EXISTENCE AND EXPONENTIAL STABILITY OF PERIODIC SOLUTION FOR CONTINUOUS-TIME AND DISCRETE-TIME GENERALIZED BIDIRECTIONAL NEURAL NETWORKS

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1 Electronic Journal of Differential Equations, Vol , No. 32, pp ISSN: URL: ttp://ejde.mat.txstate.edu or ttp://ejde.mat.unt.edu ftp ejde.mat.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 te existence and global exponential stability of positive periodic solutions for a class of continuous-time generalized bidirectional neural networks wit variable coefficients and delays. Discrete-time analogues of te continuous-time networks are formulated and te existence and global exponential stability of positive periodic solutions are studied using te continuation teorem of coincidence degree teory and Lyapunov functionals. It is sown tat te existence and global exponential stability of positive periodic solutions of te continuous-time networks are preserved by te discrete-time analogues under some restriction on te discretization step-size. An example is given to illustrate te results obtained. 1. Introduction In recent years, te stability of te following bidirectional associative neural networks wit or witout delays as been extensively studied: dx i t = a i x i t p ji f j y j t τ ji I i, i = 1, 2,..., l, 1.1 dy j t = b j y j t q ij g i x i t σ ij J j, j = 1, 2,..., m. Also some of its generalizations ave been studied and various stability conditions ave been obtained [3, 4, 5, 6, 11, 12, 14, 15, 16, 17. Here p ji, q ij, i = 1, 2,..., l, j = 1, 2,..., m are te connection weigts troug te neurons in two layers: I-layer and J-layer. On I-layer, te neurons wose states denoted by x i t receive te inputs I i and te inputs outputted by tose neurons in J-layer via activation functions output-input functions f j, wile on J-layer, te neurons wose associated states denoted by y j t receive te inputs J i and te inputs outputted from tose neurons in I-layer via activation functions output-input functions g i. And τ ji, σ ij, i = 2000 Matematics Subject Classification. 34K13, 34K25. Key words and prases. Bidirectional neural networks; global exponential stability; periodic solution; Fredolm mapping; Lyapunov functionals; discrete-time analogues. c 2006 Texas State University - San Marcos. Submitted February 16, Publised Marc 16, Supported by grants from te National Natural Sciences Foundation of Cina, and 2003A0001M from te Natural Sciences Foundation of Yunnan Province. 1

2 2 Y. LI EJDE-2006/32 1, 2,..., l, j = 1, 2,..., m are te associated delays due to te finite transmission speed among neurons in different layers. Wen tere is no delay present, 1.1 reduces to a system of ordinary differential equations wic was investigated by Kosko [7, 8, 9 and it produces many nice properties due to te special structure of connection weigts and as practical applications in storing paired patterns or memories and te ability to searc te desired patterns via bot directions: forward and backward directions. See [3, 14, 7, 8, 9 for details about te applications on learning and associative memories. It is well known tat in te study of neural dynamical systems, te periodic oscillatory beavior of te systems is an important aspect. In tis paper, we are concerned wit te following generalized BAM networks wit variable coefficients and delays dx i t dy j t = a i tx i t a ji tf j y j t p ji tg j y j t τ ji t I i t, = b j ty j t b ij t ˆf i x i t for i = 1, 2,..., l, j = 1, 2,..., m. In tis paper, we assume tat q ij tĝ i x i t σ ij t J j t, 1.2 S1 a i, b j CR, 0,, τ ji, σ ij CR, [0,, I i, J j, p ji, q ij CR, R, i = 1, 2,..., l, j = 1, 2,..., m are all ω-periodic functions. S2 f j, g j, ˆf i, ĝ i CR, R i = 1, 2,..., l, j = 1, 2,..., m are bounded on R. S3 Tere exist positive number L f j, Lg j, L ˆf i, Lĝ j suc tat f j x f j y L f j x y for all x, y R, j = 1, 2,..., m, g j x g j y L g j x y for all x, y R, j = 1, 2,..., m, ˆf i x ˆf i y L ˆf i x y for all x, y R, i = 1, 2,..., l, ĝ i x ĝ i y Lĝ i x y for all x, y R, i = 1, 2,..., l. Our purpose of tis paper is by using Mawin s continuation teorem of coincidence degree teory [2, 13 and by constructing suitable Lyapunov functions to investigate te stability and existence of periodic solutions of 1.2; ten, we sall use a novel metod in formulating discrete-time analogues of te continuous time networks. It is sown tat te existence and global exponential stability of positive periodic solutions of te continuous-time networks are preserved by te discrete-time analogues under some restriction on te discretization step-size. 2. Existence of periodic solutions In tis section, based on te Mawin s continuation teorem, we sall study te existence of at least one positive periodic solution of 1.2. First, we sall 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. Te mapping L will be called a Fredolm mapping of index zero if dim ker L = codim Im L < and Im L is closed in Y. If L is a Fredolm mapping of index zero, tere exist continuous projectors P : X X

