻¹𝑝=∅. (6)

Remark 2. Generally speaking, the topological degree of ℱ(𝑢) relative to Ω and 𝑝 can be regarded as the algebraic number of solutions of ℱ(𝑢)=𝑝 in Ω if ℱ(𝜕Ω)≠0. For instance, deg(ℱ,Ω,0)=±1 implies that ℱ(𝑢)=0 has at least one solution in Ω.

Lemma 3 (see [38]). Given any real matrix 𝑀=𝑀^𝑇>0 of appropriate dimension and a vector function 𝜔(⋅)∶[𝑎,𝑏]→ℝ^𝑛, such that the integrations concerned are well defined, then ^𝑏_𝑎𝜔(𝑠)𝑑𝑠^𝑇𝑀^𝑏_𝑎𝜔(𝑠)≤()𝑑𝑠𝑏−𝑎^𝑏_𝑎𝜔^𝑇(𝑠)𝑀𝜔(𝑠)𝑑𝑠. (7)

Lemma 4 (see [12]). Given any real matrices Σ₁, Σ₂, Σ₃ of appropriate dimensions and a scalar 𝜖>0 such that 0<Σ₃=Σ^𝑇₃, then the following inequality holds: Σ^𝑇₁Σ₂+Σ^𝑇₂Σ₁≤𝜖Σ^𝑇₁Σ₃Σ₁+𝜖⁻¹Σ^𝑇₂Σ₃⁻¹Σ₂. (8)

Lemma 5 (see [27]). Supposing that ℧, ℧_𝑖𝑗≥0(𝑖,𝑗=1,2) are symmetric matrices of appropriate dimensions, 𝛼∈[0,1] and 𝛽∈[0,1], then ℧+[(1−𝛼)℧₁₁+𝛼℧₁₂]+[(1−𝛽)℧₂₁+𝛽℧₂₂]<0 holds if the following four inequalities ℧+℧₁₁+℧₂₁<0, ℧+℧₁₁+℧₂₂<0, ℧+℧₁₂+℧₂₁<0, and ℧+℧₁₂+℧₂₂<0 hold simultaneously.

For presentation convenience, in the following, we denote Σ₁=diag𝜎⁻₁𝜎⁺₁,…,𝜎⁻_𝑛𝜎⁺_𝑛,Σ₂𝜎=diag⁻¹+𝜎⁺₁2𝜎,…,^−𝑛+𝜎⁺_𝑛2,Σ₃=diag𝛿⁻₁𝛿⁺₁,…,𝛿⁻_𝑛𝛿⁺_𝑛,Σ₄𝛿=diag⁻¹+𝛿⁺₁2𝛿,…,^−𝑛+𝛿⁺_𝑛2,Σ₅=diag𝜁⁻₁𝜁⁺₁,…,𝜁⁻_𝑛𝜁⁺_𝑛,Σ₆𝜁=diag⁻¹+𝜁⁺₁2𝜁,…,^−𝑛+𝜁⁺_𝑛2. (9)

3. Global Existence and Uniqueness of Solution

In this section, by using the contraction mapping theorem, we give a delay-independent sufficient condition to guarantee the global existence and uniqueness of the solution for models (1) and (4).

Theorem 6. Assume that the assumptions (𝐻₁)--(𝐻₅) hold; then the solution 𝑥=𝑥(𝑡,0,𝜑) of models (1) and (4) exists uniquely on [−𝜂,∞).

