Hindawi Publishing Corporation
Discrete Dynamics in Nature and Society
Volume 2009, Article ID 573534, 18 pages
doi:10.1155/2009/573534
Research Article
Existence and Exponential Stability of
Periodic Solution of High-Order Hopfield Neural
Network with Delays on Time Scales
Yongkun Li, Lili Zhao, and Ping Liu
Department of Mathematics, Yunnan University, Kunming, Yunnan 650091, China
Correspondence should be addressed to Yongkun Li, yklie@ynu.edu.cn
Received 17 April 2009; Accepted 3 August 2009
Recommended by Binggen Zhang
On time scales, by using the continuation theorem of coincidence degree theory and constructing
some suitable Lyapunov functions, the periodicity and the exponential stability are investigated
for a class of delayed high-order Hopfield neural networks HHNNs, which are new and
complement of previously known results. Finally, an example is given to show the effectiveness
of the proposed method and results.
Copyright q 2009 Yongkun Li et al. This is an open access article distributed under the Creative
Commons Attribution License, which permits unrestricted use, distribution, and reproduction in
any medium, provided the original work is properly cited.
1. Introduction
Consider the following HHNNs with time-varying delays:
n
dxi t
−ci txi t aij tfj xj t − γij t
dt
j1
n
n bijl tgj xj t − σijl t gl xl t − vijl t
1.1
j1 l1
Ii t,
i 1, 2, . . . , n,
where 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 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, aij t and bijl t are the first- and second-order connection weights of the
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Discrete Dynamics in Nature and Society
neural network, γij t ≥ 0, σijl t ≥ 0, and vijl t ≥ 0 correspond to the transmission delays,
Ii t denote the external inputs at time t, and fj and gj are the activation functions of signal
transmission.
Due to the fact that high-order neural networks have stronger approximation property,
faster convergence rate, greater storage capacity, and higher fault tolerance than lower-order
neural networks, high-order neural networks have been the object of intensive analysis by
numerous authors in the recent years. In particular, there have been extensive results on the
problem of the existence and stability of equilibrium points and periodic solutions of HHNNs
1.1 in the literature. We refer the reader to 1–9 and the references cited therein. In fact,
both continuous and discrete systems are very important in implementing and applications.
The theory of calculus on time scales see 10, 11 and references cited therein was initiated
by Stefan Hilger in his Ph.D. thesis in 1988 12 in order to unify continuous and discrete
analysis, and it has a tremendous potential for application and has recently received much
attention since his foundational work. Therefore, it is meaningful to study that on time scales
which can unify the continuous and discrete situations.
Our purpose of this paper is to consider the model
xiΔ t −ci txi t n
aij tfj xj t − γij t
j1
n
n bijl tgj xj t − σijl t gl xl t − vijl t
1.2
j1 l1
Ii t,
i 1, 2, . . . , n, t ∈ T,
where T is an ω-periodic time scale which has the subspace topology inherited from the
standard topology on R, It I1 t, I2 t, . . . , In tT is an input periodic vector function
with period ω; that is, there exists ω > 0 such that Ii t ω Ii t i 1, 2, . . . , n for all
t ∈ 0, ∞ T, and x x1 , x2 , . . . , xn T ∈ Rn , fx f1 x1 , f2 x2 , . . . , fn xn T , and gx g1 x1 , g2 x2 , . . . , gn xn T is the activation function of the neurons.
System 1.2 is supplemented with initial values given by
xi s ϕi s,
s ∈ −θ, 0
T, i 1, 2, . . . , n,
1.3
where ϕi · denotes continuous ω-periodic function defined on −θ, 0 T, θ }, v max1≤i,j,l≤n {vijl
}, γij max{γ, σ, v}, γ max1≤i,j≤n {γij }, σ max1≤i,j,l≤n {σijl
maxt∈0,ω T γij t, σijl maxt∈0,ω T σijl t, vijl maxt∈0,ω T vijl t, i, j, l 1, 2, . . . , n. To
the best of our knowledge, this is the first paper to study the stability and existence of periodic
solutions of 1.2.
Throughout this paper, we assume the following.
H1 For i, j, l 1, 2, . . . , n, ci t, aij t, bijl t, Ii t, γij t, σijl t, vijl t are positive continuous periodic functions with period ω > 0, and ci t is regressive.
