Stabilitty of Anti-periodic Solutions for Certain Shunting Inhibitory Cellular Neural Networks

Published Online August 2011 in MECS

Recently, cellular neural networks (CNNs) have shown great potential as information-processing systems, and many researchers have paid much attention to the research on the theory and application of the CNNs . Some sufficient conditions are given to ensure the existence and stability of the equilibrium point for the CNNs . The shunting inhibitory cellular neural networks (SICNNs) are a new class of CNNs , which were introduced by Bouzerdoum and Pinter in [13], and have been extensively applied in psychophysics, speech, perception, robotics, adaptive pattern recognition, vision, and image processing. Hence, they have been the object of intensive analysis by numerous researchers in recent years. In particular , there have been extensive results on the problem of the existence and stability of periodic and almost periodic solutions of SICNNs in the literature (see, e.g., [4-13] and the references therein). In contrast, however, very few results are available on a generic, in-depth, existence and exponential stability of anti-periodic solutions for SICNNs (1.1). Moreover, it is well known that the existence of anti-periodic solutions play a key role in characterizing the behavior of nonlin-

Footnotes: 8-point Times New Roman font;

Manuscript received September 13, 2010; revised March 15, 2011;

accepted July 1,2011.

Corresponding author at: School of Mathematics and Physics, Changzhou University , Changzhou 213016 ,Jiangsu, China

ear differential equations (see [14-17]). Since SICNNs can be analog voltage transmission, and voltage transmission process is often an anti-periodic process. Thus, it is worth while to continue to investigate the existence and stability of anti-periodic solutions of SICNNs.

Consider shunting inhibitory cellular neural networks (SICNNs) with continuously distributed delays described by xi′j(t) = -aij(t)xij(t)- ∑Cikjl(t)f(xkl(t-τ(t))) Ckl∈Nr(i,j)

x ij ( t ) + L ij ( t ), (1.1) where i = 1,2, L , m , j = 1,2, L , n , C_ij denotes the cell at the ( i , j ) position of the lattice, the r -neighborhood N_r ( i , j ) of C_ij is given by

N _r ( i , j ) = { C _i ^k _j ^l : max( k - i , l - j ) ≤ r ,1 ≤ k ≤ m ,1 ≤ l ≤ n } . x _ij acts as the activity of the cell C _ij , L_ij ( t ) is the external Input to C _ij , the variable a_ij ( t ) >  0 represent the passive decay rate of the cell activity, C _ij ( _t ) _≥ 0 is the connection or coupling strength of postsynaptic activity of the cell transmitted to the cell C _ij , and the activity function f ( ⋅ ) is a continuous function representing the output or firing rate of the cell C _kl , and τ ( t ) ≥ 0 corresponds to the transmission delay.

However, we note that, in most of the abovementioned literature, the activity function f(·) is assumed to be global Lipschitz continuous and bounded, that is, there exist constants µ _f and M _f such that for all x , y ∈ R

( T 0 ) I f ( x ) - f ( y ) ≤ µ f x - y ^, f ( x ) ≤ M_f .

To the best of our knowledge, few researchers have considersed SICNNs without ( T ₀ ). Thus, it is worthwhile to continue to investigate system (1.1).

In this paper, we will establish new sufficient conditions ensuring the existence, uniqueness and exponential stability of anti-periodic solutions of system (1.1) without ( T ₀ ). Moreover, an example is provided to illustrate the effectiveness of our results.

Let u(t) : R → R be continuous in t . u(t) is said to be T -anti-periodic on R if u(t + T) = -u(t) for all t ∈ R .

