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Existence and global exponential stability of periodic solution to self-connection BAM neural networks with delays
Physics Letters A 328 (2004) 127–143 www.elsevier.com/locate/pla
Existence and global exponential stability of periodic solution to self-connection BAM neural networks with delays Zhigang Liu a , Anping Chen a,∗ , Lihong Huang b a Department of Mathematics, Xiangnan University, Chenzhou 423000, China b College of Mathematics and Econometrics, Hunan University, Changsha 410082, China
Received 20 December 2003; received in revised form 23 May 2004; accepted 25 May 2004 Available online 15 June 2004 Communicated by A.P. Fordy
Abstract By using the continuation theorem of Mawhin’s coincidence degree theory, Lyapunov functional method and some analytical techniques, some sufficient conditions are obtained ensuring existence and global exponential stability of periodic solution of the self-connection BAM neural networks with periodic coefficients and delays. These results are more effective than the ones in [IEEE Trans. Circuits Systems 50 (2003) 1162] for some neural networks, which has an important leading significance in the designing globally exponentially stable and periodic oscillatory BAM neural networks with self-connection. 2004 Elsevier B.V. All rights reserved. MSC: 34K13; 34K20; 92B20 Keywords: BAM neural networks; Exponential stability; Periodic solution; Delays
1. Introduction In this Letter, we consider the self-connection BAM networks with periodic coefficients and delays: xi (t) = −ai xi (t) + cii (t)s1,i xi t − dii (t) + pj i (t)fj yj t − τj i (t) + Ii (t), p
(1.1a)
j =1 n yj (t) = −bj yj (t) + hjj (t)s2,j yj t − mjj (t) + qij (t)gi xi t − σij (t) + Jj (t),
(1.1b)
i=1
where ai > 0, bj > 0 are constants (i = 1, 2, . . . , n; j = 1, 2, . . . , p). xi (t) and yj (t) are the activations of the ith neuron and the j th neuron, respectively. pj i (t), qij (t) are the connection weights at the time t. cii (t) and hjj (t) * Corresponding author.
E-mail addresses: [email protected] (Z. Liu), [email protected] (A. Chen), [email protected] (L. Huang). 0375-9601/$ – see front matter 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2004.05.055
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Z. Liu et al. / Physics Letters A 328 (2004) 127–143
are the self-connection weights at the time t. Ii (t), Jj (t) denote the external inputs at time t. τj i (t) and σij (t) correspond to the finite speed of the axonal signal transmission at time t. dii (t) and mjj (t) are the associated delays, s1,i (t) and s2,j (t) are self-connection functions. An analog circuit implementing of BAM equation with axonal signal transmission delays can be seen in Refs. [1,2]. Clearly, if one assume that cii = 0, hjj = 0, (i = 1, 2, . . . , n; j = 1, 2, . . . , p), then system (1.1a), (1.1b) reduces to xi (t) = −ai xi (t) +
p
pj i (t)fj yj t − τj i (t) + Ii (t),
(1.2a)
j =1
yj (t) = −bj yj (t) +
n
qij (t)gi xi t − σij (t) + Jj (t),
(1.2b)
i=1
whose global asymptotic stability was investigated in [2–8,22–26]. In [1–4,27–31], some sufficient conditions have been obtained for global asymptotic stability of delayed BAM neural networks via Liapunov functional method. For the existence and global exponential stability of the almost periodic solution, we refer to [9,10]. Moreover, authors in [1,11,25] research the existence and global exponential stability of periodic oscillatory solution of system (1.2a), (1.2b) with delays. In [12–14], the authors have been studied stabilization role of inhibitory self-connections of a delayed neural networks. In addition, we also refer to [15–19,27–31]. To the best of our knowledge, few authors have considered periodic oscillatory solutions for self-connection BAM networks with periodic coefficient and delays. The main aim of this Letter is to investigate the existence and global exponential stability of the periodic solution of systems (1.1) by using Mawhin’s coincidence degree theory, Lyapunov functional method and some analytical techniques. Some simple sufficient conditions are obtained ensuring the existence and global exponential stability of periodic solution of systems (1.1), which improve and extend the known results in the literature. These results are more effective than the ones in Ref. [11] for some neural networks which has an important leading significance in the design and applications of periodic oscillatory neural circuits for the self-connection BAM with delays. Our results are not interrelated with the ones in Ref. [11]. The initial conditions associated with (1.1a), (1.1b) are of the form xi (s) = φi (s), yj (s) = ψj (s),
s ∈ [−, 0], = max(σ, d), σ =
max
1in; 1j p
s ∈ [−, 0], = max(τ, m), τ =
σij+ , d = max dii+ ,
max
1in
1j p;1in
τj+i , m = max m+ jj , 1j p
where φi (s), ψj (s) are continuous ω-periodic functions, dii+ = max dii (t), 0t ω
m+ jj = max mjj (t), 0t ω
σij+ = max σij (t), 0t ω
τj+i = max τj i (t). 0t ω
For any solution T T z(t) x T (t), y T (t) x1 (t), x2 (t), . . . , xn (t), y1 (t), y2 (t), . . . , yp (t) and periodic solution T T z∗ (t) x ∗T (t), y ∗T (t) x1∗ (t), x2∗ (t), . . . , xn∗ (t), y1∗ (t), y2∗ (t), . . . , yp∗ (t) of system (1.1a), (1.1b), define (φ T , ψ T )T − (x ∗T , y ∗T )T as p n T T T ∗T ∗T T ∗ φ ,ψ − x ,y max φi (t) − xi (t) + max ψj (t) − yj∗ (t). = i=1
−t 0
j =1
− t 0
(1.3a) (1.3b)
Z. Liu et al. / Physics Letters A 328 (2004) 127–143
129
Definition 1 [20]. Matrix A = (aij ) is said to be nonsingular M-matrix, if (i) aii > 0,
(ii)
(iii) A−1 0,
aij 0,
for i = j, i, j = 1, 2, . . . , n.
Definition 2. The periodic solution (x ∗T (t), y ∗T (t))T of system (1.1a), (1.1b) is said to be globally exponentially stable, if there exist constants α > 0 and M 1 such that p n xi (t) − x ∗ (t) + yj (t) − y ∗ (t) M φ T , ψ T T − x ∗T , y ∗T T e−αt , i j j =1
i=1
for all t > 0. Throughout this Letter, we always assume that cii (t), hjj (t), pj i (t), qij (t), Ii (t), Jj (t) are continuous ωperiodic functions. dii (t) 0, mjj (t) 0, τj i (t) 0 and σij (t) 0 are continuously differentiable ω-periodic functions and 0 dii (t) < 1, 0 mjj (t) < 1, 0 τj i (t) < 1, 0 σij (t) < 1. ai , bj are positive constants, lj i =
1 0t ω 1 − τj i (t)
1/2
kij =
1 (t) 0t ω 1 − σij
1/2
max
1/2 1 , (t) 0t ω 1 − dii 1/2 1 = max . 0t ω 1 − mjj (t)
ζii =
,
max
,
ηjj
max
(1.4) (1.5)
For every continuous ω-periodic function ϕ, we define 1 ϕ˜ = ω
ω ϕ(t) dt
and ϕ + = max ϕ(t). 0t ω
0
Furthermore, the signal functions gi , fj and self-connection functions s1,i , s2,j possess the following properties: (H1 ) There exist positive constants µj > 0, νi > 0, κ1,i > 0 and κ2,j > 0 such that fj (x) − fj (y) µj , x −y gi (x) − gi (y) νi , 0 x−y 0
s2,j (x) − s2,j (y) κ2,j , s2,j (0) = 0, fj (0) = 0, x−y s1,i (x) − s1,i (y) κ1,i , s1,i (0) = 0, gi (0) = 0, 0 x−y 0
for each x, y ∈ R, x = y, i = 1, 2, . . . , n; j = 1, 2, . . . , p. The remaining part of this Letter is organized as follows: in Section 2, we prove the existence of the periodic solution and point out that the delays have effect on the existence of the periodic solution; in Section 3, we establish the results that periodic solutions are the globally exponentially stable by using Lyapunov functional method; in Section 4, an example is given.