Proof. Transform the global existence and uniqueness of solution of the models (1) and (4) into a fixed point problem. Let ‖⋅‖^∗ be the norm in 𝐶([0,𝑡₁],ℝ^𝑛) defined by ‖𝑢‖^∗≐max_{𝑡∈0,𝑡1}𝑒^−𝜆𝑡⋅max_{[]𝑠∈0,𝑡}{‖𝑢(𝑠)‖},𝑢∈𝐶0,𝑡₁,ℝ^𝑛, (10) where 𝜆=^𝑛_𝑗=1𝑑²_𝑗+𝜎⋅^𝑛𝑛_𝑖=1𝑗=1𝑎²_𝑖𝑗+𝛿⋅^𝑛𝑛_𝑖=1𝑗=1𝑏²_𝑖𝑗+𝜁⋅𝜅⋅^𝑛𝑛_𝑖=1𝑗=1𝑤²_𝑖𝑗+1,𝜎=max_𝑗∈Λ𝜎⁻_𝑗,𝜎⁺_𝑗,𝛿=max_𝑗∈Λ𝛿⁻_𝑗,𝛿⁺_𝑗,𝜁=max_𝑗∈Λ𝜁⁻_𝑗,𝜁⁺_𝑗. (11)(12) Then it is easy to see that 𝐶([0,𝑡₁],ℝ^𝑛) is a Banach space endowed with the norm ‖⋅‖^∗. Let 𝑢∈𝐶([0,𝑡₁],ℝ^𝑛) and consider the operator ℒ₁∶𝐶([0,𝑡₁],ℝ^𝑛)→𝐶([0,𝑡₁],ℝ^𝑛) defined by ℒ₁𝑢(𝑡)=𝜑(0)+^𝑡₀()(𝑢(𝑠(𝑢((𝑠−𝐷𝑢𝑠−𝜎+𝐴𝑓))+𝐵𝑔𝑠−𝜏)))+𝑊₀^𝜌(𝑡)𝐾(𝜃)ℎ(𝑢(𝑠−𝜃))𝑑𝜃+𝐼𝑑𝑠, (13) where 𝑢(𝑠)=𝜑(𝑠), 𝑠∈[−𝜂,0].
First we show that ℒ₁ is a contraction on 𝐶([0,𝑡₁],ℝ^𝑛). Let 𝑢,𝑣∈𝐶([0,𝑡₁],ℝ^𝑛); we have ℒ₁𝑢(𝑡)−ℒ₁𝑣(𝑡=−_𝑡0)𝐷[𝑢()(+𝑠−𝜎−𝑣𝑠−𝜎)]𝑑𝑠_𝑡0𝐴𝑓(𝑢(𝑠))−𝑓(𝑣(𝑠))𝑑𝑠+^𝑡₀𝐵𝑓(𝑢(𝑠−𝜏(𝑠)))−𝑓(𝑣(𝑠−𝜏(𝑠)))𝑑𝑠+^𝑡₀𝑊₀^𝜌(𝑡)𝐾(𝜃ℎ(𝑢((𝑣(≤)[𝑠−𝜃))−ℎ𝑠−𝜃))]𝑑𝜃𝑑𝑠_𝑡0‖𝐷[𝑢()(+𝑠−𝜎−𝑣𝑠−𝜎)]‖𝑑𝑠_𝑡0𝐴𝑓(𝑢(𝑠))−𝑓(𝑣(𝑠))𝑑𝑠+^𝑡₀𝐵𝑓(𝑢(𝑠−𝜏(𝑠)))−𝑓(𝑣(𝑠−𝜏(𝑠)))𝑑𝑠+^𝑡₀𝑊₀^𝜌(𝑡)𝐾(𝜃ℎ(𝑢((𝑣(≤)[𝑠−𝜃))−ℎ𝑠−𝜃))]𝑑𝜃𝑑𝑠_𝑛𝑗=1𝑑²_𝑗^𝑡₀‖𝑢(𝑠−𝜎)−𝑣(𝑠−𝜎)‖𝑑𝑠+𝜎⋅^𝑛𝑛_𝑖=1𝑗=1𝑎²_𝑖𝑗^𝑡₀‖𝑢(𝑠)−𝑣(𝑠)‖𝑑𝑠+𝛿⋅^𝑛𝑛_𝑖=1𝑗=1𝑏²_𝑖𝑗^𝑡₀‖𝑢((𝑠((𝑠+𝑠−𝜏))−𝑣𝑠−𝜏))‖𝑑𝑠_𝑡0𝑊₀^𝜌(𝑡)𝐾(𝜃)[ℎ(𝑢(𝑠−𝜃))−ℎ(𝑣(𝑠−𝜃))]𝑑𝜃𝑑𝑠. (14) In view of (𝐻₂), we get ^𝑡₀𝑊₀^𝜌(𝑡)𝐾(𝜃)[ℎ(𝑢(𝑠−𝜃))−ℎ(𝑣(𝑠−𝜃))]𝑑𝜃𝑑𝑠≤𝜁⋅^𝑛𝑛_𝑖=1𝑗=1𝑤²_𝑖𝑗^𝑡₀₀^𝜌(𝑡)𝐾(𝜃)[𝑢(𝑠−𝜃)−𝑣(𝑠−𝜃)]𝑑𝜃𝑑𝑠≤𝜁⋅^𝑛𝑛_𝑖=1𝑗=1𝑤²_𝑖𝑗^𝑡₀₀^𝜌(𝑡)‖𝐾(𝜃)[𝑢(𝑠−𝜃)−𝑣(𝑠−𝜃)]‖𝑑𝜃𝑑𝑠≤𝜁⋅^𝑛𝑛_𝑖=1𝑗=1𝑤²_𝑖𝑗^𝑡₀₀^𝜌(𝑡)|𝑘(𝜃)|‖𝑢(𝑠−𝜃)−𝑣(𝑠−𝜃)‖𝑑𝜃𝑑𝑠≤𝜁⋅^𝑛𝑛_𝑖=1𝑗=1𝑤²_𝑖𝑗^𝑡₀₀^𝜌(𝑡)|𝑘(𝜃)|𝑑𝜃max_{𝑟∈[0,𝑠]}‖𝑢(𝑟)−𝑣(𝑟)‖𝑑𝑠≤𝜁⋅𝜅⋅^𝑛𝑛_𝑖=1𝑗=1𝑤²_𝑖𝑗^𝑡₀max_{𝑟∈[0,𝑠]}‖𝑢(𝑟)−𝑣(𝑟)‖𝑑𝑠. (15) Substituting the above inequality to (14), we obtain ℒ₁𝑢(𝑡)−ℒ₁𝑣(𝑡≤_𝑛𝑗=1)𝑑²_𝑗^𝑡₀max_{𝑟∈[0,𝑠]}‖𝑢(𝑟)−𝑣(𝑟)‖𝑑𝑠+𝜎⋅^𝑛𝑛_𝑖=1𝑗=1𝑎²_𝑖𝑗^𝑡₀max_{𝑟∈[0,𝑠]}‖𝑢(𝑟)−𝑣(𝑟)‖𝑑𝑠+𝛿⋅^𝑛𝑛_𝑖=1𝑗=1𝑏²_𝑖𝑗^𝑡₀max_{𝑟∈[0,𝑠]}‖𝑢(𝑟)−𝑣(𝑟)‖𝑑𝑠+𝜁⋅𝜅⋅^𝑛𝑛_𝑖=1𝑗=1𝑤²_𝑖𝑗^𝑡₀max_{𝑟∈[0,𝑠]}‖𝑢(𝑟)(𝑟=()−𝑣)‖𝑑𝑠𝜆−1^𝑡₀max_{𝑟∈[0,𝑠]}‖𝑢(𝑟)(𝑟=()−𝑣)‖𝑑𝑠𝜆−1^𝑡₀𝑒^𝜆𝑠𝑒^−𝜆𝑠max_{𝑟∈[0,𝑠]}‖𝑢(𝑟)(𝑟≤()−𝑣)‖𝑑𝑠𝜆−1^𝑡₀𝑒^𝜆𝑑𝑠max_{𝑠∈[0,𝑡1]}𝑒^−𝜆𝑠max_{𝑟∈[0,𝑠]}‖𝑢(𝑟)(𝑟=()−𝑣)‖𝑑𝑠𝜆−1^𝑡₀𝑒^𝜆𝑠𝑑𝑠‖𝑢−𝑣‖^∗≤𝜆−1𝜆𝑒^𝜆𝑡‖𝑢−𝑣‖^∗. (16) Since 𝑒^𝜆𝑡 is increasing in 𝑡, max_{𝑠∈[0,𝑡]}ℒ₁𝑢(𝑠)−ℒ₁𝑣(𝑠)≤𝜆−1𝜆𝑒^𝜆𝑡‖𝑢−𝑣‖^∗. (17) Then 𝑒^−𝜆𝑡max_{𝑠∈[0,𝑡]}ℒ₁𝑢(𝑠)−ℒ₁𝑣(𝑠)≤𝜆−1𝜆‖𝑢−𝑣‖^∗. (18) Note the definition of ‖⋅‖^∗; we have ℒ₁𝑢−ℒ₁𝑣^∗≤𝜆−1𝜆‖𝑢−𝑣‖^∗. (19) Thus ℒ₁ is a contraction on 𝐶([0,𝑡₁],ℝ^𝑛), and it has a unique fixed point 𝑢^∗₁∈𝐶([0,𝑡₁],ℝ^𝑛). Thus we get the fact that 𝑢^∗₁(𝑡₁) exists finitely. It implies that 𝑢^∗₁(𝑡₁)+𝐽₁(𝑢^∗₁(𝑡₁),𝑢^∗_1𝑡1) also exists finitely, since assumption (𝐻₄) holds. Then we replace 𝑢^∗₁(𝑡₁) with 𝑢^∗₁(𝑡₁)+𝐼₁(𝑢^∗₁(𝑡₁),𝑢^∗_1𝑡1) and define 𝜂₁=𝑢^∗₁(𝑡₁)+𝐼₁(𝑢^∗₁(𝑡₁),𝑢^∗_1𝑡1) for later use.
Next we show that ℒ₂ is a contraction on 𝐶([𝑡₁,𝑡₂],ℝ^𝑛). For 𝑢∈𝐶([𝑡₁,𝑡₂],ℝ^𝑛), let ‖𝑢‖^∗≐max_{𝑡∈[𝑡1,𝑡2]}𝑒^{−𝜆(𝑡−𝑡1)}⋅max_{𝑠∈[𝑡1,𝑡]}{‖𝑢(𝑠)‖}, (20) where 𝜆 is defined in (11). Let 𝑢∈𝐶([𝑡₁,𝑡₂],ℝ^𝑛) and consider the operator ℒ₂∶𝐶([𝑡₁,𝑡₂],ℝ^𝑛)→𝐶([𝑡₁,𝑡₂],ℝ^𝑛) defined by ℒ₂𝑢(𝑡)=𝜂₁+^𝑡_𝑡1()(𝑢(𝑠(𝑢((𝑠−𝐷𝑢𝑠−𝜎+𝐴𝑓))+𝐵𝑔𝑠−𝜏)))+𝑊₀^𝜌(𝑡)𝐾(𝜃)ℎ(𝑢(𝑠−𝜃))𝑑𝜃+𝐼𝑑𝑠, (21) where 𝜂₁=𝑢^∗₁(𝑡₁)+𝐼₁(𝑢^∗₁(𝑡₁)) and 𝑢(𝑠)=𝜑(𝑠),𝑢,𝑠∈−𝜂,0_∗1(𝑠),𝑠∈0,𝑡₁. (22) By virtue of the definition of ℒ₂, similar to the proof of (19), we get, for 𝑢,𝑣∈𝐶([𝑡₁,𝑡₂],ℝ^𝑛), ℒ₂𝑢−ℒ₂𝑣^∗≤𝜆−1𝜆‖𝑢−𝑣‖^∗. (23) Thus ℒ₂ is a contraction on 𝐶([𝑡₁,𝑡₂],ℝ^𝑛), and it has a unique fixed point 𝑢^∗₂∈𝐶([𝑡₁,𝑡₂],ℝ^𝑛). Moreover, we know that 𝑢^∗₂(𝑡₂) exists finitely, which implies that 𝑢^∗₂(𝑡₂)+𝐼₂(𝑢^∗₂(𝑡₂),𝑢^∗_2𝑡2) exists finitely in view of assumption (𝐻₄). Then we replace 𝑢^∗₂(𝑡₂) with 𝑢^∗₂(𝑡₂)+𝐼₂(𝑢^∗₂(𝑡₂),𝑢^∗_2𝑡2) and define 𝜂₂=𝑢^∗₂(𝑡₂)+𝐼₂(𝑢^∗₂(𝑡₂),𝑢^∗_2𝑡2) for later use.
Finally we show that ℒ_𝑛+1 is a contraction on 𝐶([𝑡_𝑛,𝑡_𝑛+1],ℝ^𝑛). For 𝑢∈𝐶([𝑡_𝑛,𝑡_𝑛+1],ℝ^𝑛), let ‖𝑢‖^∗≐max_{𝑡∈[𝑡𝑛,𝑡𝑛+1]}𝑒^{−𝜆(𝑡−𝑡𝑛)}⋅max_{𝑠∈[𝑡𝑛,𝑡]}{‖𝑢(𝑠)‖}, (24) where 𝜆 is defined in (11). Let 𝑢∈𝐶([𝑡_𝑛,𝑡_𝑛+1],ℝ^𝑛); then we can similarly consider the operator ℒ_𝑛+1∶𝐶([𝑡_𝑛,𝑡_𝑛+1],ℝ^𝑛)→𝐶([𝑡_𝑛,𝑡_𝑛+1],ℝ^𝑛) defined by ℒ_𝑛+1𝑢(𝑡)=𝜂_𝑛+^𝑡_𝑡𝑛()(𝑢(𝑠(𝑢((𝑠−𝐷𝑢𝑠−𝜎+𝐴𝑓))+𝐵𝑔𝑠−𝜏)))+𝑊₀^𝜌(𝑡)𝐾(𝜃)ℎ(𝑢(𝑠−𝜃))𝑑𝜃+𝐼𝑑𝑠, (25) where 𝜂_𝑛=𝑢^∗_𝑛(𝑡_𝑛)+𝐼_𝑛(𝑢^∗_𝑛(𝑡_𝑛)) and 𝑢(𝑠)=𝜑(𝑠),𝑢,𝑠∈−𝜂,0_∗1(𝑠),𝑠∈0,𝑡₁,𝑢^∗₂(𝑠),𝑠∈𝑡₁,𝑡₂,⋮𝑢^∗_𝑛(𝑠),𝑠∈𝑡_𝑛−1,𝑡_𝑛. (26) Then repeating the argument with ℒ_𝑛+1 replacing ℒ₁, similar to the proof of (19), we see that, for 𝑢,𝑣∈𝐶([𝑡_𝑛,𝑡_𝑛+1],ℝ^𝑛), ℒ_𝑛+1𝑢−ℒ_𝑛+1𝑣^∗≤𝜆−1𝜆‖𝑢−𝑣‖^∗. (27) Thus ℒ_𝑛+1 is a contraction on 𝐶([𝑡_𝑛,𝑡_𝑛+1],ℝ^𝑛), and it has a unique fixed point 𝑢^∗_𝑛+1∈𝐶([𝑡_𝑛,𝑡_𝑛+1],ℝ^𝑛).
Continuing in this manner, we construct 𝑢^∗(𝑡)=𝜑(𝑡),𝑢,𝑡∈−𝜂,0_∗1(𝑡),𝑡∈0,𝑡₁,𝑢^∗₂(𝑡),𝑡∈𝑡₁,𝑡₂,⋮𝑢^∗_𝑛+1(𝑡),𝑡∈𝑡_𝑛,𝑡_𝑛+1,⋮ (28) Then 𝑢^∗(𝑡) is the global solution of models (1) and (4). If 𝑣^∗(𝑡) is another solution of models (1) and (4), then it is easy to check from the above argument that 𝑢^∗(𝑡)=𝑣^∗(𝑡). Hence, the solution 𝑢^∗(𝑡)=𝑢^∗(𝑡,0,𝜑) of models (1) and (4) exists uniquely on [−𝜂,∞). This completes the proof.