Throughout this paper, for i = 1,2, L , m , j = 1, 2,L ,n , it will be assumed that aij ,Cij : R → [0,+∞ ), Lij : R → R are continuous functions and τ (t) : R → [0,τ] , and aij(t+T) = aij(t), Cij(t+ T) = Cij(t)

f ( - u ) = f ( u ) ,      τ ( t + T ) = τ ( t ),

L_ij ( t + T ) = - L_ij ( t ), ∀ t , u ∈ R .                (1.2)

Set

{ x ij ( t )} = ( x 11 ( t ), L , x 1 n ( t ), L , x m 1 ( t ), L , x mn ( t )) ^T ∈ R^{m × n} .

For ∀ x ( t ) = { x_ij ( t )} ∈ R ^{m × n} , we define the norm x ( t ) = max { x_ij ( t ) }.

( i , j )

The initial conditions associated with system (1.1) are of the form xij(s) =ϕij(s) , s ∈ [-τ,0] ,                       (1.3)

where ij = 11 ,12 , L , mn and ϕ _ij ( ⋅ ) denotes real-valued continuous function defined on [ - τ ,0] .

We also assume that the following conditions hold.

(T1 ) There exist continuous functions µ : R + → R + such that for each D f (u) - f (v) ≤ µ(D) u - v , u , v ≤ D . (1.4) ( T2 ) There exist constants λ > 0, D1 ≥ D 0 > 0 , and ij = 11 ,12 ,L ,mn , such that aijD0-     ∑Cikjl[f(0)+µ(D0)D0]D0-L+ij >0,

C kl ∈ N r ( i , j )

( λ - a ij ) +      ∑ C i^kj^l [ µ ( D 1 ) D 1 e ^λτ + f (0) + µ ( D 0 ) D 0 ] < 0,

C kl ∈ N r ( i , j )

(1.5)

where kl aij = inf aij(t), Cij = sup Cikjl(t) ij t∈R                      t∈R ij

L ⁺ ij

> sup L_ij ( t ) . t ∈ R

Definition 1. Let x∗ (t) = { xi∗j (t)} be an anti-periodic solution of system (1.1) with initial value ϕ∗ (t) = {ϕi∗j (t)} . If there exist constants λ > 0 and

M > 1 such that for every solution x(t) = {xij (t)} of system (1.1) with initial value ϕ(t) = {ϕij (t)} , xij(t) -xi∗j(t) ≤ M ϕ-ϕ∗ 1e-λt,∀t > 0,ij = 11,12,L ,mn. where ϕ -ϕ∗ 1 = -τs≤ups≤0{m(ia,jx) ϕij(s) -ϕi∗j(s) } .Then x∗ (t) is said to be globally exponentially stable.

The remaining parts of this Letter are organized as follows. In Section 2, sufficient conditions are derived for the boundedness of solution of (1.1). In Section 3, we present new sufficient conditions for the existence , uniqueness and exponential stability of anti-periodic solutionof (1.1) . In Section 4 , an illustrative example is given to show the effectiveness of the proposed theory and method . In Section 5, we give several remarks.

• II.    Preliminary results

Lemma 2.1. Let (T1 ) and (T2 ) hold. Suppose that x∗ (t) = {xi∗j(t)} is a solution of system (1.1) with initial Conditions xi∗j(s) =ϕi∗j(s) , ϕi∗j(s) < D0 , s ∈ [-τ,0] . (2.1)

Then xi∗j(t) < D0 , t≥0 , ij =11,12,L,mn. (2.2) Proof. Assume, by way of contradiction, that (2.2) does not hold. Then , there exist ij ∈ {11 ,12 , L , mn } and δ > 0 such that xi∗j(δ) = D0, and xi∗j(t) < D0, ∀t ∈ [-τ,δ], (2.3) where ij ∈ {11 ,12 , L , mn } . Calculating the upper right derivative of x ∗ (δ) , together with (T1) , (T2) and (2.3) , we can obtain

0 ≤ D ⁺ ( x i ^∗ j ( δ )) = sgn( x i ^∗ j ( δ )) D ⁺ ( x i ^∗ j ( δ ))

