Electronic Journal of Differential Equations, Vol. 2010(2010), No. 50, pp. 1–11.
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
ftp ejde.math.txstate.edu
ANTI-PERIODIC SOLUTIONS FOR RECURRENT NEURAL
NETWORKS WITHOUT ASSUMING GLOBAL LIPSCHITZ
CONDITIONS
HONG ZHANG, YUANHENG WU
Abstract. In this paper we study recurrent neural networks with time-varying delays and continuously distributed delays. Without assuming global Lipschitz conditions on the activation functions, we establish the existence and
local exponential stability of anti-periodic solutions.
1. Introduction
We consider the following model for recurrent neural networks(RNNs) with timevarying delays and continuously distributed delays
x0i (t) = −ci (t)αi (xi (t)) +
n
X
aij (t)g̃j (xj (t − τij (t)))
j=1
+
n
X
j=1
Z
(1.1)
∞
Kij (u)gj (xj (t − u))du + Ii (t),
bij (t)
i = 1, 2, . . . , n,
0
in which n corresponds to the number of units in a neural network, xi (t) corresponds
to the state vector of the ith unit at the time t, ci (t) > 0 represents the rate with
which the ith unit will reset its potential to the resting state in isolation when
disconnected from the network and external inputs at the time t. aij (t) and bij (t)
are the connection weights at the time t, τij (t) ≥ 0 corresponds to the transmission
delay of the ith unit along the axon of the jth unit at the time t, and Ii (t) denote the
external inputs at time t. g̃j and gj are activation functions of signal transmission.
As we know, RNNs is very general and includes Hopfield neural networks, cellular neural networks and BAM neural networks. The RNNs have been successfully applied to signal and image processing, pattern recognition and optimization.
Hence, they have been the object of intensive analysis by numerous authors in recent years. In particular, there have been extensive results on the problem of the
existence and stability of periodic and almost periodic solutions for RNNs in the
2000 Mathematics Subject Classification. 34C25, 34D40.
Key words and phrases. Recurrent neural networks; anti-periodic; exponential stability; delay.
c
2010
Texas State University - San Marcos.
Submitted December 11, 2009. Published April 9, 2010.
Supported by grants 06JJ2063, 07JJ6001 from the Scientific Research Fund of Hunan
Provincial Natural Science Foundation, and 08C616, 09B072 from the Scientific Research Fund
of Hunan Provincial Education Department of China.
1
2
H. ZHANG, Y. WU
EJDE-2010/50
literature. We refer the reader to [5, 6, 3, 10] and the references cited therein.
Recently, [11, 12, 14] obtained some sufficient conditions for the existence and stability of the anti-periodic solutions of RNNs. Moreover, in the above-mentioned
literature [11, 12, 14], we observe that the assumption
(T0) For each j ∈ {1, 2, . . . , n}, the activation function g̃j , gj : R → R is global
Lipschitz with Lipschitz constants L̃j and Lj ; i.e.,
|g̃j (uj ) − g̃j (vj )| ≤ L̃j |uj − vj |, |gj (uj ) − gj (vj )| ≤ Lj |uj − vj |,
(1.2)
for all uj , vj ∈ R.
has been considered as fundamental for the considered existence and stability of
anti-periodic solutions of RNNs. However, to the best of our knowledge, few
authors have considered the problems of anti-periodic solutions of RNNs without the assumptions (T0). Since the existence of anti-periodic solutions play
a key role in characterizing the behavior of nonlinear differential equations (See
[1, 2, 7, 8, 13, 15]). It is worth while to continue to investigate the existence and
stability of anti-periodic solutions of RNNs.
The main purpose of this paper is to give the conditions for the existence and
exponential stability of the anti-periodic solutions for system (1.1). We derive some
new sufficient conditions ensuring the existence and local exponential stability of the
anti-periodic solution for system (1.1), which are new and complement to previously
known results. In particular, we do not need the assumption (T0) . Moreover, an
example is also provided to illustrate the effectiveness of our results.
Let u(t) : R → R be continuous in t. u(t) is said to be T -anti-periodic on R if,
u(t + T ) = −u(t)
for all t ∈ R.
