Periodic Solutions for Impulsive Stochastic BAM Neural Networks with Time-Varying Delays in Leakage Terms

Abstract

By using an integral inequality, we establish some sufficient conditions for the existence and p-exponential stability of periodic solutions for a class of impulsive stochastic BAM neural networks with time-varying delays in leakage terms. Moreover, we present an example to illustrate the feasibility of our results.

1. Introduction

Since it was proposed by Kosko (see [1]), the bidirectional associative memory (BAM) neural networks have attracted considerable attentions due to their extensive applications in classification of patterns, associative memories, image processing, and other areas. In the past few years, many scholars have obtained lots of good results on the dynamical behaviors analysis of BAM neural networks. The reader may see [2–8] and the references therein.

But in a real nervous system, it is usually unavoidably affected by external perturbations which are in many cases of great uncertainty and hence may be treated as random. As pointed out by Haykin [9], in real nervous systems, synaptic transmission is a noisy process brought on by random fluctuations from the release of neurotransmitters and other probabilistic causes. And the stability of neural networks could be stabilized or destabilized by some stochastic inputs [10]. Therefore, it is significant and of prime importance to consider the dynamics of stochastic neural networks. With respect to stochastic neural networks, there are many works on the stability. For example, in  [11–17], the scholars studied the stability of different classes of stochastic neural networks. For other results on stochastic neural networks, the reader may see  [18–23] and the references therein.

However, the above results are mainly on the stability of considered stochastic neural networks. And it is well known that studies on neural dynamical systems not only involve a discussion of stability properties, but also involve many dynamic behaviors such as periodic oscillatory behavior. On the other hand, the neural networks are often subject to impulsive effects that in turn affect dynamical behaviors of the systems. Moreover, a leakage delay, which is the time delay in the 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. However, so far, very little attention has been paid to neural networks with time delay in the leakage (or “forgetting”) term. Such time delays in the leakage term are difficult to handle but have great impact on the dynamical behavior of neural networks. Therefore, it is meaningful to consider neural networks with time delays in the leakage term [24–32].

But to the best of our knowledge, there are few papers published on studying the existence of periodic solutions of impulsive stochastic neural networks with time delay in the leakage term. Motivated by the previous discussions, in this paper, we consider the following impulsive stochastic BAM neural networks: 𝑑𝑥_𝑖(𝑡)=−𝑎_𝑖(𝑡)𝑥_𝑖𝑡−𝜏_𝑖(𝑡)+^𝑚_𝑗=1𝑐_𝑖𝑗(𝑡)𝑓_𝑗𝑦_𝑗(𝑡)+𝐼_𝑖(𝑡)+𝑑𝑡_𝑚𝑗=1𝜎_1𝑖𝑗𝑦_𝑗(𝑡)𝑑𝑤_1𝑗(𝑡),𝑡≥𝑡₀,𝑡≠𝑡_𝑘,𝑥_𝑖𝑡⁺_𝑘=𝛼_𝑖𝑘𝑥_𝑖𝑡_𝑘,𝑡>0,𝑡=𝑡_𝑘,𝑑𝑦_𝑗(𝑡)=−𝑏_𝑗(𝑡)𝑦_𝑗𝑡−𝜎_𝑗(𝑡)+^𝑛_𝑖=1𝑑_𝑗𝑖(𝑡)𝑔_𝑖𝑥_𝑖(𝑡)+𝐽_𝑗(𝑡+)𝑑𝑡_𝑛𝑖=1𝜎_2𝑗𝑖𝑥_𝑖(𝑡)𝑑𝑤_2𝑖(𝑡),𝑡≥𝑡₀,𝑡≠𝑡_𝑘,𝑦_𝑗𝑡⁺_𝑘=𝛾_𝑗𝑘𝑦_𝑗𝑡_𝑘,𝑡>0,𝑡=𝑡_𝑘, (1) where 𝑖=1,2,…,𝑛, 𝑗=1,2,…,𝑚 (𝑛 and 𝑚 are the number of neurons in layers) and 𝑥_𝑖(𝑡) and 𝑦_𝑗(𝑡) denote the activations of the 𝑖th neuron and the 𝑗th neuron at time 𝑡; 𝑎_𝑖 and 𝑏_𝑗 represent the rate with which the 𝑖th neuron and 𝑗th neuron will reset their potential to the resting state in isolation when they are disconnected from the network and the external inputs at time 𝑡; 0<𝜏_𝑖<𝜏_𝑖 and 0<𝜎_𝑗<𝜎_𝑗 denote the leakage delays; 𝑓_𝑗 and 𝑔_𝑖 are the input-output functions (the activation functions); 𝑐_𝑖𝑗 and 𝑑_𝑗𝑖 are elements of feedback templates at time 𝑡; 𝐼_𝑖 and 𝐽_𝑗 denote biases of the 𝑖th neuron and the 𝑗th neuron at time 𝑡, 𝑖=1,2,…,𝑛,𝑗=1,2,…,𝑚; 𝑤₁(𝑡)=(𝑤₁₁(𝑡),…,𝑤_1𝑚(𝑡))^𝑇 and 𝑤₂(𝑡)=(𝑤₂₁(𝑡),…,𝑤_2𝑛(𝑡))^𝑇 are 𝑛-dimensional Brownian motions defined on complete probability space (Ω,𝐹,{𝐹_𝑡}_𝑡>0,𝑃); here, we denote by 𝐹 the associated 𝜎-algebra generated by {𝑤_𝑘(𝑡)} with the probability measure 𝑃,  𝑘=1,2;  𝜎_1𝑖𝑗 and 𝜎_2𝑗𝑖 are Borel measurable functions; 𝜎₁=(𝜎_1𝑖𝑗)_𝑛×𝑚 and 𝜎₂=(𝜎_2𝑗𝑖)_𝑚×𝑚 are diffusion coefficient matrices; 𝑥_𝑖(𝑡⁺_𝑘)=𝛼_𝑖𝑘𝑥_𝑖(𝑡_𝑘) and 𝑦_𝑗(𝑡⁺_𝑘)=𝛾_𝑗𝑘𝑦_𝑗(𝑡_𝑘) are impulses at moment 𝑡, which describe that the evolution processes experience abrupt change of state at the moments of time 𝑡_𝑘, where 𝑡₀<𝑡₁<𝑡₂<⋯ and lim_𝑘→∞𝑡_𝑘=∞.

Our main purpose in this paper is using an integral inequality, which is from a lemma in  [33], to establish some sufficient conditions on the existence and 𝑝-exponential stability of the periodic solutions of (1).

Let (Ω,𝐹,{𝐹_𝑡}_𝑡>0,𝑃) be a complete probability space with a filtration {𝐹_𝑡}_𝑡>0 satisfying the usual conditions; that is, {𝐹_𝑡}_𝑡>0 is right continuous and 𝐹₀ contains all 𝑃-null sets. Denote by PC^𝑏_𝐹0(𝑅,𝑅^𝑛+𝑚) the family of 𝐹₀-measurable, 𝑅^𝑛 valued random variables 𝑧(𝑡), where 𝑧(𝑡) is an piecewise-continuous stochastic process; that is, 𝑧(𝑠) is continuous for all but at most countable points 𝑡∈𝑅 and at these points 𝑠∈𝑅, 𝑧(𝑠⁺) and 𝑧(𝑠⁻) exist, 𝑧(𝑠⁻)=𝑧(𝑠). For 𝑧∈PC^𝑏_𝐹0(𝑅,𝑅^𝑛+𝑚), define the norm ‖𝑧‖=max_{0≤𝑡≤𝜔}(𝐸|𝑧(𝑡)|^𝑝₁)^1/𝑝, where 𝑝>1 is an integer, |𝑧(𝑡)|₁=(∑^𝑛+𝑚_𝑖=1|𝑧_𝑖(𝑡)|²)^1/2; 𝐸(⋅) stands for the correspondent expectation operator with respect to the given probability measure 𝑃.