2. The existence of a periodic solution In order to obtain the existence of periodic solutions of (1.1a), (1.1b), we shall use some concepts and results from the book by Gaines and Mawhin [21]. Let X, Z be normed vector spaces, L : Dom L ⊂ X → Z be a linear mapping and N : X → Z be a continuous mapping. The mapping L is called a Fredholm mapping of index zero if dim Ker L = codim Im L < +∞ and Im L is closed in Z. If L is a Fredholm mapping of index zero, there exist continuous projectors P : X → X and Q : Z → Z such that Im P = Ker L and Im L = Ker Q = Im(I − Q). It follows that L|Dom L∩Ker P : (I − P )X → Im L is
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invertible. we denote the inverse of the map by Kp . If Ω is a bounded open subset of X, the mapping N is called L-compact on Ω¯ if QN(Ω¯ ) is bounded and Kp (I − Q)N : Ω¯ → X is compact. Because Im Q is isomorphic to Ker L, there exists an isomorphism J : Im Q → Ker L. Consider an abstract equation in a Banach space X Lz = λNz,
λ ∈ (0, 1),
(2.0)
where L : Dom L ∩ X → X is a linear operator and λ is a parameter. Let P and Q denote two projectors P : X ∩ Dom L → Ker L and Q : X → X/ Im L. For convenience of use, we introduce Mawhin’s continuation theorem [21, p. 40] as follows. Lemma 2.1 (Mawhin’s continuation theorem [21, p. 40]). Let X be a Banach space and L be a Fredholm mapping of index zero. Assume that N : Ω¯ → X is a L-compact on Ω¯ with Ω bounded in X. Furthermore assume that (a) for each λ ∈ (0, 1), x ∈ ∂Ω ∩ Dom L, Lz = λNz; (b) for each z ∈ ∂Ω ∩ Ker L, QNz = 0, and deg{QNz, Ω ∩ Ker L, 0} = 0. ¯ Then Lz = Nz has at least one solution in Ω. Lemma 2.2 [20]. Assume that A is a nonsingular M-matrix and Aw d, then w A−1 d. Theorem 1. Assume that (H1 ) holds and (i) A is a nonsingular M-matrix, p (ii) ai − |c˜ii |κ1,i − νi j =1 |q˜ij | > 0, bj − |h˜ jj |κ2,j − µj ni=1 |p˜j i | > 0, i = 1, 2, . . . , n; j = 1, 2, . . . , p. Then system (1.1a), (1.1b) has at least one ω-periodic solution, where A = (Ars )2×2 ,
A11 = diag(a11 , a22, . . . , ann ), aii = ai − cii+ κ1,i ζii2 + ωai ζii ,
2 A22 = diag(an+1,n+1 , an+2,n+2 , . . . , an+p,n+p ), an+j,n+j = bj − h+ jj κ2,j ηjj + ωbj ηjj , A12 = (ai,n+j )n×p , ai,n+j = −pj+i µj lj2i + lj i ωai , A21 = (an+j,i )p×n , an+j,i = −qij+ νi kij2 + ωbj kij . Proof. Let x(t) = (x1 (t), x2 (t), . . . , xn (t))T , y(t) = (y1 (t), y2 (t), . . . , yp (t))T . In order to apply Lemma 2.1 to systems (1.1a), (1.1b), we take
T X = x T (t), y T (t) ∈ C R, R n+p : x(t + ω) = x(t), y(t + ω) = y(t), ω > 0 equipped with the norm p n T T T = x ,y max xi (t) + max yj (t), i=1
t ∈[0,ω]
j =1
t ∈[0,ω]
Z. Liu et al. / Physics Letters A 328 (2004) 127–143
131
then X is a Banach space. Let N(x1 , . . . , xn , y1 , . . . , yp )T p −a1 x1 (t) + c11 (t)s1,1 (x1 (t − d11 (t))) + j =1 pj 1 (t)fj (yj (t − τj 1 (t))) + I1 (t) ··· −a x (t) + c (t)s (x (t − d (t))) + p p (t)f (y (t − τ (t))) + I (t) n n nn 1,n n nn j j jn n j =1 j n = , −b1 y1 (t) + h11 (t)s2,1 (y1 (t − m11 (t))) + n qi1 (t)gi (xi (t − σi1 (t))) + J1 (t) i=1 ··· −bp yp (t) + hpp (t)s2,p (yp (t − mpp (t))) + ni=1 qip (t)gi (xi (t − σip (t))) + Jp (t) 1 ω x1 (t) x1 (t) x1 (t) x1 ω 0 x1 (t) dt ··· ··· ··· ··· ω· · · 1 xn (t) ω 0 xn (t) dt xn (t) xn xn (t) P L , y1 (t) = Q y1 (t) = y1 = y (t) , 1 ω 1 ω 0 y1 (t) dt ··· ··· ··· ··· ··· 1 ω yp yp (t) yp (t) yp (t) y (t) dt ω 0 p where (x1 (t), . . . , xn (t), y1 (t), . . . , yp (t))T ∈ X. It is easy to prove that L is a Fredholm mapping of index zero, that P : X ∩ dom L → Ker L and Q : X → X/ Im L are two projectors, and N is L-compact on Ω¯ for any given open bounded set. Corresponding to Eq. (2.0), we have xi t − dii (t) + λ pj i (t)fj yj t − τj i (t) + λIi (t), p
xi (t) = −λai xi (t) + λcii (t)s1,i
(2.1a)
j =1 n yj (t) = −λbj yj (t) + λhjj (t)s2,j yj t − mjj (t) + λ qij (t)gi xi t − σij (t) + λJj (t),
(2.1b)
i=1
where i = 1, 2, . . . , n; j = 1, 2, . . . , p. Suppose that (x1 (t), . . . , xn (t), y1 (t), . . . , yp (t))T ∈ X is a solution of system (2.1a), (2.1b) for a certain λ ∈ (0, 1). For the sake of convenience, we define x2 by ω 1/2 2 x2 = , for x ∈ C(R, R). x(t) dt 0
From (H1 ) it implies that fj (x) µj |x|, gi (x) νi |x|,
s2,j (x) κ2,j |x|, j = 1, 2, . . . , p, s1,i (x) κ1,i |x|, i = 1, 2, . . . , n.
Multiplying both sides of equation (2.1a) by xi and integrating over [0, ω], we obtain xi 22
ω =λ
xi t − dii (t) dt + λ p
cii (t)xi (t)s1,i
pj i (t)xi (t)fj
yj t − τj i (t) dt + λ
j =1 0
0
cii+ κ1,i
ω
ω
x (t)xi t − dii (t) dt +
p
i
0
j =1
pj+i µj
ω
xi (t)Ii (t) dt
0
ω 0
x (t)yj t − τj i (t) dt + I + i i
ω
x (t) dt i
0
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cii+ κ1,i
ω
2 x (t) dt i
0
+
p
1/2 ω
pj+i µj
ω
j =1
xi t − dii (t) 2 dt
1/2
0
2 x (t) dt
1/2 ω
i
0
yj t − τj i (t) 2 dt
1/2 +
√
ωIi+
ω
p
1/2
i
0
cii+ ζii κ1,i xi 2 xi 2 +
2 x (t) dt
0
pj+i µj lj i xi 2 yj 2 +
√
ωIi+ xi 2 .
j =1
It implies that xi 2
cii+ ζii κ1,i xi 2
+
p
pj+i µj lj i yj 2 +
√ + ωIi ,
(2.2)
j =1
where lj i , ζii are defined in (1.4). Multiplying both sides of Eq. (2.1b) by yj (t) and integrating from 0 to ω, a similar argument to (2.2), we have yj 2
h+ jj ηjj κ2,j yj 2
+
n
kij qij+ νi xi 2 +
√ + ωJj ,
(2.3)
i=1
where kij , ηjj are defined in (1.5). Integrating Eq. (2.1a) over [0, ω], we obtain ω
ω xi (t) dt =
ai 0
cii (t)s1,i xi t − dii (t) dt + p
ω
pj i (t)fj yj t − τj i (t) dt +
j =1 0
0
ω Ii (t) dt. 0
Thus there exists a point ξi ∈ (0, ω) such that ω ωai xi (ξi ) =
cii (t)s1,i xi t − dii (t) dt + p
ω
pj i (t)fj yj t − τj i (t) dt +
j =1 0
0
ω Ii (t) dt. 0
It follows that ωai xi (ξi ) c+ κ1,i
ω
ii
p xi t − dii (t) dt + p + µj ji
j =1
0
cii+ ζii2 κ1,i
ω
ji
j =1
yj t − τj i (t) dt + ωI + i
0
p xi (t) dt + p + µj l 2
0
√
ω
ω
ji
yj (t) dt + ωI + i
0
√ + + ω pj i µj lj2i yj 2 + ωIi+ . p
ωcii+ ζii2 κ1,i xi 2
j =1
That is √ √ ωai xi (ξi ) cii+ ζii2 κ1,i xi 2 + pj+i µj lj2i yj 2 + ωIi+ . p
j =1
(2.4)
Z. Liu et al. / Physics Letters A 328 (2004) 127–143
133
Since, for t ∈ [0, ω], xi (t) xi (ξi ) +
ω
√ x (t) dt xi (ξi ) + ωx 2 , i i
(2.5)
0
and so
ω
ai xi 2 = ai
xi (t)2 dt
1/2
√
√ ωai max xi (t) ωai xi (ξi ) + ωai xi 2 0t ω
0
cii+ ζii2 κ1,i xi 2
+
p
pj+i µj lj2i yj 2 +
√ + ωIi + ωai xi 2 .
pj+i µj lj2i yj 2 +
√ + ωIi
(2.6)
j =1
Substituting (2.2) into (2.6), we have ai xi 2 cii+ κ1,i ζii2 xi 2 +
p j =1
+ ωai cii+ ζii κ1,i xi 2
+
p
pj+i µj lj i yj 2
+
√
ωIi+
j =1
√ pj+i µj lj2i + lj i ωai yj 2 + ω(1 + ωai )Ii+ , = cii+ κ1,i ζii2 + ωai ζii xi 2 + p
j =1
i.e.,
√ ai − cii+ κ1,i ζii2 + ωai ζii xi 2 − pj+i µj lj2i + lj i ωai yj 2 ω(1 + ωai )Ii+ di . p
(2.7)
j =1
Integrating Eq. (2.1b) from 0 to ω, similar argument to (2.6), we get 2 bj yj 2 h+ jj ηjj κ2,j yj 2 +
n
qij+ νi kij2 xi 2 +
√ + ωJj + ωbj yj 2 .
qij+ νi kij2 xi 2 +
√ + ωJj
(2.8)
i=1
Substituting (2.3) into (2.8), we get 2 bj yj 2 h+ jj ηjj κ2,j yj 2 +
n i=1
+ ωbj h+ jj ηjj κ2,j yj 2
+
n
kij qij+ νi xi 2
√ + ωJj+
i=1
2 = h+ jj κ2,j ηjj
+ ωbj ηjj yj 2 +
n
√ qij+ νi kij2 + ωbj kij xi 2 + (1 + ωbj ) ωJj+ .
i=1
That is, −
n i=1
2
√ + qij+ νi kij2 + ωbj kij xi 2 + bj − h+ jj κ2,j ηjj + ωbj ηjj yj 2 (1 + ωbj ) ωJj dn+j .
(2.9)
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Z. Liu et al. / Physics Letters A 328 (2004) 127–143
(2.7) and (2.9) may be rewritten as the following form Aw d,
(2.10)
where A = (Ars )2×2 , d = (d1 , d2 , . . . , dn+p )T , T w = x1 2 , x2 2 , . . . , xn 2 , y1 2 , y2 2 , . . . , yp 2 . From condition (i) and Lemma 2.2 it follows that T ∗ w A−1 d R1∗ , R2∗ , . . . , Rn+p . That is xi 2 Ri∗ ,
∗ yj 2 Rn+j ,
i = 1, 2, . . . , n; j = 1, 2, . . . , p.
