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Indian Journal of Science and Technology                                                                    Vol. 3 No. 1 (Jan 2010)                     ISSN: 0974- 6846


                         The existence of anti-periodic solution for a class of cellular neural networks

                                                       Zhouhong Li, Chenxi Yang and Kaihong Zhao
                          Department of Mathematics, Yuxi Normal University, Yuxi, Yunnan, China-653 100
                                                                            zhouhli@yeah.net


                                                                                  Abstract

   In this paper, we use the Lyapunov function to establish new results on the existence and uniqueness of anti-
periodic solutions for a class of cellular neural networks with time-varying delays and continuously distributed delays of
                                          .                                        n                                        n
                                          x i (t ) = − di (t )hi (t , xi (t )) + ∑ aij (t ) f j ( x j (t − τ ij (t ))) + ∑ bij (t )
                                                                                   j =1                                     j =1
                                                         ∞
                                                     ×∫ Kij ( s ) g j ( x j (t − s))ds + I i (t ), i = 1, 2,..., n .
                                                        0
Moreover, we also present an example to illustrate the feasibility and effectiveness of our results.

Keywords: Cellular neural networks, distributed delays, anti-periodic solution, exponential stability, delays.


1. Introduction                                                                            at time t, f j (⋅) and g j (⋅) are activation functions of signal
     Cellular neural networks (CNNs) have been widely
studied both in theory and applications (Roska &                                           transmission. Our main purpose of this paper is by
Vandewalle, 1995; Chua & Yang, 1988; Li, 2004; Cheng                                       constructing Lyapunov functions to investigate the
et al., 2006; Chen 2002; Cao & Wang, 2005; Mohamad,                                        stability and existence of anti-periodic solutions of (1.1)
2007). They have been successfully applied to signal                                       and to give the conditions for the existence and
processing, pattern recognition, optimization and                                          exponential stability of the anti-periodic solutions for
associative memories, especially in image processing                                       system (1.1), which are new and complement previously
and solving nonlinear algebraic equations. Recently,                                       known results. Moreover, an example is also provided to
Peng & Huang (2009) and Shao (2008) have studied the                                       illustrate the effectiveness of our results.
existence and stability of anti-periodic solution of the                                   2. Notations and preliminaries
following shunting inhibitory cellular neural networks with                                      For convenience, we consider model (1.1) under
time-varying delays. In this paper, we consider the                                        some following assumptions.
cellular neural network with continuously distributed                                            Let u (t ) : R → R be continuous in t. u(t) is said to be
delays                                                                                     T-anti-periodic on R , if
 .                                  n                                   n
                                                                                            u (t + T ) = −u (t ) for all t ∈ R.
xi (t) = −di (t)hi (t, xi (t)) + ∑aij (t) f j (xj (t −τij (t))) + ∑bij (t)
                                   j =1                                j =1                If a system is T-anti-periodic ( x(t + T ) = − x(t )) , then it is
      ∞                                                                                    2T-periodic (x(t+2T) = -x(t+T) = x(t)).
× ∫ K ij ( s ) g j ( x j ( t − u )) d s + I i ( t )                           (1.1)             Throughout this paper, for i, j = 1, 2, n and for all
     0
                                                                                            t , u ∈ R , it will be assumed that
where di (t ) > 0 represents the rate with which the                             i th
                                                                                                 T > 0, d j (t + T ) hi (t + T , u ) = − d i (t )hi (t , −u ).         (2.1)
unit will reset its potential to the resting state in isolation
when disconnected from the network and external inputs
at the time t. hi (t , u ) is continuous function on                                        aij (t +T) fi (u) =−aij (t) f j (−u), bij (t +T)gj (u) =−bij (t)gj (−u). (2.2)

 R 2 , f j , g j , aij , bij , τ ij and I i are continuous function on                                  I i (t + T ) = − I i (t ),τ ij (t + T ) = τ ij (t ).           (2.3)

