Existence and p-exponential stability of periodic solution for stochastic shunting inhibitory cellular neural networks with time-varying delays

Changjin Xu; Maoxin Liao; Yicheng Pang

doi:10.1080/18756891.2016.1237192

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Volume 9, Issue 5, September 2016, Pages 945 - 956

Existence and p-exponential stability of periodic solution for stochastic shunting inhibitory cellular neural networks with time-varying delays

Authors

Changjin Xu¹^{, *}^,xcj403@126.com, Maoxin Liao²^,maoxinliao@163.com, Yicheng Pang³^,ypanggy@outlook.com

¹Guizhou Key Laboratory of Economics System Simulation, Guizhou University of Finance and Economics, Guiyang 550004, PR China

²School of Mathematics and Physics, University of South China, Hengyang 421001, PR China

³School of Mathematics and Statistics, Guizhou University of Finance and Economics, Guiyang 550004, PR China

^* Corresponding author, Email: xcj403@126.com (C.Xu).

Corresponding Author

Changjin Xuxcj403@126.com

Received 31 October 2014, Accepted 20 June 2016, Available Online 1 September 2016.

DOI: 10.1080/18756891.2016.1237192 How to use a DOI?
Keywords: Stochastic shunting inhibitory cellular neural networks; periodic solution; p-exponential stability; delay
Abstract: In this paper, we investigate a class of stochastic shunting inhibitory cellular neural networks with time-varying delays. Applying integral inequality, some sufficient conditions on the existence and p-exponential stability of periodic solutions for stochastic shunting inhibitory cellular neural networks with time-varying delays are established. An example is presented to illustrate our main theoretical findings. Our results are new and complementary to previously known studies.
Copyright: © 2016. the authors. Co-published by Atlantis Press and Taylor & Francis
Open Access: This is an open access article under the CC BY-NC license (http://creativecommons.org/licences/by-nc/4.0/).

1. Introduction

Since shunting inhibitory cellular neural networks have been successfully applied to pattern recognition, image and signal processing, vision, and optimization, their dynamics has attracted many attentions. Numerous important results on the existence and uniqueness of equilibrium point, periodic solution, almost periodic solution, pseudo almost periodic solution, almost automorphic solution and antiperiodic solution have been reported. For example, Gao et al.¹ studied the existence and stability almost periodic solutions for cellular neural networks with time-varying delays in leakage terms on time scales, Liu and Shao² analyzed the almost periodic solutions for SICNNs with time-varying delays in the leakage terms, Liu³ focused on the pseudo almost periodic solutions for neutral type CNNs with continuously distributed leakage delays, Armmaret al.⁴ made a detailed discussion on the existence and uniqueness of pseudo almost periodic solutions of recurrent neural networks with time-varying coefficients and mixed delays, Abbas and Xia⁵ investigated the almost automorphic solutions of impulsive cellular neural networks with piecewise constant argument, Li and Shu⁶ dealt with the antiperiodic solutions to impulsive shunting inhibitory cellular neural networks with distributed delays on time scales. For details, we refer readers to papers^{7,8,9,10,11,12,13,14}.

In 1994, Haykin¹⁵ pointed out that in real nervous systems, synaptic transmission is a noisy process brought on by random fluctuations from the release of neurotransmitters and other probabilistic causes. Neural networks could be stabilized or destabilized by some stochastic inputs¹⁶. Considering that neural networks are inevitably affected by the random fluctuations from the release of neurotransmitters and other probabilistic causes which is an important component in neural networks, we think that it is worth while to investigate the stochastic neural networks. Recently, there are many papers that deal with this aspect^{17,18,19,20,21}. In this paper, we will consider the following stochastic shunting inhibitory cellular neural networks with timevarying delays