3 EJDE-2006/32 EXISTENCE AND EXPONENTIAL STABILITY 3 and Q : Y Y suc tat Im P = ker L, ker Q = Im L = ImI Q. It follows tat mapping L Dom L T ker P : I P X Im L is invertible. We denote te inverse of tat mapping by K P. If Ω is an open bounded subset of X, te mapping N will be called L- compact on Ω if QN Ω is bounded and K P I QN : Ω X is compact. Since Im Q is isomorpic to ker L, tere exists an isomorpism J : Im Q ker L. For convenience of use, we introduce Mawin s continuation teorem [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 wic is L-compact on Ω i.e., QN : Ω Y and K P I QN : Ω Y are compact. Assume a for eac λ 0, 1, x Ω Dom LLx λnx, b for eac x Ω ker L. QNx 0, c degjnq, Ω ker L, 0 0. Ten Lx = Nx as at least one solution in Ω Dom L. Our result about te existence of periodic solutions of 1.2 is as follows. Teorem 2.2. Assume tat S1 and S2 old. Ten system 1.2 as at least one ω-periodic solution. Proof. To use te continuation teorem of coincidence degree teory to establis te existence of an ω-periodic solution of 1.2, we take X = Y = {u CR, R lm : ut ω = ut and u = lm max t [0,ω u i t, ten X is a Banac space. Set L : Dom L X, Lu = ut, u X, were Dom L = {u C 1 R, R lm and N : X X, N[x 1,..., x l, y 1,..., y m T = a 1 tx 1 t m a j1tf j y j t m p j1tg j y j t τ j1 t I 1 t. a l tx l t m a jltf j y j t m p jltg j y j t τ jl t I l t b 1 ty 1 t l b i1t ˆf i x i t l q i1tĝ i x i t σ i1 t J 1 t. b m ty m t l b imt ˆf i x i t l q imtĝ i x i t σ im t J m t Define two projectors P and Q as P u = Qu = 1 ω ω 0 ut, u X. Clearly, ker L = R lm, Im L = {x 1, x 2,..., x l, y 1,..., y m T X : ω 0 x it = 0, ω 0 y jt = 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 Fredolm mapping of index zero. Furtermore, similar to te proof of [10, Teorem 1, one can easily sow tat N is L-compact on Ω wit any open bounded set Ω X. Corresponding to operator equation.

4 4 Y. LI EJDE-2006/32 Lu = λnu, λ 0, 1, we ave dx i t [ = λ a i tx i t a ji tf j y j t p ji tg j y j t τ ji t I i t, dy j t [ = λ b j ty j t b ij t ˆf i x i t q ij tĝ i x i t σ ij t J j t 2.1 for i = 1, 2,..., l, j = 1, 2,..., m. Suppose tat x 1, x 2,..., x l, y 1, y 2,..., y m X is a solution of 2.1 for some λ 0, 1. Let ξ i, ζ j [0, ω suc tat x i ξ i = max t [0,ω x i t and y j ζ j = max t [0,ω y j t, i = 1, 2,..., l, j = 1, 2..., m, ten and a i ξ i x i ξ i = a ji ξ i f j y j ξ i p ji ξ i g j y j ξ i τ ji ξ i I i ξ i b j ζ j y j ζ j = b ij ζ j ˆf i x i ζ j q ij ζ j ĝ i x i ζ j σ ij ζ j J j ζ j, for i = 1, 2,..., l, j = 1, 2,..., m. Hence, a i ξ i x i ξ i a ji ξ i f j y j ξ i p ji ξ i g j y j ξ i τ ji ξ i I i ξ i, b j ζ j y j ζ j b ij ζ j ˆf i x i ζ j q ij ζ j ĝ i x i ζ j σ ij ζ j J j ζ j for i = 1, 2,..., l, j = 1, 2,..., m. Terefore, were x i ξ i 1 a [mm f a M mm g p M I M, i = 1, 2,..., l, y j ζ j 1 b [lm ˆf b M lmĝq M J M, j = 1, 2,..., m, a = min t [0,ω {a it, i = 1, 2,..., n, M f = sup{ f j u, j = 1, 2,..., m, u R b = min t [0,ω {b jt, i = 1, 2,..., l, M g = sup{ g j u, j = 1, 2,..., m, u R M ˆf = sup{ i u, i = 1, 2,..., l, Mĝ = sup{ ĝ i u, i = 1, 2,..., l, u R u R a M = max t [0,ω { a jit, i = 1, 2,..., l, j = 1, 2,..., m, b M = max t [0,ω { b ijt, i = 1, 2,..., l, j = 1, 2,..., m, p M = max t [0,ω { p jit, i = 1, 2,..., l, j = 1, 2,..., m, q M = max t [0,ω { q ijt, i = 1, 2,..., l, j = 1, 2,..., m, I M = max t [0,ω { I it, i = 1, 2,..., l, J M = max t [0,ω { J it, j = 1, 2,..., m.

