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

Authors: Li Yang; Yongkun Li
Journal: International Journal of Differential Equations
 (2013)

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## Abstract

By using an integral inequality, we establish some sufficient conditions for the existence andp-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.


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## Body

## 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:(1)dxi(t)=[∑j=1m‍-ai(t)xi(t-τi(t))+∑j=1m‍cij(t)fj(yj(t))+Ii(t)]dt+∑j=1m‍σ1ij(yj(t))dw1j(t),t≥t0,t≠tk,xi(tk+)=αikxi(tk),t>0,t=tk,dyj(t)=[-bj(t)yj(t-σj(t))+∑i=1n‍dji(t)gi(xi(t))+Jj(t)(t-σj(t))]dt+∑i=1n‍σ2ji(xi(t))dw2i(t),t≥t0,t≠tk,yj(tk+)=γjkyj(tk),t>0,t=tk,
where i=1,2,…,n, j=1,2,…,m (n and m are the number of neurons in layers) and xi(t) and yj(t) denote the activations of the ith neuron and the jth neuron at time t; ai and bj 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<τi<τ¯i and 0<σj<σ¯j denote the leakage delays; fj and gi are the input-output functions (the activation functions); cij and dji are elements of feedback templates at time t; Ii and Jj 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),…,w1m(t))T and w2(t)=(w21(t),…,w2n(t))T are n-dimensional Brownian motions defined on complete probability space (Ω,F,{Ft}t>0,P); here, we denote by F the associated σ-algebra generated by {wk(t)} with the probability measure P,  k=1,2;  σ1ij and σ2ji are Borel measurable functions; σ1=(σ1ij)n×m and σ2=(σ2ji)m×m are diffusion coefficient matrices; xi(tk+)=αikxi(tk) and yj(tk+)=γjkyj(tk) are impulses at moment t, which describe that the evolution processes experience abrupt change of state at the moments of time tk, where t0<t1<t2<⋯ and limk→∞tk=∞.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,{Ft}t>0,P) be a complete probability space with a filtration {Ft}t>0 satisfying the usual conditions; that is, {Ft}t>0 is right continuous and F0 contains all P-null sets. Denote by PCF0b(R,Rn+m) the family of F0-measurable, Rn 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∈PCF0b(R,Rn+m), define the norm ∥z∥=max0≤t≤ω(E|z(t)|1p)1/p, where p>1 is an integer, |z(t)|1=(∑i=1n+m|zi(t)|2)1/2; E(·) stands for the correspondent expectation operator with respect to the given probability measure P.For convenience, for anω-periodic function f:R→R, denote f¯=max0≤t≤ω|f(t)|,  f_=min0≤t≤ω|f(t)|. The initial condition of (1) is
(2)xi(s)=φi(s),yj(s)=ψj(s),s≤t0,
where φi,ψj∈PCF0b((-δ,t0],R), δ=max{max1≤i≤nτ¯i,max1≤j≤mσ¯j},  i=1,2,…,n,  j=1,2,…,m.Throughout this paper, we assume that the following conditions hold:(H1)
ai(t)>0,  bj(t)>0,  τi(t), σj(t), cij(t), dji(t), Ii(t), and Jj(t) are all periodic continuous functions with period ω for t∈R,  i=1,2,…,n,  j=1,2,…,m;(H2)
fj, gi, σ1ij, and σ2ji are Lipschitz-continuous with Lipschitz constants Ljf>0, Lig>0, l1ij>0, l2ji>0 and fj(0)=gi(0)=σ1ij(0)=σ2ji(0)=0, i=1,2,…,n,  j=1,2,…,m;(H3)
{αik} and {γjk} are real sequences and ∏t0<tk<t‍αik and ∏t0<tk<t‍γjk 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
(3)0≤α_≤|∏t0<tk<t‍αik|≤α¯,0≤γ_≤|∏t0<tk<t‍γjk|≤γ¯,
where i=1,2,…,n,  j=1,2,…,m,  and k=1,2,….This paper is organized as follows. In Section2, 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 processxt(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,…,tm, the joint distribution of the random variables xt1+kω(s),xt2+kω(s),…,xtm+kω(s) is independent of k,  k=±1,±2,….Lemma 3 (see [34]).
Ifx(t) is an ω-periodic stochastic process, then its mathematical expectation and variance are ω-periodic.Definition 4.