R; in which n corresponds to the number of units in a                                          Then, we suppose that there exits constants I i , aij , bij ,
neural network,                 xi (t ) corresponds to the state of the                    and     τ   such that
 i th unit at the time t, τ ij (t ) ≥ 0 corresponds to the                                    0 < I i = sup I i (t ) , aij = sup aij (t ) , bij = sup bij (t ) ,
                                                                                                           t∈R                        t∈R                      t∈R


                                                                                                                                     {             }
transmission delay of the i th unit along the axon of the
jth unit at the time t, and I i (t ) denote the external inputs                                                      τ = max max τ ij (t )
                                                                                                                           1≤i , j ≤ n t∈[ 0,T ]

Research article                                                     “Cellular neural networks”                                                          Zhouhong Li et al.
Indian Society for Education and Environment (iSee)                           http://www.indjst.org                                                      Indian J.Sci.Technol.
                                                                                                                                                                                                                                                      2

Indian Journal of Science and Technology                                                                                           Vol. 3 No. 1 (Jan 2010)                                                              ISSN: 0974- 6846

      We also assume that the following conditions hold.
                                                                                                               xi (t ) − xi* (t ) ≤ M ϕ − ϕ * 1 e−λt , ∀t > 0, i = 1, 2,..., n ,
        (T0) for   i ∈ {1, 2,...n} , hi (t , u ) : R → R are                       2
                                                                                                             where
continuous function, and there exist nonnegative constant
hi > 0 such that                                                                                               ϕ − ϕ * 1 = sup −∞≤ s ≤ n max1≤i ≤ n ϕi ( s ) − ϕi* ( s )                                                                               ,

hi (t,0) = 0, hi u − v ≤ sgn(u − v)(hi (t, u) − hi (t, v)) ,
                                                                                                                         *
                                                                                                             then x (t ) is said to be globally exponentially stable.
                                                                                                                The following lemmas will be used to prove our main
for all u , v ∈ R .                                                                                          results in Section 3.
(T1) for each            j ∈ {1, 2,..., n} ,there exist nonnegative                                          Lemma 2.1. [15] Let (T0), (T1), (T2) and (T3) hold,
                                                                                                             suppose that x (t ) = ( x1 (t ), x2 (t ),..., xn (t ))T is a solution
                                                                                                                           %         %        %            %
constants Fi , Gi such that
                                                                                                             of system (1.1) with initial conditions
 f j (0) = 0, f j (u) − f j (v) ≤ Fj u −v , gj (0) = 0, gj (u) − gj (v) ≤ Gj u −v                             xi ( s ) = ϕi ( s ), ϕi ( s ) < ζ i β , s ∈ R− , i = 1, 2,..., n. (2.4)
                                                                                                              %          %         %              I

,for all u , v ∈ R .                                                                                          Then
(T2) for     i, j ∈ {1, 2,...n} , take delay kernel                                                            xi (t ) < ζ i β for all t > 0, i = 1, 2,..., n .
                                                                                                                             I
                                                                                                                                                                                                                                           (2.5)
                   +
Kij ∈ C ( R , R) are continuous integrable, and satisfies                                                    Proof. Assume, by way of contradiction, that (2.2) does
                                                                                                             not hold. Then, there must exist
    ∞
∫       Kij ( s ) ds ≤kij .                                                                                   i ∈ {1, 2,..., n} and t0 > 0 such that
  0
      (T3)     there             exist       β > 0, λ > 0 and
                                             constants                                                          xi (t0 ) = ζ i β , and ϕi ( s ) < ζ i β for all
                                                                                                                %              I
                                                                                                                                       %              I


