LuoandLong Advances in Difference Equations (2016) 2016:9 DOI 10.1186/s13662-015-0697-y

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A new inequality of ^-operator and its application to stochastic non-autonomous impulsive neural networks with delays

CrossMark

Tianqi Luo and Shujun Long*

Correspondence: longer207@sina.com College of Mathematics and Information Science, Leshan Normal University, Leshan, 614004, P.R. China

Abstract

In this paper, based on the properties of £-operator and ^-matrix, we develop a new inequality of £-operator to be effective for non-autonomous stochastic systems. From the new inequality obtained above, we derive the sufficient conditions ensuring the global exponential stability of the stochastic non-autonomous impulsive cellular neural networks with delays. Our conclusions generalize some works published before. One example is provided to illustrate the superiority of the proposed results.

MSC: 34D23; 34K20

Keywords: exponential stability; inequality; non-autonomous; stochastic; delays; neural networks; impulses

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1 Introduction

Recently, the dynamical behaviors for cellular neural networks have been popular with many researchers because of their extensive applications in signal processing, pattern recognition, optimization problems and many other fields. The stability of networks is one of the crucial properties in such applications. Delay and impulsive effects exist widely in many practical models such as population models and neural networks. There are many works on the dynamic behaviors of various kinds of neural networks with delays or impulses [1-9]. Furthermore, some real systems are usually affected by external disturbance with great uncertainty which may be treated as random. In real nervous systems and in the implementation of artificial networks, Haykin [10] has pointed out that the synaptic transmission is a noisy process which is caused by random fluctuations from the release of neurotransmitters and other probabilistic factors. Hence, noise must be taken into consideration in the model construction. Among them, stability analysis of different stochastic systems has been a focused research subject in the literature. Stochastic perturbation is the main factor that affect the stability of systems including neural networks in performing the computation. In addition, the results in [11] suggested that certain stochastic inputs can stabilize or destabilize one neural network. This implies that the stability analysis of stochastic neural networks has primary significance in its design and applications. Therefore, some results on the stability of neural networks with stochastic perturbations have been reported [11-21]. In addition, when we consider long-time dynamic behaviors of a

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system, the parameters of the system frequently vary with time due to the environmental disturbances. In this case, a non-autonomous neural network model is the best choice for accurately depicting evolutionary processes of networks. Therefore, it is of great significance to study the dynamic behaviors of non-autonomous neural networks [22-38].

By using a Lyapunov function, the authors of [34, 35] have investigated the stability of non-autonomous systems without impulses and obtained the determinant conditions for asymptotic stability or exponential stability of the corresponding system. However, the conditions are true only for all t > 0. Besides, many authors used the linear matrix inequality (LMI) and the Lyapunov-Krasovskii functional to study the dynamic behaviors of various kinds of neural networks and obtained many interesting new results [6,16, 21, 25]. However, the results given in LMI form are commonly dependent on the delays. Particularly, for time-varying delays system [6, 16], one must require constraint conditions such as the differentiability of delay functions. The authors of [25] considered the stability and existence of periodic solution to bidirectional associative memory non-autonomous neural networks with delay and obtained some new results which are given in a function matrix inequality, but it is not easy to compute by Matlab LMI Control Toolbox. Thus, LMI technique is ineffective for dealing with the non-autonomous system. In addition to the methods mentioned before, a differential inequality is also a very useful tool for studying the dynamic behaviors of differential dynamical systems [24, 30, 32, 37-44], but many of the inequalities obtained before cannot be used to investigate the non-autonomous systems. The authors in [32] considered the periodic attractor and dissipativity of non-autonomous cellular neural networks with delays. The authors of [30] investigated the invariant and attracting sets of neural networks with reaction-diffusion terms. However, the results in [30, 32] require the time-varying coefficients to have a common factor. In practical applications, this condition is very strict and not easy to meet. The authors of [38] developed an inequality to investigate the stability of non-autonomous cellular neural networks with impulse and time-varying delays, but this inequality cannot handle stochastic non-autonomous neural networks. The authors of [37] investigated the exponential ^-stability of stochastic Takagi-Sugeno non-autonomous neural networks with impulses and time-varying delays, but the conditions imposed on the diffusion coefficient matrix are very strict. As far as we know, there are no results on the stability of non-autonomous stochastic neural networks with time delays and impulses except for [37].

Motivated by the previous analysis, in this paper, applying the properties of ^-operator and ^-matrix, we develop a new inequality of ^-operator that is effective for stochastic non-autonomous system. Based on the new inequality of ^-operator, we study the stochastic non-autonomous impulsive cellular neural networks with time-varying delays and obtain the sufficient conditions for the pth moment exponential stability of the corresponding systems. Our main results do not require common factors of the coefficients of the system, relax the conditions imposed on the diffusion coefficient matrix, and generalize some early results. One example is provided to demonstrate the effectiveness of the proposed results.