For convenience, for an 𝜔-periodic function 𝑓∶𝑅→𝑅, denote 𝑓=max_{0≤𝑡≤𝜔}|𝑓(𝑡)|,  𝑓=min_{0≤𝑡≤𝜔}|𝑓(𝑡)|. The initial condition of (1) is 𝑥_𝑖(𝑠)=𝜑_𝑖(𝑠),𝑦_𝑗(𝑠)=𝜓_𝑗(𝑠),𝑠≤𝑡₀, (2) where 𝜑_𝑖,𝜓_𝑗∈PC^𝑏_𝐹0((−𝛿,𝑡₀],𝑅), 𝛿=max{max_{1≤𝑖≤𝑛}𝜏_𝑖,max_{1≤𝑗≤𝑚}𝜎_𝑗},  𝑖=1,2,…,𝑛,  𝑗=1,2,…,𝑚.

Throughout this paper, we assume that the following conditions hold:(𝐻₁)𝑎_𝑖(𝑡)>0,  𝑏_𝑗(𝑡)>0,  𝜏_𝑖(𝑡), 𝜎_𝑗(𝑡), 𝑐_𝑖𝑗(𝑡), 𝑑_𝑗𝑖(𝑡), 𝐼_𝑖(𝑡), and 𝐽_𝑗(𝑡) are all periodic continuous functions with period 𝜔 for 𝑡∈𝑅,  𝑖=1,2,…,𝑛,  𝑗=1,2,…,𝑚; (𝐻₂)𝑓_𝑗, 𝑔_𝑖, 𝜎_1𝑖𝑗, and 𝜎_2𝑗𝑖 are Lipschitz-continuous with Lipschitz constants 𝐿^𝑓_𝑗>0, 𝐿^𝑔_𝑖>0, 𝑙_1𝑖𝑗>0, 𝑙_2𝑗𝑖>0 and 𝑓_𝑗(0)=𝑔_𝑖(0)=𝜎_1𝑖𝑗(0)=𝜎_2𝑗𝑖(0)=0, 𝑖=1,2,…,𝑛,  𝑗=1,2,…,𝑚; (𝐻₃){𝛼_𝑖𝑘} and {𝛾_𝑗𝑘} are real sequences and ∏_{𝑡0<𝑡𝑘<𝑡}𝛼_𝑖𝑘 and ∏_{𝑡0<𝑡𝑘<𝑡}𝛾_𝑗𝑘 are 𝜔-periodic, 𝑖=1,2,…,𝑛, 𝑗=1,2,…,𝑚,  𝑘=1,2,….

Remark 1. From (𝐻₃), there exist constants  𝛼,𝛼,𝛾, and 𝛾 such that 0≤𝛼≤_{𝑡0<𝑡𝑘<𝑡}𝛼_𝑖𝑘≤𝛼,0≤𝛾≤_{𝑡0<𝑡𝑘<𝑡}𝛾_𝑗𝑘≤𝛾, (3) where 𝑖=1,2,…,𝑛,  𝑗=1,2,…,𝑚,  and 𝑘=1,2,….

This paper is organized as follows. In Section 2, we introduce some definitions and state some preliminary results which are needed in later sections. In Section 3, we state and prove our results. In Section 4, we give an example to illustrate the feasibility of our results obtained in the previous section.

2. Preliminaries

In this section, we introduce some definitions and state some preliminary results.

Definition 2. A stochastic process 𝑥_𝑡(𝑠) is said to be periodic with period 𝜔 if its finite dimensional distributions are periodic with period 𝜔; that is, for any positive integer 𝑚 and any moments of time 𝑡₁,𝑡₂,…,𝑡_𝑚, the joint distribution of the random variables 𝑥_{𝑡1+𝑘𝜔}(𝑠),𝑥_{𝑡2+𝑘𝜔}(𝑠),…,𝑥_{𝑡𝑚+𝑘𝜔}(𝑠) is independent of 𝑘,  𝑘=±1,±2,….

Lemma 3 (see [34]). If 𝑥(𝑡) is an 𝜔-periodic stochastic process, then its mathematical expectation and variance are 𝜔-periodic.

Definition 4. The solution 𝑥_𝑡(𝑡₀,𝜑) of (1) is said to be (i)𝑝-uniformly bounded if for each 𝛼>0, 𝑡₀∈𝑅, there exists a positive constant 𝜃=𝜃(𝛼) which is independent of 𝑡₀ such that ‖𝜑‖^𝑝≤𝛼 implies 𝐸(‖𝑥_𝑡(𝑡₀,𝜑)‖^𝑝)≤𝜃,  𝑡≥𝑡₀, (ii)𝑝-point dissipative if there exists a constant 𝑁>0 such that for any point 𝜑∈PC^𝑏_𝐹0([−𝜏,0],𝑅^𝑛), there exists 𝑇(𝑡₀,𝜑) such that for each 𝛼>0, 𝑡₀∈𝑅, there exists a positive constant 𝜃=𝜃(𝛼) which is independent of 𝑡₀ such that 𝐸(‖𝑥_𝑡(𝑡₀,𝜑)‖^𝑝)≤𝑁,  𝑡≥𝑡₀+𝑇(𝑡₀,𝜑).

Lemma 5 (see [35]). Under conditions (𝐻₁)--(𝐻₃), assume that the solution of (1) is 𝑝-uniformly bounded and 𝑝-point dissipative for 𝑝>2; then (1) has an 𝜔-periodic solution.

Lemma 6 (see [36]). For any 𝑥∈𝑅^𝑛₊ and 𝑝>0, |𝑥|^𝑝≤𝑛^{𝑛((𝑝/2)−1)∨0}_𝑖=1𝑥^𝑝_𝑖,_𝑛𝑖=1𝑥_𝑖^𝑝≤𝑛^{𝑛(𝑝−1)∨0}_𝑖=1𝑥^𝑝_𝑖. (4)

Definition 7. The periodic solution 𝑥(𝑡,𝑡₀,𝜑) with initial value 𝜑∈PC^𝑏_𝐹0([−𝜏,0],𝑅^𝑛) of (1) is said to be 𝑝-exponential stable if there are constants 𝜆>0 and 𝑀>1 such that any solution 𝑦(𝑡,𝑡₀,𝜑₁) with initial value 𝜑₁∈PC^𝑏_𝐹0([−𝜏,0],𝑅^𝑛) of (1) satisfies 𝐸𝑥−𝑦^𝑝₁≤𝑀𝜑−𝜑₁^𝑝e^{−𝜆(𝑡−𝑡0)},𝑡≥𝑡₀. (5)

Lemma 8 (see [33]). Let 𝑢(𝑡)∈𝐶(𝑅,𝑅^𝑛₊) be a solution of the delay integral inequality 𝑢(𝑡)≤𝑀₁e^{−𝛿(𝑡−𝑡0)}𝜑⁺_𝜏+^𝑡_𝑡0e^{−𝐶1(𝑡−𝑠)}𝐴₁𝑢(𝑠)+𝑑𝑠_𝑡𝑡0e^{−𝐶1(𝑡−𝑠)}𝐵₁[𝑢(𝑠)]⁺_𝜏𝑑𝑠+𝐽₁,𝑡≥𝑡₀,𝑢(𝑡)≤𝜑(𝑡),∀𝑡∈𝑡₀−𝜏,𝑡₀, (6) where 𝐴₁,𝐵₁,𝐶₁, and 𝑀₁∈𝑅₊^𝑛×𝑛,  𝐽₁≥0 is a constant vector, and 𝜑(𝑡)∈𝐶([𝑡₀−𝜏,𝑡₀],𝑅^𝑛₊). If 𝜌(Π=𝐶₁⁻¹(𝐴₁+𝐵₁))<1, then there are constants 0<𝜆≤𝛿 and 𝑁≥1 such that 𝑢(𝑡)≤𝑁𝑧e^{−𝜆(𝑡−𝑡0)}+(𝐼−Π)⁻¹𝐽₁,𝑡≥𝑡₀, (7) where 𝑧 satisfies [𝜑]⁺_𝜏≤𝑧.