(1)dxij(t)=[−aij(t)xij(t)+Lij(t)−∑Bkl∈Nr(i,j)Bijkl(t)fij(t,xkl(t))xij(t)−∑Ckl∈Nr(i,j)Cijkl(t)gij(t,xkl(t−τkl(t)))xij(t)]dt+∑Dkl∈Nr(i,j)Dijkl(t)σij(xij(t))dwij(t),t≥t0,

where i = 1,2, ⋯, m, j = 1,2, ⋯, n, τ_ij(t) > 0 denotes axonal signal transmission delay at time t, C_ij(t) denotes the cell at the (i, j) position of the lattice at time t, the r-neighborhood N_r(i, j) of C_ij(t) is given as Nr(i,j)={Cijkl:max(|k−i|,|l−j|≤r),1≤k≤m,1≤l≤n} , x_ij(t) stands for the activity of the cell C_ij(t), L_ij(t) denotes the external input to C_ij(t), a_ij(t) > 0 stands for the passive decay rate of the cell activity, Bijkl(t)≥0 and Cijkl(t)≥0 represent the connection or coupling strength of postsynaptic of activity of the cell transmitted to the cell C_ij(t) at time t and t − τ_kl(t), respectively, the activity functions f_ij(t, x_kl(t)) and g_ij(t, x_kl(t)) are continuous functions representing the output or firing rate of the cell C_kl(t) at time t and t −τ_kl(t), respectively, w(t) = (w₁₁(t),w₁₂(t), ⋯, w_mn(t))^T is m × n-dimensional Brownian motions defined on a complete probability space, σ_ij ∈ C(R,R) is a Borel measurable function and σ = (σ_ij)_mn×mn is a diffusion coefficient matrix.

Let Rn(R+n) be the space of n-dimensional (non-negative) real column vectors and R^mn be the space of m × n-dimensional real column vectors. We denote (Ω, F, {F_t}_t≥0, P) by a complete probability space with a filtration {F_t}_t≥0, where F is a σ-algebra on a given set Ω, P is the probability measure and the filtration Ftt. satisfies the usual conditions, that is, {F_t}_t≥0 is right continuous and F₀ contains all P-null sets. Denote by BCF0b(R,Rmn) the family of bounded F₀-measurable, R^mn valued random variables x(t), that is, the value of x(t) is an m × n-dimensional real vector and can be decided from the values of w(s) for s ≤ 0. Then BCF0b(R,Rmn) is a Banach space with the norm ||x|| =sup0≤t≤ω(E|x(t)|1p)1p, where p > 1 is an integer, |x(t)|₁ = max_(i,j)|x_ij(t)|, and E(.) stands for the correspondent expectation operator with respect to the given probability measure P. For convenience, for an ω-periodic continuous function f : R → R, denote f = max_0≤t≤ω | f (t)|, f = min_0≤t≤ω | f (t)|, for any ϕ∈BCF0b([−τ,0],Rmn), denote [ϕ(t)]τ+=(|ϕ11|τ,|ϕ12|τ,⋯,|ϕmn|τ)T, where |φ_ij|_τ = sup_{−τ≤s≤0} |φ_ij(t + s)|, i = 1,2, ⋯, m, j = 1,2, ⋯, n.

The initial value of system (1) takes the form

(2)xij(s)=φij(s),s∈[t0−τ,t0],

where φij(s)∈BCF0b([t0−τ,0],R),τ=max1≤k≤m,1≤l≤n{τkl},t0∈R.

Throughout this paper, we make the assumption as follows.

(H1) For i = 1,2, ⋯, m, j = 1,2, ⋯, n, a_ij(t), Bijkl(t), Cijkl(t), Dijkl(t) and L_ij(t) are all ω-periodic continuous functions for all t ∈ R.
(H2) For i = 1,2, ⋯, m, j = 1,2, ⋯, n, there exist positive constants L_ijf, g_ijg, σ_ijσ, M_f and M_g such that
|fij(t,u)−fij(t,v)| ≤Lijf|u−v|,|fij(u)| ≤Mf,|gij(t,u)−gij(t,v)| ≤Lijg|u−v|,|gij(u)| ≤Mg,|σij(t,u)−σij(t,v)| ≤Lijσ|u−v|
for all u, v, t ∈ R.

The remainder of the paper is organized as follows: in Section 2, several definitions and some preliminary results which are useful in later section are introduced. some sufficient conditions for the the existence of periodic solutions of system (1) are derived in Section 3. In Section 4, the p-exponential stability of periodic solutions are analyzed. An examples are given to illustrate the feasibility and effectiveness of our results obtained in previous section in Section 5. A brief conclusion is drawn in Section 6.

2. Preliminaries

In this section, we shall recall several definitions and present some preliminary results which are necessary in later sections.

Definition 2.1 ²²

A stochastic process x(t) 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), x(t₂ + kω), ⋯, x(t_m + kω)) are independent of k, k = ±1, ±2, ⋯.