4. Existence of an Equilibrium Point

In previous sections, we have showed the global existence and uniqueness of solution for models (1) and (4). In this section, without requiring the boundedness, differentiability, or monotonicity of the activation functions, we establish a delay-independent sufficient condition for the existence of an equilibrium point of model (1). As usual, we denote an equilibrium point of the model (1) by the constant vector 𝑥^∗∈ℝ^𝑛, where 𝑥^∗ satisfies −𝐷𝑥^∗+𝐴𝑓𝑥^∗+𝐵𝑔𝑥^∗+𝑊𝕂ℎ𝑥^∗𝐼+𝐼=0,_𝑘𝑥^∗,𝑥^∗=0,𝑘∈ℤ₊, (29) where 𝕂=𝕂(𝑡)=∫₀^𝜌(𝑡)𝐾(𝑠)𝑑𝑠. In this paper, it is assumed that the impulsive function 𝐼_𝑘 satisfies 𝐼_𝑘(𝑥^∗,𝑥^∗)=0 for 𝑘∈ℤ₊. Hence, to prove the existence of solution of (29), it suffices to show that the following has a solution: 𝑥^∗−𝐷⁻¹𝐴𝑓𝑥^∗−𝐷⁻¹𝐵𝑔𝑥^∗−𝐷⁻¹𝑊𝕂ℎ𝑥^∗−𝐷⁻¹𝐼=0, (30) in view of 𝐷>0.

Theorem 7. Assume that the assumption (𝐻₃) holds. Then model (1) has at least one equilibrium point if, for any 𝑡>0, 𝐷−[𝐴]⁺Σ−[𝐵]⁺Δ−[𝑊𝕂]⁺Θ is an 𝑀-matrix, where Σ=diag(𝜎₁,…,𝜎_𝑛), 𝜎_𝑗=max{|𝜎⁻_𝑗|,|𝜎⁺_𝑗|}, Δ=diag(𝛿₁,…,𝛿_𝑛), 𝛿_𝑗=max{|𝛿⁻_𝑗|,|𝛿⁺_𝑗|}, and Θ=diag(𝜁₁,…,𝜁_𝑛), 𝜁_𝑗=max{|𝜁⁻_𝑗|,|𝜁⁺_𝑗|}.

Proof. From (30), we note that it suffices to prove that the following has at least one solution: 𝑝(𝑥)=𝑥−𝑊₁𝑓(𝑥)−𝑊₂𝑔(𝑥)−𝑊₃ℎ(𝑥)−𝐽^′=0, (31) where 𝑊₁=𝐷⁻¹𝐴, 𝑊₂=𝐷⁻¹𝐵, 𝑊₃=𝐷⁻¹𝑊𝕂, 𝐽^′=𝐷⁻¹𝐽. In order to use topological degree theory, we consider the following homotopic mapping: 𝑃(𝑥,𝜆)=𝜆𝑝(𝑥)+(1−𝜆)𝑥,𝜆∈[0,1]. (32) Note that 𝐷−[𝐴]⁺Σ−[𝐵]⁺Δ−[𝑊𝕂]⁺Θ is an 𝑀-matrix; it can be deduced that 𝐼−[𝑊₁]⁺Σ−[𝑊₂]⁺Δ−[𝑊₃]⁺Θ is also an 𝑀-matrix. This implies that (𝐼−[𝑊₁]⁺Σ−[𝑊₂]⁺Δ−[𝑊₃]⁺Θ)⁻¹≥0 and there exists a positive vector 𝑋₀∈ℝ^𝑛 such that (𝐼−[𝑊₁]⁺Σ−[𝑊₂]⁺Δ−[𝑊₃]⁺Θ)𝑋₀>0. It then follows that [𝑃(𝑥,𝜆)]⁺=𝜆𝑝(𝑥)+(1−𝜆)𝑥⁺=𝑥−𝜆𝑊₁𝑓(𝑥)−𝜆𝑊₂𝑔(𝑥)−𝜆𝑊₃ℎ(𝑥)−𝜆𝐽^′+≥[𝑥]⁺−𝜆𝑊₁𝑓(𝑥)⁺−𝜆𝑊₂𝑔(𝑥)⁺−𝜆𝑊₃ℎ(𝑥)⁺−𝜆𝐽^′+≥[𝑥]⁺−𝜆𝑊₁⁺𝑓(𝑥)⁺−𝜆𝑊₂⁺𝑔(𝑥)⁺−𝜆𝑊₃⁺[ℎ(𝑥)]⁺−𝜆𝐽^′+≥[𝑥]⁺−𝜆𝑊₁⁺Σ[𝑥]⁺−𝜆𝑊₂⁺Δ[𝑥]⁺−𝜆𝑊₃⁺Θ[𝑥]⁺−𝜆𝐽^′+≥(1−𝜆)[𝑥]⁺+𝜆𝐼−𝑊₁⁺Σ−𝑊₂⁺Δ−𝑊₃⁺Θ⋅[𝑥]⁺−𝐼−𝑊₁⁺Σ−𝑊₂⁺Δ−[𝑊₃]⁺Θ⁻¹𝐽^′+. (33) Let Ω=𝑥∣[𝑥]⁺≤𝐼−𝑊₁⁺Σ−𝑊₂⁺Δ−𝑊₃⁺Θ⁻¹𝐽^′++𝑋₀. (34) It is obvious that set Ω is not empty and, for any 𝑥∈𝜕Ω, we have [𝑃(𝑥,𝜆)]⁺≥(1−𝜆)[𝑥]⁺+𝜆𝐼−𝑊₁⁺Σ−𝑊₂⁺Δ−𝑊₃⁺Θ𝑋₀>0,𝜆∈[0,1], (35) which implies that 𝑃(𝑥,𝜆)≠0 for all 𝑥∈𝜕Ω and 𝜆∈[0,1]. By topological degree invariance theory, we obtain 𝑝(𝑥)(𝑃())(𝑃())deg,Ω,0=deg𝑥,𝜆,Ω,0=deg𝑥,0,Ω,0=1. (36) Therefore, from Remark 2, we know that 𝑝(𝑥)=0 has at least one solution in Ω. This completes the proof.

