Hindawi Publishing Corporation
Journal of Inequalities and Applications
Volume 2010, Article ID 132790, 19 pages
doi:10.1155/2010/132790
Research Article
Existence and Stability of Antiperiodic Solution for
a Class of Generalized Neural Networks with
Impulses and Arbitrary Delays on Time Scales
Yongkun Li, Erliang Xu, and Tianwei Zhang
Department of Mathematics, Yunnan University, Kunming, Yunnan 650091, China
Correspondence should be addressed to Tianwei Zhang, [email protected]
Received 14 June 2010; Accepted 16 August 2010
Academic Editor: Kok Lay Teo
Copyright q 2010 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.
By using coincidence degree theory and Lyapunov functions, we study the existence and global
exponential stability of antiperiodic solutions for a class of generalized neural networks with
impulses and arbitrary delays on time scales. Some completely new sufficient conditions are
established. Finally, an example is given to illustrate our results. These results are of great
significance in designs and applications of globally stable anti-periodic Cohen-Grossberg neural
networks with delays and impulses .
1. Introduction
In this paper, we consider the following generalized neural networks with impulses and
arbitrary delays on time scales:
x
Δ
t
A
k
− x
t
−
k
I
k
x
t
k
,t t
k
,k∈ N,
1.1
where T is an ω/2-periodic time scale and if t ∈ T,θ∈ E, then t θ ∈ T,Eis a subset
of R
−
−∞, 0, At, xt diaga
1
t, x
1
t,a
2
t, x
T
,f
i
t, x
t
f
i
t, x
1t
,x
2t
, ,x
nt
,x
it
θx
i
t θ,t∈ T,θ ∈ E, i 1, 2, ,n, and xt
k
,xt
−
k
represent the right and left limits of xt
k
in the sense of time scales, {t
l
T
I ∩ T, especially, we denote that T
T ∩ 0, ∞.
System 1.1 includes many neural continuous and discrete time networks 1–9. For
examples, the high-order Hopfield neural networks with impulses and delays see 8:
x
i
t
−a
i
x
i
t
⎡
⎣
b
i
x
i
t
x
j
t − τ
j
t
−
n
j1
n
l1
b
ijl
t
g
j
x
j
t − τ
j
t
k
x
i
t
k
− x
i
t
−
k
e
ik
x
i
t
k
,i 1, 2, ,n, k 1, 2, , 1.3
the Cohen-Grossberg neural networks with bounded and unbounded delays see 9:
x
f
j
x
j
t
−
n
j1
c
ij
t
g
j
x
j
t − τ
ij
t
t
⎤
⎦
,t
/
t
k
,
1.4
Δx
i
t
k
x
i
t
k
− x
i
t
−
k
θ
φ
i
θ
,θ∈ E, i 1, 2, ,n. 1.6
Throughout this paper, we assume that
H
1
a
i
t, u ∈ CT × R, R
,a
i
t ω/2, −ua
i
t, u, and there exist positive constants
a
m
i
,a
M
i
such that 0 <a
m
i
<a
b
i
t, u
− b
i
t, v
|
≤ L
b
i
|
u − v
|
,b
i
t, 0
0
, 1.7
for all t ∈ T,u,v ∈ R,i 1, 2, ,n;
H
3
f
i
∈ CT × R
i
n
j1
x
jt
− y
jt
, 1.8
for all t, x
1t
, ,x
nt
, t, y
1t
, ,y
nt
∈ T × R
n
and f
i
t, 0, ,00,i 1, 2, ,n;
H
4
I
ik
T
|
h
t
|
,h
m
min
t∈0,ω
T
|
h
t
|
, h
2
ω
0
|
h
t
t
: sup
{
s ∈ T : s<t
}
,μ
t
σ
t
− t. 2.1
4 Journal of Inequalities and Applications
Definition 2.2 see 31.Afunctionf : T → R is called right-dense continuous provided it
is continuous at right-dense point of T and left-side limit exists finite at left-dense point of
T. The set of all right-dense continuous functions on T will be denoted by C
rd
C
rd
T
C
rd
T, R.Iff is continuous at each right-dense and left-dense point, then f is said to be a
continuous function on T, the set of continuous function will be denoted by CT.