Lemma 2.1 ²³

If x(t) is an ω-periodic stochastic process, then its mathematical expectation and variance are ω-periodic.

Definition 2.2

A function x(t) = (x₁₁(t), x₁₂(t), ⋯, x_mn(t))^T defined on [t₀ − τ, ∞) is said to be a solution of (1) with initial condition (2) if the following conditions holds.

(i)
x_ij(t) is absolutely continuous on [t₀ − τ, ∞), i = 1,2, ⋯, m, j = 1,2, ⋯, n,
(ii)
x_ij(t) satisfies (1) for almost everywhere t ∈ [t₀, ∞), i = 1,2, ⋯, m, j = 1,2, ⋯, n,
(iii)
x_ij(s) = φ_ij(s), s ∈ [t₀−τ, t₀], i=1,2, ⋯, m, j = 1,2, ⋯, n.

Throughout this paper, we assume that (1) with initial condition (2) has a unique solution. Denote the solution of (1) by x(t) = x(t, t₀, φ) for all φ∈BCF0b([t0−τ,t0],Rmn) and t₀ ∈ R.

Definition 2.3 ²²

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∈BCF0b([−τ,0],Rn), there exists T(t₀, φ) such that E(||x(t, t₀, φ)||^p) ≤ N, t ≥ t₀ + T(t₀, φ).

Lemma 2.2 ²⁴

In addition to (H1) and (H2), suppose that the solution of (1) is p-uniformly bounded and p-point dissipative for p > 2, then (1) has an ω-periodic solution.

Lemma 2.3 ²⁵

For any x∈R+n and p > 0,

|x|p≤n(p2−1)∨0∑i=1nxip,(∑i=1nxi)p≤n(p−1)∨0∑i=1nxip.

Definition 2.4 ²²

The periodic solution x(t, t₀, φ) with initial value φ∈BCF0b([−τ,0],Rn) of (1) is said to be p-exponentially stable, if there are constants λ > 0 and M > 0 such that for any solution y(t, t₀, φ₁) with initial value φ1∈BCF0b([−τ,0],Rn) of (1) satisfies

E(|x−y|1p)≤M|varphi −φ1||pe−λ(t−t0),t≥t0.

Lemma 2.4 ²²

Let u(t)∈C(R,R+n) be a solution of the delay integral inequality

(3){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, M1∈R+n×n, J₁ ⩾ 0 is a constant vector, φ(t)∈C([t−τ,t0],R+n). If ρ(Π) < 1, where Π=C1−1(A1+B1), then there are constants 0 < λ ⩽ δ and N ⩾ 1 such that

u(t)≤Nze−λ(t−t0)+(I−Π)−1J1,t≥t0,

where z satisfies [φ]τ+≤z.

Lemma 2.5 ²²

Assume that all conditions of Lemma 2.4 hold. If J₁ = 0, then all solutions of inequality of (1) exponentially convergent to zero.

In view of Lemma 2.4 and Lemma 2.5, we have the following results.

Lemma 2.6 ²⁶

Let u(t)∈C(R,R+n) be a solution of the delay integral inequality

(4){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, M1∈R+n×n, J₁ ⩾ 0 is a constant vector, φ(t)∈C([t−τ,t0],R+n). If A1+B1C1<1, then there are constants 0 < λ ⩽ δ and N ⩾ 1 such that

u(t)≤Nze−λ(t−t0)+(1−A1+B1C1)−1J1,t≥t0,

where z satisfies [φ]τ+≤z. Moreover, if J₁ = 0, then all solutions of the inequality of (4) are exponentially convergent to zero.

3. Existence of Periodic Solution

In this section, we discuss the existence of periodic solution of (1).

Theorem 3.1

In addition to (H1) and (H2), assume further that

(H3) there exists an integer p > 2 such that ρδ₋₁ < 1, where δ = min_(i,j) {a_ij}, σ=(p(p−1)2)p2,

ρ=max(i,j){5p−1(a_ij)1−p[(Mf∑Bkl∈Nr(i,j)B¯ijkl)p+(Mg∑Ckl∈Nr(i,j)C¯ijkl)p]+σ(mn)p2(2a_ij(p−1)p−2)1−p2×(∑Dkl∈Nr(i,j)D¯ijkl)p,

then (1.1) has an ω-periodic solution.