2 Preliminaries

Let Rmxn be the set of m x n real matrices. Usually E denotes an n x n unit matrix. Rn denotes the space of n-dimensional real column vectors, | • | denotes Euclidean norm, N = {1,2,..., n}, N = {1,2,...}, R+ = [0, +(»). For M,N e Rmxn or M,N e Rn, the notation

M > N (M > N) indicates that each pair of corresponding elements of M and N satisfies the inequality '> (>)'. Particularly, M e Rmxn is called a non-negative matrix if M > 0, and x e Rn is called a positive vector if x >0. Let p(M) denote the spectral radius of square matrix M.

L1(R+, R+) denotes the family of all continuous functions h : R+ ^ R+ satisfying /0+TO h(t) dt < to. C[X, Y ] denotes the space of continuous mappings from X to Y. Inparticular, let C = C[[ -t, 0], Rn] denote the family of all Rn-valued functions y which is bounded continuous and defined on [-t, 0]. The norm of C is defined by ||y|| = sup-T<e<0 |y(0)|.

PC[J,Rn] = {$ : J ^ Rn|$(v) is continuous for all but at most countable points v e J and at these points v e J, $(v+) and $(v ) exist and $(v) = $(v+)}, where $(v-) and $(v+) denote the left-hand and right-hand limits of the function $(v) at time v, respectively, and J c R is an interval. Especially, let PC = PC[[-t, 0],Rn].

For any x e Rn, $ e C or $ e PC, p > 0, we define

= (\x1\p,..., \xn\py1, [m]T = ([m]T,..., [ut)]j ,

[<fr(t)]T = sup \0i(t + s)\, i e N,

-T <s<0

and D+$(t) denotes the upper-right-hand derivative of 0(t) at time t.

(fi, F, {Ft}t>0, P) denotes a complete probability space with a filtration {Ft}t>0 satisfying the usual conditions (i.e., it is right continuous and F0 contains all P-null sets). Let Cbn[[to - t, to],Rn] (CbF[[to - t, to],Rn]) denote the family of all bounded F0(Ft)-measurable, C[[t0 - t, t0],Rn]-value random variables 0, let PCb^o [[t0 - t, t0],Rn] (PC%t [[t0 - t, t0],Rn]) denote the family of all bounded Fc>(Ft)-measurable, PC[[t0 -t, t0],Rn]-value random variables 0, satisfying \\$\\pLp = supt0-T<e<t0 E\0(0)\p < to forp > 0, where E denotes the expectation of stochastic process.

We study the following stochastic non-autonomous impulsive cellular neural networks with delays:

dxi(t) = [-fli(t)xi(t) + bij(t)fj(xj (t)) + £j=i cij(t)gj(xj(t - Tij (t))) + /i(t)] dt

+ £"=1 jt, x(t), x(t - T (t))) dwj(t), t > t0, t = tk,

Xi(tk)=Iik (x(t-)), t = tk, Xi(s) = $i(s), t0 - t < s < t0,

where i e N, and n is the number of units in a neural network; xi(t) is the state variable of the ith unit at time t; f(0 andg(-) are the activation functions of the jth unit at time t and t - Tij(t), respectively; Tij(t) is the time-varying delay satisfying 0 < Tij(t) < t and t > 0 at time t; ai(t) > 0 is the rate with which the ith unit will reset its potential to the resting state in isolation when disconnected from the network and external inputs; bij(t), cij(t) denote the strengths of the jth neuron on the ith unit at time t and t - Tij(t), respectively; Ii(t) denotes the bias of the ith unit at time t; a(t,x(t),x(t - t(t))) = (aij(t,x(t),x(t - t(t))))nxn is the diffusion coefficient matrix, and a>(t) = (rn1(t),...,rnn(t))T is an n-dimensional Brownian motion defined on a complete probability space (fi, F, {Ft}t>0, P); $(s) = ($1(s),$2(s),...,$n(s))T e PCF [[t0 - t, t0],Rn] is the initial function vector. The impulsive function Ik = (I1k,...,Ink)T e C[Rn,Rn], and the fixed impulsive moments tk (k e N) satisfy t0 < t1 < t2 < ••• and limk^TO tk = to.

Throughout this paper, we assume that f (•), g;(-), aj(t, •, •) satisfy the linear growth condition and are Lipschitz continuous as well. Therefore, we can know that system (2.1) has a unique global solution denoted by x(t) = (xi(t),...,xn(t))T on t > to and E(supto<s<t |x(t)|r) < œ for all t > to and r > 0.

Definition 2.1 System (2.1) is called globally and exponentially p-stable, if there exist constants M > 1 and X > 0 such that for any two solutions x(t, $) and x(t, ty) with $, ty e [[t0 - r, t0],Rn], respectively, one has

E|x(t,$) -x(t, ty)\p < M||$ - ty fLpe-X(t-t0), t > to.

Furthermore, if x* is an equilibrium point of system (2.1), then we call the equilibrium point x* exponentially p-stable.

Definition 2.2 ([45]) Let matrix D = (dij)nxn satisfy dj < 0, i = j, then the statement D is a nonsingular ^-matrix' is equivalent to one of the following conditions.

(i) D = B -M and p(B-1M) < 1, whereM > 0, B = diagj^,...,bn}.

