Keywords: shunting inhibitory cellular neural network, neutral delay, almost periodic solution, fixed point theorem

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Electronic Journal of Qualitative Theory of Differential Equations 2011, No. 30, 1-12;http://www.math.u-szeged.hu/ejqtde/

Existence of almost periodic solution for SICNN with a neutral delay

Zheng Fang¹ Yongqing Yang²

School of Science, Jiangnan University, Wuxi, 214122, PR China Abstract

In this paper, a kind of shunting inhibitory cellular neural network with a neutral delay was considered. By using the Banach fixed point theorem, we established a result about the existence and uniqueness of the almost periodic solution for the shunting inhibitory cellular neural network.

Keywords: shunting inhibitory cellular neural network, neutral delay, almost periodic solution, fixed point theorem.

1. Introduction

Shunting inhibitory cellular neural network (SICNN) is a kind of very important model and has been investigated by many authors (see [1, 2, 3, 4] and the reference therein) due to its wide applications in practical fields such as robotics, adaptive pattern recognition and image processing. In [1], Ding studied the following SICNN

x^′_ij =−aijxij − X

Ckl∈Nr(i,j)

C_ij^klf[xkl(t−τ(t))]xij(t) +Lij(t),

Most of the existing SICNN models are concerns with the delays in the state.

However, it is not enough for it can not describe the phenomenon precisely. It is natural and useful to consider the model with a neutral delay, it means that the system describing the model depends on not only the past information of the state but also the information of the derivative of the state, i.e., the decay rate of the cells. This kind of model is described by a differential equation with a neutral delay. The neutral type differential equations have many applications, for more details we refer to [6]. Some authors have considered the Hopfield neural networks with neutral delays, see [7, 8].

To the best of our knowledge, there is few consideration about the shunting inhibitory cellular neural network with a neutral delay. In this paper, we consider the following

1e-mail: fangzhengjd@126.com, corresponding author

2e-mail: yyq640613@gmail.com

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shunting inhibitory cellular neural network with a neutral delay:

x^′_ij =−aij(t)xij − X

C_kl∈Nr(i,j)

C_ij^kl(t)f[xkl(t−τ(t))]xij(t)

− X

Ckl∈Ns(i,j)

D^kl_ij(t)g[x^′_kl(t−σ(t))]xij(t) +Iij(t). (1.1) In this model, Cij represents the cell at the (i, j) position of the lattice, the r−neighborhood Nr(i, j) of Cij is defined as follows

Nr(i, j) =n

Ckl : max{|k−i|,|l−j |} ≤r, 1≤k ≤m,1≤l ≤no ,

xij(t) describes the state of the cell Cij, the coefficient aij(t) >0 is the passive decay rate of the cell activity, C_ij^kl(t), D_ij^kl(t) are connection weights or coupling strength of postsynaptic activity of the cell Ckl transmitted to the cellCij, and f, gare continuous activity functions, representing the output or firing rate of the cell Ckl , andτ(t), σ(t) correspond to the transmission delays.

In the following, we give some basic knowledge about the almost periodic functions and almost periodic solutions of differential equations, please refer to [9, 10] for more details.

Definition 1.1(See [10]) Letu:R→Rⁿbe continuous in t. uis said to be almost periodic on R if, for any ǫ > 0, the set T(u, ǫ) = {δ :| u(t+δ)−u(t)|< ǫ,∀ t ∈R} is relatively dense, i.e., for ∀ǫ > 0, it is possible to find a real number l = l(ǫ) > 0, for any interval with length l(ǫ), there exists a number δ=δ(ǫ) in this interval such that

|u(t+δ)−u(t)|< ǫ, for all t∈R.

Definition 1.2 ([9, 10]) If u:R→Rⁿ is continuously differentiable in t, u(t) and u^′(t) are almost periodic on R, then u(t) is said to be a continuously differentiable almost periodic function.

LetAP(R,R^m×n) andAP¹(R,R^m×n) be the set of continuous almost periodic functions, and continuously differentiable almost periodic functions from R to R^m×n, re- spectively. For each ϕ ∈AP¹(R,R^m×n), define

kϕk0 = sup

t∈R

maxi,j {|ϕi,j |}, kϕk= max{kϕk0,kϕ^′k0}.