H2 There exist positive constants Mj , Nj , j 1, 2, . . . , n such that |fj x| ≤ Mj , |gj x| ≤
Nj for j 1, 2, . . . , n, x ∈ R.
Discrete Dynamics in Nature and Society
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H3 Functions fj u, gj u j 1, 2, . . . , n satisfy the Lipschitz condition; that is, there
exist constants Lj , Hj > 0 such that |fj u1 − fj u2 | ≤ Lj |u1 − u2 |, |gj u1 − gj u2 | ≤
Hj |u1 − u2 |, j 1, 2, . . . , n.
2. Preliminaries
In this section, we will first recall some basic definitions and lemmas which are used in what
follows.
Let T be a nonempty closed subset time scale of R. The forward and backward jump
operators σ, ρ : T → T and the graininess μ : T → R are defined, respectively, by
σt inf{s ∈ T : s > t},
ρt sup{s ∈ T : s < t},
μt σt − t.
2.1
A point t ∈ T is called left-dense if t > inf T and ρt t, left-scattered if ρt < t,
right-dense if t < sup T and σt t, and right-scattered if σt > t. If T has a left-scattered
maximum m, then Tk T \ {m}; otherwise Tk T. If T has a right-scattered minimum m,
then Tk T \ {m}; otherwise Tk T.
A function f : T → R is right-dense continuous provided that it is continuous at
right-dense point in T and its left-side limits exist at left-dense points in T. If f is continuous
at each right-dense point and each left-dense point, then f is said to be continuous function
on T.
For y : T → R and t ∈ Tk , we define the delta derivative of yt, yΔ t to be the
number if it exists with the property that for a given ε > 0, there exists a neighborhood U
of t such that
yσt − ys − yΔ tσt − s < ε|σt − s|
2.2
for all s ∈ U.
If y is continuous, then y is right-dense continuous, and if y is delta differentiable at t,
then y is continuous at t.
A function r : T → R is called regressive if
1 μtrt /
0
2.3
for all t ∈ Tk .
If r is regressive function, then the generalized exponential function er is defined by
er t, s exp
t
ξμτ rτΔτ
for s, t ∈ T,
2.4
s
with the cylinder transformation
⎧
⎪
⎨ Log1 hz , if h /
0,
h
ξh z ⎪
⎩z,
if h 0.
2.5
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Discrete Dynamics in Nature and Society
Let p, q : T → R be two regressive functions, we define
p ⊕ q : p q μpq,
p : −
p
,
1 μp
p q : p ⊕ q .
2.6
Then the generalized exponential function has the following properties.
Lemma 2.1 see 10. Assume that p, q : T → R are two regressive functions, then
i e0 t, s ≡ 1 and ep t, t ≡ 1;
ii ep σt, s 1 μtptep t, s;
iii ep t, σs ep t, s/1 μsps;
iv 1/ep t, s ep t, s;
v ep t, s 1/ep s, t ep s, t;
vi ep t, sep s, r ep t, r;
vii ep t, seq t, s ep⊕q t, s;
viii ep t, s/eq t, s epq t, s.
Lemma 2.2 see 10. Assume that f, g : T → R are delta differentiable at t ∈ Tk , then
fg
Δ
t f Δ tgt fσtg Δ t ftg Δ t f Δ tgσt.
2.7
Let y be right-dense continuous. If Y Δ t yt, then one defines the delta integral by
t
ysΔs Y t − Y a.
2.8
a
Lemma 2.3. If a, b ∈ T, α, β ∈ R and f, g ∈ CT, R, then
b
b
b
1 a αft βgtΔt α a ftΔt β a gtΔt;
b
2 If ft ≥ 0 for all a ≤ t < b, then a ftΔt ≥ 0;
b
b
3 If |ft| ≤ gt on a, b : {t ∈ T : a ≤ t < b}, then | a ftΔt| ≤ a gtΔt.
In this paper, one assumes that k min{0, ∞
can obtain Lemma 2.4.
T}. Clearly, from Lemma 2.3, one
Lemma 2.4. If f, g ∈ CT, R, and ft ≤ gt on t ∈ k, k ω, then
kω
k
ftΔt ≤
kω
k
gtΔt.
