Pth Moment Exponential Stability of Impulsive Stochastic Neural Networks with Mixed Delays

Abstract

This paper investigates the problem of pth moment exponential stability for a class of stochastic neural networks with time-varying delays and distributed delays under nonlinear impulsive perturbations. By means of Lyapunov functionals, stochastic analysis and differential inequality technique, criteria on pth moment exponential stability of this model are derived. The results of this paper are completely new and complement and improve some of the previously known results (Stamova and Ilarionov (2010), Zhang et al. (2005), Li (2010), Ahmed and Stamova (2008), Huang et al. (2008), Huang et al. (2008), and Stamova (2009)). An example is employed to illustrate our feasible results.

1. Introduction

The dynamics of neural networks have drawn considerable attention in recent years due to their extensive applications in many fields such as image processing, associative memories, classification of patters, and optimization. Since the integration and communication delays are unavoidably encountered in biological and artificial neural systems, it may result in oscillation and instability. The stability analysis of delayed neural networks has been extensively investigated by many researchers, for instance, see [1–30].

In real nervous systems, there are many stochastic perturbations that affect the stability of neural networks. The result in Mao [24] suggested that one neural network could be stabilized or destabilized by certain stochastic inputs. It implies that the stability analysis of stochastic neural networks has primary significance in the design and applications of neural networks, such as [7, 12–16, 18, 20, 22–24, 26, 27, 30].

On the other hand, it is noteworthy that the state of electronic networks is often subjected to some phenomenon or other sudden noises. On that account, the electronic networks will experience some abrupt changes at certain instants that in turn affect dynamical behaviors of the systems [5, 6, 17–23, 28, 29]. Therefore, it is necessary to take both stochastic effects and impulsive perturbations into account on dynamical behaviors of delayed neural networks [18, 20, 22, 23].

Very recently, Li et al. [22] have employed the properties of M-cone and inequality technique to investigate the mean square exponential stability of impulsive stochastic neural networks with bounded delays. Wu et al. [23] studied the exponential stability of the equilibrium point of bounded discrete-time delayed dynamic systems with linear impulsive effects by using Razumikhin theorems. To the best of authors’ knowledge, however, few authors have considered the pth moment exponential stability of impulsive stochastic neural networks with mixed delays.

Motivated by the discussions above, our object in this paper is to present the sufficient conditions ensuring pth moment exponential stability for a class of stochastic neural networks with time-varying delays and distributed delays under nonlinear impulsive perturbations by virtue of Lyapunov method, inequality technique and Itô formula. The results obtained in this paper generalize and improve some of the existing results [5, 8, 18, 19, 26–28]. The effectiveness and feasibility of the developed results have been shown by a numerical example.

2. Model Description and Preliminaries

Let ℝ denote the set of real numbers, ℝ^𝑛 the 𝑛-dimensional real space equipped with the Euclidean norm |⋅|, ℤ⁺ the set of nonnegative integral numbers. 𝐸(⋅) stands for the mathematical expectation operator. ℒ denotes the well-known ℒ-operator given by the Itô formula. 𝑤(𝑡)=(𝑤₁(𝑡),…,𝑤_𝑚(𝑡)) is 𝑚-dimensional Brownian motion defined on a complete probability space (Ω,ℱ,𝑃) with a natural filtration {ℱ_𝑡}_𝑡≥0 generated by {𝑤(𝑠)∶0≤𝑠≤𝑡}, where we associate Ω with the canonical space generated by 𝑤(𝑡) and denote by ℱ the associated 𝜎-algebra generated by 𝑤(𝑡) with the probability measure  𝑃.  Let 𝜎(𝑡,𝑥,𝑦)=(𝜎_𝑖𝑙(𝑡,𝑥_𝑖,𝑦_𝑖))_𝑛×𝑚∈ℝ^𝑛×𝑚, and 𝜎_𝑖(𝑡,𝑥_𝑖,𝑦_𝑖) be 𝑖th row vector of 𝜎(𝑡,𝑥,𝑦).

In [5, 6], the researchers investigated the following impulsive neural networks with time-varying delays: 𝑥̇_𝑖(𝑡)=−𝑎_𝑖𝑥_𝑖(𝑡)+^𝑛_𝑗=1𝑏_𝑖𝑗𝑓_𝑗𝑥_𝑗(𝑡)+^𝑛_𝑗=1𝑐_𝑖𝑗𝑔_𝑗𝑥_𝑗𝑡−𝜏_𝑗(𝑡)+𝐼_𝑖,𝑡≠𝑡_𝑘,𝑥_𝑖𝑡_𝑘=𝑝_𝑖𝑘𝑥𝑡⁻_𝑘,𝑘∈ℤ⁺,𝑖∈Λ. (2.1) The authors in [7, 26, 27] studied the stochastic recurrent neural networks with time-varying delays: 𝑑𝑥_𝑖(𝑡)=−𝑎_𝑖𝑥_𝑖(𝑡)+^𝑛_𝑗=1𝑏_𝑖𝑗𝑓_𝑗𝑥_𝑗(𝑡)+^𝑛_𝑗=1𝑐_𝑖𝑗𝑔_𝑗𝑥_𝑗𝑡−𝜏_𝑗(𝑡)+𝐼_𝑖+𝑑𝑡_𝑚𝑙=1𝜎_𝑖𝑙𝑡,𝑥_𝑖(𝑡),𝑥_𝑖𝑡−𝜏_𝑖(𝑡)𝑑𝑤_𝑙(𝑡). (2.2)

In this paper, we will study the generalized stochastically perturbed neural network model with time-varying delays and distributed delays under nonlinear impulses defined by the state equations: 𝑑𝑥_𝑖(𝑡)=−𝑎_𝑖𝑥_𝑖(𝑡)+^𝑛_𝑗=1𝑏_𝑖𝑗𝑓_𝑗𝑥_𝑗(𝑡)+^𝑛_𝑗=1𝑐_𝑖𝑗𝑔_𝑗𝑥_𝑗𝑡−𝜏_𝑗(𝑡)+^𝑛_𝑗=1𝑑_𝑖𝑗^𝑡_−∞𝐾_𝑖𝑗(𝑡−𝑠)⋅ℎ_𝑗𝑥_𝑗(𝑠)𝑑𝑠+𝐼_𝑖+𝑑𝑡_𝑚𝑙=1𝜎_𝑖𝑙𝑡,𝑥_𝑖(𝑡),𝑥_𝑖𝑡−𝜏_𝑖(𝑡)𝑑𝑤_𝑙(𝑡),𝑡≠𝑡_𝑘,𝑥_𝑖𝑡_𝑘=𝑝_𝑖𝑘𝑥𝑡⁻_𝑘,𝑘∈ℤ⁺,𝑖∈Λ, (2.3) where Λ={1,2,…,𝑛}, the time sequence {𝑡_𝑘} satisfies 0=𝑡₀<𝑡₁<𝑡₂<⋯<𝑡_𝑘<𝑡_𝑘+1⋯, lim_𝑘→∞𝑡_𝑘=∞; 𝑥(𝑡)=(𝑥₁(𝑡),𝑥₂(𝑡),…,𝑥_𝑛(𝑡))^𝑇 and 𝑥_𝑖(𝑡) corresponds to the state of the 𝑖th unit at time 𝑡; 𝑏_𝑖𝑗, 𝑐_𝑖𝑗,  and 𝑑_𝑖𝑗 denote the constant connection weight; 𝜏_𝑗(𝑡) is the time-varying transmission delay and satisfies 0≤𝜏_𝑗(𝑡)≤𝜏_𝑗,0<𝜂_𝑗=inf_𝑡∈𝑅⁺{1−𝜏̇_𝑗(𝑡)}, for 𝑗∈Λ.  𝑓_𝑗(⋅),  𝑔_𝑗(⋅),  ℎ_𝑗(⋅) denote the activation functions of the 𝑗th neuron; the delay kernel 𝐾_𝑖𝑗(⋅) is the real-valued nonnegative piecewise continuous functions defined on [0,∞);  𝑛 corresponds to the numbers of units in a neural network; 𝐼_𝑖 denotes the external bias on the 𝑖th unit; 𝑝_𝑖𝑘(𝑥(𝑡_𝑘)) represents the abrupt change of the state 𝑥_𝑖(𝑡) at the impulsive moment 𝑡_𝑘.

System (2.3) is supplemented with initial condition given by 𝑥_𝑖(𝑠)=𝜑_𝑖(𝑠),𝑠∈(−∞,0],𝑖∈Λ, (2.4) where 𝜑(𝑠)=(𝜑₁(𝑠),𝜑₂(𝑠),…,𝜑_𝑛(𝑠))^𝑇∈PCB^𝑏_ℱ0((−∞,0],ℝ^𝑛)=̇BPCB^𝑏_ℱ0. Denote by PCB^𝑏_ℱ0 the family of all bounded ℱ₀-measurable, PC((−∞,0],ℝ^𝑛)-value random variables 𝜑, satisfying sup_{𝜃∈(−∞,0]}𝐸|𝜑(𝜃)|^𝑝<∞, where PC((−∞,0],ℝ^𝑛)={𝜑∶[−∞,0]→ℝ^𝑛} is continuous everywhere except at finite number of points 𝑡_𝑘,  at which 𝜑(𝑡⁺_𝑘) and 𝜑(𝑡⁻_𝑘) exist and 𝜑(𝑡⁺_𝑘)=𝜑(𝑡_𝑘).