5 EJDE-2006/32 EXISTENCE AND EXPONENTIAL STABILITY 5 Let ξ i, ζ j [0, ω be suc tat x i ξ i = min t [0,ω x i t and y j ζ j = min t [0,ω y j t, i = 1, 2,..., l, j = 1, 2,..., m, ten a i ξ i x i ξ i = a ji ξ i f j y j ξ i p ji ξ i g j y j ξ i τ ji ξ i I i ξ i b j ζ j y j ζ j = n b ij ζ j ˆf i x i ζ j for i = 1, 2,..., l, j = 1, 2,..., m. Tus, n q ij ζ j ĝ i x i ζ j σ ij ζ j J j ζ j a i ξ i x i ξ i a ji ξ i f j y j ξ i p ji ξ i g j y j ξ i τ ji ξ i I i ξ i, b j ζ j y j ζ j n b ij ζ j ˆf i x i ζ j n q ij ζ j ĝ i x i ζ j σ ij ζ j J j ζ j for i = 1, 2,..., l, j = 1, 2,..., m. Terefore, x i ξ i 1 a [mm f a M mm g p M I M, y j ζ j 1 b [lm ˆf b M lmĝq M J M for i = 1, 2,..., l, j = 1, 2,..., m. Denote C = l a [mm f a M mm g p M I M m b [lm ˆf b M lmĝq M J M D, were D is a positive constant. Ten it is clear tat C is independent of λ. Now we take Ω = {u X : u < C. Tis Ω satisfies condition a in Lemma 2.1. Wen u Ω ker L = Ω R lm, u is a constant vector in R mn wit u = C. Ten u T QNu = { ā i x 2 i ā ji x i f j y j p ji x i g j y j x i Ī i { b j y 2 j bij y j ˆfi x i q ij y j ĝ i x i y j Jj < 0, were u = x 1, x 2,..., x l, y 1, y 2,..., y m. If necessary, we can let C be large suc tat { ā i x 2 i { b j y 2 j ā ji x i f j y j p ji x i g j y j x i Ī i bij y j ˆfi x i q ij y j ĝ i x i y j Jj < 0. So for any u Ω ker L, QNu 0. Tis proves tat condition b in Lemma 2.1 is satisfied.

6 6 Y. LI EJDE-2006/32 Furtermore, let Ψγ; u = γu 1 γqnu, ten for any x Ω ker L, u T Ψγ; u < 0, we get deg{jqn, Ω ker L, 0 = deg{ u, Ω ker L, 0 0, ence condition c of Lemma 2.1 is also satisfied. Tus, by Lemma 2.1 we conclude tat Lu = Nu as at least one solution in X, tat is, 1.2 as at least one ω-periodic solution. Te proof is complete. Next, we sall construct some suitable Lyapunov functionals to derive te sufficient conditions wic ensure tat te global exponential stability of periodic solutions of te system 1.2 associated wit te initial conditions x i s = ϕ i s, s [ τ, 0, τ = max t [0,ω {τ jit, i = 1, 2,..., l, j = 1, 2,..., m, y j s = ψ j s, s [ σ, 0, σ = max t [0,ω {σ ijt, i = 1, 2,..., l, j = 1, 2,..., m, were ϕ i C[ τ, 0, R, ψ i C[ σ, 0, R, i = 1, 2,..., l, j = 1, 2,..., m. In te sequel, we will use te following notation: b M ij a m i = min t [0,ω a it, b m j = min t [0,ω b jt, a M ji = max t [0,ω p jit, = max t [0,ω q ijt, τ M ji = max t [0,ω τ jit, σ M ij = max t [0,ω σ ijt, p M ji = max t [0,ω p jit, q M ij = max t [0,ω q ijt, were i = 1, 2,..., l, j = 1, 2,..., m. Our result about te global exponential stability of periodic solutions of 1.2 is as follows. Teorem 2.3. Assume tat S1-S3 old. Furtermore, assume P1 For i = 1, 2,..., l and j = 1, 2,..., m, σ ij, τ ji C 1 R, [0, satisfy P2 σ M ij a m i > b m j > = max t [0,ω σ ijt < 1, τ ji M = max τ jit < 1, t [0,ω i qm ij Lĝ i 1 σ ij M, i = 1, 2,..., l, a M ji L f j pm ji Lg j 1 τ ji M, j = 1, 2,..., m, ten 1.2 as a unique ω-periodic solution x 1, x 2,..., x l, y 1, y2,..., ym T moreover, tere exist constants η > 0 and Λ 1 suc tat for t > 0, x i t x i t y j t yj t Λe ηt[ sup s [ τ,0 x i s x i s sup s [ σ,0 y j s yj s. and,

7 EJDE-2006/32 EXISTENCE AND EXPONENTIAL STABILITY 7 Proof. Let xt = {x 1 t, x 2 t,..., x l t, y 1 t, y 2 t,..., y m t be an arbitrary solution of 1.2, and x t = {x 1t, x 2t,..., x l t, y 1t, y 2t,..., y mt be an ω-periodic solution of 1.2. Ten d x i t x i t a m i x i t x i t a M ji L f j y jt yj t p M ji L g j y jt τ ji t yj t τ ji t, d y j t yj t b m j y j t yj t n n i x it x i t q M ij Lĝ i x it σ ij t x i t σ ij t, fori = 1, 2,..., l, j = 1, 2,..., m. Let F i and G j be defined by 2.2 F i ε i = a m i G j ζ j = b m j ε i ζ j were ε i, ζ j [0,. It is clear tat F i 0 = a m i G j 0 = b m j i qm ij Lĝ i eεiσm ij 1 σ M ij M a M ji L f j pm ji Lg j eζjτ ji 1 τ M ji, i = 1, 2,..., l,, j = 1, 2,..., m, i qm ij Lĝ i 1 σ ij M > 0, i = 1, 2,..., l, a M ji L f j pm ji Lg j 1 τ ji M > 0, j = 1, 2,..., m. Since F i and G j are continuous on [0, and F i ε i, G j ζ j as ε i, ζ j, tere exist ε i, ζ j > 0 suc tat F i ε i = 0, G j ζ j = 0 for ε i 0, ε i and F i ε i > 0, G j ζ j > 0 for ζ j 0, ζj. By coosing η = min{ε 1, ε 2,..., ε l, ζ 1, ζ 2,..., ζ m, we obtain F i η = a m i G i η = b m j Now let us define η η i qm ij Lĝ i eησm ij 1 σ M ij M a M ji L f j pm ji Lg j eητ ji 1 τ M ji 0, i = 1, 2,..., l, 0, j = 1, 2,..., m. u i t = e ηt x i t x i t, t [ τ,, i = 1, 2,..., l, v j t = e ηt y j t y j t, t [ τ,, j = 1, 2,..., m. 2.3