It is easy to check that (AP(R,R^m×n),k · k₀) and (AP¹(R,R^m×n),k · k) are all Banach spaces.

Definition 1.3 ([9, 10]) Let x ∈ Rⁿ and Q(t) be an n ×n continuous matrix defined on R. The linear system

x^′(t) =Q(t)x(t) (1.2)

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is said to admit an exponential dichotomy on R if there exist positive constants k, α, projection P and the fundamental solution matrix X(t) of (1.2) satisfying

kX(t)P X⁻¹(s)k< ke⁻^α(t⁻^s), t ≥s, kX(t)(I−P)X⁻¹(s)k< ke^−α(s−t), t≤s.

Lemma 1.1 ([9, 10]) If the linear system (1.2) admits an exponential dichotomy, then almost periodic system

x^′(t) =Q(t)x(t) +g(t) (1.3) has a unique almost periodic solutionx(t), and

x(t) = Z t

−∞

X(t)P X⁻¹(s)ds− Z +∞

t

X(t)(I −P)X(s)ds. (1.4) Lemma 1.2 ([9, 10]) Let c(t) be an almost periodic function on R and

M[ci] = lim

T→+∞

1 T

Z t+T t

ci(s)ds >0, i= 1,2,· · ·n.

Then the linear system

x^′(t) =diag(−c1(t),−c2(t),· · ·, cn(t))x(t) admits an exponential dichotomy on R.

2. Main results

Firstly, We give some assumptions.

(H1) aij(t), C_ij^kl(t), D^kl_ij(t), Iij(t), τ(t), σ(t) are all almost periodic functions, i= 1,2,· · ·, n, j = 1,2,· · ·, m;

(H2) the activity functionsf andg are Lipschtiz functions, i.e., there exist L >0, l >0 such that

|f(x)−f(y)| ≤L|x−y|, ∀x, y ∈R,

|g(x)−g(y)| ≤l|x−y|, ∀x, y ∈R; (H3) sup

t∈R

|C_ij^kl(t)| = C_ij^kl < +∞, sup

t∈R

|D^kl_ij(t)| = D_ij^kl < +∞, sup

t∈R

|aij(t)| = a⁺_ij <

+∞, aij(u)≥aij∗ >0.

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Let

ϕ₀ =n

t

Z

−∞

e⁻^R^s^t^a^ij^(u)duIij(s)dso

, kϕ₀k=R₀ θ₁ = max

i,j

n X

Ckl∈Nr(i,j)

C_ij^kl

2LR+|f(0)|

+ X

Ckl∈Ns(i,j)

D_ij^kl

2lR+|g(0)|o

; θ2 = max

i,j

n X

Ckl∈Nr(i,j)

C_ij^kl

4LR+|f(0)|

+ X

Ckl∈Ns(i,j)

D_ij^kl

4lR+|g(0)|o . where R is a constant with R ≥R₀.

Theorem 2.1 If (H¹), (H²), (H³) and the following conditions are satisfied (1) R0 ≤R <+∞;

(2) θ₁ ·max

i,j

n 1

aij∗,1 + ^a

+ ij

aij∗

o

≤ ¹₂; (3) θ =θ2·max

i,j

n 1

aij∗,1 + ^a

+ ij

aij∗

o

<1,

then Eq.(1.1) has a unique almost periodic solution.

Proof. For any given ϕ={ϕij} ∈AP¹(R,R^m×n), we consider the almost periodic solution of the following differential equation

C_kl∈Nr(i,j)

C_ij^kl(t)f(ϕkl(t−τ(t)))ϕij(t)

− X

Ckl∈Ns(i,j)

D^kl_ij(t)g(ϕ^′_kl(t−σ(t)))ϕij(t) +Iij(t). (2.1) Since ϕ_ij(t), aij(t),C_ij^kl(t), D_ij^kl(t),τ(t), σ(t) andI_ij(t) are all almost periodic functions, andM[aij]>0, according to Lemma 1 and lemma 2, we know that Eq.(2.1) has a unique almost periodic solution x^ϕ ={x^ϕ_ij}, which can be expressed as follows

x^ϕ_ij =

t

Z

−∞

e⁻^R^s^t^a^ij^(u)duh

− X

C_kl∈Nr(i,j)

C_ij^kl(s)f(ϕkl(s−τ(s)))ϕij(s)

− X

Ckl∈Ns(i,j)

D^kl_ij(s)g(ϕ^′_kl(s−σ(s)))ϕij(s) +Iij(s)i ds.

