Existence and exponential stability of almost periodic solutions for cellular neural networks with continuously distributed delays

EXISTENCE AND EXPONENTIAL STABILITY OF ALMOST PERIODIC SOLUTIONS FOR

CELLULAR NEURAL NETWORKS WITH CONTINUOUSLY DISTRIBUTED DELAYS

Bingwen Liu and Lihong Huang

Abstract. In this paper cellular neural networks with continu- ously distributed delays are considered. Sufficient conditions for the existence and exponential stability of the almost periodic solutions are established by using fixed point theorem, Lyapunov functional method and differential inequality technique. The results of this paper are new and they complement previously known results.

1. Introduction

Consider the following models for cellular neural networks(CNNs) with continuously distributed delays

(1.1)

x ⁰ _i (t) = − c _i (t)x _i (t) + X n j=1

a _ij (t)f _j (x _j (t))

+ X n j=1

b _ij (t) Z _∞

0

K _ij (u)f _j (x _j (t − u))du + I _i (t), in which n corresponds to the number of units in a neural network, x _i (t) corresponds to the state vector of the ith unit at the time t, c _i (t) > 0 represents 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 at the time t. a _ij (t), b _ij (t) are the connection weights

Received April 13, 2005.

2000 Mathematics Subject Classification: 34C25, 34K13, 92B20.

Key words and phrases: cellular neural networks, almost periodic solution, expo- nential stability, continuously distributed delays.

This work was supported by the NNSF (10371034) of China and the Project and

supported by Hunan Provincial Natural Science Foundation of China (05JJ40009).

at the time t, and I _i (t) denote the external inputs at time t. f _j (j = 1, 2, . . . , n) are signal transmission functions.

Throughout this paper, it will be assumed that c _i , I _i , a _ij , b _ij : R → R are almost periodic functions, where i, j = 1, 2, . . . , n. We suppose that

˜

c _i , a _ij , b _ij and I _i are constants such that

(1.2)

0 < ˜ c _i = inf

t∈R |c _i (t)|, sup

t∈R

|a _ij (t)| = a _ij , sup

t∈R

|b _ij (t)| = b _ij , sup

t∈R

|I _i (t)| = I _i , where i, j = 1, 2, . . . , n.

We also assume that the following conditions (T ₀ ), (T ₁ ) and (T ₂ ) hold.

(T ₀ ) For j ∈ {1, 2, . . . , n}, f _j : R → R are Lipschitz continuous with Lipschitz constants L _j , that is,

|f _j (u _j ) − f _j (v _j )| ≤ L _j |u _j − v _j |, ∀ u _j , v _j ∈ R.

(T ₁ ) For i, j ∈ {1, 2, . . . , n}, the delay kernels K _ij : [0, ∞) → R are continuous, integrable and satisfy

Z _∞

0

|K _ij (s)|ds ≤ k _ij .

(T ₂ ) For i, j ∈ {1, 2, . . . , n}, there exists a constant λ ₀ > 0 such

that Z _∞

0

|K _ij (s)|e ^λ

^s ds < +∞.

The initial conditions associated with system (1.1) are of the form (1.3) x _i (s) = ϕ _i (s), s ∈ (−∞, 0], i = 1, 2, . . . , n,

where ϕ = (ϕ ₁ (t), ϕ ₂ (t), . . . , ϕ _n (t)) ^T , ϕ _i (t) : R → R, i = 1, 2, . . . , n, are almost periodic functions.

Definition 1.1. (see [7, 11]) Let u(t) : R −→ R ⁿ be continuous in t. u(t) is said to be almost periodic on R if, for any ε > 0, the set T (u, ε) = {δ : |u(t + δ) − u(t)| < ε, ∀t ∈ R} is relatively dense, i.e., for

∀ε > 0, it is possible to find a real number l = l(ε) > 0, for any interval with length l(ε), there exists a number δ = δ(ε) in this interval such that |u(t + δ) − u(t)| < ε, for ∀t ∈ R.