8 8 Y. LI EJDE-2006/32 Ten it follows from 2.2 and 2.3 tat d u i t d v j t a m i ηu i t a M ji L f j v jt p M ji L g j eητjit v j t τ ji t, b m j ηv j t n i u it n q M ij Lĝ i eησijt u i t σ ij t, i = 1, 2,..., n, j = 1, 2,..., m. Consider a Lyapunov function defined by 2.4 V t = u i t v j t p M ji Lg j eητ M ji 1 τ M ji q M ij Lĝ i eησm ij 1 σ M ij t t τ jit t t σ ijt v j s ds u i s ds, 2.5 and we note tat V t > 0 for t > 0 and V 0 is positive and finite. Calculating te derivatives of V along te solutions of 2.4, we get dv = [ a m i ηu i t a M ji L f j v jt [ b m j ηv j t [ b m j [ a m i η η i m a M ji L f j i u it p M ji Lg j eητ M ji 1 τ M ji qij M Lĝ i eησm ij 1 σ ij M u i t p M ji Lg j eητ M ji 1 τ M ji F i ηu i t G j ηv j t 0, t > 0. v j t qij M Lĝ i eησm ij 1 σ ij M u i t v j t It follows tat V t V 0 for t > 0 and ence from 2.3 and 2.5 we obtain u i t v j t u i 0 v j 0 p M ji Lg j eητ M ji 1 τ M ji q M ij Lĝ i eησm ij 1 σ M ij 0 τ ji0 0 σ ij0 v j s ds u i s ds. 2.6

9 EJDE-2006/32 EXISTENCE AND EXPONENTIAL STABILITY 9 It follows from 2.3 and 2.6 tat x i t x i t y j t yj t e ηt Λe ηt[ were t > 0 and Λ = 1 1 sup s [ τ,0 p M ji Lg j eητ M ji 1 τ M ji qij M Lĝ i eησm ij 1 σ ij M { max 1 1 i l, 1 j m sup s [ τ,0 sup s [ σ,0 x i s x i s p M ji Lg j eητ M ji 1 τ M ji x i s x i s y j s y j s sup s [ σ,0, 1 y j s yj s, qij M Lĝ i eησm ij 1 σ ij M Te uniqueness of te periodic solution is follows from 2.7. Tis completes te proof. 3. Discrete-time analogues In tis section, we sall use a semi-discretization tecnique to obtain te discretetime analogue of 1.2. For convenience, we use te following notations. Let Z denote te set of integers; Z 0 = {0, 1, 2,... ; [a, b Z = {a, a 1,..., b 1, b, were a, b Z, a b; [a, Z = {a, a1, a2,..., were a Z. Wile tere is no unique way of obtaining a discrete-time analogue from te continuous-time network 1.2, we begin by approximating te network 1.2 by equations wit piecewise constant arguments of te form dx i t [ t [ t = a i xi t a ji fj y j t dy j t p ji [ t [ t = b j yj t n [ t q ij gj y j [ t n ĝi x i [ t τji [ t [ t b ij ˆfi x i t σij [ t [ t I i, [ t J j, 3.1 were i = 1, 2,..., n, j = 1, 2,..., m, t [n, n 1, n Z 0 and is a fixed positive real number denoting a uniform discretization step-size and [r denotes te integer part of r R. We note tat [t/ = n for t [n, n1. For convenience, we use te notations a i n = a i n, b i n = b i n, a ji n = a ji n, p ji n = p ji n, b ij n = b ij n, q ij n = q ij n, τ ji n = τ ji n, σ ij n = σ ij n, I i n = I i n, J j n = J j n, x i n = x i n, and y j n = y j n. Wit tese