ζ i > 0, i = 1, 2,..., n such that for all t > 0 , there holds                                                t ∈ ( −∞, t0 ] , j = 1, 2,..., n .                                                                                               (2.6)
                          n                        n
− d i ( t ) h iζ i +     ∑a
                          j =1
                                   ij   F jζ j +   ∑b G ζ
                                                   j =1
                                                          ij       j   j   k ij < − β < 0 ,                  Calculating the upper left derivative                                                      xi (t ) , together
                                                                                                                                                                                                        %
i, j = 1, 2,..., n .                                                                                         with (T0 ) , (T1 ) , (T2 ) and (T3 ) implies that
For convenience, we introduce some notations. We will                                                           0 ≤ D + ( xi (t0 ) )
                                                                                                                          %
use x (t ) = ( x1 (t ), x2 (t ),..., xn (t )) ∈ R to denote a
                                             T   n
                                                                                                                                                       n                                                        n                ∞
                                                                                                              = [ −di (t0 )hi (t0 , xi (t0 )) + ∑ aij (t0 ) f j ( x j (t0 − τ ij (t0 ))) + ∑ dij (t0 )∫ Kij ( s)g j ( x j (t0 − s))ds
                                                                                                                                    %                             %                                                   %
                                                                               T                                                                                                                                                0
column vector in which they symbol ()                                                  denotes the                                                     j =1                                                    j =1



transpose of a vector. We let x denote absolute-value
                                                                                                                  + I i (t0 ) ] sgn( xi (t0 ))
                                                                                                                                     %
                                                                                                                                                 n                                           n                      ∞
                                                                                                                ≤ − d i ( t 0 ) hi ζ    I
                                                                                                                                            +   ∑      a ij ( t 0 ) F j ζ        I
                                                                                                                                                                                     +    ∑         bij ( t 0 ) ∫       K ij ( s ) d s G j ζ    I
                                                                                                                                                                                                                                                    + I i (t0 )
                                   x = ( x1 , x2 ,..., xn                  )
                                                                           T                                                           iβ                                       jβ
                                                                                                                                                                                                                    0
                                                                                                                                                                                                                                               jβ
vector given by                                                                    , and define                                                 j =1                                         j =1

                                                                                                                                                n                                             n
                                                                                                                                                                                                                                     I
                                                                                                              ≤ [ − d i ( t 0 ) hiζ i + ∑ a ij ( t 0 ) F j ζ j + ∑ bij ( t 0 ) k ij G j ξ j                                             + I i (t0 )
    x = m ax xi .                                                                                                                               j =1
                                                                                                                                                                                            
                                                                                                                                                                                             j =1                                    β
              1≤ i ≤ n
                                                                                                                                                 n                                       n
    The initial conditions associated with system (1.1) are                                                                                                                                                                     I
                                                                                                              ≤ [ − d i ( t 0 ) hi ζ i +        ∑      a ij e λ τ F j ζ     j    +   ∑            b ij k ij e λ s d s G j ζ j 
                                                                                                                                                                                                                               β + I i (t0 )
of the form                                                                                                                                     j =1                                     j =1


xi ( s ) = ϕi ( s ) , s ∈ ( −∞, 0] , i = 1, 2,..., n,                                                         < − β ×β + I i (t0 ) = 0.
                                                                                                                     I


where        ϕi (⋅)    denote a real-value bounded continuous                                                Which is a contradiction and implies that (2.6) holds, the
                                                                                                             proof of Lemma 2.1 is now completed.
function defined on    ( −∞, 0] .Denote                                                                      Remark 2.1. In view of the roundedness of this solution,
                                                                                                             from the theory of functional differential equations by Hale
R + = [ 0, ∞ ) , R− = ( −∞, 0] .                                                                                                                +
                                                                                                             (1977), it follows that x (t ) on R .
                                                                                                                                     %
Definition 2.1. Let x (t ) = ( x1 (t ), x2 (t ),..., xn (t ))
                                        *          *           *                       *    T
                                                              be an                                          Lemma 2.2. Suppose that (T 0) − (T 3) are satisfied. Let
anti-periodic solution of system (1.1) with initial                                                           x* (t ) = ( x1 (t ), x2 (t ),..., xn (t ))T be
                                                                                                                           *        *            *

value ϕ * (t ) = (ϕ1 (t ), ϕ 2 (t ),..., ϕ n (t ))T . If there exist
                   *         *             *
                                                                                                             the solution of system(1.1) with initial
constants       λ > 0 and M >                 1 such that for every solution                                 value ϕ * (t ) = (ϕ1 (t ), ϕ 2 (t ),..., ϕ n (t ))T , ϕ * (t ) < ζ i β ,
                                                                                                                                *         *             *                         I