Proof

It follows from the method of variation parameter and (1) that for t ⩾ t₀, i = 1,2, ⋯, m, j = 1,2, ⋯, n,

(5)xij(t)=xij(t0)e−∫t0taij(ξ)dξ−∫t0te−∫t0taij(ξ)dξ[∑Bkl∈Nr(i,j)Bijkl(s)fij(s,xkl(s))xij(s)+∑Ckl∈Nr(i,j)Cijkl(s)gij(s,xkl(s−τkl(s)))xij(s)−Lij(s)]ds+∫t0te−∫t0taij(ξ)dξ∑Dkl∈Nr(i,j)Dijkl(s)σij(xij(s))dwij(s).

Let

Θij(1)=xij(t0)e−∫0taij(ξ)dξ,Θij(2)=∫t0te−∫t0taij(ξ)dξ∑Bkl∈Nr(i,j)Bijkl(s)fij(s,xkl(s))xij(s)ds,Θij(3)=∫t0te−∫0taij(ξ)dξ∑Ckl∈Nr(i,j)Cijkl(s)×gij(s,xkl(s−τkl(s)))xij(s)ds,Θij(4)=∫t0te−∫t0taij(ξ)dξLij(s)ds,Θij(5)=∫t0te−∫t0taij(ξ)dξ∑Dkl∈Nr(i,j)Dijkl(s)σij(xij(s))dwij(s).

Taking expectations and applying Lemma 2.3, we get

(6)E|xij(t)|p≤E(|Θij(1)|p+|Θij(2)|p+|Θij(3)|p+|Θij(4)|p+|Θij(5)|p),

where i = 1,2, ⋯, m, j = 1,2, ⋯, n. Next, we will evaluate every term of (6). For the first term of (6), we have

(7)E|Θij(1)|p=xij(t0)e−∫t0taij(ξ)dξ≤E|xij(t0)e−a_ij(t−t0)|p≤e−pa_ij(t−t0)E|xij(t0)|p.

For the second term of (6), we have

(8)E|Θij(2)|p=E|∫t0te−∫t0taij(ξ)dξ∑Bkl∈Nr(i,j)Bijkl(s)fij(s,xkl(s))xij(s)ds|p≤E(∫t0te−a_ij(t−s)∑Bkl∈Nr(i,j)B¯ijkl(s)|fij(s,xkl(s))|×|xij(s)|ds)p≤E(∫t0te−a_ij(t−s)∑Bkl∈Nr(i,j)B¯ijkl(s)Mf|xij(s)|ds)p=E(∫t0t(e−a_ij(t−s))p−1p(e−a_ij(t−s))1p×∑Bkl∈Nr(i,j)B¯ijkl(s)Mf|xij(s)|ds)p≤E((∫t0te−a_ij(t−s)ds)p−1∫t0te−a_ij(t−s)×(∑Bkl∈Nr(i,j)B¯ijkl(s)Mf|xij(s)|)pds)≤(a_ij)1−p∫t0te−a_ij(t−s)(Mf∑Bkl∈Nr(i,j)B¯ijkl)p×E||xij(s)|pds.

For the third term of (6), we have

(9)E|Θij(3)|p=E|∫t0te−∫t0taij(ξ)dξ∑Ckl∈Nr(i,j)Cijkl(s)gij(s,xkl(s−τkl(s)))xij(s)ds|p=E(∫t0te−∫0taij(ξ)dξ∑Ckl∈Nr(i,j)|Cijkl(s)||gij(s,xkl(s−τkl(s)))||xij(s)|ds|p)≤E(∫t0te−∫t0taij(ξ)dξ∑Ckl∈Nr(i,j)C¯ijklMg|xij(s)|ds)p≤E((∫t0te−aij(t−s)ds)p−1∫t0te−aij(t−s)×(∑Ckl∈Nr(i,j)C¯ijklMg|xij(s)|)pds)≤(a_ij)1−p∫t0te−aij(t−s)(∑Ckl∈Nr(i,j)C¯ijklMg)p×E|xij(s)|pds.

For the fourth term of (6), we get

(10)E|Θij(4)|p=E|∫t0te−∫0taij(ξ)dξLij(s)ds|p≤E|∫t0te−aij(t−s)Lij(s)ds|p≤(L¯ija_ij)p.