We define a nonlinear operator on AP¹(R,R^m×n) as follows T(ϕ)(t) =x^ϕ(t), ∀ϕ∈AP¹(R,R^m×n).

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It is obvious that the fixed point ofT is a solution of Eq.(1.1). In the following we will show thatT is a contract mapping, thus the Banach fixed point theorem assert thatT has a fixed point.

Let E be defined as follows

E ={ϕ∈AP¹(R,R^m^×ⁿ)| kϕ−ϕ0k ≤R}.

Firstly, we show that T(E) ⊆ E. For each ϕ ∈ E, we have kϕ − ϕ0k ≤ R, kϕk ≤ kϕ−ϕ0k+kϕ0k ≤R+R ≤2R. Thus

kT ϕ−ϕ0k0

= sup

t∈R

maxi,j

n

t

Z

−∞

e⁻^R^s^t^a^ij^(u)duh

− X

C_kl∈Nr(i,j)

− X

C_kl∈Ns(i,j)

D_ij^kl(s)g(ϕ^′_kl(s−σ(s)))ϕij(s)i ds

o

≤ sup

t∈R

maxi,j

n

t

Z

−∞

e⁻^R^s^t^a^ij^(u)du

− X

Ckl∈Nr(i,j)

− X

Ckl∈Ns(i,j)

D_ij^kl(s)g(ϕ^′_kl(s−σ(s)))ϕij(s) dso

≤ sup

t∈R

maxi,j

n

t

Z

−∞

e⁻^R^s^t^a^ij^(u)duh X

Ckl∈Nr(i,j)

|C_ij^kl(s)||f(ϕkl(s−τ(s)))||ϕij(s)|

+ X

C_kl∈Ns(i,j)

|D^kl_ij(s)||g(ϕ^′_kl(s−σ(s)))||ϕij(s)|i dso

≤ sup

t∈R

maxi,j

n

t

Z

−∞

C_kl∈Nr(i,j)

C_ij^kl

|f(ϕkl(s−τ(s)))−f(0)|+|f(0)|

+ X

Ckl∈Ns(i,j)

D_ij^kl

|g(ϕ^′_kl(s−σ(s))−g(0)|+|g(0)|)i dso

2R

≤ sup

t∈R

maxi,j

n

t

Z

−∞

h X

Ckl∈Nr(i,j)

C_ij^kl(L|ϕkl(s−τ(s))|+|f(0)|)

+ X

C_kl∈Ns(i,j)

D_ij^kl(l|ϕ^′_kl(s−σ(s))|+|g(0)|)i dso

2R

≤ sup

t∈R

maxi,j

n

t

Z

−∞

C_kl∈Nr(i,j)

C_ij^kl(2LR+|f(0)|)

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+ X

C_kl∈Ns(i,j)

D^kl_ij(2lR+|g(0)|)i dso

2R

≤ sup

t∈R

maxi,j

n

t

Z

−∞

e^−a^ij∗^(t−s)dso 2Rθ₁

≤ 2Rθ1max

i,j

n 1 aij∗

o

≤ R (2.2)

and

k(T ϕ−ϕ₀)^′k0

= sup

t∈R

maxi,j

Z t

−∞

−aij(t)e⁻^R^s^t^a^ij^(u)du· h− X

Ckl∈Nr(i,j)

C_ij^kl(s)f(ϕkl(s−τ(s)))

− X

C_kl∈Ns(i,j)

D_ij^kl(s)g(ϕ^′_kl(s−σ(s)))i

ϕij(s)ds + h

− X

Ckl∈Nr(i,j)

C_ij^kl(t)f(ϕkl(t−τ(t)))

− X

C_kl∈Ns(i,j)

D_ij^kl(t)g(ϕ^′_kl(t−σ(t)))i ϕij(t)

= sup

t∈R

maxi,j

Z t

−∞

aij(t)e⁻^R^s^t^a^ij^(u)du· h X

Ckl∈Nr(i,j)

+ X

Ckl∈Ns(i,j)

D_ij^kl(s)g(ϕ^′_kl(s−σ(s)))i

ϕij(s)ds

− h X

C_kl∈Nr(i,j)

