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Hindawi Publishing Corporation
Journal of Inequalities and Applications
Volume 2009, Article ID 461757, 13 pages
doi:10.1155/2009/461757
Research Article
A New Singular Impulsive Delay Differential
Inequality and Its Application
Zhixia Ma
1
and Xiaohu Wang
2
1
College of Computer Science and Technology, Southwest University for Nationalities,
Chengdu 610041, China
2
Yangtze Center of Mathematics, Sichuan University, Chengdu 610064, China
Correspondence should be addressed to Xiaohu Wang,
Received 10 January 2009; Accepted 5 March 2009
Recommended by Wing-Sum Cheung
A new singular impulsive delay differential inequality is established. Using this inequality, the
invariant and attracting sets for impulsive neutral neural networks with delays are obtained. Our
results can extend and improve earlier publications.
Copyright q 2009 Z. Ma and X. Wang. 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
It is well known that inequality technique is an important tool for investigating dynamical
behavior of differential equation. The significance of differential and integral inequalities
in the qualitative investigation of various classes of functional equations has been fully
illustrated during the last 40 years 1–3. Various inequalities have been established such
as the delay integral inequality in 4,thedifferential inequalities in 5, 6, the impulsive
differential inequalities in 7–10, Halanay inequalities in 11–13, and generalized Halanay
inequalities in 14–17. By using the technique of inequality, the invariant and attracting sets
for differential systems have been studied by many authors 9, 18–21.
However, the inequalities mentioned above are ineffective for studying the invariant
and attracting sets of impulsive nonautonomous neutral neural networks with time-
varying delays. On the basis of this, this article is devoted to the discussion of this
problem.
Motivated by the above discussions, in this paper, a new singular impulsive
delay differential inequality is established. Applying this equality and using the meth-
ods in 10, 22, some sufficient conditions ensuring the invariant set and the global
attracting set for a class of neutral neural networks system with impulsive effects are
obtained.
2 Journal of Inequalities and Applications
2. Preliminaries
Throughout the paper, E
n
means n-dimensional unit matrix, Rthe set of real numbers, N
the set of positive integers, and N
Δ
{1, 2, ,n}. A ≥ B A>B means that each pair of
corresponding elements of A and B satisfies the inequality “≥ >”. Especially, A is called a
nonnegative matrix if A ≥ 0.
CX, Y denotes the space of continuous mappings from the topological space X to the
topological space Y. In particular, let C
Δ
C−τ, 0, R
n
, where τ>0 is a constant.
PCa, b, R
n
denotes the space of piecewise continuous functions ψs : a, b → R
n
with at most countable discontinuous points and at this points ψs are right continuous.
Especially, let PC
Δ
PC−τ,0, R
n
. Furthermore, put PCa, b, R
n
c∈a,b
PCa, c, R
n
.
PC
1
a, b, R
n
{ψs : a, b → R
n
| ψs, ˙ψs ∈ PCa, b, R
n
}, where ˙ψs
denotes the derivative of ψs. In particular, let PC
1
Δ
PC
1
a, b, R
n
.
H {ht : R → R | ht is a positive integrable function and satisfies
sup
a≤t<b
t
t−τ
hsds σ<∞ and lim
t →∞
t
a
hsds ∞}.
For x ∈ R
n
,A ∈ R
n×n
, ϕ ∈ C or ϕ ∈ PC, we define x
|x
1
|, ,|x
n
|
T
, A
|a
ij
|
n×n
, ϕt
τ
ϕ
1
t
τ
, ,ϕ
n
t
τ
T
, ϕt
τ
ϕt
τ
, ϕ
i
t
τ
sup
−τ≤θ<0
{ϕ
i
t
θ}. And we introduce the following norm, respectively,
x max
1≤i≤n
x
i
, A max
1≤i≤n
n
j1
a
ij
, ϕ
τ
sup
−τ≤s≤0
ϕs
. 2.1
For any ϕ ∈ PC
1
, we define the following norm:
ϕ
1τ
max
ϕ
τ
,
ϕ
τ
. 2.2
For an M-matrix D defined in 23, we denote
Ω
M
D
z ∈ R
n
| Dz > 0,z>0
. 2.3
It is a cone without conical surface in R
n
. We call it an “M-cone”.