For the fifth term of (6), we get

It follows from (8)–(11) that

(12)E|xij(t)|p≤5p−1{e−pa_ij(t−t0)E|xij(t0)|p+(L¯ija_ij)p+(a_ij)1−p∫t0te−a_ij(t−s)(Mf∑Bkl∈Nr(i,j)B¯ijkl)pE||xij(s)|pds+(a_ij)1−p∫t0te−aij(t−s)(Mg∑Ckl∈Nr(i,j)C¯ijkl)p×E|xij(s)|pds+σ(mn)p2(2a_ij(p−1)p−2)1−p2×(∫t0te−a_ij(t−s)(∑Dkl∈Nr(i,j)D¯ijkl)p×E|xij(s)|pds)}.

Define

(13)V(t)=(v11(t),V12(t),⋯,Vmn(t))T,

where V_ij(t) = E|x_ij(t)|_p, i = 1,2, ⋯, m, j = 1,2, ⋯, n. It follows from (12) that

(14)Vij(t)≤5p−1e−δ(t−t0)Vij(t0)+∫t0te−δ(t−s)Vij(s)ds+ι,

where l=max(i,j){(L¯ija_ij)p}. In view of (H3) and Lemma 2.4, we know that the solutions of (1) are p-uniformly bounded and the family of all solutions of (1) is p-point dissipative. By Lemma 2.2, we can conclude that (1) has an ω-periodic solution. The proof of Theorem 3.1 is completed.

4. p-exponential Stability of Periodic Solution

In this section, we will consider the p-exponential stability of periodic solutions of (1).

Theorem 4.1

In addition to (H1)-(H2), assume further that

(H4) there exists an integer p > 2 such that (ρ₁ + ρ₂)δ⁻¹ < 1, where

ρ1=max(i,j){6p−1((a_ij)1−p[(Mf∑Bkl∈Nr(i,j)B¯ijkl)p+(NLijf∑Bkl∈Nr(i,j)B¯ijkl)p]+σ(mn)p2(2a_ij(p−1)p−2)1−p2×(∑Dkl∈Nr(i,j)D¯ijkl)p)}

and

ρ2=max(i,j){6p−1(a_ij)1−p[(Mg∑Ckl∈Nr(i,j)C¯ijkl)p+(NLijg∑Ckl∈Nr(i,j)C¯ijkl)p]},

then the periodic solution of (1) is p-exponentially stable.

Proof

Obviously, if (H4) holds, then (H3) is fulfilled. In view of Theorem 3.1, we know that (1) has an ω-periodic solution x*(t)={xij*(t)} with the initial condition φ(t) = {φ_ij(t)}, i = 1,2, ·,m, j = 1,2, ⋯, n. Thus x^*(t) is p-uniform, namely, there is a constant C₀ > 0 such that E|xij*(t)|p<C0, i = 1,2, ·,m, j = 1,2, ⋯, n. Assume that x(t) = {x_ij(t) is an arbitrary solution of (1) with the initial condition ψ(t) = {ψ_ij(t)}, i = 1,2, ·,m, j = 1,2, ⋯, n. Let

(15)u(t)={uij(t)}={xij(t)−xij*(t)},

where i = 1,2, ·,m, j = 1,2, ⋯, n. Then for i = 1,2, ·,m, j = 1,2, ⋯, n and t ⩾ t₀, we get

(16)duij(t)=[−aij(t)uij(t)−∑Bkl∈Nr(i,j)Bijkl(t)(fij(t,xkl(t))xij(t)−fij(t,xkl*(t))xij*(t))−∑Ckl∈Nr(i,j)Cijkl(t)(gij(t,xkl(t−τkl(t)))xij(t)−gij(t,xkl*(t−τkl(t)))xij*(t))]dt+∑Dkl∈Nr(i,j)Dijkl(t)(σij(xij(t))−σij(xij*(t)))dwij(t)

with this initial condition

ϕij(s)=ψij(s)−φij(s),s∈[−τ,t0],

where i = 1,2, ·, m, j = 1,2, ⋯, n. Applying the method of variation parameter, we have