In the proof of our main result, one will use the following three lemmas which can be
found in 13, 14.
Lemma 2.5 see 13. Let t1 , t2 ∈ k, k ω T, and t ∈ T. If f : T → R is ω-periodic, then
kω
kω
ft ≤ ft1 k |f Δ s|Δs, and ft ≥ ft2 − k |f Δ s|Δs.
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Lemma 2.6 14, Cauchy-Schwarz inequality on time scales. Let a, b ∈ T. For rd-continuous
b
b
b
2
functions f, g : a, b → R, one has a |ftgt|Δt ≤ a |ft|2 Δt a |gt|2 Δt.
Lemma 2.7. Let r : T → R be right-dense continuous and regressive. a ∈ T, and ya ∈ R. The
unique solution of the initial value problem
yΔ t rtyt ht,
2.9
ya ya
is given by
yt er t, aya t
2.10
er t, σshsΔs.
a
For the sake of convenience, one introduces the following notations:
ci bijl aij 1
ω
kω
1
ω
kω
aij ci tΔt,
k
bijl tΔt,
k
max t∈ k,kω
T
aij t,
Ai t −ci txi t n
ci 1
ω
kω
aij tΔt,
Ii k
max |ci t|,
t∈ k,kω
T
bijl
1
ω
ci− max bijl t,
t∈k,kω
T
kω
Ii tΔt,
k
min |ci t|,
t∈ k,kω
T
Ii max |Ii t|,
t∈ k,kω
T
2.11
aij tfj xj t − γij t
j1
n
n bijl tgj xj t − σijl t gl xl t − vijl t Ii t,
i, j, l 1, 2, . . . n.
j1 l1
Obliviously, for system 1.2, finding the periodic solutions is equivalent to finding those of
the following boundary-value problem:
xiΔ t −ci txi t n
aij tfj xj t − γij t
j1
n
n bijl tgj xj t − σijl t gl xl t − vijl t Ii t,
j1 l1
t ∈ k, k ω ∩ T,
xi k xi k ω ,
i 1, 2, . . . , n.
Now, one states Mawhin’s continuous theorem 15.
i 1, 2, . . . , n,
2.12
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Discrete Dynamics in Nature and Society
Theorem 2.8. Let X and Z be two Banach spaces and let L be a Fredholm mapping of index zero. let
Ω ⊂ X be an open bounded set and let N : Ω → Z be a continuous operator which is L-compact on
Ω. Assume that
1 for each λ ∈ 0, 1, x ∈ ∂Ω ∩ Dom L, Lx /
λNx;
2 for each x ∈ ∂Ω ∩ Ker L, QNx /
0;
3 degJNQx, Ω ∩ Ker L, 0 /
0, where JQN : Ker L → Ker L.
Then Lx Nx has at least one solution in Ω ∩ Dom L.
In order to apply Theorem 2.8 to system 2.12, we first define
X Z x x1 , x2 , . . . , xn T ∈ CT, Rn : xt ω xt, t ∈ T ,
n
x x1 , x2 , . . . , xn T 2.13
max |xi t|
T
i1 t∈ k,kω
for any x ∈ Xor Z. Then X and Z are Banach spaces with the norm ·
Nx A1 t, A2 t, . . . , An tT ,
. Let
2.14
x∈X
T
Lx x1Δ , x2Δ , . . . , xnΔ
Px Qz 1
ω
kω
1
ω
kω
k
kω
1
x1 tΔt,
ω
z1 tΔt,
k
1
ω
kω
k
2.15
kω
1
x2 tΔt, . . . ,
ω
z2 tΔt, . . . ,
k
1
ω
kω
T
xn tΔt
,
k
2.16
T
zn tΔt
z ∈ Z.
,
k
Then it follows that
Ker L x1 , x2 , . . . , xn T ∈ X : x1 , x2 , . . . , xn T w1 t, w2 t, . . . , wn tT ∈ Rn , t ∈ T ,
Im L ⎧
⎨
z∈Z:
⎩
1
ω
kω
k
1
z1 tΔt,
ω
kω
k
1
z2 tΔt, . . . ,
ω
kω
k
T
zn tΔt
⎫
⎬
0
⎭
2.17
is closed in Z, dim Ker L n codim Im L. It is not difficult to show that P and Q are
continuous and satisfy Im P Ker L, Im L Ker Q Im I − Q. It is easy to see that Im L is
closed in Z, which leads to the following lemma.