Z (t ) = ( x1 (t ), x2 (t ),..., xn (t )) of system(1.1) with any
                                                   T

                                                                                                             and x (t ) = ( x1 (t ), x2 (t ),..., xn (t ))T be the solution of
initial value ϕ (t ) = (ϕ1 (t ), ϕ 2 (t ),..., ϕ n (t )) ,                     T
                                                                                                             system (1.1) with initial value

Research article                                                                           “Cellular neural networks”                                                                                                    Zhouhong Li et al.
Indian Society for Education and Environment (iSee)                                             http://www.indjst.org                                                                                                     Indian J.Sci.Technol.
                                                                                                                                                                                                                                                                           3

Indian Journal of Science and Technology                                                                                                     Vol. 3 No. 1 (Jan 2010)                                                           ISSN: 0974- 6846

ϕ (t ) = (ϕ1 (t ), ϕ 2 (t ),..., ϕ n (t ))T .Assume also the                                                                                               Vi (t ) = yi (t ) eλt < mξi ,for all
following condition is satisfied.                                                                                        t > 0 , i = 1, 2,..., n .                                          (2.10)
 (T 4) there exist constantsη > 0 , λ > 0 and ξi > 0 ,
                                                                                                                        Otherwise, there must exist i ∈ {1, 2,..., n} and ti > 0 such
i = 1, 2,..., n such that for all t > 0 , there holds
                           n                                     n
                                                                                                                        that
                                                                                                                                                     Vi (ti ) = mξi and V j (t ) < mξ j , for all
                                                                              ∞
   (λ − di (ti )hi )ξi + ∑ aij (ti ) e Fjξ j + ∑ bij (ti )∫
                                                 λτ
                                                                                  Kij (s) e dsGjξ j < −η < 0 .
                                                                                         λs
                                                                              0
                           j =1                                  j =1

Then there exist constants M ϕ > 1 such that                                                                             t ∈ (−∞, ti ) , j = 1, 2,..., n .                                      (2.11)
                                                                                                                        It follows from (2.8) that
                  xi (t ) − xi* (t ) ≤ M ϕ ϕ − ϕ * 1 e − λt , for all                                                                        Vi (t ) − mξ j = 0 and
                                         t > 0 , i = 1, 2,..., n .                                                       V j (t ) − mξ j < 0 , ∀t ∈ ( −∞, ti ) , j = 1, 2,..., n .                                                                 (2.12)
Proof.                                                                                                                  Together with (2.9) and (2.12), we obtain
                   {              }
Let y (t ) = y j (t ) = x j (t ) − x j (t ) = x (t ) − x (t ) .Then
                                                                 *                           *
                                                                                                                         0 ≤ D + (Vi (ti ) − mξi )
                                                                     n
yi′(t) =−di (t) hi (t, xi (t)) −hi (t, xi*(t)) + ∑aij (t) f j ( yj (t −τij (t)) + x*(t −τij (t)))                          = D + (Vi (ti )
                                                                                   j
                                                                  j=1                                                                        n                                                          n                   ∞
                                  n                   ∞                                                              ≤ [ λ − di (ti ) hi + ∑ aij (ti ) Fj y j (ti −τij (ti )) + ∑ bij (ti ) ∫ Kij (s) Gj yi (ti − s) ds]eλti
− f j (x (t −τij (t)))) + ∑bij (t)∫ Kij (s)(gj ( yj (t − s) + x (t − s))
          *                                                                              *                                                                                                                                 0
                                                                                                                                             j =1                                                       j =1
          j                                           0                                  j
                                                                                                                                                                 n
                                  j =1
                                                                                                                                                                 ∑
                                                                                                                                                                                                                                λ ( ti −τ ij ( ti ))       λτ ij ( t i )
                                                                                                                       = ( λ − d i ( t i ) h i ) e λ ti +                a ij ( t i ) F j y i ( t i − τ ij ( t i )) e                                  e
− g j ( x (t − s)))ds
              *
              j                                                                                    (2.7)                                                         j =1
                                                                                                                              n                      ∞
where i = 1, 2,..., n .                                                                                                  + ∑ b ij ( t i )        ∫       K ij ( s ) G         j       y j ( t i − s ) d s e λ ( ti − s ) e λ s d s
                                                                                                                                                     0
                                                                                                                             j =1
     We consider the Lyapunov functional                                                                                                                  n                                      n                     ∞
                                                                                  λt                                     ≤ [ (λ − d i (ti ) h i )ξ i + ∑ aij (ti ) e λτ F jξ j + ∑ bij (ti ) ∫ K ij ( s ) e λ s dsG jξ j ]m
                                                 Vi (t ) = yi (t ) e , i = 1, 2,..., n .                                                                  j =1                                   j =1
                                                                                                                                                                                                                      0