3. Singular Impulsive Delay Differential Inequality
For convenience, we introduce the following conditions.
C
1
Let the r-dimensional diagonal matrix K diag{k
1
, ,k
r
} satisfy
k
i
> 0,i∈ S ⊂N
∗
Δ
{1, ,r}, k
i
0,i∈ S
∗
Δ
N
∗
− S. 3.1
C
2
Let U −P Q be an M-matrix, where Q q
ij
r×r
≥ 0andP p
ij
r×r
satisfies
p
ij
≥ 0,i
/
j, p
ij
0,i
/
j, i ∈N
∗
,j∈ S
∗
. 3.2
Journal of Inequalities and Applications 3
Theorem 3.1. Assume the conditions C
1
and C
2
hold. Let L L
1
, ,L
r
and ut
u
1
t, ,u
r
t
T
be a solution of the following singular delay differential inequality with the initial
conditions ut ∈ PCa − τ, a, R
r
:
KD
ut ≤ ht
PutQ
ut
τ
L
,t∈ a, b, 3.3
where τ>0,a<b≤ ∞, and u
i
t ∈ Ca, b, R,i∈ S, u
i
t ∈ PCa, b, R,i∈ S
∗
, ht ∈H.
Then
ut ≤ dze
−λ
t
a
hsds
− P Q
−1
L, t ∈ a, b, 3.4
provided that the initial conditions satisfy
ut ≤ dze
−λ
t
a
hsds
− P Q
−1
L, t ∈ a − τ,a, 3.5
where d ≥ 0, z z
1
, ,z
r
T
∈ Ω
M
U and the positive number λ satisfies the following inequality:
λK P Qe
λσ
z<0,t∈ a, b. 3.6
Proof. By the conditions C
2
and the definition of M-matrix, there is a constant vector z
z
1
, ,z
r
T
such that P Qz<0, −P Q
−1
exists and −P Q
−1
≥ 0.
By using continuity, we obtain that there must exist a positive constant λ satisfying the
inequality 3.6,thatis,
r
j1
p
ij
q
ij
e
λσ
z
j
< −λk
i
z
i
,i∈N
∗
. 3.7
Denote by
vt
v
1
t, ,v
r
t
T
utP Q
−1
L, t ∈ a − τ, b. 3.8
It follows from 3.3 and 3.5 that
KD
vt ≤ ht
PutQ
ut
τ
L
≤ ht
PvtQ
vt
τ
,t∈ a, b,
vt ≤ dze
−λ
t
a
hsds
,t∈ a − τ, a.
3.9
In the following, we will prove that for any positive constant ε,
v
i
t ≤ d εz
i
e
−λ
t
a
hsds
ω
i
t,t∈ a, b,i∈N
∗
. 3.10
4 Journal of Inequalities and Applications
Let
℘
i ∈N
∗
| v
i
t >w
i
t for some t ∈ a, b
,
θ
i
inf
t ∈ a, b | v
i
t >w
i
t,i∈ ℘
.
3.11
If inequality 3.10 is not true, then ℘ is a nonempty set and there must exist some integer
m ∈ ℘ such that θ
m
min
i∈℘
{θ
i
}∈a, b.
If m ∈ S,byv
m
t ∈ Ca, b, R and the inequality 3.5, we can get
θ
m
>a, v
m
θ
m
w
m
θ
m
,D
v
m
θ
m
≥ ˙w
m
θ
m
, 3.12
v
i
t ≤ w
i
t,t∈ a − τ, θ
m
,i∈N
∗
,v
i
θ
m
≤ w
i
θ
m
,i∈ S. 3.13
By using C
2
, 3.3, 3.7, 3.12, 3.13,andv
i
t
τ
sup
−τ≤θ<0
{u
i
t θ},i∈N
∗
,weobtain
that
k
m
D
v
m
θ
m
≤ h
θ
m
r
j1
p
mj
v
j
θ
m
q
mj
v
j
θ
m
τ
h
θ
m
p
mm
v
m
θ
m
j
/
m, j∈S
p
mj
v
j
θ
m
j∈S
∗
p
mj
v
j
θ
m
j1
q
mj
v
j
θ
m
τ
≤ hθ
m
j∈S
p
mj
d εz
j
e
−λ
θ
m
a
hsds
r
j1
q
mj
d εz
j
e
−λ
θ
m
−τ
a
hsds
≤ d εh
θ
m
r
j1
p
mj
q
mj
e
λσ
z
j
e
−λ
θ
m
a
hsds
< −d ελk
m
z
m
h
θ
m
e
−λ
θ
m
a
hsds
.
