Electronic Journal of Differential Equations, Vol. 2006(2006), No. 32, pp. 1–21. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu (login: ftp) EXISTENCE AND EXPONENTIAL STABILITY OF PERIODIC SOLUTION FOR CONTINUOUS-TIME AND DISCRETE-TIME GENERALIZED BIDIRECTIONAL NEURAL NETWORKS YONGKUN LI Abstract. We study the existence and global exponential stability of positive periodic solutions for a class of continuous-time generalized bidirectional neural networks with variable coefficients and delays. Discrete-time analogues of the continuous-time networks are formulated and the existence and global exponential stability of positive periodic solutions are studied using the continuation theorem of coincidence degree theory and Lyapunov functionals. It is shown that the existence and global exponential stability of positive periodic solutions of the continuous-time networks are preserved by the discrete-time analogues under some restriction on the discretization step-size. An example is given to illustrate the results obtained. 1. Introduction In recent years, the stability of the following bidirectional associative neural networks with or without delays has been extensively studied: m X dxi (t) = −ai xi (t) + pji fj (yj (t − τji )) + Ii , dt j=1 l X dyj (t) qij gi (xi (t − σij )) + Jj , = −bj yj (t) + dt i=1 i = 1, 2, . . . , l, (1.1) j = 1, 2, . . . , m. Also some of its generalizations have been studied and various stability conditions have been obtained [3, 4, 5, 6, 11, 12, 14, 15, 16, 17]. Here pji , qij , i = 1, 2, . . . , l, j = 1, 2, . . . , m are the connection weights through the neurons in two layers: I-layer and J-layer. On I-layer, the neurons whose states denoted by xi (t) receive the inputs Ii and the inputs outputted by those neurons in J-layer via activation functions (output-input functions) fj , while on J-layer, the neurons whose associated states denoted by yj (t) receive the inputs Ji and the inputs outputted from those neurons in I-layer via activation functions (output-input functions) gi . And τji , σij , i = 2000 Mathematics Subject Classification. 34K13, 34K25. Key words and phrases. Bidirectional neural networks; global exponential stability; periodic solution; Fredholm mapping; Lyapunov functionals; discrete-time analogues. c 2006 Texas State University - San Marcos. Submitted February 16, 2006. Published March 16, 2006. Supported by grants 10361006 from the National Natural Sciences Foundation of China, and 2003A0001M from the Natural Sciences Foundation of Yunnan Province. 1 2 Y. LI EJDE-2006/32 1, 2, . . . , l, j = 1, 2, . . . , m are the associated delays due to the finite transmission speed among neurons in different layers. When there is no delay present, (1.1) reduces to a system of ordinary differential equations which was investigated by Kosko [7, 8, 9] and it produces many nice properties due to the special structure of connection weights and has practical applications in storing paired patterns or memories and the ability to search the desired patterns via both directions: forward and backward directions. See [3, 14, 7, 8, 9] for details about the applications on learning and associative memories. It is well known that in the study of neural dynamical systems, the periodic oscillatory behavior of the systems is an important aspect. In this paper, we are concerned with the following generalized BAM networks with variable coefficients and delays m m X X dxi (t) = −ai (t)xi (t) + aji (t)fj (yj (t)) + pji (t)gj (yj (t − τji (t))) + Ii (t), dt j=1 j=1 l l X X dyj (t) = −bj (t)yj (t) + bij (t)fˆi (xi (t)) + qij (t)ĝi (xi (t − σij (t))) + Jj (t), dt i=1 i=1 (1.2) for i = 1, 2, . . . , l, j = 1, 2, . . . , m. In this paper, we assume that (S1) ai , bj ∈ C(R, (0, ∞)), τji , σij ∈ C(R, [0, ∞)), Ii , Jj , pji , qij ∈ C(R, R), i = 1, 2, . . . , l, j = 1, 2, . . . , m are all ω-periodic functions. (S2) fj , gj , fˆi , ĝi ∈ C(R, R) i = 1, 2, . . . , l, j = 1, 2, . . . , m are bounded on R. ˆ (S3) There exist positive number Lfj , Lgj , Lfi , Lĝj such that |fj (x) − fj (y)| ≤ Lfj |x − y| for all x, y ∈ R, j = 1, 2, . . . , m, |gj (x) − gj (y)| ≤ Lgj |x − y| for all x, y ∈ R, j = 1, 2, . . . , m, ˆ |fˆi (x) − fˆi (y)| ≤ Lfi |x − y| for all x, y ∈ R, i = 1, 2, . . . , l, |ĝi (x) − ĝi (y)| ≤ Lĝi |x − y| for all x, y ∈ R, i = 1, 2, . . . , l. Our purpose of this paper is by using Mawhin’s continuation theorem of coincidence degree theory [2, 13] and by constructing suitable Lyapunov functions to investigate the stability and existence of periodic solutions of (1.2); then, we shall use a novel method in formulating discrete-time analogues of the continuous time networks. It is shown that