5. Global Asymptotic Stability

It should be noted that Theorem 7 can guarantee the existence of an equilibrium point but not the uniqueness. In this section, we will derive some sufficient conditions to guarantee not only the global asymptotic stability but also the uniqueness of the equilibrium point. For this purpose, the impulsive function 𝐼_𝑘 which is viewed as a perturbation of the equilibrium point 𝑥^∗ of models (1) and (4) without impulses is defined by 𝑥𝑡_𝑘−𝑥𝑡⁻_𝑘=𝐼_𝑘𝑥𝑡⁻_𝑘,𝑥_𝑡⁻_𝑘=−𝐽_𝑘𝑥𝑡⁻_𝑘−𝑥^∗−𝐷^𝑡𝑘_{𝑡𝑘−𝜎}𝑥(𝑢)−𝑥^∗𝑑𝑢,𝑘∈ℤ₊, (37) where 𝐽_𝑘,𝑘∈ℤ₊ are some 𝑛×𝑛 real matrices. It is clear that 𝐼_𝑘(𝑥^∗,𝑥^∗)=0, 𝑘∈ℤ₊. Such type of impulses describes the fact that the encountered instantaneous perturbations depend on not only the state of neurons at impulse times 𝑡_𝑘 but also the state of neurons in its recent history, which reflects more realistic dynamics. The similar nonlinear impulsive perturbations, which include linear impulsive perturbations and nonimpulsive perturbations as their special cases, have also been investigated by some researchers recently [17–21, 39].

Let 𝑦(𝑡)=𝑥(𝑡)−𝑥^∗; then we rewrite the models (1) and (4) as follows: 𝑦̇(𝑡)=−𝐷𝑦(𝑡−𝜎)+𝐴𝑓𝑦(𝑡)+𝐵𝑔𝑦(𝑡−𝜏(𝑡))+𝑊^𝑡_{𝑡−𝜌(𝑡)}𝐾(𝑡−𝑠)ℎ𝑦(𝑠)𝑑𝑠,𝑡>0,𝑡≠𝑡_𝑘,Δ𝑦𝑡_𝑘=𝑦𝑡_𝑘−𝑦𝑡⁻_𝑘=−𝐽_𝑘𝑦𝑡⁻_𝑘−𝐷^𝑡𝑘_{𝑡𝑘−𝜎}𝑦(𝑢)𝑑𝑢,𝑘∈ℤ₊,𝑦(𝑠)=𝜑(𝑠)−𝑥^∗,𝑠∈−𝜂,0, (38) where 𝑓(𝑦(⋅))=𝑓(𝑦(⋅)+𝑥^∗)−𝑓(𝑥^∗), 𝑔(𝑦(⋅))=𝑔(𝑦(⋅)+𝑥^∗)−𝑔(𝑥^∗) and ℎ(𝑦(⋅))=ℎ(𝑦(⋅)+𝑥^∗)−ℎ(𝑥^∗). For convenience in our discussion, in the following, we replace 𝑓 with 𝑓, replace 𝑔 with 𝑔, and replace ℎ with ℎ. Then using a simple transformation, model (38) has an equivalent form as follows: 𝑑𝑦(𝑡)𝑑𝑡−𝐷^𝑡_𝑡−𝜎𝑦(𝑢)(𝑡)𝑦(𝑡)𝑦((𝑡𝑑𝑢=−𝐷𝑦+𝐴𝑓+𝐵𝑔𝑡−𝜏))+𝑊^𝑡_{𝑡−𝜌(𝑡)}𝐾(𝑡−𝑠)ℎ𝑦(𝑠)𝑑𝑠,𝑡>0,𝑡≠𝑡_𝑘,Δ𝑦𝑡_𝑘=𝑦𝑡_𝑘−𝑦𝑡⁻_𝑘=−𝐽_𝑘𝑦𝑡⁻_𝑘−𝐷^𝑡𝑘_{𝑡𝑘−𝜎}𝑦(𝑢)𝑑𝑢,𝑘∈ℤ₊,𝑦(𝑠)=𝜑(𝑠)−𝑥^∗,𝑠∈−𝜂,0. (39)

Theorem 8. Under the conditions in Theorem 7, model (1) has a unique equilibrium point which is globally asymptotically stable if there exist a constant 𝛾>0, an 𝑛×𝑛 inverse matrix 𝑄₁, six 𝑛×𝑛 matrices 𝑃>0, 𝑄₂>0, 𝑄₃>0, 𝑍>0, 𝑇₁₁>0, and 𝑇₂₂>0, three 𝑛×𝑛 diagonal matrices 𝑈₁>0, 𝑈₂>0, and 𝑈₃>0, and real matrices 𝑇₁₂, 𝑋_𝑙(𝑙=1,…,6) with appropriate dimension such that 𝑇₁₁𝑇₁₂⋆𝑇₂₂>0,𝑃𝐼−𝐽_𝑘^𝑇𝑃⋆𝑃≥0,𝑘∈ℤ₊,Ξ𝜒_𝑗⋆−𝑍<0,𝑗=1,2, (40)(41)(42) where ΠΞ=¹¹−𝑄₁𝛾𝑇^𝑇₁₂−𝑄₁𝐷𝑋₁𝐷𝑃𝐷Π₁₇Π₁₈𝑈₃Σ₆Π_1,10⋆Π₂₂0−𝛾𝑄₁𝐷𝑋₂0𝛾𝑄₁𝐴𝛾𝑄₁𝐵0Π_2,10⋆⋆Π₃₃0000𝑈₂Σ₄00⋆⋆⋆−𝑄₃000000⋆⋆⋆⋆Π₅₅00000⋆⋆⋆⋆⋆−𝑄₂−𝐷𝑃𝐴−𝐷𝑃𝐵0−𝐷𝑃𝑊⋆⋆⋆⋆⋆⋆−𝑈₁000⋆⋆⋆⋆⋆⋆⋆−𝑈₂00⋆⋆⋆⋆⋆⋆⋆⋆Π₉₉0⋆⋆⋆⋆⋆⋆⋆⋆⋆Π_10,10,𝜒₁=𝑐𝑜𝑙𝑋₁,𝑋₂,𝑂,𝑂,𝑋₃,𝜒,𝑂,𝑂,𝑂,𝑂,𝑂₂=𝑐𝑜𝑙𝑋₄,𝑋₅,𝑂,𝑂,𝑂,𝑂,𝑂,𝑂,𝑂,𝑋₆, (43) with Π₁₁=−𝑃𝐷−𝐷𝑃+𝜎²𝑄₂+𝑄₃−𝑈₁Σ₁−𝑈₃Σ₅,Π₁₇=𝑃𝐴+𝑄₁𝐴+𝑈₁Σ₂,Π₁₈=𝑃𝐵+𝑄₁Π𝐵,_1,10=𝑃𝑊+𝑄₁𝑊+𝑋₄,Π₂₂=−𝛾𝑄₁−𝛾𝑄^𝑇₁+𝛾²𝜏𝑇₂₂,Π_2,10=𝛾𝑄₁𝑊+𝑋₅,Π₃₃=𝜏𝑇₁₁−𝛾𝑇^𝑇₁₂−𝛾𝑇₁₂−𝑈₂Σ₃,Π₅₅=𝑋₃+𝑋^𝑇₃,Π₉₉=𝜌²𝐾(0)𝑍𝐾(0)−𝑈₃,Π_10,10=𝑋₆+𝑋^𝑇₆. (44)