The norms are defined by the following norms, respectively: 𝜑_𝑝=sup_{𝑠∈(−∞,0]}^𝑛_𝑖=1𝜑_𝑖(𝑠)^𝑝1/𝑝,‖𝑥‖_𝑝=^𝑛_𝑖=1𝑥_𝑖^𝑝1/𝑝. (2.5)

Throughout this paper, the following standard hypothesis are needed.

(H1) Functions 𝑎_𝑖(⋅)∶ℝ→ℝ are continuous and monotone increasing, that is, there exist real numbers 𝑎_𝑖>0 such that 𝑎_𝑖(𝑢)−𝑎_𝑖(𝑣)𝑢−𝑣≥𝑎_𝑖, (2.6) for all 𝑢,𝑣∈ℝ,  𝑢≠𝑣,  𝑖∈Λ.

(H2)  Functions 𝑓_𝑗, 𝑔_𝑗,andℎ_𝑗 are Lipschitz-continuous on ℝ with Lipschitz constants 𝐿^𝑓_𝑗>0,𝐿^𝑔_𝑗>0,  and 𝐿^ℎ_𝑗>0, respectively. That is, 𝑓_𝑗(𝑢)−𝑓_𝑗(𝑣)≤𝐿^𝑓_𝑗|𝑢−𝑣|,𝑔_𝑗(𝑢)−𝑔_𝑗(𝑣)≤𝐿^𝑔_𝑗|𝑢−𝑣|,ℎ_𝑗(𝑢)−ℎ_𝑗(𝑣)≤𝐿^ℎ_𝑗|𝑢−𝑣|, (2.7) for all 𝑢,𝑣∈ℝ,  𝑖∈Λ.

(H3)  The delay kernels 𝐾_𝑖𝑗∶[0,∞)→ℝ⁺ satisfy 𝐾_𝑖𝑗(𝑠)≤𝒦(𝑠)∀𝑖,𝑗∈Λ,𝑠∈[0,∞),^∞₀𝒦(𝑠)𝑒^𝜇0𝑠𝑑𝑠<∞, (2.8) where 𝒦(𝑠)∶[0,∞)→ℝ⁺ is continuous and integrable, and the constant 𝜇₀ denotes some positive number.

(H4)  There exist nonnegative constants 𝑒_𝑖, 𝑙_𝑖 such that 𝜎_𝑖𝑡,𝑢^′,𝑣^′−𝜎_𝑖(𝑡,𝑢,𝑣)𝜎_𝑖𝑡,𝑢^′,𝑣^′−𝜎_𝑖(𝑡,𝑢,𝑣)^𝑇≤𝑒_𝑖𝑢^′−𝑢²+𝑙_𝑖𝑣^′−𝑣², (2.9) for all 𝑢,𝑣,𝑢^′,𝑣^′∈ℝ,  𝑖∈Λ.

(H5)  There exist nonnegative matrixes 𝑃_𝑘=(𝑝^𝑘_𝑖𝑗)_𝑛×𝑛 such that 𝑝_𝑖𝑘𝑢₁,𝑢₂,…,𝑢_𝑛−𝑝_𝑖𝑘𝑣₁,𝑣₂,…,𝑣_𝑛^𝑝≤^𝑛_𝑗=1𝑝^𝑘_𝑖𝑗𝑢_𝑗−𝑣_𝑗^𝑝, (2.10) for any (𝑢₁,𝑢₂,…,𝑢_𝑛)^𝑇,(𝑣₁,𝑣₂,…,𝑣_𝑛)^𝑇∈ℝ^𝑛, where 𝑝≥1 is an integer.

We end this section by introducing three definitions.

Definition 2.1. A constant vector 𝑥^∗=(𝑥^∗₁,𝑥^∗₂,…,𝑥^∗_𝑛)^𝑇∈ℝ^𝑛 is said to be an equilibrium point of system (2.3) if 𝑥^∗ is governed by the algebraic system 𝑎_𝑖𝑥^∗_𝑖=^𝑛_𝑗=1𝑏_𝑖𝑗𝑓_𝑗𝑥^∗_𝑗+^𝑛_𝑗=1𝑐_𝑖𝑗𝑔_𝑗𝑥^∗_𝑗+^𝑛_𝑗=1𝑑_𝑖𝑗^𝑡_−∞𝐾_𝑖𝑗(𝑡−𝑠)ℎ_𝑗𝑥^∗_𝑗𝑑𝑠+𝐼_𝑖, (2.11) where it is assumed that impulse functions 𝑝_𝑖𝑘(⋅) satisfy 𝑝_𝑖𝑘(𝑥^∗₁,𝑥^∗₂,…,𝑥^∗_𝑛)=𝑥^∗_𝑖 for all 𝑖∈Λ and 𝜎(𝑡,𝑥^∗_𝑖,𝑥^∗_𝑖)=0.

Definition 2.2. The equilibrium 𝑥^∗=(𝑥^∗₁,𝑥^∗₂,…,𝑥^∗_𝑛)^𝑇∈ℝ^𝑛 of system (2.3) is said to be pth moment exponentially stable if there exist 𝜆>0 and 𝑀≥1 such that 𝐸𝑥(𝑡)−𝑥^∗𝑝≤𝐸𝜑−𝑥^∗𝑝𝑀𝑒^−𝜆𝑡for𝑡≥0, (2.12) where 𝑥(𝑡) is an any solution of system (2.3) with initial value 𝜑∈PCB^𝑏_𝐹0((−∞,0],ℝ^𝑛).

Definition 2.3 (Forti and Tesi, 1995 [25]). A map 𝐻∶ℝ^𝑛→ℝ^𝑛 is a homeomorphism of ℝ^𝑛 onto itself if 𝐻 is continuous and one-to-one, and its inverse map 𝐻⁻¹ is also continuous.

3. Main Result

For convenience, we denote that 𝜙_𝑖=𝑝𝑎_𝑖−^𝑛_𝑗=1^𝑝−1_𝑙=1𝑏_𝑖𝑗^{𝑝𝛼𝑙,𝑖𝑗}𝐿^𝑓_𝑗^{𝑝𝛽𝑙,𝑖𝑗}+𝑐_𝑖𝑗^{𝑝𝛾𝑙,𝑖𝑗}𝐿^𝑔_𝑗^{𝑝𝛿𝑙,𝑖𝑗}+𝑑_𝑖𝑗^{𝑝𝜉𝑙,𝑖𝑗}𝐿^ℎ_𝑗^{𝑝𝜂𝑙,𝑖𝑗∞}₀𝒦(𝑠)−𝑑𝑠_𝑛𝑗=1𝜇_𝑗𝜇_𝑖𝑏_𝑗𝑖^{𝑝𝛼𝑝,𝑗𝑖}𝐿^𝑓_𝑖^{𝑝𝛽𝑝,𝑗𝑖},𝜓_𝑖=^𝑛_𝑗=1𝜇_𝑗𝜇_𝑖𝑐_𝑗𝑖^{𝑝𝛾𝑝,𝑗𝑖}𝐿^𝑔_𝑖^{𝑝𝛿𝑝,𝑗𝑖},Φ_𝑖=𝜙_𝑖−𝑝𝑝−12𝑒_𝑖−𝑝−1𝑝−22𝑙_𝑖,Ψ_𝑖=𝜓_𝑖+𝑝−1𝑙_𝑖,𝜁_𝑖𝑗=𝜇_𝑖𝜇_𝑗𝑑_𝑖𝑗^{𝑝𝜉𝑝,𝑖𝑗}𝐿^ℎ_𝑗^{𝑝𝜂𝑝,𝑖𝑗},𝜇=max_𝑖𝜇_𝑖,𝜇=min_𝑖𝜇_𝑖, (3.1) where 𝜇_𝑖 are positive constant, 𝛼_{𝑙,𝑖𝑗}, 𝛽_{𝑙,𝑖𝑗}, 𝛾_{𝑙,𝑖𝑗}, 𝛿_{𝑙,𝑖𝑗}, 𝜉_{𝑙,𝑖𝑗}, and 𝜂_{𝑙,𝑖𝑗} are real numbers and satisfy ^𝑝_𝑙=1𝛼_{𝑙,𝑖𝑗}=1,^𝑝_𝑙=1𝛽_{𝑙,𝑖𝑗}=1,^𝑝_𝑙=1𝛾_{𝑙,𝑖𝑗}=1,^𝑝_𝑙=1𝛿_{𝑙,𝑖𝑗}=1,^𝑝_𝑙=1𝜉_{𝑙,𝑖𝑗}=1,^𝑝_𝑙=1𝜂_{𝑙,𝑖𝑗}=1. (3.2)

Lemma 3.1. If 𝑎_𝑖(𝑖=1,2,…,𝑝) denote 𝑝 nonnegative real numbers, then 𝑎₁𝑎₂⋯𝑎_𝑝≤𝑎^𝑝₁+𝑎^𝑝₂+⋯+𝑎^𝑝_𝑝𝑝, (3.3) where 𝑝≥1 denotes an integer.
A particular form of (3.3), namely, 𝑎₁^𝑝−1𝑎₂≤𝑝−1𝑎^𝑝₁𝑝+𝑎^𝑝₂𝑝,for𝑝=1,2,3,…. (3.4)

Lemma 3.2. If 𝐻∶ℝ^𝑛→ℝ^𝑛 is a continuous function and satisfies the following conditions. (1)𝐻(𝑥) is injective on ℝ^𝑛, that is, 𝐻(𝑥)≠𝐻(𝑦) for all 𝑥≠𝑦. (2)‖𝐻(𝑥)‖→∞ as ‖𝑥‖→∞.
Then, 𝐻(𝑥) is homeomorphism of ℝ^𝑛.