(ii) All the leading principal minors of D are positive.

(iii) The diagonal elements of D are all positive and there exists a positive vector d such that Dd > 0 or DTd > 0.

For a ^-matrix D, from (iii) of Definition 2.1, we know Qm(D) = {z e Rn\Dz > 0,z > 0} = $ and satisfies k1z1 + k2z2 e &m(D) for any vectors zi, z2 e &m(D) and scalars ki, k2 > 0.

For A e Rnxn and \A\ =0, we denote Qp(A) = {z e Rn\Az = p(A)z}, where p(A) is an eigenvalue of A. Then (A) includes all positive eigenvectors of A provided that the matrix A has at least one positive eigenvector (see Ref. [46]).

Lemma 2.1 ([47]) For ai > 0 and YTi=i a = 1, xi > 0, we have

rK <£

where the sign of equality holds if and only ifxi = xjfor all i, j e N.

Lemma 2.2 ([12]) For ai > 0, xi > 0, i e N and any integral number p > 0, we have

(n \P / n \p-1 n

J2aixn an J2aixp.

i=1 ' \ i=1 ' i=1

Lemma 2.3 ([12]) For an integral numberp > 2, there exists ep(n) > 0 such that

(n \ 2 n

J2lxi\2) <£|x;lp, Vx = (x1,...,xn)r e Rn.

i=1 ' i=1

3 Anew ^-operator inequality

Let C2,1 [Rn x R+; R+] denote the family of non-negative functions V(x, t) on Rn x R+ which are once continuously differentiate in t and twice continuously differentiate in x. Associated with the system (2.1), for each V(x, t) e C2,1 [Rn x R+;R+], we define an operator LV from Rn x Rn x R+ to R by

LV (x, t) = Vt (x, t) + Vx(x, t)[-A(t)x(t)+B(t)f (x(t)) + C(t)g(y) +/(t)] + 2 trace [a T(t, x, y)Vxxa (t, x, y)],

< / \ \ , , d V (x, t)

y = x(t - r (t)), Vt (x, t)= d ,

d V 2(x, t)

VxxoJd^M.....¡^M), Vxx = (

\ dx1 d xn J \

dxidxj / nxn

The differential inequality is the main tool for investigating differential equations. Therefore, by using the properties of the L-operator and the M-matrix, we introduce a new inequality of the L-operator that is effective for a stochastic non-autonomous system.

Theorem 3.1 Let a < b < +to, P = (pij)nxn, Pij > 0 for i = j, Q = (qij)nxn > 0, a(t) = (aij(t))nxn > 0, p(t) = (Pij(t))nxn > 0, r(t) = (ri(t),...,rn(t))T > 0 andfor any t > a, there exists a constant S >0 such that eS(t-a)ri(t), aij(t), pij(t), i, j e N are integrablefunctions on [a, t]. The functions Vi(x) e C2[Rn, R+] satisfy

LVi(x) < £ [(Pij + aij(t)) Vj(x(t)) + (qij + Pj(t)) Vj(x(t - r (t)))] + n(t),

t e [a, b), Vi e N. (.1)

Suppose that n = -(P + Q) is an M-matrix, we obtain

EVi(x(t)) < zie-x(t-aQ(s)ds, t e [a, b), Vi e N, (3.2)

provided that EVi(x(t)) < to for all t e [a, b) and

EVi(x(t)) < Zie-k(t-a), t e [a - r,a], Vi e N, (3.3)

where X e (0, S], z e fi^(n) with zi > 1, Vi e N, satisfy

(XE + P + QeXr)z < 0, (3.4)

and Q (s) = maX1<i<nlE;i1(aij (s) + Pij(s)eXr) Zj + eX(s-a)r;(s)}.

Proof By using the Ito formula, for the solution process x(t) of(2.1) and V;(x) e C2[Rn, R+], we can obtain

Vi(x(t)) = Vi(x(a)) + j LVi(x(s)) ds

+ fat 9 Vidl(s)) a (s,x(s),x(s - r (s))) da>(s), t > a, Vi e N. (3.5)

Then we get

EVi(x(t)) = EVi(x(a)) + i ECVi(x(s)) ds, t > a, Vi e N. (3.6) J a

For small enough At > 0, we have

f t+At

EVi(x(t + At)) = EVi(x(a)) + / E£Vi(x(s)) ds, t > a, Vi e N. (3.7)

Therefore, from (3.1), (3.6), and (3.7), we have EVi(x(t + At)) - EVi(x(t))

E£Vi(x(s)) ds

J^ipij + j))Ejx(s)) + £+ j))EVj(x(s - r(s))) + n(s)

.j=1 j=1

J^ipij + aij(s) EVj(x(s)) . j=1

+ ^ (s^ [EVj(x(s^] r + n (s)

ds, t > a, Vi e N. (3.8)

Because EVi(x(t)) < to for all t e [a, b), we know EVi(x(t)) is continuous. Thus, from (3.8), we can obtain

D+EVi(x(t)) < £ [(Pj + aij(t)) EVj(x(t)) + (qij + pt) (t)) [EVj(x(t))] J + n(t),

t e [a, b), Vi e N. (.9)

Let vi(t) = EVi(x(t)). For proving Theorem 3.1, we only need to prove

Vi(t) < Zie-X(t-a<>sds, t e [a, b), Vi e N, (.10)

provided that

vi(t) < Zie-X(t-a), t e [a - r, a], Vi e N, (3.1)

holds.