                 (2.8)                                                                                                  Thus,
                                                                                                                                                                     n                                         n                         ∞
Calculating the left upper derivative of Vi (t ) along the                                                               0 ≤ ( λ − d i ( t i ) h i )ξ i +        ∑j =1
                                                                                                                                                                          a ij ( t i ) e λ τ F j ξ j +         ∑
                                                                                                                                                                                                               j =1
                                                                                                                                                                                                                      b ij ( t i )   ∫   0
                                                                                                                                                                                                                                             K ij ( s ) e λ s d sG j ξ         j


solution y (t ) = yi (t ) of system (2.4) with the initial value                                                         Which contradicts (T4), Hence (2.10) hold. Letting
ϕ = ϕ − ϕ * , form (2.7), we have                                                                                        M ϕ > 1 such that
                                                          n
D+ (Vi (t)) ≤−di (t)hi yi (t) eλt + ∑ aij (t)( f j ( yj (t −τij (t)) + x*(t −τij (t)))
                                                                                                                         max {mξi } ≤ M ϕ ϕ − ϕ * , i = 1, 2,L , n.                                                                               (2.13)
                                                                        j                                                 1≤ i ≤ n                                                          1
                                                          j =1
                                                                                                                        In view of (2.10) and (2.13), we get
                                             n
                                                                                                                          xi (t ) − xi* (t ) = yi (t ) ≤ max {mξi } e− λt ≤ M ϕ ϕ − ϕ * e− λt ,
                                                                 ∞
− f j ( x (t −τ ij (t )))) e + ∑ bij (t )∫ Kij (s)(g j ( y j (t − s) + x (t − s))
          *
          j
                                   λt                                                                   *
                                                                                                        j
                                                                 0                                                                                                                1≤i ≤ n                                                                             1
                                            j =1
                                                                                                                        where i = 1,2,…, n, t > 0. This completes the proof of
                   − g j ( x (t − s ))) ds e + λ yi (t ) e
                                  *
                                  j
                                                                         λt                   λt
                                                                                                                        Lemma 2.2.
                                                                                                                        Remark 2.2. If x (t ) = ( x1 (t ), x 2 (t ),L , x n (t )) is the T-
                                                           n                                                                                              *                       *              *                         *             T
   ≤ (λ − di (t)hi ) yi (t) e + ∑ aij (t) Fj yi (t −τij (t)) e
                                            λt                                                     λt

                                                          j =1                                                          anti-periodic solution of system (1.1), it follows from
                                                                                                                                                              *
    n                  ∞                                                                                                Lemma 2.2 and Definition 1.1 that x (t ) is globally
+ ∑ bij (t ) ∫ Kij ( s) G j y j (t − s) dseλt ,                                                     (2.9)               exponentially stable.
                       0
   j =1