Definition 2.3 see 31. For x : T → R, one defines the delta derivative of xt,x
Δ
≤ ε
|
σ
t
− s
|
, 2.2
for all s ∈ U.
Definition 2.4 see 31.IfF
Δ
tft, then one defines the delta integral by
t
a
f
s
Δs F
t
− F
a
.
2.4
for each s ∈ Nε, s>t, where μt, sσt − s.
In case t is right-scattered and ut is continuous at t, this reduces to
D
u
Δ
t
u
σ
t
− u
t
σ
t
− t
.
2.5
Similar to 34, we will give the definition of anti-periodic function on a time scale as
following.
τ
Δτ
, 2.6
Journal of Inequalities and Applications 5
for s, t ∈ T, with the cylinder transformation
ξ
h
z
⎧
⎪
⎨
⎪
⎩
Log
1 hz
h
, if h
/
0,
z, if h 0.
2.7
p
t, s;
v e
p
t, s1/e
p
s, te
p
s, t;
vi e
p
t, se
p
s, re
p
t, r;
vii e
Δ
p
·,spe
p
·,s.
Lemma 2.9 see 31. Assume that f, g : T → R are delta differentiable at t ∈ T
k
.Then
fg
Δ
f
Δ
t
g
σ
t
.
2.9
The following lemmas can be found in 35, 36, respectively.
Lemma 2.10. Let t
1
,t
2
∈ 0,w
T
.Ifx : T → R is ω-periodic, then
x
t
≤ x
t
1
x
Δ
s
Δs.
2.10
Lemma 2.11. Let a, b ∈ T. For rd-continuous functions f, g : a, b → R, one has
b
a
f
t
g
t
Δt ≤
1/2
.
2.11
Definition 2.12. The anti-periodic solution x
∗
tx
∗
1
t,x
∗
2
t, ,x
∗
n
t
T
of system 1.1 is
said to be globally exponentially stable if there exist positive constants and M M ≥ 1,
6 Journal of Inequalities and Applications
for any solution xtx
1
t,x
2
t, ,x
n
t
T
of system 1.1 with the initial value φt
φ
1
e
t, α
φ − x
∗
,
2.12
where
φ − x
∗
n
i1
sup
s∈E
T
Then the equation Lx Nx has at least one solution on DL ∩
Ω.
3. Existence of Antiperiodic Solutions
In this section, by using Lemma 2.13, we will study the existence of at least one anti-periodic
solution of 1.1.
Theorem 3.1. Assume that H
1
–H
4
hold. Suppose further that
H
5
E e
ij
n×n
is a nonsingular M matrix, where, for i, j 1, 2, ,n,
e
ij
⎧
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎩
ωa
I
ik
−
1
μ
i
ωa
m
i
a
M
i
ωc
i
,i j,
−
1
μ
i
ωa
m
i
a
M
i
ωc
point it is continuous on the left}. Take
X
x ∈ C
0,ω; t
1
, ,t
q
,t
q1
, ,t
2q
T
: x
t
ω
2
−x
t
, ∀t ∈
0,
ω
2
y
,
3.3
respectively, where |x
i
|
0
max
t∈0,ω
T
|x
i
t|, i 1, ,n, ·is any norm of R
n×q
.
Set
L :Dom L ∩ X −→ Y,x−→
x
Δ
, Δx
t
1
, ,Δx
0,
ω
2
T
,
N : X −→ Y,
Nx
⎛
⎜
⎜
⎜
⎝
⎛
⎜
⎜
⎜
⎝
A
1
t
.
.
.
A
n
t
1
⎞
⎟
⎟
⎟
⎠
, ,
⎛
⎜
⎜
⎜
⎝
I
1q
x
1
t
q
.
.
.
I
nq
t
b
i
t, x
i
t
f
i
t, x
t
,i 1, 2, ,n.
3.6
It is easy to see that
Ker L
{
0
}
, Im L
z
f, C
f, C
1
, ,C
q
1
ω
ω
0
f
s
Δs, 0, ,0
,
3.9
8 Journal of Inequalities and Applications
respectively. It is not difficult to show that P and Q are continuous projectors such that
Im P Ker L, Im L Ker Q Im
I − Q
. 3.10
Further, let L
−1
P
f
s
Δs −
1
2
q
k1
C
k
,
3.11
in which C
qi
−C
i
for all 1 ≤ i ≤ q.