= - a_ij ( δ )sgn( x_i ^∗ _j ( δ )) x_i ^∗ _j ( δ ) - sgn( x_i ^∗ _j )[ ∑ C_i^k_j^l ( δ )

C kl ∈ N r ( i , j )

f ( x i ^∗ j ( δ - τ ( δ ))) x i ^∗ j ( δ ) - L ij ( δ )]

≤

- a ij ( δ ) x i ^∗ j ( δ ) +

∑ C i^kj^l ( δ ) f ( x i ^∗ j ( δ - τ ( δ ))) x i ^∗ j ( δ ) C kl ∈ N r ( i , j )

+ L ij ( δ )

≤ - a ij ( δ ) x i ^∗ j ( δ ) +    ∑ C i ^k j ^l ( δ ) f ( x i ^∗ j ( δ - τ ( δ ))) x i ^∗ j ( δ )

C kl ∈ N r ( i , j )

+ L ij ( δ )

≤- a _ij D 0 +     ∑     C i^kj^l [ f (0) + µ ( D 0 ) x i ^∗ j ( δ - τ ( δ )] D 0 + L ⁺ ij

C kl ∈ N r ( i , j )

≤ - a _ijD ₀ +      ∑ C ij [ f (0) + µ ( D ₀) D ₀] D ₀ + L ⁺ _ij

C kl ∈ N r ( i , j )

<0

This is a contradiction and hence (2.2) holds. This completes the proof.

Remark 2.1. In view of the boundedness of this solution, from the theory of functional differential equations in [18], it follows that x ^∗ ( t ) can be defined on [ - τ , +∞ ) . Lemma 2.2. Suppose that ( T ₁ ) and ( T ₂ ) are satisfied. Let x ^∗ ( t ) = ( x ₁ ^∗ ₁( t ), x ₁ ^∗ ₂( t ), L , x_m ^∗ _n ( t )) ^T be the solution of system (1.1) with initial value (2.1) , and x ( t ) = ( x ₁₁( t ), x ₁₂( t ), L , x_mn ( t )) ^T be the solution of system (1.1) with initial value ϕ ( t ) = ( ϕ ₁₁( t ), ϕ ₁₂( t ), L ϕ _mn ( t )) ^T and

ϕij (s) < D1 for s ∈ [-τ,0] . Then there exist constants λ > 0 and M > 1 such that xij(t) - xi∗j(t) ≤ M ϕ-ϕ∗ e-λt, ∀t > 0,ij = 11,12,L, mn. Proof. Set y(t) = {yij (t)} = {xij (t) - xi∗j (t)} Then yi′j(t)=-aij(t)yij(t)-       ∑Cikjl(t)[f(xkl(t-τ(t)))xij(t)

C kl ∈ N r ( i , j )

• - f ( x k ^∗ l ( t - τ ( t ))) x i ^∗ j ( t )],                    (2.4)

where ij ∈ {11 ,12 , L , mn } .

We consider the Lyapunov functional

V ij ( t ) = y ij ( t ) e ^{λ t} , ij = 11,12, L mn .          (2.5)

Let M >  1 denotes an arbitrary real number and set ϕ - ϕ ^∗ = sup {max ϕ _ij ( s ) - ϕ _i ^∗ _j ( s )} > 0 .

• ¹    - τ ≤ s ≤ 0 ^{( i , j )}

It follows from (2.5) that

V_ij ( t ) = y_ij ( t ) e ^{λ t} <  M ϕ - ϕ ^∗ ₁, t ∈ [ - τ ,0], where ij ∈ {11 ,12 , L , mn } .We claim that V_ij ( t ) = y_ij ( t ) e ^{λ t} <  M ϕ - ϕ ^∗ ₁, t >  0, ij = 11,12, L mn .

(2.6) Contrarily, there must exist ij ∈ {11 ,12 , L , mn } and t_ij >  0 such that

V_ij ( t_ij ) = M ϕ - ϕ ^∗ ₁ ,and V_ij ( t ) < M ϕ - ϕ ^∗ ₁, t ∈ [ - τ , t_ij ) .