(17)uij(t)=uij(t0)e−∫0taij(ξ)dξ−∫t0te−∫t0taij(ξ)dξ×[∑Bkl∈Nr(i,j)Bijkl(s)(fij(s,xkl(s))xij(s)−fij(s,xkl*(s))xij*(s))+∑Ckl∈Nr(i,j)Cijkl(s)(gij(s,xkl(s−τkl(s)))xij(s)−gij(s,xkl*(s−τkl(s)))xij*(s)]ds+∫t0te−∫t0taij(ξ)dξ∑Dkl∈Nr(i,j)Dijkl(s)(σij(xij(s))−σij(xij*(s)))dwij(s)=uij(t0)e−∫0taij(ξ)dξ−∫t0te−∫0taij(ξ)dξ×[∑Bkl∈Nr(i,j)Bijkl(s)(fij(s,xkl(s))uij(s)(fij(s,xkl(s))−fij(s,xkl*(s))xij*(s))+∑Ckl∈Nr(i,j)Cijkl(s)(gij(s,xkl(s−τkl(s)))uij(s)+(gij(s,xkl(s−τkl(s)))−gij(s,xkl*(s−τkl(s))))xij*(s))]ds+∫t0te−∫t0taij(ξ)dξ∑Dkl∈Nr(i,j)Dijkl(s)(σij(xij(s))−σij(xij*(s)))dwij(s),

where i = 1,2, ·, m, j = 1,2, ⋯, n and t ⩾ t₀. Let

Φij(1)=uij(t0)e−∫0taij(ξ)dξ,Φij(2)=∫t0te−∫t0taij(ξ)dξ×∑Bkl∈Nr(i,j)Bijkl(s)fij(s,xkl(s))uij(s)ds,Φij(3)=∫t0te−∫t0taij(ξ)dξ∑Bkl∈Nr(i,j)Bijkl(s)×(fij(s,xkl(s))−fij(s,xkl*(s))xij*(s)ds,Φij(4)=∫t0te−∫t0taij(ξ)dξ×∑Ckl∈Nr(i,j)Cijkl(s)gij(s,xkl(s−τkl(s)))uij(s)ds,Φij(5)=∫t0te−∫t0taij(ξ)dξ×∑Ckl∈Nr(i,j)Cijkl(s)(gij(s,xkl(s−τkl(s))) −gij(s,xkl*(s−τkl(s))))xij*(s)ds,Φij(6)=∫t0te−∫t0taij(ξ)dξ×∑Dkl∈Nr(i,j)Dijkl(s)(σij(xij(s))−σij(xij*(s)))dwij(s).

Taking expectations and applying Lemma 2.3, we have

(18)E|uij(t)|p≤6p−1E(|Φij(1)|p+|Φij(2)|p+|Φij(3)|p+|Φij(4)|p+|Φij(5)|p+|Φij(6)|p).

where i = 1,2, ⋯, m, j = 1,2, ⋯, n. Applying the similar method in the proof of Theorem 3.1, we get

E|Φij(1)|p≤e−pa_ij(t−t0)E|uij(t0)|pE|Φij(2)|p=E|∫t0te−∫t0taij(ξ)dξ×∑Bkl∈Nr(i,j)Bijkl(s)fij(s,xkl(s))uij(s)ds|p≤(a_ij)1−p∫t0te−a_ij(t−s)(Mf∑Bkl∈Nr(i,j)B¯ijkl)p×E||uij(s)|pds,E|Φij(3)|p≤E|∫t0te−∫t0taij(ξ)dξ×∑Bkl∈Nr(i,j)Bijkl(s)(fij(s,xkl(s))−fij(s,xkl*(s))xij*(s)ds|p≤E(∫t0te−a_ij(t−s)×∑Bkl∈Nr(i,j)B¯ijklC0Lijf|ukl(s)|ds)p≤(a_ij)1−p∫t0te−a_ij(t−s)×(C0Lijf∑Bkl∈Nr(i,j)B¯ijkl)pE|ukl(s)|pds,E|Φij(4)|p≤E|∫t0te−∫0taij(ξ)dξ∑Ckl∈Nr(i,j)Cijkl(s)×gij(s,xkl(s−τkl(s)))uij(s)ds|p≤(a_ij)1−p∫t0te−aij(t−s)(Mg∑Ckl∈Nr(i,j)C¯ijkl)pE||ukl(s−τkl(s))|pds,E|Φij(5)|p≤E|∫t0te−∫t0taij(ξ)dξ∑Ckl∈Nr(i,j)Cijkl(s)(gij(s,xkl(s−τkl(s)))−gij(s,xkl*(s−τkl(s))))xij*(s)ds|p≤(a_ij)1−p∫t0te−a_ij(t−s)×(C0Lijg∑Ckl∈Nr(i,j)C¯ijkl)p×E|ukl(s−τkl(s))|pds,E|Φij(6)|p≤E|∫t0te−∫t0taij(ξ)dξ×∑Dkl∈Nr(i,j)Dijkl(s)(σij(xij(s))−σij(xij*(s)))dwij(s)|p≤σ(mn)p2(2aij(p−1)p−2)1−p2(∫t0te−a_ij(t−s)(∑Dkl∈Nr(i,j)D¯ijkl)p×E|uij(s)|pds).