Definition 1.2. Let Z ^∗ (t) = (x ^∗ ₁ (t), x ^∗ ₂ (t), . . . , x ^∗ _n (t)) ^T be an almost

periodic solution of system (1.1) with initial value ϕ ^∗ = (ϕ ^∗ ₁ (t), ϕ ^∗ ₂ (t), . . . ,

ϕ ^∗ _n (t)) ^T . If there exist constants λ > 0 and M _ϕ > 1 such that for every

solution Z(t) = (x ₁ (t), x ₂ (t), . . . , x _n (t)) ^T of system (1.1) with any initial value ϕ = (ϕ ₁ (t), ϕ ₂ (t), . . . , ϕ _n (t)) ^T ,

|x _i (t) − x ^∗ _i (t)| ≤ M _ϕ kϕ − ϕ ^∗ ke ^−λt , ∀t > 0, i = 1, 2, . . . , n, where ϕ _i (t), ϕ ^∗ _i (t) : R → R, i = 1, 2, . . . , n, are almost periodic func- tions, kϕ − ϕ ^∗ k = sup

−∞≤s≤0 max

1≤i≤n |ϕ _i (s) − ϕ ^∗ _i (s)|. Then Z ^∗ (t) is said to be global exponential stable.

It is well known that the delayed cellular neural networks (DCNNs) have been successfully applied to signal and image processing, pattern recognition and optimization. Hence, they have been the object of inten- sive analysis by numerous authors in recent years. In particular, there have been extensive results on the problem of the existence and stability of periodic solutions of system (1.1) in the literature. We refer the reader to [2, 3, 6, 8, 9, 12, 16] and the references cited therein. However, there exist few results on the existence and exponential stability of the almost periodic solutions of system (1.1). We only find that Liu [15] and Chen [4] studied the existence and exponential stability of the almost periodic solutions of CNNs with continuously distributed delays. In [15], we also find that the conditions (T ₂ ) is a sufficient condition for the functions G _i (µ) and F _j (µ) of (4.2) to be continuous functions. Therefore, we can introduce the main results of [15] and [4] as follows.

Theorem A. Suppose that (T ₀ ) and (T ₁ ) hold, K _ij (s) ≥ 0 for all s ∈ [0, ∞), M [c _i ] > 0, and there exist constants λ ₀ > 0 and r < 1 such that

(1.4) Z _∞

0

K _ij (s)e ^λ

^s ds < +∞, r = max

1≤i≤n

© X ⁿ

j=1

˜

c _i ⁻¹ (a _ij + b _ij k _ij )L _j ª ,

where i, j = 1, 2, . . . , n, M [c _i ] = lim

T →+∞

T 1

R _t+T

t c _i (s)ds. Then system (1.1) has exactly one almost periodic solution. Moreover, the almost periodic solution is globally exponentially stable.

The main purpose of this paper is to give the conditions for the ex-

istence and exponentially stability of the almost periodic solutions for

system (1.1). By applying fixed point theorem and differential inequality

technique, we derive some new sufficient conditions ensuring the exis-

tence, uniqueness and exponential stability of the almost periodic solu-

tion, which impose a less restrictive constraint than those in [15] and [4],

and improve and extend some previous result. Moreover, an example is

also provided to illustrate the effectiveness of the new results.

For convenience, we introduce some notations. We will use x = (x ₁ , x ₂ , . . . , x _n ) ^T ∈ R ⁿ to denote a column vector, in which the symbol ( ^T ) denotes the transpose of a vector. For matrix D = (d _ij ) _n×n , D ^T denotes the transpose of D, and E _n denotes the identity matrix of size n. A matrix or vector D ≥ 0 means that all entries of D are greater than or equal to zero. D > 0 can be defined similarly. For matrices or vectors D and E, D ≥ E (resp. D > E) means that D − E ≥ 0 (resp.

D − E > 0).

The following lemmas and definitions will be useful to prove our main results in Section 2.