(2.7) where ij ∈ {11 ,12 , L , mn } . It follows from (2.7) that V_ij ( t_ij ) - M ϕ - ϕ ^∗ ₁ = 0 , V_ij ( t ) - M ϕ - ϕ ^∗ ₁ <  0, t ∈ [ - τ , t_ij ) ^.

(2.8) Calculating the upper right derivative of V_ij ( t_ij ) along the solution y ( t_ij ) = { y_ij ( t_ij )} of system (2.4) with the initial value ϕ = ϕ - ϕ ^∗ and with (2.2) and (2.8), we obtain

0 ≤ D ⁺ ( V ij ( t ij ) - M ϕ - ϕ ^∗ ₁) = D ⁺ ( V ij ( t ij ))

= sgn( y_ij ( t_ij )) D ⁺ ( y_ij ( t_ij )) e ^{λ t ij} + λ y_ij ( t_ij ) e ^{λ t ij}

= sgn( y ij ( t ij ))( D ⁺ ( x ij ( t ij )) - D ⁺ ( x i ^∗ j ( t ij ))) e ^{λτ ij}

+ λ y_ij ( t_ij ) e ^{λτ ij}

= -aij(tij)sgn(yij(tij))[xij(tij) -xi∗j(tij)]eλτij -sgn(yij(tij)) kl                                                   ∗                    ∗ ij ij kl ij ij ij ij ij ij ij ij ij Ckl∈Nr(i, j)

eλτij + λyij (tij) eλτij

≤ - a ij ( t ij ) y ij ( t ij ) e ^{λ t ij} +         ∑ C i^kj^l ( t ij )

C kl ∈ N r ( i , j )

f ( x_kl ( t_ij - τ ( t_ij ))) x_ij ( t_ij ) - f ( x_k ^∗ _l ( t_ij - τ ( t_ij ))) x_i ^∗ _j ( t_ij )

e ^{λ tij} + λ y_ij ( t_ij ) e ^{λ tij}

≤ ( λ - a ij ) y ij ( t ij ) e ^{λ tij} +         ∑ C i^kj^l e ^{λ tij}

C _kl ∈ N _r ( i , j )

[ f ( x_kl ( t_ij - τ ( t_ij ))) x_ij ( t_ij ) - f ( x_k ^∗ _l ( t_ij - τ ( t_ij ))) x_ij ( t_ij )

+ f ( x_k ^∗ _l ( t_ij - τ ( t_ij ))) x_ij ( t_ij ) - f ( x_k ^∗ _l ( t_ij - τ ( t_ij ))) x_i ^∗ _j ( t_ij )]

≤ ( λ - a _ij ) y_ij ( t_ij ) e ^{λ tij} +         ∑ C i^kj^l e ^{λ tij}

C kl ∈ N r ( i , j )

[ µ ( D ₁) x_kl ( t_ij - τ ( t_ij )) - x_k ^∗ _l ( t_ij - τ ( t_ij )) x_ij ( t_ij )

+ f ( x_k ^∗ _l ( t_ij - τ ( t_ij ))) - f (0) x_ij ( t_ij ) - x_i ^∗ _j ( t_ij )]

≤ ( λ - a ij ) y ij ( t ij ) e ^{λ tij} +         ∑ C i^kj^l [ µ ( D 1 ) D 1

C kl ∈ N r ( i , j )

y_kl ( t_{ij -} τ ( t_ij )) e λ ( t ij - τ ( t ij )) e λτ ( t ij )

+ ( f (0) + µ ( D 0 ) D 0 ) y ij ( t ij ) e ^{λ tij} ]

≤ {( λ - a ij ) +       ∑ C i^kj^l [ µ ( D 1 ) D 1 e ^λτ + f (0)

C kl ∈ N r ( i , j )