Throughout this article, for i, j = 1, 2, . . . , n, it will be assumed that ci , Ii , aij , bij :
R → R and τij : R → [0, +∞) are continuous 2T −periodic functions, and
ci (t + T )αi (u) = −ci (t)αi (−u),
aij (t + T )g̃j (u) = −aij (t)g̃j (−u),
∀t, u ∈ R,
bij (t + T ) = −bij (t),
gj (u) = gj (−u))
bij (t + T ) = bij (t), gj (u) = −gj (−u),
τij (t + T ) = τij (t),
or
∀t, u ∈ R,
Ii (t + T ) = −Ii (t),
∀u ∈ R .
(1.3)
(1.4)
(1.5)
Then, we can choose constants I and τ such that
I = max sup |Ii (t)|,
1≤i≤n t∈R
τ = max { max τij (t)}.
1≤i,j≤n t∈[0,T ]
(1.6)
We also assume that the following conditions:
(H0) For each i ∈ {1, 2, . . . , n}, αi : R → R are continuous function, and there
exist constants di such that αi (0) = 0, di |u−v| ≤ sign(u−v)(αi (u)−αi (v)),
for all u, v ∈ R.
(H1) For each j ∈ {1, 2, . . . , n}, there exist f˜j , h̃j , fj , hj ∈ C(R, R) and constants
˜
Lfj , Lh̃j , Lfj , Lhj ∈ [0, +∞) such that the following conditions are satisfied:
(1) f˜j (0) = 0, h̃j (0) = 0, g̃j (u) = f˜j (u)h̃j (u), for all u ∈ R;
˜
(2) |f˜j (u) − f˜j (v)| ≤ Lf |u − v|, |h̃j (u) − h̃j (v)| ≤ Lh̃ |u − v|, for all u, v ∈ R.
j
j
(3) fj (0) = 0, hj (0) = 0, gj (u) = fj (u)hj (u), for all u ∈ R;
(4) |fj (u) − fj (v)| ≤ Lfj |u − v|, |hj (u) − hj (v)| ≤ Lhj |u − v|, for all u, v ∈ R.
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ANTI-PERIODIC SOLUTIONS
3
(H2) For i, j ∈ {1, 2, . . . , n}, the delay kernels Kij : [0, ∞) → R are continuous,
integrable.
(H3) There exist constants η > 0, λ > 0 and ξi > 0, i = 1, 2, . . . , n, such that for
all t > 0, there holds 0 < ξi ηI ≤ 1 and
(λ − ci (t))di ξi +
n
X
˜
3|aij (t)|eλτ Lfj Lh̃j ξj
j=1
+
n
X
∞
Z
|Kij (u)|eλu duLfj Lhj ξj < −η.
3|bij (t)|
0
j=1
We will use the following notation: x = {xj } = (x1 , x2 , . . . , xn )T ∈ Rn to denote a
column vector, in which the symbol (T ) denotes the transpose of a vector. We let
|x| denote the absolute-value vector given by |x| = (|x1 |, |x2 |, . . . , |xn |)T , and define
kxk = max1≤i≤n |xi |.
The initial conditions associated with system (1.1) are of the form
s ∈ (−∞, 0], i = 1, 2, . . . , n,
xi (s) = ϕi (s),
(1.7)
where ϕi (·) denotes real-valued bounded continuous function defined on (−∞, 0].
The remaining part of this paper is organized as follows. In Section 2, we shall
derive new sufficient conditions for checking the existence of bounded solutions of
(1.1). In Section 3, we present some new sufficient conditions for the existence and
local exponential stability of the anti-periodic solution of (1.1). In Section 4, we
shall give an example and some remarks to illustrate our results obtained in the
previous sections.
2. Preliminary Results
The following lemmas will be used to prove our main results in Section 3.
Lemma 2.1. Let (H0)–(H3) hold. Suppose that x
e(t) = (e
x1 (t), x
e2 (t), . . . , x
en (t))T is
a solution of (1.1) with initial conditions
x
ei (s) = ϕ
ei (s),
|ϕ
ei (s)| < ξi
I
≤ 1,
η
s ∈ (−∞, 0], i = 1, 2, . . . , n.
(2.1)
Then
I
|e
xi (t)| < ξi ,
η
for all t ≥ 0, i = 1, 2, . . . , n.