+ X

Ckl∈Ns(i,j)

D^kl_ij(t)g(ϕ^′_kl(t−σ(t)))i ϕij(t)

≤ sup

t∈R

maxi,j

Z t

−∞

|aij(t)|e⁻^R^s^t^a^ij^(u)du· h X

Ckl∈Nr(i,j)

|C_ij^kl(s)| |f(ϕkl(s−τ(s)))|

+ X

Ckl∈Ns(i,j)

|D_ij^kl(s)| |g(ϕ^′_kl(s−σ(s)))|i

|ϕij(s)|ds

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+ h X

C_kl∈Nr(i,j)

|C_ij^kl(t)| |f(ϕkl(t−τ(t)))|

+ X

Ckl∈Ns(i,j)

|D_ij^kl(t)| |g(ϕ^′_kl(t−σ(t)))|i

|ϕij(t)|

≤ sup

t∈R

maxi,j

Z t

−∞

a⁺_ije⁻^a^ij∗^(t⁻^s)ds· h X

Ckl∈Nr(i,j)

C_ij^kl

2LR+|f(0)|

+ X

Ckl∈Ns(i,j)

D_ij^kl

2lR+|g(0)|i

+h X

C_kl∈Nr(i,j)

C_ij^kl

2LR+|f(0)|

+ X

Ckl∈Ns(i,j)

D_ij^kl

2lR+|g(0)|i

·2R

≤ 2Rθ1max

i,j

n1 + a⁺_ij aij∗

o

≤ R. (2.3)

From (2.2) and (2.3), we have kT ϕ−ϕ0k ≤R, thus T(E)⊆E. Let ϕ, ψ∈E, denote by

I1(s) = X

Ckl∈Nr(i,j)

C_ij^kl(s)h f

ϕkl(s−τ(s))

ϕij(s)−f

ψkl(s−σ(s))

ψij(s)i ,

I2(s) = X

C_kl∈Ns(i,j)

D_ij^kl(s)h g

ϕ^′_kl(s−σ(s))

ϕij(s)−g

ψ_kl^′ (s−σ(s))

ψij(s)i .

We have

|I₁(s)|

=

X

C_kl∈Nr(i,j)

C_ij^kl(s)h f

ϕkl(s−τ(s))

ϕij(s)−f

ψkl(s−σ(s))

ψij(s)i

≤ X

C_kl∈Nr(i,j)

C_ij^klh f

ϕkl(s−τ(s))

ϕij(s)−f

ϕkl(s−τ(s))

ψij(s)

+ f

ϕkl(s−τ(s))

ψij(s)−f

ψkl(s−σ(s)) ψij(s)

i

= X

Ckl∈Nr(i,j)

C_ij^klh f

ϕkl(s−τ(s)) ·

ϕij(s)−ψij(s)

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+ f

ϕkl(s−τ(s))

−f

ψkl(s−σ(s)) ·

ψij(s)

i

≤ X

Ckl∈Nr(i,j)

C_ij^klh f

ϕ_kl(s−τ(s)) ·

ϕ_ij(s)−ψ_ij(s)

+L·

ϕkl(s−τ(s))−ψkl(s−σ(s)) ·

ψij(s)

i

≤ X

C_kl∈Nr(i,j)

C_ij^klh f

ϕkl(s−τ(s))

−f(0) +

f(0)

·

ϕij(s)−ψij(s) +L·

ϕkl(s−τ(s))−ψkl(s−σ(s)) ·

ψij(s)

i

≤ X

Ckl∈Nr(i,j)

C_ij^klh

2LR+|f(0)|

·

ϕij(s)−ψij(s)

+2LR·

ϕkl(s−τ(s))−ψkl(s−σ(s)) i

≤ X

C_kl∈Nr(i,j)

C_ij^kl

4LR+|f(0)|

· kϕ−ψk Similarly,

|I₂(s)|

=

X

Ckl∈Ns(i,j)

D_ij^kl(s)h g

ϕ^′_kl(s−τ(s))

ϕij(s)−g

ψ_kl^′ (s−σ(s))

ψij(s)i

≤ X

C_kl∈Ns(i,j)