+ µ ( D 0 ) D 0 ]} M ϕ - ϕ ^∗ ₁ .              (2.9)

Thus,

0 ≤ ( λ - a ij ) +     ∑ C i^kj^l [ µ ( D 1 ) D 1 e ^λτ + f (0) + µ ( D 0 ) D 0 ],

C kl ∈ N r ( i , j )

which contradicts ( T ₂ ) Hence, (2.6) holds. It follows that y_ij ( t ) <  M ϕ - ϕ ^∗ ₁ e ^{- λ t} , t >  0, ij = 11,12, L mn ^. (2.10)

This completes the proof . The proof of Lemma 2.2 is completed.

Remark 2.2. If x ^∗ ( t ) = ( x ₁ ^∗ ₁( t ), x ₁ ^∗ ₂( t ), L , x_m ^∗ _n ( t )) ^T is the T-anti-periodic solution of system (1.1), it follows from Lemma 2.2 and Definition 1 that x ^∗ ( t ) is globally exponentially stable.

The following is our main result.

Theorem 3.1. Suppose that ( T ₁ ) and ( T ₂ ) are satisfied.

Then system (1.1) has exactly one T-anti-periodic solution x ^∗ ( t ) . Moreover, x ^∗ ( t ) is globally exponentially stable.

Proof. Let x(t) =(x11(t), x12(t),L, xmn(t))T be a solution of system (1.1) with initial conditions xj(s) = ф,(s),|ф,(s)| < Do,s g [-T,0\,ij = 11,12,l,mn.

(3.1)

By Lemma 2.1, the solution x(t) is bounded and

|xj ( s )| <  D 0 , t g [ - т , +» ), ij = 11,12, l , mn . (3.2)

From (1.1) and (1.2), we have

(( - 1) k ^{+ 1} x₄( t + ( k + 1) T )) ' = ( - 1) ^{k + 1} x j ( t + ( k + 1) T )

= ( - 1) ^{k + 1} { - a j ( t + ( k + 1) T ) x_y ( t + ( k + 1) T ) -

Z C " ( t + ( k + 1) T ) f ( x ki ( t + ( k + 1) T - t( t + ( k + 1) T ))) C ki g N r (i, j )

X j ( t + ( k + 1) T ) + L j (t + ( k + 1) T )}

= ( - 1) ^{k + 1}{ - a j ( t ) X j (t + ( k + 1) T ) -       £ C " ( t )

C ki g N r ( i , j j)

f ( x_M ( t + ( k + 1) T - T (t )))x_y ( t + ( k + 1) T ) + ( - 1) ^{k + 1} L j ( t )}

= - a j ( t )( - 1) ^{k + 1} X j ( t + ( k + 1) T ) -      Z C " ( t )

C» g N r ( i , j )

f (( - 1) ^{k + 1} x_kl ( t + ( k + 1) T - т ( t )))( - 1) ^{k + 1} X j ( t + ( k + 1) T ))

+ L j ( t ) ,                                            (3.3)

where ij g {11,12, l , mn }. Thus, for any natural number k , ( - 1) ^{k + 1} x ( t + ( k + 1) T ) are the solutions of system (1.1). Then, by Lemma 2.2, there exists a constant M >  0 such that

|( - 1) ^{k + 1} x j ( t + ( k + 1) T )) - ( - 1) k x j ( t + kT )|

= | x ij ( t + kt + T )) + x j ( t + kT )|

• <    Me ~ ^{X (} t ^{+ kT )}    sup {maxi x j ( s + T ) + x j ( s )|}

  - т < s < 0 ⁽‘, j ^)l

• <    2 MD ₀ e - ^{k t} ( e - ” ) ^k , V t + kT >  0 ,               (3.4)

Thus, we can choose a sufficiently large constant N >  0 and a positive constant a such that

|( - 1) ^{k + 1} x j ( t + ( k + 1) T )) - ( - 1) ^k x _ц ( t + kT )|

• < a (e - " ) ^k , V k >  N ,                      (3.5)

on any compact set of R. For any natural number p , we obtain

( - 1) p ^{+ 1} x j ( t + ( p + 1) T )