(2.2)
Proof. Assume, by way of contradiction, that (2.2) does not hold. Then, there must
exist i ∈ {1, 2, . . . , n} and ρ > 0 such that
I
|e
xi (ρ)| = ξi ,
η
|e
xj (t)| < ξj
I
η
for all t ∈ (−∞, ρ), j = 1, 2, . . . , n.
(2.3)
Calculating the upper left derivative of |e
xi (t)|, together with (H0), (H1), (H2) and
(H3), (2.3) implies
0 ≤ D+ (|e
xi (ρ)|)
= −ci (ρ)αi (e
xi (ρ)) sgn(e
xi (ρ)) +
n
hX
j=1
aij (ρ)g̃j (e
xj (ρ − τij (ρ)))
4
H. ZHANG, Y. WU
+
n
X
∞
Z
bij (ρ)
EJDE-2010/50
i
Kij (u)gj (e
xj (ρ − u))du + Ii (ρ) sgn(e
xi (ρ))
0
j=1
n
X
≤ −ci (ρ)|αi (e
xi (ρ))| + aij (ρ)g̃j (e
xj (ρ − τij (ρ)))
j=1
+
n
X
∞
Z
bij (ρ)
0
j=1
Kij (u)gj (e
xj (ρ − u))du + Ii (ρ)
n
I X
≤ −ci (ρ)di ξi + aij (ρ)f˜j (e
xj (ρ − τij (ρ)))h̃j (e
xj (ρ − τij (ρ)))
η
j=1
Z ∞
n
X
+
Kij (u)fj (e
xj (ρ − u))hj (e
xj (ρ − u))du + Ii (ρ)
bij (ρ)
0
j=1
n
I X
I
˜ I
|aij (ρ)|Lfj ξj Lh̃j ξj
+
η j=1
η
η
Z
n
∞
X
I
I
+
|bij (ρ)|
|Kij (u)|duLfj ξj Lhj ξj + |Ii (ρ)|
η
η
0
j=1
≤ −ci (ρ)di ξi
n
I X
I
˜
+
|aij (ρ)|Lfj Lh̃j ξj
η j=1
η
Z
n
∞
X
I
+
|bij (ρ)|
|Kij (u)|duLfj Lhj ξj + |Ii (ρ)|
η
0
j=1
Z
n
n
X
X
˜
≤ [−ci (ρ)di ξi +
|aij (ρ)|Lfj Lh̃j ξj +
|bij (ρ)|
≤ −ci (ρ)di ξi
j=1
∞
|Kij (s)|dsLfj Lhj ξj ]
0
j=1
I
η
+ |Ii (ρ)|
n
h
X
˜
≤ − ci (ρ)di ξi +
|aij (ρ)|eλτ |Lfj Lh̃j ξj
j=1
+
n
X
j=1
< −η ×
Z
|bij (ρ)|
∞
|Kij (s)|eλs dsLfj Lhj ξj
0
iI
η
+ |Ii (ρ)|
I
+ |Ii (ρ)| ≤ 0,
η
which is a contradiction and implies that (2.2) holds. The proof is complete.
Remark 2.2. In view of the boundedness of this solution, from [9, Theorems
2.3-2.4], it follows that x
e(t) can be defined on (−∞, ∞).
Lemma 2.3. Suppose that (H0), (H1), (H2), (H3) are satisfied. Let x∗ (t) =
(x∗1 (t), x∗2 (t), . . . , x∗n (t))T be the solution of (1.1) with initial value ϕ∗ = (ϕ∗1 (t),
ϕ∗2 (t), . . . , ϕ∗n (t))T , where
|ϕ∗i (s)| < ξi
I
≤ 1,
η
s ∈ (−∞, 0], i = 1, 2, . . . , n.