Theorem 3.3. System (2.3) exists a unique equilibrium 𝑥^∗ under the assumptions (H1)–(H3) if the following condition is also satisfied:
(H6)  𝜙_𝑖>𝜓_𝑖+∑^𝑛_𝑗=1𝜁_𝑗𝑖∫^∞₀𝒦(𝑠)𝑑𝑠.

Proof. Defining a map 𝐻(𝑥)=(ℎ₁(𝑥),ℎ₂(𝑥),…,ℎ_𝑛(𝑥))^𝑇∈𝐶⁰(ℝ^𝑛,ℝ^𝑛), where ℎ_𝑖(𝑥)=−𝑎_𝑖𝑥_𝑖+^𝑛_𝑗=1𝑏_𝑖𝑗𝑓_𝑗𝑥_𝑗+^𝑛_𝑗=1𝑐_𝑖𝑗𝑔_𝑗𝑥_𝑗+^𝑛_𝑗=1𝑑_𝑖𝑗^𝑡_−∞𝐾_𝑖𝑗(𝑡−𝑠)ℎ_𝑗𝑥_𝑗𝑑𝑠+𝐼_𝑖, (3.5) the map 𝐻 is a homeomorphism on ℝ^𝑛 if it is injective on ℝ^𝑛 and satisfies ‖𝐻(𝑥)‖→∞ as ‖𝑥‖→∞.
In the following, we will prove that 𝐻(𝑥) is a homeomorphism.
Firstly, we claim that 𝐻(𝑥) is an injective map on ℝ^𝑛. Otherwise, there exist 𝑥^𝑇,𝑦^𝑇∈ℝ^𝑛, and 𝑥^𝑇≠𝑦^𝑇 such that 𝐻(𝑥)=𝐻(𝑦), then 𝑎_𝑖𝑥_𝑖−𝑎_𝑖𝑦_𝑖=^𝑛_𝑗=1𝑏_𝑖𝑗𝑓_𝑗𝑥_𝑗−𝑓_𝑗𝑦_𝑗+^𝑛_𝑗=1𝑐_𝑖𝑗𝑔_𝑗𝑥_𝑗−𝑔_𝑗𝑦_𝑗+^𝑛_𝑗=1𝑑_𝑖𝑗^𝑡_−∞𝐾_𝑖𝑗(𝑡−𝑠)ℎ_𝑗𝑥_𝑗−ℎ_𝑗𝑦_𝑗𝑑𝑠. (3.6) It follows from (H1)–(H3) that 𝑎_𝑖𝑥_𝑖−𝑦_𝑖≤^𝑛_𝑗=1𝑏_𝑖𝑗𝐿^𝑓_𝑗𝑥_𝑗−𝑦_𝑗+^𝑛_𝑗=1𝑐_𝑖𝑗𝐿^𝑔_𝑗𝑥_𝑗−𝑦_𝑗+^𝑛_𝑗=1𝑑_𝑖𝑗𝐿^ℎ_𝑗𝑥_𝑗−𝑦_𝑗^𝑡_−∞𝒦(𝑡−𝑠)𝑑𝑠. (3.7) Therefore, ^𝑛_𝑖=1𝑝𝜇_𝑖𝑎_𝑖𝑥_𝑖−𝑦_𝑖^𝑝≤^𝑛𝑛_𝑖=1𝑗=1𝑝𝜇_𝑖𝑏_𝑖𝑗𝐿^𝑓_𝑗𝑥_𝑖−𝑦_𝑖^𝑝−1𝑥_𝑗−𝑦_𝑗+^𝑛𝑛_𝑖=1𝑗=1𝑝𝜇_𝑖𝑐_𝑖𝑗𝐿^𝑔_𝑗𝑥_𝑖−𝑦_𝑖^𝑝−1𝑥_𝑗−𝑦_𝑗+^𝑛𝑛_𝑖=1𝑗=1𝑝𝜇_𝑖𝑑_𝑖𝑗𝐿^ℎ_𝑗𝑥_𝑖−𝑦_𝑖^𝑝−1𝑥_𝑗−𝑦_𝑗^𝑡_−∞𝒦()≤𝑡−𝑠𝑑𝑠_{𝑛𝑛𝑖=1𝑗=1}𝜇_𝑖^𝑝−1_𝑙=1𝑏_𝑖𝑗^{𝑝𝛼𝑙,𝑖𝑗}𝐿^𝑓_𝑗^{𝑝𝛽𝑙,𝑖𝑗}𝑥_𝑖−𝑦_𝑖^𝑝+𝑏_𝑖𝑗^{𝑝𝛼𝑝,𝑖𝑗}𝐿^𝑓_𝑗^{𝑝𝛽𝑝,𝑖𝑗}𝑥_𝑗−𝑦_𝑗^𝑝+^𝑛𝑛_𝑖=1𝑗=1𝜇_𝑖^𝑝−1_𝑙=1𝑐_𝑖𝑗^{𝑝𝛾𝑙,𝑖𝑗}𝐿^𝑔_𝑗^{𝑝𝛿𝑙,𝑖𝑗}𝑥_𝑖−𝑦_𝑖^𝑝+𝑐_𝑖𝑗^{𝑝𝛾𝑝,𝑖𝑗}𝐿^𝑔_𝑗^{𝑝𝛿𝑝,𝑖𝑗}𝑥_𝑗−𝑦_𝑗^𝑝+^𝑛𝑛_𝑖=1𝑗=1𝜇_𝑖^𝑡_−∞𝒦(𝑡−𝑠)𝑑𝑠^𝑝−1_𝑙=1𝑑_𝑖𝑗^{𝑝𝜉𝑙,𝑖𝑗}𝐿^ℎ_𝑗^{𝑝𝜂𝑙,𝑖𝑗}𝑥_𝑖−𝑦_𝑖^𝑝+^𝑛𝑛_𝑖=1𝑗=1𝜇_𝑖^𝑡_−∞𝒦(𝑡−𝑠)𝑑𝑠𝑑_𝑖𝑗^{𝑝𝜉𝑝,𝑖𝑗}𝐿^ℎ_𝑗^{𝑝𝜂𝑝,𝑖𝑗}𝑥_𝑗−𝑦_𝑗^𝑝=^𝑛_𝑖=1𝜇_𝑖𝑝𝑎_𝑖−𝜙_𝑖+𝜓_𝑖+^𝑛_𝑗=1𝜁_𝑗𝑖^∞₀𝒦(𝑠)𝑑𝑠𝑥_𝑖−𝑦_𝑖^𝑝. (3.8) From (H6), it leads to a contradiction with our assumption. Therefore, 𝐻(𝑥) is an injective map on ℝ^𝑛.
To demonstrate the property ‖𝐻(𝑥)‖→∞ as ‖𝑥‖→∞, we have sgn𝑥^𝑛_𝑖=1𝑖𝑝𝜇_𝑖ℎ_𝑖(𝑥)−ℎ_𝑖(0)𝑥_𝑖^𝑝−1=^𝑛_𝑖=1sgn𝑥_𝑖𝑝𝜇_𝑖𝑥_𝑖^𝑝−1−𝑎_𝑖𝑥_𝑖+^𝑛_𝑗=1𝑏_𝑖𝑗𝑓_𝑗𝑥_𝑗+^𝑛_𝑗=1𝑐_𝑖𝑗𝑔_𝑗𝑥_𝑗+^𝑛_𝑖=1sgn𝑥_𝑖𝑝𝜇_𝑖𝑥_𝑖^𝑛𝑝−1_𝑗=1𝑑_𝑖𝑗^𝑡_−∞𝐾_𝑖𝑗(𝑡−𝑠)ℎ_𝑗𝑥_𝑗𝑑𝑠≤−^𝑛_𝑖=1𝑝𝑎_𝑖𝜇_𝑖𝑥_𝑖^𝑝+^𝑛_𝑖=1𝜇_𝑖𝑝𝑎_𝑖−𝜙_𝑖+𝜓_𝑖+^𝑛_𝑗=1𝜁_𝑗𝑖^∞₀𝒦(𝑠)𝑑𝑠𝑥_𝑖^𝑝=−^𝑛_𝑖=0𝜇_𝑖𝜙_𝑖−𝜓_𝑖−^𝑛_𝑗=1𝜁_𝑗𝑖^∞₀𝒦(𝑠)𝑑𝑠𝑥_𝑖^𝑝≤−𝜔‖𝑥‖^𝑝_𝑝, (3.9) where 𝜔=min_{1≤𝑖≤𝑛}{𝜇_𝑖(𝜙_𝑖−𝜓_𝑖−∑^𝑛_𝑗=1𝜁_𝑗𝑖∫^∞₀𝒦(𝑠)𝑑𝑠)}. Then, we have 𝜔‖𝑥‖^𝑝_𝑝≤𝑝𝜇^𝑛_𝑖=1ℎ_𝑖(𝑥)−ℎ_𝑖(0)𝑥_𝑖^𝑝−1. (3.10) Using the Hölder inequality, we obtain ‖𝑥‖^𝑝_𝑝≤𝑝𝜇𝜔^𝑛_𝑖=1𝑥_𝑖^{𝑝1−1/𝑝𝑛}_𝑖=1ℎ_𝑖(𝑥)−ℎ_𝑖(0)^𝑝1/𝑝, (3.11) which leads to ‖𝑥‖_𝑝≤𝑝𝜇𝜔‖𝐻(𝑥)‖_𝑝+‖𝐻(0)‖_𝑝. (3.12) From (3.12), we see that ‖𝐻(𝑥)‖→∞ as ‖𝑥‖→∞. Thus, the map 𝐻(𝑥) is a homeomorphism on ℝ^𝑛 under the sufficient condition (H6), and hence it has a unique fixed point 𝑥^∗. This fixed point is the unique solution of the system (2.3). The proof is now complete.