the existence and global exponential stability of positive periodic solutions of the continuous-time networks are preserved by the discrete-time analogues under some restriction on the discretization step-size. 2. Existence of periodic solutions In this section, based on the Mawhin’s continuation theorem, we shall study the existence of at least one positive periodic solution of (1.2). First, we shall make some preparations. Let X, Y be normed vector spaces, L : Dom L ⊂ X → Y be a linear mapping, and N : X → Y be a continuous mapping. The mapping L will be called a Fredholm mapping of index zero if dim ker L = codim Im L < +∞ and Im L is closed in Y. If L is a Fredholm mapping of index zero, there exist continuous projectors P : X → X EJDE-2006/32 EXISTENCE AND EXPONENTIAL STABILITY 3 and Q : Y → Y such that Im P = ker L, ker Q = Im L = Im(I − Q). It follows that mapping L|Dom L T ker P : (I − P )X → Im L is invertible. We denote the inverse of that mapping by KP . If Ω is an open bounded subset of X, the mapping N will be called L- compact on Ω̄ if QN (Ω̄) is bounded and KP (I − Q)N : Ω̄ → X is compact. Since Im Q is isomorphic to ker L, there exists an isomorphism J : Im Q → ker L. For convenience of use, we introduce Mawhin’s continuation theorem [2, P. 40] as follows. Lemma 2.1. Let Ω ⊂ X be an open bounded set and let N : X → Y be a continuous operator which is L-compact on Ω̄ (i.e., QN : Ω̄ → Y and KP (I − Q)N : Ω̄ → Y are compact). Assume T (a) for each λ ∈ (0, 1), T x ∈ ∂Ω Dom LLx 6= λN x, (b) for each x ∈ T ∂Ω ker L. QN x 6= 0, (c) deg(JN Q, Ω ker L, 0) 6= 0. T Then Lx = N x has at least one solution in Ω̄ Dom L. Our result about the existence of periodic solutions of (1.2) is as follows. Theorem 2.2. Assume that (S1) and (S2) hold. Then system (1.2) has at least one ω-periodic solution. Proof. To use the continuation theorem of coincidence degree theory to establish the existence of an ω-periodic solution of (1.2), we take X = Y = {u ∈ C(R, Rl+m ) : Pl+m u(t + ω) = u(t)} and kuk = i=1 maxt∈[0,ω] |ui (t)|, then X is a Banach space. Set L : Dom L ∩ X, Lu = u̇(t), u ∈ X, where Dom L = {u ∈ C 1 (R, Rl+m )} and N : X → X, N [x1 , . . . , xl , y1 , . . . , ym ]T = Pm aj1 (t)fj (yj (t)) + j=1 pj1 (t)gj (yj (t − τj1 (t))) + I1 (t) .. . −a (t)x (t) + Pm a (t)f (y (t)) + Pm p (t)g (y (t − τ (t))) + I (t) l l j j j j jl l j=1 jl j=1 jl P P . −b1 (t)y1 (t) + li=1 bi1 (t)fˆi (xi (t)) + li=1 qi1 (t)ĝi (xi (t − σi1 (t))) + J1 (t) .. . Pl P l bim (t)fˆi (xi (t)) + qim (t)ĝi (xi (t − σim (t))) + Jm (t) −bm (t)ym (t) + −a1 (t)x1 (t) + Pm j=1 i=1 i=1 Define two projectors P and Q as 1 P u = Qu = ω Z ω u(t) dt, u ∈ X. 0 Rω Clearly, ker L = Rl+m , Im L = {(x1 , x2 , . . . , xl , y1 , . . . , ym )T ∈ X : 0 xi (t) dt = Rω 0, 0 yj (t) dt = 0, i = 1, 2, . . . , l, j = 1, 2, . . . , m} is closed in X and dim ker L = codim Im L = l + m. Hence, L is a Fredholm mapping of index zero. Furthermore, similar to the proof of [10, Theorem 1], one can easily show that N is L-compact on Ω̄ with any open bounded set Ω ⊂ X. Corresponding to operator equation 4 Y. LI EJDE-2006/32 Lu = λN u, λ ∈ (0, 1), we have m m h i X X dxi (t) = λ − ai (t)xi (t) + aji (t)fj (yj (t)) + pji (t)gj (yj (t − τji (t))) + Ii (t) , dt j=1 j=1 l l h i X X dyj (t) = λ − bj (t)yj (t) + bij (t)fˆi (xi (t)) + qij (t)ĝi (xi (t − σij (t))) + Jj (t) dt i=1 i=1 (2.1) for i = 1, 2, . . . , l, j = 1, 2, . . . , m. Suppose that (x1 , x2 , . . . , xl , y1 , y2 , . . . , ym ) ∈ X is a solution of (2.1) for some λ ∈ (0, 1). Let ξ¯i , ζ̄j ∈ [0, ω] such that xi (ξ¯i ) = maxt∈[0,ω] xi (t) and yj (ζ̄j ) = maxt∈[0,ω] yj (t), i = 1, 2, . . . , l, j = 1, 2 . . . , m, then m X ai (ξ¯i )xi (ξ¯i ) = aji (ξ¯i )fj (yj (ξ¯i )) + j=1 m X pji (ξ¯i )gj (yj (ξ¯i − τji (ξ¯i ))) + Ii (ξ¯i ) j=1 and l X bj (ζ̄j )yj (ζ̄j ) = l X bij (ζ̄j )fˆi (xi (ζ̄j )) + i=1 qij (ζ̄j )ĝi (xi (ζ̄j − σij (ζ̄j ))) + Jj (ζ̄j ), i=1 for i = 1, 2, . . . , l, j = 1, 2, . . . , m. Hence, ai (ξ¯i )xi (ξ¯i ) ≤ m X m X |aji (ξ¯i )||fj (yj (ξ¯i ))| + j=1 bj (ζ̄j )yj (ζ̄j ) ≤ l X |pji (ξ¯i )||gj (yj (ξ¯i − τji (ξ¯i )))| + |Ii (ξ¯i )|, j=1 |bij (ζ̄j )||fˆi (xi (ζ̄j ))| + l X i=1 |qij (ζ̄j )||ĝi (xi (ζ̄j − σij (ζ̄j )))| + |Jj (ζ̄j )| i=1 for i = 1, 2, . . . , l, j = 1, 2, . . . , m. Therefore, 1 xi (ξ¯i ) ≤ [mMf aM + mMg pM + I M ], i = 1, 2, . . . , l, a 1 yj (ζ̄j ) ≤ [lMfˆbM + lMĝ q M + J M ], j = 1, 2, . . . , m, b where a = min {ai (t), i = 1, 2, . . . , n}, t∈[0,ω] Mf = sup {|fj (u)|, j = 1, 2, . . . , m}, u∈R b = min {bj (t), i = 1, 2, . . . , l}, t∈[0,ω] Mg = sup {|gj (u)|, j = 1, 2, . . . , m}, u∈R Mfˆ = sup {|fˆi (u)|, i = 1, 2, . . . , l}, u∈R M a Mĝ = sup {|ĝi (u)|, i = 1, 2, . . . , l}, u∈R = max {|aji (t)|, i = 1, 2, . . . , l, j = 1, 2, . . . , m}, t∈[0,ω] b M = max {|bij (t)|, i = 1, 2, . . . , l, j = 1, 2, . . . , m}, t∈[0,ω] pM = max {|pji (t)|, i = 1, 2, . . . , l, j = 1, 2, . . . , m}, t∈[0,ω] q M = max {|qij (t)|, i = 1, 2, . . . , l, j = 1, 2, . . . , m}, t∈[0,ω] I M = max {|Ii (t)|, i = 1, 2, . . . , l}, t∈[0,ω] J M = max {|Ji (t)|, j = 1, 2, . . . , m}. t∈[0,ω] EJDE-2006/32 EXISTENCE AND EXPONENTIAL STABILITY 5 Let ξ i , ζ j ∈ [0, ω] be such that xi (ξ i ) = mint∈[0,ω] xi (t) and yj (ζ j ) = mint∈[0,ω] yj (t), i = 1, 2, . . . , l, j = 1, 2, . . . , m, then m X ai (ξ i )xi (ξ i ) = aji (ξ i )fj (yj (ξ i )) + j=1 n X j=1 n X i=1 i=1 bij (ζ j )fˆi (xi (ζ j )) + bj (ζ j )yj (ζ j ) = m X pji (ξ i )gj (yj (ξ i − τji (ξ i ))) + Ii (ξ i ) qij (ζ j )ĝi (xi (ζ j − σij (ζ j ))) + Jj (ζ j ) for i = 1, 2, . . . , l, j = 1, 2, . . . , m. Thus, ai (ξ i )xi (ξ i ) ≥− m X m X |aji (ξ i )||fj (yj (ξ i ))| − j=1 |pji (ξ i )||gj (yj (ξ i − τji (ξ i )))| − |Ii (ξ i )|, j=1 bj (ζ j )yj (ζ j ) ≥− n X |bij (ζ j )||fˆi (xi (ζ j ))| − i=1 n X |qij (ζ j )||ĝi (xi (ζ j − σij (ζ j )))| − |Jj (ζ j )| i=1 for i = 1, 2, . . . , l, j = 1, 2, . . . , m. Therefore, 1 xi (ξ¯i ) ≥ − [mMf aM + mMg pM + I M ], a 1 yj (ζ̄j ) ≥ − [lMfˆbM + lMĝ q M + J M ] b for i = 1, 2, . . . , l, j = 1, 2, . . . , m. Denote C = al [mMf aM + mMg pM + I M ] + m M + lMĝ q M + J M ] + D, where D is a positive constant. Then it is clear b [lMfˆb that C is independent of λ. Now we take Ω T = {u ∈ X : kuk T < C}. This Ω satisfies condition (a) in Lemma 2.1. When u ∈ ∂Ω ker L = ∂Ω Rl+m , u is a constant vector in Rm+n with kuk = C. Then uT QN u = l n X − āi x2i + i=1 + m X āji xi fj (yj ) + j=1 m n X − b̄j yj2 + j=1 l X m X p̄ji xi gj (yj ) + xi I¯i o j=1 b̄ij yj fˆi (xi ) + i=1 l X q̄ij yj ĝi (xi ) + yj J¯j o < 0, i=1 where u = (x1 , x2 , . . . , xl , y1 , y2 , . . . , ym ). If necessary, we can let C be large such that l n X − āi x2i + i=1 + m n X āji xi fj (yj ) + j=1 − b̄j yj2 + j=1 So for any u ∈ ∂Ω is satisfied. m X l X i=1 T m X p̄ji xi gj (yj ) + xi I¯i o j=1 b̄ij yj fˆi (xi ) + l X q̄ij yj ĝi (xi ) + yj J¯j o < 0. i=1 ker L, QN u 6= 0. This proves that condition (b) in Lemma 2.1 6 Y. LI EJDE-2006/32 Furthermore, let Ψ(γ; u) = −γu + (1 − γ)QN u, then for any x ∈ ∂Ω ∩ ker L, uT Ψ(γ; u) < 0, we get deg{JQN, Ω ∩ ker L, 0} = deg{−u, Ω ∩ ker L, 0} = 6 0, hence condition (c) of Lemma 2.1 is also satisfied. Thus, by Lemma 2.1 we conclude that Lu = N u has at least one solution in X, that is, (1.2) has at least one ω-periodic solution. The proof is complete. Next, we shall construct some suitable Lyapunov functionals to derive the sufficient conditions which ensure that the global exponential stability of periodic solutions of the system (1.2) associated with the initial conditions xi (s) = ϕi (s), s ∈ [−τ, 0], yj (s) = ψj (s), s ∈ [−σ, 0], τ = max {τji (t), i = 1, 2, . . . , l, j = 1, 2, . . . , m}, t∈[0,ω] σ = max {σij (t), i = 1, 2, . . . , l, j = 1, 2, . . . , m}, t∈[0,ω] where ϕi ∈ C([−τ, 0], R), ψi ∈ C([−σ, 0], R), i = 1, 2, . . . , l, j = 1, 2, . . . , m. In the sequel, we will use the following notation: am i = min ai (t), bm j = min bj (t), t∈[0,ω] bM ij = max |qij (t)|, t∈[0,ω] M τji = max |τji (t)|, t∈[0,ω] pM ji aM ji = max |pji (t)|, t∈[0,ω] t∈[0,ω] = max |pji (t)|, t∈[0,ω] M qij M σij = max |σij (t)|, t∈[0,ω] = max |qij (t)|, t∈[0,ω] where i = 1, 2, . . . , l, j = 1, 2, . . . , m. Our result about the global exponential stability of periodic solutions of (1.2) is as follows. Theorem 2.3. Assume that (S1)-(S3) hold. Furthermore, assume (P1) For i = 1, 2, . . . , l and j = 1, 2, . . . , m, σij , τji ∈ C 1 (R, [0, ∞)) satisfy 0M 0 σij = max σij (t) < 1, t∈[0,ω] 0M 0 τji = max τji (t) < 1, t∈[0,ω] (P2) am i > m X ˆ f bM ij Li + j=1 bm j > l X f aM ji Lj + i=1 M ĝ qij Li , 0M 1 − σij g pM ji Lj 0M 1 − τji , i = 1, 2, . . . , l, j = 1, 2, . . . , m, ∗ T then (1.2) has a unique ω-periodic solution (x∗1 , x∗2 , . . . , x∗l , y1∗ , y2∗ , . . . , ym ) and, moreover, there exist constants η > 0 and Λ ≥ 1 such that for t > 0, l X |xi (t) − x∗i (t)| + i=1 ≤ Λe−ηt m X |yj (t) − yj∗ (t)| j=1 l hX sup |xi (s) − x∗i (s)| + i=1 s∈[−τ,0] m X i sup |yj (s) − yj∗ (s)| . j=1 s∈[−σ,0] EJDE-2006/32 EXISTENCE AND EXPONENTIAL STABILITY 7 Proof. Let x(t) = {x1 (t), x2 (t), . . . , xl (t), y1 (t), y2 (t), . . . , ym (t)} be an arbitrary so∗ lution of (1.2), and x∗ (t) = {x∗1 (t), x∗2 (t), . . . , x∗l (t), y1∗ (t), y2∗ (t), . . . , ym (t)} be an ω-periodic solution of (1.2). Then m X d+ (xi (t) − x∗i (t)) f ∗ ∗ ≤ −am |x (t) − x (t)| + aM i i i ji Lj |yj (t) − yj (t)| dt j=1 + m X g ∗ pM ji Lj |yj (t − τji (t)) − yj (t − τji (t))|, j=1 n X d+ (yj (t) − yj∗ (t)) fˆ ∗ ∗ ≤ −bm |y (t) − y (t)| + bM j j j ij Li |xi (t) − xi (t)| dt i=1 + n X (2.2) M ĝ qij Li |xi (t − σij (t)) − x∗i (t − σij (t))|, i=1 fori = 1, 2, . . . , l, j = 1, 2, . . . , m. Let Fi and Gj be defined by Fi (εi ) = am i m X − εi − fˆ bM ij Li + j=1 Gj (ζj ) = bm j − ζj − l X f aM ji Lj + i=1 M M ĝ εi σij qij Li e , 0M 1 − σij M g ζj τji pM ji Lj e 0M 1 − τji i = 1, 2, . . . , l, , j = 1, 2, . . . , m, where εi , ζj ∈ [0, ∞). It is clear that Fi (0) = am i − m X M ĝ qij Li > 0, 0M 1 − σij ˆ f bM ij Li + j=1 Gj (0) = bm j − l X f aM ji Lj + i=1 g pM ji Lj 0M 1 − τji > 0, i = 1, 2, . . . , l, j = 1, 2, . . . , m. Since Fi (·) and Gj (·) are continuous on [0, ∞) and Fi (εi ), Gj (ζj ) → ∞ as εi , ζj → ∞, there exist ε∗i , ζj∗ > 0 such that Fi (εi ) = 0, Gj (ζj ) = 0 for εi ∈ (0, ε∗i ) and Fi (εi ) > 0, Gj (ζj ) > 0 for ζj ∈ (0, ζj∗ ). By choosing ∗ η = min{ε∗1 , ε∗2 , . . . , ε∗l , ζ1∗ , ζ2∗ , . . . , ζm }, we obtain Fi (η) = am i −η− m X fˆ bM ij Li + j=1 Gi (η) = bm j −η− l X f aM ji Lj + i=1 M M ĝ ησij qij Li e ≥ 0, 0M 1 − σij M g ητji pM ji Lj e 0M 1 − τji ≥ 0, i = 1, 2, . . . , l, j = 1, 2, . . . , m. Now let us define ui (t) = eηt |xi (t) − x∗i (t)|, ηt vj (t) = e |yj (t) − yj∗ (t)|, t ∈ [−τ, ∞), i = 1, 2, . . . , l, t ∈ [−τ, ∞), j = 1, 2, . . . , m. (2.3) 8 Y. LI EJDE-2006/32 Then it follows from (2.2) and (2.3) that m m X X d+ ui (t) g ητji (t) m M f ≤ −(ai − η)ui (t) + aji Lj vj (t) + pM vj (t − τji (t)), ji Lj e dt j=1 j=1 n n X X d+ vj (t) M fˆ M ĝ ησij (t) ≤ −(bm − η)v (t) + b L u (t) + qij Li e ui (t − σij (t)), j j ij i i dt i=1 i=1 (2.4) i = 1, 2, . . . , n, j = 1, 2, . . . , m. Consider a Lyapunov function defined by V (t) = l X ui (t) + i=1 + M Z g ητji m X pM ji Lj e j=1 m X 1− 0M τji vj (t) + vj (s) ds t−τji (t) M M ĝ ησij qij Li e 0M 1 − σij i=1 l X j=1 t (2.5) Z t ui (s) ds , t−σij (t) and we note that V (t) > 0 for t > 0 and V (0) is positive and finite. Calculating the derivatives of V along the solutions of (2.4), we get M g ητji l h m m i X X X pM dV + ji Lj e m M f v (t) ≤ − (ai − η)ui (t) + aji Lj vj (t) + j 0M dt 1 − τji i=1 j=1 j=1 + m h X − (bm j − η)vj (t) + j=1 ≤− l h X am i m X −η− fˆ bM ij Li − j=1 m h X bm j −η− j=1 =− l M ĝ ησij X qij Li e + l X Fi (η)ui (t) − i=1 i=1 M m M ĝ ησij X qij Li e j=1 f aM ji Lj − i=1 l X M fˆ bM ij Li ui (t) i=1 i=1 − l X M g ητji l X pM ji Lj e i=1 m X 0M 1 − σij 0M 1 − τji Gj (η)vj (t) ≤ 0, 0M 1 − σij i ui (t) i ui (t) i vj (t) t > 0. j=1 It follows that V (t) ≤ V (0) for t > 0 and hence from (2.3) and (2.5) we obtain l X i=1 ui (t) + m X j=1 vj (t) ≤ l X ui (0) + i=1 + m X j=1 M Z g ητji m X pM ji Lj e j=1 vj (0) + 1− 0M τji M M ĝ ησij qij Li e 0M 1 − σij i=1 l X 0 vj (s) ds −τji (0) (2.6) Z 0 −σij (0) ui (s) ds . EJDE-2006/32 EXISTENCE AND EXPONENTIAL STABILITY 9 It follows from (2.3) and (2.6) that l X |xi (t) − x∗i (t)| + i=1 |yj (t) − yj∗ (t)| j=1 ≤ e−ηt l X 1+ i=1 + m X m X 1+ j=1 −ηt ≤ Λe l hX m X M g ητji pM ji Lj e j=1 0M 1 − τji M M ĝ ησij Li e qij 0M 1 − σij i=1 l X s∈[−τ,0] (2.7) sup |xi (s) − i=1 s∈[−τ,0] sup |xi (s) − x∗i (s)| sup |yj (s) − yj∗ (s)| s∈[−σ,0] x∗i (s)| + m X i sup |yj (s) − yj∗ (s)| , j=1 s∈[−σ,0] where t > 0 and M M g ητji l m M ĝ ησij o n X X pM qij Li e ji Lj e ,1 + ≥ 1. Λ= max 1+ 0M 0M 1≤i≤l, 1≤j≤m 1 − τji 1 − σij i=1 j=1 The uniqueness of the periodic solution is follows from (2.7). This completes the proof. 3. Discrete-time analogues In this section, we shall use a semi-discretization technique to obtain the discretetime analogue of (1.2). For convenience, we use the following notations. Let Z denote the set of integers; Z+ 0 = {0, 1, 2, . . . }; [a, b]Z = {a, a + 1, . . . , b − 1, b}, where a, b ∈ Z, a ≤ b; [a, ∞)Z = {a, a+1, a+2, . . . }, where a ∈ Z. While there is no unique way of obtaining a discrete-time analogue from the continuous-time network (1.2), we begin by approximating the network (1.2) by equations with piecewise constant arguments of the form m X t t dxi (t) = −ai h xi (t) + aji h fj (yj (t)) dt h h j=1 + m X pji j=1 t t t t h gj yj h − τji h h , + Ii h h h h n X t t dyj (t) = −bj h yj (t) + bij h fˆi (xi (t)) dt h h i=1 + n X i=1 qij (3.1) t t t t h ĝi xi h − σij h + Jj h , h h h h where i = 1, 2, . . . , n, j = 1, 2, . . . , m, t ∈ [nh, (n + 1)h), n ∈ Z+ 0 and h is a fixed positive real number denoting a uniform discretization step-size and [r] denotes the integer part of r ∈ R. We note that [t/h] = n for t ∈ [nh, (n+1)h). For convenience, we use the notations ai (n) = ai (nh), bi (n) = bi (nh), aji (n) = aji (nh), pji (n) = pji (nh), bij (n) = bij (nh), qij (n) = qij (nh), τji (n) = τji (nh), σij (n) = σij (nh), Ii (n) = Ii (nh), Jj (n) = Jj (nh), xi (n) = xi (nh), and yj (n) = yj (nh). With