(2.4)
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ANTI-PERIODIC SOLUTIONS
5
Then there exist constants λ > 0 and Mϕ > 1 such that for every solution x(t) =
(x1 (t), x2 (t), . . . , xn (t))T of (1.1) with initial value ϕ = (ϕ1 (t), ϕ2 (t), . . . , ϕn (t))T ∈
G1 (x∗ ),
|xi (t) − x∗i (t)| ≤ Mϕ kϕ − ϕ∗ k1 e−λt ,
∀t > 0, i = 1, 2, . . . , n,
where kϕ − ϕ∗ k1 = sup−∞≤s≤0 max1≤i≤n |ϕi (s) − ϕ∗i (s)|, and
G1 (x∗ ) = ϕ ∈ C((−∞, 0]; Rn ) :
sup
−∞≤s≤0
|ϕi (t) − ϕ∗i (t)| <
ξi
,
max1≤j≤n {ξj }
i = 1, 2, . . . , n .
Proof. In view (2.4) and Lemma 2.1,
|x∗i (t)| < ξi
I
≤ 1,
η
for all t ∈ R, i = 1, 2, . . . , n.
(2.5)
Let y(t) = {yj (t)} = {xj (t) − x∗j (t)} = x(t) − x∗ (t). Then
yi0 (t) = −ci (t)[αi (xi (t)) − αi (x∗i (t))]
+
n
X
aij (t)(g̃j (yj (t − τij (t)) + x∗j (t − τij (t))) − g̃j (x∗j (t − τij (t))))
j=1
+
n
X
∞
Z
Kij (u)(gj (yj (t − u) + x∗j (t − u)) − gj (x∗j (t − u)))du,
bij (t)
0
j=1
(2.6)
where i = 1, 2, . . . , n. We consider the Lyapunov functional
Vi (t) = |yi (t)|eλt ,
i = 1, 2, . . . , n.
(2.7)
Calculating the left right derivative of Vi (t) along the solution y(t) = {yj (t)} of
system (2.6) with the initial value ϕ̄ = ϕ − ϕ∗ , from (2.6), we have
D+ (Vi (t))
≤ −ci (t)di |yi (t)|eλt +
n
X
|aij (t)(g̃j (yj (t − τij (t)) + x∗j (t − τij (t)))
j=1
−
g̃j (x∗j (t
λt
− τij (t))))|e
+
n
X
Z
∞
Kij (u)(gj (yj (t − u) + x∗j (t − u))
|bij (t)
j=1
0
− gj (x∗j (t − u)))du|eλt + λ|yi (t)|eλt
≤ (λ − ci (t)di )|yi (t)|eλt +
n
X
|aij (t)|(|f˜j (yj (t − τij (t))
j=1
+ x∗j (t − τij (t)))h̃j (yj (t − τij (t)) + x∗j (t − τij (t)))
− f˜j (x∗ (t − τij (t)))h̃j (yj (t − τij (t)) + x∗ (t − τij (t)))|
j
j
+ |f˜j (x∗j (t − τij (t)))h̃j (yj (t − τij (t)) + x∗j (t − τij (t)))
− f˜j (x∗j (t − τij (t))))h̃j (x∗j (t − τij (t)))|)eλt
Z ∞
n
X
+
|bij (t)|
|Kij (u)|(|fj (yj (t − u) + x∗j (t − u))hj (yj (t − u) + x∗j (t − u))
j=1
0
6
H. ZHANG, Y. WU
EJDE-2010/50
− fj (yj (t − u) + x∗j (t − u))hj (x∗j (t − u))| + |fj (yj (t − u)
+ x∗j (t − u))hj (x∗j (t − u)) − fj (x∗j (t − u))hj (x∗j (t − u))|)dueλt
≤ (λ − ci (t)di )|yi (t)|eλt +
n
X
˜
|aij (t)|Lfj Lh̃j |yj (t − τij (t))|(|yj (t − τij (t))
j=1
+
+
x∗j (t
n
X
|x∗j (t
− τij (t))| +
− τij (t))|)eλt
Z ∞
|bij (t)|
Kij (u)Lfj Lhj |yj (t − u)|(|yj (t − u) + x∗j (t − u)|
0
j=1
+
|x∗j (t
− u)|)dueλt ,
(2.8)
∗
where i = 1, 2, . . . , n. In view of the definition of ϕ ∈ G1 (Z ),
ξi
, for all t ∈ (−∞, 0], j = 1, 2, . . . , n.
Vi (t) = |yi (t)|eλt <
max1≤j≤n {ξj }
We claim that
Vi (t) = |yi (t)|eλt <
ξi
max1≤j≤n {ξj }
,
for all t > 0,
i = 1, 2, . . . , n.