Lemma 2.9. Let L and N be defined by 2.14 and 2.15, respectively, then L is a Fredholm operator
of index zero.
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Lemma 2.10. Let L and N be defined by 2.14 and 2.15, respectively, suppose that Ω is an open
bounded subset of Dom L, then N is L-compact on Ω.
Proof. Through an easy computation, we find that the inverse KP : Im L → Ker P
of LP has the form
KP zt t
zsΔs −
k
1
ω
kω
t
k
k
Dom L
2.18
zsΔsΔt.
Thus, the expression of QNx is
QNx 1
ω
kω
k
kω
1
A1 tΔt,
ω
k
kω
1
A2 tΔt, . . . ,
ω
T
An tΔt
,
2.19
k
and then
⎛
⎜
⎜
⎜
⎜
KP I − QNx ⎜
⎜
⎜
⎝
⎞
t
A1 sΔs ⎟
⎟
⎟
..
⎟
⎟
.
⎟
⎟
t
⎠
An sΔs
k
k
⎛
1
⎜
⎜ω
⎜
⎜
−⎜
⎜
⎜
⎝1
ω
⎛
kω
k
kω
k
⎞
t
A1 sΔsΔt ⎟
⎟
k
⎟
..
⎟
⎟
.
⎟
⎟
t
⎠
An sΔsΔt
2.20
k
t−k
1
−
ω
ω2
⎜
⎜
⎜
⎜
−⎜
⎜
⎜
⎜
⎝ t−k
1
−
ω
ω2
kω t − k Δt
kω
⎞
A1 sΔs ⎟
⎟
⎟
⎟
..
⎟.
.
⎟
⎟
kω
⎟
kω ⎠
t − k Δt
An sΔs
k
k
k
k
Thus, QN and KP I − QN are continuous. Since X is a Banach space, it is not difficult to
show that KP I − QNΩ is compact. Moreover, QNΩ is bounded. Thus, N is L-compact
on Ω for any open bounded set Ω ⊂ X. The proof of Lemma 2.10 is completed.
3. Existence of Periodic Solution
In this section, we study the existence of periodic solution of 1.2 based on Mawhin’s
continuation theorem.
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Discrete Dynamics in Nature and Society
Theorem 3.1. Assume that H1 -H2 hold, then system 1.2 has at least one ω-periodic solution.
Proof. Based on the Lemma 2.9 and Lemma 2.10, now, what we need to do is just to search
for an appropriate open, bounded subset Ω for the application of the continuation theorem.
Corresponding to the operator equation Lx λNx, λ ∈ 0, 1, we have
⎡
xiΔ t λ⎣−ci txi t n
aij tfj xj t − γij t
j1
n
n bijl tgj xj t − σijl t gl xl t − vijl t
3.1
j1 l1
⎤
Ii t, t ∈ T
k, k ω ⎦,
xi k xi k ω ,
i 1, 2, . . . , n.
For the sake of convenience, defined x
2
by
x
i 1, 2, . . . , n,
2
kω
1/2
2
|xt| Δt
3.2
k
for x ∈ CT, R. Suppose that x1 t, x2 t, . . . , xn tT ∈ X is a solution of system 3.1 for a
certain λ ∈ 0, 1. Integrating 3.1 over k, k ω, we obtain
kω
Ai tΔt 0.
3.3
k
Hence
kω
ci sxi sΔs
k
⎡
⎣
kω n
k
j1
⎤
n
n aij tfj xj t−γij t bijl tgj xj t−σijl t gl xl t−vijl t Ii t⎦ Δt,
j1 l1
i 1, 2, . . . , n.
3.4
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Let ζi , ηi ∈ T k, k ω, such that xi ζi inft∈T k,kω xi t, xi ηi supt∈T k,kω xi t.