these 10 Y. LI EJDE-2006/32 preparations, (3.1) can be rewritten as dxi (t) dt = −ai (n)xi (t) + m X aji (n)fj (yj (n)) + j=1 m X pji (t)gj (yj (n − τji (n))) + Ii (n), j=1 dyj (t) dt = −bj (n)yj (n) + l X bij (n)fˆi (xi (n)) + i=1 l X qij (n)ĝi (xi (n − σij (n))) + Jj (n), i=1 (3.2) where i = 1, 2, . . . , l, j = 1, 2, . . . , m, t ∈ [nh, (n+1)h), n ∈ Z+ 0 . The initial values of (3.2) will be given below in (3.6). Integrating (3.2) over the interval [nh, t), where t < (n + 1)h, we get m eai (n)t − eai (n)nh n X aji (n)fj (yj (n)) ai (n) j=1 xi (t)eai (n)t − xi (n)e−ai (n)nh = + m X o pji (t)gj (yj (n − τji (n))) + Ii (n) , j=1 (3.3) l bj (n)t yj (t)e bj (n)nh − yj (n)e ebj (n)t − ebj (n)nh n X = bij (n)fˆi (xi (n)) bj (n) i=1 + l X o qij (n)ĝi (xi (n − σij (n))) + Jj (n) , i=1 i = 1, 2, . . . , l, j = 1, 2, . . . , m. By allowing t → (n + 1)h in the above expression, we obtain m X −ai (n)h aji (n)fj (yj (n)) xi (n + 1) = xi (n)e + αi (h) j=1 + αi (h) m X pji (n)gj (yj (n − τji (n))) + αi (h)Ii (n), i = 1, 2, . . . , l, j=1 yj (n + 1) = yj (n)e−bj (n)h + βj (h) l X bij (n)fˆi (xi (n)) i=1 + βj (h) l X qij (n)ĝi (xi (n − σij (n))) + βj (h)Jj (n), j = 1, 2, . . . , m, i=1 (3.4) where αi (h) = 1 − e−ai (n)h 1 − e−bj (n)h , βj (h) = , ai (n) bj (n) i = 1, 2, . . . , l, i = 1, 2, . . . , m, n ∈ Z+ 0 . It is not difficult to verify that αi (h) > 0, βi (h) > 0 if ai , bi , h > 0 and αi (h) ≈ h + O(h2 ), βi (h) ≈ h + O(h2 ) for small h > 0. Also, one can show that (3.4) converges towards (1.2) when h → 0+ . In studying EJDE-2006/32 EXISTENCE AND EXPONENTIAL STABILITY 11 the discrete-time analogue (3.4), we assume that h ∈ (0, ∞), ai , bi : Z → (0, ∞), τji , σij : Z → Z+ 0, aji , pji , bij , qij , Ii : Z → R, i = 1, 2, . . . , l, j = 1, 2, . . . , m (3.5) and the nonlinear activation functions fj , gj , fˆi , ĝi satisfy (S2) and (S3). The system (3.4) is supplemented with initial values given by xi (s) = ϕi (s), s ∈ [−τ, 0]Z , τ = yj (s) = ψj (s), s ∈ [−σ, 0]Z , σ = max 1≤i≤l,1≤j≤m max 1≤i≤l,1≤j≤m sup{τji (n), n ∈ Z}, i = 1, 2, . . . , l, sup{σij (n), n ∈ Z}, j = 1, 2, . . . , m, (3.6) where ϕi (·) and ψj (·) denote real-valued continuous functions defined on [−τ, 0] and [−σ, 0], respectively. In what follows, for convenience, we will use the following notation: am i = aM ji = pM ji = M τji = {ai (n)}, bm j = {|aji (n)|}, bM ij = {|pji (n)|}, M qij = {|τji (n)|}, M σij = min n∈[0,ω−1]Z max n∈[0,ω−1]Z max n∈[0,ω−1]Z max n∈[0,ω−1]Z max n∈[0,ω−1]Z {bj (n)}, max n∈[0,ω−1]Z max n∈[0,ω−1]Z max n∈[0,ω−1]Z {|bij (n)|}, {|qij (n)|}, {|σij (n)|}, for i = 1, 2, . . . , l and j = 1, 2, . . . , m. Also define aM = max {aM ji (n)}, bM = 1≤i≤l, 1≤j≤m Mf = sup{|fj (u)|, j = 1, 2, . . . , m}, u∈R Mfˆ = sup{|fˆi (u)|, i = 1, 2, . . . , l}, Mĝ = sup{|ĝi (u)|, i = 1, 2, . . . , l}, u∈R I = max n∈[0,ω−1]Z {bM ij }, Mg = sup{|gj (u)|, j = 1, 2, . . . , m}, u∈R M max 1≤i≤l, 1≤j≤m {|Ii (n)|, i = 1, 2, . . . , l}, u∈R J M = max n∈[0,ω−1]Z {|Jj (n)|, j = 1, 2, . . . , m}. The following result was given in [1, Lemma 3.2]. Lemma 3.1. Let f : Z → R be ω-periodic, i.e., f (k + ω) = f (k). Then for any fixed n1 , n2 ∈ Iω , and any k ∈ Z, one has f (n) ≤ f (n1 ) + ω−1 X |f (s + 1) − f (s)|, s=0 f (n) ≥ f (n2 ) − ω−1 X |f (s + 1) − f (s)|. s=0 Using Lemma 2.1, we shall show the following result about the existence of at least one periodic solution of (3.4). Theorem 3.2. Assume that (S2), (S3) and (3.5) hold. Furthermore, assume that (S4) ai , aji , pji , Ii , bj , bij , qij , Jj , i = 1, 2, . . . , l, j = 1, 2, . . . , m are all ω-periodic functions, where ω > 1 is a positive integer. (S5) ω − 1 ω − 1 1 1 h≤ min − m ln , − m ln . 1≤i≤l, 1≤j≤m ai ω bj ω 12 Y. LI EJDE-2006/32 Then system (3.4) has at least one ω-periodic solution. Proof. Define ld = {u = {u(n)} : u(n) ∈ Rl+m , n ∈ Z}. Let lω ⊂ ld denotes the subspace of all ω periodic sequences equipped with the norm || · ||, i.e., ||u|| = ||(u1 , u2 . . . , ul+m )T || = l+m X i=1 max n∈[0,ω−1]Z |ui (n)|, where u = {(u1 (n), u2 (n), . . . , ul+m (n)), n ∈ Z} ∈ lω . It is not difficult to show that lω is a finite-dimensional Banach space. Let l0ω = {u = {u(n)} ∈ lω : ω−1 X u(n) = 0}, n=0 lcω = {u = {u(n)} ∈ lω : u(n) = c ∈ Rm , n ∈ Z}. Then it is easy to check that l0ω and lcω are both closed linear subspaces of lω and lω = l0ω ⊕ lcω , dim lcω = l + m. Take X = Y = lω and let Pm x1 (n)(e−a1 (n)h − 1) + α1 (h) j=1 aj1 (n)fj (yj (n)) x1 .. .. ., . P xl xl (n)(e−al (n)h − 1) + αl (h) m a (n)f (y (n)) jl j j j=1 P N l y1 = −b (n)h ˆ 1 y (n)(e − 1) + β (h) b (n) f (x (n)) 1 i i 1 i=1 i1 . . .. .. P l ym ym (n)(e−bm (n)h − 1) + βm (h) i=1 bim (n)fˆi (xi (n)) Pm α1 (h) j=1 pj1 (t)gj (yj (n − τj1 (n))) + α1 (h)I1 (n) .. . P α (h) m p (t)g (y (n − τ (n))) + α (h)I (n) l j j jl l l j=1 jl P + , β1 (h) li=1 qi1 (n)ĝi (xi (n − σi1 (n))) + β1 (h)J1 (n) .. . Pl βm (h) i=1 qim (n)ĝi (xi (n − σim (n))) + βm (h)Jm (n) x ∈ X, n ∈ Z, (Lu)(n) = u(n + 1) − u(n), u ∈ X, n ∈ Z. It is easy to see that L is a bounded linear operator with ker L = lcω , Im L = l0ω , dim ker L = l + m = codim Im L, then it follows that L is a Fredholm mapping of index zero. Define Pu = ω−1 1 X u(s), ω s=0 u ∈ X, Qv = ω−1 1 X v(s), ω s=0 v ∈ Y. It is not difficult to show that P and Q are continuous projectors such that Im P = ker L, Im L = ker Q = Im(I − Q). EJDE-2006/32 EXISTENCE AND EXPONENTIAL STABILITY Furthermore, the generalized inverse (to L) KP : Im L → ker P which is given by ω−1 ω−1 X 1 X KP (y) = y(s) − (ω − s)y(s). ω s=0 s=0 T 13 Dom L exists, Obviously, QN and KP (I − Q)N are continuous. Since X is a finite-dimensional Banach space, one can easily show that KP (I − Q)N (Ω̄) is compact for any open bounded set Ω ⊂ X. Moreover, QN (Ω̄) is bounded, and hence N is L-compact on Ω̄ with any open bounded set Ω ⊂ X. We now are in a position to search for an appropriate open, bounded subset Ω ⊂ X for the continuation theorem. Corresponding to operator equation Lu = λN u, λ ∈ (0, 1), we have m n X xi (n + 1) − xi (n) = λ − xi (n)(1 − e−ai (n)h ) + αi (h) aji (n)fj (yj (n)) j=1 + αi (h) m X o pji (n)gj (yj (n − τji (n))) + αi (h)Ii (n) , j=1 n −bj (n)h yj (n + 1) − yj (n) = λ − yj (n)(1 − e ) + βj (h) l X (3.7) bij (n)fˆi (xi (n)) i=1 + βj (h) l X o qij (n)ĝi (xi (n − σij (n))) + βj (h)Jj (n) , i=1 where i = 1, 2, . . . , l, j = 1, 2, . . . , m. Suppose that {(x1 (n), x2 (n), . . . , xl (n), y1 (n), y2 (n), . . . , ym (n))T } ∈ X is a solution of system (3.7) for a certain λ ∈ (0, 1). Summing on both sides of (3.7) from 0 to ω − 1 with respect to n, we obtain ω−1 X ω−1 X xi (s)(1 − e−ai (s)h ) = s=0 m X αi (h) s=0 + aji (s)fj (yj (s)) j=1 ω−1 X αi (h) s=0 m X pji (s)gj (yj (s − τji (s))) + ω−1 X αi (h)Ii (s), s=0 j=1 (3.8) for i = 1, 2, . . . , l, and ω−1 X ω−1 X yj (s)(1 − e−bj (s)h ) = s=0 βj (h) s=0 + l X bij (s)fˆi (xi (s)) i=1 ω−1 X βj (h) s=0 l X qij (s)ĝi (xi (s − σij (s))) + ω−1 X βj (h)Jj (s), s=0 i=1 (3.9) for j = 1, 2, . . . , m. Let ni , n̄j ∈ [0, ω − 1]Z such that xi (ni ) = max n∈[0,ω−1]Z {xi (n)}, yj (n̄j ) = max n∈[0,ω−1]Z {yj (n)}, for i = 1, 2, . . . , l, j = 1, 2, . . . , m, then it follows from (3.8) and (3.9) that xi (ni ) ω−1 X s=0 (1 − e−ai (s)h ) ≥ ω−1 X s=0 αi (h) m X j=1 aji (s)fj (yj (s)) 14 Y. LI ω−1 X + αi (h) s=0 EJDE-2006/32 m X pji (s)gj (yj (s − τji (s))) + ω−1 X αi (h)Ii (s), s=0 j=1 for i = 1, 2, . . . , l and yj (n̄j ) ω−1 X (1 − e−bj (s)h ) ≥ ω−1 X s=0 βj (h) s=0 + l X bij (s)fˆi (xi (s)) i=1 ω−1 X βj (h) s=0 l X qij (s)ĝi (xi (s − σij (s))) + ω−1 X βj (h)Jj (s), s=0 i=1 for j = 1, 2, . . . , m. Hence xi (ni ) ≥ h ω−1 X m i−1 h ω−1 X X (1 − e−ai (s)h ) αi (h) aji (s)fj (yj (s)) s=0 ω−1 X + s=0 αi (h) s=0 ≥− m X pji (s)gj (yj (s − τji (s))) + ω−1 X ω−1 X i αi (h)Ii (s) s=0 j=1 h ω−1 X m i−1 h ω−1 X X (1 − e−ai (s)h ) αi (h) |aji (s)||fj (yj (s))| s=0 + j=1 s=0 αi (h) s=0 m X j=1 |pji (s)||gj (yj (s − τji (s)))| + ≥ −αi (h)ω i αi (h)|Ii (s)| s=0 j=1 h ω−1 X ω−1 X (3.10) i−1 (1 − e−ai (s)h ) [maM Mf + mpM Mg + I M ] s=0 := −Ai , i = 1, 2, . . . , l and yj (n̄j ) ≥ h ω−1 X l i−1 h ω−1 X X βj (h) bij (s)fˆi (xi (s)) (1 − e−bj (s)h ) s=0 + ω−1 X s=0 βj (h) s=0 l X i=1 qij (s)ĝi (xi (s − σij (s))) + h ω−1 X i βj (h)Jj (s) s=0 i=1 ≥ −βj (h)ω ω−1 X i−1 (1 − e−bj (s)h ) [lbM Mfˆ + lq M Mĝ + J M ] s=0 := −Bj , j = 1, 2, . . . , m. Let n∗i , n∗j ∈ [0, ω − 1]Z such that xi (n∗i ) = min n∈[0,ω−1]Z {xi (n)}, yj (n∗j ) = min n∈[0,ω−1]Z {yj (n)}, for i = 1, 2, . . . , l, j = 1, 2, . . . , m. Again, it follows from (3.8) and (3.9) that xi (n∗i ) ω−1 X s=0 (1 − e−ai (s)h ) ≤ ω−1 X s=0 αi (h) m X j=1 aji (s)fj (yj (s)) (3.11) EJDE-2006/32 EXISTENCE AND EXPONENTIAL STABILITY + ω−1 X αi (h) s=0 m X pji (s)gj (yj (s − τji (s))) + 15 ω−1 X αi (h)Ii (s), s=0 j=1 for i = 1, 2, . . . , l and yj (n∗j ) ω−1 X ω−1 X (1 − e−bj (s)h ) ≤ s=0 βj (h) l X s=0 + bij (s)fˆi (xi (s)) i=1 ω−1 X βj (h) s=0 l X qij (s)ĝi (xi (s − σij (s))) + ω−1 X βj (h)Jj (s), s=0 i=1 for j = 1, 2, . . . , m. Hence xi (n∗i ) ≤ h ω−1 X −ai (s)h (1 − e m i−1 h ω−1 X X aji (s)fj (yj (s)) αi (h) ) s=0 + s=0 ω−1 X m X αi (h) s=0 j=1 pji (s)gj (yj (s − τji (s))) + ω−1 X i αi (h)Ii (s) s=0 j=1 (3.12) i−1 h ω−1 X [maM Mf + mpM Mg + I M ] (1 − e−ai (s)h ) ≤ ωαi (h) s=0 = Ai , i = 1, 2, . . . , l and yj (n∗j ) ≤ h ω−1 X l i−1 h ω−1 X X bij (s)fˆi (xi (s)) βj (h) (1 − e−bj (s)h ) s=0 s=0 + ω−1 X βj (h) s=0 l X i=1 qij (s)ĝi (xi (s − σij (s))) + ω−1 X i βj (h)Jj (s) s=0 i=1 (3.13) h ω−1 i−1 X ≤ ωβj (h) (1 − e−bj (s)h ) [lbM Mfˆ + lq M Mĝ + MJ ] s=0 = Bj , j = 1, 2, . . . , m. According to (3.7), we have m n X |xi (n + 1) − xi (n)| ≤ λ |xi (n)|(1 − e−ai (n)h ) + αi (h) |aji (n)||fj (yj (n))| j=1 + αi (h) m X o |pji (n)||gj (yj (n − τji (n)))| + αi (h)|Ii (n)| j=1 m ≤ |xi (n)|(1 − e−ai h ) + αi (h)[maM Mf + mpM Mg + I M ] (3.14) 16 Y. LI EJDE-2006/32 and l n X |yj (n + 1) − yj (n)| ≤ λ |yj (n)|(1 − e−bj (n)h ) + βj (h) |bij (n)||fˆi (xi (n))| i=1 + βj (h) l X o |qij (n)||ĝi (xi (n − σij (n)))| + βj (h)|Jj (n)| i=1 m ≤ |yj (n)|(1 − e−bi h ) + βj (h)[lbM Mfˆ + lpM Mĝ + J M ], (3.15) where i = 1, 2, . . . , l, j = 1, 2, . . . , m. It follows from (3.10), (3.12), (3.14) and Lemma 3.1 that for i = 1, 2, . . . , l, xi (n) ≤ xi (n∗i ) + ω−1 X |xi (s + 1) − xi (s)| s=0 m ≤ Ai + (1 − e−ai h ) ω−1 X |xi (s)| + ωαi (h)[maM Mf + mpM Mg + I M ] s=0 and xi (n) ≥ xi (ni ) − ω−1 X |xi (s + 1) − xi (s)| s=0 m ≥ −Ai − (1 − e−ai h ) ω−1 X |xi (s)| − ωαi (h)[maM Mf + mpM Mg + I M ]. s=0 Thus, m |xi (n)| ≤ Ai + (1 − e−ai h ) ω−1 X |xi (s)| + ωαi (h)[maM Mf + mpM Mg + I M ] (3.16) s=0 for i = 1, 2, . . . , l. Summing on both sides of (3.16) from 0 to ω − 1 with respect to n, we obtain ω−1 X |xi (s)| s=0 m ≤ ωAi + ω(1 − e−ai h ) ω−1 X |xi (s)| + ω 2 αi (h)[maM Mf + mpM Mg + I M ] s=0 for i = 1, 2, . . . , l. Since 0 < h ≤ − a1m ln i ω−1 X ω−1 ω for i = 1, 2, . . . , l, we have |xi (s)| s=0 h i m ≤ [1 − ω(1 − e−ai h )]−1 ωAi + ω 2 αi (h)(maM Mf + mpM Mg + I M ) := Ci , (3.17) i = 1, 2, . . . , l. In view of (3.16) and (3.17), we get m |xi (n)| ≤ Ai + (1 − e−ai h )Ci + ωαi (h)[maM Mf + mpM Mg + I M ] := Ei , EJDE-2006/32 EXISTENCE AND EXPONENTIAL STABILITY 17 for i = 1, 2, . . . , l. Similarly, it follows from (3.11), (3.13), (3.15) and Lemma 3.1 that m |yj (n)| ≤ Bi + (1 − e−bj h )Dj + ωβj (h)[lbM Mfˆ + lq M Mĝ + J M ] := Fj , for j = 1, 2, . . . , m. where h i m Dj = [1 − ω(1 − e−bj h )]−1 ωBj + ω 2 βj (h)(lbM Mfˆ + lq M Mĝ + J M ) , Pl Pm for j = 1, 2, . . . , m. Denote C = i=1 Ei + j=1 Fj +H, where H > 0 is a constant. Clearly, C is independent of λ. Now we take Ω = {x ∈ X : ||x|| < C}. The rest of the proof is similar to that of the proof of Theorem 2.2 and will be omitted. The proof is complete. Theorem 3.3. Assume that (S2)-(S3) and (3.5) hold. Furthermore, assume that τji (n) ≡ τji , σij (n) ≡ σij ∈ Z + , n ∈ Z, i, j = 1, 2, . . . , m are constants and (P3) am i > m X fˆ M ĝ bM ij Li + qij Lj , i = 1, 2, . . . , l, j=1 bm j > l X f M g aM L + p L ji j ji i , j = 1, 2, . . . , m, i=1 ∗ (n))T } of then the ω-periodic solution {(x∗1 (n), x∗2 (n), . . . , x∗l (n), y1∗ (n), y2∗ (n), . . . , ym (3.4) is unique and is globally exponentially stable in the sense that there exist constants λ > 1 and δ ≥ 1 such that l X |xi (n) − x∗i (n)| + i=1 ≤δ m X |yj (n) − yj∗ (n)| i=1 l 1 n X λ sup i=1 s∈[−τ,0]Z |xi (s) − x∗i (s)| + m X (3.18) sup |yj (s) − yj∗ (s)| , j=1 s∈[−σ,0]Z for n ∈ Z+ . Proof. Let x(n) = {(x1 (n), x2 (n), . . . , xl (n), y1 (n), y2 (n), . . . , ym (n))T } be an arbitrary solution of (3.4), and ∗ x∗ (n) = {(x∗1 (n), x∗2 (n), . . . , x∗m (n), y1∗ (n), y2∗ (n), . . . , ym (n))T } 18 Y. LI EJDE-2006/32 be an ω-periodic solution of (3.4). Then |xi (n + 1) − x∗i (n + 1)| m ≤ |xi (n) − x∗i (n)|e−ai h + αi (h) m X f ∗ aM ji Lj |yj (n) − yj (n)| j=1 + αi (h) m X g ∗ pM ji Lj |yj (n − τji (n))) − yj (n − τji (n)))|, j=1 m |yj (n + 1) − yj∗ (n + 1)| ≤ |yj (n) − yj∗ (n)|e−bj h + βj (h) l X ˆ f ∗ bM ij Li |xi (n) − xi (n)| i=1 + βj (h) l X M ĝ qij Li |xi (n − σij (n)) − x∗i (n − σij (n))|, i=1 (3.19) where i = 1, 2, . . . , l, j = 1, 2, . . . , m. Now we consider functions Γi (·, ·) and Θj (·, ·), i = 1, 2, . . . , l, j = 1, 2, . . . , m, defined by m X Γi (µi , n) = 1 − µi e−ai (n)h − µi αi (h) ˆ f bM ij Li − j=1 Θj (νj , n) = 1 − νj e−bj (n)h − νj αj (h) l X m X σ M +1 M ĝ qij Lj θi (h)µi ij j=1 f aM ji Lj − i=1 l X τ M +1 g ji pM ji Lj θj (h)νj , i=1 where µi , νj ∈ [1, ∞), n ∈ [0, ω − 1]Z , i = 1, 2, . . . , l, j = 1, 2, . . . , m. Since Γi (1, n) = 1 − e−ai (n)h − αi (h) m X ˆ f bM ij Li − αi (h) j=1 m X M ĝ qij Lj j=1 m m h i X X fˆ M ĝ = αi (h) ai (n) − bM qij Lj ij Li − j=1 j=1 m m h i X X M fˆ M ĝ ≥ αi (h) am − b L − qij Lj > 0, i ij i j=1 j=1 for n ∈ [0, ω − 1]Z , i = 1, 2, . . . , l, and Θj (1, n) = 1 − e−bj (n)h − βj (h) l X f aM ji Lj − βj (h) i=1 l X g pM ji Li i=1 l l h i X X f g = βj (h) bj (n) − aM pM ji Lj − ji Li i=1 i=1 l l i h X X g M f ≥ βj (h) bm − a L − pM j ji j ji Li > 0, i=1 i=1 for n ∈ [0, ω − 1]Z , j = 1, 2, . . . , m. Using the continuity of Γi (µi , n) and Θj (νj , n) on [1, ∞) with respect to µi and νj , respectively, for every n ∈ [0, ω − 1]Z and the fact that