Definition 1.3. (see [7, 11]) Let x ∈ R ⁿ and Q(t) be an n × n continuous matrix defined on R. The linear system

(1.5) x ⁰ (t) = Q(t)x(t)

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 (1.5) satisfying

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

Lemma 1.1. (see [7, 11]) If the linear system (1.5) admits an expo- nential dichotomy, then almost periodic system

(1.6) x ⁰ (t) = Q(t)x + g(t)

has a unique almost periodic solution x(t), and (1.7)

x(t) = Z _t

−∞

X(t)P Z ⁻¹ (s)g(s)ds − Z _+∞

t

X(t)(I − P )X ⁻¹ (s)g(s)ds.

Lemma 1.2. (see [7, 11]) Let c _i (t) be an almost periodic function on R and

M [c _i ] = lim

T →+∞

1 T

Z _t+T

t

c _i (s)ds > 0, i = 1, 2, . . . , n.

Then the linear system

x ⁰ (t) = diag(−c ₁ (t), −c ₂ (t), . . . , −c _n (t))x(t) admits an exponential dichotomy on R.

Definition 1.4. (see [1, 14]) A real n × n matrix W = (w _ij ) _n×n

is said to be an M -matrix if w _ij ≤ 0, i, j = 1, 2, . . . , n, i 6= j, and

W ⁻¹ ≥ 0, where W ⁻¹ denotes the inverse of W .

Lemma 1.3. (see [1, 14]) Let W = (w _ij ) _n×n with w _ij ≤ 0, i, j = 1, 2, . . . , n, i 6= j. Then the following statements are equivalent.

(1) W is an M -matrix.

(2) There exists a vector η = (η ₁ , η ₂ , . . . , η _n ) > (0, 0, . . . , 0) such that ηW > 0.

(3) There exists a vector ξ = (ξ ₁ , ξ ₂ , . . . , ξ _n ) ^T > (0, 0, . . . , 0) ^T such that W ξ > 0.

Lemma 1.4. (see [1, 14]) Let A ≥ 0 be an n×n matrix and ρ(A) < 1.

Then (E _n − A) ⁻¹ ≥ 0, where ρ(A) denotes the spectral radius of A.

2. Existence of almost periodic solutions

Theorem 2.1. Suppose that (T ₀ ) and (T ₁ ) hold, and ρ(D ⁻¹ EL) < 1, where

D = diag( ˜ c ₁ , ˜ c ₂ , . . . , ˜ c _n ), ¯ A = (a _ij ) _n×n , B = (b ¯ _ij k _ij ) _n×n , L = diag(L ₁ , L ₂ , . . . , L _n ),

I = (I ₁ , I ₂ , . . . , I _n ), E = (a _ij + b _ij k _ij ) _n×n = ¯ A + ¯ B.

Then, there exists at least one almost periodic solution of system (1.1).

Proof. Let

X = {φ|φ = (φ ₁ (t), φ ₂ (t), . . . , φ _n (t)) ^T },

where φ _i : R → R is an almost periodic function, i = 1, 2, . . . , n. Then, X is a Banach space with the norm defined by kφk _X = sup

t∈R

1≤i≤n max |φ _i (t)|.

To proceed further, we need to introduce an auxiliary equation

(2.1)

x ⁰ _i (t) = − c _i (t)x _i (t) + X n j=1

a _ij (t)f _j (φ _j (t))

+ X n j=1

b _ij (t) Z _∞

0

K _ij (u)f _j (φ _j (t − u))du + I _i (t), where i = 1, 2, . . . , n, φ(t) = (φ ₁ (t), φ ₂ (t), . . . , φ _n (t)) ^T ∈ X. Notice that M [c _i ] > 0, i = 1, 2, . . . , n, it follows from Lemma 1.2 that the linear system

(2.2) x ⁰ (t) = −c _i (t)x(t), i = 1, 2, . . . , n,

admits an exponential dichotomy on R. Thus, by Lemma 1.1, we obtain that the system (2.1) has exactly one almost periodic solution:

(2.3)

x ^φ (t) = (x ^φ ₁ (t), x ^φ ₂ (t), . . . , x ^φ _n (t)) ^T

= µ Z _t

−∞

e ^{− R}

^c

^(u)du [ X n j=1

a _1j (s)f _j (φ _j (s))

+ X n j=1

b _1j (s) Z _∞

0

K _1j (u)f _j (φ _j (s − u))du

+ I ₁ (s)]ds, . . . , Z _t

−∞

e ^{− R}

^c

^(u)du [ X n j=1

a _nj (s)f _j (φ _j (s))

+ X n j=1

b _nj (s) · Z _∞

0

K _nj (u)f _j (φ _j (s − u))du + I _n (s)]ds

¶ _T .