Proof. Consider the following Lyapunov-Krasovskii functional as 𝑉𝑡,𝑦(𝑡)=𝑉₁𝑡,𝑦(𝑡)+𝑉₂𝑡,𝑦(𝑡)+𝑉₃𝑡,𝑦(𝑡)+𝑉₄𝑡,𝑦(𝑡)+𝑉₅𝑡,𝑦(𝑡), (45) where 𝑉₁𝑡,𝑦(𝑡)=𝑦(𝑡)−𝐷^𝑡_𝑡−𝜎𝑦(𝑢)𝑑𝑢^𝑇×𝑃𝑦(𝑡)−𝐷^𝑡_𝑡−𝜎𝑦(𝑢),𝑉𝑑𝑢₂𝑡,𝑦(𝑡)=𝜎^𝑡_𝑡−𝜎^𝑡_𝑠𝑦^𝑇(𝑢)𝑄₂𝑦(𝑢)+𝑑𝑢𝑑𝑠_{𝑡𝑡−𝜎}𝑦^𝑇(𝑠)𝑄₃𝑦(𝑠)𝑉𝑑𝑠,₃𝑡,𝑦(𝑡)=^𝑡₀^𝑢_{(𝑢)𝑢−𝜏}𝑦((𝑢𝛾𝑦̇(𝑠)𝑢−𝜏))^𝑇𝑇₁₁𝑇₁₂⋆𝑇₂₂×𝑦((𝑢𝛾𝑦̇(𝑠)𝑉𝑢−𝜏))𝑑𝑠𝑑𝑢,₄𝑡,𝑦(𝑡)=𝛾²⁰_−𝜏^𝑡_𝑡+𝑢𝑦̇^𝑇(𝑠)𝑇₂₂𝑦̇(𝑠)𝑉𝑑𝑠𝑑𝑢,₅𝑡,𝑦(𝑡)=𝜌^𝑡_𝑡−𝜌^𝑡_𝑢ℎ^𝑇𝑦(𝑠)𝐾()()𝑦(𝑠)𝑡−𝑠𝑍𝐾𝑡−𝑠×ℎ𝑑𝑠𝑑𝑢. (46) Calculating the upper right derivative of 𝑉 along the trajectories of model (39) at the continuous interval [𝑡_𝑘−1,𝑡_𝑘),𝑘∈ℤ₊, we obtain 𝐷⁺𝑉₁=2𝑦(𝑡)−𝐷^𝑡_𝑡−𝜎𝑦(𝑢)𝑑𝑢^𝑇×𝑃−𝐷𝑦(𝑡)+𝐴𝑓𝑦(𝑡)+𝐵𝑔𝑦(𝑡−𝜏(𝑡))+𝑊^𝑡_{𝑡−𝜌(𝑡)}𝐾(𝑡−𝑠)ℎ𝑦(𝑠)𝑑𝑠=−2𝑦^𝑇(𝑡)𝑃𝐷𝑦(𝑡)+2𝑦^𝑇(𝑡)𝑃𝐴𝑓𝑦(𝑡)+2𝑦^𝑇(𝑡)𝑃𝐵𝑔𝑦(𝑡−𝜏(𝑡))+2𝑦^𝑇(𝑡)𝑃𝑊^𝑡_{𝑡−𝜌(𝑡)}𝐾(𝑡−𝑠)ℎ𝑦(𝑠)𝑑𝑠+2𝑦^𝑇(𝑡)𝐷𝑃𝐷^𝑡_𝑡−𝜎𝑦(𝑢)𝑑𝑢−2^𝑡_𝑡−𝜎𝑦(𝑢)𝑑𝑢^𝑇𝐷𝑃𝐴𝑓𝑦(𝑡)−2^𝑡_𝑡−𝜎𝑦(𝑢)𝑑𝑢^𝑇𝐷𝑃𝐵𝑔𝑦(𝑡−𝜏(𝑡))−2^𝑡_𝑡−𝜎𝑦(𝑢)𝑑𝑢^𝑇𝐷𝑃𝑊^𝑡_{𝑡−𝜌(𝑡)}𝐾(𝑡−𝑠)ℎ𝑦(𝑠)𝑑𝑠. (47) It follows from Lemma 3 that 𝐷⁺𝑉₂=𝜎²𝑦^𝑇(𝑡)𝑄₂𝑦(𝑡)−𝜎^𝑡_𝑡−𝜎𝑦^𝑇(𝑢)𝑄₂𝑦(𝑢)𝑑𝑢+𝑦^𝑇(𝑡)𝑄₃𝑦(𝑡)−𝑦^𝑇(𝑡−𝜎)𝑄₃𝑦(𝑡−𝜎)≤𝜎²𝑦^𝑇(𝑡)𝑄₂𝑦(𝑡)−^𝑡_𝑡−𝜎𝑦(𝑢)𝑑𝑢^𝑇𝑄₂^𝑡_𝑡−𝜎𝑦(𝑢)𝑑𝑢+𝑦^𝑇(𝑡)𝑄₃𝑦(𝑡)−𝑦^𝑇(𝑡−𝜎)𝑄₃𝑦(),𝐷𝑡−𝜎₊𝑉₃=^𝑡_{𝑡−𝜏(𝑡)}𝛾̇𝑦(𝑡−𝜏(𝑡))𝑦(𝑠)^𝑇𝑇₁₁𝑇₁₂⋆𝑇₂₂𝑦((𝑡𝛾𝑦̇(𝑠)(𝑡)𝑦𝑡−𝜏))𝑑𝑠=𝜏^𝑇(𝑡−𝜏(𝑡))𝑇₁₁𝑦(𝑡−𝜏(𝑡))+2𝛾𝑦^𝑇(𝑡)𝑇^𝑇₁₂𝑦(𝑡−𝜏(𝑡))−2𝛾𝑦^𝑇(𝑡−𝜏(𝑡))𝑇^𝑇₁₂𝑦(𝑡−𝜏(𝑡))+𝛾^2𝑡_{𝑡−𝜏(𝑡)}𝑦̇^𝑇(𝑠)𝑇₂₂𝑦̇(𝑠)𝑑𝑠≤𝑦^𝑇(𝑡−𝜏(𝑡))𝜏𝑇₁₁−2𝛾𝑇^𝑇₁₂𝑦(𝑡−𝜏(𝑡))+2𝛾𝑦^𝑇(𝑡)𝑇^𝑇₁₂𝑦(𝑡−𝜏(𝑡))+𝛾^2𝑡_𝑡−𝜏𝑦̇^𝑇(𝑠)𝑇₂₂𝑦̇(𝑠)𝐷𝑑𝑠,₊𝑉₄=𝛾²𝜏𝑦̇^𝑇(𝑡)𝑇₂₂𝑦̇(𝑡)−𝛾²⁰_−𝜏𝑦̇^𝑇(𝑡+𝑢)𝑇₂₂𝑦̇(𝑡+𝑢)𝑑𝑢=𝛾²𝜏𝑦̇^𝑇(𝑡)𝑇₂₂𝑦̇(𝑡)−𝛾^2𝑡_𝑡−𝜏𝑦̇^𝑇(𝑠)𝑇₂₂𝑦̇(𝑠)𝐷𝑑𝑠,₊𝑉₅=𝜌²ℎ^𝑇𝑦(𝑡)𝐾(0)𝑍𝐾(0)ℎ𝑦(𝑡)−𝜌^𝑡_𝑡−𝜌ℎ^𝑇𝑦(𝑠)𝐾(𝑡−𝑠)𝑍𝐾(𝑡−𝑠)ℎ𝑦(𝑠)𝑑𝑠=𝜌²ℎ^𝑇𝑦(𝑡)𝐾(0)𝑍𝐾(0)ℎ𝑦(𝑡)−𝜌^{𝑡−𝜌(𝑡)}_𝑡−𝜌ℎ^𝑇𝑦(𝑠)𝐾()()ℎ𝑦(𝑠)+𝑡−𝑠𝑍𝐾𝑡−𝑠𝑑𝑠_{𝑡𝑡−𝜌(𝑡)}ℎ^𝑇𝑦(𝑠)𝐾(𝑡−𝑠)𝑍𝐾(𝑡−𝑠)ℎ𝑦(𝑠)𝑑𝑠≤𝜌²ℎ^𝑇𝑦(𝑡𝐾(0)(0)ℎ𝑦(𝑡−𝜌)𝑍𝐾)𝜌−𝜌(𝑡)^{𝑡−𝜌(𝑡)}_𝑡−𝜌𝐾(𝑡−𝑠)ℎ𝑦(𝑠)𝑑𝑠^𝑇×𝑍^{𝑡−𝜌(𝑡)}_𝑡−𝜌𝐾()ℎ𝑦(𝑠)−𝜌𝑡−𝑠𝑑𝑠𝜌(𝑡)^𝑡_{(𝑡)𝑡−𝜌}𝐾(𝑡−𝑠)ℎ𝑦(𝑠)𝑑𝑠^𝑇×𝑍^𝑡_{(𝑡)𝑡−𝜌}𝐾(𝑡−𝑠)ℎ𝑦(𝑠)𝑑𝑠. (48) In addition, we note that 0=2𝑦(𝑡)+𝛾𝑦(̇𝑡)^𝑇𝑄₁−𝑦̇(𝑡)+𝑦̇(𝑡̇)=2𝑦(𝑡)+𝛾𝑦(𝑡)^𝑇𝑄₁×−𝑦̇(𝑡)()𝑦(𝑡𝑦((𝑡−𝐷𝑦𝑡−𝜎+𝐴𝑓)+𝐵𝑔𝑡−𝜏))+𝑊^𝑡_{𝑡−𝜌(𝑡)}𝐾(𝑡−𝑠)ℎ𝑦(𝑠)𝑑𝑠=−2𝑦^𝑇(𝑡)𝑄₁𝑦̇(𝑡)−2𝑦^𝑇(𝑡)𝑄₁𝐶𝑦(𝑡−𝜎)+2𝑦^𝑇(𝑡)𝑄₁𝐴𝑓𝑦(𝑡)+2𝑦^𝑇(𝑡)𝑄₁𝐵𝑔𝑦(𝑡−𝜏(𝑡))+2𝑦^𝑇(𝑡)𝑄₁𝑊^𝑡_{𝑡−𝜌(𝑡)}𝐾()ℎ𝑦(𝑠)𝑦̇𝑡−𝑠𝑑𝑠−2𝛾^𝑇(𝑡)𝑄₁𝑦̇(𝑡)−2𝛾𝑦̇^𝑇(𝑡)𝑄₁𝐶𝑦(𝑡−𝜎)+2𝛾𝑦̇^𝑇(𝑡)𝑄₁𝑦(𝑡)𝑦̇𝐴𝑓+2𝛾^𝑇(𝑡)𝑄₁𝑦((𝑡𝑦̇𝐵𝑔𝑡−𝜏))+2𝛾^𝑇(𝑡)𝑄₁𝑊^𝑡_{𝑡−𝜌(𝑡)}𝐾(𝑡−𝑠)ℎ𝑦(𝑠)𝑑𝑠. (49) For simplicity, we denote 𝜉₁(𝑡)=^𝑡_{𝑡−𝜌(𝑡)}𝐾()ℎ𝑦(𝑠)𝜉𝑡−𝑠𝑑𝑠,₂(𝑡)=^{𝑡−𝜌(𝑡)}_𝑡−𝜌𝐾(𝑡−𝑠)ℎ𝑦(𝑠)𝑑𝑠. (50) By applying Lemma 4, for any 𝑡>0 with 0≤𝜏(𝑡)≤𝜏, 0≤𝜌(𝑡)≤𝜌, we have −𝜌(𝑡)𝜉𝜌−𝜌²(𝑡)^𝑇𝑍𝜉₂(𝑡)−𝜌𝜌(𝑡)𝜉₁(𝑡)^𝑇𝑍𝜉₁(𝑡)≤2𝜉^𝑇(𝑡)𝜒₁𝜉₂(𝑡)+𝜌−𝜌(𝑡)𝜌𝜉^𝑇(𝑡)𝜒₁𝑍⁻¹𝜒^𝑇₁𝜉(𝑡)+𝜌(𝑡)𝜌𝜉^𝑇(𝑡)𝜒₂𝑍⁻¹𝜒^𝑇₂𝜉(𝑡)+2𝜉^𝑇(𝑡)𝜒₂𝜉₁(𝑡), (51) where 𝜉(𝑡)=𝑦(𝑡),𝑦̇(𝑡),𝑦(𝑡−𝜏(𝑡)),𝑦(𝑡−𝜎),𝜉₂(𝑡),_{𝑡𝑡−𝜎}𝑦(𝑠)𝑦(𝑡𝑦((𝑡,ℎ𝑔(𝑡)𝑑𝑠,𝑓),𝑔𝑡−𝜏)),𝜉₁(𝑡)𝑑𝑠^𝑇. (52) Moreover, for any 𝑛×𝑛 diagonal matrices, 𝑈₁>0, 𝑈₂>0, and 𝑈₃>0, the following inequality holds by the methods in [40]: 𝑦(𝑡)𝑓(𝑦(𝑡))^𝑇−𝑈₁Σ₁𝑈₁Σ₂⋆−𝑈₁𝑦(𝑡)𝑓𝑦(𝑡)+𝑦((𝑡𝑔𝑦((𝑡𝑡−𝜏))𝑡−𝜏))^𝑇−𝑈₂Σ₃𝑈₂Σ₄⋆−𝑈₂⋅𝑦((𝑡𝑔𝑦((𝑡+𝑦(𝑡)ℎ𝑦(𝑡)𝑡−𝜏))𝑡−𝜏))^𝑇×−𝑈₃Σ₅𝑈₃Σ₆⋆−𝑈₃𝑦(𝑡)ℎ𝑦(𝑡)≥0. (53) Combining (45)–(53), one may deduce that 𝐷⁺𝑉≤𝜉^𝑇(𝑡)Ξ(𝑡)𝜉(𝑡),𝑡∈𝑡_𝑘−1,𝑡_𝑘,𝑘∈ℤ₊, (54) whereΞ(𝑡)(𝑡)=Ξ+𝜌−𝜌𝜌𝜒₁𝑍⁻¹𝜒^𝑇₁+𝜌(𝑡)𝜌𝜒₂𝑍⁻¹𝜒^𝑇₂,ΠΞ=¹¹−𝑄₁𝛾𝑇^𝑇₁₂−𝑄₁𝐷𝑋₁𝐷𝑃𝐷Π₁₇Π₁₈𝑈₃Σ₆Π_1,10⋆Π₂₂0−𝛾𝑄₁𝐷𝑋₂0𝛾𝑄₁𝐴𝛾𝑄₁𝐵0Π_2,10⋆⋆Π₃₃0000𝑈₂Σ₄00⋆⋆⋆−𝑄₃000000⋆⋆⋆⋆Π₅₅00000⋆⋆⋆⋆⋆−𝑄₂−𝐷𝑃𝐴−𝐷𝑃𝐵0−𝐷𝑃𝑊⋆⋆⋆⋆⋆⋆−𝑈₁000⋆⋆⋆⋆⋆⋆⋆−𝑈₂00⋆⋆⋆⋆⋆⋆⋆⋆Π₉₉0⋆⋆⋆⋆⋆⋆⋆⋆⋆Π_10,10,𝜉(𝑡)=𝑦(𝑡),𝑦̇(𝑡),𝑦(𝑡−𝜏(𝑡)),𝑦(𝑡−𝜎),𝜉₂(𝑡),∫^𝑡_𝑡−𝜎𝑦(𝑠)𝑑𝑠,𝑓𝑦(𝑡),𝑔𝑦(𝑡−𝜏(𝑡)),ℎ𝑔(𝑡),𝜉₁(𝑡)𝑑𝑠^𝑇. (55)From Lemma 5 we obtain that Ξ(𝑡)<0 if the following inequalities hold simultaneously: Ξ+𝜒₁𝑍⁻¹𝜒^𝑇₁<0,Ξ+𝜒₂𝑍⁻¹𝜒^𝑇₂<0. (56) Based on the well-known Schur complements [36], we get the fact that (56) is equivalent to (42). Therefore, Ξ(𝑡)<0, 𝑡∈[𝑡_𝑘−1,𝑡_𝑘), 𝑘∈ℤ₊.