(2.9)
Contrarily, there must exist i ∈ {1, 2, . . . , n} and ti > 0 such that
ξj
ξi
, Vj (t) <
,
Vi (ti ) =
max1≤j≤n {ξj }
max1≤j≤n {ξj }
for all t ∈ (−∞, ti ), j = 1, 2, . . . , n, which, together with ϕ ∈ G1 (Z ∗ ), implies
ξj
ξi
= 0, Vj (t) −
< 0,
(2.10)
Vi (ti ) −
max1≤j≤n {ξj }
max1≤j≤n {ξj }
for all t ∈ (−∞, ti ), j = 1, 2, . . . , n, and
|yj (t)| ≤ 1,
∀t ∈ (−∞, ti ), j = 1, 2, . . . , n.
(2.11)
Together with (2.5), (2.8), (2.10) and (2.11), we obtain
0 ≤ D+ (Vi (ti ) − mξi )
= D+ (Vi (ti ))
λti
≤ (λ − ci (ti )di )|yi (ti )|e
+
n
X
˜
|aij (ti )|Lfj Lh̃j |yj (ti − τij (ti ))|(|yj (ti − τij (ti ))
j=1
+
+
x∗j (ti
n
X
|x∗j (ti
− τij (ti ))| +
− τij (ti ))|)eλti
Z ∞
|bij (ti )|
|Kij (u)|Lfj Lhj |yj (ti − u)|
0
j=1
× |yj (ti − u) + x∗j (ti − u)| + |x∗j (ti − u)| dueλti
λti
≤ (λ − ci (ti )di )|yi (ti )|e
+
n
X
˜
3|aij (ti )|Lfj Lh̃j |yj (ti − τij (ti ))|eλti
j=1
+
n
X
j=1
Z
3|bij (ti )|
∞
|Kij (u)|Lfj Lhj |yj (ti − u)|dueλti
0
= (λ − ci (ti )di )|yi (ti )|eλti
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+
n
X
ANTI-PERIODIC SOLUTIONS
7
˜
3|aij (ti )|Lfj Lh̃j ||yj (ti − τij (ti ))|eλ(ti −τij (ti )) eλτij (ti )
j=1
+
n
X
∞
Z
|Kij (u)|Lfj Lhj |yj (ti − u)|eλ(ti −u) eλu du
3|bij (ti )|
0
j=1
n
X
h
≤ (λ − ci (ti )di )ξi +
˜
3|aij (ti )|eλτ Lfj Lh̃j ξj
j=1
+
n
X
∞
Z
|Kij (u)|eλu duLfj Lhj ξj
3|bij (ti )|
0
j=1
i
1
.
max1≤j≤n {ξj }
(2.12)
Thus,
0 ≤ (λ − ci (ti )di )ξi +
n
X
˜
3|aij (ti )|eλτ Lfj Lh̃j ξj
j=1
+
n
X
Z
(2.13)
∞
λu
|Kij (u)|e
3|bij (ti )|
duLfj Lhj ξj ,
0
j=1
which contradicts (H3). Hence, (2.9) holds. Letting
kϕ − ϕ∗ k1 =
sup
max |ϕj (s) − ϕ∗j (s)| > 0,
−∞≤s≤0 1≤j≤n
i = 1, 2, . . . , n,
and Mϕ > 1 such that
ξi
≤ Mϕ kϕ − ϕ∗ k1 ,
max1≤j≤n {ξj }
i = 1, 2, . . . , n.
(2.14)
In view of (2.9) and (2.13),
|xi (t) − x∗i (t)| = |yi (t)| ≤ max {mξi }e−λt ≤ Mϕ kϕ − ϕ∗ k1 e−λt ,
1≤i≤n
where i = 1, 2, . . . , n, t > 0. This completes the proof.
Remark 2.4. If x∗ (t) = (x∗1 (t), x∗2 (t), . . . , x∗n (t))T be the T -anti-periodic solution
of system (1.1), it follows from Lemma 2.3 that x∗ (t) is locally exponentially stable.
3. Main Results
The following is our main result.
Theorem 3.1. Suppose that (H0)-(H3) are satisfied. Then (1.1) has at least one
T -anti-periodic solution x∗ (t). Moreover, x∗ (t) is locally exponentially stable.