(2.11)
From (2.2)–(2.5), and (2.11), there exist n + p positive constants Rl such that xi (t) Ri , yj (t) Rn+j , i = 1, 2, . . . , n; j = 1, 2, . . . , p, for ∀t ∈ [0, ω]. Clearly, Rl (l = 1, 2, . . . , n + p) are independent of λ, Denote ∗
M =
n+p
Rl + C,
l=1
where C > 0 is taken sufficiently large such that ∗
min(a, b)M >
n i=1
where
|I˜i | +
p
1in
(2.12)
j =1
a = min
|J˜j |,
ai − |c˜ii |κ1,i − νi
p
|q˜ij | ,
b = min
1j p
j =1
bj − |h˜ jj |κ2,j − µj
n
|p˜ j i | .
i=1
When Now we take Ω = {(x T (t), y T (t))T ∈ X: (x T , y T )T < M ∗ }, this satisfies condition (a) of Lemma 2.1. p z = (x T , y T )T ∈ ∂Ω ∩ Ker L = ∂Ω ∩ R n+p , (x T , y T )T is a constant vector in R n+p with ni=1 |xi | + j =1 |yj | = M ∗ . Then −a x + c˜ s (x ) + p p˜ f (y ) + I˜ 1 1 11 1,1 1 1 x1 j =1 j 1 j j · · · ... p ˜ xn −an xn + c˜nn s1,n (xn ) + j =1 p˜ j n fj (yj ) + In QN . n y1 = −b1 y1 + h˜ 11 s2,1 (y1 ) + i=1 q˜i1 gi (xi ) + J˜1 ... ··· n yp q˜ip gi (xi ) + J˜p −bp yp + h˜ pp s2,p (yp ) + i=1
Therefore, QN(x1 , x2 , . . . , xn , y1 , y2 , . . . , yp )T p p n n p˜j i fj (yj ) + I˜i + q˜ij gi (xi ) + J˜j = −ai xi + c˜ii s1,i (xi ) + −bj yj + h˜ jj s2,j (yj ) + i=1
j =1
j =1
i=1
Z. Liu et al. / Physics Letters A 328 (2004) 127–143
n
ai |xi | −
i=1
−
n
p p p n n |c˜ii |s1,i (xi ) − |p˜ j i |fj (yj ) − |I˜i | + bj |yj | − |h˜ jj |s2,j (yj ) i=1 j =1
i=1
p n
|q˜ij |gi (xi ) −
j =1 i=1
n
−
n
νi |q˜ij ||xi | −
n
ai − κ1,i |c˜ii | − νi
−
p
n i=1
j =1
|I˜i | +
p
j =1 p
|q˜ij | |xi | +
n
|I˜i | +
i=1
p
p
bj |yj | −
j =1
bj − κ2,j |h˜ jj | − µj
j =1
p
κ2,j |h˜ jj ||yj |
j =1
n
|p˜ j i | |yj |
i=1
|J˜j |
j =1
min(a, b)M ∗ −
µj |p˜ j i ||yj | −
|J˜j |
j =1
i=1
p n i=1 j =1
i=1
p n
j =1
i=1
|J˜j |
κ1,i |c˜ii ||xi | −
j =1 i=1
=
p j =1
ai |xi | −
i=1
135
n
|I˜i | +
i=1
p
|J˜j | > 0.
j =1
Consequently, QNz = 0,
for z ∈ ∂Ω ∩ Ker L.
Define Φ : Ker L × [0, 1] → R n+p by Φ(x1 , x2 , . . . , xn , y1 , y2 , . . . , yp , µ) p −a1 x1 + c˜11 s1,1 (x1 ) + j =1 p˜j 1 fj (yj ) + I˜1 −a1 x1 ··· ... p −an xn −an xn + c˜nn s1,n (xn ) + j =1 p˜ j n fj (yj ) + I˜n = µ , n −b1 y1 + (1 − µ) ˜ ˜ −b1 y1 + h11 s2,1 (y1 ) + i=1 q˜i1 gi (xi ) + J1 ... ··· n −bp yp ˜ ˜ −bp yp + hpp s2,p (yp ) + i=1 q˜ip gi (xi ) + Jp where z = (x1 , x2 , . . . , xn , y1 , y2 , . . . , yp )T ∈ R n+p , µ ∈ [0, 1]. When (x1 , . . . , xn , y1 , . . . , yp )T ∈ ∂Ω ∩ Ker L
p and µ ∈ [0, 1], the vector (x1 , . . . , xn , y1 , . . . , yp )T is a constant vector in R n+p with ni=1 |xi | + j =1 |yj | = M ∗ . Thus Φ(x1 , x2 , . . . , xn , y1 , y2 , . . . , yp , µ) p n p˜ j i fj (yj ) + I˜i = −µai xi + (1 − µ) −ai xi + c˜ii s1,i (xi ) + i=1 j =1 p n + q˜ij gi (xi ) + J˜j −µbj yj + (1 − µ) −bj yj + h˜ jj s2,j (yj ) + j =1
i=1
136
Z. Liu et al. / Physics Letters A 328 (2004) 127–143
p n = p˜j i fj (yj ) + I˜i ai xi − (1 − µ) c˜ii s1,i (xi ) + i=1 j =1 p n + q˜ij gi (xi ) + J˜j bj yj − (1 − µ) h˜ jj s2,j (yj ) + j =1 i=1 n p ∗ ˜ ˜ min(a, b)M − |Ii | + |Jj | > 0. i=1
j =1
Therefore, Φ(x1 , x2 , . . . , xn , y1 , y2 , . . . , yp , µ) = (0, 0, . . . , 0, 0, 0, . . ., 0)T , for (x1 , x2 , . . . , xn , y1 , y2 , . . . , yp )T ∈ ∂Ω ∩ Ker L. As a result, we have deg QN(x1 , . . . , xn , y1 , . . . , yp )T , Ω ∩ Ker L, (0, . . . , 0, 0, . . . , 0)T = deg (−a1 x1 , . . . , −an xn , −b1 y1 , . . . , −bp yp )T , Ω ∩ Ker L, (0, . . . , 0)T = 0. By now, we know that Ω satisfies all conditions of Lemma 2.1. Therefore, system (1.1a), (1.1b) has at least one ω-periodic solution. This completes the proof. 2
3. Global exponential stability of periodic solution In this section, we shall discuss the global exponential stability of BAM neural networks with self-connection terms. Suppose that T z∗ = x1∗ (t), . . . , xn∗ (t), y1∗ (t), . . . , yp∗ (t) is a periodic solution of (1.1a), (1.1b) as described in Theorem 1. T z = x1 (t), . . . , xn (t), y1 (t), . . . , yp (t) is any solution of system (1.1a), (1.1b). Set ui (t) = xi (t) − xi∗ (t), vj (t) = yj (t) − yj∗ (t), Fj vj (t) = fj vj (t) + yj∗ (t) − fj yj∗ (t) , Gi ui (t) = gi ui (t) + xi∗ (t) − gi xi∗ (t) , S2,j vj (t) = s2,j vj (t) + yj∗ (t) − s2,j yj∗ (t) , S1,i ui (t) = s1,i ui (t) + xi∗ (t) − s1,i xi∗ (t) , where i = 1, 2, . . . , n; j = 1, 2, . . . , p. It is easy to see that system (1.1a), (1.1b) can be reduced to the following system pj i (t)Fj vj t − τj i (t) , ui (t) = −ai ui (t) + cii (t)S1,i ui t − dii (t) + p
(3.1a)
j =1 n vj (t) = −bj vj (t) + hjj (t)S2,j vj t − mjj (t) + qij (t)Gi ui t − σij (t) . i=1
(3.1b)
Z. Liu et al. / Physics Letters A 328 (2004) 127–143
137
By (H1 ), we have Gi (u) − Gi (v) νi , u−v Fj (u) − Fj (v) µj , 0 u−v
S1,i (u) − S1,i (v) κ1,i , Gi (0) = S1,i (0) = 0, u−v S2,j (u) − S2,j (v) κ2,j , Fj (0) = S2,j (0) = 0, 0 u−v
0
0
(3.2)
for i = 1, 2, . . . , n; j = 1, 2, . . . , p. Theorem 2. Assume that all conditions of Theorem 1 hold. If there exist positive constants λi and λn+j (i = 1, 2, . . . , n; j = 1, 2, . . . , p) which satisfy the following condition: (H2 )
p qij+ νi cii+ κ1,i + λi −ai + λn+j < 0, 1 − dii+ 1 − σij+ j =1
λn+j −bj +
h+ jj κ2,j 1 − m+ jj
+
n i=1
λi
pj+i µj 1 − τj+ i
< 0.
Then there exists a unique ω-periodic solution (x ∗T (t), y ∗T (t))T of the system (1.1a), (1.1b) which is globally exponentially stable. That is, all other solutions (x T (t), y T (t))T of (1.1a), (1.1b) satisfy p n xi (t) − x ∗ (t) + yj (t) − y ∗ (t) M φ T , ψ T T − x ∗T , y ∗T T e−αt , i j
(3.3)
j =1
i=1
for t 0. Proof. The uniqueness of the ω-periodic solution will follow from (3.3). Therefore, it suffices to prove the ω-periodic solution is globally exponentially stable. That is, every solutions (x T (t), y T (t))T of (1.1a), (1.1b) satisfy (3.3). In view of (H2 ), we can choose a suitable constant α ∈ (0, min1in; 1j p {ai , bj }) such that λi α − ai +
cii+ κ1,i
e 1 − dii+
αdii+
+
p j =1
λn+j
qij+ νi
e 1 − σij+
ασij+
< 0,
n pj+i µj ατ + h+ jj κ2,j αm+ jj e λ e ji < 0. λn+j α − bj + + i 1 − m+ 1 − τj+ jj i i=1
(3.4)
By (3.1), (3.2), we have p + D + ui (t) −ai ui (t) + cii+ κ1,i ui t − dii (t) + pj i µj vj t − τj i (t)
(3.5a)
j =1
and n D + vj (t) −bj vj (t) + h+ t − m κ (t) + qij+ νi ui t − σij (t) , v jj jj 2,j j i=1
where D + denotes the upper right Dini derivative. Consider a Lyapunov functional V (t) defined by V (t) = V1 (t) + V2 (t),
(3.5b)
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Z. Liu et al. / Physics Letters A 328 (2004) 127–143
V1 (t) =
n
λi e
αt
i=1
+
n c+ κ1,i ui (t) + λi ii + 1 − dii i=1
p n
λi
i=1 j =1
V2 (t) =
p
λn+j e
αt
j =1
+
p n
t
pj+i µj 1 − τj+ i
t
ui (r)eα(r+dii+) dr
t −dii (t )
+ vj (r)eα(r+τji ) dr,
t −τji (t )
p h+ jj κ2,j λn+j vj (t) + 1 − m+ jj j =1
λn+j
j =1 i=1
t
qij+ νi 1 − σij+
t
+ vj (t)eα(r+mjj ) dr
t −mjj (t )
+ ui (r)eα(r+σij ) dr.