For arbitrary 𝑡>0, without loss of generality, we set 𝑡∈[𝑡_𝑛−1,𝑡_𝑛), for some 𝑛∈ℤ₊. Then integrating inequality (54) at each interval [𝑡_𝑘−1,𝑡_𝑘), 1≤𝑘≤𝑛−1, and [𝑡_𝑛−1,𝑡), we derive 𝑉𝑡⁻₁≤𝑉(0)+^𝑡1₀𝜉^𝑇(𝑢)Ξ(𝑡)𝜉(𝑢)𝑉𝑡𝑑𝑢,₋₂≤𝑉𝑡₁+^𝑡2_𝑡1𝜉^𝑇(𝑢)Ξ(𝑡)𝜉(𝑢)⋮𝑉𝑡𝑑𝑢,_−𝑛−1≤𝑉𝑡_𝑛−2+^𝑡𝑛−1_𝑡𝑛−2𝜉^𝑇(𝑢)Ξ(𝑡)𝜉(𝑢)𝑉(𝑡)𝑡𝑑𝑢,≤𝑉_𝑛−1+^𝑡_𝑡𝑛−1𝜉^𝑇(𝑢)Ξ(𝑡)𝜉(𝑢)𝑑𝑢, (57) which implies that 𝑉(𝑡)≤𝑉(0)+^𝑡₀𝜉^𝑇(𝑢)Ξ(𝑡)𝜉(𝑢)+𝑑𝑢_{0<𝑡𝑘≤𝑡}𝑉𝑡_𝑘−𝑉𝑡⁻_𝑘,𝑡≥0. (58) Now, in order to analyze (58), we consider the change of 𝑉 at impulse times 𝑡_𝑘, 𝑘∈ℤ₊.
Firstly, it follows from (41) that 𝑃𝐼−𝐽_𝑘𝑃⟺⋆𝑃≥0𝐼−𝐼−𝐽_𝑘^𝑇𝑃0𝐼𝐼−𝐽_𝑘𝑃⋆𝑃−𝐼0𝐼−𝐽_𝑘𝐼⟺≥0𝑃−𝐼−𝐽_𝑘^𝑇𝑃𝐼−𝐽_𝑘0⋆𝑃≥0⟺𝑃−𝐼−𝐽_𝑘^𝑇𝑃𝐼−𝐽_𝑘>0. (59) Secondly, from model (39), it can be obtained that 𝑦𝑡_𝑘−𝐷^𝑡𝑘_{𝑡𝑘−𝜎}𝑦(𝑢)𝑡𝑑𝑢=𝑦⁻_𝑘−𝐽_𝑘𝑦𝑡⁻_𝑘−𝐷^𝑡𝑘_{𝑡𝑘−𝜎}𝑦(𝑢)𝑑𝑢−𝐷^𝑡𝑘_{𝑡𝑘−𝜎}𝑦(𝑢)=𝑑𝑢𝐼−𝐽_𝑘𝑦𝑡⁻_𝑘−𝐷^𝑡𝑘_{𝑡𝑘−𝜎}𝑦(𝑢)𝑑𝑢, (60) which together with (59) yields 𝑉₁𝑡_𝑘=𝑦(𝑡_𝑘)−𝐷^𝑡𝑘_{𝑡𝑘−𝜎}𝑦(𝑢)𝑑𝑢^𝑇×𝑃𝑦𝑡_𝑘−𝐷^𝑡𝑘_{𝑡𝑘−𝜎}𝑦(𝑢)=𝑑𝑢𝑦(𝑡⁻_𝑘)−𝐷^𝑡𝑘_{𝑡𝑘−𝜎}𝑦(𝑢)𝑑𝑢^𝑇𝐼−𝐽_𝑘^𝑇𝑃𝐼−𝐽_𝑘×𝑦𝑡⁻_𝑘−𝐷^𝑡𝑘_{𝑡𝑘−𝜎}𝑦(𝑢)≤𝑦𝑡𝑑𝑢_−𝑘−𝐷^𝑡𝑘_{𝑡𝑘−𝜎}𝑦(𝑢)𝑑𝑢^𝑇×𝑃𝑦𝑡⁻_𝑘−𝐷^𝑡𝑘_{𝑡𝑘−𝜎}𝑦(𝑢)𝑑𝑢=𝑉₁𝑡⁻_𝑘. (61) Thus, we can deduce that 𝑉𝑡_𝑘≤𝑉𝑡⁻_𝑘,𝑘∈ℤ₊. (62) Substituting the above inequality to (58), it yields 𝑉(𝑡)−^𝑡₀𝜉^𝑇(𝑢)Ξ(𝑡)𝜉(𝑢)𝑑𝑢≤𝑉(0),𝑡≥0. (63) Applying Lemma 3 and (63), we have 𝐷^𝑡_𝑡−𝜎𝑦(𝑢)𝑑𝑢²=𝐷^𝑡_𝑡−𝜎𝑦(𝑢)𝑑𝑢^𝑇𝐷^𝑡_𝑡−𝜎𝑦(𝑢)𝑑𝑢≤𝜆_max𝐷^2𝑡_𝑡−𝜎𝑦(𝑠)𝑑𝑠^𝑇𝑡_𝑡−𝜎𝑦(𝑠)≤𝜆𝑑𝑠_max𝐷²𝜆_min𝑄₃^𝑡_𝑡−𝜎𝑦(𝑠)𝑑𝑠^𝑇𝑄₃^𝑡_𝑡−𝜎𝑦(𝑠)𝜆𝑑𝑠≤𝜎^max𝐷²𝜆_min𝑄₃^𝑡_𝑡−𝜎𝑦^𝑇(𝑠)𝑄₃𝑦(𝑠)𝜆𝑑𝑠≤𝜎^max𝐷²𝜆_min𝑄₃𝑉₂(𝑡)𝜆≤𝜎^max𝐷²𝜆_min𝑄₃𝑉(𝑡)𝜆≤𝜎^max𝐷²𝜆_min𝑄₃𝑉(0),𝑡≥0. (64) Similarly, 𝑦(𝑢)−𝐷^𝑡_𝑡−𝜎𝑦(𝑢)𝑑𝑢²=𝑦(𝑢)−𝐷^𝑡_𝑡−𝜎𝑦(𝑢)𝑑𝑢^𝑇𝑦(𝑢)−𝐷^𝑡_𝑡−𝜎𝑦(𝑢)≤𝑉𝑑𝑢₁(𝑡)𝜆_min(𝑃)≤𝑉(𝑡)𝜆_min(𝑃)≤𝑉(0)𝜆_min(𝑃),𝑡≥0. (65) Hence, it can be obtained that 𝑦(𝑡)≤𝐷^𝑡_𝑡−𝜎𝑦(𝑢)𝑑𝑢+𝑦(𝑢)−𝐷^𝑡_𝑡−𝜎𝑦(𝑢)≤𝑑𝑢𝜎𝜆_max𝐷²𝜆_min𝑄₃𝑉(0)+𝑉(0)𝜆_min(𝑃)<∞,𝑡≥0, (66) where 𝑉(0)=𝑦(0)−𝐷⁰_−𝜎𝑦(𝑢)𝑑𝑢^𝑇𝑃𝑦(0)−𝐷⁰_−𝜎𝑦(𝑢)𝑑𝑢+𝜎⁰_−𝜎⁰_𝑠𝑦^𝑇(𝑢)𝑄₂𝑦(𝑢)𝑑𝑢𝑑𝑠+⁰_−𝜎𝑦^𝑇(𝑠)𝑄₃𝑦(𝑠)𝑑𝑠+𝛾²⁰_−𝜏⁰_𝑢𝑦̇^𝑇(𝑠)𝑇₂₂𝑦̇(𝑠)𝑑𝑠𝑑𝑢+𝜌⁰_−𝜌⁰_𝑢ℎ^𝑇𝑦(𝑠)𝐾()()ℎ𝑦(𝑠)≤−𝑠𝑍𝐾−𝑠𝑑𝑠𝑑𝑢2𝜆_max(𝑃)1+𝜎²max_𝑖∈Λ𝑑_𝑖+12𝜎³𝜆_max𝑄₂+𝜎𝜆_max𝑄₃+12𝛾²𝜏²𝜆_max𝑇₂₂+12𝜌³𝜆_max(𝑍)𝜑²_𝜂<∞. (67) So the solution 𝑦(𝑡) of models (1) and (4) is uniformly bounded on [0,∞). Thus, considering the continuity of activation function 𝑓 (i.e., (𝐻₃)), it can be deduced from system (38) that there exists some constant 𝑀>0 such that ‖𝑦(̇𝑡)‖≤𝑀, 𝑡∈[𝑡_𝑘−1,𝑡_𝑘), 𝑘∈ℤ₊. It implies that, |𝑦̇_𝑖(𝑡)|≤𝑀, 𝑡∈[𝑡_𝑘−1,𝑡_𝑘), 𝑘∈ℤ₊, and 𝑖∈Λ, where 𝑦̇ denotes the right hand derivative of 𝑦 at impulsive times.
Finally, we can prove that ‖𝑦(𝑡)‖→0 as 𝑡→∞, which is similar to the corresponding proof in the literature [25]. Here we omit it. Therefore, the zero solution of (38) or (39) is globally asymptotically stable, which implies that models (1) and (4) have a unique equilibrium point which is globally asymptotically stable. This completes the proof.