Lemma 9 (see [33]). Assume that all conditions of Lemma 9 hold. If 𝐽₁=0, then all solutions of inequality of (6) exponentially converge to zero.

By Lemmas 8 and 9, we have the following corollary.

Corollary 10. Let 𝑢(𝑡)∈𝐶(𝑅,𝑅₊) be a solution of the delay integral inequality 𝑢(𝑡)≤𝑀₁e^{−𝛿(𝑡−𝑡0)}𝜑⁺_𝜏+^𝑡_𝑡0e^{−𝐶1(𝑡−𝑠)}𝐴₁𝑢(𝑠)+𝑑𝑠_𝑡𝑡0e^{−𝐶1(𝑡−𝑠)}𝐵₁[𝑢(𝑠)]⁺_𝜏𝑑𝑠+𝐽₁,𝑡≥𝑡₀,𝑢(𝑡)≤𝜑(𝑡),∀𝑡∈𝑡₀−𝜏,𝑡₀, (8) where 𝐴₁,𝐵₁,𝐶₁, and 𝑀₁∈𝑅₊,  𝐽₁≥0 is a constant, and 𝜑(𝑡)∈𝐶([𝑡₀−𝜏,𝑡₀],𝑅₊). If (𝐴₁+𝐵₁)/𝐶₁<1, then there are constants 0<𝜆≤𝛿 and 𝑁≥1 such that 𝑢(𝑡)≤𝑁𝑧e^{−𝜆(𝑡−𝑡0)}+𝐴1−¹+𝐵₁𝐶₁⁻¹𝐽₁,𝑡≥𝑡₀, (9) where 𝑧 satisfies [𝜑]⁺_𝜏≤𝑧. Moreover, if 𝐽₁=0, then all solutions of inequality of (8) exponentially converge to zero.

Under our assumptions, we consider the following system: 𝑑𝑢_𝑖(𝑡)=−𝑎_𝑖(𝑡)𝑢_𝑖𝑡−𝜏_𝑖(𝑡)+_{𝑡0<𝑡𝑘<𝑡}𝛼^𝑚−1_{𝑖𝑘𝑗=1}𝑐_𝑖𝑗(𝑡)𝑓_{𝑗𝑡0<𝑡𝑘<𝑡}𝛾_𝑗𝑘𝑣_𝑗(𝑡)+_{𝑡0<𝑡𝑘<𝑡}𝛼⁻¹_𝑖𝑘𝐼_𝑖(𝑡)𝑑𝑡+_{𝑡0<𝑡𝑘<𝑡}𝛼^𝑚−1_{𝑖𝑘𝑗=1}𝜎_{1𝑖𝑗𝑡0<𝑡𝑘<𝑡}𝛾_𝑗𝑘𝑣_𝑗(𝑡)𝑑𝑤_1𝑗(𝑡),𝑡≥𝑡₀,𝑑𝑣_𝑗(𝑡)=−𝑏_𝑗(𝑡)𝑣_𝑗𝑡−𝜎_𝑗(𝑡)+_{𝑡0<𝑡𝑘<𝑡}𝛾^𝑛−1_{𝑗𝑘𝑖=1}𝑑_𝑗𝑖(𝑡)𝑔_{𝑖𝑡0<𝑡𝑘<𝑡}𝛼_𝑖𝑘𝑢_𝑖(𝑡)+_{𝑡0<𝑡𝑘<𝑡}𝛾⁻¹_𝑗𝑘𝐽_𝑗(𝑡)𝑑𝑡+_{𝑡0<𝑡𝑘<𝑡}𝛾^𝑛−1_{𝑗𝑘𝑖=1}𝜎_{2𝑗𝑖𝑡0<𝑡𝑘<𝑡}𝛼_𝑖𝑘𝑢_𝑖(𝑡)𝑑𝑤_2𝑖(𝑡),𝑡≥𝑡₀, (10) where 𝑖=1,2,…,𝑛,  𝑗=1,2,…,𝑚.

Similar to Lemma 2.1 in paper  [33], we have the following lemma.

Lemma 11. Let (𝐻₁)--(𝐻₃) hold. Then, (i) if (𝑢₁(𝑡),…,𝑢_𝑛(𝑡),𝑣₁(𝑡),…,𝑣_𝑚(𝑡))^𝑇 is a solution of (10), then (∏_{𝑡0<𝑡𝑘<𝑡}𝛼_1𝑘𝑢₁(𝑡),…,∏_{𝑡0<𝑡𝑘<𝑡}𝛼_𝑛𝑘𝑢_𝑛(𝑡),∏_{𝑡0<𝑡𝑘<𝑡}𝛾_1𝑘𝑣₁(𝑡),…,∏_{𝑡0<𝑡𝑘<𝑡}𝛾_𝑚𝑘𝑣_𝑚(𝑡))^𝑇 is a solution of (1); (ii) if (𝑥₁(𝑡),…,𝑥_𝑛(𝑡),𝑦₁(𝑡),…,𝑦_𝑚(𝑡))^𝑇 is a solution of (1), then (∏_{𝑡0<𝑡𝑘<𝑡}𝛼⁻¹_𝑖𝑘𝑥₁(𝑡),…,∏_{𝑡0<𝑡𝑘<𝑡}𝛼⁻¹_𝑖𝑘𝑥_𝑛(𝑡), ∏_{𝑡0<𝑡𝑘<𝑡}𝛾⁻¹_𝑗𝑘𝑦₁(𝑡),…,∏_{𝑡0<𝑡𝑘<𝑡}𝛾⁻¹_𝑚𝑘𝑦_𝑚(𝑡))^𝑇 is a solution of (10).

3. Main Results

In this section, we will state and prove the sufficient conditions for the existence and 𝑝-exponential stability of periodic solution of (1).