(3.6)
t −σij (t )
Calculating the derivative of the Vk (t) (k = 1, 2) along solutions of (3.1), we have n n
αt c+ κ1,i + αt + D V1 (t) (3.1) = λi αe ui (t) + e D ui (t) + λi ii + ui (t)eα(t +dii ) 1 − dii i=1 i=1 +
−
n
λi
i=1
+
pj+i µj α(t +τ +) ji λi + vj (t) e 1 − τ j i j =1
p n i=1
−
cii+ κ1,i + ui t − dii (t) 1 − dii (t) eα(t −dii (t )+dii ) 1 − dii+
p n
λi
i=1 j =1
pj+i µj α(t −τji (t )+τ +) ji . + vj t − τj i (t) 1 − τj i (t) e 1 − τj i +
+
α(t −τji (t )+τji ) + By (3.5a) and noting that 1 − τj i (t) 1 − τj+ eαt , eα(t −dii (t )+dii ) eαt , i , 1 − dii (t) 1 − dii and e we obtain n n
D + V1 (t)(3.1) eαt λi α ui (t) − ai ui (t) + eαt λi cii+ κ1,i ui t − dii (t) i=1
+ eαt
i=1
i=1
−e
αt
n c+ κ1,i + λi pj+i µj vj t − τj i (t) + eαt λi ii + eαdii ui (t) 1 − dii j =1 i=1
p n
n
λi cii+ κ1,i ui t
i=1
− eαt
p n
p n pj+i µj ατ + αt ji v (t) − dii (t) + e λi j + e 1 − τj i i=1 j =1
λi pj+i µj vj t − τj i (t)
i=1 j =1
=e
αt
n i=1
λi α − ai +
cii+ κ1,i
1 − dii+
e
αdii+
p n pj+i µj ατ + ui (t) + eαt ji v (t). λi j + e 1 − τj i i=1 j =1
(3.7)
Z. Liu et al. / Physics Letters A 328 (2004) 127–143
139
Similar to (3.7), we get p p n qij+ νi ασ + h+ jj κ2,j αm+ αt vj (t) + eαt jj ij u (t). D V2 (t) (3.1) e λn+j α − bj + e λ n+j i + + e 1 − mjj 1 − σij j =1 j =1 i=1 (3.8) It follows from (3.4), (3.6)–(3.8) that p n qij+ νi ασ + cii+ κ1,i αd + + αt ui (t) ij ii λn+j λi α − ai + + D V (t) (3.1) e + e + e 1 − dii 1 − σij i=1 j =1 p n pj+i µj ατ + h+ jj κ2,j αm+ αt jj ji v (t) < 0. +e (3.9) λi λn+j α − bj + + j + e + e 1 − mjj 1 − τj i j =1 i=1 +
Namely, V (t) V (0), for t 0. Again by (3.6), we obtain n p αt − ui (t) + vj (t) , V (t) e λ
(3.10)
j =1
i=1
where λ− = min1ln+p {λl }. On the other hand, from (3.6), we have n c+ κ1,i V (0) = λi ui (0) + λi ii + 1 − dii i=1 i=1
0
n
+
ui (r)eα(r+dii+) dr
−dii (0)
0 pj+i µj + vj (r)eα(r+τji ) dr λi + 1 − τj i i=1 j =1 −τji (0)
p n
p h+ jj κ2,j λn+j vj (0) + λn+j + 1 − m+ jj j =1 j =1 p
+
p n
λn+j
j =1 i=1
+
λ
0
qij+ νi 1 − σij+
0
+ vj (r)eα(r+mjj ) dr
−mjj (0)
+ ui (r)eα(r+σij ) dr
−σij (0)
0 p n n + c κ 1,i ii + ui (0) + vj (0) + λ ui (r)eα(r+d) dr + 1 − dii i=1 j =1 i=1 −d
+
+λ
p n pj+i µj i=1 j =1
1 − τj+ i
0 −τ
0 p h+ jj κ2,j vj (r)eα(r+τ ) dr + λ+ vj (r)eα(r+m) dr + 1 − mjj j =1 −m
0 n qij+ νi + ui (r)eα(r+σ ) dr +λ + 1 − σij j =1 i=1 −σ p
+
λ
p n n cii+ κ1,i ui (0) + vj (0) + λ+ deαd 1 − dii+ i=1 j =1 i=1
max ui (r)
−dr0
140
Z. Liu et al. / Physics Letters A 328 (2004) 127–143 p n pj+i µj
+ λ+ τ eατ
i=1
+ λ+ σ eασ
p h+ jj κ2,j vj (r) + λ+ meαm max + −τ r0 1 − τj i 1 − m+ jj j =1 j =1
p n qij+ νi
max 1 − σij+ −σ r0 j =1 i=1
max vj (r)
−mr0
ui (r)
T T λ+ M1 φ T , ψ T − x ∗T , y ∗T ,
(3.11)
where σ=
max
1in; 1j p
σij+ ,
m = max m+ jj ,
1 − dii+
M1(2)
τe
ατ
p n pj+i µj i=1 j =1
1 − τj+ i
+ meαm + σe
ασ
τj+i ,
(1)
M1 = M1 + M2 ,
p h+ jj κ2,j j =1
max
1j p;1in (1)
max {λl },
1ln+p
n cii+ κ1,i i=1
τ=
1in
λ+ =
1j p
M1(1) 1 + deαd
d = max dii+ ,
1 − m+ jj
,
p n qij+ νi j =1 i=1
1 − σij+
.
Set M=
M1 λ+ . λ−
It follows from (3.10) and (3.11) that p n ui (t) + vj (t) M φ T , ψ T T − x ∗T , y ∗T T e−αt , i=1
(3.12)
j =1
for t 0. This completes the proof. 2 Remark. When self-connection terms cii = 0, hjj = 0 (i = 1, 2, . . . , n; j = 1, 2, . . . , p), system (1.1) reduces to the following BAM neural networks model: xi (t) = −ai xi (t) +
p
pj i (t)fj yj t − τj i (t) + Ii (t),
(3.13a)
j =1
yj (t) = −bj yj (t) +
n
qij (t)gi xi t − σij (t) + Jj (t),
(3.13b)
i=1
Corollary. Assume that all conditions of Theorem 1 hold. If there exist positive constants λi and λn+j (i = 1, 2, . . . , n; j = 1, 2, . . . , p) satisfy the following conditions: (H3 )
− λi ai +
p j =1
λn+j
qij+ νi 1 − σij+
< 0,
−λn+j bj +
n i=1
λi
pj+i µj 1 − τj+ i
< 0.
Then ω-periodic solution of the system (3.13a), (3.13b) is globally exponentially stable.
Z. Liu et al. / Physics Letters A 328 (2004) 127–143
141
4. An example In this section, we give an example to illustrate that our results are feasible, which shows our new results are more effective than ones in [11] for some neural networks. Consider the following simple BAM networks: xi (t) = −ai xi (t) +
2
pj i (t)fj yj t − τj i (t) + Ii (t),
j =1
yj (t) = −bj yj (t) +
2
qij (t)fi xi t − σij (t) + Jj (t),
(4.1)
i=1
i = 1, 2; j = 1, 2, where Ii (t) = sin 40πt, τj i =
1 , 20
Jj (t) = cos 40πt,
σij =
1 ; 20
ω=
τj i+ = 0,
1 , 20
σij + = 0
(i, j = 1, 2).
Taking fj (x) = gi (x) = 12 (|x + 1| − |x − 1|), we have νi = µj = 1 (i, j = 1, 2). Again taking (a1 , a2 )T = (1, 1)T , Let
Then
q11 (t) q12 (t) q21 (t) q22 (t) + + q12 q11
+ + q21 q22
Moreover,
=
(b1 , b2 )T = (2, 3)T .
=
5 5 11 11 10 10 23 23
1 0 −1 0 1 −1 A= −0.5 −0.5 2 −0.5 −0.5 0
5 5 11 cos 40πt 11 sin 40πt 10 10 23 sin 40πt 23 cos 40πt
,
+ + p12 p11 + + p21 p22
−1 −1 , 0 3
,
=
3.5 2.5 A−1 = 1.5 1
20 21 20 21
2.5 3.5 1.5 1
20 21 20 21
3 3 2 1
p11 (t) p12 (t) p21 (t) p22 (t)
=
20 21 sin 40πt 20 21 cos 40πt
lj i = kij = 1.
,
2 2 > 0. 1 1
Hence, A is a nonsingular M-matrix. Set λl = 1 (l = 1, 2, 3, 4), we get q˜11 q˜12 p˜11 p˜ 12 0 0 = = , 0 0 p˜21 p˜ 22 q˜21 q˜22 ai − νi
2
|q˜ij | = 1 > 0,
b j − µj
j =1
−λ1 a1 +
2 j =1
2
|p˜ j i | = 1 > 0,
i=1
λ2+j
+ q1j ν1 + 1 − σ1j
=−
1 < 0, 11
−λ2 a2 +
2 j =1
λ2+j
+ q2j ν2 + 1 − σ2j
=−
3 < 0, 23
20 21 cos 40πt 20 21 sin 40πt
.
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Z. Liu et al. / Physics Letters A 328 (2004) 127–143
−λ3 b1 +
2
λi
i=1
+ p1i µ1
1 − τ1i +
fj (x) − fj (y) 1, 0 x −y
=−
2 < 0, 21
fj (0) = 0,
−λ4 b2 +
2 i=1
λi
+ p2i µ2
1 − τ2i +
gi (x) − gi (y) 1, 0 x−y
=−
23 < 0, 21
gi (0) = 0,
for i, j = 1, 2. Thus, it follows from corollary of Theorem 2 that systems (4.1) has a unique and all other solutions of systems (4.1) converge exponentially to the periodic solution. But + 2 2 q1j 188 + λ1 −2a1 + µ1 pj 1 + µ1 λ2+j + = 231 > 0, 1 − σ1j j =1 j =1 + 2 2 q2j 374 λ2 −2a2 + µ2 pj+2 + µ2 λ2+j + = 483 > 0. 1 − σ2j j =1
1 20 -periodic
solution
j =1
So, the assumption (H3) in Ref. [11] is not available. It is easy to see that the results in this Letter and the ones in Ref. [11] are not interrelated, i.e., there always exists a networks which satisfies one and does not satisfy another.
5. Conclusion In this Letter, By using the continuation theorem of Mawhin’s coincidence degree theory, Lyapunov functional method and some analytical techniques, some sufficient conditions are obtained ensuring existence and global exponential stability of periodic solution of the self-connection BAM neural networks with periodic coefficients and delays. We would like to point out that these results are more effective than the ones in [11] for some neural networks, which has an important leading significance in designing globally exponentially stable and periodic oscillatory BAM neural networks with self-connection.
Acknowledgements The authors would like to thank the reviewers and the editor for their valuable suggestions and comments. This work was supported by the Natural Science Foundation of China under Grant 10371034 and by the Foundation for University Excellent Teacher by Chinese Ministry of Education (No. [2002] 78), the Scientific Research Foundation of Hunan Provincial Education Department (01C009, 03C009).