where i =1,2,..., n. Let                    m > 1 denote an arbitrary real                                              Lemma 2.3. Let                    T = (Tij ) n×n be a M-matrix, then
number such that
                                                                                                                         Tii > 0, i = 1, 2,L , n, Tij ≤ 0(i ≠ j , i, j = 1, 2,L , n),
        mξi > ϕ − ϕ * = sup max ϕ j ( s) − ϕ j * ( s) > 0 ,
                                        1        −∞< s ≤ 0 1≤ j ≤ n                                                     matrix G = ( gij ) n×n .
                                                 i = 1, 2,..., n .
                                                                                                                               0,          i = j,
It following from (2.8) that                                                                                             gij = 
   Vi (t ) = yi (t ) eλt < mξi , for all t ∈ R + , i = 1, 2,..., n .                                                           Tij / Tii , i, j = 1, 2,L , n.
We claim that

Research article                                                                                   “Cellular neural networks”                                                                                                   Zhouhong Li et al.
Indian Society for Education and Environment (iSee)                                                        http://www.indjst.org                                                                                                Indian J.Sci.Technol.
                                                                                                                                                                                                               4

Indian Journal of Science and Technology                                                                                          Vol. 3 No. 1 (Jan 2010)                              ISSN: 0974- 6846

3. Results                                                                                                      (−1)k+1vi (t + (k +1)T) −(−1)k vi (t + kT) ≤ α(e−λT )k =αe−λτk ,
The following is our main result.
Theorem 3.1. Suppose that (T0) - (T4) are satisfied. Then                                                      ∀k > P, i = 1, 2,L , n, (3.6)
system (1.1) has exactly are                                                                                   on any compact set of R. It follows from (3.5),(3.6) and
                             *                     *
T-anti-periodic solution x (t ) . Moreover, x (t ) is globally                                                 (3.7) that ( −1) v(t + pT ) uniformly converges to a
                                                                                                                                         p

exponentially stable.
                                                                                                               continuums function x (t ) = ( x1 (t ), x2 (t ),L , xn (t )) on
                                                                                                                                                    *              *           *              *       T
Proof. Let v(t ) = (v1 (t ), v2 (t ),L , vn (t )) be a solution of
                                                 T
                                                                                                               any compact set of R.
system(1.1) with initial conditions                                                                                                                          *
                                                                                                                    Now we will show that x (t ) is T-anti-periodic solution
vi ( s ) = ϕi v ( s ) ,                                                                                                                                *
                                                                                                               of system(1.1). First, x (t ) is T-anti-periodic, since,
                        I
ϕi v ( s ) < ζ i            , s ∈ (−∞, 0], i = 1, 2,L , n.                           (3.1)                          x*(t +T) = lim(−1)p v(t +T + pT) =− lim (−1)p+1v(t +( p+1)T) =−x*(t)
                        β                                                                                                          p→∞                              ( p+1)→∞
                                                                                                                                                   *
According to Remark 2.1, v (t ) exists on ( −∞, 0] .                                                           Next, we prove that x (t ) is a solution of system (1.1). In
Moreover, by Lemma 2.1, the solution v (t ) is bounded                                                         fact, together with the continuity of the right side of
                                                                                                               system (1.1) and (3.3) implies that
and
                    I                                                                                          {((−1)     p +1
                                                                                                                                 v(t + ( p + 1)T ))′} uniformly converges to a
 vi (t ) < ζ i           , for all t ∈ R, i = 1, 2,L , n.                                 (3.2)                continuous function on any compact set of R. Thus,
                    β                                                                                          letting p → ∞ , we obtain
Form (2.1)-(2.3), we have                                                                                                                            n                                n
                                                                                                               d *
                                                                                                                  {xi (t)} =−di (t)hi (t, x*i (t)) +∑aij (t) f j (x*j (t −τij (t))) +∑bij (t)∫0 Kij (s)gj (x*j (t −s))ds
                                                                                                                                                                                              ∞
    ((−1) k +1 vi (t + (k + 1)T ))′                                                                            dt                                   j =1                             j=1
= (−1)k +1vi′(t + (k +1)T )
                                                  n                                               n
                                                                                                                    + I i (t ), i = 1, 2,..., n .                              (3.7)
= −di (t )hi (t,(−1)k +1vi (t + (k +1)T )) + ∑aij (t) f j ((−1)k +1 v j (t + (k +1)T −τ ij (t))) + ∑bij (t)                         *
                                                  j =1                                            j =1         Therefore, x (t ) is a solution of (1.1). Then, by Lemma
       ∞                                                                                                                                 *
  ×∫ K ij ( s )g j (−1)          k +1
                                        (v j (t + (k + 1)T − s ))ds + I i (t ),                                2.2, we can prove that x (t ) is globally exponentially
      0
                                                                                                               stable. This completes the proof.
i = 1, 2,L , n.
   Thus, for any natural number k , ( −1) k +1 v(t + ( k + 1)T )                                               4. An example
                                                                                                                  In this section, we will give an example illustrate the
are the solution of system (1.1) on R . Then, by Lemma
                                                                                                               feasibility and effectiveness of our results obtained in
2.2, there exists a constant Q > 0 such that
                                                                                                               section 3.
           (−1) k +1 vi (t + (k + 1)T ) − (−1) k vi (t + kT )                                                  Let n = 2. Consider the following cellular neural networks
                                                                                                              ⋅                   sinu             sint           sint ∞
≤ Qe − λ ( t + kT ) sup max vi ( s + T ) + vi ( s)
                                                                                                                                                                     8 ∫0
                                                                                                              x1(t) =−(1+ sinu)+         x1(t −2)+      x2(t −1)+        sinse−sx1(t −s)ds
                        −∞< s ≤ 0 1≤ i ≤ n                                                                                           8               72
                                                                                                              