Define a mapping Φ : X −→ X by setting Φ(φ(t)) = x ^φ (t), ∀ φ ∈ X.

Let φ, ψ ∈ X. Then, by (T ₀ ) and (T ₁ ), we have

|Φ(φ(t)) − Φ(ψ(t))|

= (|(Φ(φ(t)) − Φ(ψ(t))) ₁ |, . . . , |(Φ(φ(t)) − Φ(ψ(t))) _n |) ^T

= µ¯ ¯

¯ ¯ Z _t

−∞

e ^{− R}

^c

^(u)du ( X n j=1

a _1j (s)(f _j (φ _j (s)) − f _j (ψ _j (s)))

+ X n j=1

b _1j (s) Z _∞

0

K _1j (u) · (f _j (φ _j (s − u)) − f _j (ψ _j (s − u)))du)ds

¯ ¯

¯ ¯,

. . . ,

¯ ¯

¯ ¯ Z _t

−∞

e ^{− R}

^c

^(u)du ( X n j=1

a _nj (s)(f _j (φ _j (s)) − f _j (ψ _j (s)))

+ X n j=1

b _nj (s) Z _∞

0

K _nj (u)(f _j (φ _j (s − u)) − f _j (ψ _j (s − u)))du)ds

¯ ¯

¯ ¯

¶ _T

≤ µ Z _t

−∞

e ^{− ˜ c}

^(t−s) ( X n j=1

a _1j L _j |φ _j (s) − ψ _j (s)|

+ X n j=1

b _1j Z _∞

0

|K _1j (u)|L _j |φ _j (s − u) − ψ _j (s − u)|du)ds,

. . . , Z _t

−∞

e ^{− ˜ c}

^(t−s) µ X _n

j=1

a _nj L _j |φ _j (s) − ψ _j (s)| + X n j=1

b _nj

· Z _∞

0

|K _nj (u)|L _j |φ _j (s − u) − ψ _j (s − u)|du)ds

¶ _T

≤ µ Z _t

−∞

e ^{− ˜ c}

^(t−s) µ X _n

j=1

a _1j L _j sup

t∈R

|φ _j (t) − ψ _j (t)|

+ X n j=1

b _1j Z _∞

0

|K _1j (u)|L _j sup

t∈R

|φ _j (t) − ψ _j (t)|du)ds,

. . . , Z _t

−∞

e ^{− ˜ c}

^(t−s) µ X _n

j=1

a _nj L _j sup

t∈R |φ _j (t) − ψ _j (t)| + X n j=1

b _nj

· Z _∞

0

|K _nj (u)|L _j sup

t∈R

|φ _j (t) − ψ _j (t)|du)ds

¶ _T

≤ µ X _n

j=1

˜

c ₁ ⁻¹ (a _1j + b _1j k _1j )L _j sup

t∈R

|φ _j (t) − ψ _j (t)|,

. . . , X n j=1

˜

c ₁ ⁻¹ (a _nj + b _nj k _nj )L _j · sup

t∈R

|φ _j (t) − ψ _j (t)|

¶ _T , which implies that

(2.4) µ sup

t∈R

|(Φ(φ(t)) − Φ(ψ(t))) ₁ |, . . . , sup

t∈R

|(Φ(φ(t)) − Φ(ψ(t))) _n |

¶ _T

≤ µ X _n

j=1

˜

c ₁ ⁻¹ (a _1j + b _1j k _1j )L _j sup

t∈R

|φ _j (t) − ψ _j (t)|,

. . . , X n j=1

˜

c _n ⁻¹ (a _nj + b _nj k _nj )L _j · sup

t∈R |φ _j (t) − ψ _j (t)|

¶ _T

≤ F µ

sup

t∈R

|φ ₁ (t) − ψ ₁ (t)|, . . . , sup

t∈R

|φ _n (t) − ψ _n (t)|

¶ _T

= F µ

sup

t∈R

|(φ(t) − ψ(t)) ₁ |, . . . , sup

t∈R

|(φ(t) − ψ(t)) _n |

¶ _T

,