Then by 3.4 and Lemma 2.4, we have
kω
ωci xi ζi ≤
k
⎡
n
⎣ aij sfj xj s − γij s
j1
⎤
bijl sgj xj s − σijl s gl xl s − vijl s Ii s⎦Δs
j1 l1
n n
kω n aij sfj xj s − γij s Δs
≤
k
j1
kω n n kω
k
k
bijl sgj xj s − σijl s gl xl s − vijl s Δs j1 l1
⎤
⎡
n
n
n bijl Nj Nl Ii ⎦,
≤ ω⎣ aij Mj j1
Ii sΔs
i 1, 2, . . . , n.
j1 l1
3.5
(
( (
Hence xi ζi ≤ 1/ci { nj1 aij Mj nj1 nl1 bijl
Nj Nl Ii } : Bi , i 1, 2, . . . , n. By 3.4
and Lemma 2.4, we can also have
ωci xi ηi ≥ −
kω
k
⎡
n
⎣ aij sfj xj s − γij s
j1
⎤
bijl sgj xj s − σijl s gl xl s − vijl s Ii s⎦Δs
j1 l1
n
n ≥−
kω n aij sfj xj s − γij s Δs
k
−
j1
kω n n kω
k
k
bijl sgj xj s − σijl s gl xl s − vijl s Δs −
j1 l1
⎤
⎡
n
n
n ≥ −ω⎣ aij Mj bijl
Nj Nl Ii ⎦,
j1
|Ii s|Δs
i 1, 2, . . . , n.
j1 l1
3.6
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Discrete Dynamics in Nature and Society
(
( (
Nj Nl Ii } −Bi , i 1, 2, . . . , n. From
Hence xi ηi ≥ −1/ci { nj1 aij Mj nj1 nl1 bijl
3.1, 3.4, and Lemma 2.6, we have
Δ xi tΔt
kω k
kω
≤
|ci t||xi t|Δt k
aij tfj xj t − γij t Δt
k
j1
kω n n kω
k
k
bijl tgj xj t − σijl t gl xl t − vijl t Δt j1 l1
kω
≤
1/2 |ci t| Δt
2
k
kω n n
kω
|Ii t|Δt
1/2
2
|xi t| Δt
k
aij t2 Δt
kω 1/2 k
j1
n
n bijl
1/2
fj xj t − γij t 2 Δt
kω k
j1 l1
≤ ω1/2 ci xi
1/2 gj xj t − σijl t 2 Δt
kω k
2
k
n
ωaij Mj j1
n
n 1/2
gl xl t − σijl t 2 Δt
kω Ii ω
Nj Nl Ii ω.
ωbijl
j1 l1
3.7
From Lemma 2.7 and 3.1, for i 1, 2, . . . , n, we can obtain
xi t e−λci t t, k xi k
t
k
⎡
n
λe−λci t t, σs⎣ aij sfj xj s − γij s
j1
n
n ⎤
bijl sgj xj s − σijl s gl xl s − vijl s Ii s⎦Δs.
j1 l1
3.8
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11
Hence
n
|xi t| ≤ xi k j1
n
n kω j1 l1
k
kω
aij sfj xj s − γij s Δs
kω k
bijl sgj xj s − σijl s gl xl s − vijl s Δs
3.9
|Ii s|Δs
k
n
n
n ≤ xi k aij Nj ω bijl Nj Nl ω Ii ω
j1
: ui ,
j1 l1
i 1, 2, . . . , n,
that is,
xi
2
kω
1/2
2
≤ ui ω1/2 ,
|xi t| Δt
i 1, 2, . . . , n.
3.10
k
Substituting 3.10 into 3.7, we have
n
n
n Δ Δt ≤ ωc u ωa M Nj Nl Ii ω,
ωbijl
i
j
xi t
i
ij
kω k
j1
i 1, 2, . . . , n.
3.11
j1 l1
From Lemma 2.5, we have
xi t ≤ xi ζi k
xi t ≥ xi ηi −
Δ xi tΔt,
kω Δ xi tΔt,
kω k
i 1, 2, . . . , n,
3.12
i 1, 2, . . . , n.
From 3.5, 3.6 and 3.11, there exist positive constants ξi i 1, 2, . . . , n such that for
t ∈ k, k ω T, |xi t| ≤ ξi , i 1, 2, . . . , n. Clearly, ξi i 1, 2, . . . , n is independent of λ.
(
Denote H ∗ ni1 ξi C, where C > 0 is taken sufficiently large so that
⎛
⎞
n n n min ci H ∗ > n max⎝Ii aij Mj bijl Nj Nl ⎠.