To establish some sufficient conditions ensuring the pth moment exponential stability of the equilibrium point of 𝑥^∗ of system (2.3), we transform 𝑥^∗ to the origin by using the transformation 𝑦_𝑖(𝑡)=𝑥_𝑖(𝑡)−𝑥^∗ for 𝑖∈Λ. Then system (2.3) can be rewritten as the following form: 𝑑𝑦_𝑖(𝑡)=−𝑎_𝑖𝑦_𝑖(𝑡)+^𝑛_𝑗=1𝑏_𝑖𝑗𝑓_𝑗𝑦_𝑗(𝑡)+^𝑛_𝑗=1𝑐_𝑖𝑗𝑔_𝑗(𝑦_𝑗𝑡−𝜏_𝑗(𝑡)+^𝑛_𝑗=1𝑑_𝑖𝑗^𝑡_−∞𝐾_𝑖𝑗(𝑡−𝑠)ℎ_𝑗𝑦_𝑗(𝑠)+𝑑𝑠𝑑𝑡_𝑚𝑙=1𝜎_𝑖𝑙(𝑡,𝑦_𝑖(𝑡),𝑦_𝑖𝑡−𝜏_𝑖(𝑡)𝑑𝑤_𝑙(𝑡),𝑡≠𝑡_𝑘,𝑦_𝑖𝑡_𝑘=𝑝_𝑖𝑘𝑦𝑡⁻_𝑘,𝑘∈𝑍⁺,𝑖∈Λ, (3.13) where 𝑎_𝑖𝑦_𝑖(𝑡)=𝑎_𝑖𝑦_𝑖(𝑡)+𝑥^∗_𝑖−𝑎_𝑖𝑥^∗_𝑖,𝑓_𝑗𝑦_𝑗(𝑡)=𝑓_𝑗𝑦_𝑗(𝑡)+𝑥^∗_𝑗−𝑓_𝑗𝑥^∗_𝑗,𝑔_𝑗𝑦_𝑗𝑡−𝜏_𝑗(𝑡)=𝑔_𝑗𝑦_𝑗𝑡−𝜏_𝑗(𝑡)+𝑥^∗_𝑗−𝑔_𝑗𝑥^∗_𝑗,ℎ_𝑗𝑦_𝑗(𝑠)=ℎ_𝑗𝑦_𝑗(𝑠)+𝑥^∗_𝑗−ℎ_𝑗𝑥^∗_𝑗,𝜎_𝑖𝑗𝑡,𝑦_𝑗(𝑡),𝑦_𝑗𝑡−𝜏_𝑗(𝑡)=𝜎_𝑖𝑗𝑡,𝑦_𝑗(𝑡)+𝑥^∗_𝑗,𝑦_𝑗𝑡−𝜏_𝑗(𝑡)+𝑥^∗_𝑗−𝜎_𝑖𝑗𝑡,𝑥^∗_𝑗,𝑥^∗_𝑗,𝑝_𝑖𝑘𝑦𝑡_𝑘=𝑝_𝑖𝑘𝑦𝑡_𝑘+𝑥^∗−𝑝_𝑖𝑘𝑥^∗. (3.14) In order to obtain our results, the following assumptions are necessary.

(H7)  When 𝜂_𝑖≥1,  Φ_𝑖>Ψ_𝑖+∑^𝑛_𝑗=1𝜁_𝑗𝑖∫^∞₀𝒦(𝑠)𝑑𝑠 for any 𝑖∈Λ.

When 0<𝜂_𝑖≤1,  Φ_𝑖>(Ψ_𝑖/𝜂_𝑖)+∑^𝑛_𝑗=1𝜁_𝑗𝑖∫^∞₀𝒦(𝑠)𝑑𝑠 for any 𝑖∈Λ.

(H8)  There exist constant 𝜆∈(0,𝜇₀) and 𝛼∈[0,𝜆), 𝜌_𝑘=max_𝑖∈Λ{1,∑^𝑛_𝑗=1(𝜇_𝑗/𝜇_𝑖)𝑝^𝑘_𝑗𝑖}≤𝑒^{𝛼(𝑡𝑘−𝑡𝑘−1)} such that 𝜆<Φ_𝑖−Ψ_𝑖𝜂_𝑖𝑒^𝜆𝜏−^𝑛_𝑗=1𝜁_𝑗𝑖^∞₀𝒦(𝑠)𝑒^𝜆𝑠𝑑𝑠_∞0𝒦(𝑠)𝑒^𝜆𝑠𝑠𝑑𝑠≤^∞₀𝒦(𝑠)𝑒^𝜇0𝑠𝑑𝑠. (3.15)

Theorem 3.4. Assume that (H1)–(H5) and (H7)-(H8) hold, then system (2.3) exists a unique equilibrium 𝑥^∗, and the equilibrium point is pth moment exponentially stable.