Similar to the proof of Theorem 3.1in38,itisnotdifficult to show that QN
Ω,
K
P
I − QNΩ are relatively compact for any open bounded set Ω ⊂ X. Therefore, N is
L-compact on
Ω for any open bounded set Ω ⊂ X.
Corresponding to the operator equation Lx − Nx λ−Lx − N−x,λ ∈ 0, 1,we
have
x
Δ
1
1 λ
I
k
x
t
k
−
λ
1 λ
I
k
−x
t
k
,t t
k
,k∈ N,
3.12
or
x
Δ
i
1
1 λ
I
ik
x
i
t
k
−
λ
1 λ
I
ik
−x
i
t
k
,t t
k
,i 1, 2, ,n, k ∈ N,
3.13
where
G
i
t, −x
a
i
t, −x
i
t
b
i
t, −x
i
t
f
i
t, −x
t
,i 1, 2, ,n.
2q1
k1
t
k
t
k−1
x
Δ
i
t
Δt
2q
k1
|
Δx
i
t
Δt
2q
k1
1
1 λ
I
ik
x
i
t
k
−
λ
1 λ
I
ik
−x
i
t
t, −x
|}
Δt
1
1 λ
λ
1 λ
2q
k1
max
{|
I
ik
x
i
t
k
|
,
|
I
i
t
f
i
t, x
t
,
a
i
t, −x
i
t
b
i
t, −x
i
,
|
I
ik
−x
i
t
k
|}
≤ a
M
i
ω
0
max
{|
b
i
t, x
i
t
− b
f
i
t, x
1t
, ,x
nt
− f
i
t, 0, ,0
,
f
i
t, −x
1t
, ,−x
nt
− f
i
t, 0, ,0
ik
−x
i
t
k
− I
ik
0
|}
2q
k1
|
I
ik
0
|
≤ a
M
i
⎡
⎣
⎦
2q
k1
L
I
ik
|
x
i
|
0
2q
k1
|
I
ik
0
|
≤ a
M
i
L
b
i
|
x
i
|
0
2q
k1
|
I
ik
0
|
,
3.15
where i 1, 2, ,n. Integrating 3.13 from0toω, we have from H
1
–H
4
that
ω
0
t, −x
i
t
1 λ
Δt
10 Journal of Inequalities and Applications
ω
0
a
i
t, x
i
t
ω
0
a
i
t, x
i
t
b
i
t, x
i
t
Δt
ω
0
a
i
t, −x
i
t
f
i
t, −x
t
Δt
1
1 λ
2q
k1
I
ik
x
i
t
f
i
t, x
1t
, ,x
nt
−
f
i
t, 0, ,0
,
f
i
t, −x
1t
, ,−x
nt
0
|
,
|
I
ik
−x
i
t
k
− I
ik
0
|}
2q
k1
|
I
ik
0
2q
k1
|
I
ik
0
|
,i 1, 2, ,n,
3.16
by H
2
,weobtainthat
ω
0
a
i
t, x
i
t
ω
1
μ
i
2q
k1
L
I
ik
|x
i
|
0
1
μ
i
2q
k1
|
I
ik
0
|
,
3.17
i
t, x
i
t
x
i
t
i
1
Δt
ω
0
a
i
t, x
i
t
ω
0
t
Δt ≥
ω
0
a
i
t, x
i
t
x
i
t
i
2
Δt −
ω
0
a
i
t, x
i
i
tΔt on the both sides of 3.18 and 3.19, respectively, we obtain that
x
i
t
i
1
≥
1
ω
0
a
i
t, x
i
t
Δt
ω
0
a
i
t, x
i
2
≤
1
ω
0
a
i
t, x
i
t
Δt
ω
0
a
i
t, x
i
t
x
i
i
t
i
max
t∈0,ω
T
x
i
t,x