The solutionxt(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(∥xt(t0,φ)∥p)≤θ,  t≥t0,(ii)
p-point dissipative if there exists a constant N>0 such that for any point φ∈PCF0b([-τ,0],Rn), 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(∥xt(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 anyx∈R+n and p>0,
(4)|x|p≤n((p/2)-1)∨0∑i=1n‍xip,(∑i=1n‍xi)p≤n(p-1)∨0∑i=1n‍xip.Definition 7.
The periodic solutionx(t,t0,φ) with initial value φ∈PCF0b([-τ,0],Rn) 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∈PCF0b([-τ,0],Rn) of (1) satisfies
(5)E(|x-y|1p)≤M∥φ-φ1∥pe-λ(t-t0),t≥t0.Lemma 8 (see [33]).
Letu(t)∈C(R,R+n) be a solution of the delay integral inequality
(6)u(t)≤M1e-δ(t-t0)[φ]τ++∫t0t‍e-C1(t-s)A1u(s)ds+∫t0t‍e-C1(t-s)B1[u(s)]τ+ds+J1,u(t)u(t)u(t)u(t)u1(t)t≥t0,u(t)≤φ(t),∀t∈[t0-τ,t0],
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
(7)u(t)≤Nze-λ(t-t0)+(I-Π)-1J1,t≥t0,
where z satisfies [φ]τ+≤z.Lemma 9 (see [33]).
Assume that all conditions of Lemma9 hold. If J1=0, then all solutions of inequality of (6) exponentially converge to zero.By Lemmas8 and 9, we have the following corollary.Corollary 10.
Letu(t)∈C(R,R+) be a solution of the delay integral inequality
(8)u(t)≤M1e-δ(t-t0)[φ]τ++∫t0t‍e-C1(t-s)A1u(s)ds+∫t0t‍e-C1(t-s)B1[u(s)]τ+ds+J1,t≥t0,u(t)≤φ(t),∀t∈[t0-τ,t0],
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
(9)u(t)≤Nze-λ(t-t0)+(1-A1+B1C1)-1J1,u(t)u(t)u(t)u(t)u(t)t≥t0,
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:(10)dui(t)=[∏t0<tk<t‍-ai(t)ui(t-τi(t))+∏t0<tk<t‍αik-1∑j=1m‍cij(t)fj(∏t0<tk<t‍γjkvj(t))+∏t0<tk<t‍αik-1Ii(t)]dt+∏t0<tk<t‍αik-1∑j=1m‍σ1ij(∏t0<tk<t‍γjkvj(t))dw1j(t),u(t)u(t)u(t)u(t)u(t)u(t)u(t)2222t≥t0,dvj(t)=[∏t0<tk<t‍-bj(t)vj(t-σj(t))+∏t0<tk<t‍γjk-1∑i=1n‍dji(t)gi(∏t0<tk<t‍αikui(t))+∏t0<tk<t‍γjk-1Jj(t)]dt+∏t0<tk<t‍γjk-1∑i=1n‍σ2ji(∏t0<tk<t‍αikui(t))dw2i(t),u(t)u(t)u(t)u(t)u(t)u(t)u(t)(t)(t)t≥t0,
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),…,un(t),v1(t),…,vm(t))T is a solution of (10), then (∏t0<tk<tα1ku1(t),…,∏t0<tk<tαnkun(t),∏t0<tk<tγ1kv1(t),…,∏t0<tk<tγmkvm(t)∏t0<tk<t‍)T is a solution of (1);(ii)
if(x1(t),…,xn(t),y1(t),…,ym(t))T is a solution of (1), then (∏t0<tk<tαik-1x1(t),…,∏t0<tk<tαik-1xn(t), ∏t0<tk<tγjk-1y1(t),…,∏t0<tk<tγmk-1ym(t)∏t0<tk<t‍)T is a solution of (10).


## 3. Main Results

In this section, we will state and prove the sufficient conditions for the existence andp-exponential stability of periodic solution of (1).Theorem 12.