Proof. From (H7), if 0<𝜂_𝑖≤1, then 𝜙_𝑖>Φ_𝑖>Ψ_𝑖𝜂_𝑖+^𝑛_𝑗=1𝜁_𝑗𝑖^∞₀𝒦(𝑠)𝜓𝑑𝑠>^𝑖𝜂_𝑖+^𝑛_𝑗=1𝜁_𝑗𝑖^∞₀𝒦(𝑠)𝑑𝑠>𝜓_𝑖+^𝑛_𝑗=1𝜁_𝑗𝑖^∞₀𝒦(𝑠)𝑑𝑠, (3.16) if 𝜂_𝑖≥1, 𝜙_𝑖>Φ_𝑖>Ψ_𝑖+^𝑛_𝑗=1𝜁_𝑗𝑖^∞₀𝒦(𝑠)𝑑𝑠>𝜓_𝑖+^𝑛_𝑗=1𝜁_𝑗𝑖^∞₀𝒦(𝑠)𝑑𝑠, (3.17) for any 𝑖∈Λ.
Then regardless of cases, (H6) and Φ_𝑖>Ψ_𝑖𝜂_𝑖+^𝑛_𝑗=1𝜁_𝑗𝑖^∞₀𝒦(𝑠)𝑑𝑠 (3.18) are satisfied. Therefore, system (2.3) exists a unique equilibrium point 𝑥^∗.
Now, we define 𝑢_𝑖𝑡,𝑦(𝑡)=𝑒^𝜆𝑡𝜇_𝑖𝑦_𝑖(𝑡)^𝑝=𝑢_𝑖(𝑡),𝑈𝑡,𝑦(𝑡)=^𝑛_𝑖=1𝑢_𝑖(𝑡). (3.19) For 𝑡≠𝑡_𝑘, 𝑘∈ℤ⁺, we obtain ℒ𝑈=𝑈_𝑡+12trace𝜎^𝑇𝑡,𝑦_𝑖(𝑡),𝑦_𝑖𝑡−𝜏_𝑖(𝑡)𝑈_{𝑦𝑖𝑦𝑖}𝑡,𝑦𝜎𝑡,𝑦_𝑖(𝑡),𝑦_𝑖𝑡−𝜏_𝑖(𝑡+_𝑛𝑖=1)𝑈_𝑦𝑖−𝑎_𝑖𝑦_𝑖(𝑡)+^𝑛_𝑗=1𝑏_𝑖𝑗𝑓_𝑗𝑦_𝑗(𝑡)+^𝑛_𝑗=1𝑐_𝑖𝑗𝑔_𝑗𝑦_𝑗𝑡−𝜏_𝑗(𝑡)+^𝑛𝑛_𝑖=1𝑗=1𝑈_𝑦𝑖𝑑_𝑖𝑗^𝑡_−∞𝐾_𝑖𝑗(𝑡−𝑠)ℎ_𝑗𝑦_𝑗(𝑠)≤𝑑𝑠_𝑛𝑖=1𝜆−𝑝𝑎_𝑖+𝑝𝑝−12𝑒_𝑖𝑢_𝑖(𝑡)+𝑝𝑒^{𝜆𝑡𝑛𝑛}_𝑖=1𝑗=1𝜇_𝑖𝑏_𝑖𝑗𝐿^𝑓_𝑗𝑦_𝑖(𝑡)^𝑝−1𝑦_𝑗(𝑡+_{𝑛𝑛𝑖=1𝑗=1})𝜇_𝑖𝑐_𝑖𝑗𝐿^𝑔_𝑗𝑦_𝑖(𝑡)^𝑝−1𝑦_𝑗𝑡−𝜏_𝑗(𝑡)+𝑝−12𝑙_𝑖𝑦_𝑖(𝑡)^𝑝−2𝑦_𝑖𝑡−𝜏_𝑖(𝑡)²+^𝑛𝑛_𝑖=1𝑗=1𝜇_𝑖𝑑_𝑖𝑗𝐿^ℎ_𝑗𝑦_𝑖(𝑡)^𝑝−1∞₀𝒦(𝑠)𝑦_𝑗(≤𝑡−𝑠)𝑑𝑠_𝑛𝑖=1𝜆−𝑝𝑎_𝑖+𝑝𝑝−12𝑒_𝑖𝑢_𝑖(𝑡)+𝑝𝑝−12𝑒^𝑛𝜆𝑡_𝑖=1𝜇_𝑖𝑙_𝑖𝑦_𝑖(𝑡)^𝑝−2𝑦_𝑖𝑡−𝜏_𝑖(𝑡)²+𝑒^{𝑛𝜆𝑡𝑛}_𝑖=1𝑗=1𝜇_𝑖^𝑝−1_𝑙=1𝑏_𝑖𝑗^{𝑝𝛼𝑙,𝑖𝑗}𝐿^𝑓_𝑗^{𝑝𝛽𝑙,𝑖𝑗}𝑦_𝑖(𝑡)^𝑝+𝑏_𝑖𝑗^{𝑝𝛼𝑝,𝑖𝑗}𝐿^𝑓_𝑗^{𝑝𝛽𝑝,𝑖𝑗}𝑦_𝑗(𝑡)^𝑝+𝑒^{𝑛𝜆𝑡𝑛}_𝑖=1𝑗=1𝜇_𝑖^𝑝−1_𝑙=1𝑐_𝑖𝑗^{𝑝𝛾𝑙,𝑖𝑗}𝐿^𝑔_𝑗^{𝑝𝛿𝑙,𝑖𝑗}𝑦_𝑖(𝑡)^𝑝+𝑐_𝑖𝑗^{𝑝𝛾𝑝,𝑖𝑗}𝐿^𝑔_𝑗^{𝑝𝛿𝑝,𝑖𝑗}𝑦_𝑗(𝑡)^𝑝+𝑒^{𝑛𝜆𝑡𝑛}_𝑖=1𝑗=1𝜇_𝑖^∞₀𝒦(𝑠)^𝑝−1_𝑙=1𝑑_𝑖𝑗^{𝑝𝜉𝑙,𝑖𝑗}𝐿^ℎ_𝑗^{𝑝𝜂𝑙,𝑖𝑗}𝑦_𝑖(𝑡)^𝑝+𝑑_𝑖𝑗^{𝑝𝜉𝑙,𝑖𝑗}𝐿^ℎ_𝑗^{𝑝𝜂𝑝,𝑖𝑗}𝑦_𝑗(𝑡−𝑠)^𝑝≤𝑑𝑠_{𝑛𝑖=1𝑛𝑗=1𝑝−1𝑙=1}𝑏_𝑖𝑗^{𝑝𝛼𝑙,𝑖𝑗}𝐿^𝑓_𝑗^{𝑝𝛽𝑙,𝑖𝑗}+𝑐_𝑖𝑗^{𝑝𝛾𝑙,𝑖𝑗}𝐿^𝑔_𝑗^{𝑝𝛿𝑙,𝑖𝑗}+𝑑_𝑖𝑗^{𝑝𝜉𝑙,𝑖𝑗}𝐿^ℎ_𝑗^{𝑝𝜂𝑙,𝑖𝑗∞}₀𝒦(𝑠)𝑑𝑠𝑢_𝑖(𝑡)+^𝑛_𝑖=1𝜆−𝑝𝑎_𝑖+𝑝𝑝−12𝑒_𝑖+𝑝−1𝑝−22𝑙_𝑖+^𝑛_𝑗=1𝜇_𝑗𝜇_𝑖𝑏_𝑗𝑖^{𝑝𝛼𝑝,𝑗𝑖}𝐿^𝑓_𝑗^{𝑝𝛽𝑝,𝑗𝑖}𝑢_𝑖(𝑡)+^𝑛𝑛_𝑖=1𝑗=1𝜇_𝑖𝜇_𝑗𝑑_𝑖𝑗^{𝑝𝜉𝑝,𝑖𝑗}𝐿^ℎ_𝑗^{𝑝𝜂𝑝,𝑖𝑗∞}₀𝒦(𝑠)𝑒^𝜆𝑠𝑢_𝑗()+𝑡−𝑠𝑑𝑠_𝑛𝑖=1𝑒^{𝜆𝜏𝑖(𝑡)𝑛}_𝑗=1𝜇_𝑗𝜇_𝑖𝑐_𝑗𝑖^{𝑝𝛾𝑝,𝑗𝑖}𝐿^𝑔_𝑖^{𝑝𝛿𝑝,𝑗𝑖}+𝑝−1𝑙_𝑖𝑢_𝑖𝑡−𝜏_𝑖(𝑡)=^𝑛_𝑖=1𝜆−Φ_𝑖𝑢_𝑖(𝑡)+Ψ_𝑖𝑒^{𝜆𝜏𝑖(𝑡)}𝑢_𝑖𝑡−𝜏_𝑖(𝑡)+^𝑛𝑛_𝑖=1𝑗=1𝜁_𝑖𝑗^∞₀𝒦(𝑠)𝑒^𝜆𝑠𝑢_𝑗(𝑡−𝑠)𝑑𝑠. (3.20) while Lemma 3.1 is used in the second inequality. Let 𝑉𝑡,𝑦(𝑡)=𝑈𝑡,𝑦(𝑡)+^𝑛_𝑖=1Ψ_𝑖𝑒^{𝜆𝜏𝑖𝑡}_{𝑡−𝜏𝑖(𝑡)}𝑢_𝑖(𝑠)1−𝜏̇_𝑖𝜓_𝑖⁻¹(𝑠)+𝑑𝑠_{𝑛𝑛𝑖=1𝑗=1}𝜁_𝑖𝑗^∞₀𝒦(𝑠)𝑒^𝜆𝑠𝑡_𝑡−𝑠𝑢_𝑗(𝑟)𝑑𝑟𝑑𝑠, (3.21) for 𝑡≥0, where 𝜓_𝑖(𝑠)=𝑠−𝜏_𝑖(𝑠), applying It𝑜̂ formula to (3.21), we can get ℒ𝑉=ℒ𝑈+^𝑛_𝑖=1Ψ_𝑖𝑒^𝜆𝜏𝑖𝑢_𝑖(𝑡)1−𝜏̇_𝑖𝜓_𝑖⁻¹(𝑡)−𝑢_𝑖𝑡−𝜏_𝑖(𝑡)1−𝜏̇_𝑖(𝑡)𝑡−𝜏_𝑖(𝑡)^′+^𝑛𝑛_𝑖=1𝑗=1𝜁_𝑖𝑗^∞₀𝒦(𝑠)𝑒^𝜆𝑠𝑢_𝑗(𝑡)−𝑢_𝑗(𝑡−𝑠)𝑑𝑠≤−^𝑛_𝑖=1Φ_𝑖Ψ−𝜆−^𝑖𝜂_𝑖𝑒^𝜆𝜏−^𝑛_𝑗=1𝜁_𝑗𝑖^∞₀𝒦(𝑠)𝑒^𝜆𝑠𝑑𝑠𝑢_𝑖(𝑡). (3.22) From (H8), we have ℒ𝑉<0. (3.23) On the other hand, we have 𝐸𝑈𝑡_𝑘,𝑦𝑡_𝑘=𝐸𝑒^{𝜆𝑡𝑘𝑛}_𝑖=1𝜇_𝑖𝑦_𝑖𝑡_𝑘^𝑝≤𝐸𝑒^{𝜆𝑡𝑘𝑛}_𝑖=1𝜇_𝑖^𝑛_𝑗=1𝑝^𝑘_𝑖𝑗𝑦_𝑗𝑡⁻_𝑘^𝑝≤𝜌_𝑘𝐸𝑈𝑡⁻_𝑘,𝑦𝑡⁻_𝑘,𝐸𝑉𝑡_𝑘,𝑦𝑡_𝑘=𝐸𝑈𝑡_𝑘,𝑦𝑡_𝑘+𝐸^𝑛_𝑖=1Ψ_𝑖𝑒^{𝜆𝜏𝑖𝑡𝑘}_{𝑡𝑘−𝜏𝑖𝑡𝑘}𝑢_𝑖(𝑠)1−𝜏̇_𝑖𝜓_𝑖⁻¹(𝑠)𝑑𝑠+𝐸^𝑛𝑛_𝑖=1𝑗=1𝜁_𝑖𝑗^∞₀𝒦(𝑠)𝑒^{𝜆𝑠𝑡𝑘}_{𝑡𝑘−𝑠}𝑢_𝑗(𝑟)𝑑𝑟𝑑𝑠≤𝜌_𝑘𝐸𝑉𝑡⁻_𝑘,𝑦𝑡⁻_𝑘, (3.24)(3.25) for 𝑡∈[𝑡_𝑘−1,𝑡_𝑘),  𝑘∈𝑍⁺, by (3.23), (3.25), and (H8), we have 𝐸𝑈𝑡,𝑦(𝑡)≤𝐸𝑉𝑡,𝑦(𝑡)≤𝐸𝑉𝑡_𝑘−1,𝑦𝑡_𝑘−1≤𝜌_𝑘−1𝐸𝑉𝑡⁻_𝑘−1,𝑦𝑡⁻_𝑘−1≤𝜌₀𝜌₁⋯𝜌_𝑘−1𝐸𝑉𝑡₀,𝑦𝑡₀≤𝑒^𝛼𝑡1⋯𝑒^{𝛼(𝑡𝑘−1−𝑡𝑘−2)}𝐸𝑉𝑡₀,𝑦𝑡₀≤𝑒^𝛼𝑡𝐸𝑉0,𝑦(0), (3.26) where 𝜌₀=1.
On the other hand, we observe that 𝑉0,𝑦(0)≤^𝑛_𝑖=1𝜇_𝑖𝑦_𝑖(0)^𝑝+^𝑛_𝑖=1Ψ_𝑖𝑒^{𝜆𝜏𝑖0}_−𝜏𝑢_𝑖(𝑠)1−𝜏̇_𝑖𝜓_𝑖⁻¹(𝑠)+𝑑𝑠_{𝑛𝑛𝑖=1𝑗=1}𝜁_𝑖𝑗^∞₀𝒦(𝑠)𝑒^𝜆𝑠0_−𝑠𝑢_𝑗(𝑟)≤𝑑𝑟𝑑𝑠_𝑛𝑖=1𝜇_𝑖1+^𝑛_𝑗=1𝜁_𝑗𝑖^∞₀𝒦(𝑠)𝑒^𝜆𝑠Ψ𝑠𝑑𝑠+^𝑖𝜏𝜂_𝑖𝑒^𝜆𝜏𝑖sup_{−∞<𝑠≤0}𝑦_𝑖(𝑠)^𝑝. (3.27) It follows that 𝐸^𝑛_𝑖=1𝑦_𝑖(𝑡)^𝑝≤𝑀𝑒^{−(𝜆−𝛼)𝑡}sup^𝑛_{−∞<𝑠≤0𝑖=1}𝑦_𝑖(𝑠)^𝑝, (3.28) for 𝑡≥0, where 1≤𝑀=𝜇𝜇1+max_𝑖∈Λ^𝑛_𝑗=1𝜁_𝑗𝑖^∞₀𝒦(𝑠)𝑒^𝜆𝑠Ψ𝑠𝑑𝑠+^𝑖𝜏𝜂_𝑖𝑒^𝜆𝜏𝑖<∞, (3.29) which means that 𝐸𝑥(𝑡)−𝑥^∗𝑝≤𝐸𝜑−𝑥^∗𝑝𝑀𝑒^{−(𝜆−𝛼)𝑡}for𝑡≥𝑡₀. (3.30) Therefore, the equilibrium point of system (2.3) is pth moment exponentially stable.