i
t
i
min
t∈0,ω
T
x
i
t, by the arbitrariness
of t
i
1
,t
i
2
in view of 3.15, 3.17, 3.20, we have
x
i
t
i
t
Δt −
ω
0
x
Δ
i
t
Δt
≥−
1
ω
0
a
i
t, x
i
ω
0
x
Δ
i
t
Δt
≥−
1
ωa
m
i
⎡
⎣
1
μ
i
a
M
i
n
2q
k1
|
I
ik
0
|
⎤
⎦
−
⎡
⎣
a
M
i
L
b
i
√
ω
x
j
2
k1
|
I
ik
0
|
⎤
⎦
,
x
i
t
i
2
≤
1
ω
0
a
i
t, x
i
t
Δt
≤
1
ω
0
a
i
t, x
i
t
Δt
ω
0
a
i
Δt
≤
1
ωa
m
i
⎡
⎣
1
μ
i
a
M
i
c
i
n
j1
x
j
2
√
ω
1
⎡
⎣
a
M
i
L
b
i
√
ω
x
j
2
a
M
i
c
i
n
j1
x
j
|
x
i
|
0
max
t∈0,ω
T
|
x
i
t
|
≤
1
ωa
m
i
⎡
⎣
1
μ
i
a
M
i
c
i
2q
k1
|
I
ik
0
|
⎤
⎦
⎡
⎣
a
M
i
L
b
i
√
ω
x
i
2
a
M
|
I
ik
0
|
⎤
⎦
,
3.22
where i 1, 2, ,n. In addition, we have that
x
i
2
ω
0
|
x
i
s
|
Δs
x
i
|
0
≤
⎡
⎣
1
μ
i
a
M
i
c
i
n
j1
x
j
2
√
ω
1
μ
i
⎡
⎣
a
M
i
L
b
i
√
ω
x
i
2
a
M
i
c
i
n
j1
x
j
2
μ
i
a
M
i
ωc
i
n
j1
x
j
0
1
μ
i
2q
k1
L
I
ik
|
x
i
|
x
i
|
0
a
M
i
ωc
i
n
j1
x
j
0
2q
k1
L
I
ik
|
x
i
L
b
i
− ωa
m
i
2q
k1
L
I
ik
−
1
μ
i
2q
k1
L
I
ik
|x
i
|
0
−
1
0
|
ωa
m
i
2q
k1
|
I
ik
0
|
D
i
,i 1, 2, ,n.
3.25
Denote that,
|
x
|
0
|
x
1
|
0
≤ D. 3.27
From the conditions of Theorem 3.1, E is a nonsingular M matrix, therefore,
|
x
|
0
≤ E
−1
D M
1
,M
2
, ,M
n
T
.
3.28
Let
M
n
i1
M
i
1
Clearly,Mis independent of λ
t
T
is an ω/2-anti-periodic solution of system 1.1.
In this section, we will construct some suitable Lyapunov functions to study the global
exponential stability of this anti-periodic solution.
Theorem 4.1. Assume that H
1
–H
5
hold. Suppose further that
H
6
there exist positive constants L
a
i
such that
|
a
i
t, u
− a
i
t, v
|
≤ L
a
t, v
u − v
≤ 0,i 1, 2, ,n,
|
a
i
t, u
b
i
t, u
− a
i
t, v
b
i
t, v
|
≥ L
ab
i
t
−L
ab
i
L
a
i
r
M
i
n
j1
1 μ
t − θ
e
t − θ, t
a
t
k
, 0 <γ
ik
< 2,i 1, ,n, k ∈ N. 4.4
Then the ω/2-anti-periodic solution of system 1.1 is globally exponentially stable.