Let(H1)–(H3) hold. Suppose further that  (H4)
there exists an integerp>2 such that (ϵ+c)/θ<1, where
(11)θ=min{min1≤i≤na_i,min1≤j≤mb_j},c=6p-1max{max1≤i≤na¯ip,max1≤j≤mb¯jp},ϵ=max{c,ϵ1,ϵ2},ϵ1=6p-1∑i=1n‍[a_i1-p(γ¯α_∑j=1m‍(c¯ijLjf)p/(p-1))p-122222222+lpn(p/2)-1(2a_i(p-1)p-2)1-(p/2)22222222×(γ¯α_∑j=1m‍(l1ij)p/(p-1))p-1(γ¯α_∑j=1m‍(c¯ijLjf)p/(p-1))p-1],ϵ2=6p-1∑j=1m‍[b_j1-p(α¯γ_∑i=1n‍(d¯jiLig)p/(p-1))p-122222222+lpm(p/2)-1(2b_j(p-1)p-2)1-(p/2)22222222×(α¯γ_∑i=1n‍(l2ji)p/(p-1))p-1].
Then (1) has an ω-periodic solution, which is p-exponentially stable.Proof.
We can rewrite (10) as follows:
(12)dui(t)=[(∏t0<tk<t‍γjkvj(t))-ai(t)ui(t)+ai(t)ui(t)-ai(t)ui(t-τi(t))+∏t0<tk<t‍αik-1∑j=1m‍cij(t)fj×(∏t0<tk<t‍γjkvj(t))+∏t0<tk<t‍αik-1Ii(t)]dt+∏t0<tk<t‍αik-1∑j=1m‍σ1ij(∏t0<tk<t‍γjkvj(t))dw1j(t),u(t)u(t)u(t)u(t)2t≥t0,i=1,2,…,n,dvj(t)=[(∏t0<tk<t‍αikui(t))-bj(t)vj(t)+bj(t)vj(t)-bj(t)vj(t-σj(t))+∏t0<tk<t‍γjk-1∑i=1n‍dji(t)gi×(∏t0<tk<t‍αikui(t))+∏t0<tk<t‍γjk-1Jj(t)]dt+∏t0<tk<t‍γjk-1∑i=1n‍σ2ji(∏t0<tk<t‍αikui(t))dw2i(t),u(t)u(t)u(t)u(t)t≥t0,j=1,2,…,m.
By the method of variation parameter, for t≥t0, i=1,2,…,n, from the first equation of (12), we have the following:
(13)ui(t)=ui(t0)e-∫t0t‍ai(ϑ)dϑ+∫t0t‍e-∫st‍ai(ϑ)dϑ×[(∏t0<tk<t‍γjkvj(s))ai(s)ui(s)-ai(s)ui(s-τi(s))+∏t0<tk<t‍αik-1∑j=1m‍cij(s)fj×(∏t0<tk<t‍γjkvj(s))+∏t0<tk<t‍αik-1Ii(s)]ds+∫t0t‍e-∫st‍ai(ϑ)dϑ∏t0<tk<t‍αik-1∑j=1m‍σ1ij×(∏t0<tk<t‍γjkvj(s))dw1j(s).
For i=1,2,…,n, denote
(14)F1i=ui(t0)e-∫t0t‍ai(ϑ)dϑ,F2i=∫t0t‍e-∫st‍ai(ϑ)dϑai(s)ui(s)ds,F3i=∫t0t‍e-∫st‍ai(ϑ)dϑai(s)ui(s-τi(s))ds,F4i=∫t0t‍e-∫st‍ai(ϑ)dϑ∏t0<tk<t‍αik-1Ii(s)ds,F5i=∫t0t‍e-∫st‍ai(ϑ)dϑ∏t0<tk<t‍αik-1×∑j=1m‍cij(s)fj(∏t0<tk<t‍γjkvj(s))ds,F6i=∫t0t‍e-∫st‍ai(ϑ)dϑ∏t0<tk<t‍αik-1×∑j=1mσ1ij(∏t0<tk<t‍γjkvj(s))dw1j(s)‍.
Considering expectations, using Lemma 6, for i=1,2,…,n, we have
(15)E|ui(t)|p≤6p-1E(|F1i|p+|F2i|p+|F3i|p+|F4i|p+|F5i|p+|F6i|p).
For i=1,2,…,n, we evaluate the first term of (15) as follows:
(16)E|F1i|p=E|ui(t0)e-∫t0t‍ai(ϑ)dϑ|p≤E|ui(t0)e-a_i(t-t0)|p≤e-pa_i(t-t0)E|ui(t0)|p.