Theorem 12. Let (𝐻₁)--(𝐻₃) hold. Suppose further that (𝐻₄) there exists an integer 𝑝>2 such that (𝜖+𝑐)/𝜃<1, where 𝜃=minmin_{1≤𝑖≤𝑛}𝑎_𝑖,min_{1≤𝑗≤𝑚}𝑏_𝑗,𝑐=6^𝑝−1maxmax_{1≤𝑖≤𝑛}𝑎^𝑝_𝑖,max_{1≤𝑗≤𝑚}𝑏^𝑝_𝑗,𝜖=max𝑐,𝜖₁,𝜖₂,𝜖₁=6^𝑛𝑝−1_𝑖=1𝑎_𝑖^1−𝑝𝛾𝛼^𝑚_𝑗=1𝑐_𝑖𝑗𝐿^𝑓_𝑗^{𝑝/(𝑝−1)𝑝−1}+𝑙_𝑝𝑛^(𝑝/2)−12𝑎_𝑖𝑝−1𝑝−2^1−(𝑝/2)×𝛾𝛼^𝑚_𝑗=1𝑙_1𝑖𝑗^{𝑝/(𝑝−1)𝑝−1},𝜖₂=6^𝑚𝑝−1_𝑗=1𝑏_𝑗^1−𝑝𝛼𝛾^𝑛_𝑖=1𝑑_𝑗𝑖𝐿^𝑔_𝑖^{𝑝/(𝑝−1)𝑝−1}+𝑙_𝑝𝑚^(𝑝/2)−12𝑏_𝑗𝑝−1𝑝−2^1−(𝑝/2)×𝛼𝛾^𝑛_𝑖=1𝑙_2𝑗𝑖^{𝑝/(𝑝−1)𝑝−1}. (11)
Then (1) has an 𝜔-periodic solution, which is 𝑝-exponentially stable.