Because n is an ^-matrix, we can get a vector z e (n)with zi > 1, i e N and nz >0, that is, (P + Q)z < 0. From the continuity of the function, we know there exists a constant X e (0,5] satisfying (3.4).

For proving (3.10), we first of all prove, for any given e >0,

vi(t)<(1 + €)ze-X(t-a)eitaB(s)ds = 6(t), t e [a, b), Vi e N. (3.12)

If (3.12) is not true, given that Vi(t) is continuous on [a, b) and the fact (3.11) holds, then there must be a constant t* e (a, b) and m e N such that

Vm(t*) = Hm (t*), D+Vm(t*) > Hm (t*), (3.13)

vi(t) < Hi(t), t e [a - r, t*], Vi e N. (.14)

By using (3.9), (3.11)-(3.14), zm > 1, p.) > 0 (i = j), and Q > 0, we can get

D+Vm{t*) < £[(pm; + 0!mj(i*))Vj(t*) + (qmj + Pmj{t*)) [jt*)] J + Tm(t*)

< £[ipmj + ajt0)(1 + e)Zje-A(t*-a)e^ 0(s)ds j=1

+ (?mj + Pmj(t*^)eAr (1 + e)Zje-x(tt-a0(s)ds] + r^t*)

< £(pmj + qmjexr)(1 + e)Zje-x(t*-a^ 0(s)ds j=1

+ £(amj(t*) + Pmj(t*yr) ^(1 + e)Zme-k(tt-a)ef° 0(s)ds j=1 Zm

+ eA(t*-a)Tm(t*)(1 + e)Zme-A(t*-a)e^*0(s)ds + Tm(t*)[1 -(1 + e)zmetf 0(s)ds]

< -X(1 + e)Zme-A(t*-a)e^*0(s)ds + 0(t*)(1 + e)Zme-A(t*-a)e^*0(s)ds

= H'm (t*), (3.15)

which contradicts the second inequality in (3.13). Thus, (3.12) holds. Letting e ^ 0+ in (3.12), we obtain (3.10). □

Remark 3.1 If a(t) = 0 and p(t) = 0 in (3.1), we can get Theorem 1 in [44]. 4 Application to neural networks

For system (2.1), some assumptions are given in the following:

(Ai) For i, j e N, ai(t) > 0, bj(t), cij(t) andIi(t) are bounded continuous functions defined on R+.

(A2) There are positive constants l; and kj, j e N such that

f (t) -fj(s) \ < lj\T - s|, \|j(T)-jj(s)\ < kj\T - s|, Vt,s e R.

(A3) There exist non-negative bounded functions mij(t), nij(t), and hi(t), i, j e N such that

[aij(t, v, u) - aij(t, V, U)]2 < mij(t)(vj - Vj)2 + %(t)(uj - Uj)2 + hi(t), Vu, U, v, V e Rn, t > t0.

(A4) There exist non-negative integrable functions <%(f), fa(t) on [t0, t] such that

P(t) < P + a(t), Q(t) < Q + fat) and n = -(P + Q) is a nonsingular ^-matrix,

where P(t) = p(t))„x„, p(t) = -put(t) + (p - l)EJ=i(i^<;(t)lj + Mt)|kj) + ±p - l)(p -2) T,U(mij(t) + nv(t)) + lMt)|/; + (p -l)mü(t) + i(p -l)(p -2)p (t) = |bj(t)|/j + (p -l)mj(t),

i = j, Q(t) = (qij (t))nxn, qij (t) = ICj (t)|kj +(p - l)«,; (t), Ó (t) = (aij (t))nxn, fat) = (/3,y (t))nxn, P = (^ij)nxn,pij > 0, i = j, Q = (<7ij)nxn > 0, i,j e N,p > 2. Let r,(t) = (p - l)(nhi(t))p, i e N, and there exists a constant 5 >0 such that e'r(t-to)ri(t) is an integrable function on [t0, t]. (A5) For any u, v e Rn, there exist matrices Rk = (rj)nxn > 0 such that

[Ik(u)-Ik(v)]+ < Rk[u - v]+, k e N.

Let Rk = (^)nxn, rj > rf^ r^V^

(A6) The set & = (Rk))n is nonempty (i.e., & = 0), for a given z e the

scalar X e (0, 5] satisfies

(XE + p + QeXT)z <0. (4.l)

(A7) There are constants 0 < ¡ < X and b > 0 which satisfy

í &(s) ds < ¡(t - tü) + b, (.2)

where &(s) = maxl<i<n(EjLl(a/j (s) + Aj(s)eXT) f + eX(s-t0)ri(s)}. (Ag) There exists a constant y such that

ln Yk < Y < X - ¡, k e N, (.3)

tk - tk-1 where yk > max{1, p(Rk)}.