                              I                                                                                       sint ∞
                                                                                                                          72 ∫0
≤ 2e − λ (t + kT )Q max ζ i  , for all t+kT>0, i=1,2,…,n.                                                           +         coss e−sx2(t −s)ds −sint,
                                                                                                              
                    1≤ i ≤ n
                              β                                                                             ⋅
(3.3)                                                                                                         x (t) =−(1+ cosu)+ cosu x (t −6)+ cost x (t −4)+ cost ∞ cosse−sx (t −s)ds
Thus, for any natural number p, we obtain                                                                     2                                                      2 ∫0
                                                                                                                                            1              2                         1
                                                                                                                                      2                8
                                              p
                                                                                                              
(−1) p +1 vi (t + ( p + 1)T ) = vi (t ) + ∑ (−1) p +1 vi (t + ( p + 1)T ) − (−1) p vi (t + pT ) 
                                                                                                                      cost ∞
                                                                                                                              ∫0 sins e x2(t −s)ds+sint,
                                                                                                                                        −s
                                             k =0                                                                      +
                                                         (3.4)                                                
                                                                                                                          8
Then,                                                                                                                                                                                              (4.1)
                                                  p                                                            where
(−1) p+1 vi (t + ( p +1)T ) ≤ vi (t ) + ∑ (−1) p+1 vi (t + ( p + 1)T ) − (−1) p vi (t + pT )
                                                                                            
                                               k =0                                                            d1(t) = d2 (t) = 1, h1(t, u) = 1+ | sin u |, h2 (t, u) = 1+ | cos u | ,
                          , (3.5)                                                                                                   1
where i = 1, 2,L , n.                                                                                          a11 (t ) = b11 (t ) = | sin t |,
                                                                                                                                    8
   In view of (3.4), we can choose a sufficiently large
constant P > 0 and a positive constant α > 0 such that
                                                                                                                               1                       1                        1
                                                                                                               a12(t) =b (t) = |sint |,a21(t) =b21(t) = |cost |,a22(t) =b22(t) = |cost |,
                                                                                                                        12
                                                                                                                               72                      2                        8

Research article                                                                     “Cellular neural networks”                                                                        Zhouhong Li et al.
Indian Society for Education and Environment (iSee)                                           http://www.indjst.org                                                                     Indian J.Sci.Technol.
                                                                                                                                                                         5