For the second term of (15), we have
(17)E|F2i|p=E|∫t0t‍e-∫st‍ai(ϑ)dϑai(s)ui(s)ds|p≤E(∫t0t‍e-∫st‍ai(ϑ)dϑ|ai(s)||ui(s)|ds)p≤E(∫t0t‍a¯ie-a_i(t-s)|ui(s)|ds)p=a¯ip∫t0t‍e-pa_i(t-s)E|ui(s)|pds,u(t)u(t)u(t)i=1,2,…,n.
For the third term of (15), we have
(18)E|F3i|p=E|∫t0t‍e-∫st‍ai(ϑ)dϑai(s)ui(s-τi(s))ds|p≤E(∫t0t‍e-∫st‍ai(ϑ)dϑ|ai(s)||ui(s-τi(s))|ds)p≤E(∫t0t‍a¯ie-a_i(t-s)|ui(s-τi(s))|ds)p=a¯ip∫t0t‍e-pa_i(t-s)E|ui(s-τi(s))|pds,u(t)u(t)u(t)u(t)ui=1,2,…,n.
As for the fourth term of (15), for i=1,2,…,n, we have
(19)E|F4i|p=E|∫t0t‍e-∫st‍ai(ϑ)dϑ∏t0<tk<t‍αik-1Ii(s)ds|p≤E|∫t0t‍e-a_i(t-s)∏t0<tk<t‍αik-1Ii(s)ds|p≤(I¯ia_iα_)p.
For the fifth term of (15), by Hölder inequality, we have
(20)E|F5i|p=E|(∏t0<tk<t‍γjkvj(s))∫t0t‍e-∫st‍ai(ϑ)dϑ∏t0<tk<t‍αik-1×∑j=1m‍cij(s)fj(∏t0<tk<t‍γjkvj(s))ds|p≤E(∏t0<tk<t‍∫t0t‍e-a_i(t-s)|∏t0<tk<t‍αik-1|∑j=1m‍|cij(s)|u(t)×|fj(∏t0<tk<t‍γjkvj(s))|ds)p≤E(∏t0<tk<t‍∫t0t‍e-a_i(t-s)u(t)×|∏t0<tk<t‍αik-1|∑j=1m‍|cij(s)|Ljfu(t)×|∏t0<tk<t‍γjkvj(s)|ds)p≤E(∫t0t‍e-a_i(t-s)γ¯α_∑j=1m‍|cij(s)|u(t)×Ljf|vj(s)|ds∑j=1m‍)p=E(∑j=1m‍∫t0t‍(e-a_i(t-s))(p-1)/p(e-a_i(t-s))1/pu(t)×γ¯α_∑j=1m‍|cij(s)|Ljf|vj(s)|ds)p≤E(∑j=1m‍(∫t0t‍e-a_i(t-s)ds)p-1u(t)×∫t0t‍e-a_i(t-s)(γ¯α_∑j=1m‍c¯ijLjf|vj(s)|)pds)≤a_i1-p∫t0t‍e-a_i(t-s)(γ¯α_∑j=1m‍(c¯ijLjf)p/(p-1))p-1u(t)u×∑j=1m‍E|vj(s)|pds,i=1,2,…,n.