1≤i≤n
1≤i≤n
j1
j1 l1
3.13
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Discrete Dynamics in Nature and Society
Now we take Ω {x1 t, x2 t , . . . , xn tT : x1 t, x2 t, . . . , xn tT < H ∗ } .
Thus 1 of Theorem 2.8 is satisfied. When x1 t, x2 t, . . . , xn tT ∈ ∂Ω Rn , x1 t,
x2 t, . . . , xn tT is a constant vector in Rn with |x1 | |x2 | · · · |xn | H ∗ , then
⎛
QNx1 t, x2 t, . . . , xn tT ⎝−ci xi t n
aij fj xj t − γij t
j1
n
n ⎞
bijl gj xj t − σijl t gl xl t − vijl t Ii ⎠
j1 l1
.
n×1
3.14
Therefore
n n
ci xi t − aij fj xj t − γij t
QNx1 t, x2 t, . . . , xn tT i1 j1
−
bijl gj xj t − σijl t gl xl t − vijl t − Ii j1 l1
n
n ≥
n
ci |xi t| −
i1
−
n n aij Mj
i1 j1
n n
n bijl Nj Nl −
i1 j1 l1
≥
n
i1
ci |xi t|
i1
−
n Ii n
⎛
⎞
n
n n ⎝Ii aij Mj bijl Nj Nl ⎠
i1
≥ min ci 1≤i≤n
j1
j1 l1
n
|xi t|
i1
⎛
⎞
n n n − n max⎝Ii aij Mj bijl Nj Nl ⎠
1≤i≤n
j1
j1 l1
> 0.
3.15
Consequently,
QNx1 t, x2 t, . . . , xn tT /
0, 0, . . . , 0T ,
3.16
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for x1 t, x2 t, . . . , xn tT ∈ ∂Ω
Ψ : Ker L × 0, 1 → X by
13
Ker L . This satisfies condition 2 of Theorem 2.8. Define
Ψ x1 , x2 , . . . , xn , μ −μx1 , x2 , . . . , xn T 1 − μ QNx1 t, x2 t, . . . , xn tT
3.17
When x1 t, x2 t, . . . , xn tT ∈ ∂Ω Ker L, x1 , x2 , . . . , xn T is a constant vector in Rn with
(n
∗
0, 0, . . . , 0T . Therefore
i1 |xi | H , we easily have Ψx1 , x2 , . . . , xn , μ /
deg QNx1 t, x2 t, . . . , xn tT , Ω Ker L, 0, 0, . . . , 0T
deg −x1 t, −x2 t, . . . , −xn tT , Ω Ker L, 0, 0, . . . , 0T / 0.
3.18
Condition 3 of Theorem 2.8 is also satisfied. Thus, by Theorem 2.8 we can obtain that Lx Nx has at least one solution in X. That is, system 1.2 has at least one ω-periodic solution.
The proof is complete.
4. Global Exponential Stability of Periodic Solution
In this section, we will construct suitable Lyapunov functions to study the global exponential
stability of the periodic solution of 1.2 on time scales. So first we will introduce some
definitions.
Definition 4.1 see 12. A function f from T to R is positively regressive if 1 μtft > 0
for every t ∈ T.
Denote R is the set of positively regressive functions from T to R, and denote T 0, ∞ T.
Definition 4.2. The periodic solution x∗ t of system 1.2 is said to be exponentially stable
if there exists a positive constant α with −α ∈ R such that for every δ ∈ T, there exists
N Nδ ≥ 1 such that the solution xt of 1.2 through δ, ϕδ satisfies xt − x∗ t ≤
(
N ϕδ − x∗ δ e−α t, δ, t ∈ T where ϕδ − x∗ δ ni1 maxδ∈−θ,0 T |ϕi δ − xi∗ δ|.
Theorem 4.3. Assume that H1 –H3 hold. Suppose further that there exists n positive constants
i > 0, i 1, 2, . . . , n such that
H4 −i ci− n
n
n aij Mj j bijl
Hj Nl j Hl Nj l < 0,
j1
i 1, 2, . . . , n,
j1 l1
then the ω-periodic solution of system 1.2 is globally exponentially stable.