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Existence and global exponential stability of periodic solution to self-connection BAM neural networks with delays Zhigang Liu a , Anping Chen a,∗ , Lihong Huang b a Department of Mathematics, Xiangnan University, Chenzhou 423000, China b College of Mathematics and Econometrics, Hunan University, Changsha 410082, China
Received 20 December 2003; received in revised form 23 May 2004; accepted 25 May 2004 Available online 15 June 2004 Communicated by A.P. Fordy
Abstract By using the continuation theorem of Mawhin’s coincidence degree theory, Lyapunov functional method and some analytical techniques, some sufficient conditions are obtained ensuring existence and global exponential stability of periodic solution of the self-connection BAM neural networks with periodic coefficients and delays. These results are more effective than the ones in [IEEE Trans. Circuits Systems 50 (2003) 1162] for some neural networks, which has an important leading significance in the designing globally exponentially stable and periodic oscillatory BAM neural networks with self-connection. 2004 Elsevier B.V. All rights reserved. MSC: 34K13; 34K20; 92B20 Keywords: BAM neural networks; Exponential stability; Periodic solution; Delays
1. Introduction In this Letter, we consider the self-connection BAM networks with periodic coefficients and delays: xi (t) = −ai xi (t) + cii (t)s1,i xi t − dii (t) + pj i (t)fj yj t − τj i (t) + Ii (t), p
(1.1a)
j =1 n yj (t) = −bj yj (t) + hjj (t)s2,j yj t − mjj (t) + qij (t)gi xi t − σij (t) + Jj (t),
(1.1b)
i=1
where ai > 0, bj > 0 are constants (i = 1, 2, . . . , n; j = 1, 2, . . . , p). xi (t) and yj (t) are the activations of the ith neuron and the j th neuron, respectively. pj i (t), qij (t) are the connection weights at the time t. cii (t) and hjj (t) * Corresponding author.
E-mail addresses: [email protected] (Z. Liu), [email protected] (A. Chen), [email protected] (L. Huang). 0375-9601/$ – see front matter 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2004.05.055
128
Z. Liu et al. / Physics Letters A 328 (2004) 127–143
are the self-connection weights at the time t. Ii (t), Jj (t) denote the external inputs at time t. τj i (t) and σij (t) correspond to the finite speed of the axonal signal transmission at time t. dii (t) and mjj (t) are the associated delays, s1,i (t) and s2,j (t) are self-connection functions. An analog circuit implementing of BAM equation with axonal signal transmission delays can be seen in Refs. [1,2]. Clearly, if one assume that cii = 0, hjj = 0, (i = 1, 2, . . . , n; j = 1, 2, . . . , p), then system (1.1a), (1.1b) reduces to xi (t) = −ai xi (t) +
p
pj i (t)fj yj t − τj i (t) + Ii (t),
(1.2a)
j =1
yj (t) = −bj yj (t) +
n
qij (t)gi xi t − σij (t) + Jj (t),
(1.2b)
i=1
whose global asymptotic stability was investigated in [2–8,22–26]. In [1–4,27–31], some sufficient conditions have been obtained for global asymptotic stability of delayed BAM neural networks via Liapunov functional method. For the existence and global exponential stability of the almost periodic solution, we refer to [9,10]. Moreover, authors in [1,11,25] research the existence and global exponential stability of periodic oscillatory solution of system (1.2a), (1.2b) with delays. In [12–14], the authors have been studied stabilization role of inhibitory self-connections of a delayed neural networks. In addition, we also refer to [15–19,27–31]. To the best of our knowledge, few authors have considered periodic oscillatory solutions for self-connection BAM networks with periodic coefficient and delays. The main aim of this Letter is to investigate the existence and global exponential stability of the periodic solution of systems (1.1) by using Mawhin’s coincidence degree theory, Lyapunov functional method and some analytical techniques. Some simple sufficient conditions are obtained ensuring the existence and global exponential stability of periodic solution of systems (1.1), which improve and extend the known results in the literature. These results are more effective than the ones in Ref. [11] for some neural networks which has an important leading significance in the design and applications of periodic oscillatory neural circuits for the self-connection BAM with delays. Our results are not interrelated with the ones in Ref. [11]. The initial conditions associated with (1.1a), (1.1b) are of the form xi (s) = φi (s), yj (s) = ψj (s),
s ∈ [−, 0], = max(σ, d), σ =
max
1in; 1j p
s ∈ [−, 0], = max(τ, m), τ =
σij+ , d = max dii+ ,
max
1in
1j p;1in
τj+i , m = max m+ jj , 1j p
where φi (s), ψj (s) are continuous ω-periodic functions, dii+ = max dii (t), 0t ω
m+ jj = max mjj (t), 0t ω
σij+ = max σij (t), 0t ω
τj+i = max τj i (t). 0t ω
For any solution T T z(t) x T (t), y T (t) x1 (t), x2 (t), . . . , xn (t), y1 (t), y2 (t), . . . , yp (t) and periodic solution T T z∗ (t) x ∗T (t), y ∗T (t) x1∗ (t), x2∗ (t), . . . , xn∗ (t), y1∗ (t), y2∗ (t), . . . , yp∗ (t) of system (1.1a), (1.1b), define (φ T , ψ T )T − (x ∗T , y ∗T )T as p n T T T ∗T ∗T T ∗ φ ,ψ − x ,y max φi (t) − xi (t) + max ψj (t) − yj∗ (t). = i=1
−t 0
j =1
− t 0
(1.3a) (1.3b)
Z. Liu et al. / Physics Letters A 328 (2004) 127–143
129
Definition 1 [20]. Matrix A = (aij ) is said to be nonsingular M-matrix, if (i) aii > 0,
(ii)
(iii) A−1 0,
aij 0,
for i = j, i, j = 1, 2, . . . , n.
Definition 2. The periodic solution (x ∗T (t), y ∗T (t))T of system (1.1a), (1.1b) is said to be globally exponentially stable, if there exist constants α > 0 and M 1 such that p n xi (t) − x ∗ (t) + yj (t) − y ∗ (t) M φ T , ψ T T − x ∗T , y ∗T T e−αt , i j j =1
i=1
for all t > 0. Throughout this Letter, we always assume that cii (t), hjj (t), pj i (t), qij (t), Ii (t), Jj (t) are continuous ωperiodic functions. dii (t) 0, mjj (t) 0, τj i (t) 0 and σij (t) 0 are continuously differentiable ω-periodic functions and 0 dii (t) < 1, 0 mjj (t) < 1, 0 τj i (t) < 1, 0 σij (t) < 1. ai , bj are positive constants, lj i =
1 0t ω 1 − τj i (t)
1/2
kij =
1 (t) 0t ω 1 − σij
1/2
max
1/2 1 , (t) 0t ω 1 − dii 1/2 1 = max . 0t ω 1 − mjj (t)
ζii =
,
max
,
ηjj
max
(1.4) (1.5)
For every continuous ω-periodic function ϕ, we define 1 ϕ˜ = ω
ω ϕ(t) dt
and ϕ + = max ϕ(t). 0t ω
0
Furthermore, the signal functions gi , fj and self-connection functions s1,i , s2,j possess the following properties: (H1 ) There exist positive constants µj > 0, νi > 0, κ1,i > 0 and κ2,j > 0 such that fj (x) − fj (y) µj , x −y gi (x) − gi (y) νi , 0 x−y 0
s2,j (x) − s2,j (y) κ2,j , s2,j (0) = 0, fj (0) = 0, x−y s1,i (x) − s1,i (y) κ1,i , s1,i (0) = 0, gi (0) = 0, 0 x−y 0
for each x, y ∈ R, x = y, i = 1, 2, . . . , n; j = 1, 2, . . . , p. The remaining part of this Letter is organized as follows: in Section 2, we prove the existence of the periodic solution and point out that the delays have effect on the existence of the periodic solution; in Section 3, we establish the results that periodic solutions are the globally exponentially stable by using Lyapunov functional method; in Section 4, an example is given.