where F = D ⁻¹ EL. Let m be a positive integer. Then, from (2.4), we get

(2.5) µ

sup

t∈R

|(Φ ^m (φ(t)) − Φ ^m (ψ(t))) ₁ |, . . . , sup

t∈R

|(Φ ^m (φ(t)) − Φ ^m (ψ(t))) _n |

¶ _T

= µ

sup

t∈R |(Φ(Φ ^m−1 (φ(t))) − Φ(Φ ^m−1 (ψ(t)))) ₁ |, . . . , sup

t∈R |(Φ(Φ ^m−1 (φ(t)))

− Φ(Φ ^m−1 (ψ(t)))) _n |

¶ _T

≤ F µ

sup

t∈R

|(Φ ^m−1 (φ(t)) − Φ ^m−1 (ψ(t))) ₁ |,

. . . , sup

t∈R

|(Φ ^m−1 (φ(t)) − Φ ^m−1 (ψ(t))) _n |

¶ _T

.. .

≤ F ^m µ

sup

t∈R

|(φ(t) − ψ(t)) ₁ |, . . . , sup

t∈R

|(φ(t) − ψ(t)) _n |

¶ _T

= F ^m µ

sup

t∈R

|φ ₁ (t) − ψ ₁ (t)|, . . . , sup

t∈R

|φ _n (t) − ψ _n (t)|

¶ _T .

Since ρ(F ) < 1, we obtain

m−→+∞ lim F ^m = 0,

which implies that there exist a positive integer N and a positive con- stant r < 1 such that

(2.6) F ^N = (D ⁻¹ EL) ^N = (h _ij ) _n×n , and X n j=1

h _ij ≤ r, i = 1, 2, . . . , n.

In view of (2.5) and (2.6), we have

|(Φ ^N (φ(t)) − Φ ^N (ψ(t))) _i | ≤ sup

t∈R |(Φ ^N (φ(t)) − Φ ^N (ψ(t))) _i |

≤ X n j=1

h _ij sup

t∈R |φ _j (t) − ψ _j (t)|

≤ (sup

t∈R

1≤j≤n max |φ _j (t) − ψ _j (t)|) X n j=1

h _ij

≤ rkφ(t) − ψ(t)k _X , for all t ∈ R, i = 1, 2, . . . , n. It follows that (2.7) kΦ ^N (φ(t)) − Φ ^N (ψ(t))k _X = sup

t∈R

1≤i≤n max |(Φ ^N (φ(t)) − Φ ^N (ψ(t))) _i |

≤ rkφ(t) − ψ(t)k _X .

This implies that the mapping Φ ^N : X −→ X is a contraction mapping.

By the fixed point theorem of Banach space, Φ possesses a unique fixed point Z ^∗ in X such that ΦZ ^∗ = Z ^∗ . We know from (2.1) and (2.3) that Z ^∗ satisfies system (1.1), and therefore, it is an almost periodic solution of system (1.1). The proof of Theorem 2.1 is now complete.

3. Uniqueness and exponential stability

In this section, we establish some results for the uniqueness and ex- ponential stability of the almost periodic solution of (1.1).

Theorem 3.1. Let (T ₂ ) hold. Suppose that all the conditions of Theorem 2.1 are satisfied. Then system (1.1) has exactly one almost periodic solution Z ^∗ (t). Moreover, Z ^∗ (t) is globally exponentially stable.