Corollary 3.5. If 𝑝≥2, under the assumptions (H1)–(H5), system (2.3) exists a unique equilibrium point 𝑥^∗, and 𝑥^∗ is pth moment exponentially stable if the following two conditions are satisfied:
(H9) 𝑝𝑎_𝑖−𝑝−1^𝑛_𝑗=1𝑏_𝑖𝑗𝐿^𝑓_𝑗+𝑐_𝑖𝑗𝐿^𝑔_𝑗+𝑑_𝑖𝑗𝐿^ℎ_𝑗^∞₀𝒦(𝑠)𝑑𝑠−^𝑛_𝑗=1𝑏_𝑗𝑖𝐿^𝑓_𝑖>^𝑛_𝑗=1𝑐_𝑗𝑖𝐿^𝑔_𝑖+𝑑_𝑗𝑖𝐿^ℎ_𝑖^∞₀𝒦(𝑠)+𝑝𝑑𝑠𝑝−12𝑙_𝑖+𝑒_𝑖. (3.31)
(H10)  There exist constant 𝜆∈(0,𝜇₀) and 𝛼∈[0,𝜆),  𝜌_𝑘=max_𝑖∈Λ{1,∑^𝑛_𝑗=1(𝜇_𝑗/𝜇_𝑖)𝑝^𝑘_𝑗𝑖}≤𝑒^{𝛼(𝑡𝑘−𝑡𝑘−1)} such that 𝜆<𝑝𝑎_𝑖−𝑝−1^𝑛_𝑗=1𝑏_𝑖𝑗𝐿^𝑓_𝑗+𝑐_𝑖𝑗𝐿^𝑔_𝑗+𝑑_𝑖𝑗𝐿^ℎ_𝑗^∞₀𝒦(𝑠)𝑑𝑠−^𝑛_𝑗=1𝑏_𝑗𝑖𝐿^𝑓_𝑖−𝑝𝑝−12𝑒_𝑖−𝑝−2𝑝−12𝑙_𝑖−^𝑛_𝑗=1𝑐_𝑗𝑖𝐿^𝑔_𝑖+𝑝−1𝑙_𝑖𝑒^𝜆𝜏−^𝑛_𝑗=1𝑑_𝑗𝑖𝐿^ℎ_𝑖^∞₀𝒦(𝑠)𝑒^𝜆𝑠𝑑𝑠. (3.32)

Proof. In Theorem 3.4, let 𝛼_{𝑙,𝑖𝑗}=𝛽_{𝑙,𝑖𝑗}=𝛾_{𝑙,𝑖𝑗}=𝛿_{𝑙,𝑖𝑗}=𝜉_{𝑙,𝑖𝑗}=𝜂_{𝑙,𝑖𝑗}=1/𝑝,  𝜇_𝑖=1 for all 𝑖,𝑗,𝑙∈Λ.  The result is obtained directly.

Remark 3.6. In Corollary 3.5, if 𝑝=2, the conditions (H9) and (H10) are less weak than the following conditions: (𝐻₉)2min_𝑖∈Λ𝑎_𝑖−max_𝑖∈Λ^𝑛_𝑗=1𝑏_𝑖𝑗𝐿^𝑓_𝑗−max_𝑖∈Λ^𝑛_𝑗=1𝑐_𝑖𝑗𝐿^𝑔_𝑗−max_𝑖∈Λ^𝑛_𝑗=1𝑑_𝑖𝑗𝐿^ℎ_𝑗^∞₀𝒦(𝑠)>𝑑𝑠_𝑛𝑗=1max_𝑖∈Λ𝑏_𝑗𝑖𝐿^𝑓_𝑖+^𝑛_𝑗=1max_𝑖∈Λ𝑐_𝑗𝑖𝐿^𝑔_𝑖+^𝑛_𝑗=1max_𝑖∈Λ𝑑_𝑗𝑖𝐿^ℎ_𝑖^∞₀𝒦(𝑠)𝑑𝑠+max_𝑖∈Λ𝑙_𝑖+^𝑛_𝑗=1max_𝑖∈Λ𝑒_𝑖. (3.33)(𝐻₁₀) There exist constant 𝜆∈(0,𝜇₀) and 𝛼∈[0,𝜆), 𝜌_𝑘=max_𝑖∈Λ{1,∑^𝑛_𝑗=1(𝜇_𝑗/𝜇_𝑖)𝑝^𝑘_𝑗𝑖}≤𝑒^{𝛼(𝑡𝑘−𝑡𝑘−1)} such that 𝜆<2min_𝑖∈Λ𝑎_𝑖−max_𝑖∈Λ^𝑛_𝑗=1𝑏_𝑖𝑗𝐿^𝑓_𝑗−max_𝑖∈Λ^𝑛_𝑗=1𝑐_𝑖𝑗𝐿^𝑔_𝑗−max_𝑖∈Λ^𝑛_𝑗=1𝑑_𝑖𝑗𝐿^ℎ_𝑗^∞₀𝒦(𝑠)−𝑑𝑠_𝑛𝑗=1max_𝑖∈Λ𝑏_𝑗𝑖𝐿^𝑓_𝑖−max_𝑖∈Λ𝑒_𝑖−^𝑛_𝑗=1max_𝑖∈Λ𝑐_𝑗𝑖𝐿^𝑔_𝑖+max_𝑖∈Λ𝑙_𝑖𝑒^𝜆𝜏−^𝑛_𝑗=1max_𝑖∈Λ𝑑_𝑗𝑖𝐿^ℎ_𝑖^∞₀𝒦(𝑠)𝑒^𝜆𝑠𝑑𝑠, (3.34) while (𝐻₉) and (𝐻₁₀) were asked in Theorem  3.2 in [18].