3.14
Since m ∈ S, we have k
m
> 0byH
1
. Then 3.14 becomes
D
u
m
θ
m
< −d ελz
m
h
θ
m
e
−λ
θ
m
a
rsds
˙w
m
θ
m
, 3.15
which contradicts the second inequality in 3.12.
If m ∈ S
∗
, then k
m
0byC
1
and v
m
t ∈ PCa, b, R. From the inequality 3.5,we
can get
θ
m
>a, v
m
θ
m
≥ w
m
θ
m
,v
i
θ
m
≤ w
i
θ
m
,i∈ S,
v
i
t ≤ w
i
t,t∈ a − τ, θ
m
,i∈N
∗
.
3.16
Journal of Inequalities and Applications 5
By using C
2
, 3.3, 3.7, 3.16,andv
i
t
τ
sup
−τ≤θ<0
{v
i
t θ},i∈N
∗
,weobtainthat
0 ≤
r
j1
p
mj
v
j
θ
m
j1
q
mj
v
j
θ
m
τ
j∈S
p
mj
v
j
θ
m
p
mm
v
m
θ
m
j
/
m, j∈S
∗
p
mj
v
j
θ
m
r
j1
q
mj
v
j
θ
m
τ
≤ d ε
j∈S
p
mj
z
j
p
mm
z
m
j1
q
mj
z
j
e
λ
θ
m
θ
m
−τ
hsds
e
−λ
θ
m
a
hsds
≤ d ε
r
j1
p
mj
z
j
q
mj
z
j
e
λσ
e
−λ
θ
m
a
hsds
< −d εk
m
z
m
h
θ
m
e
−λ
θ
m
a
hsds
0.
3.17
This is a contradiction. Thus the inequality 3.10 holds. Therefore, letting ε → 0in3.10,
we have
vtutP Q
−1
L ≤ dze
−λ
t
a
hsds
,t∈ a, b. 3.18
The proof is complete.
Remark 3.2. In order to overcome the difficulty that ut in 3.3 may be discontinuous, we
introduce the notation u
i
t
τ
sup
−τ≤s<0
{u
i
t s} which is different from the notation
u
i
t
τ
sup
−τ≤s≤0
{u
i
t s} in 7. However, when u
i
t is continuous in t, we have
u
i
t
τ
sup
−τ≤s<0
u
i
t s
sup
−τ≤s≤0
u
i
t s
,i∈N
∗
. 3.19
So we can get 7, Lemma 1 when we choose K E
r
, S N
∗
, ht ≡ 1inTheorem 3.1.
Remark 3.3. Suppose that L L
1
, ,L
r
T
0andht ≡ 1inTheorem 3.1, then we can get
10, Theorem 3.1.
4. Applications
The singular impulsive delay differential inequality obtained in Section 3 can be widely
applied to study the dynamics of impulsive neutral differential equations. To illustrate the
theory, we consider the following nonautonomous impulsive neutral neural networks with
delays
˙xt−DtxtAtF
xt
BtG
x
t − τt
CtH
˙x
t − rt
Jt,t
/
t
k
,
xtI
k
t, x
t
−
,t t
k
,
4.1
6 Journal of Inequalities and Applications
where x x
1
, ,x
n
T
∈ R
n
is the neural state vector; Dtdiag{d
1
t, ,d
n
t} >
0,Ata
ij
t
n×n
, Btb
ij
t
n×n
, Ctc
ij
t
n×n
are the interconnection matrices
representing the weighting coefficients of the neurons; Fxf
1
x
1
, ,f
n
x
n
T
,Gx
g
1
x
1
, ,g
n
x
n
T
,Hxh
1
x
1
, ,h
n
x
n
T
are activation functions; τt
τ
ij
t
n×n
,rtr
ij
t
n×n
are transmission delays; JtJ
1
t, ,J
n
t
T
denotes
the external inputs at time t. I
k
t, yI
1k
t, y, ,I
nk
t, y
T
represents impulsive
perturbations; the fixed moments of time t
k
satisfy t
k
<t
k1
, lim
k → ∞
t
k
∞,k∈ N.
