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1 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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