Corollary 3.7. Under the assumptions (H2) and (H8), system (2.1) exists a unique equilibrium point 𝑥^∗, and 𝑥^∗ is globally exponentially stable if 0<𝜂_𝑖≤1, and the following condition is also satisfied: 𝑝𝑎_𝑖−𝑝−1^𝑛_𝑗=1𝐿^𝑓_𝑗𝑏_𝑖𝑗^{𝑝𝛼𝑖𝑗/(𝑝−1)}+𝐿^𝑔_𝑗𝑐_𝑖𝑗^{𝑝𝛾𝑖𝑗/(𝑝−1)}−^𝑛_𝑗=1𝜇_𝑗𝜇_𝑖𝑏_𝑗𝑖^{𝑝(1−𝛼𝑗𝑖)}𝐿^𝑓_𝑖>1𝜂_𝑖^𝑛_𝑗=1𝜇_𝑗𝜇_𝑖𝑐_𝑗𝑖^{𝑝(1−𝛾𝑗𝑖)}𝐿^𝑔_𝑖. (3.35)

Proof. In Theorem 3.4, when 𝑝>1, choosing 𝛼_{𝑙,𝑖𝑗}=𝛼_𝑖𝑗/(𝑝−1),  𝛾_{𝑙,𝑖𝑗}=𝛾_𝑖𝑗/(𝑝−1) for 𝑙=1,2,…,𝑝−1, and 𝑑_𝑖𝑗=0,  𝛽_{𝑙,𝑖𝑗}=𝛿_{𝑙,𝑖𝑗}=1/𝑝 for all 𝑙∈Λ, then the result is obtained.

Remark 3.8. If 0<𝜂_𝑖≤1,  𝐿^𝑓_𝑖=𝐿^𝑔_𝑖=𝐿_𝑖,  it follows from (3.35) that 𝑝𝑎_𝑖−𝑝−1^𝑛_𝑗=1𝐿_𝑗𝑏_𝑖𝑗^{𝑝𝛼𝑖𝑗/(𝑝−1)}+𝐿_𝑗𝑐_𝑖𝑗^{𝑝𝛾𝑖𝑗/(𝑝−1)}−^𝑛_𝑗=1𝜇_𝑗𝜇_𝑖𝑏_𝑗𝑖^{𝑝(1−𝛼𝑗𝑖)}𝐿_𝑖>^𝑛_𝑗=1𝜇_𝑗𝜇_𝑖𝑐_𝑗𝑖^{𝑝(1−𝛾𝑗𝑖)}𝐿_𝑖, (3.36) this is less conservative than the following inequality: min_𝑖∈Λ𝑝𝑎_𝑖−𝑝−1^𝑛_𝑗=1𝐿_𝑗𝑏_𝑖𝑗^{𝑝𝛼𝑖𝑗/(𝑝−1)}+𝐿_𝑗𝑐_𝑖𝑗^{𝑝𝛾𝑖𝑗/(𝑝−1)}−^𝑛_𝑗=1𝜇_𝑗𝜇_𝑖𝑏_𝑗𝑖^{𝑝(1−𝛼𝑗𝑖)}𝐿_𝑖>max_𝑖∈Λ^𝑛_𝑗=1𝜇_𝑗𝜇_𝑖𝑐_𝑗𝑖^{𝑝(1−𝛾𝑗𝑖)}𝐿_𝑖, (3.37) while (3.37) was required in Theorem  3.1 in [5] and in the only theorem in [8].
When 𝑝=2,  𝛼_𝑖𝑗=𝛾_𝑖𝑗=1/2,  𝜇_𝑖=1,  (3.37) is equal to min_𝑖∈Λ2𝑎_𝑖−^𝑛_𝑗=1𝐿_𝑗𝑏_𝑖𝑗+𝐿_𝑗𝑐_𝑖𝑗−^𝑛_𝑗=1𝑏_𝑗𝑖𝐿_𝑖>max_𝑖∈Λ^𝑛_𝑗=1𝑐_𝑗𝑖𝐿_𝑖, (3.38) while (3.38) was required in Theorem  3.2 in [19].

Similar to Corollaries 3.5 and 3.7, the following results are directly gained from Theorem 3.4.

Corollary 3.9. Under the assumptions (H2), (H4), and (H8), system (2.2) exists a unique equilibrium point 𝑥^∗, and 𝑥^∗ is pth moment exponentially stable if 0<𝜂_𝑖≤1, and the following condition is satisfied: 𝑝𝑎_𝑖−𝑝−1^𝑛_𝑗=1𝑏_𝑖𝑗𝐿^𝑓_𝑗+𝑐_𝑖𝑗𝐿^𝑔_𝑗−^𝑛_𝑗=1𝜇_𝑗𝜇_𝑖𝑏_𝑗𝑖𝐿^𝑓_𝑖−𝑝𝑝−12𝑒_𝑖−𝑝−1𝑝−22𝑙_𝑖>1𝜂_𝑖𝜇_𝑗𝜇_𝑖^𝑛_𝑗=1𝑐_𝑗𝑖𝐿^𝑔_𝑖+𝑝−1𝑙_𝑖. (3.39) Especially, if 𝑝=2, then the equilibrium 𝑥^∗ is exponentially stable in mean square if the following condition is also satisfied: 2𝑎_𝑖−^𝑛_𝑗=1𝑏_𝑖𝑗^{2𝛼𝑖𝑗}𝐿^𝑓_𝑗^{2𝛽𝑖𝑗}+𝑐_𝑖𝑗^{2𝛾𝑖𝑗}𝐿^𝑔_𝑗^{2𝛿𝑖𝑗}−^𝑛_𝑗=1𝜇_𝑗𝜇_𝑖𝑏_𝑗𝑖^{2−2𝛼𝑗𝑖}𝐿^𝑓_𝑖^{2−2𝛽𝑗𝑖}−𝑒_𝑖>1𝜂_𝑖𝜇_𝑗𝜇_𝑖^𝑛_𝑗=1𝑐_𝑗𝑖^{2−2𝛾𝑗𝑖}𝐿^𝑔_𝑖^{2−2𝛿𝑗𝑖}+𝑙_𝑖. (3.40)

Remark 3.10. If 0<𝜂_𝑖≤1, it follows from (3.39) that 𝑝𝑎_𝑖−𝑝−1^𝑛_𝑗=1𝑏_𝑖𝑗𝐿^𝑓_𝑗+𝑐_𝑖𝑗𝐿^𝑔_𝑗−^𝑛_𝑗=1𝜇_𝑗𝜇_𝑖𝑏_𝑗𝑖𝐿^𝑓_𝑖−𝑝𝑝−12𝑒_𝑖−𝑝−1𝑝−22𝑙_𝑖>𝜇_𝑗𝜇_𝑖^𝑛_𝑗=1𝑐_𝑗𝑖𝐿^𝑔_𝑖+𝑝−1𝑙_𝑖. (3.41) This is less conservative than the following inequality: min_{1≤𝑖≤𝑛}𝑝𝑎_𝑖−𝑝−1^𝑛_𝑗=1𝑏_𝑖𝑗𝐿^𝑓_𝑗+𝑐_𝑖𝑗𝐿^𝑔_𝑗−^𝑛_𝑗=1𝜇_𝑗𝜇_𝑖𝑏_𝑗𝑖𝐿^𝑓_𝑖−^𝑛_𝑗=1𝑝−1𝑝−22𝑒_𝑖+𝑙_𝑖−^𝑛_𝑗=1𝜇_𝑗𝜇_𝑖𝑝−1𝑒_𝑖>max_{1≤𝑖≤𝑛}𝜇_𝑗𝜇_𝑖^𝑛_𝑗=1𝑐_𝑗𝑖𝐿^𝑔_𝑖+𝜇_𝑗𝜇_𝑖𝑝−1𝑙_𝑖, (3.42) while (3.42) was required in Theorem 3.3 in [26].

Remark 3.11. If 0<𝜂_𝑖≤1, it follows from (3.40) that 2𝑎_𝑖−^𝑛_𝑗=1𝑏_𝑖𝑗^{2𝛼𝑖𝑗}𝐿^𝑓_𝑗^{2𝛽𝑖𝑗}+𝑐_𝑖𝑗^{2𝛾𝑖𝑗}𝐿^𝑔_𝑗^{2𝛿𝑖𝑗}−^𝑛_𝑗=1𝜇_𝑗𝜇_𝑖𝑏_𝑗𝑖^{2−2𝛼𝑗𝑖}𝐿^𝑓_𝑖^{2−2𝛽𝑗𝑖}−𝑒_𝑖>𝜇_𝑗𝜇_𝑖^𝑛_𝑗=1𝑐_𝑗𝑖^{2−2𝛾𝑗𝑖}𝐿^𝑔_𝑖^{2−2𝛿𝑗𝑖}+𝑙_𝑖, (3.43) this is less conservative than the following inequality: min_{1≤𝑖≤𝑛}2𝑎_𝑖−^𝑛_𝑗=1𝑏_𝑖𝑗^{2𝛼𝑖𝑗}𝐿^𝑓_𝑗^{2𝛽𝑖𝑗}+𝑐_𝑖𝑗^{2𝛾𝑖𝑗}𝐿^𝑔_𝑗^{2𝛿𝑖𝑗}−^𝑛_𝑗=1𝜇_𝑗𝜇_𝑖𝑏_𝑗𝑖^{2−2𝛼𝑗𝑖}𝐿^𝑓_𝑖^{2−2𝛽𝑗𝑖}−max_{1≤𝑖≤𝑛}𝜇_𝑖𝜇_𝑖𝑒_𝑖>max_{1≤𝑖≤𝑛}𝜇_𝑗𝜇_𝑖^𝑛_𝑗=1𝑐_𝑗𝑖^{2−2𝛾𝑗𝑖}𝐿^𝑔_𝑖^{2−2𝛿𝑗𝑖}+max_{1≤𝑖≤𝑛}𝜇_𝑖𝜇_𝑖𝑙_𝑖, (3.44) while (3.44) was required in Theoerm 3.3 in [27].