4.1
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Discrete Dynamics in Nature and Society
Proof. According to Theorem 3.1, we know that 1.2 has an ω-periodic solution x∗ t x1∗ t, x2∗ t, . . . , xn∗ tT . Suppose that xt x1 t, x2 t, . . . , xn tT is an arbitrary solution
of 1.2. Then it follows from system 1.2 that
xi t − xi∗ t
Δ
n
−ci t xi t − xi∗ t aij t fj xj t − γij t − fj∗ xj t − γij t
j1
n
n bijl t gj xj t − σijl t gl xl t − vijl t
j1 l1
− gj xj∗ t − σijl t gl xl∗ t − vijl t ,
i 1, 2, . . . , n,
4.2
with initial values given by
xi s ϕi s,
s ∈ −θ, 0
T, i 1, 2, . . . , n,
4.3
where θ is defined as in1.3 . If H4 holds, it can always find a small enough constant α > 0
satisfying ∀t ∈ T, 1 − μtα > 0, namely, −α ∈ R such that
n
−ci− α i aij Mj eα t, t − γij t j
j1
n
n bijl
Hj Nl j eα t, t − σijl t Hl Nj l eα t, t − vijl t < 0,
4.4
i 1, 2, . . . , n.
j1 l1
We define a Lyapunov function H h1 , h2 , . . . , hn T by hi eα t, δ|xi t − xi∗ t|, δ ∈
−θ, 0 T, i 1, 2, . . . , n. In view of 4.2, we obtain
αeα t, δxi t − xi∗ t eα σt, δ sign xi t − xi∗ t
hΔ
i t
×
⎧
⎨
⎩
− ci t xi t − xi∗ t
n
aij t fj xj t − γij t − fj xj∗ t − γij t
j1
n
n bijl t gj xj t − σijl t gl xl t − vijl t
j1 l1
⎫
⎬
−gj xj∗ t − σijl t gl xl∗ t − vijl t
⎭
Discrete Dynamics in Nature and Society
15
≤ αeα t, δxi t − xi∗ t eα σt, δ
⎧
n ⎨
× − ci txi t − xi∗ t aij txj t − γij t − xj∗ t − γij t Mj
⎩
j1
n n bijl t Hj Nl xj t − σijl t − x∗ t − σijl t j
j1 l1
⎫
⎬
Hl Nj xl t − vijl t − xl∗ t − vijl t ⎭
≤ eα σt, δ
⎧
n ⎨
× −ci t αxi t − xi∗ t aij txj t − γij t − xj∗ t − γij t Mj
⎩
j1
n n bijl t Hj Nl xj t − σijl t − x∗ t − σijl t j
j1 l1
⎫
⎬
∗
Hl Nj xl t − vijl t − xl t − vijl t ⎭
⎧
n
⎨
−ci− α hi t aij Mj eα t, t − γij t hj t − γij t
≤ 1 μtα
⎩
j1
n
n bijl
× Hj Nl eα t, t − σijl t hj t − σijl t
j1 l1
⎫
⎬
,
Hl Nj eα t, t − vijl t hl t − vijl t
⎭
i 1, 2 . . . , n.
4.5
Defining the curve ρ {ωl : ωi i l, l > 0, i 1, 2, . . . , n} and the set Ωω {u :
0 ≤ u ≤ ω}, Si ω {u ∈ Ωω : ui ω}, i 1, 2 . . . , n. It is obvious that if l < l, then
Ωωl ⊂ Ωωl. We will prove that the zero solution of 4.2 is exponential stable, namely,
there exists a constant β > 0 such that
xt − x∗ t
≤ Nδe−β t, δϕt − x∗ t,
t ≥ 0.
4.6
Let M max1≤i≤n {i }, m min1≤i≤n {εi }, l0 1 − δ|ϕi − xi∗ |0 /m , where −δ ≥ 0 is a
constant, |ϕi − xi∗ |0 max1≤i≤n {maxδ∈−θ,0 T |ϕi δ − xi∗ δ|}. Then, {|H| : |H| eα t, δ|ϕδ −
x∗ δ|, −θ ≤ t ≤ δ ≤ 0} ⊂ Ωωl0 , namely,
|hi δ| eα t, δϕi δ − xi∗ δ < i l0 ,
−θ ≤ δ ≤ 0, i 1, 2, . . . , n.