Proof. We can rewrite (10) as follows: 𝑑𝑢_𝑖(𝑡)=−𝑎_𝑖(𝑡)𝑢_𝑖(𝑡)+𝑎_𝑖(𝑡)𝑢_𝑖(𝑡)−𝑎_𝑖(𝑡)𝑢_𝑖𝑡−𝜏_𝑖(𝑡)+_{𝑡0<𝑡𝑘<𝑡}𝛼^𝑚−1_{𝑖𝑘𝑗=1}𝑐_𝑖𝑗(𝑡)𝑓_𝑗×_{𝑡0<𝑡𝑘<𝑡}𝛾_𝑗𝑘𝑣_𝑗(𝑡)+_{𝑡0<𝑡𝑘<𝑡}𝛼⁻¹_𝑖𝑘𝐼_𝑖(𝑡)𝑑𝑡+_{𝑡0<𝑡𝑘<𝑡}𝛼^𝑚−1_{𝑖𝑘𝑗=1}𝜎_{1𝑖𝑗𝑡0<𝑡𝑘<𝑡}𝛾_𝑗𝑘𝑣_𝑗(𝑡)𝑑𝑤_1𝑗(𝑡),𝑡≥𝑡₀,𝑖=1,2,…,𝑛,𝑑𝑣_𝑗(𝑡)=−𝑏_𝑗(𝑡)𝑣_𝑗(𝑡)+𝑏_𝑗(𝑡)𝑣_𝑗(𝑡)−𝑏_𝑗(𝑡)𝑣_𝑗𝑡−𝜎_𝑗(𝑡)+_{𝑡0<𝑡𝑘<𝑡}𝛾^𝑛−1_{𝑗𝑘𝑖=1}𝑑_𝑗𝑖(𝑡)𝑔_𝑖×_{𝑡0<𝑡𝑘<𝑡}𝛼_𝑖𝑘𝑢_𝑖(𝑡)+_{𝑡0<𝑡𝑘<𝑡}𝛾⁻¹_𝑗𝑘𝐽_𝑗(𝑡)𝑑𝑡+_{𝑡0<𝑡𝑘<𝑡}𝛾^𝑛−1_{𝑗𝑘𝑖=1}𝜎_{2𝑗𝑖𝑡0<𝑡𝑘<𝑡}𝛼_𝑖𝑘𝑢_𝑖(𝑡)𝑑𝑤_2𝑖(𝑡),𝑡≥𝑡₀,𝑗=1,2,…,𝑚. (12) By the method of variation parameter, for 𝑡≥𝑡₀, 𝑖=1,2,…,𝑛, from the first equation of (12), we have the following: 𝑢_𝑖(𝑡)=𝑢_𝑖𝑡₀e^{−∫𝑡𝑡0𝑎𝑖(𝜗)𝑑𝜗}+^𝑡_𝑡0e^{−∫𝑡𝑠𝑎𝑖(𝜗)𝑑𝜗}×𝑎_𝑖(𝑠)𝑢_𝑖(𝑠)−𝑎_𝑖(𝑠)𝑢_𝑖𝑠−𝜏_𝑖(𝑠)+_{𝑡0<𝑡𝑘<𝑡}𝛼^𝑚−1_{𝑖𝑘𝑗=1}𝑐_𝑖𝑗(𝑠)𝑓_𝑗×_{𝑡0<𝑡𝑘<𝑡}𝛾_𝑗𝑘𝑣_𝑗(𝑠)+_{𝑡0<𝑡𝑘<𝑡}𝛼⁻¹_𝑖𝑘𝐼_𝑖(𝑠)+𝑑𝑠_𝑡𝑡0e^{−∫𝑡𝑠𝑎𝑖(𝜗)𝑑𝜗}_{𝑡0<𝑡𝑘<𝑡}𝛼^𝑚−1_{𝑖𝑘𝑗=1}𝜎_1𝑖𝑗×_{𝑡0<𝑡𝑘<𝑡}𝛾_𝑗𝑘𝑣_𝑗(𝑠)𝑑𝑤_1𝑗(𝑠). (13) For 𝑖=1,2,…,𝑛, denote 𝐹_1𝑖=𝑢_𝑖𝑡₀e^{−∫𝑡𝑡0𝑎𝑖(𝜗)𝑑𝜗},𝐹_2𝑖=^𝑡_𝑡0e^{−∫𝑡𝑠𝑎𝑖(𝜗)𝑑𝜗}𝑎_𝑖(𝑠)𝑢_𝑖(𝑠)𝐹𝑑𝑠,_3𝑖=^𝑡_𝑡0e^{−∫𝑡𝑠𝑎𝑖(𝜗)𝑑𝜗}𝑎_𝑖(𝑠)𝑢_𝑖𝑠−𝜏_𝑖(𝑠𝐹)𝑑𝑠,_4𝑖=^𝑡_𝑡0e^{−∫𝑡𝑠𝑎𝑖(𝜗)𝑑𝜗}_{𝑡0<𝑡𝑘<𝑡}𝛼⁻¹_𝑖𝑘𝐼_𝑖(𝑠)𝐹𝑑𝑠,_5𝑖=^𝑡_𝑡0e^{−∫𝑡𝑠𝑎𝑖(𝜗)𝑑𝜗}_{𝑡0<𝑡𝑘<𝑡}𝛼⁻¹_𝑖𝑘×^𝑚_𝑗=1𝑐_𝑖𝑗(𝑠)𝑓_{𝑗𝑡0<𝑡𝑘<𝑡}𝛾_𝑗𝑘𝑣_𝑗(𝑠)𝐹𝑑𝑠,_6𝑖=^𝑡_𝑡0e^{−∫𝑡𝑠𝑎𝑖(𝜗)𝑑𝜗}_{𝑡0<𝑡𝑘<𝑡}𝛼⁻¹_𝑖𝑘×^𝑚_𝑗=1𝜎_{1𝑖𝑗𝑡0<𝑡𝑘<𝑡}𝛾_𝑗𝑘𝑣_𝑗(𝑠)𝑑𝑤_1𝑗(𝑠). (14) Considering expectations, using Lemma 6, for 𝑖=1,2,…,𝑛, we have 𝐸𝑢_𝑖(𝑡)^𝑝≤6^𝑝−1𝐸𝐹_1𝑖^𝑝+𝐹_2𝑖^𝑝+𝐹_3𝑖^𝑝+𝐹_4𝑖^𝑝+𝐹_5𝑖^𝑝+𝐹_6𝑖^𝑝. (15) For 𝑖=1,2,…,𝑛, we evaluate the first term of (15) as follows: 𝐸𝐹_1𝑖^𝑝=𝐸𝑢_𝑖𝑡₀e^{−∫𝑡𝑡0𝑎𝑖(𝜗)𝑑𝜗𝑝}≤𝐸𝑢_𝑖𝑡₀e^{−𝑎𝑖(𝑡−𝑡0)𝑝}≤e^{−𝑝𝑎𝑖(𝑡−𝑡0)}𝐸𝑢_𝑖𝑡₀^𝑝. (16) For the second term of (15), we have 𝐸𝐹_2𝑖^𝑝=𝐸^𝑡_𝑡0e^{−∫𝑡𝑠𝑎𝑖(𝜗)𝑑𝜗}𝑎_𝑖(𝑠)𝑢_𝑖(𝑠)𝑑𝑠^𝑝≤𝐸^𝑡_𝑡0e^{−∫𝑡𝑠𝑎𝑖(𝜗)𝑑𝜗}𝑎_𝑖(𝑠)𝑢_𝑖(𝑠)𝑑𝑠^𝑝≤𝐸^𝑡_𝑡0𝑎_𝑖e^{−𝑎𝑖(𝑡−𝑠)}𝑢_𝑖(𝑠)𝑑𝑠^𝑝=𝑎^𝑝_𝑖^𝑡_𝑡0e^{−𝑝𝑎𝑖(𝑡−𝑠)}𝐸𝑢_𝑖(𝑠)^𝑝𝑑𝑠,𝑖=1,2,…,𝑛. (17) For the third term of (15), we have 𝐸𝐹_3𝑖^𝑝=𝐸^𝑡_𝑡0e^{−∫𝑡𝑠𝑎𝑖(𝜗)𝑑𝜗}𝑎_𝑖(𝑠)𝑢_𝑖𝑠−𝜏_𝑖(𝑠)𝑑𝑠^𝑝≤𝐸^𝑡_𝑡0e^{−∫𝑡𝑠𝑎𝑖(𝜗)𝑑𝜗}𝑎_𝑖(𝑠)𝑢_𝑖𝑠−𝜏_𝑖(𝑠)𝑑𝑠^𝑝≤𝐸^𝑡_𝑡0𝑎_𝑖e^{−𝑎𝑖(𝑡−𝑠)}𝑢_𝑖𝑠−𝜏_𝑖(𝑠)𝑑𝑠^𝑝=𝑎^𝑝_𝑖^𝑡_𝑡0e^{−𝑝𝑎𝑖(𝑡−𝑠)}𝐸𝑢_𝑖𝑠−𝜏_𝑖(𝑠)^𝑝𝑑𝑠,𝑖=1,2,…,𝑛. (18) As for the fourth term of (15), for 𝑖=1,2,…,𝑛, we have 𝐸𝐹_4𝑖^𝑝=𝐸^𝑡_𝑡0e^{−∫𝑡𝑠𝑎𝑖(𝜗)𝑑𝜗}_{𝑡0<𝑡𝑘<𝑡}𝛼⁻¹_𝑖𝑘𝐼_𝑖(𝑠)𝑑𝑠^𝑝≤𝐸^𝑡_𝑡0e^{−𝑎𝑖(𝑡−𝑠)}_{𝑡0<𝑡𝑘<𝑡}𝛼⁻¹_𝑖𝑘𝐼_𝑖(𝑠)𝑑𝑠^𝑝≤𝐼_𝑖𝑎_𝑖𝛼^𝑝. (19) For the fifth term of (15), by Hölder inequality, we have 