= x _y( t ) + £ [( - 1) ^{k + 1} x _y( t + ( k + 1) T ) k = 0

• - ( - 1) k x j ( t + kT )\ .                           (3.6)

Then

| ( - 1) ^{p + 1} x j ( t + ( p + 1) T )|

p

• <| x j (t ) + Z ( - 1) k ^{+ 1} x j (t + ( k + 1) 7 )-( - 1) ^kx j (t + kT) | , k = 0

(3.7) where j = 11,12, l , mn . It follows from (3.5) and

• (3.7)    that {( - 1) ^px ( t + pT)} uniformly converges to a continuous function x *( t ) = ( x ^^ t ), x 1* ₂( t ), l , x*_mn(t))^T on any compact set of R.

Now we will show that x *( t ) is the T-anti-periodic solution of system (1.1). First, x *( t ) is T-anti-periodic, since

x*(t + T) = lim(-1) px(t + T+pT)

p ^»

= - lim(-1)p+1 x(t+(p+1)T) = -x*(t). p+1^»

Next , we prove that x*(t) is a solution of (1.1) . In fact , together with the continuity of the right side of (1.1), (3.3) implies that {((-1)px(t + pT))'}uniformly converges to a continuous function on any compact set of R. Thus, letting p ^», we obtain dd^ {x*i (t)}=- ay (t) xy( t)

-

Z C " ( t )

C kl G N_r ( i , j )

f ( x * ( t - t ( t ))) x- j ( t ) + Ц(t ). (3.8)

Therefore, x *( t ) is a solution of (1.1).

Finally, by Lemma 2.2, we can prove that x * (t ) is globally exponentially stable. This completes the proof.

If we take aij(t) = ajj, Cj-(t) = Cij,   then the system (1.1) can be modified to the following form:

x j , ( t ) = - a_yx_y ( t ) - Z C " f ( x ki ( t - т ( t ))) C ki g N r ( i , j )

x j ( t ) + Ц ( t ),                         (3.9)

We can make the following conclusion:

Corollary 3.2 Assume that (T1) and (T2)' hold, where (T2)' :There exist constants я > 0,D 1 > D0 > 0 , and ij = 11,12, l , mn , such that ajD0 -     Z Ckl[|f (0)| + ^(D0)D0]D0 - L+ > 0,

C ki g N r ( i , j )

( Я - a ij ) +     Z C k^l [ ^ ( D 1 ) D 1 e ^ + 1 f (0)| + ^ ( D _y) D 0 ] <  0 ^.

C ki g N_r ( i , j )

then system (3.9) has one T-anti-periodic solution. Moreover, the T-anti-periodic solution is globally exponentially stable.

In [14], J.Shao also discussed existence and exponential stability of anti-periodic solutions of system (3.9) and The following result was proved.

Theorem A (J.Shao[14]) Assume that

( T 1 ) there exist constants p f and m _f such that for all u , v g R

I f ( u ) - f ( v )| <  p f\ u - v |, | f ( u )| <  M f ■

( T 2 ) there exist constants § .. >  0, n >  0 and X >  0 , ij = 11,12, l , mn , such that

6 ij	= a	- X CM. , C kl G N r ( i , j )
( X -	a ij ) +	X C " M f + C tl G N r ( i , j )	_          +  - у Ck ц, -^eXT ij f C tl G N r ( i , j )                ° ij

< -n < 0 , where - + > max Li (t)|, T = max {t(t)} . then syst-J         t G R I j I             t G R em (3.9) has one T-anti-periodic solution. Moreover, the T-anti-periodic solution is globally exponentially stable.