4.7
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Discrete Dynamics in Nature and Society
We can claim that |hi t| < i l0 for t > 0, i 1, 2, . . . , n. If it is not true, then there exist some
i and t1 ti > 0 such that |hi t1 | i l0 , hΔ
i t1 ≥ 0 and |hi t| ≤ i l0 for −θ ≤ t ≤ t1 , i 1, 2, . . . , n. However, from 4.4 and 4.5, we get
⎡
n
≤ 1 μtα ⎣ −ci− α i aij Mj eα t, t − γij t j
hΔ
i t1 j1
n n
bijl
⎤
Hj Nl eα t, t − σijl t j Hl Nj eα t, t − vijl t l ⎦l0 < 0
j1 l1
4.8
for t > 0, i 1, 2, . . . , n, this is a contradiction. So |hi t| < i l0 , for t ≥ 0. Also
xi t − x∗ t < eΘα t, δi l0 ≤ 1 eΘα t, δi 1 − δϕi − x∗ ,
i
i 0
m
i 1, 2, . . . , n, t ≥ 0,
4.9
which means that
xt − x∗ t ≤
1
eΘα t, δM 1 − δϕδ − x∗ δ.
m
4.10
Denote −β Θα −α/1 μα ∈ R , N Nδ M /m 1 − δ > 1, in view of 4.10, we
have
xt − x∗ t ≤ Ne−β t, δϕs − x∗ s.
4.11
From Definition 4.2, the periodic solution of system 1.2 is globally exponentially stable. The
proof is complete.
5. An Example
Let T R, in this case, xΔ t dxt/dt. Consider the following equation
2
dxi t
−ci txi t aij tfj xj t − γij t
dt
j1
2
2 j1 l1
bijl tgj xj t − σijl t gl xl t − vijl t Ii t,
5.1
i 1, 2,
Discrete Dynamics in Nature and Society
17
where
*
x
,
1
21/2
*
)
1
x
,
g1 x1 arctan
1
21/2
)
f1 x1 sin
1
*
x
,
2
23/2
*
)
1
x
g2 x2 arctan
.
2
23/2
)
f2 x2 sin
1
5.2
Obviously, fi xi , gi xi i 1, 2 satisfy H1 and H2 , and
H1 H2 M1 M2 1,
N1 N2 π
.
2
5.3
Take
a11 t 1 cos2πt,
c1 t 20 5 sin2πt,
a12 t 2 cos2πt,
a21 t 2 cos2πt,
c2 t 33 16 sin2πt,
I1 t 1 sin2πt,
a22 t 3 cos2πt,
I2 t 1 cos2πt,
b111 t b222 t 1 1
sin2πt,
4 4
b112 t b212 t 1 1
cos2πt,
3 3
b121 t b221 t 1 1
cos2πt,
5 5
b122 t b211 t 1 1
sin2πt.
6 6
5.4
One can verifies that H1 is satisfied, and ω 1, c1− 15, c2− 17, a11 2, a12 3, a21 b222
1/2, b112
b212
2/3, b121
b221
2/5, b122
b211
1/3, so, if we
3, a22 4, b111
take 1 2 1, we can obtain
−1 c1− 2
a1j Mj j j1
−2 c2−
2
j1
a2j Mj j
2
2 19
b1jl
Hj Nl j Hl Nj l −15 5 π < 0,
10
j1 l1
2
2 19
b2jl
Hj Nl j Hl Nj l −17 7 π < 0.
10
j1 l1
5.5
Condition 4.2 is satisfied. From Theorem 3.1 and 4.3, we know that 5.1 has at least one
1-periodic solution and this solution is exponential stability.
6. Conclusion
Sufficient conditions are derived to guarantee the stability and existence of periodic solutions
for a class of delayed high-order Hopfield neural networks on time scales. To the best of our
knowledge, the results presented here have been not appeared in the related literature. In fact,
both continuous and discrete systems, are very important in implementing and applications.
But it is troublesome to study the existence and stability of periodic solutions for continuous
and discrete systems respectively. Therefore, it is meaningful to study that on time scales
which can unify the continuous and discrete situations. Also, our methods used in this paper
may be applied to some other systems.
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Discrete Dynamics in Nature and Society
Acknowledgments
This work is supported by the National Natural Sciences Foundation of People’s Republic of
China under Grant 1097118.
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