𝐸𝐹_5𝑖^𝑝=𝐸^𝑡_𝑡0e^{−∫𝑡𝑠𝑎𝑖(𝜗)𝑑𝜗}_{𝑡0<𝑡𝑘<𝑡}𝛼⁻¹_𝑖𝑘×^𝑚_𝑗=1𝑐_𝑖j(𝑠)𝑓_{𝑗𝑡0<𝑡𝑘<𝑡}𝛾_𝑗𝑘𝑣_𝑗(𝑠)𝑑𝑠^𝑝≤𝐸^𝑡_𝑡0e^{−𝑎𝑖(𝑡−𝑠)}_{𝑡0<𝑡𝑘<𝑡}𝛼⁻¹_𝑖𝑘^𝑚_𝑗=1𝑐_𝑖𝑗(𝑠×𝑓_{𝑗𝑡0<𝑡𝑘<𝑡})𝛾_𝑗𝑘𝑣_𝑗(𝑠)𝑑𝑠^𝑝≤𝐸^𝑡_𝑡0e^{−𝑎𝑖(𝑡−𝑠)}×_{𝑡0<𝑡𝑘<𝑡}𝛼⁻¹_𝑖𝑘^𝑚_𝑗=1𝑐_𝑖𝑗(𝑠)𝐿^𝑓_𝑗×_{𝑡0<𝑡𝑘<𝑡}𝛾_𝑗𝑘𝑣_𝑗(𝑠)𝑑𝑠^𝑝≤𝐸^𝑡_𝑡0e^{−𝑎𝑖(𝑡−𝑠)}𝛾𝛼^𝑚_𝑗=1𝑐_𝑖𝑗(𝑠)×𝐿^𝑓_𝑗𝑣_𝑗(𝑠)𝑑𝑠^𝑝=𝐸^𝑡_𝑡0e^{−𝑎𝑖(𝑡−𝑠)(𝑝−1)/𝑝}e^{−𝑎𝑖(𝑡−𝑠)1/𝑝}×𝛾𝛼^𝑚_𝑗=1𝑐_𝑖𝑗(𝑠)𝐿^𝑓_𝑗𝑣_𝑗(𝑠)𝑑𝑠^𝑝≤𝐸^𝑡_𝑡0e^{−𝑎𝑖(𝑡−𝑠)}𝑑𝑠^𝑝−1×^𝑡_𝑡0e^{−𝑎𝑖(𝑡−𝑠)}𝛾𝛼^𝑚_𝑗=1𝑐_𝑖𝑗𝐿^𝑓_𝑗𝑣_𝑗(𝑠)^𝑝𝑑𝑠≤𝑎_𝑖^1−𝑝𝑡_𝑡0e^{−𝑎𝑖(𝑡−𝑠)}𝛾𝛼^𝑚_𝑗=1𝑐_𝑖𝑗𝐿^𝑓_𝑗^{𝑝/(𝑝−1)𝑝−1}×^𝑚_𝑗=1𝐸𝑣_𝑗(𝑠)^𝑝𝑑𝑠,𝑖=1,2,…,𝑛. (20) As for the last term of (15), using Proposition  1.9 in  [37] and Hölder inequality, for 𝑖=1,2,…,𝑛, we have 𝐸𝐹_6𝑖^𝑝=𝐸^𝑡_𝑡0e^{−∫𝑡𝑠𝑎𝑖(𝜗)𝑑𝜗}_{𝑡0<𝑡𝑘<𝑡}𝛼⁻¹_𝑖𝑘×^𝑚_𝑗=1𝜎_{1𝑖𝑗𝑡0<𝑡𝑘<𝑡}𝛾_𝑗𝑘𝑣_𝑗(𝑠)𝑑𝑤_1𝑗(𝑠)^𝑝≤𝑙_𝑝^𝑡_𝑡0e^{−𝑝𝑎𝑖(𝑡−𝑠)}𝐸_{𝑡0<𝑡𝑘<𝑡}𝛼^𝑛−2_{𝑖𝑘𝑗=1}𝜎²_1𝑖𝑗×_{𝑡0<𝑡𝑘<𝑡}𝛾_𝑗𝑘𝑣_𝑗(𝑠)^𝑝/22/𝑝𝑑𝑠^𝑝/2≤𝑙_𝑝𝑛^{(𝑝/2)−1𝑡}_𝑡0e^{−𝑝𝑎𝑖(𝑡−𝑠)}×𝐸𝛾𝛼^𝑛_𝑗=1𝑙_1𝑖𝑗𝑣_𝑗(𝑠)^𝑝2/𝑝𝑑𝑠^𝑝/2=𝑙_𝑝𝑛^{(𝑝/2)−1𝑡}_𝑡0e^{−(𝑝−1)𝑎𝑖(𝑡−𝑠)}e^{−𝑎𝑖(𝑡−𝑠)}×𝐸𝛾𝛼^𝑛_𝑗=1𝑙_1𝑖𝑗𝑣_𝑗(𝑠)^𝑝2/𝑝𝑑𝑠^𝑝/2≤𝑙_𝑝𝑛^{(𝑝/2)−1𝑡}_𝑡0e^{(−2𝑎𝑖(𝑝−1)/(𝑝−2))(𝑡−𝑠)}𝑑𝑠^(𝑝/2)−1×^𝑡_𝑡0e^{−𝑎𝑖(𝑡−𝑠)}𝐸𝛾𝛼^𝑛_𝑗=1𝑙_1𝑖𝑗𝑣_𝑗(𝑠)^𝑝𝑑𝑠≤𝑙_𝑝𝑛^(𝑝/2)−12𝑎_𝑖𝑝−1𝑝−2^1−(𝑝/2)×^𝑡_𝑡0e^{−𝑎𝑖(𝑡−𝑠)}×𝐸𝛾𝛼^𝑚_𝑗=1𝑙_1𝑖𝑗𝑣_𝑗(𝑠)^𝑝𝑑𝑠≤𝑙_𝑝𝑛^(𝑝/2)−12𝑎_𝑖𝑝−1𝑝−2^1−(𝑝/2)×^𝑡_𝑡0e^{−𝑎𝑖(𝑡−𝑠)}𝛾𝛼^𝑚_𝑗=1𝑙_1𝑖𝑗^{𝑝/(𝑝−1)𝑝−1}×^𝑚_𝑗=1𝐸𝑣_𝑗(𝑠)^𝑝𝑑𝑠, (21) where 𝑙_𝑝=(𝑝(𝑝−1)/2)^𝑝/2. Therefore, for 𝑖=1,2,…,𝑛, we have 𝐸𝑢_𝑖(𝑡)^𝑝≤6^𝑝−1e^{−𝑝𝑎𝑖(𝑡−𝑡0)}𝐸𝑢_𝑖𝑡₀^𝑝+𝑎^𝑝_𝑖^𝑡_𝑡0e^{−𝑝𝑎𝑖(𝑡−𝑠)}𝐸𝑢_𝑖(𝑠)^𝑝+𝑑𝑠𝑎^𝑝_𝑖^𝑡_𝑡0e^{−𝑝𝑎𝑖(𝑡−𝑠)}𝐸𝑢_𝑖𝑠−𝜏_𝑖(𝑠)^𝑝𝑑𝑠+𝑎_𝑖^1−𝑝𝑡_𝑡0e^{−𝑎𝑖(𝑡−𝑠)}×𝛾𝛼^𝑚_𝑗=1𝑐_𝑖𝑗𝐿^𝑓_𝑗^{𝑝/(𝑝−1)𝑝−1}×^𝑛_𝑗=1𝐸𝑣_𝑗(𝑠)^𝑝𝑑𝑠+𝑙_𝑝𝑛^(𝑝/2)−12𝑎_𝑖𝑝−1𝑝−2^1−(𝑝/2)×^𝑡_𝑡0e^{−𝑎𝑖(𝑡−𝑠)}𝛾𝛼^𝑚_𝑗=1𝑙_1𝑖𝑗^{𝑝/(𝑝−1)𝑝−1}×^𝑚_𝑗=1𝐸𝑣_𝑗(𝑠)^𝑝𝑑𝑠+𝐼_𝑖𝑎_𝑖𝛼^𝑝=6^𝑝−1e^{−𝑝𝑎𝑖(𝑡−𝑡0)}𝐸𝑢_𝑖𝑡₀^𝑝+𝑎^𝑝_𝑖^𝑡_𝑡0e^{−𝑝𝑎𝑖(𝑡−𝑠)}𝐸𝑢_𝑖(𝑠)^𝑝+𝑑𝑠𝑎^𝑝_𝑖^𝑡_𝑡0e^{−𝑝𝑎𝑖(𝑡−𝑠)}𝐸𝑢_𝑖𝑠−𝜏_𝑖(𝑠)^𝑝+𝑑𝑠_𝑡𝑡0e^{−𝑎𝑖(𝑡−𝑠)}×𝑎_𝑖^1−𝑝𝛾𝛼^𝑚_𝑗=1𝑐_𝑖𝑗𝐿^𝑓_𝑗^{𝑝/(𝑝−1)𝑝−1}+𝑙_𝑝𝑛^(𝑝/2)−12𝑎_𝑖𝑝−1𝑝−2^1−(𝑝/2)×𝛾𝛼^𝑚_𝑗=1𝑙_1𝑖𝑗^{𝑝/(𝑝−1)𝑝−1}×^𝑚_𝑗=1𝐸𝑣_𝑗(𝑠)^𝑝𝑑𝑠+𝐼_𝑖𝑎_𝑖𝛼^𝑝≤6^𝑝−1e^{−𝑎𝑖(𝑡−𝑡0)}𝐸𝑢_𝑖𝑡₀^𝑝+𝑎^𝑝_𝑖^𝑡_𝑡0e^{−𝑎𝑖(𝑡−𝑠)}𝐸𝑢_𝑖(𝑠)^𝑝+𝑑𝑠𝑎^𝑝_𝑖^𝑡_𝑡0e^{−𝑎𝑖(𝑡−𝑠)}𝐸𝑢_𝑖𝑠−𝜏_𝑖(𝑠)^𝑝+𝑑𝑠_𝑡𝑡0e^{−𝑎𝑖(𝑡−𝑠)}×𝑎_𝑖^1−𝑝𝛾𝛼^𝑚_𝑗=1𝑐_𝑖𝑗𝐿^𝑓_𝑗^{𝑝/(𝑝−1)𝑝−1}+𝑙_𝑝𝑛^(𝑝/2)−12𝑎_𝑖𝑝−1𝑝−2^1−(𝑝/2)×𝛾𝛼^𝑚_𝑗=1𝑙_1𝑖𝑗^{𝑝/(𝑝−1)𝑝−1}×^𝑚_𝑗=1𝐸𝑣_𝑗(𝑠)^𝑝𝑑𝑠+𝐼_𝑖𝑎_𝑖𝛼^𝑝. (22) Similarly, for 𝑡≥𝑡₀,  𝑗=1,2,…,𝑚, from the second equation of (12), we can obtain the following: 𝐸𝑣_𝑗(𝑡)^𝑝≤6^𝑝−1e^{−𝑏𝑗(𝑡−𝑡0)}𝐸𝑣_𝑗𝑡₀^𝑝+𝑏^𝑝_𝑗^𝑡_𝑡0e^{−𝑏𝑗(𝑡−𝑠)}𝐸𝑣_𝑗(𝑠)^𝑝+𝑑𝑠𝑏^𝑝_𝑗^𝑡_𝑡0e^{−𝑏𝑗(𝑡−𝑠)}𝐸𝑣_𝑗𝑠−𝜎_𝑗(𝑠)^𝑝+𝑑𝑠_𝑡𝑡0e^{−𝑏𝑗(𝑡−𝑠)}×𝑏_𝑗^1−𝑝𝛼𝛾^𝑛_𝑖=1𝑑_𝑗𝑖𝐿^𝑔_𝑖^{𝑝/(𝑝−1)𝑝−1}+𝑙_𝑝𝑚^(𝑝/2)−12𝑏_𝑗𝑝−1𝑝−2^1−(𝑝/2)×𝛼𝛾^𝑛_𝑖=1𝑙_2𝑗𝑖^{𝑝/(𝑝−1)𝑝−1}×^𝑛_𝑖=1𝐸𝑢_𝑖(𝑠)^𝑝𝑑𝑠+𝐽_𝑗𝑏_𝑗𝛾^𝑝. (23) Hence, by (22) and (23), we have that ^𝑛_𝑖=1𝐸𝑢_𝑖(𝑡)^𝑝+^𝑚_𝑗=1𝐸𝑣_𝑗(𝑡)^𝑝≤6^𝑛𝑝−1_𝑖=1e^{−𝑎𝑖(𝑡−𝑡0)}𝐸𝑢_𝑖𝑡₀^𝑝+𝑎^𝑝_𝑖^𝑡_𝑡0e^{−𝑎𝑖(𝑡−𝑠)}𝐸𝑢_𝑖(𝑠)^𝑝+𝑑𝑠𝑎^𝑝_𝑖^𝑡_𝑡0e^{−𝑎𝑖(𝑡−𝑠)}𝐸𝑢_𝑖𝑠−𝜏_𝑖(𝑠)^𝑝+𝑑𝑠_𝑡𝑡0e^{−𝑎𝑖(𝑡−𝑠)}×𝑎_𝑖^1−𝑝𝛾𝛼^𝑚_𝑗=1𝑐_𝑖𝑗𝐿^𝑓_𝑗^{𝑝/(𝑝−1)𝑝−1}+𝑙_𝑝𝑛^(𝑝/2)−12𝑎_𝑖𝑝−1𝑝−2^1−(𝑝/2)×𝛾𝛼^𝑚_𝑗=1𝑙_1𝑖𝑗^{𝑝/(𝑝−1)𝑝−1}×^𝑚_𝑗=1𝐸𝑣_𝑗(𝑠)^𝑝𝑑𝑠+𝐼_𝑖𝑎_𝑖𝛼^𝑝+6^𝑚𝑝−1_𝑗=1e^{−𝑏𝑗(𝑡−𝑡0)}𝐸𝑣_𝑗𝑡₀^𝑝+𝑏^𝑝_𝑗^𝑡_𝑡0e^{−𝑏𝑗(𝑡−𝑠)}𝐸𝑣_𝑗(𝑠)^𝑝+𝑑𝑠𝑏^𝑝_𝑗^𝑡_𝑡0e^{−𝑏𝑗(𝑡−𝑠)}𝐸𝑣_𝑗𝑠−𝜎_𝑗(𝑠)^𝑝+𝑑𝑠_𝑡𝑡0e^{−𝑏𝑗(𝑡−𝑠)}×𝑏_𝑗^1−𝑝𝛼𝛾^𝑛_𝑖=1𝑑_𝑗𝑖𝐿^𝑔_𝑖^{𝑝/(𝑝−1)𝑝−1}+𝑙_𝑝𝑚^(𝑝/2)−12𝑏_𝑗𝑝−1𝑝−2^1−(𝑝/2)×𝛼𝛾^𝑛_𝑖=1𝑙_2𝑗𝑖^{𝑝/(𝑝−1)𝑝−1}×^𝑛_𝑖=1𝐸𝑢_𝑖(𝑠)^𝑝𝑑𝑠+𝐽_𝑗𝑏_𝑗𝛾^𝑝. (24) Set 𝑉(𝑡)=∑^𝑛_𝑖=1𝐸|𝑢_𝑖(𝑡)|^𝑝+∑^𝑚_𝑗=1𝐸|𝑣_𝑗(𝑡)|^𝑝. By (24), we have that 𝑉(𝑡)≤6^𝑝−1e^{−𝜃(𝑡−𝑡0)}𝑉𝑡₀+^𝑡_𝑡0𝜖e^{−𝜃(𝑡−𝑠)}𝑉(𝑠)+𝑑𝑠_𝑡𝑡0𝑐e^{−𝜃(𝑡−𝑠)}𝑉⁺_𝛿(𝑠)𝑑𝑠+𝐻, (25) where 𝑉⁺_𝛿(𝑠)=max_{−𝛿≤𝑠≤0}|𝑉(𝑡+𝑠)|, 𝐻=6^𝑝−1max{{max_{1≤𝑖≤𝑛}(𝐼_𝑖/𝑎_𝑖𝛼)^𝑝},{max_{1≤𝑗≤𝑚}(𝐽_𝑗/𝑏_𝑗𝛾)^𝑝}}. By (𝐻₄) and Corollary 10, the solutions of (10) are 𝑝-uniformly bounded and it also shows that the family of all solutions of (10) is 𝑝-point dissipative. Then, it follows from Lemma 5 that (10) has an 𝜔-periodic solution (𝑢^∗₁(𝑡),…,𝑢^∗_𝑛(𝑡),𝑣^∗₁(𝑡),…,𝑣^∗_𝑚(𝑡))^𝑇.
Suppose that (𝑢₁(𝑡),…,𝑢_𝑛(𝑡),𝑣₁(𝑡),…,𝑣_𝑚(𝑡))^𝑇 is an arbitrary solution of (10). Then it follows from (10) that 𝑑𝑢_𝑖(𝑡)−𝑢^∗_𝑖(𝑡)=−𝑎_𝑖(𝑡)𝑢_𝑖𝑡−𝜏_𝑖(𝑡)−𝑢^∗_𝑖𝑡−𝜏_𝑖(𝑡)+_{𝑡0<𝑡𝑘<𝑡}𝛼^𝑚−1_{𝑖𝑘𝑗=1}𝑐_𝑖𝑗(𝑡)×𝑓_{𝑗𝑡0<𝑡𝑘<𝑡}𝛾_𝑗𝑘𝑣_𝑗(𝑡)−𝑓_{𝑗𝑡0<𝑡𝑘<𝑡}𝛾_𝑗𝑘𝑣^∗_𝑗(𝑡)𝑑𝑡+_{𝑡0<t𝑘<𝑡}𝛼⁻¹_𝑖𝑘×^𝑚_𝑗=1𝜎_{1𝑖𝑗𝑡0<𝑡𝑘<𝑡}𝛾_𝑗𝑘𝑣_𝑗(𝑡−_𝑚𝑗=1)𝜎_{1𝑖𝑗𝑡0<𝑡𝑘<𝑡}𝛾_𝑗𝑘𝑣^∗_𝑗(𝑡)𝑑𝑤_1𝑗(𝑡),𝑡≥𝑡₀𝑑𝑣,𝑖=1,2,…,𝑛,_𝑗(𝑡)−𝑣^∗_𝑗(𝑡)=−𝑏_𝑗(𝑡)𝑣_𝑗𝑡−𝜎_𝑗(𝑡)−𝑣^∗_𝑗𝑡−𝜎_𝑗(𝑡)+_{𝑡0<𝑡𝑘<𝑡}𝛾^𝑛−1_{𝑗𝑘𝑖=1}𝑑_𝑗𝑖(𝑡)×𝑔_{𝑖𝑡0<𝑡𝑘<𝑡}𝛼_𝑖𝑘𝑢_𝑖(𝑡)−𝑔_{𝑖𝑡0<𝑡𝑘<𝑡}𝛼_𝑖𝑘𝑢^∗_𝑖(𝑡)𝑑𝑡+_{𝑡0<𝑡𝑘<𝑡}𝛾^𝑛−1_{𝑗𝑘𝑖=1}𝜎_{2𝑗𝑖𝑡0<𝑡𝑘<𝑡}𝛼_𝑖𝑘𝑢_𝑖(𝑡)−𝜎_{2𝑗𝑖𝑡0<𝑡𝑘<𝑡}𝛼_𝑖𝑘𝑢^∗_𝑖(𝑡)𝑑𝑤_2𝑖(𝑡),𝑡≥𝑡₀,𝑗=1,2,…,𝑚. (26) Let 𝑈(𝑡)=∑^𝑛_𝑖=1|𝑢_𝑖(𝑡)−𝑢^∗_𝑖(𝑡)|^𝑝+∑^𝑚_𝑗=1|𝑣_𝑗(𝑡)−𝑣^∗_𝑗(𝑡)|^𝑝. Proceeding as the proof of the existence of periodic solution of (10), from (26), we obtain that 𝑈(𝑡)≤6^𝑝−1e^{−𝜃(𝑡−𝑡0)}𝑈𝑡₀+^𝑡_𝑡0𝜖e^{−𝜃(𝑡−𝑠)}𝑈(𝑠)+𝑑𝑠_𝑡𝑡0𝑐e^{−𝜃(𝑡−𝑠)}𝑈⁺_𝛿(𝑠)𝑑𝑠. (27) By (𝐻₄) and Corollary 10, the periodic solution of (10) is 𝑝-exponentially stable; that is, the periodic solution of (1) is 𝑝-exponentially stable. Therefore, (1) has an 𝜔-periodic solution, which is 𝑝-exponentially stable. This completes the proof of Theorem 12.