Indian Journal of Science and Technology                                                                             Vol. 3 No. 1 (Jan 2010)        ISSN: 0974- 6846

f1(u) = f2(u) = g1(u) = g2(u) =u, K11(s) = K22(s) =|sins| e−s, K (s) = K21(s) =|coss| e−s,
                                                                12
                                                                                                              solutions of discrete delayed cellular neural networks.
                                                                                                              Phy. Lett. A. 333, 51-61.
I1(t) = −sin t, I2 (t) = sin t,τ11(t) = 2,τ12 (t) =1,τ21(t) = 6,τ22 (t) = 4. 8.                               Mohamad S (2007) Global exponential stability in
                                                                                                              DCNNs with distributed delays and unbounded
Note that                                                                                                     activations. J. Comput. Appl. Math. 205 161-173.
                                               1             1                                         9.     Peng G and Huang L (2009) Anti-periodic solutions
h1 = h2 =1, F = F2 =1, G1 = G2 = 1, a11 = b11 = , a12 = b12 = ,
             1                                                                                                for shunting inhibitory cellular neural networks with
                                               8             72                                               continuously distributed delays. Nonlin. Anal. 10,
             1               1                                                                                2434-2440.
a21 = b21 = , a22 = b22 = ,                                                                            10.    Roska T Vandewalle J (1995) Cellular neural
             2               8
                       ∞
                                                                                                              networks, Wiley, New York.
I1 = I 2 = 1, ∫ K ij ( s )ds ≤ kij = 1, i, j = 1, 2. Therefore, it                                     11.    Shao J (2008) Anti-periodic solutions for shunting
                   0                                                                                          inhibitory cellular neural networks with time-varying
follows from the theory of M-matrix in [16] that there exist                                                  delays. Phy. Lett. A. 372, 5011-5016.
                       1
constant      β=         > 0, and ζ 1 = ζ 2 = 1 such that for all t
                       6
>0, there holds
                           2                       2
                                                                                         1
− d i ( t ) h iζ i +   ∑
                       j =1
                               a1 j F jζ   j   +   ∑
                                                   j =1
                                                          b1 j G j ζ j k 1 j < − β = −
                                                                                         6
                                                                                           < 0,


                           2                        2
                                                                                           1
− d i ( t ) h iζ i +   ∑a
                       j =1
                                2 j   F jζ j +     ∑b
                                                   j =1
                                                            2 j   G jζ j k 2 j < − β = −
                                                                                           6
                                                                                             < 0,


where i = 1, 2, which implies that system(4.1) satisfies all
the conditions in Theorem 3.1.
Hence, system (4.1) has exactly one T-anti-periodic
solution. Moreover, the T-anti-periodic solution is globally
exponentially stable.

Conclusions
   In this paper, cellular neural networks with time-
varying delays and continuously distributed delays have
been studied. New sufficient conditions for the existence
and exponential stability of anti-periodic solutions have
been established which extend and improve some
previously known results.

References
1. Cao J and Wang J (2005) Global asymptotic and
    robust stability of recurrent neural networks with time
    delays. IEEE Trans. Circ. Syst. I. 52 (2) 417-426.
2. Chen YM (2002) Global stability of neural networks
    with distributed delays. Neural Networks.15, 867-871.
3. Cheng CY, Lin KH and Shih CW (2006) Multi stability
    in recurrent neural networks. SIAM J.Appl. Math. 66
    (4), 1301-1320.
4. Chua LO and Yang L (1988) Cellular neural
    networks: theory and applications. IEEE Trans.Circ.
    Syst. 35, 1257-1290.
5. Hale JK (1977) Theory of functional differential
    equations. Springer-Verlag, NY.
6. Johnson CR and Smith RL (1996) The completion
    problem for M-matrices and inverse M-matrices,
    linear algebra and its applications. Lin. Alg. Appl.
    241-243, 655-667.
7. Li Y (2004) Global stability and existence of periodic
Research article                                 “Cellular neural networks”                                                                          Zhouhong Li et al.
Indian Society for Education and Environment (iSee)                                       http://www.indjst.org                                      Indian J.Sci.Technol.

				
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