10 10 Y. LI EJDE-2006/32 preparations, 3.1 can be rewritten as dx i t = a i nx i t a ji nf j y j n p ji tg j y j n τ ji n I i n, dy j t = b j ny j n b ij n ˆf i x i n q ij nĝ i x i n σ ij n J j n, 3.2 were i = 1, 2,..., l, j = 1, 2,..., m, t [n, n1, n Z 0. Te initial values of 3.2 will be given below in 3.6. Integrating 3.2 over te interval [n, t, were t < n 1, we get x i te aint x i ne ainn = e aint e ainn { m a ji nf j y j n a i n p ji tg j y j n τ ji n I i n, y j te bjnt y j ne bjnn = e bjnt e bjnn { b ij n b j n ˆf i x i n q ij nĝ i x i n σ ij n J j n, 3.3 i = 1, 2,..., l, j = 1, 2,..., m. By allowing t n 1 in te above expression, we obtain x i n 1 = x i ne ain α i a ji nf j y j n α i p ji ng j y j n τ ji n α i I i n, i = 1, 2,..., l, y j n 1 = y j ne bjn β j were β j b ij n ˆf i x i n q ij nĝ i x i n σ ij n β j J j n, j = 1, 2,..., m, α i = 1 e ain a i n, β j = 1 e bjn, b j n 3.4 i = 1, 2,..., l, i = 1, 2,..., m, n Z 0. It is not difficult to verify tat α i > 0, β i > 0 if a i, b i, > 0 and α i O 2, β i O 2 for small > 0. Also, one can sow tat 3.4 converges towards 1.2 wen 0. In studying

11 EJDE-2006/32 EXISTENCE AND EXPONENTIAL STABILITY 11 te discrete-time analogue 3.4, we assume tat 0,, a i, b i : Z 0,, a ji, p ji, b ij, q ij, I i : Z R, τ ji, σ ij : Z Z 0, i = 1, 2,..., l, j = 1, 2,..., m 3.5 and te nonlinear activation functions f j, g j, ˆf i, ĝ i satisfy S2 and S3. Te system 3.4 is supplemented wit initial values given by x i s = ϕ i s, s [ τ, 0 Z, τ = max sup{τ jin, n Z, i = 1, 2,..., l, 1 i l,1 j m y j s = ψ j s, s [ σ, 0 Z, σ = max sup{σ ijn, n Z, j = 1, 2,..., m, 1 i l,1 j m 3.6 were ϕ i and ψ j denote real-valued continuous functions defined on [ τ, 0 and [ σ, 0, respectively. In wat follows, for convenience, we will use te following notation: a m i = min n [0, Z {a i n, b m j = max n [0, Z {b j n, a M ji = p M ji = max { a ji n, b M ij = max { b ij n, n [0, Z n [0, Z max { p ji n, qij M = max { q ij n, n [0, Z n [0, Z τ M ji = max n [0, Z { τ ji n, σ M ij = max n [0, Z { σ ij n, for i = 1, 2,..., l and j = 1, 2,..., m. Also define I M = a M = max 1 i l, 1 j m {am ji n, b M = max 1 i l, 1 j m {bm ij, M f = sup{ f j u, j = 1, 2,..., m, u R M g = sup{ g j u, j = 1, 2,..., m, u R M ˆf = sup{ i u, i = 1, 2,..., l, Mĝ = sup{ ĝ i u, i = 1, 2,..., l, u R u R max { I i n, i = 1, 2,..., l, J M = max { J j n, j = 1, 2,..., m. n [0, Z n [0, Z Te following result was given in [1, Lemma 3.2. Lemma 3.1. Let f : Z R be ω-periodic, i.e., fk ω = fk. Ten for any fixed n 1, n 2 I ω, and any k Z, one as fn fn 1 fs 1 fs, fn fn 2 fs 1 fs. Using Lemma 2.1, we sall sow te following result about te existence of at least one periodic solution of 3.4. Teorem 3.2. Assume tat S2, S3 and 3.5 old. Furtermore, assume tat S4 a i, a ji, p ji, I i, b j, b ij, q ij, J j, i = 1, 2,..., l, j = 1, 2,..., m are all ω-periodic functions, were ω > 1 is a positive integer. S5 { 1 min 1 i l, 1 j m a m i ln ω 1 1, ω b m j ln ω 1. ω

12 12 Y. LI EJDE-2006/32 Ten system 3.4 as at least one ω-periodic solution. Proof. Define l d = {u = {un : un R lm, n Z. Let l ω l d denotes te subspace of all ω periodic sequences equipped wit te norm, i.e., u = u 1, u 2..., u lm T = lm max u i n, n [0, Z were u = {u 1 n, u 2 n,..., u lm n, n Z l ω. It is not difficult to sow tat l ω is a finite-dimensional Banac space. Let l0 ω = {u = {un l ω : un = 0, n=0 l ω c = {u = {un l ω : un = c R m, n Z. Ten it is easy to ceck tat l ω 0 and l ω c are bot closed linear subspaces of l ω and l ω = l ω 0 l ω c, dim l ω c = l m. Take X = Y = l ω and let x 1 x 1 ne a1n 1 α 1 m a j1nf j y j n.,. N x l x l ne aln 1 α l m y 1 = a jlnf j y j n y 1 ne b1n 1 β 1 l b i1n ˆf i x i n.. y m y m ne bmn 1 β m l b imn ˆf i x i n α 1 m p j1tg j y j n τ j1 n α 1 I 1 n. α l m p jltg j y j n τ jl n α l I l n β 1 l q, i1nĝ i x i n σ i1 n β 1 J 1 n. β m l q imnĝ i x i n σ im n β m J m n x X, n Z, Lun = un 1 un, u X, n Z. It is easy to see tat L is a bounded linear operator wit ker L = l ω c, Im L = l ω 0, dim ker L = l m = codim Im L, ten it follows tat L is a Fredolm mapping of index zero. Define P u = 1 us, u X, Qv = 1 vs, v Y. ω ω It is not difficult to sow tat P and Q are continuous projectors suc tat Im P = ker L, Im L = ker Q = ImI Q.