Theorem 4.1 Assume that (A1)-(A8) are all true. Then we know system (2.1) is exponentially p-stable and the exponential convergent rate is no less than X - ¡x - y.

Proof For any two solutions x(t) and y(t) of system (2.1) corresponding to initial values 0(s), <p(s) e PCF [[t0 - r, t0],Rn], respectively. Let zi(t) = xi(t) -yi(t), i e N. Then from (2.1), we get

dzi(t) = [-ai(t)zi(t) + £;=l bj(t)(fj(xj(t)) -fj(yj(t)))

+ EJ=l %(t)jj(t - Tij(t))) -#(yj(t - Tj(t))))] dt

Jrr,,(t x(t) x(t - T (f)))-a,,(f y(t) y(t - T (t)))) drn,(t)

+ E^K (t, x(t), x(t - t (t))) - aij (t, y(t), y(t - t (t)))) djt),

t > t0, t = tk,

Zi(tk) = Xi(tk) -yi(tk) = Iik(x(t-)) -Iik(y(t-)), t = tk, Zi(s) = 0i(s) - ^i(s), t0 - T < s < t0.

Let Vi(Z(t)) = \Zi(t)\p, p > 2, i e N. Then we get

^ = P\Zi\p-1 sgn(Zi) =p\Zi\p-2Zi, ^^ = P(P -1)\Zi \p-2, dZi d ZJ

where sgn(-) denotes sign function. Therefore, from (A1)-(A4), Lemma 2.1, and (4.4), we get

LVi(Z) = p \ Zi (t) |p-2Zi(t) -ai(t)Zi(t) + £ by(t)[fi(xj(t)) -fjj))]

+ £Cij(t)[gj(Xj(t - rij(t))) -gj(yj(t - rij(t)))]

+ -p(p -1)\Zi\p-^[aij(t,x(t),x(t - r(t)))- aq{t,y(t),y(t - r (t)))]2

< -pai(t)\Zi(t)\p + p\Zi(t)\p-^\bij(t)\lj\Zj(t)\

+ p\Zi(t)\p-^\Cij (t)\kj\Zj(t - rij(t)) \

+ -p(p - 1) \ Zi (t) \p-2 £ m (t)\Zj(t)\2 + nij (t) jt - rij(t)) \2 + hi(t)] 2 j=1

< -pai(t)\Zi(t)\p + £\bij\lj[(p-1)\Zi(t)\p + \Zj(t)\p]

+ £ \ Cj\kj[(p - 1) \Zi(t) \p + jt - rjj(t)) \p]

+ 2(p -1)(p -2)£ mij (t)\Zi(t)\p j=1

+ (p -1) £mij(t)\Zj(t)\p + 2(p - 1)(p - 2) £nij(t)\Zi(t)\p j=1 j=1 n i

+ (p -1) £nij(t) \Zj(t - rij(t)) \p + 2(p - 1)(p - 2)\Zi(t) \p + (p -1)(nhi(t))2 j=1

< £[pij(t)V(Z(t)) + qij(t)Vj(Z(t- r(t)))] + Ti(t)

< £[Cp0 + 0/(t)) V<Z(t^ + {qij + pij(t)) Vj(^t - r(t)))] + Ti(t). (4.5)

For the initial conditions <^(s), ^(s) e PCF0 [[t0 - r, t0], Rn], we know z(s) = <^(s)- ^(s) e PCF0 [[t0 - r, t0],Rn]. From the assumption that, for any initial value in PCb^o [[t0 -r, t0],Rn], model (2.1) has a global solution satisfying E(supt0<s<t \x(t)\T) < to for all t > t0 and t > 0, we know E(supt0<s<t \Z(t)\T) < to for all t > t0 and t >0. Thus, we know

EVi(z(t)) < to. Since n = -(P + Q) is an ^-matrix and ^ is nonempty, there must be a positive vector z e ^ and a constant X e (0,5] such that (4.1) holds and

EVi(z(i)) < . Z r 1 k- ^e-X(t-t0), t e [to - r, to], (4.6)

mini« {zj}

where k > 0 is a constant such that k||0 - vIIlp — 1. From (A4), (4.5), (4.6), and Theorem 3.1, we get

EVi(z(t)) < . z\ l kU - vf^^e^0(s)ds, t e [to, ti). (4.7)

mmi<j<„ {z,-}

Assume that the inequalities

EVi(z(t)) < yon ■■■Ym-i . z f , k U - vl^e-*^/* ®(s) ds, mini<j<„{zj}

tm-1 < t < tm, (.8)

hold for all m = i, 2,..., k, where yo = i. Then, from (4.8), (A5), and Lemma 2.2, we obtain EVi(z(tk)) = E|xi(tk)-ji(tk)|p

= E|/ik(^t-)) -Iik{y{t-))