As for the last term of (15), using Proposition  1.9 in  [37] and Hölder inequality, for i=1,2,…,n, we have
(21)E|F6i|p=E|(∏t0<tk<t‍γjkvj(s))∫t0t‍e-∫st‍ai(ϑ)dϑ∏t0<tk<t‍αik-1×∑j=1m‍σ1ij(∏t0<tk<t‍γjkvj(s))dw1j(s)|p≤lp[(∏t0<tk<t‍γjkvj(s))|p/2)2/p∫t0t‍((∏t0<tk<t‍γjkvj(s)(∏t0<tk<t‍γjkvj(s))|p/2)e-pa_i(t-s)E|(∏t0<tk<t‍γjkvj(s))∏t0<tk<t‍αik-2∑j=1n‍σ1ij2×(∏t0<tk<t‍γjkvj(s))|p/2)2/pds]p/2≤lpn(p/2)-1[(γ¯α_∑j=1n‍l1ij|vj(s)|)p)2/p∫t0t‍(E(γ¯α_∑j=1n‍l1ij|vj(s)|)p)2/pe-pa_i(t-s)u(t)u(t)u×E(γ¯α_∑j=1nl1ij|vj(s)|)p)2/pds]p/2=lpn(p/2)-1[(γ¯α_∑j=1n‍l1ij|vj(s)|)p)2/p∫t0t‍((γ¯α_∑j=1n‍l1ij|vj(s)|)p)2/pe-(p-1)a_i(t-s)e-a_i(t-s)u(t)u(t)u×E(γ¯α_∑j=1n‍l1ij|vj(s)|)p)2/pds]p/2≤lpn(p/2)-1(∫t0t‍e(-2a_i(p-1)/(p-2))(t-s)ds)(p/2)-1×(∫t0t‍e-a_i(t-s)E(γ¯α_∑j=1n‍l1ij|vj(s)|)pds)≤lpn(p/2)-1(2a_i(p-1)p-2)1-(p/2)×((γ¯α_∑j=1m‍l1ij|vj(s)|)p∫t0t‍e-a_i(t-s)u(t)u×E(γ¯α_∑j=1m‍l1ij|vj(s)|)pds)≤lpn(p/2)-1(2a_i(p-1)p-2)1-(p/2)×(∑j=1m‍∑j=1m‍∫t0t‍e-a_i(t-s)(γ¯α_∑j=1m‍(l1ij)p/(p-1))p-1u(t)u×∑j=1m‍E|vj(s)|pds),
where lp=(p(p-1)/2)p/2. Therefore, for i=1,2,…,n, we have
(22)E|ui(t)|p≤6p-1{∑j=1m‍e-pa_i(t-t0)E|ui(t0)|pu(t)u(t)u(t)+a¯ip∫t0t‍e-pa_i(t-s)E|ui(s)|pdsu(t)u(t)u(t)+a¯ip∫t0t‍e-pa_i(t-s)E|ui(s-τi(s))|pdsu(t)u(t)u(t)+a_i1-p∫t0t‍e-a_i(t-s)u(t)u(t)u(t)×(γ¯α_∑j=1m‍(c¯ijLjf)p/(p-1))p-1u(t)u(t)u(t)×∑j=1n‍E|vj(s)|pdsu(t)u(t)u(t)+lpn(p/2)-1(2a_i(p-1)p-2)1-(p/2)u(t)u(t)u(t)×(∑j=1m‍∫t0t‍e-a_i(t-s)(γ¯α_∑j=1m‍(l1ij)p/(p-1))p-1u(t)u(t)u(t)×∑j=1m‍E|vj(s)|pds)+(I¯ia_iα_)p}=6p-1{∑j=1m‍e-pa_i(t-t0)E|ui(t0)|pu(t)u(t)u(t)+a¯ip∫t0t‍e-pa_i(t-s)E|ui(s)|pdsu(t)u(t)u(t)+a¯ip∫t0t‍e-pa_i(t-s)E|ui(s-τi(s))|pdsu(t)u(t)u(t)+∫t0t‍e-a_i(t-s)u(t)u(t)u(t)×[a_i1-p(γ¯α_∑j=1m‍(c¯ijLjf)p/(p-1))p-1u(t)u(t)u(t)u(t)222+lpn(p/2)-1(2a_i(p-1)p-2)1-(p/2)u(t)u(t)u(t)u(t)222×(γ¯α_∑j=1m‍(l1ij)p/(p-1))p-1]u(t)u(t)u(t)×∑j=1m‍E|vj(s)|pds+(I¯ia_iα_)p}≤6p-1{∑j=1m‍e-a_i(t-t0)E|ui(t0)|pu(t)u(t)u(t)+a¯ip∫t0t‍e-a_i(t-s)E|ui(s)|pdsu(t)u(t)u(t)+a¯ip∫t0t‍e-a_i(t-s)E|ui(s-τi(s))|pdsu(t)u(t)u(t)+∫t0t‍e-a_i(t-s)u(t)u(t)u(t)222×[a_i1-p(γ¯α_∑j=1m‍(c¯ijLjf)p/(p-1))p-1u(t)u(t)u(t)u(t)222+lpn(p/2)-1(2a_i(p-1)p-2)1-(p/2)u(t)u(t)u(t)u(t)222×(γ¯α_∑j=1m‍(l1ij)p/(p-1))p-1]u(t)u(t)u(t)222×∑j=1m‍E|vj(s)|pds+(I¯ia_iα_)p}.