When there is no leakage delay, that is, 𝜎=0, models (1) and (4) become 𝑥̇(𝑡)=−𝐷𝑥(𝑡)+𝐴𝑓(𝑥(𝑡))+𝐵𝑔(𝑥(𝑡−𝜏(𝑡)))+𝑊^𝑡_{(𝑡)𝑡−𝜌}𝐾(𝑡−𝑠)ℎ(𝑥(𝑠))𝑑𝑠+𝐼,𝑡>0,𝑡≠𝑡_𝑘,Δ𝑥𝑡_𝑘=𝑥𝑡_𝑘−𝑥𝑡⁻_𝑘=−𝐼_𝑘𝑥𝑡⁻_𝑘−𝑥^∗,𝑘∈ℤ₊(𝑠)(𝑠).,𝑥=𝜑,𝑠∈−𝜂,0 (68) For model (68), we have the following result by Theorem 8.

Corollary 9. Under the conditions in Theorem 7, model (68) has a unique equilibrium point which is globally asymptotically stable if there exist a constant 𝛾>0, an 𝑛×𝑛 inverse matrix 𝑄₁, four 𝑛×𝑛 matrices 𝑃>0, 𝑍>0, 𝑇₁₁>0, and 𝑇₂₂>0, three 𝑛×𝑛 diagonal matrices 𝑈₁>0, 𝑈₂>0, and 𝑈₃>0, and real matrices 𝑇₁₂, 𝑋_𝑙(𝑙=1,…,6) with appropriate dimension such that 𝑇₁₁𝑇₁₂⋆𝑇₂₂𝑃>0,𝐼−𝐽_𝑘^𝑇𝑃⋆𝑃≥0,𝑘∈ℤ₊,Ξ𝜒_𝑗⋆−𝑍<0,𝑗=1,2, (69) where ΠΞ=¹¹−𝑄₁𝛾𝑇^𝑇₁₂𝑋₁Π₁₅Π₁₆𝑈₃Σ₆Π₁₈⋆Π₂₂0𝑋₂𝛾𝑄₁𝐴𝛾𝑄₁𝐵0Π₂₈⋆⋆Π₃₃00𝑈₂Σ₄00⋆⋆⋆Π₄₄0000⋆⋆⋆⋆−𝑈₁000⋆⋆⋆⋆⋆−𝑈₂00⋆⋆⋆⋆⋆⋆Π₇₇0⋆⋆⋆⋆⋆⋆⋆Π₈₈,𝜒₁=col𝑋₁,𝑋₂,𝑂,𝑋₃,𝜒,𝑂,𝑂,𝑂,𝑂₂=col𝑋₄,𝑋₅,𝑂,𝑂,𝑂,𝑂,𝑂,𝑋₆, (70) with Π₁₁=−PD−DP−𝑈₁Σ₁−𝑈₃Σ₅,Π₁₅=PA+𝑄₁𝐴+𝑈₁Σ₂,Π₁₆=PB+𝑄₁Π𝐵,₁₈=PW+𝑄₁𝑊+𝑋₄,Π₂₂=−𝛾𝑄₁−𝛾𝑄^𝑇₁+𝛾²𝜏𝑇₂₂,Π₂₈=𝛾𝑄₁𝑊+𝑋₅,Π₃₃=𝜏𝑇₁₁−𝛾𝑇^𝑇₁₂−𝛾𝑇₁₂−𝑈₂Σ₃,Π₄₄=𝑋₃+𝑋^𝑇₃,Π₇₇=𝜌²𝐾(0)ZK(0)−𝑈₃,Π₈₈=𝑋₆+𝑋^𝑇₆. (71)