2. The existence of a periodic solution In order to obtain the existence of periodic solutions of (1.1a), (1.1b), we shall use some concepts and results from the book by Gaines and Mawhin [21]. Let X, Z be normed vector spaces, L : Dom L ⊂ X → Z be a linear mapping and N : X → Z be a continuous mapping. The mapping L is called a Fredholm mapping of index zero if dim Ker L = codim Im L < +∞ and Im L is closed in Z. If L is a Fredholm mapping of index zero, there exist continuous projectors P : X → X and Q : Z → Z such that Im P = Ker L and Im L = Ker Q = Im(I − Q). It follows that L|Dom L∩Ker P : (I − P )X → Im L is
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Z. Liu et al. / Physics Letters A 328 (2004) 127–143
invertible. we denote the inverse of the map by Kp . If Ω is a bounded open subset of X, the mapping N is called L-compact on Ω¯ if QN(Ω¯ ) is bounded and Kp (I − Q)N : Ω¯ → X is compact. Because Im Q is isomorphic to Ker L, there exists an isomorphism J : Im Q → Ker L. Consider an abstract equation in a Banach space X Lz = λNz,
λ ∈ (0, 1),
(2.0)
where L : Dom L ∩ X → X is a linear operator and λ is a parameter. Let P and Q denote two projectors P : X ∩ Dom L → Ker L and Q : X → X/ Im L. For convenience of use, we introduce Mawhin’s continuation theorem [21, p. 40] as follows. Lemma 2.1 (Mawhin’s continuation theorem [21, p. 40]). Let X be a Banach space and L be a Fredholm mapping of index zero. Assume that N : Ω¯ → X is a L-compact on Ω¯ with Ω bounded in X. Furthermore assume that (a) for each λ ∈ (0, 1), x ∈ ∂Ω ∩ Dom L, Lz = λNz; (b) for each z ∈ ∂Ω ∩ Ker L, QNz = 0, and deg{QNz, Ω ∩ Ker L, 0} = 0. ¯ Then Lz = Nz has at least one solution in Ω. Lemma 2.2 [20]. Assume that A is a nonsingular M-matrix and Aw d, then w A−1 d. Theorem 1. Assume that (H1 ) holds and (i) A is a nonsingular M-matrix, p (ii) ai − |c˜ii |κ1,i − νi j =1 |q˜ij | > 0, bj − |h˜ jj |κ2,j − µj ni=1 |p˜j i | > 0, i = 1, 2, . . . , n; j = 1, 2, . . . , p. Then system (1.1a), (1.1b) has at least one ω-periodic solution, where A = (Ars )2×2 ,
A11 = diag(a11 , a22, . . . , ann ), aii = ai − cii+ κ1,i ζii2 + ωai ζii ,
2 A22 = diag(an+1,n+1 , an+2,n+2 , . . . , an+p,n+p ), an+j,n+j = bj − h+ jj κ2,j ηjj + ωbj ηjj , A12 = (ai,n+j )n×p , ai,n+j = −pj+i µj lj2i + lj i ωai , A21 = (an+j,i )p×n , an+j,i = −qij+ νi kij2 + ωbj kij . Proof. Let x(t) = (x1 (t), x2 (t), . . . , xn (t))T , y(t) = (y1 (t), y2 (t), . . . , yp (t))T . In order to apply Lemma 2.1 to systems (1.1a), (1.1b), we take
T X = x T (t), y T (t) ∈ C R, R n+p : x(t + ω) = x(t), y(t + ω) = y(t), ω > 0 equipped with the norm p n T T T = x ,y max xi (t) + max yj (t), i=1
t ∈[0,ω]
j =1
t ∈[0,ω]
Z. Liu et al. / Physics Letters A 328 (2004) 127–143
131
then X is a Banach space. Let N(x1 , . . . , xn , y1 , . . . , yp )T p −a1 x1 (t) + c11 (t)s1,1 (x1 (t − d11 (t))) + j =1 pj 1 (t)fj (yj (t − τj 1 (t))) + I1 (t) ··· −a x (t) + c (t)s (x (t − d (t))) + p p (t)f (y (t − τ (t))) + I (t) n n nn 1,n n nn j j jn n j =1 j n = , −b1 y1 (t) + h11 (t)s2,1 (y1 (t − m11 (t))) + n qi1 (t)gi (xi (t − σi1 (t))) + J1 (t) i=1 ··· −bp yp (t) + hpp (t)s2,p (yp (t − mpp (t))) + ni=1 qip (t)gi (xi (t − σip (t))) + Jp (t) 1 ω x1 (t) x1 (t) x1 (t) x1 ω 0 x1 (t) dt ··· ··· ··· ··· ω· · · 1 xn (t) ω 0 xn (t) dt xn (t) xn xn (t) P L , y1 (t) = Q y1 (t) = y1 = y (t) , 1 ω 1 ω 0 y1 (t) dt ··· ··· ··· ··· ··· 1 ω yp yp (t) yp (t) yp (t) y (t) dt ω 0 p where (x1 (t), . . . , xn (t), y1 (t), . . . , yp (t))T ∈ X. It is easy to prove that L is a Fredholm mapping of index zero, that P : X ∩ dom L → Ker L and Q : X → X/ Im L are two projectors, and N is L-compact on Ω¯ for any given open bounded set. Corresponding to Eq. (2.0), we have xi t − dii (t) + λ pj i (t)fj yj t − τj i (t) + λIi (t), p
xi (t) = −λai xi (t) + λcii (t)s1,i
(2.1a)
j =1 n yj (t) = −λbj yj (t) + λhjj (t)s2,j yj t − mjj (t) + λ qij (t)gi xi t − σij (t) + λJj (t),
(2.1b)
i=1
where i = 1, 2, . . . , n; j = 1, 2, . . . , p. Suppose that (x1 (t), . . . , xn (t), y1 (t), . . . , yp (t))T ∈ X is a solution of system (2.1a), (2.1b) for a certain λ ∈ (0, 1). For the sake of convenience, we define x2 by ω 1/2 2 x2 = , for x ∈ C(R, R). x(t) dt 0
From (H1 ) it implies that fj (x) µj |x|, gi (x) νi |x|,
s2,j (x) κ2,j |x|, j = 1, 2, . . . , p, s1,i (x) κ1,i |x|, i = 1, 2, . . . , n.
Multiplying both sides of equation (2.1a) by xi and integrating over [0, ω], we obtain xi 22
ω =λ
xi t − dii (t) dt + λ p
cii (t)xi (t)s1,i
pj i (t)xi (t)fj
yj t − τj i (t) dt + λ
j =1 0
0
cii+ κ1,i
ω
ω
x (t)xi t − dii (t) dt +
p
i
0
j =1
pj+i µj
ω
xi (t)Ii (t) dt
0
ω 0
x (t)yj t − τj i (t) dt + I + i i
ω
x (t) dt i
0
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Z. Liu et al. / Physics Letters A 328 (2004) 127–143
cii+ κ1,i
ω
2 x (t) dt i
0
+
p
1/2 ω
pj+i µj
ω
j =1
xi t − dii (t) 2 dt
1/2
0
2 x (t) dt
1/2 ω
i
0
yj t − τj i (t) 2 dt
1/2 +
√
ωIi+
ω
p
1/2
i
0
cii+ ζii κ1,i xi 2 xi 2 +
2 x (t) dt
0
pj+i µj lj i xi 2 yj 2 +
√
ωIi+ xi 2 .
j =1
It implies that xi 2
cii+ ζii κ1,i xi 2
+
p
pj+i µj lj i yj 2 +
√ + ωIi ,
(2.2)
j =1
where lj i , ζii are defined in (1.4). Multiplying both sides of Eq. (2.1b) by yj (t) and integrating from 0 to ω, a similar argument to (2.2), we have yj 2
h+ jj ηjj κ2,j yj 2
+
n
kij qij+ νi xi 2 +
√ + ωJj ,
(2.3)
i=1
where kij , ηjj are defined in (1.5). Integrating Eq. (2.1a) over [0, ω], we obtain ω
ω xi (t) dt =
ai 0
cii (t)s1,i xi t − dii (t) dt + p
ω
pj i (t)fj yj t − τj i (t) dt +
j =1 0
0
ω Ii (t) dt. 0
Thus there exists a point ξi ∈ (0, ω) such that ω ωai xi (ξi ) =
cii (t)s1,i xi t − dii (t) dt + p
ω
pj i (t)fj yj t − τj i (t) dt +
j =1 0
0
ω Ii (t) dt. 0
It follows that ωai xi (ξi ) c+ κ1,i
ω
ii
p xi t − dii (t) dt + p + µj ji
j =1
0
cii+ ζii2 κ1,i
ω
ji
j =1
yj t − τj i (t) dt + ωI + i
0
p xi (t) dt + p + µj l 2
0
√
ω
ω
ji
yj (t) dt + ωI + i
0
√ + + ω pj i µj lj2i yj 2 + ωIi+ . p
ωcii+ ζii2 κ1,i xi 2
j =1
That is √ √ ωai xi (ξi ) cii+ ζii2 κ1,i xi 2 + pj+i µj lj2i yj 2 + ωIi+ . p
j =1
(2.4)
Z. Liu et al. / Physics Letters A 328 (2004) 127–143
133
Since, for t ∈ [0, ω], xi (t) xi (ξi ) +
ω
√ x (t) dt xi (ξi ) + ωx 2 , i i
(2.5)
0
and so
ω
ai xi 2 = ai
xi (t)2 dt
1/2
√
√ ωai max xi (t) ωai xi (ξi ) + ωai xi 2 0t ω
0
cii+ ζii2 κ1,i xi 2
+
p
pj+i µj lj2i yj 2 +
√ + ωIi + ωai xi 2 .
pj+i µj lj2i yj 2 +
√ + ωIi
(2.6)
j =1
Substituting (2.2) into (2.6), we have ai xi 2 cii+ κ1,i ζii2 xi 2 +
p j =1
+ ωai cii+ ζii κ1,i xi 2
+
p
pj+i µj lj i yj 2
+
√
ωIi+
j =1
√ pj+i µj lj2i + lj i ωai yj 2 + ω(1 + ωai )Ii+ , = cii+ κ1,i ζii2 + ωai ζii xi 2 + p
j =1
i.e.,
√ ai − cii+ κ1,i ζii2 + ωai ζii xi 2 − pj+i µj lj2i + lj i ωai yj 2 ω(1 + ωai )Ii+ di . p
(2.7)
j =1
Integrating Eq. (2.1b) from 0 to ω, similar argument to (2.6), we get 2 bj yj 2 h+ jj ηjj κ2,j yj 2 +
n
qij+ νi kij2 xi 2 +
√ + ωJj + ωbj yj 2 .
qij+ νi kij2 xi 2 +
√ + ωJj
(2.8)
i=1
Substituting (2.3) into (2.8), we get 2 bj yj 2 h+ jj ηjj κ2,j yj 2 +
n i=1
+ ωbj h+ jj ηjj κ2,j yj 2
+
n
kij qij+ νi xi 2
√ + ωJj+
i=1
2 = h+ jj κ2,j ηjj
+ ωbj ηjj yj 2 +
n
√ qij+ νi kij2 + ωbj kij xi 2 + (1 + ωbj ) ωJj+ .
i=1
That is, −
n i=1
2
√ + qij+ νi kij2 + ωbj kij xi 2 + bj − h+ jj κ2,j ηjj + ωbj ηjj yj 2 (1 + ωbj ) ωJj dn+j .
(2.9)
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Z. Liu et al. / Physics Letters A 328 (2004) 127–143
(2.7) and (2.9) may be rewritten as the following form Aw d,
(2.10)
where A = (Ars )2×2 , d = (d1 , d2 , . . . , dn+p )T , T w = x1 2 , x2 2 , . . . , xn 2 , y1 2 , y2 2 , . . . , yp 2 . From condition (i) and Lemma 2.2 it follows that T ∗ w A−1 d R1∗ , R2∗ , . . . , Rn+p . That is xi 2 Ri∗ ,
∗ yj 2 Rn+j ,
i = 1, 2, . . . , n; j = 1, 2, . . . , p.