< ^ 4 j-) |j

(n \ p-i n

j=i / j=i nn

<£ rfE\Zj (t-)|p = £ fE^zfe)) j=i j=i

< YoYi ■ ■ ■ Yk-i £ j-n-z—-k U - ^ll^^*s)ds

j=i —lni</<n{z/}

= YoYi ■ ■ ■ Yk-iPR) . * f , k№ -^e^-to)0(s)ds —ini<j<n {z,} L

< YoYi ■ ■ ■ Yk-iYk . z\ , k||0 -^e^"«V*0(s)ds. (4.9)

—ini<j<n {z,}

This, together with (4.8) and Yk — i, leads to

EVi(z(t)) < YoYi ■■■Yk-iYk . z\ , k U - ^llPpe^e^ ®(s) ds mini<,<n{z,}

= YoYi ■ ■ ■ Yke^ok °(s)dse-X(tk. kU - ^l^e-X(t-tk),

—ini<j<n{zj} L

t e [tk - r, tk]. (4.io)

z = Y0Y1 ■■■Ykeft Q(s ds , z K110 - v\\plp, z =(^1,...,zn)T, min1<j<n{zj} L

Ui(z(t)) = eX(tk-t0) Vi(z(t)), then we know the vector z e fi^(n) with z; > 1, i e N. From (4.10), we get

EUi(z(t)) < ~zie-X(t-tk), t e [tk - r, tk]. (4.11)

From (4.5), we obtain

LUi(z(t)) < £[p + caij(t^Uj(z(t)) + (qij + Pij(t))Uj(z(t-r(t)))] + eX(tk"^(t). (4.12)

Furthermore, we can easily get

Q(s) = max j (aj(s) + Pij(s)eXr) z + eX(s-tk)eX(tk"^(s) [

" " I j=1

= max^ (aij (s) + Pij(s)eXr)z + e^^s) = 0(s).

1<i<n j=1 ij ij zi i

Therefore, from (A4), (4.11), (4.12), and Theorem 3.1, we get

EUi(z(t)) < zie-X(t-tk)/tk§{s)ds, t e [tk, tk+1), (4.13)

that is,

EVi(z(t)) < Y0Y1 ■ ■ ■ Yk-1Yk • Zt, x k M - ^\Lpe-X(t-t0)e/t'Q(s)ds, min1<j<n{zj}

t e [tk, tk+1). (4.14)

By using the mathematical induction method, we know

EVi(z(t)) < Y0Y1 ■ ■ ■ Yk-1 —:—~—;—rk\\0 - ^Lpe"^^Q(s)ds,

min1<j<n{zj} L

t e [tk-1, tk), k e N. (4.15)

From (4.3), we know Yk < eY(tk-tk-1). Then we can use (4.2) and (4.15) to get |xi(t)-yi(t)|p

= EVi(z(t)) < eY(t1-t0) ■ ■ ■ eY-z-k\\0 - ^„e-*-^ A ®(s)ds

min1<j<n{zj} L

K\\0 - v\\Lj,e-(X-n-y)(t-t0), Vt e [tc tk),k e N, (4.16)

mln1<j<n{zj} where K > k is a proper constant.

From (4.16) and Lemma 2.3, we get

E|x(t)-y(t)\P E|xi(t)-yi(t)\p = EV^z(t))

ep(n) i=i ep(n) i=i

1 ^ Zi • ||P „-(X-X-y)(t-to)

ep(n) ^ minijnizj} ± M\\4 - V fpe-^-v)(t-t0), t > i0. (4.17)

Therefore, the conclusion of Theorem 4.1 holds.

If Ii(t) = 0, j 0,0) = 0 for t > to, Iik(0) = 0,_/j(0) = #(0) = 0, i,j e N, k e N then the system (2.1) becomes the following system:

dxi(t) = [-«i(i)xi(i) + E/=1 bij(t)fj(xj(t)) + EU Cij(t)gj(xj(t - Tij(t)))] dt

(4.18)

+ EJ=1 jt, x(t), x(t - t (t))) dj), t > to, t = tt,

xi(tk)=Iik (x(t-)), t = tk, xi(s) = 0i(s), t0 - r < s < t0,

with an equilibrium point x* = 0. From Theorem 4.1, we can get the following conclusion. □

Corollary 4.1 Assume that the conditions (A1)-(A8) are all true. Then the zero solution x* = 0 of (4.18) is exponentiallyp-stable and the exponential convergent rate is no less than X - ¡ - Y.

Remark 4.1 The authors in [24] obtained some new results on p-moment exponential stability of non-autonomous stochastic differential equation with delay. The model (4.18) without impulses is a special case of equation (2) in [24]. However, the results in [24] require the coefficients to have a common factor and hi(t) = 0 (i e N, t > to) in assumption (A3) to be true.

If Iik (x) = x;, i e N, k e N and 0(s) = (01(s),...,0n(s))T e C^o [[t0- r, t0], Rn],from system (2.1), we can get the following model without impulses:

dxi(t) = [-ai(t)xi(t) + E;=! bij(t)fj(xj (t)) + Em cij(t)gj(xj(t - rj (t))) + Ii(t)] dt

+ Em jx(t),x(t - r (t))) d«j(t), t > t0, (4.19)

xi(s) = 0i(s), t0 - r < s < t0.