Γi (µi , n) → −∞ as µi → ∞ and Θj (νj , n) → −∞ as νj → ∞ uniformly in n ∈ [0, ω − 1]Z , i = 1, 2, . . . , l, j = 1, 2, . . . , m, we see that there exist νi∗ (n), νj∗ (n) ∈ (1, ∞) such that Γi (µ∗i (n), n) = 0 and Θj (νj∗ (n), n) = 0 for n ∈ EJDE-2006/32 EXISTENCE AND EXPONENTIAL STABILITY 19 [0, ω − 1]Z , i = 1, 2, . . . , l, j = 1, 2, . . . , m. By choosing λ = min{µ∗i (n), νj∗ (n), n ∈ [0, ω − 1]Z , i = 1, 2, . . . , l, j = 1, 2, . . . , m}, where λ > 1, we obtain Γi (λ, n) ≥ 0 and Θj (λ, n) ≥ 0 for all n ∈ [0, ω − 1]Z , i = 1, 2, . . . , l, j = 1, 2, . . . , m, that is, m X λe−ai (n)h + λαi (h) ˆ f bM ij Li + j=1 λe−bj (n)h + λαj (h) l X m X M M ĝ qij Lj θi (h)λσij +1 ≤ 1, n ∈ [0, ω − 1]Z , j=1 f aM ji Lj + i=1 l X M g τji +1 pM ≤ 1, n ∈ [0, ω − 1]Z , ji Lj θj (h)λ i=1 for i = 1, 2, . . . , l, j = 1, 2, . . . , m. Hence m λe−ai h + λαi (h) m X ˆ f bM ij Li + j=1 −bm j h λe + λαj (h) l X m X M M ĝ qij Lj θi (h)λσij +1 ≤ 1, j=1 f aM ji Lj + i=1 l X (3.20) M g τji +1 pM ji Lj θj (h)λ ≤ 1, i=1 for i = 1, 2, . . . , l, j = 1, 2, . . . , m. Now let us consider |xi (n) − x∗i (n)| , n ∈ [−τ, ∞)Z , i = 1, 2, . . . , l, αi (h) |yj (n) − yj∗ (n)| , n ∈ [−τ, ∞)Z , j = 1, 2, . . . , m. vj (n) = λn βj (h) ui (n) = λn (3.21) Using (3.4) and (3.21), we derive that m ∆ui (n) ≤ −(1 − λe−ai h )ui (n) + λ m X f βj (h)aM ji Lj vj (n) j=1 + m X g τji +1 βj (h)pM vj (n − τji ), ji Lj λ i = 1, 2, . . . , l, j=1 −bm j h ∆vj (n) ≤ −(1 − λe )vj (n) + λ l X (3.22) fˆ αi (h)bM ij Li ui (n) i=1 + l X M ĝ σij +1 αi (h)qij Li λ ui (n − σij ), j = 1, 2, . . . , m. i=1 We consider the Lyapunov function l X V (n) = ui (n) + i=1 + m X m X vi (n) + l X vj (s) s=n−τji j=1 j=1 n−1 X M g τji +1 βj (h)pM ji Lj λ M M ĝ σij αi (h)qij Li λ +1 n−1 X (3.23) ui (s) . s=n−σij i=1 Calculating the difference ∆V (n) = V (n + 1) − V (n) along (3.22), we obtain ∆V (n) ≤ − l X i=1 m (1 − λe−ai h )ui (n) − λ m X j=1 f βj (h)aM ji Lj vj (n) 20 Y. LI − m X EJDE-2006/32 m X M m g τji +1 βj (h)pM vj (n) − (1 − λe−bj h )vj (n) ji Lj λ j=1 −λ j=1 l X ˆ f αi (h)bM ij Li ui (n) − i=1 =− − l X M M ĝ σij αi (h)qij Lj λ +1 ui (n) i=1 l X m 1 − λe−ai h − λαi (h) m X i=1 j=1 m X l X −bm j h 1 − λe − λβj (h) j=1 ˆ f bM ij Li − αi (h) m X M M ĝ σij qij Lj λ +1 ui (n) j=1 f aM ji Lj − βj (h) i=1 l X M g τji +1 pM vj (n). ji Lj λ i=1 Using (3.20) in the above we deduce that ∆V (n) ≤ 0 for n ∈ Z+ 0 . From this result and (3.23) it follows that l X ui (n) + i=1 m X vj (n) ≤ V (n) ≤ V (0) for n ∈ Z+ . (3.24) j=1 Thus l X ui (n) + i=1 m X = λn l X |xi (n) − x∗ (n)| i αi (h) i=1 ≤ l X ui (0) + i=1 + ≤ vj (n) j=1 m X m X |yi (n) − y ∗ (n)| i j=1 vi (0) + l X m X i=1 m X ui (s) s=−σij M M ĝ σij +1 qij Li λ σij l X M g τji +1 pM τji ji Lj λ i=1 j=1 −1 X M M ĝ σij αi (h)qij Li λ +1 j=1 1 + βj (h) vj (s) s=−τji l X i=1 1 + αi (h) −1 X M j=1 m X βj (h) g τji +1 βj (h)pM ji Lj λ j=1 + + λn 1 sup {|xi (s) − x∗i (s)|} αi (h) s∈[−σ,0] 1 sup {|yj (s) − yj∗ (s)|}. βj (h) s∈[−τ,0] Therefore, we obtain the assertion (3.18), where δ= max{max1≤i≤l αi (h), max1≤j≤m βj (h)} σ≥1 min{min1≤i≤l αi (h), min1≤j≤m βj (h)} and σ= max 1≤i≤l, 1≤j≤m m l n o X X M M g τji M ĝ σij +1 1 + αi (h) qij Li λ +1 σij , 1 + βj (h) pM L λ τ ≥ 1. ji ji j j=1 i=1 We conclude from (3.18) that the unique periodic solution of (3.4) is globally exponentially stable and this completes the proof. EJDE-2006/32 EXISTENCE AND EXPONENTIAL STABILITY 21 4. An Example Consider the following BAM neural networks system with discrete delays 2 X dxi = −ai (t)xi (t) + pji (t)fj (yj (t − τji )) + Ii (t), dt j=1 i = 1, 2, 2 X dyj = −bj (t)yj (t) + qij (t)gi (xi (t − σij )) + Jj (t), dt i=1 j = 1, 2, (4.1) u in which for i, j = 1, 2, fj (u) = arctan(u + j), gi (u) = (i+u 2 ) , ai (t) = 5(cos t + 2), i bj (t) = sin t + 4, pji (t) = − sin(i + j)t, qij (t) = j sin t, τji , σij are positive constants, Ii (t), Jj (t) are any continuous 2π-periodic functions. Then it is easy to show that (4.1) satisfies all the conditions of Theorem 2.3, hence (4.1) has a unique 2π-periodic solution which is globally exponential stable. References [1] Fan, M. and Wang, K.; Periodic solutions of a discrete time nonautonomous ratio-dependent predator-prey system. Mathematical and Computer Modelling, vol. 35 (2002), 951-961. [2] Gaines, R.E. and Mawhin, J.L. Coincidence Degree and Nonlinear Differential Equations. Berlin: Springer-Verlag, 1977. [3] K. Gopalsamy, X.Z. 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Global stability of bidirectional associative memory neural networks with distributed delays. Physics Letters A, vol 297 (2002), 182-190. Department of Mathematics, Yunnan University, Kunming, Yunnan 650091, China E-mail address: yklie@ynu.edu.cn