Proof. Let v(t) = (v1 (t), v2 (t), . . . , vn (t))T be a solution of (1.1) with initial conditions
I
vi (s) = ϕvi (s), |ϕvi (s)| < ξi , s ∈ (−∞, 0], i = 1, 2, . . . , n.
(3.1)
η
By Lemma 2.1, the solution v(t) is bounded and
I
|vi (t)| < ξi ,
η
for all t ∈ R, i = 1, 2, . . . , n.
(3.2)
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H. ZHANG, Y. WU
EJDE-2010/50
From (1.1)–(1.5),
((−1)k+1 vi (t + (k + 1)T ))0
= (−1)k+1 vi0 (t + (k + 1)T )
= (−1)k+1 {−ci (t + (k + 1)T )αi (vi (t + (k + 1)T ))
+
+
n
X
j=1
n
X
aij (t + (k + 1)T )g̃j (vj (t + (k + 1)T − τij (t + (k + 1)T )))
Z
∞
Kij (u)gj (vj (t + (k + 1)T − u))du
bij (t + (k + 1)T )
0
j=1
+ Ii (t + (k + 1)T )}
= (−1)k+1 {−ci (t + (k + 1)T )αi (vi (t + (k + 1)T ))
+
n
X
aij (t + (k + 1)T )g̃j (vj (t + (k + 1)T − τij (t)))
(3.3)
j=1
+
n
X
Z
∞
Kij (u)gj (vj (t + (k + 1)T − u))du
bij (t + (k + 1)T )
0
j=1
+ Ii (t + (k + 1)T )}
= −ci (t)αi ((−1)k+1 vi (t + (n + 1)T ))
+
+
n
X
j=1
n
X
j=1
aij (t)g̃j ((−1)k+1 vj (t + (k + 1)T − τij (t)))
Z
bij (t)
∞
Kij (u)gj ((−1)k+1 vj (t + (k + 1)T − u))du + Ii (t),
0
i = 1, 2, . . . , n. Thus, for any natural number k, (−1)k+1 v(t + (k + 1)T ) are the
solutions of (1.1) on R. Then, by Lemma 2.3, there exists a constant M > 0 such
that
|(−1)k+1 vi (t + (k + 1)T ) − (−1)k vi (t + kT )|
≤ M e−λ(t+kT )
max |vi (s + T ) + vi (s)|
sup
−∞≤s≤0 1≤i≤n
I
≤ e−λ(t+kT ) M 2 max {ξi }, ∀t + kT > 0,
1≤i≤n
η
Thus, for any natural number m, we obtain
(−1)m+1 vi (t + (m + 1)T ) = vi (t) +
m
X
(3.4)
i = 1, 2, . . . , n.
[(−1)k+1 vi (t + (k + 1)T ) − (−1)k vi (t + kT )].
k=0
(3.5)
Then,
|(−1)m+1 vi (t+(m+1)T )| ≤ |vi (t)|+
m
X
|(−1)k+1 vi (t+(k +1)T )−(−1)k vi (t+kT )|,
k=0
(3.6)
where i = 1, 2, . . . , n.
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ANTI-PERIODIC SOLUTIONS
9
In view of (3.4), we can choose a sufficiently large constant N > 0 and a positive
constant α such that
|(−1)k+1 vi (t + (k + 1)T ) − (−1)k vi (t + kT )| ≤ α(e−λT )k ,
∀k > N, i = 1, 2, . . . , n,
(3.7)
on any compact subset of R. It follows from (3.5)–(3.7) that {(−1)m v(t + mT )}
converges uniformly to a continuous function x∗ (t) = (x∗1 (t), x∗2 (t), . . . , x∗n (t))T on
any compact subset of R.
Now we show that x∗ (t) is T -anti-periodic solution of system (1.1). First, x∗ (t)
is T -anti-periodic, since
x∗ (t + T ) = lim (−1)m v(t + T + mT )
m→∞
=−
lim
(−1)m+1 v(t + (m + 1)T ) = −x∗ (t).