Then we can get the following conclusion.

Theorem 4.2 Assume that (A1)-(A4) hold, (A7) holds for X e (0,5] which satisfies

(XE + n + QeXr)z <0, z e ^M(n). (4.20)

Then the system (4.19) is exponentially p-stable and the exponential convergent rate is no less than X - ¡¡.

If Ii(t) = 0, atj(t,0,0) = 0 for t > t0,f(0) = ,§,(0) = 0, i,j e N, then the system (4.19) becomes the following model:

dxi(t) = [-fli(t)xi(t) + j bij(t)fj(xj(t)) + ^ cij(t)gj(xj(t - Tj(t)))] dt

+ E/U jx(t),x(t - t(t))) da,(t), t > t0, (4.21)

Xi(s) = 0i(s), t0 - T < s < t0,

with an equilibrium point x* = 0. From Theorem 4.2, we get the following corollary.

Corollary 4.2 Assume that (A1)-(A4) hold, (A7) holds for X e (0,5] which satisfies the inequality (4.20). Then the zero solution x* =0 of (4.21) is exponentially p-stable and the exponential convergent rate is no less than X - ¡¡.

Remark 4.2 The models investigated in [14, 21] can be considered as special cases of model (4.21), but they require the differentiability of delay functions and supt>t0 Tj(t) < 1. In addition, combining ai,(t, 0,0) = 0 for t > t0 with (A3), we can get

trace[ar (t, v, u)a(t, v, u)] = ££ aj (t, v, u)

1=1 7=1

' n \ / n \ n

J2mij(t)) v2 + ( E nu (t)) U + E hi(t)

L \ i=l / \ i=l / i=l

However, the authors of [14, 21] require that hi(t) = 0 (i e N, t > to) in assumption (A3) is true.

Remark 4.3 The authors in [23] used the methods in [34, 35] to study the p-moment exponential stability of non-autonomous stochastic Cohen-Grossberg neural networks and obtained some new results. It is well known that the model (4.21) is a special case of equation (3) in [23], however, the condition (12) in [23] is equivalent to requiring that -(P(t) + Q(t)) is a nonsingular ^-matrix for all t > to. In addition, the results in [23] require that hi(t) = 0 (i e N, t > to) in assumption (A3) is true.

5 Examples

Example 5.1 Consider the following system:

dxi(t) = [-(3 - §| cos t|)x1(t) + (1 + sin te-°2t)f1(x1(t)) + (f + e-0'1t)f2(xi(t)) + 01 + 2e-01t)g1 (xx(t - 0.11 sin 20t|)) + (22 + e-0,1t)g2(x2(t - 0.1| sin 30t|))] dt

+ (# x1(t) + e-t sin2 x1(t)) da1(t) + (# x2(t) + e-t sin2 x2(t)) da2(t), • dx2(t) = [-(200 - 11 cos t|)x2(t) + (1f + e-0,2t^(t)) + (1 + cos te-0,2tfxsW) (f.1) + (12 + e-0,2t )g1 (x1 (t - 0.11 sin 30t|)) + (22 + e-0,2t)g2(x2(t - 0.1| sin40t|))] dt

+ (^f x1(t) + e-t sin2 x1(t)) d«1(t) + (^f x2 (t) + e-t sin2 x2(t)) d«2(t), xi(s) = &(s), -0.1 < s < 0,

wheref1(s) =f2(s) = ¿(|s + 1| - |s -1|), g1(s) =g2(s) = s. We can easily know r = 0.1, l1 = l2 = k1 = k2 = 1,f1(0) = f2(0) = g1(0) = g2(0) = 0, aij(t,0,0) = 0, i,j = 1,2. Evidently, model (5.1) has an equilibrium point zero.

For aij(t,x(t),x(t - r (t))) = xj(t) + e-t sin2 x;(t), i, j = 1,2, we can derive |aiy(t,x(t),x(t -r(t)))|2 < 10xf(t) + 2e-2t, that is, mij(t) = ^ nv(t) = 0, hi(t) = 2e-2t, r(t) = 4e-2t, i, j = 1,2.

Case 1. Let p = 2, by simple computation, we can get the parameters of (A4) as follows:

-23 + 41 cos ^ + 21 sin t|e-0.2t + 4e-01t 10 + e-01

45 + 3 | 20+4 |

p(f\-t 10 1 4 . ^ 10

i(t) = l 30+ e-02 -10 + 441 cos t| +-2| cos i|e-°.2i + Be-02

/ 1 + 2e-0.1t 1+ e-0.1t \ ^ /23

Q(t) = I 2 + 2e 2+ e P =10 10

Q(t) \ li p-0.2t 1 n 0.2t I , P I 11 _45

\ 12 + e 2 + 2e ) \ 30 20/

n=(2 a n=-(n+Q)=(-0:95 -72),

a (t) = (4(t))2x2 + (a l(t))2,2-\ n 3

41 cos t| 0 ' 0 41 cos t|

' 2| sin t|e-0,2t + 4e-0-1t e-0'1t

+ \ e-0,2t 2| cos t|e-0,2t + 3e-0'2^ ,

2e-0.1t e-0.1t P (t)=Pj(t^2x2= e-0.2t 2e-0.2t

We can easily know that n is a nonsingular ^-matrix, and we obtain

^M(n) = jfe,z2) > 0 — z1 < z2 < 2z1 Apparently, z = (1,1)T e ^^(n), and X = 0.54 satisfies

(XE + n + neXr)z = (-0.0045, -0.1999)T < (0,0)T. We compute

J(t) = I | cos s| ds.