13 EJDE-2006/32 EXISTENCE AND EXPONENTIAL STABILITY 13 Furtermore, te generalized inverse to L K P : Im L ker P Dom L exists, wic is given by K P y = ys 1 ω sys. ω Obviously, QN and K P I QN are continuous. Since X is a finite-dimensional Banac space, one can easily sow tat K P I QN Ω is compact for any open bounded set Ω X. Moreover, QN Ω is bounded, and ence N is L-compact on Ω wit any open bounded set Ω X. We now are in a position to searc for an appropriate open, bounded subset Ω X for te continuation teorem. Corresponding to operator equation Lu = λnu, λ 0, 1, we ave { x i n 1 x i n = λ x i n1 e ain α i a ji nf j y j n α i p ji ng j y j n τ ji n α i I i n, { y j n 1 y j n = λ y j n1 e bjn β j β j b ij n ˆf i x i n q ij nĝ i x i n σ ij n β j J j n, 3.7 were i = 1, 2,..., l, j = 1, 2,..., m. Suppose tat {x 1 n, x 2 n,..., x l n, y 1 n, y 2 n,..., y m n T X is a solution of system 3.7 for a certain λ 0, 1. Summing on bot sides of 3.7 from 0 to ω 1 wit respect to n, we obtain x i s1 e ais = α i a ji sf j y j s for i = 1, 2,..., l, and α i y j s1 e bjs = β j p ji sg j y j s τ ji s b ij s ˆf i x i s β j for j = 1, 2,..., m. Let n i, n j [0, ω 1 Z suc tat x i n i = α i I i s, 3.8 q ij sĝ i x i s σ ij s β j J j s, max {x i n, y j n j = max {y j n, n [0, Z n [0, Z for i = 1, 2,..., l, j = 1, 2,..., m, ten it follows from 3.8 and 3.9 tat x i n i 1 e ais α i a ji sf j y j s 3.9

14 14 Y. LI EJDE-2006/32 for i = 1, 2,..., l and α i p ji sg j y j s τ ji s α i I i s, y j n j 1 e bjs for j = 1, 2,..., m. Hence x i n i [ β j b ij s ˆf i x i s β j q ij sĝ i x i s σ ij s β j J j s, 1 [ 1 e ais α i a ji sf j y j s α i [ p ji sg j y j s τ ji s α i I i s 1 [ 1 e ais α i a ji s f j y j s α i α i ω [ p ji s g j y j s τ ji s α i I i s 1[ma 1 e ais M M f mp M M g I M := A i, i = 1, 2,..., l 3.10 and y j n j 1 [ 1 e bjs β j [ β j β j ω [ b ij s ˆf i x i s q ij sĝ i x i s σ ij s β j J j s 1[lb 1 e bjs M M ˆf lqm Mĝ J M := B j, j = 1, 2,..., m Let n i, n j [0, ω 1 Z suc tat x i n i = min {x i n, y j n j = min {y j n, n [0, Z n [0, Z for i = 1, 2,..., l, j = 1, 2,..., m. Again, it follows from 3.8 and 3.9 tat x i n i 1 e ais α i a ji sf j y j s

15 EJDE-2006/32 EXISTENCE AND EXPONENTIAL STABILITY 15 α i p ji sg j y j s τ ji s α i I i s, for i = 1, 2,..., l and y j n j 1 e bjs for j = 1, 2,..., m. Hence β j b ij s ˆf i x i s β j q ij sĝ i x i s σ ij s β j J j s, x i n i 1 [ 1 e ais α i a ji sf j y j s [ α i ωα i [ p ji sg j y j s τ ji s α i I i s 1[ma 1 e ais M M f mp M M g I M = A i, i = 1, 2,..., l 3.12 and y j n j 1 [ 1 e bjs β j [ β j ωβ j [ b ij s ˆf i x i s q ij sĝ i x i s σ ij s β j J j s 1[lb 1 e bjs M M ˆf lqm Mĝ M J = B j, j = 1, 2,..., m According to 3.7, we ave { x i n 1 x i n λ x i n 1 e ain α i α i a ji n f j y j n p ji n g j y j n τ ji n α i I i n x i n 1 e am i α i [ma M M f mp M M g I M 3.14

16 16 Y. LI EJDE-2006/32 and { y j n 1 y j n λ y j n 1 e bjn β j β j b ij n ˆf i x i n q ij n ĝ i x i n σ ij n β j J j n y j n 1 e bm i β j [lb M M ˆf lp M Mĝ J M, 3.15 were i = 1, 2,..., l, j = 1, 2,..., m. It follows from 3.10, 3.12, 3.14 and Lemma 3.1 tat for i = 1, 2,..., l, and Tus, x i n x i n i x i s 1 x i s A i 1 e am i x i s ωα i [ma M M f mp M M g I M x i n x i n i x i s 1 x i s A i 1 e am i x i s ωα i [ma M M f mp M M g I M. x i n A i 1 e am i x i s ωα i [ma M M f mp M M g I M 3.16 for i = 1, 2,..., l. Summing on bot sides of 3.16 from 0 to ω 1 wit respect to n, we obtain x i s ωa i ω1 e am i x i s ω 2 α i [ma M M f mp M M g I M for i = 1, 2,..., l. Since 0 < 1 a m i x i s ln ω for i = 1, 2,..., l, we ave [1 ω1 e am i 1[ ωa i ω 2 α i ma M M f mp M M g I M := C i, i = 1, 2,..., l. In view of 3.16 and 3.17, we get x i n A i 1 e am i C i ωα i [ma M M f mp M M g I M := E i, 3.17