D_ij^kl

4lR+|g(0)|

· kϕ−ψk

kT ϕ−T ψk0

= sup

t∈R

maxi,j

t

Z

−∞

e⁻^R^s^t^a^ij^(u)du· h

− X

C_kl∈Nr(i,j)

C_ij^kl(s)

[f(ϕkl(s−τ(s)))ϕij(s)−f(ψkl(s−σ(s)))ψij(s)

− X

Ckl∈Ns(i,j)

D_ij^kl(s)

g(ϕ^′_kl(s−σ(s)))ϕij(s)−g(ψ_kl^′ (s−σ(s)))ψij(s)i ds

≤ sup

t∈R

maxi,j

n

t

Z

−∞

|I1(s)|+|I2(s)|

dso

≤ sup

t∈R

maxi,j

n

t

Z

−∞

e⁻^R^s^t^a^ij^(u)dudsh X

C_kl∈Nr(i,j)

C_ij^kl

4LR+|f(0)|

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+ X

C_kl∈Ns(i,j)

D^kl_ij

4lR+|g(0)|io

· kϕ−ψk

≤ sup

t∈R

maxi,j

n

t

Z

−∞

e^−a^ij∗^(t−s)dso

·θ₂ · kϕ−ψk

≤ θ2·max

i,j { 1 aij∗

} · kϕ−ψk (2.4)

And

k(T ϕ−T ψ)^′k0

= sup

t∈R

maxi,j

Z t

−∞

−aij(t)e⁻^R^s^t^a^ij^(u)du· nh− X

C_kl∈Nr(i,j)

− X

Ckl∈Ns(i,j)

D_ij^kl(s)g(ϕ^′_kl(s−σ(s)))i ϕij(s)

+ h X

Ckl∈Nr(i,j)

C_ij^kl(s)f(ψkl(s−τ(s)))

+ X

Ckl∈Ns(i,j)

D_ij^kl(s)g(ψ^′_kl(s−σ(s)))i

ψij(s)o ds

+nh

− X

C_kl∈Nr(i,j)

− X

C_kl∈Ns(i,j)

D_ij^kl(t)g(ϕ^′_kl(t−σ(t)))i ϕij(t)

+h X

Ckl∈Nr(i,j)

C_ij^kl(t)f(ψkl(t−τ(t)))

+ X

C_kl∈Ns(i,j)

D_ij^kl(t)g(ψ_kl^′ (t−σ(t)))i

ψij(t)o

≤ sup

t∈R

maxi,j

nZ t

−∞

|aij(t)|e⁻^R^s^t^a^ij^(u)du

|I1(s)|+|I2(s)|

ds +

|I1(t)|+|I2(t)|o

≤ sup

t∈R

maxi,j

nZ t

−∞

|aij(t)|e⁻^R^s^t^a^ij^(u)duds+ 1o

·θ₂· kϕ−ψk

≤ sup

t∈R

maxi,j

nZ t

−∞

a⁺_ije^−a^ij∗^(t−s)ds+ 1o

·θ2· kϕ−ψk

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≤ θ₂ ·max

i,j

n

1 + a⁺_ij a_ij∗

o

· kϕ−ψk (2.5)

From (2.4) and (2.5), we have kT ϕ−T ψk ≤θ2 ·max

i,j

n 1

aij∗, 1 + ^a

+ ij

aij∗

o· kϕ−ψk= θ· kϕ−ψk. According to the condition of this theorem we know θ < 1, therefore T has a unique fixed point.

Example We consider the following SICCN with neutral a delay:

C_kl∈Nr(ij)

C_ij^kl(t)f[xkl(t−τ(t))]xij(t)

+ X

Ckl∈Ns(i,j)

D_ij^kl(t)g[x^′_kl(t−σ(t))]xij(t) +Iij(t), (2.6)

where i= 1,2,3, j= 1,2,3, τ(t) = cos²t, σ(t) = sin 2t,f(x) = ⁴₅sinx, g(x) = ³₄|x|,







a11(t) a12(t) a13(t) a21(t) a22(t) a23(t) a₃₁(t) a₃₂(t) a₃₃(t)





=







5 +|sint| 5 +|sin 2t| 9 +|sint|

6 +|cost| 6 +|sint| 7 +|cost|

8 +|cost| 8 +|sint| 5 +|sin 2t|













c₁₁(t) c₁₂(t) c₁₃(t) c₂₁(t) c₂₂(t) c₂₃(t) c₃₁(t) c₃₂(t) c₃₃(t)