For any t > 0, there must be an integer n > 0 satisfying nn - | < t < nn + Let t ■■ nn - n2 + u, 0 < u < n, then we get

J(t) = I | coss| ds

r 2 ^ , r 'n+2 rt

= I cossds+ ^ (-1) I cossds+(-1)n I cossds

J0 k=1 'kn - 2 Jnn - nj-

= 1 + ]C(-1)k(si^kn - ^ - sin^kn + + (-1)n^sin t - sin^nn -

= 1 + ]T((-1)2* - (_1)2*-1) + (-!)«/"n

l— l v v

nn _ — + u ) _ (-1)

k=1 v x 2 ( n \

= (2n - cos u) = — I nn--)+1- c°s u

n V 2/

< — t + (1 - cos u), 0 < u < n. (.2)

Since 0(s) = 41 coss| + ma^i^tE^^s) + N(s)eXT) + e0'54^)} = f1 coss| + 0*(s) and a2, ft, e L1[R+,R+], e0-54sri(s) = 4e-L46s e L1[R+,R+], i,j = 1,2, we easily know 0*(s) e

L1[R+, R+]. Combined with (5.2), we obtain

e/0 e(s) ds = ef0 e*(s) dse |/01 c°s s| ds

< e/0 6*(s) dseteId-cosu)

3 t , -

< Met, (.3)

where M > 1 isa constant.

Thus, from Corollary 4.2, we know the zero solution x* = 0 of (5.1) is exponentially 2-stable (see Figure 1), the exponential convergent rate is no less than 0.54 = 0.0625.

Remark 5.1 Apparently, -(P(t) + Q(t)) is not a nonsingular ^-matrix for all t > 0, and hi(t) = 2e-2t = 0 is not true for all t > 0, thus the results in [23, 24] are invalid for (5.1). In addition, the delay functions Ti,(t) do not satisfy supt>0 jt) < 1, therefore, when model (5.1) is autonomous, the results in [14, 21] are invalid for it.

State trajectory of x^t)

State trajectory of x2(t)

time t

Figure 1 The state trajectories of model (5.1) without impulses.

30 20 10 0

30 20 10 0

State trajectory of x1 (t)

20 timet

State trajectory of x (t)

20 timet

Figure 2 The state trajectories of model (5.1) with impulses (5.2).

Case 2. If

xi(tk) = hk (x(t-)) = 0.2e°,02x1(t-) + 0.8ea02x2(t-),

x2(tk) = I2k(x(t-)) = 0.6e0-02x^tk) + 0.4ea02x2(tk), tk - tk-1 = 1,k e N, then we can get the following parameters of (A5), (A6), (A8):

kk = e0,0^0'2 0^ , p(Rk) = e0-04, Qp(Rk) = {(zi,Z2) > 0|Z1 = Z2}.

0.6 0.4

Therefore, Q = P|^(Qp(Rk)) n Q^(n) = {(Z1,Z2) > 0|z1 = Z2} is not empty. Let z = (1,1)T e Q and yk = e0,04, we can obtain for k e N

ln yk ln e004

rk -= 0.04 = y < X - x = 0.0625.

tk - tk-1

From Corollary 4.1, we know the zero solution to (5.1) with impulses (5.4) is exponentially 2-stable (see Figure 2).

6 Conclusion

In this paper, we have analyzed the stochastic non-autonomous impulsive cellular neural networks with delays. Based on the properties of ^-operator and ^-matrix, we have developed a new inequality of ^-operator. We have applied the new inequality to stochastic non-autonomous neural networks and derived the sufficient conditions for the pth moment exponential stability of the considered system without impulses or with impulses. Our results do not require differentiability of the delay functions and have relaxed the conditions imposed on the diffusion coefficient matrix. The results obtained have generalized some early works.

Competing interests

The authors declare that they have no competing interests.

Authors' contributions

All authors have made equal and significant contributions in writing this paper. All authors read and approved the final

manuscript.

Acknowledgements

This work is supported by National Natural Science Foundation of China under Grant 11271270, Fundamental Research

Fund of Sichuan Provincial Science and Technology Department under Grants 2013JYZ014,2012JYZ019, Scientific

Research Fund of Sichuan Provincial Education Department under Grant 16TD0029and Project of Leshan Normal

University under Grant Z1324.

Received: 7 May 2015 Accepted: 16 November 2015 Published online: 13 January 2016

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