17 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 tat y j n B i 1 e bm j D j ωβ j [lb M M ˆf lq M Mĝ J M := F j, for j = 1, 2,..., m. were D j = [1 ω1 e bm j 1[ ωb j ω 2 β j lb M M ˆf lq M Mĝ J M, for j = 1, 2,..., m. Denote C = l E i m F j H, were H > 0 is a constant. Clearly, C is independent of λ. Now we take Ω = {x X : x < C. Te rest of te proof is similar to tat of te proof of Teorem 2.2 and will be omitted. Te proof is complete. Teorem 3.3. Assume tat S2-S3 and 3.5 old. Furtermore, assume tat τ ji n τ ji, σ ij n σ ij Z, n Z, i, j = 1, 2,..., m are constants and P3 a m i > b m j > i qm ij Lĝ j, i = 1, 2,..., l, a M ji L f j pm ji L g i, j = 1, 2,..., m, ten te ω-periodic solution {x 1n, x 2n,..., x l n, y 1n, y 2n,..., y mn T of 3.4 is unique and is globally exponentially stable in te sense tat tere exist constants λ > 1 and δ 1 suc tat x i n x i n y j n yj n 1 n{ δ sup x i s x λ i s sup y j s yj s, s [ τ,0 Z s [ σ,0 Z 3.18 for n Z. Proof. Let xn = {x 1 n, x 2 n,..., x l n, y 1 n, y 2 n,..., y m n T be an arbitrary solution of 3.4, and x n = {x 1n, x 2n,..., x mn, y 1n, y 2n,..., y mn T

18 18 Y. LI EJDE-2006/32 be an ω-periodic solution of 3.4. Ten x i n 1 x i n 1 x i n x i n e am i α i a M ji L f j y jn yj n α i p M ji L g j y jn τ ji n yj n τ ji n, y j n 1 y j n 1 y j n y j n e bm j β j β j i x in x i n q M ij Lĝ i x in σ ij n x i n σ ij n, 3.19 were i = 1, 2,..., l, j = 1, 2,..., m. Now we consider functions Γ i, and Θ j,, i = 1, 2,..., l, j = 1, 2,..., m, defined by Γ i µ i, n = 1 µ i e ain µ i α i Θ j ν j, n = 1 ν j e bjn ν j α j m i a M ji L f j q M ij Lĝ j θ iµ σm ij 1 i p M ji L g j θ jν τ M ji 1 j, were µ i, ν j [1,, n [0, ω 1 Z, i = 1, 2,..., l, j = 1, 2,..., m. Since Γ i 1, n = 1 e ain α i i α i qij M Lĝ j [ = α i a i n [ α i a m i m i m i for n [0, ω 1 Z, i = 1, 2,..., l, and Θ j 1, n = 1 e bjn β j [ = β j b j n [ β j b m j qij M Lĝ j q M ij Lĝ j > 0, a M ji L f j β j a M ji L f j a M ji L f j p M ji L g i p M ji L g i p M ji L g i > 0, for n [0, ω 1 Z, j = 1, 2,..., m. Using te continuity of Γ i µ i, n and Θ j ν j, n on [1, wit respect to µ i and ν j, respectively, for every n [0, ω 1 Z and te fact tat Γ i µ i, n as µ i and Θ j ν j, n as ν j uniformly in n [0, ω 1 Z, i = 1, 2,..., l, j = 1, 2,..., m, we see tat tere exist νi n, ν j n 1, suc tat Γ iµ i n, n = 0 and Θ jνj n, n = 0 for n

19 EJDE-2006/32 EXISTENCE AND EXPONENTIAL STABILITY 19 [0, ω 1 Z, i = 1, 2,..., l, j = 1, 2,..., m. By coosing λ = min{µ i n, ν j n, n [0, ω 1 Z, i = 1, 2,..., l, j = 1, 2,..., m, were λ > 1, we obtain Γ i λ, n 0 and Θ j λ, n 0 for all n [0, ω 1 Z, i = 1, 2,..., l, j = 1, 2,..., m, tat is, λe ain λα i m i qij M Lĝ j θ iλ σm ij 1 1, n [0, ω 1 Z, λe bjn λα j a M ji L f j p M ji L g j θ jλ τ M ji 1 1, n [0, ω 1 Z, for i = 1, 2,..., l, j = 1, 2,..., m. Hence λe am i λα i m i qij M Lĝ j θ iλ σm ij 1 1, λe bm j λα j a M ji L f j p M ji L g j θ jλ τ M ji 1 1, for i = 1, 2,..., l, j = 1, 2,..., m. Now let us consider u i n = λ n x in x i n, n [ τ, Z, α i i = 1, 2,..., l, v j n = λ n y jn yj n, n [ τ, Z, β j j = 1, 2,..., m. Using 3.4 and 3.21, we derive tat u i n 1 λe am i u i n λ β j a M ji L f j v jn β j p M ji L g j λτji1 v j n τ ji, i = 1, 2,..., l, v j n 1 λe bm j v j n λ We consider te Lyapunov function V n = α i i u in α i q M ij Lĝ i λσij1 u i n σ ij, j = 1, 2,..., m. u i n β j p M ji L g M j λτ ji 1 v i n α i q M ij Lĝ i λσm ij 1 n 1 v j s s=n τ ji n 1 u i s. s=n σ ij Calculating te difference V n = V n 1 V n along 3.22, we obtain V n 1 λe am i u i n λ β j a M ji L f j v jn