(2.11)
From (2.2)–(2.5), and (2.11), there exist n + p positive constants Rl such that xi (t) Ri , yj (t) Rn+j , i = 1, 2, . . . , n; j = 1, 2, . . . , p, for ∀t ∈ [0, ω]. Clearly, Rl (l = 1, 2, . . . , n + p) are independent of λ, Denote ∗
M =
n+p
Rl + C,
l=1
where C > 0 is taken sufficiently large such that ∗
min(a, b)M >
n i=1
where
|I˜i | +
p
1in
(2.12)
j =1
a = min
|J˜j |,
ai − |c˜ii |κ1,i − νi
p
|q˜ij | ,
b = min
1j p
j =1
bj − |h˜ jj |κ2,j − µj
n
|p˜ j i | .
i=1
When Now we take Ω = {(x T (t), y T (t))T ∈ X: (x T , y T )T < M ∗ }, this satisfies condition (a) of Lemma 2.1. p z = (x T , y T )T ∈ ∂Ω ∩ Ker L = ∂Ω ∩ R n+p , (x T , y T )T is a constant vector in R n+p with ni=1 |xi | + j =1 |yj | = M ∗ . Then −a x + c˜ s (x ) + p p˜ f (y ) + I˜ 1 1 11 1,1 1 1 x1 j =1 j 1 j j · · · ... p ˜ xn −an xn + c˜nn s1,n (xn ) + j =1 p˜ j n fj (yj ) + In QN . n y1 = −b1 y1 + h˜ 11 s2,1 (y1 ) + i=1 q˜i1 gi (xi ) + J˜1 ... ··· n yp q˜ip gi (xi ) + J˜p −bp yp + h˜ pp s2,p (yp ) + i=1
Therefore, QN(x1 , x2 , . . . , xn , y1 , y2 , . . . , yp )T p p n n p˜j i fj (yj ) + I˜i + q˜ij gi (xi ) + J˜j = −ai xi + c˜ii s1,i (xi ) + −bj yj + h˜ jj s2,j (yj ) + i=1
j =1
j =1
i=1
Z. Liu et al. / Physics Letters A 328 (2004) 127–143
n
ai |xi | −
i=1
−
n
p p p n n |c˜ii |s1,i (xi ) − |p˜ j i |fj (yj ) − |I˜i | + bj |yj | − |h˜ jj |s2,j (yj ) i=1 j =1
i=1
p n
|q˜ij |gi (xi ) −
j =1 i=1
n
−
n
νi |q˜ij ||xi | −
n
ai − κ1,i |c˜ii | − νi
−
p
n i=1
j =1
|I˜i | +
p
j =1 p
|q˜ij | |xi | +
n
|I˜i | +
i=1
p
p
bj |yj | −
j =1
bj − κ2,j |h˜ jj | − µj
j =1
p
κ2,j |h˜ jj ||yj |
j =1
n
|p˜ j i | |yj |
i=1
|J˜j |
j =1
min(a, b)M ∗ −
µj |p˜ j i ||yj | −
|J˜j |
j =1
i=1
p n i=1 j =1
i=1
p n
j =1
i=1
|J˜j |
κ1,i |c˜ii ||xi | −
j =1 i=1
=
p j =1
ai |xi | −
i=1
135
n
|I˜i | +
i=1
p
|J˜j | > 0.
j =1
Consequently, QNz = 0,
for z ∈ ∂Ω ∩ Ker L.
Define Φ : Ker L × [0, 1] → R n+p by Φ(x1 , x2 , . . . , xn , y1 , y2 , . . . , yp , µ) p −a1 x1 + c˜11 s1,1 (x1 ) + j =1 p˜j 1 fj (yj ) + I˜1 −a1 x1 ··· ... p −an xn −an xn + c˜nn s1,n (xn ) + j =1 p˜ j n fj (yj ) + I˜n = µ , n −b1 y1 + (1 − µ) ˜ ˜ −b1 y1 + h11 s2,1 (y1 ) + i=1 q˜i1 gi (xi ) + J1 ... ··· n −bp yp ˜ ˜ −bp yp + hpp s2,p (yp ) + i=1 q˜ip gi (xi ) + Jp where z = (x1 , x2 , . . . , xn , y1 , y2 , . . . , yp )T ∈ R n+p , µ ∈ [0, 1]. When (x1 , . . . , xn , y1 , . . . , yp )T ∈ ∂Ω ∩ Ker L
p and µ ∈ [0, 1], the vector (x1 , . . . , xn , y1 , . . . , yp )T is a constant vector in R n+p with ni=1 |xi | + j =1 |yj | = M ∗ . Thus Φ(x1 , x2 , . . . , xn , y1 , y2 , . . . , yp , µ) p n p˜ j i fj (yj ) + I˜i = −µai xi + (1 − µ) −ai xi + c˜ii s1,i (xi ) + i=1 j =1 p n + q˜ij gi (xi ) + J˜j −µbj yj + (1 − µ) −bj yj + h˜ jj s2,j (yj ) + j =1
i=1
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Z. Liu et al. / Physics Letters A 328 (2004) 127–143
p n = p˜j i fj (yj ) + I˜i ai xi − (1 − µ) c˜ii s1,i (xi ) + i=1 j =1 p n + q˜ij gi (xi ) + J˜j bj yj − (1 − µ) h˜ jj s2,j (yj ) + j =1 i=1 n p ∗ ˜ ˜ min(a, b)M − |Ii | + |Jj | > 0. i=1
j =1
Therefore, Φ(x1 , x2 , . . . , xn , y1 , y2 , . . . , yp , µ) = (0, 0, . . . , 0, 0, 0, . . ., 0)T , for (x1 , x2 , . . . , xn , y1 , y2 , . . . , yp )T ∈ ∂Ω ∩ Ker L. As a result, we have deg QN(x1 , . . . , xn , y1 , . . . , yp )T , Ω ∩ Ker L, (0, . . . , 0, 0, . . . , 0)T = deg (−a1 x1 , . . . , −an xn , −b1 y1 , . . . , −bp yp )T , Ω ∩ Ker L, (0, . . . , 0)T = 0. By now, we know that Ω satisfies all conditions of Lemma 2.1. Therefore, system (1.1a), (1.1b) has at least one ω-periodic solution. This completes the proof. 2
3. Global exponential stability of periodic solution In this section, we shall discuss the global exponential stability of BAM neural networks with self-connection terms. Suppose that T z∗ = x1∗ (t), . . . , xn∗ (t), y1∗ (t), . . . , yp∗ (t) is a periodic solution of (1.1a), (1.1b) as described in Theorem 1. T z = x1 (t), . . . , xn (t), y1 (t), . . . , yp (t) is any solution of system (1.1a), (1.1b). Set ui (t) = xi (t) − xi∗ (t), vj (t) = yj (t) − yj∗ (t), Fj vj (t) = fj vj (t) + yj∗ (t) − fj yj∗ (t) , Gi ui (t) = gi ui (t) + xi∗ (t) − gi xi∗ (t) , S2,j vj (t) = s2,j vj (t) + yj∗ (t) − s2,j yj∗ (t) , S1,i ui (t) = s1,i ui (t) + xi∗ (t) − s1,i xi∗ (t) , where i = 1, 2, . . . , n; j = 1, 2, . . . , p. It is easy to see that system (1.1a), (1.1b) can be reduced to the following system pj i (t)Fj vj t − τj i (t) , ui (t) = −ai ui (t) + cii (t)S1,i ui t − dii (t) + p
(3.1a)
j =1 n vj (t) = −bj vj (t) + hjj (t)S2,j vj t − mjj (t) + qij (t)Gi ui t − σij (t) . i=1
(3.1b)
Z. Liu et al. / Physics Letters A 328 (2004) 127–143
137
By (H1 ), we have Gi (u) − Gi (v) νi , u−v Fj (u) − Fj (v) µj , 0 u−v
S1,i (u) − S1,i (v) κ1,i , Gi (0) = S1,i (0) = 0, u−v S2,j (u) − S2,j (v) κ2,j , Fj (0) = S2,j (0) = 0, 0 u−v
0
0
(3.2)
for i = 1, 2, . . . , n; j = 1, 2, . . . , p. Theorem 2. Assume that all conditions of Theorem 1 hold. If there exist positive constants λi and λn+j (i = 1, 2, . . . , n; j = 1, 2, . . . , p) which satisfy the following condition: (H2 )
p qij+ νi cii+ κ1,i + λi −ai + λn+j < 0, 1 − dii+ 1 − σij+ j =1
λn+j −bj +
h+ jj κ2,j 1 − m+ jj
+
n i=1
λi
pj+i µj 1 − τj+ i
< 0.
Then there exists a unique ω-periodic solution (x ∗T (t), y ∗T (t))T of the system (1.1a), (1.1b) which is globally exponentially stable. That is, all other solutions (x T (t), y T (t))T of (1.1a), (1.1b) satisfy p n xi (t) − x ∗ (t) + yj (t) − y ∗ (t) M φ T , ψ T T − x ∗T , y ∗T T e−αt , i j
(3.3)
j =1
i=1
for t 0. Proof. The uniqueness of the ω-periodic solution will follow from (3.3). Therefore, it suffices to prove the ω-periodic solution is globally exponentially stable. That is, every solutions (x T (t), y T (t))T of (1.1a), (1.1b) satisfy (3.3). In view of (H2 ), we can choose a suitable constant α ∈ (0, min1in; 1j p {ai , bj }) such that λi α − ai +
cii+ κ1,i
e 1 − dii+
αdii+
+
p j =1
λn+j
qij+ νi
e 1 − σij+
ασij+
< 0,
n pj+i µj ατ + h+ jj κ2,j αm+ jj e λ e ji < 0. λn+j α − bj + + i 1 − m+ 1 − τj+ jj i i=1
(3.4)
By (3.1), (3.2), we have p + D + ui (t) −ai ui (t) + cii+ κ1,i ui t − dii (t) + pj i µj vj t − τj i (t)
(3.5a)
j =1
and n D + vj (t) −bj vj (t) + h+ t − m κ (t) + qij+ νi ui t − σij (t) , v jj jj 2,j j i=1
where D + denotes the upper right Dini derivative. Consider a Lyapunov functional V (t) defined by V (t) = V1 (t) + V2 (t),
(3.5b)
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Z. Liu et al. / Physics Letters A 328 (2004) 127–143
V1 (t) =
n
λi e
αt
i=1
+
n c+ κ1,i ui (t) + λi ii + 1 − dii i=1
p n
λi
i=1 j =1
V2 (t) =
p
λn+j e
αt
j =1
+
p n
t
pj+i µj 1 − τj+ i
t
ui (r)eα(r+dii+) dr
t −dii (t )
+ vj (r)eα(r+τji ) dr,
t −τji (t )
p h+ jj κ2,j λn+j vj (t) + 1 − m+ jj j =1
λn+j
j =1 i=1
t
qij+ νi 1 − σij+
t
+ vj (t)eα(r+mjj ) dr
t −mjj (t )
+ ui (r)eα(r+σij ) dr.