Proof. From Theorem 2.1, system (1.1) has at least one almost pe- riodic solution Z ^∗ (t) = (x ^∗ ₁ (t), x ^∗ ₂ (t), . . . , x ^∗ _n (t)) ^T with initial value ϕ ^∗ = (ϕ ^∗ ₁ (t), ϕ ^∗ ₂ (t), . . . , ϕ ^∗ _n (t)) ^T . Let Z(t) = (x ₁ (t), x ₂ (t), . . . , x _n (t)) ^T be an ar- bitrary solution of system (1.1) with initial value ϕ = (ϕ ₁ (t), ϕ ₂ (t), . . . , ϕ _n (t)) ^T , Set y(t) = (y ₁ (t), y ₂ (t), . . . , y _n (t)) ^T = Z(t) − Z ^∗ (t). Then

(3.1)

y _i ⁰ (t) = − c _i (t)y _i (t) + X n j=1

a _ij (t)f _j ^∗ (y _j (t))

+ X n j=1

b _ij (t) Z _∞

0

K _ij (u)f _j ^∗ (y _j (t − u))du, where i = 1, 2, . . . , n,

f _j ^∗ (y _j (t)) = f _j (y _j (t) + x ^∗ _j (t)) − f _j (x ^∗ _j (t)), Z _∞

0

K _ij (u)f _j ^∗ (y _j (t − u))du = Z _∞

0

K _ij (u)(f _j (y _j (t − u) + x ^∗ _j (t − u))

− f _j (x ^∗ _j (t − u)))du.

Since ρ(F ) = ρ(D ⁻¹ EL) < 1, it follows from Lemma 1.4 that E _n − D ⁻¹ EL is an M -matrix. In view of Lemma 1.3, there exists a constant vector ¯ ξ = (¯ ξ ₁ , ¯ ξ ₂ , . . . , ¯ ξ _n ) ^T > (0, 0, . . . , 0) ^T such that

(E _n − D ⁻¹ EL)¯ ξ > (0, 0, . . . , 0) ^T . Then,

− ˜ c _i ξ ¯ _i + X n j=1

ξ ¯ _j (a _ij + b _ij k _ij )L _j < 0, i = 1, 2, . . . , n.

Therefore, we can choose a constant d > 1 such that (3.2) ξ _i = d¯ ξ _i > sup

−∞<t≤0

|y _i (t)|, i = 1, 2, . . . , n, and

(3.3)

− ˜ c _i ξ _i + X n j=1

ξ _j (a _ij + b _ij k _ij )L _j

= [− ˜ c _i ξ ¯ _i + X n j=1

ξ ¯ _j (a _ij + b _ij k _ij )L _j ]d < 0, i = 1, 2, . . . , n.

Set

(3.4) Γ _i (ω) = ωξ _i − ˜ c _i ξ _i + X n j=1

ξ _j (a _ij + b _ij Z _∞

0

|K _ij (s)|e ^ωs ds)L _j , i = 1, 2, . . . , n.

Clearly, Γ _i (ω), i = 1, 2, . . . , n, are continuous functions on [0, λ ₀ ]. Since Γ _i (0) = − ˜ c _i ξ _i +

X n j=1

ξ _j (a _ij + b _ij Z _∞

0

|K _ij (s)|ds)L _j

≤ − ˜ c _i ξ _i + X n j=1

ξ _j (a _ij + b _ij k _ij )L _j < 0, i = 1, 2, . . . , n, we can choose a positive constant λ ∈ [0, λ ₀ ] such that

(3.5) Γ _i (λ) = (λ − ˜ c _i )ξ _i + X n j=1

ξ _j (a _ij + b _ij Z _∞

0

|K _ij (s)|e ^λs ds)L _j

< 0, i = 1, 2, . . . , n.

We consider the Lyapunov functional

(3.6) V _i (t) = |y _i (t)|e ^λt , i = 1, 2, . . . , n.

Obviously, for any y _i (t) 6= 0, V _i (t) > 0. Calculating the upper right derivative of V _i (t) along the solution y(t) = (y ₁ (t), y ₂ (t), . . . , y _n (t)) ^T of system (3.1) with the initial value ¯ ϕ = ϕ − ϕ ^∗ , we have

(3.7)

D ⁺ (V _i (t)) ≤ − ˜ c _i |y _i (t)|e ^λt + X n j=1

a _ij L _j |y _j (t)|e ^λt

+ X n j=1

b _ij L _j e ^λt Z _∞

0

|K _ij (u)||y _j (t − u)|du + λ|y _i (t)|e ^λt

= [(λ − ˜ c _i )|y _i (t)| + X n j=1

a _ij L _j |y _j (t)|

+ X n j=1

b _ij L _j Z _∞

0

|K _ij (u)||y _j (t − u)|du]e ^λt ,

where i = 1, 2, . . . , n.