4. An Example

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

Let 𝑛=𝑚=2. Consider the following impulsive stochastic BAM neural network: 𝑑𝑥_𝑖(𝑡)=−𝑎_𝑖(𝑡)𝑥_𝑖𝑡−𝜏_𝑖(𝑡)+²_𝑗=1𝑐_𝑖𝑗(𝑡)𝑓_𝑗𝑦_𝑗(𝑡)+𝐼_𝑖(𝑡)+𝑑𝑡_2𝑗=1𝜎_1𝑖𝑗𝑦_𝑗(𝑡)𝑑𝑤_1𝑗(𝑡),𝑡≥𝑡₀,𝑡≠𝑡_𝑘,𝑥_𝑖𝑡⁺_𝑘=𝛼_𝑖𝑘𝑥_𝑖𝑡_𝑘,𝑡>0,𝑡=𝑡_𝑘,𝑑𝑦_𝑗(𝑡)=−𝑏_𝑗(𝑡)𝑦_𝑗𝑡−𝜎_𝑗(𝑡)+²_𝑖=1𝑑_𝑗𝑖(𝑡)𝑔_𝑖𝑥_𝑖(𝑡)+𝐽_𝑗(𝑡+)𝑑𝑡_2𝑖=1𝜎_2𝑗𝑖𝑥_𝑖(𝑡)𝑑𝑤_2𝑖(𝑡),𝑡≥𝑡₀,𝑡≠𝑡_𝑘,𝑦_𝑗𝑡⁺_𝑘=𝛾_𝑗𝑘𝑦_𝑗𝑡_𝑘,𝑡>0,𝑡=𝑡_𝑘, (28) where 𝑖,𝑗=1,2 and the coefficients are as follows: 𝑎₁(𝑡)𝜋=1.15+0.05sin2𝑎𝑡,₂(𝑡)𝜋=0.85+0.01cos2𝑏𝑡,₁(𝑡)𝜋=0.86+0.01sin2𝑏𝑡,₂(𝑡)𝜋=0.74+0.01cos2𝑐𝑡,₁₁(𝑡)𝜋=0.04sin2𝑐𝑡,₁₂(𝑡)𝜋=0.02cos2𝑐𝑡,₂₁(𝑡)𝜋=0.03sin2c𝑡,₂₂(𝑡)𝜋=0.01sin2𝑑𝑡,₁₁(𝑡)𝜋=0.04+0.03cos2𝑑𝑡,₁₂(𝑡)𝜋=0.01sin2𝑑𝑡,₂₁(𝑡)𝜋=0.05+0.01sin2𝑑𝑡,₂₂(𝑡)𝜋=0.02cos2𝑓𝑡,₁(𝑢)𝑓=0.03sin𝑢,₂(𝑢)𝑔=0.04cos𝑢,₁(𝑢)𝑔=0.05cos𝑢,₂(𝑢)𝜎=0.07sin𝑢,₁₁₁(𝑢)𝜎=0.02sin𝑢,₁₁₂(𝑢)𝜎=0.03cos𝑢,₁₂₁(𝑢)𝜎=0.01sin𝑢,₁₂₂(𝑢)𝜎=0.04cos𝑢,₂₁₁(𝑢)𝜎=0.01cos𝑢,₂₁₂(𝑢)𝜎=0.04sin𝑢,₂₂₁(𝑢)𝜎=0.03sin𝑢,₂₂₂(𝑢)𝜏=0.04cos𝑢,₁(𝑡)=𝜏₂(𝑡)=𝜎₁(𝑡)=𝜎₂(𝑡)𝜋=0.2+0.1sin2𝐼𝑡,₁(𝑡)=𝐼₂(𝑡)=0.6sin√𝐽2𝑡,₁(𝑡)=𝐽₂(𝑡)𝛼=0.5cos𝑡,𝑖,𝑗=1,2,_1𝑘=𝛾_1𝑘=3^{sin(𝑘𝜋/2)},𝛼_2𝑘=𝛾_2𝑘=3^{cos(𝑘𝜋/2)},𝑘=1,2,…. (29)

By calculating, we have 𝑎₁=1.2,𝑎₁=1.1,𝑎₂=0.86,𝑎₂=0.84,𝑏₁=0.87,𝑏₁=0.85,𝑏₂=0.75,𝑏₂=0.73,𝑐₁₁=0.04,𝑐₁₂=0.02,𝑐₂₁=0.03,𝑐₂₂=0.01,𝑑₁₁=0.07,𝑑₁₂=0.01,𝑑₂₁=0.06,𝑑₂₂𝐿=0.02,_𝑓1=0.03,𝐿^𝑓₂𝐿=0.04,_𝑔1=0.05,𝐿^𝑔₂𝑙=0.07,₁₁₁=0.02,𝑙₁₁₂𝑙=0.03,₁₂₁=0.01,𝑙₁₂₂𝑙=0.04,₂₁₁=0.01,𝑙₂₁₂𝑙=0.04,₂₂₁=0.03,𝑙₂₂₂=0.04,𝛼=𝛾=3,𝛼=𝛾=1. (30) Taking 𝑝=3, we can verify that all conditions of Theorem 12 are satisfied. Hence, (28) has a 4-periodic solution, which is 3-exponentially stable.

By the numerical simulation in Figures 1, 2, and 3 we can show that our results are plausible and effective.

Figure 1 

Transient states of stochastic BAM neural networks.

Figure 2 

State response of 𝑥1 and 𝑥2.

Figure 3 

State response of 𝑦1 and 𝑦2.

Acknowledgment

This work is supported by the National Natural Science Foundation of People’s Republic of China under Grant 10971183.