20 20 Y. LI EJDE-2006/32 λ = β j p M ji L g M j λτ ji 1 v j n α i i u in 1 λe am i λα i 1 λe bm j λβ j 1 λe bm j v j n α i qij M Lĝ j λσm ij 1 u i n i α i a M ji L f j β j q M ij Lĝ j λσm ij 1 u i n p M ji L g M 1 j λτ ji v j n. Using 3.20 in te above we deduce tat V n 0 for n Z 0. From tis result and 3.23 it follows tat Tus u i n v j n V n V 0 for n Z u i n v j n = λ n x i n x i n α i u i 0 λ n m β j p M ji L g M j λτ ji 1 v i 0 1 α i 1 β j α i q M ij Lĝ i λσm ij 1 y i n y i n β j 1 v j s s= τ ji 1 u i s s= σ ij qij M Lĝ i λσm ij 1 1 σ ij sup { x i s x i s α i s [ σ,0 Terefore, we obtain te assertion 3.18, were and σ = { max 1 α i 1 i l, 1 j m p M ji L g M j λτ ji 1 1 τ ji sup { y j s yj s. β j s [ τ,0 δ = max{max 1 i l α i, max 1 j m β j min{min 1 i l α i, min 1 j m β j σ 1 q M ij Lĝ i λσm ij 1 σ ij, 1 β j p M ji L g M j λτ ji 1 τ ji 1. We conclude from 3.18 tat te unique periodic solution of 3.4 is globally exponentially stable and tis completes te proof.

21 EJDE-2006/32 EXISTENCE AND EXPONENTIAL STABILITY An Example Consider te following BAM neural networks system wit discrete delays dx i 2 = a itx i t p ji tf j y j t τ ji I i t, i = 1, 2, dy j = b jty j t 2 q ij tg i x i t σ ij J j t, j = 1, 2, 4.1 in wic for i, j = 1, 2, f j u = arctanu j, g i u = u iu 2, a it = 5cos t 2, b j t = sin t4, p ji t = sinijt, q ij t = i j sin t, τ ji, σ ij are positive constants, I i t, J j t are any continuous 2π-periodic functions. Ten it is easy to sow tat 4.1 satisfies all te conditions of Teorem 2.3, ence 4.1 as a unique 2π-periodic solution wic is globally exponential stable. References [1 Fan, M. and Wang, K.; Periodic solutions of a discrete time nonautonomous ratio-dependent predator-prey system. Matematical and Computer Modelling, vol , [2 Gaines, R.E. and Mawin, J.L. Coincidence Degree and Nonlinear Differential Equations. Berlin: Springer-Verlag, [3 K. Gopalsamy, X.Z. He, Delay-independent stability in bi-directional associative memory networks. IEEE Trans Neural Networks, 1994;5: [4 Gopalsamy, K. and He, XZ. Delay-independent stability in bi-directional associative memory networks. IEEE Trans Neural Networks, vol , [5 Hopfield, J.J. Neuron wit graded response ave collective computational properties like tose of two-state neurons. Proc Natl Acad Sci USA, vol , [6 Kelly, D.G. Stability in contractive nonlinear neural networks. IEEE Trans Biomed Eng vol , [7 Kosko, B. Adaptive bidirectional associative memories. Appl Opt, vol , [8 Kosko, B. Bidirectional associative memories, IEEE Trans Systems Man Cybernet, vol , [9 Kosko, B. Unsupervised learning in noise. IEEE Trans Neural Networks, vol , [10 Li, YK. and Kuang, Y.; Periodic solutions of periodic delay Lotka-Volterra equations and systems. J Mat Anal Appl, vol , [11 Liao, XF. and Yu, JB. Qualitative analysis of bidrectional associative memory wit time delays. Int J Circuit Teory Appl, vol , [12 Liao, XX. Stability of te Hopfield neural networks. Sci. Cina, vol no. 5, [13 Mawin, J.L. Topological degree metods in nonlinear boundary value problems. CBMS Regional Conf. Ser. in Mat., No. 40, Providence, RI: Amer Mat Soc, [14 Moamad, S. Global exponential stability in continuous-time and discrete-time delay bidirectional neural networks. Pysica D vol , [15 Rao, VSH. and Paneendra, BRM. Global dynamics of bidirectional associative memory neural networks involving transmission delays and dead zones. Neural networks, vol , [16 Zang, Y. and Yang, YR. Global stability analysis of bi-directional associative memory neural networks wit time delay. Int J Circ Teor Appl, Vol , [17 Zao, H. Global stability of bidirectional associative memory neural networks wit distributed delays. Pysics Letters A, vol , Department of Matematics, Yunnan University, Kunming, Yunnan , Cina address: yklie@ynu.edu.cn