Remark 10. In [25], Li et al. have investigated a class of recurrent neural networks with discrete time-varying delay and constant delay in the leakage term under impulsive perturbations. However, the bounded time-varying distributed delays were not taken into account in network models. In fact, neural networks usually have a spatial extent due to the presence of a multitude of parallel pathways with a variety of axon sizes and lengths, and hence there is a distribution of propagation delays over a period of time. Thus, the time-varying distributed delays should be taken into account to reflect more realistic dynamics. The models studied in the present paper have wider range of applications. In the future, we will further investigate the relevant models with such mixed time delays by other current approaches, such as constructing Lyapunov-Krasovskii functional with triple integral terms and introducing free-weighting matrices.

Remark 11. In [27], Wang et al. have dealt with the global asymptotic stability of neural networks with mixed time delays under impulsive perturbations. But there is a derivative restriction of transmission delay 𝜏̇(𝑡)≤𝑐<1, which is a conservative condition. In our results, based on a superior Lyapunov-Krasovskii functional, the criteria do not require the differentiability even differential boundedness of time-varying transmission delay 𝜏(𝑡). In addition, we have considered the basic properties, such as existence and uniqueness of solutions and equilibrium. So our results are less conservative and may be applied effectively in a wider range.

Remark 12. Recently, more researchers have begun to take into account the effect of time-varying leakage delay, which has essential difference from constant leakage delay, on dynamics of models. We would like to think that it may lead to more technical difficulties. How to improve the dynamics, especially the stability properties of neural networks with (time-varying) leakage delay, may be an interesting problem and requires further research.

6. Illustrative Example

In this section, an example is given to demonstrate the effectiveness of our results.

Example 1. Consider the following recurrent neural networks model: 𝑥̇(𝑡)=−𝐷𝑥(𝑡−𝜎)+𝐴𝑓(𝑥(𝑡))+𝐵𝑔(𝑥(𝑡−𝜏(𝑡)))+𝑊^𝑡_{(𝑡)𝑡−𝜌}𝐾(𝑡−𝑠)ℎ(𝑥(𝑠))𝑑𝑠+𝐼,𝑡>0,𝑡≠𝑡_𝑘,Δ𝑥𝑡_𝑘=𝑥𝑡_𝑘−𝑥𝑡⁻_𝑘=𝐼_𝑘𝑥𝑡⁻_𝑘,𝑥_𝑡⁻_𝑘,𝑘∈ℤ₊, (72) where 𝑓=𝑔=ℎ=|𝑠|, 𝜎=0.08, 𝜏(𝑡)=0.09−0.01[sin𝑡]^∗, 𝜌(𝑡)=0.08−0.02[sin𝑡]^∗, 𝐾(𝑠)=𝑒^−𝑠, 𝐼=(0,0)^𝑇, 𝑡_𝑘=0.1𝑘, and 𝐼_𝑘=diag(0.5,0.5), 𝑘∈ℤ₊, and parameter matrices 𝐷,𝐴,𝐵, and 𝑊 are given as follows: ,.𝐷=9008,𝐴=1.5−0.3−0.10.2𝐵=−0.60.41.7−1.8,𝑊=0.20.3−0.50.15 (73)
In this case, we know that 𝜎⁻_𝑗=𝛿⁻_𝑗=𝜁⁻_𝑗=−1, 𝜎⁺_𝑗=𝛿⁺_𝑗=𝜁⁺_𝑗=1, 𝜎=0.08, 𝜏=0.1, and 𝜌=0.1. Let 𝛾=2.6; via Matlab LMI toolbox, the feasible solution for the LMIs in Theorem 8 is derived as follows: 𝑃=0.0017000.0026,𝑄₁=,𝑄0.00300.00010.00010.0037₂=0.80110.00020.00020.8061,𝑄₃=,0.59820.00090.00090.6026𝑍=0.8887000.8887,𝑈₁=,𝑈1.0046001.0046₂=0.8793000.8793,𝑈₃=,𝑇1.0117001.0117₁₁=1.3262−0.0001−0.00011.3260,𝑇₁₂=,𝑇0.0388000.0387₂₂=,𝑋0.00960.00020.00020.0113₁=10⁻³×,𝑋0.05340.0807−0.13110.0400₂=10⁻³×,𝑋0.11170.1773−0.35540.1106₃=−0.317600−0.3176,𝑋₄=,𝑋−0.0004−0.00070.0012−0.0004₅=,𝑋−0.0014−0.00220.0044−0.0014₆=.−0.317700−0.3177 (74) Hence, from Theorem 8, the unique equilibrium point 𝑥^∗=(0,0)^𝑇 of system (72) is globally asymptotically stable.

7. Conclusion

In this paper, we have investigated a class of impulsive neural networks with mixed time delays and generalized activation functions. Firstly, by using the contraction mapping theorem, we have given a sufficient condition to guarantee the global existence and uniqueness of the solution for the addressed neural networks. Then, a delay-independent sufficient condition for existence of the equilibrium point and some delay-dependent sufficient conditions for stability have been derived, respectively, by using topological degree theory and suitable Lyapunov-Krasovskii functional. The obtained results require neither the boundedness, monotonicity, and differentiability of the activation functions nor the differentiability of time-varying delay. Finally, an example has been given to show the effectiveness and less conservativeness of the obtained results. In the future, we will do some further research on impulsive neural network models with leakage time-varying delay and continuously distributed delay.

Conflict of Interests

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

Acknowledgment

This work was partially supported by the National Natural Science Foundation of China Grant no. 11301304.