We claim that

(3.8) V _i (t) = |y _i (t)|e ^λt < ξ _i , t > 0, i = 1, 2, . . . , n.

Contrarily, there must exist i ∈ {1, 2, . . . , n} and t _i > 0 such that (3.9) V _i (t _i ) = ξ _i and V _j (t) < ξ _j , ∀ t ∈ (−∞, t _i ), j = 1, 2, . . . , n, which implies that

(3.10)

V _i (t _i ) − ξ _i = 0 and V _j (t) − ξ _j < 0, ∀ t ∈ (−∞, t _i ), j = 1, 2, . . . , n.

Together with (3.7) and (3.10), we obtain

(3.11)

0 ≤ D ⁺ (V _i (t _i ) − ξ _i )

= D ⁺ (V _i (t _i ))

≤ [(λ − ˜ c _i )|y _i (t _i )| + X n j=1

a _ij L _j |y _j (t _i )|

+ X n j=1

b _ij L _j Z _∞

0

|K _ij (u)||y _j (t _i − u)|du]e ^λt

= (λ − ˜ c _i )ξ _i + X n j=1

a _ij L _j |y _j (t _i )|e ^λt

+ X n j=1

b _ij L _j Z _∞

0

e ^λu |K _ij (u)||y _j (t _i − u)|e ^λ(t

^−u) du

≤ (λ − ˜ c _i )ξ _i + X n j=1

ξ _j (a _ij + b _ij Z _∞

0

|K _ij (s)|e ^λs ds)L _j . Thus,

0 ≤ (λ − ˜ c _i )ξ _i + X n j=1

ξ _j (a _ij + b _ij Z _∞

0

|K _ij (s)|e ^λs ds)L _j ,

which contradicts (3.5). Hence, (3.8) holds. It follows that (3.12) |y _i (t)| < max

1≤i≤n {ξ _i }e ^−λt , t > 0, i = 1, 2, . . . , n.

Letting k ¯ ϕk = kϕ − ϕ ^∗ k > 0, it follows from (3.12) that we can choose a constant M _ϕ > 1 such that

(3.13) max

1≤i≤n {ξ _i } ≤ M _ϕ kϕ − ϕ ^∗ k, i = 1, 2, . . . , n.

In view of (3.11) and (3.12), we get

|x _i (t) − x ^∗ _i (t)| = |y _i (t)| ≤ max

1≤i≤n {ξ _i }e ^−λt ≤ M _ϕ kϕ − ϕ ^∗ ke ^−λt , where i = 1, 2, . . . , n, t > 0. This completes the proof.

Remark 3.1. Theorem A has been obtained under the assumption that the row norm of matrix D ⁻¹ EL is less than 1. Clearly, this implies that ρ(D ⁻¹ EL) < 1. Therefore, the existing results in [15] and [4] are direct corollaries of Theorem 3.1 of this paper.

Corollary 3.1. Let (T ₀ ), (T ₁ ), and (T ₂ ) hold. Suppose that E _n − D ⁻¹ EL is an M -matrix. Then system (1.1) has at least exactly one almost periodic solution Z ^∗ (t). Moreover, Z ^∗ (t) is globally exponentially stable.

Proof. Noticing that E _n − D ⁻¹ EL is an M -matrix, it follows that there exists a vector η = (η ₁ , η ₂ , . . . , η _n ) ^T > 0 such that

(3.13) (E _n − D ⁻¹ EL)η > 0,

that is,

(3.14) −c _i η _i + X n j=1

(a _ij + b _ij )L _j η _j < 0, i = 1, 2, . . . , n.

For any matrix norm k · k and nonsingular matrix B, kAk _B = kB ⁻¹ ABk also defines a matrix norm. Let B = diag(η ₁ , η ₂ , . . . , η _n ). Then (3.14) implies that the row norm of matrix B ⁻¹ (D ⁻¹ EL)B is less than 1.