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

Example 4.1. Consider the following SICNNs with continuously distributed delays:

x j ( t ) = - a ij ( t ) X j ( t ) - X C " ( t ) f ( x_H ( t - t ( t ))) C tl g N r ( i , j )

X j ( t ) + - j ( t ),                              < 4.1 )

^ a 11( t )	a 12( t )	a 13( t )	a 14( t )
a 21( t )	a 22( t )	a 23( t )	a 24( t )
a 31( t )	a 32( t )	a 33( t )	a 34( t )
I a 41( t )	a 42( t )	a 43( t )	a 44( t ) J

	' 2 +	sin t	2 +	sin t	2 +	cos t	3 +	cos t	^
	3 +	cos t	2 +	cos t	3 +	sin t	3 +	sin t
=	3 +	sin t	2 +	cos t	2 +	sin t	2 +	sin t
	3 +	cos t	2 +	sin t	3 +	cos t	2 +	sin t	7

' C 11 ( t )	C 12 ( t )	C 13 ( t )	C 14 ( t ) "
C 21 ( t )	C 22 ( t )	C 23 ( t )	C 24 ( t )
C 31 ( t )	C 32 ( t )	C 33 ( t )	C 34 ( t )
1 C 41 ( t )	C 42 ( t )	C 43 ( t )	C 44 ( t ) 7
' 0.07	0.05	0.04	0.02"
0.13	0	0.15	0.05
0.01	0.12	0	0.14
v 0.03	0.14	0.05	0.01 7

л 0.03	sin t	0.15sin t	0.06	cos	t 0.08	cos t	)
0.07	cos t	0	0.05	sin	0.05	sin t
0.09	sin t	0.08 cos t	0		0.06	sin t
( 0.07	cos t	0.06sin t	0.05	cos	t 0.09	sin t	7
' - 11 ( t )		L 12 ( t )    L 13 ( t )	L 14	( t ) )
- 21 ( t )		L 22( t ) L 23 ( t )	L 24	( t )
- 31 ( t )		L 32 ( t )   L 33 ( t )	L 34	( t )
I - 41 ( t )		L 42 ( t ) L 43 ( t )	L 44	( t > J

^ 0.4 sin t	0.4cos t	0.1sin t	0.2cos t2
0.3cos t	0.1sin t	0.2cos t	0.3sin t
0.2 cos t	0.4sin t	0.1cos t	0.2sin t
v 0.1sin t	0.1cos t	0.2sin t	0.1cos 1 7

₂                   x ² + 1

Set r = 1,T(t) = cost, and f(x) =----- , clearly, f'(x) = 2 x , then ЦD) = 2D, kl                i                     kl

C„ . N VU) ^{C 11} = ⁰-⁵’   ^X C ki G N _l11,2) ^{C 12} = ⁰-⁸’

X C. . N_lM C ^{k 3} = ⁰-7.   X C . G «1,4, ^{C 4 4} = «-5.

klkl

^X C tl g N > (2,1) ^{C 21} = ^{0.8,   X} C ki g N > (2,2) ^{C 22} = ^1.1,

E— kl—kl

C ₂3= 1.0, У C ₂4 = 0.7,

Ctl GN1(2,3)               ,   Ck-GCtl gN.(2,4)

~ kl _        i

C₃₁ = 0.8,  у         C₃₂ = 1.1,

Ckl GN 1(3,1)              ,   Д.!СИ gN 1(3,2)

— kl             ,—kl

C₃₃ = 1.1,  У         C₃₄ = 0.7,

Ckl GN 1(3,3)               ,    C-4Ctl gN 1(3,4)

~ kl                    i c41 = 0.6, У        c42 = 0.7,

Ckl GN 1(4,1)              ,   ^Ctl gN 1(4,2)