Then we have

(19)E|uij|p≤6p−1{e−pa_ij(t−t0)E|uij(t0)|p+(a_ij)1−p∫t0te−a_ij(t−s)(Mf∑Bkl∈Nr(i,j)B¯ijkl)p×E||uij(s)|pds+(a_ij)1−p∫t0te−a_ij(t−s)(C0Lijf∑Bkl∈Nr(i,j)B¯ijkl)p×E|ukl(s)|pds+(a_ij)1−p∫t0te−a_ij(t−s)(Mg∑Ckl∈Nr(i,j)C¯ijkl)p×E||ukl(s−τkl(s))|pds+(a_ij)1−p∫t0te−a_ij(t−s)(C0Lijg∑Ckl∈Nr(i,j)C¯ijkl)p×E|ukl(s−τkl(s))|pds+σ(mn)p2(2a_ij(p−1)p−2)1−p2(∫t0te−a_ij(t−s)(∑Dkl∈Nr(i,j)D¯ijkl)p×E|uij(s)p|ds)}.

Define

(20)W(t)=(W11(t),W12(t),⋯,Wmn(t))T,

where W_ij(t) = E|u_ij(t)|^p, i = 1,2, ⋯, m, j = 1,2, ⋯, n. It follows from (19) that

(21)Wij(t)≤6p−1e−δ(t−t0)Wij(t0)+∫t0te−δ(t−s)ρ1Wij(s)ds∫t0te−δ(t−s)ρ2|Wij(s)|τ+ds,

where

ρ1=max(i,j){6p−1((a_ij)1−p[(Mf∑Bkl∈Nr(i,j)B¯ijkl)p+(C0Lijf∑Bkl∈Nr(i,j)B¯ijkl)p]+σ(mn)p2(2a_ij(p−1)p−2)1−p2×(∑Dkl∈Nr(i,j)D¯ijkl)p)}

and

ρ2=max(i,j){6p−1(a_ij)1−p[(Mg∑Ckl∈Nr(i,j)C¯ijkl)p+(C0Lijg∑Ckl∈Nr(i,j)C¯ijkl)p]}.

In view of (H4) and Lemma 2.5, we can conclude that the periodic solution x^*(t) of (1) is pexponentially stable. The proof of Theorem 4.1 is completed.

Remark 4.1

In Zhao and Zhang²⁷, Zhao and Zhang investigated the almost periodic solution of the following shunting inhibitory cellular neural networks with variable coefficients and time-varying delays

(22)x˙ij(t)=−aij(t)xij(t)−∑Bkl∈Nr(i,j)Bijkl(t)fij(t,xkl(t))xij(t)−∑Ckl∈Nr(i,j)Cijkl(t)gij(t,xkl(t−τkl(t)))xij(t)+Lij(t).

By applying Dini derivative, they established some criteria on the existence and local exponential stability of (22). All the results in Zhao and Zhang²⁷ can not be applicable to system (1) to obtain the the existence and p-exponential stability of periodic solutions. This implies that the results of this article are essentially new.