Therefore, ρ(D ⁻¹ EL) < 1. Corollary 3.1 follows immediately from The- orem 3.1.

4. An example

In this section, we give an example to demonstrate the results ob- tained in previous sections.

Example 4.1. Consider the following CNNs with continuously dis- tributed delays:

(4.1)

 

 

 

 

 

 

 



 

 

 

 

 

 

 

x ⁰ ₁ (t) = −x ₁ (t) + ¹ ₄ (sin t)f ₁ (x ₁ (t)) + ₃₆ ¹ (cos t)f ₂ (x ₂ (t)) + ¹ ₄ (sin t) R _∞

0 (sin u)e ^−u · f ₁ (x ₁ (t − u))du + ₃₆ ¹ (cos t) R _∞

0 (sin u)e ^−u f ₁ (x ₂ (t − u))du + I ₁ (t), x ⁰ ₂ (t) = −x ₂ (t) + (sin 2t)f ₁ (x ₁ (t)) + ¹ ₄ (cos 4t)f ₂ (x ₂ (t))

+(sin 2t) R _∞

0 (sin u)e ^−u · f ₁ (x ₁ (t − u))du + ¹ ₄ (cos 4t) R _∞

0 (sin u)e ^−u f ₂ (x ₂ (t − u))du + I ₂ (t), x ⁰ ₃ (t) = −x ₃ (t) + ¹ ₄ (cos 4t)f ₃ (x ₃ (t))

+ ¹ ₄ (sin 2t) R _∞

0 (sin u)e ^−u f ₃ (x ₁ (t − u))du + I ₃ (t), wheref ₁ (x) = f ₂ (x) = f ₃ (x) = ¹ ₂ (|x + 1| − |x − 1|), I ₁ (t), I ₂ (t) and I ₃ (t) are almost periodic functions on R.

Notice that ˜ c ₁ = ˜ c ₂ = ˜ c ₃ = L ₁ = L ₂ = L ₃ = 1, a ₁₁ = b ₁₁ = ¹ ₄ , a ₁₂ = b ₁₂ = ₃₆ ¹ , a ₁₃ = b ₁₃ = 0, a ₂₁ = b ₂₁ = 1, a ₂₂ = b ₂₂ = ¹ ₄ , a ₂₃ = b ₂₃ = a ₃₁ = b ₃₁ = a ₃₂ = b ₃₂ = 0, a ₃₃ = b ₃₃ = ¹ ₄ , k ₁₁ = k ₁₂ = k ₂₁ = k ₂₂ = k ₃₃ = 1.

Then, we get

D ⁻¹ EL = D ⁻¹ ( ¯ A + ¯ B)L =





1 2 1 18 0 2 ¹ ₂ 0 0 0 ¹ ₂



 . Hence, we have

ρ(D ⁻¹ EL) = ρ(D ⁻¹ ( ¯ A + ¯ B)L) = 5

6 < 1.

Therefore, it follows from Lemma 3.1 that system (4.1) has at least exactly one almost periodic solution. Moreover, The almost periodic solution is globally exponentially stable.

Remark. (4.1) is a very simple form of DCNNs equations. One can observe that kD ⁻¹ ( ¯ A+ ¯ B)Lk ₁ = ⁵ ₂ , where k·k ₁ is the row norm of matrix.

Therefore, all the results in [2-6, 8, 13, 15, 16] and the references therein can not be applicable to system (4.1). This implies that the results of this paper are essentially new.

5. Conclusion

In this paper, cellular neural networks with continuously distributed delays have been studied. Some sufficient conditions for the existence and exponential stability of the almost periodic solutions have been es- tablished. These obtained results are new and they complement pre- viously known results. Moreover, an example is given to illustrate the effectiveness of the new results.

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Bingwen Liu

College of Mathematics and Information Science Jiaxing University

Jiaxing, Zhejiang 314001, P. R. China E-mail: [email protected] Lihong Huang

College of Mathematics and Econometrics Hunan University

Changsha 410082, P. R. China