— kl—kl

C ₄3= 0.8,  у C 44 = 0.4,

Ctl g N1(4,3)   43         ,   Z.J Ctl g N1(4,4)   44

		Г a 11    a 12	a	13 a	4		Г 2	2	2 3 ^
		a 21 a 22	a	23 a 24		=	3	2	33	,
		a 31 a 32	a	33    a 34			3	2	22	,
		V a 41   a 42	a	43 a 44 >			V 3	2	3 2
	Г -	+ / + 11      12	+ L 13	L + L 14		Г 0.5		0.5	0.2	0.3 2
	- l l V L	+    / + 21     22 +    / + 31     32 +    / + 41     42	J + L 23 I + L 33 I + L 43	I + L 24 I + L 34 I, + 44 7	>	0.4 0.3 V 0.2		0.2 0.5 0.2	0.3 0.2 0.3	0.4 0.3 0.2 7	,
Take D0 = Д			= 1	and	Define			continuous			function

F j ( to ) by setting

F j ( to ) = ( to - a_у ) +     X C [ ц ( D 1 ) D 1 e   + 1 f (0)|

^C kl ^{G N} r ⁽ i , ^j )

+ M( D 0) D о], where ®e[0,1], ij = 11,12, ^ , mn . Then, we obtain kl

C ij [ ^ ( D 1 ) D 1

(" j )               '              C ki e N ( i , j )

+ | f (0)| + д ( D o ) D o ]} <- 0.9.

Thus, there exists X > 0 such that Fij (2) < 0 for ij = 11,12, l , mn and min{ayD0 -    XCk[If (0)| + Ц(D0)D0]D0 - L+j} > Ц,

⁽ i ■ j )            C„ e N r ( i , j )                                    ²⁵

So ( T ₂ ) holds. By Theorem 3.1, all the solution of system (4.1) with initial value |^j ( s )| <  D 1 for s e [ - r ,0] converge exponentially to one П -anti-periodic solution.

If we take a j (t ) = a j ,

' C 11 (t ) C 12 ( t ) C 13 ( t ) C 14 (t )'

C 21 ( t ) C 22 (t ) C 2з ( t ) C 2₄ ( t )

C 31 ( t ) C 32 ( t ) C ₃ з ( t ) C ₃4 ( t )

( C41( t) C42(t) C43( t) C44( t )v л 0.1  0.2  0.10.P

0.2   0   0.20.1

=

0.1  0.2   00.2

(0.1  0.2  0.10.1

and other conditions of the above example also hold, then, by Corollary 3.2 , all the solution of system (4.1) with initial value | ^ j ( s )| <  D 1 for s e [ - r ,0] converge exponentially to one П -anti-periodic solution.

Remark 4.1. The function J ( • ) in Example 4.1 does not satisfy the conditions of Theorem A (J.Shao[14]) . Thus , the results in Theorem A (J.Shao[14]) can not be applied to Example 4.1. This implies that the results of this paper are essentially new.

• V . Conclusion

In this paper , shunting inhibitory cellular neural networks with continuously distributed delays have been studied. New Sufficient conditions for the existence and global exponential stability of anti-periodic solutions have been established , which complement previously known results. Moreover, an example is given to illustrate the effectiveness of our results.

This work is supported by National Natural Science Foundation of China (10461006) and Basic Subject Foundation of Changzhou University (JS 201004).

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  Huiyan Kang was born in Baotou, Inner Mongolia, China in 1973. She received the B.S. and the M.S. degree in applied mathematics from Inner Mongolia Normal University, Huhhot, China in 1996 and 2002, respectively. Her main research interests include nonlinear system,neural

networks, stability theory, and control theory.

She is now a lecturer at the school of ChangzhouUniversity. Recently she has published 8 research papers in the area of Control and Mathematics.

Ligeng Si was born in Huhhot, Inner Mongolia, China in 1931. He received the B.S. degree in mathematics from Hebei Normal University, Shijiazhuang, China in 1958. His main research interests include stability theory and applied mathematics.

Now he is a professer and Masteral Adviser of Inner Mongolia Normal University. He is the author or coauther of many journal paper and three edited books and a revie-wer of Mathematical Reviews.