5. An Example with Its Numerical Simulations

Example 5.1

Let i = j = 2. Consider the following stochastic shunting inhibitory cellular neural networks with time-varying delays

(23)dxij(t)=[−aij(t)xij(t)+Lij(t)−∑Bkl∈Nr(i,j)Bijkl(t)fij(t,xkl(t))xij(t)−∑Ckl∈Nr(i,j)Cijkl(t)gij(t,xkl(t−τkl(t)))xij(t)]dt+∑Dkl∈Nr(i,j)Dijkl(t)σij(xij(t))dwij(t),

where fij(t,x)=sin15x+45x, gij(t,x)=sin13x+23x and

Then L_{ij f} = L_ijg = M_f = M_g = 1(i, j = 1,2), τ = 0.02, L₁₁_σ = L₂₂_σ = 0.1, L₁₂_σ = L₂₁_σ = 0.2 and

[∑Bkl∈N1(1,1)B¯11kl∑Bkl∈N1(1,2)B¯12kl∑Bkl∈N1(2,1)B¯21kl∑Bkl∈N1(2,2)B¯22kl]=[0.080.120.040.08],[∑Ckl∈N1(1,1)C¯11kl∑Ckl∈N1(1,2)C¯12kl∑Bkl∈N1(2,1)C¯21kl∑Ckl∈N1(2,2)C¯22kl]=[0.120.120.080.08],[a_11a_12a_21a_22]=[0.280.380.390.29],[L¯11L¯12L¯21L¯22]=[0.20.10.30.2].

Take p = 3, r = 1. Then we have δ ≈ 0.2302, σ₁ ≈ 0.0704, σ₂ ≈ 0.0462. It is easy to check that all the conditions in Theorem 3.1 and Theorem 4.2 are fulfilled. Hence we can conclude that then (23) has a 4-periodic solution, which is 3-exponentially stable. The results are shown in Figs. 1–2.

6. Conclusions

In this paper, a class of stochastic shunting inhibitory cellular neural networks with time-varying delays are considered. We establish some sufficient conditions ensuring the existence and p-exponential stability of periodic solutions for stochastic shunting inhibitory cellular neural networks with timevarying delays by using integral inequalities. Comparisons between our results and the previous results show that our results complement the earlier publications and are completely new. An example is presented to illustrate our main theoretical findings. Our results play an important key in designing of shunting inhibitory cellular neural networks. The obtained results show that under some appropriate circumstances, stochastic shunting inhibitory cellular neural networks with time-varying delays can display sustainable periodic oscillatory phenomenon. These periodic oscillatory phenomenon can help us to process visual information quickly and effectively^28,29. Also periodic oscillatory phenomenon can be helpful for us to predict pathological brain states, which is important to diagnose disease in medical science^30,31,32.

Acknowledgments

This work is supported by National Natural Science Foundation of China (No.61673008, No.11261010 and No.11526063), Natural Science and Technology Foundation of Guizhou Province(J[2015]2025 and J[2015]2026), 125 Special Major Science and Technology of Department of Education of Guizhou Province ([2012]011) and Natural Science Foundation of the Education Department of Guizhou Province(KY[2015]482).

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Journal: International Journal of Computational Intelligence Systems
Volume-Issue: 9 - 5
Pages: 945 - 956
Publication Date: 2016/09/01
ISSN (Online): 1875-6883
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TY  - JOUR
AU  - Changjin Xu
AU  - Maoxin Liao
AU  - Yicheng Pang
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DA  - 2016/09/01
TI  - Existence and p-exponential stability of periodic solution for stochastic shunting inhibitory cellular neural networks with time-varying delays
JO  - International Journal of Computational Intelligence Systems
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EP  - 956
VL  - 9
IS  - 5
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International Journal of Computational Intelligence Systems

Existence and p-exponential stability of periodic solution for stochastic shunting inhibitory cellular neural networks with time-varying delays

1. Introduction

2. Preliminaries

Definition 2.1 22

Lemma 2.1 23

Definition 2.2

Definition 2.3 22

Lemma 2.2 24

Lemma 2.3 25

Definition 2.4 22

Lemma 2.4 22

Lemma 2.5 22

Lemma 2.6 26

3. Existence of Periodic Solution

Theorem 3.1

Proof

4. p-exponential Stability of Periodic Solution

Theorem 4.1

Proof

Remark 4.1

5. An Example with Its Numerical Simulations

Example 5.1

6. Conclusions

Acknowledgments

References

Cite this article

Definition 2.1 ²²

Lemma 2.1 ²³

Definition 2.3 ²²

Lemma 2.2 ²⁴

Lemma 2.3 ²⁵

Definition 2.4 ²²

Lemma 2.4 ²²

Lemma 2.5 ²²

Lemma 2.6 ²⁶