(3.6)
t −σij (t )
Calculating the derivative of the Vk (t) (k = 1, 2) along solutions of (3.1), we have n n
αt c+ κ1,i + αt + D V1 (t) (3.1) = λi αe ui (t) + e D ui (t) + λi ii + ui (t)eα(t +dii ) 1 − dii i=1 i=1 +
−
n
λi
i=1
+
pj+i µj α(t +τ +) ji λi + vj (t) e 1 − τ j i j =1
p n i=1
−
cii+ κ1,i + ui t − dii (t) 1 − dii (t) eα(t −dii (t )+dii ) 1 − dii+
p n
λi
i=1 j =1
pj+i µj α(t −τji (t )+τ +) ji . + vj t − τj i (t) 1 − τj i (t) e 1 − τj i +
+
α(t −τji (t )+τji ) + By (3.5a) and noting that 1 − τj i (t) 1 − τj+ eαt , eα(t −dii (t )+dii ) eαt , i , 1 − dii (t) 1 − dii and e we obtain n n
D + V1 (t)(3.1) eαt λi α ui (t) − ai ui (t) + eαt λi cii+ κ1,i ui t − dii (t) i=1
+ eαt
i=1
i=1
−e
αt
n c+ κ1,i + λi pj+i µj vj t − τj i (t) + eαt λi ii + eαdii ui (t) 1 − dii j =1 i=1
p n
n
λi cii+ κ1,i ui t
i=1
− eαt
p n
p n pj+i µj ατ + αt ji v (t) − dii (t) + e λi j + e 1 − τj i i=1 j =1
λi pj+i µj vj t − τj i (t)
i=1 j =1
=e
αt
n i=1
λi α − ai +
cii+ κ1,i
1 − dii+
e
αdii+
p n pj+i µj ατ + ui (t) + eαt ji v (t). λi j + e 1 − τj i i=1 j =1
(3.7)
Z. Liu et al. / Physics Letters A 328 (2004) 127–143
139
Similar to (3.7), we get p p n qij+ νi ασ + h+ jj κ2,j αm+ αt vj (t) + eαt jj ij u (t). D V2 (t) (3.1) e λn+j α − bj + e λ n+j i + + e 1 − mjj 1 − σij j =1 j =1 i=1 (3.8) It follows from (3.4), (3.6)–(3.8) that p n qij+ νi ασ + cii+ κ1,i αd + + αt ui (t) ij ii λn+j λi α − ai + + D V (t) (3.1) e + e + e 1 − dii 1 − σij i=1 j =1 p n pj+i µj ατ + h+ jj κ2,j αm+ αt jj ji v (t) < 0. +e (3.9) λi λn+j α − bj + + j + e + e 1 − mjj 1 − τj i j =1 i=1 +
Namely, V (t) V (0), for t 0. Again by (3.6), we obtain n p αt − ui (t) + vj (t) , V (t) e λ
(3.10)
j =1
i=1
where λ− = min1ln+p {λl }. On the other hand, from (3.6), we have n c+ κ1,i V (0) = λi ui (0) + λi ii + 1 − dii i=1 i=1
0
n
+
ui (r)eα(r+dii+) dr
−dii (0)
0 pj+i µj + vj (r)eα(r+τji ) dr λi + 1 − τj i i=1 j =1 −τji (0)
p n
p h+ jj κ2,j λn+j vj (0) + λn+j + 1 − m+ jj j =1 j =1 p
+
p n
λn+j
j =1 i=1
+
λ
0
qij+ νi 1 − σij+
0
+ vj (r)eα(r+mjj ) dr
−mjj (0)
+ ui (r)eα(r+σij ) dr
−σij (0)
0 p n n + c κ 1,i ii + ui (0) + vj (0) + λ ui (r)eα(r+d) dr + 1 − dii i=1 j =1 i=1 −d
+
+λ
p n pj+i µj i=1 j =1
1 − τj+ i
0 −τ
0 p h+ jj κ2,j vj (r)eα(r+τ ) dr + λ+ vj (r)eα(r+m) dr + 1 − mjj j =1 −m
0 n qij+ νi + ui (r)eα(r+σ ) dr +λ + 1 − σij j =1 i=1 −σ p
+
λ
p n n cii+ κ1,i ui (0) + vj (0) + λ+ deαd 1 − dii+ i=1 j =1 i=1
max ui (r)
−dr0
140
Z. Liu et al. / Physics Letters A 328 (2004) 127–143 p n pj+i µj
+ λ+ τ eατ
i=1
+ λ+ σ eασ
p h+ jj κ2,j vj (r) + λ+ meαm max + −τ r0 1 − τj i 1 − m+ jj j =1 j =1
p n qij+ νi
max 1 − σij+ −σ r0 j =1 i=1
max vj (r)
−mr0
ui (r)
T T λ+ M1 φ T , ψ T − x ∗T , y ∗T ,
(3.11)
where σ=
max
1in; 1j p
σij+ ,
m = max m+ jj ,
1 − dii+
M1(2)
τe
ατ
p n pj+i µj i=1 j =1
1 − τj+ i
+ meαm + σe
ασ
τj+i ,
(1)
M1 = M1 + M2 ,
p h+ jj κ2,j j =1
max
1j p;1in (1)
max {λl },
1ln+p
n cii+ κ1,i i=1
τ=
1in
λ+ =
1j p
M1(1) 1 + deαd
d = max dii+ ,
1 − m+ jj
,
p n qij+ νi j =1 i=1
1 − σij+
.
Set M=
M1 λ+ . λ−
It follows from (3.10) and (3.11) that p n ui (t) + vj (t) M φ T , ψ T T − x ∗T , y ∗T T e−αt , i=1
(3.12)
j =1
for t 0. This completes the proof. 2 Remark. When self-connection terms cii = 0, hjj = 0 (i = 1, 2, . . . , n; j = 1, 2, . . . , p), system (1.1) reduces to the following BAM neural networks model: xi (t) = −ai xi (t) +
p
pj i (t)fj yj t − τj i (t) + Ii (t),
(3.13a)
j =1
yj (t) = −bj yj (t) +
n
qij (t)gi xi t − σij (t) + Jj (t),
(3.13b)
i=1
Corollary. Assume that all conditions of Theorem 1 hold. If there exist positive constants λi and λn+j (i = 1, 2, . . . , n; j = 1, 2, . . . , p) satisfy the following conditions: (H3 )
− λi ai +
p j =1
λn+j
qij+ νi 1 − σij+
< 0,
−λn+j bj +
n i=1
λi
pj+i µj 1 − τj+ i
< 0.
Then ω-periodic solution of the system (3.13a), (3.13b) is globally exponentially stable.
Z. Liu et al. / Physics Letters A 328 (2004) 127–143
141
4. An example In this section, we give an example to illustrate that our results are feasible, which shows our new results are more effective than ones in [11] for some neural networks. Consider the following simple BAM networks: xi (t) = −ai xi (t) +
2
pj i (t)fj yj t − τj i (t) + Ii (t),
j =1
yj (t) = −bj yj (t) +
2
qij (t)fi xi t − σij (t) + Jj (t),
(4.1)
i=1
i = 1, 2; j = 1, 2, where Ii (t) = sin 40πt, τj i =
1 , 20
Jj (t) = cos 40πt,
σij =
1 ; 20
ω=
τj i+ = 0,
1 , 20
σij + = 0
(i, j = 1, 2).
Taking fj (x) = gi (x) = 12 (|x + 1| − |x − 1|), we have νi = µj = 1 (i, j = 1, 2). Again taking (a1 , a2 )T = (1, 1)T , Let
Then
q11 (t) q12 (t) q21 (t) q22 (t) + + q12 q11
+ + q21 q22
Moreover,
=
(b1 , b2 )T = (2, 3)T .
=
5 5 11 11 10 10 23 23
1 0 −1 0 1 −1 A= −0.5 −0.5 2 −0.5 −0.5 0
5 5 11 cos 40πt 11 sin 40πt 10 10 23 sin 40πt 23 cos 40πt
,
+ + p12 p11 + + p21 p22
−1 −1 , 0 3
,
=
3.5 2.5 A−1 = 1.5 1
20 21 20 21
2.5 3.5 1.5 1
20 21 20 21
3 3 2 1
p11 (t) p12 (t) p21 (t) p22 (t)
=
20 21 sin 40πt 20 21 cos 40πt
lj i = kij = 1.
,
2 2 > 0. 1 1
Hence, A is a nonsingular M-matrix. Set λl = 1 (l = 1, 2, 3, 4), we get q˜11 q˜12 p˜11 p˜ 12 0 0 = = , 0 0 p˜21 p˜ 22 q˜21 q˜22 ai − νi
2
|q˜ij | = 1 > 0,
b j − µj
j =1
−λ1 a1 +
2 j =1
2
|p˜ j i | = 1 > 0,
i=1
λ2+j
+ q1j ν1 + 1 − σ1j
=−
1 < 0, 11
−λ2 a2 +
2 j =1
λ2+j
+ q2j ν2 + 1 − σ2j
=−
3 < 0, 23
20 21 cos 40πt 20 21 sin 40πt
.
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Z. Liu et al. / Physics Letters A 328 (2004) 127–143
−λ3 b1 +
2
λi
i=1
+ p1i µ1
1 − τ1i +
fj (x) − fj (y) 1, 0 x −y
=−
2 < 0, 21
fj (0) = 0,
−λ4 b2 +
2 i=1
λi
+ p2i µ2
1 − τ2i +
gi (x) − gi (y) 1, 0 x−y
=−
23 < 0, 21
gi (0) = 0,
for i, j = 1, 2. Thus, it follows from corollary of Theorem 2 that systems (4.1) has a unique and all other solutions of systems (4.1) converge exponentially to the periodic solution. But + 2 2 q1j 188 + λ1 −2a1 + µ1 pj 1 + µ1 λ2+j + = 231 > 0, 1 − σ1j j =1 j =1 + 2 2 q2j 374 λ2 −2a2 + µ2 pj+2 + µ2 λ2+j + = 483 > 0. 1 − σ2j j =1
1 20 -periodic
solution
j =1
So, the assumption (H3) in Ref. [11] is not available. It is easy to see that the results in this Letter and the ones in Ref. [11] are not interrelated, i.e., there always exists a networks which satisfies one and does not satisfy another.
5. Conclusion In this Letter, By using the continuation theorem of Mawhin’s coincidence degree theory, Lyapunov functional method and some analytical techniques, some sufficient conditions are obtained ensuring existence and global exponential stability of periodic solution of the self-connection BAM neural networks with periodic coefficients and delays. We would like to point out that these results are more effective than the ones in [11] for some neural networks, which has an important leading significance in designing globally exponentially stable and periodic oscillatory BAM neural networks with self-connection.
Acknowledgements The authors would like to thank the reviewers and the editor for their valuable suggestions and comments. This work was supported by the Natural Science Foundation of China under Grant 10371034 and by the Foundation for University Excellent Teacher by Chinese Ministry of Education (No. [2002] 78), the Scientific Research Foundation of Hunan Provincial Education Department (01C009, 03C009).
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