=







0.004|sin 3t| 0.002|sin 3t| 0.001|sin 3t|

0.002|sin 3t| 0.001|sin 3t| 0.001|sin 3t|

0.001|sin 3t| 0.002|sin 3t| 0.001|sin 3t|













d₁₁(t) d₁₂(t) d₁₃(t) d₂₁(t) d₂₂(t) d₂₃(t) d31(t) d32(t) d33(t)





=







0.001|cos 2t| 0.001|cos 2t| 0.002|cos 2t|

0.001|cos 2t| 0.002|cos 2t| 0.003|cos 2t|

0.002|cos 2t| 0.002|cos 2t| 0.001|cos 2t|













I11(t) I12(t) I13(t) I21(t) I22(t) I23(t) I31(t) I32(t) I33(t)





=







sint sint cost sint cost cost cost cost cost







In the following, we will check that all assumptions of the theorem are satisfied. By computing,we have







a⁺₁₁ a⁺₁₂ a⁺₁₃ a⁺₂₁ a⁺₂₂ a⁺₂₃ a⁺₃₁ a⁺₃₂ a⁺₃₃





=







6 6 10 7 7 8 9 9 6





,







a11∗ a12∗ a13∗

a21∗ a22∗ a23∗

a31∗ a32∗ a33∗





=







5 5 9 6 6 7 8 8 5







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





c₁₁ c₁₂ c₁₃ c₂₁ c₂₂ c₂₃ c₃₁ c₃₂ c₃₃





=







0.004 0.002 0.001 0.002 0.001 0.001 0.001 0.002 0.001





,







d₁₁ d₁₂ d₁₃ d₂₁ d₂₂ d₂₃ d₃₁ d₃₂ d₃₃





=







0.001 0.001 0.002 0.001 0.002 0.003 0.002 0.002 0.001





.

Note that f and g are Lipschtz functions with f(0) = g(0) = 0, the Lipschtz constants of f and g, L, l, are less than 1, we take L=l= 1.

X

ckl∈N1(1,1)

ckl+ X

ckl∈N1(1,1)

dkl = 0.014, P

ckl∈N1(1,2)

ckl+ P

ckl∈N1(1,2)

dkl= 0.021, X

c_kl∈N1(1,3)

ckl+ X

c_kl∈N1(1,3)

dkl = 0.013, P

ckl∈N1(2,1)

ckl+ P

ckl∈N1(2,1)

dkl= 0.021, X

ckl∈N1(2,2)

ckl+ X

ckl∈N1(2,2)

dkl = 0.030, P

ckl∈N1(2,3)

ckl+ P

ckl∈N1(2,3)

dkl= 0.019, X

c_kl∈N1(3,1)

ckl+ X

c_kl∈N1(3,1)

dkl = 0.013, P

ckl∈N1(3,2)

ckl+ P

ckl∈N1(3,2)

dkl= 0.019, X

ckl∈N1(3,3)

ckl+ X

ckl∈N1(3,3)

dkl = 0.013,

From computing we know kϕ0k ≤ ¹¹₅, we choose R = 3. Obviously max

i,j

n 1 aij∗,1 +

a⁺_ij aij∗

o = ¹¹₅ . θ₁ = 0.18, θ₂ = 0.36, θ₁ ·max

i,j {_a¹

ij∗,1 + ^a

+ ij

aij∗} = 0.396 ≤ ¹₂, θ = θ₂ · maxi,j

n 1

aij∗,1 + ^a

+ ij

aij∗

o = 0.792 < 1. So all the conditions of theorem 2.1 are satisfied, hence by the theorem 2.1, Eq.(1.1) has a unique almost periodic solution.

Acknowledgements

The authors would like to thank the referees for their valuable suggestions and comments.

The first author was supported by Fund for Young Teachers of Jiangnan University (2008LQN010) and Program for Innovative Research Team of Jiangnan University.

The second author was supported by the National Natural Science Foundation of China under Grant 60875036, the Key Research Foundation of Science and Technology of the Ministry of Education of China under Grant 108067, and Program for Innovative Research Team of Jiangnan University.

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References

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(Received July 1, 2010)