Similarly, for t≥t0,  j=1,2,…,m, from the second equation of (12), we can obtain the following:
(23)E|vj(t)|p≤6p-1{(J¯jb_jγ_)pe-b_j(t-t0)E|vj(t0)|pu(t)u+b¯jp∫t0t‍e-b_j(t-s)E|vj(s)|pdsu(t)u+b¯jp∫t0t‍e-b_j(t-s)E|vj(s-σj(s))|pdsu(t)u+∫t0t‍e-b_j(t-s)u(t)u(t)(t)×[b_j1-p(α¯γ_∑i=1n‍(d¯jiLig)p/(p-1))p-1u(t)u(t)(t)u2+lpm(p/2)-1(2b_j(p-1)p-2)1-(p/2)u(t)u(t)(t)u2×(α¯γ_∑i=1n‍(l2ji)p/(p-1))p-1]u(t)u(t)(t)×∑i=1n‍E|ui(s)|pds+(J¯jb_jγ_)p}.
Hence, by (22) and (23), we have that
(24)∑i=1n‍E|ui(t)|p+∑j=1m‍E|vj(t)|p≤6p-1∑i=1n{∑j=1m‍e-a_i(t-t0)E|ui(t0)|p‍u(t)u(t)(t)(t)(t)u+a¯ip∫t0t‍e-a_i(t-s)E|ui(s)|pdsu(t)u(t)u(t)(t)(t)+a¯ip∫t0t‍e-a_i(t-s)E|ui(s-τi(s))|pdsu(t)u(t)u(t)(t)(t)+∫t0t‍e-a_i(t-s)u(t)u(t)u(t)u22(t)(t)×[a_i1-p(γ¯α_∑j=1m‍(c¯ijLjf)p/(p-1))p-1u(t)u(t)u(t)u(t)u(t)(t)2+lpn(p/2)-1(2a_i(p-1)p-2)1-(p/2)u(t)u(t)u(t)u(t)u(t)(t)2×(γ¯α_∑j=1m‍(l1ij)p/(p-1))p-1]u(t)u(t)u(t)u22(t)(t)(t)×∑j=1m‍E|vj(s)|pds+(I¯ia_iα_)p}+6p-1∑j=1m‍{(J¯jb_jγ_)pe-b_j(t-t0)E|vj(t0)|pu(t)u(t)u(t)u(t)(t)+b¯jp∫t0t‍e-b_j(t-s)E|vj(s)|pdsu(t)u(t)u(t)u(t)(t)+b¯jp∫t0t‍e-b_j(t-s)E|vj(s-σj(s))|pdsu(t)u(t)u(t)u(t)(t)+∫t0t‍e-b_j(t-s)u(t)u(t)u(t)u(t)u22(t)×[b_j1-p(α¯γ_∑i=1n‍(d¯jiLig)p/(p-1))p-1u(t)u(t)u(t)u(t)u22(t)22+lpm(p/2)-1(2b_j(p-1)p-2)1-(p/2)u(t)u(t)u(t)u(t)u22(t)22×(α¯γ_∑i=1n‍(l2ji)p/(p-1))p-1]u(t)u(t)u(t)u(t)u22(t)×∑i=1n‍E|ui(s)|pds+(J¯jb_jγ_)p}.
Set V(t)=∑i=1nE|ui(t)|p+∑j=1mE|vj(t)|p. By (24), we have that
(25)V(t)≤6p-1e-θ(t-t0)V(t0)+∫t0t‍ϵe-θ(t-s)V(s)ds+∫t0t‍ce-θ(t-s)Vδ+(s)ds+H,
where Vδ+(s)=max-δ≤s≤0|V(t+s)|, H=6p-1max{{max1≤i≤n(I¯i/a_iα_)p},{max1≤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 (u1*(t),…,un*(t),v1*(t),…,vm*(t))T.