Proof. According to Theorem 3.1, w e know that system 1.1 has an ω/2-anti-periodic
solution x
∗
tx
∗
1
t,x
∗
2
t, ,x
∗
n
t
T
with initial value x
∗
s,s ∈ E
T
, suppose that
14 Journal of Inequalities and Applications
xtx
1
t,x
b
i
t, x
i
t
− a
i
t, x
∗
i
t
b
i
t, x
∗
i
t
a
,t∈ T
,t
/
t
k
,
Δ
x
i
t
k
− x
∗
i
t
k
−γ
ik
x
i
t
t, x
i
t
b
i
t, x
i
t
− a
i
t, x
∗
i
t
b
i
t, x
∗
i
t, x
∗
t
a
i
t, x
i
t
b
i
t, x
i
t
− a
i
t, x
∗
i
t
f
i
t, x
t
a
i
t, x
∗
i
t
f
i
t, x
t
− f
i
t, x
∗
i
x
i
t
− x
∗
i
t
L
a
i
r
M
i
x
i
t
− x
−L
ab
i
L
a
i
r
M
i
x
i
t
− x
∗
i
t
a
x
i
t
k
− x
∗
i
t
k
1 − γ
ik
x
i
1
t
n
i1
e
t, α
x
i
t
− x
∗
i
t
,
V
x
j
s
− x
∗
j
s
Δs.
4.9
Journal of Inequalities and Applications 15
For t ∈ T
,t
/
t
k
,k ∈ N, calculating the delta derivative D
V t
Δ
n
i1
e
σ
t
,α
D
x
i
t
− x
∗
i
t
e
σ
t
,α
×
⎡
⎣
−L
ab
i
L
a
i
r
M
i
x
i
t
⎤
⎦
⎫
⎬
⎭
n
i1
1 μ
t
−L
ab
i
L
a
i
r
M
i
t, α
c
i
n
i1
n
j1
a
M
i
x
j
t θ
− x
∗
j
t θ
c
i
x
j
t
− x
∗
j
t
−
n
i1
n
j1
1 μ
t
4.11
By assumption H
8
, it concludes that
D
V t
Δ
D
V
1
t
Δ
D
V
2
t
Δ
≤
n
i1
1 μ
t
n
i1
n
j1
1 μ
t − θ
e
t − θ, α
a
M
i
c
i
x
a
i
r
M
i
n
j1
1 μ
t − θ
e
t − θ, t
a
M
i
c
i
e
k
V
1
t
k
V
2
t
k
n
i1
e
t
k
,α
k
θ
1 μ
s − θ
e
s − θ, α
a
M
i
c
i
x
j
s
− x
∗
j
t
k
n
i1
n
j1
t
k
t
k
θ
1 μ
s − θ
e
s − θ, α
a
.
On the other hand, we have
V
0
V
1
0
V
2
0
n
i1
e
0,α
x
i
s − θ, α
a
M
i
c
i
x
j
s
− x
∗
j
s
Δs
≤
n
i1
⎧
Δs
⎫
⎬
⎭
sup
s∈E
T
x
i
s
− x
∗
i
s
≤ M
n
i1
sup
α∈E
T
⎛
⎝
e
0,α
n
j1
0
θ
1 μ
s − θ
e
s − θ, α
a
M
i
≤ V
t
≤ V
0
≤ M
sup
s∈E
T
n
i1
φ
i
s
− x
∗
e
0,α
sup
s∈E
T
n
i1
φ
i
s
− x
∗
i
s
M
B
t, x
t
F
t, x
t
,t∈ T,t
/
t
k,
Δx
t
k
x
t
k
− x
t
−
π
arctan
|
u
|
,
B
t, u
1
100
u
u
,F
t, x
t
⎛
⎜
⎜
⎜
⎜
⎝
⎟
⎟
⎟
⎠
,
g
j
2×1
1
1000
sin u
sin u
,
c
i
2×1
1
1000
sin t
cos t
5.2
when T R,system5.1 has at least one exponentially stable π-anti-periodic solution.
Proof. By calculation, we have a
m
1
10,a
M
1
11,a
m
2
11,a
m
2
12,L
a
1
L
a
2
2/π, L
b
1
L
b
2
1/100,c
M
1
are satisfied. Furthermore, we can easily
calculate that
E ≈
7.52 −12.74
−11.25 3.61
5.3
is a nonsingular M matrix, thus H
5
is satisfied.
When T R,μt0. Take 0.01,θ −1, we have that
Ψ
1
, t
≈−0.04 < 0, Ψ
2
, t
≈−0.03 < 0. 5.4
18 Journal of Inequalities and Applications
Hence H
10
holds. By Theorems 3.1 and 4.1,system5.1 has at least one exponentially stable
π-anti-periodic solution. This completes the proof.
6. Conclusions
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