(m+1)→∞
Next, we prove that x∗ (t) is a solution of (1.1). In fact, together with the continuity
of the right side of (1.1), (3.3) implies that {((−1)m+1 v(t + (m + 1)T ))0 } uniformly
converges to a continuous function on any compact set of R. Thus, letting m → ∞,
we obtain
n
X
d ∗
{xi (t)} = −ci (t)αi (x∗i (t)) +
aij (t)g̃j (x∗j (t − τij (t)))
dt
j=1
(3.8)
Z
n
∞
X
∗
+
bij (t)
Kij (u)gj (xj (t − u))du + Ii (t).
j=1
0
Therefore, x∗ (t) is a solution of (1.1). Finally, by Lemma 2.3, we can prove that
x∗ (t) is globally exponentially stable. This completes the proof.
4. An Example
In this section, we give an example to demonstrate the results obtained in previous sections. Consider the recurrent neutral network
1
1
x01 (t) = −(x1 (t) + x31 (t)) + (cos t)g̃1 (x1 (t − 22)) +
(cos t)g̃2 (x2 (t − 11))
24
108
Z ∞
1
+ (sin t)g1 (
| sin u|e−u x1 (t − u)du)
24
0
Z ∞
1
+
(sin t)g2 (
| cos u|e−u x2 (t − u)du) + I1 (t),
108
0
1
1
0
3
x2 (t) = −(x2 (t) + x2 (t)) + (cos t)g̃1 (x1 (t − 6)) + (cos t)g̃(x2 (t − 8))
24
Z ∞ 12
1
−u
+ (sin t)g1 (
| cos u|e x1 (t − u)du)
12
0
Z ∞
1
+ (sin t)g2 (
| sin u|e−u x2 (t − u)du) + I2 (t),
24
0
(4.1)
1
where g̃1 (x) = g̃2 (x) = | sin x|x, g1 (x) = g2 (x) = x2 = x × x, I1 (t) = 24
cos t,
1
I2 (t) = 24
sin t.
˜
+
+
+
1
1
Noting that c1 = c2 = Lfj = Lh̃j = Lfj = Lhj = 1, a+
11 = b11 = 24 , a12 = b12 = 108 ,
R
∞
+
+
+
+
1
1
+
a21 = b21 = 12 , a22 = b22 = 24 , di = 1, 0 Kij (s)ds ≤ 1, where i, j = 1, 2, β =
10
H. ZHANG, Y. WU
EJDE-2010/50
supt∈R β(t). Then
˜
+ f h̃
+ f h
dij = c−1
i di (3aij Lj Lj + 3bij Lj Lj ) i, j = 1, 2,
D = (dij )2×2 =
1/4 1/18
.
1/2 1/4
From the theory of M -matrices in [4], we can choose constants η = 41 and ξi = 1
such that for all t > 0,
Z ∞
2
2
X
X
f˜ h̃
+
− ci di ξi +
L
3a+
L
ξ
+
3b
Kij (s)dsLfj Lhj ξj
j
ij j j
ij
j=1
< −ci di ξi +
2
X
j=1
0
j=1
˜
f h̃
a+
ij Lj Lj ξj +
2
X
f h
b+
ij Lj Lj ξj < −η,
j=1
where i = 1, 2, which implies that (4.1) satisfy all the conditions in Theorem 3.1.
Hence, (4.1) has at least one π-anti-periodic solution x∗ (t). Moreover, x∗ (t) is
locally exponentially stable, the domain of attraction of Z ∗ (t) is the set G1 (x∗ ).
Remark 4.1. System (4.1) is a very simple form of recurrent neural networks with
mixed delays. Since hi (u) 6= u, i = 1, 2, g̃1 (x) = g̃2 (x) = | sin x|x, g1 (x) = g2 (x) =
x2 = x × x. One can observe that condition (T0) is not satisfied. Therefore, the
results in in this article and their references can not be applied to (4.1).
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ANTI-PERIODIC SOLUTIONS
11
Hong Zhang
Department of Mathematics, Hunan University of Arts and Science, Changde, Hunan
415000, China
E-mail address: hongzhang320@yahoo.com.cn
Yuanheng Wu
College of Mathematics and Information Sciences, Guangzhou University, Guangzhou,
Guangdong 510006, China
E-mail address: wyhcd2006@yahoo.com.cn