Suppose that(u1(t),…,un(t),v1(t),…,vm(t))T is an arbitrary solution of (10). Then it follows from (10) that
(26)d(ui(t)-ui*(t))=-ai(t)(ui(t-τi(t))-ui*(t-τi(t)))+∏t0<tk<t‍αik-1∑j=1m‍cij(t)×[fj(∏t0<tk<t‍γjkvj(t))u(t)u(t)u-fj(∏t0<tk<t‍γjkvj*(t))]dt+∏t0<tk<t‍αik-1×[∑j=1m‍σ1ij(∏t0<tk<t‍γjkvj(t))u(t)u(t)u-∑j=1m‍σ1ij(∏t0<tk<t‍γjkvj*(t))]dw1j(t),u(t)u(t)u(t)u(t)u(t)t≥t0,i=1,2,…,n,d(vj(t)-vj*(t))=-bj(t)(vj(t-σj(t))-vj*(t-σj(t)))+∏t0<tk<t‍γjk-1∑i=1n‍dji(t)×[gi(∏t0<tk<t‍αikui(t))u(t)u(t)u-gi(∏t0<tk<t‍αikui*(t))]dt+∏t0<tk<t‍γjk-1∑i=1n‍[σ2ji(∏t0<tk<t‍αikui(t))u(t)u(t)uu(t)u(t)u-σ2ji(∏t0<tk<t‍αikui*(t))]dw2i(t),u(t)u(t)uu(t)u(t)uu(t)u(t)t≥t0,j=1,2,…,m.
Let U(t)=∑i=1n|ui(t)-ui*(t)|p+∑j=1m|vj(t)-vj*(t)|p. Proceeding as the proof of the existence of periodic solution of (10), from (26), we obtain that
(27)U(t)≤6p-1e-θ(t-t0)U(t0)+∫t0t‍ϵe-θ(t-s)U(s)ds+∫t0t‍ce-θ(t-s)Uδ+(s)ds.
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.Letn=m=2. Consider the following impulsive stochastic BAM neural network:
(28)dxi(t)=[∑j=12‍-ai(t)xi(t-τi(t))+∑j=12‍cij(t)fj(yj(t))+Ii(t)]dt+∑j=12‍σ1ij(yj(t))dw1j(t),u(t)u(t)ut≥t0,t≠tk,xi(tk+)=αikxi(tk),t>0,t=tk,dyj(t)=[∑i=12‍-bj(t)yj(t-σj(t))+∑i=12‍dji(t)gi(xi(t))+Jj(t)]dt+∑i=12‍σ2ji(xi(t))dw2i(t),u(t)u(t)ut≥t0,t≠tk,yj(tk+)=γjkyj(tk),t>0,t=tk,
where i,j=1,2 and the coefficients are as follows:
(29)a1(t)=1.15+0.05sinπ2t,a2(t)=0.85+0.01cosπ2t,b1(t)=0.86+0.01sinπ2t,b2(t)=0.74+0.01cosπ2t,c11(t)=0.04sinπ2t,c12(t)=0.02cosπ2t,c21(t)=0.03sinπ2t,c22(t)=0.01sinπ2t,d11(t)=0.04+0.03cosπ2t,d12(t)=0.01sinπ2t,d21(t)=0.05+0.01sinπ2t,d22(t)=0.02cosπ2t,f1(u)=0.03sinu,f2(u)=0.04cosu,g1(u)=0.05cosu,g2(u)=0.07sinu,σ111(u)=0.02sinu,σ112(u)=0.03cosu,σ121(u)=0.01sinu,σ122(u)=0.04cosu,σ211(u)=0.01cosu,σ212(u)=0.04sinu,σ221(u)=0.03sinu,σ222(u)=0.04cosu,τ1(t)=τ2(t)=σ1(t)=σ2(t)=0.2+0.1sinπ2t,I1(t)=I2(t)=0.6sin2t,J1(t)=J2(t)=0.5cost,i,j=1,2,α1k=γ1k=3sin(kπ/2),α2k=γ2k=3cos(kπ/2),k=1,2,….By calculating, we have(30)a¯1=1.2,a_1=1.1,a¯2=0.86,a_2=0.84,b¯1=0.87,b_1=0.85,b¯2=0.75,b_2=0.73,c¯11=0.04,c¯12=0.02,c¯21=0.03,c¯22=0.01,d¯11=0.07,d¯12=0.01,d¯21=0.06,d¯22=0.02,L1f=0.03,L2f=0.04,L1g=0.05,L2g=0.07,l111=0.02,l112=0.03,l121=0.01,l122=0.04,l211=0.01,l212=0.04,l221=0.03,l222=0.04,α¯=γ¯=3,α_=γ_=1.
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 Figures1, 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 ofx1 and x2.Figure 3
State response ofy1 and y2.



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*Source: 543947-2013-09-17.xml*