4. Illustrative Example

In this section, we will give an example to show that the conditions given in the previous sections are less weak than those given in some of the earlier literatures, such as [18].

Consider the following stochastic neural networks with mixed time delays 𝑑𝑥_𝑖(𝑡)=−𝑎_𝑖𝑥_𝑖(𝑡)+²_𝑗=1𝑏_𝑖𝑗𝑓_𝑗𝑥_𝑗(𝑡)+²_𝑗=1𝑐_𝑖𝑗𝑔_𝑗𝑥_𝑗𝑡−𝜏_𝑗(𝑡)+²_𝑗=1𝑑_𝑖𝑗⋅^𝑡_−∞𝐾_𝑖𝑗(𝑡−𝑠)ℎ_𝑗𝑥_𝑗(𝑠)𝑑𝑠+𝐼_𝑖𝑑𝑡+𝜎_𝑖𝑡,𝑥_𝑖(𝑡),𝑥_𝑖𝑡−𝜏_𝑖(𝑡)𝑑𝑤_𝑖(𝑡), (4.1) where 𝑡≠𝑡_𝑘,  𝑖∈Λ={1,2},  𝑡_𝑘=𝑘,  𝑓_𝑗(𝑠)=𝑔_𝑗(𝑠)=ℎ_𝑗(𝑠)=(1/2)[|𝑥+1|−|𝑥−1|], and  𝐾_𝑖𝑗(𝑠)=𝑒^−𝑠. 𝑎_𝑖2×1=3,𝑏1.7_𝑖𝑗2×2=,𝑐0.10.20.40.3^𝑖𝑗2×2=,𝑑0.20.30.10.4_𝑖𝑗2×2=,𝐼0.10.20.20.3^𝑖2×1=,𝜎0.942.5^𝑖2×1=0.1𝑥₁𝑡−𝜏₁(𝑡)𝑥−1.20.1₂𝑡−𝜏₂(𝑡).−1.4 (4.2)

In the following, we introduce the following nonlinear impulsive controllers: 𝑥₁𝑡_𝑘=0.05sin𝑥₁𝑡⁻_𝑘−1.2−0.02𝑥₂𝑡_𝑘+1.48,𝑘∈ℤ⁺,𝑥₂𝑡_𝑘=−0.03𝑥₁𝑡_𝑘+0.04cos𝑥₂𝑡⁻_𝑘−1.4+1.72,𝑘∈ℤ⁺. (4.3) In this case, we have 𝐿^𝑓_𝑗=𝐿^𝑔_𝑗=𝐿^ℎ_𝑗=1,  𝒦(𝑠)=𝑒^−𝑠,  𝑒_𝑗=0,  𝑙_𝑗=0.01 for 𝑗=1,2, and  𝜇₀=0.9.

For 𝑝=2, we can compute that 2𝑎₁−²_𝑗=1𝑏_1𝑗𝐿^𝑓_𝑗+𝑐_1𝑗𝐿^𝑔_𝑗+𝑑_1𝑗𝐿^ℎ_𝑗^∞₀𝒦(𝑠)𝑑𝑠−²_𝑗=1𝑏_𝑗1𝐿^𝑓₁>=1.8_2𝑗=1𝑐_𝑗1𝐿^𝑔₁+𝑑_𝑗1𝐿^ℎ₁^∞₀𝒦(𝑠)𝑑𝑠+𝑙₁+𝑒₁=0.61,2𝑎₂−²_𝑗=1𝑏_2𝑗𝐿^𝑓_𝑗+𝑐_2𝑗𝐿^𝑔_𝑗+𝑑_2𝑗𝐿^ℎ_𝑗^∞₀𝒦(𝑠)𝑑𝑠−²_𝑗=1𝑏_𝑗2𝐿^𝑓₂>=3.8_2𝑗=1𝑐_𝑗2𝐿^𝑔₁+𝑑_𝑗2𝐿^ℎ₁^∞₀𝒦(𝑠)𝑑𝑠+𝑙₂+𝑒₂𝑃=1.21,_𝑘=.0.20.080.120.16 (4.4) Choosing 𝜇_𝑖=1 for 𝑖=1,2,  we have 𝜌_𝑘=max_𝑖∈Λ{1,∑²_𝑗=1𝑝^𝑘_𝑗𝑖}=1  and 𝜆=0.6and𝛼=0.5,  then 0.6=𝜆<2𝑎₁−²_𝑗=1𝑏_1𝑗𝐿^𝑓_𝑗+𝑐_1𝑗𝐿^𝑔_𝑗+𝑑_1𝑗𝐿^ℎ_𝑗^∞₀𝒦(𝑠)𝑑𝑠−²_𝑗=1𝑏_𝑗1𝐿^𝑓₁−𝑒₁−²_𝑗=1𝑐_𝑗1𝐿^𝑔₁+𝑙₁𝑒^𝜆𝜏−²_𝑗=1𝑑_𝑗1𝐿^ℎ₁^∞₀𝒦(𝑠)𝑒^𝜆𝑠𝑑𝑠=1.05−0.31𝑒^0.12=0.7,0.6=𝜆<2𝑎₂−²_𝑗=1𝑏_2𝑗𝐿^𝑓_𝑗+𝑐_2𝑗𝐿^𝑔_𝑗+𝑑_2𝑗𝐿^ℎ_𝑗^∞₀𝒦(𝑠)𝑑𝑠−²_𝑗=1𝑏_𝑗2𝐿^𝑓₂−𝑒₂−²_𝑗=1𝑐_𝑗2𝐿^𝑔₂+𝑙₂𝑒^𝜆𝜏−²_𝑗=1𝑑_𝑗2𝐿^ℎ₂^∞₀𝒦(𝑠)𝑒^𝜆𝑠𝑑𝑠=2.55−0.71𝑒^0.12=1.749,_∞0𝒦(𝑠)𝑒^𝜆𝑠𝑠𝑑𝑠=6.25≤^∞₀𝒦(𝑠)𝑒^𝜇0𝑠𝑑𝑠=10. (4.5) Thus, all conditions of Theorem 3.4 in this paper are satisfied; the equilibrium solution is exponentially stable in mean square. From above discussion, it is easy to see that 2min_𝑖∈Λ𝑎_𝑖−max_𝑖∈Λ²_𝑗=1𝑏_𝑖𝑗𝐿^𝑓_𝑗−max_𝑖∈Λ²_𝑗=1𝑐_𝑖𝑗𝐿^𝑔_𝑗−max_𝑖∈Λ²_𝑗=1𝑑_𝑖𝑗𝐿^ℎ_𝑗^∞₀𝒦(𝑠)−𝑑𝑠_𝑛𝑗=1max_𝑖∈Λ𝑏_𝑗𝑖𝐿^𝑓_𝑖<=1.0_2𝑗=1max_𝑖∈Λ𝑐_𝑗𝑖𝐿^𝑔_𝑖+²_𝑗=1max_𝑖∈Λ𝑑_𝑗𝑖𝐿^ℎ_𝑖^∞₀𝒦(𝑠)𝑑𝑠+max_𝑖∈Λ𝑙_𝑖+²_𝑗=1max_𝑖∈Λ𝑒_𝑖=1.11, (4.6) which implies that the condition (𝐻₉) in [18] do not hold for this example. So our results are less weaker than some previous results.

5. Conclusion

In this paper, we investigate the 𝑝th moment exponential stability for stochastic neural networks with mixed delays under nonlinear impulsive effects. By means of Lyapunov functionals, stochastic analysis, and differential inequality technique, some sufficient conditions for the pth moment exponential stability of this system are derived. The results of this paper are new, and they supplement and improve some of the previously known results [5, 8, 18, 19, 26, 27]. Moreover, examples are given to illustrate the effectiveness of our results. Furthermore, the method given in this paper may be extended to study other neural networks, such as the model in [29] and stochastic Cohen-Grossberg neural networks in [30], and we can get improved results too.

Acknowledgments

The authors would like to thank the editor and the anonymous referees for their detailed comments and valuable suggestions which considerably improved the presentation of this paper. This work was supported in part by the National Natural Science Foundation of China under Grants no. 11101054, Hunan Provincial Natural Science Foundation of China under Grant no. 12jj4005, the Open Fund Project of Key Research Institute of Philosophies and Social Sciences in Hunan Universities under Grants no. 11FEFM11 and the Scientific Research Funds of Hunan Provincial Science and Technology Department of China under Grants no. 2012SK3096.