The initial condition for 4.1 is given by
x
t
0
s
ϕs ∈ PC
1
,t
0
∈ R, − τ ≤ s ≤ 0. 4.2
We always assume that for any ϕ ∈ PC
1
, 4.1 has at least one solution through t
0
,ϕ,
denoted by xt, t
0
,ϕ or x
t
t
0
,ϕsimply xt or x
t
if no confusion should occur.
Definition 4.1. The set S ⊂ PC
1
is called a positive invariant set of 4.1, if for any initial value
ϕ ∈ S, we have the solution x
t
t
0
,ϕ ∈ S for t ≥ t
0
.
Definition 4.2. The set S ⊂ PC
1
is called a global attracting set of 4.1, if for any initial value
ϕ ∈ PC
1
,thesolutionx
t
t
0
,ϕ converges to S as t → ∞.Thatis,
distx
t
,S −→ 0ast −→ ∞, 4.3
where distφ, Sinf
ψ∈S
distφ, ψ,distφ, ψsup
s∈−τ,0
|φs − ψs|, for φ ∈ PC
1
.
Throughout this section, we suppose the following.
H
1
Dt ∈ PCR, R
n
,At,Bt,Ct,τt,rt are continuous. Moreover, 0 ≤ τ
ij
t ≤ τ
and 0 <r
ij
t ≤ τ i, j ∈N.
H
2
There exist nonnegative matrices
D
1
diag{
d
11
, ,
d
1n
},
D
2
diag{
d
21
, ,
d
2n
},
J
J
1
, ,
J
n
T
, hs ∈Hand a constant δ>0 such that
D
1
ht ≤ Dt ≤
D
2
ht, 0 <ht ≤
1
δ
,
Jt
≤
Jht. 4.4
H
3
There exist nonnegative matrices
A a
ij
n×n
,
B
b
ij
n×n
,
C c
ij
n×n
such that
At
≤
Aht,
Bt
≤
Bht,
Ct
≤
Cht. 4.5
H
4
There exist nonnegative matrices
F diag{α
1
, ,α
n
},
G diag{β
1
, ,β
n
},
H
diag{γ
1
, ,γ
n
} such that for all u ∈ R
n
the activating functions F·,G· and H·
satisfy
Fu
≤
Fu
,
Gu
≤
Gu
,
Hu
≤
Hu
. 4.6
Journal of Inequalities and Applications 7
H
5
There exists nonnegative matrix
I
k
I
k
ij
n×n
, such that for all u ∈ R
n
, i ∈Nand
k ∈ N
I
k
t, u
≤
I
k
u
. 4.7
H
6
Denote by
U
A
F,
V
B
G,
W
C
H,
K
E
n
0
00
diag
k
1
, ,
k
2n
,
L
J,
J
T
,
P
−
D
1
U 0
D
2
U −δE
n
Δ
p
ij
t
2n×2n
, Q
V
W
V
W
Δ
q
ij
t
2n×2n
,
4.8
and let
D −P Q be an M-matrix, and −P Q
−1
L
1
,
2
T
≥ 0,
1
,
2
∈
R
n
.
H
7
There exists a constant ν such that
ln η
k
≤ ν
t
k
t
k−1
hsds, k ∈ N,μ
∞
k1
ln μ
k
< ∞, 4.9
where ν<λ, and the scalar λ>0 is determined by the inequality
λ
K P Qe
λσ
z
∗
< 0, 4.10
where z
∗
z
1
, ,z
2n
T
∈ Ω
M
D,and
η
k
,μ
k
≥ 1,η
k
z
∗
x
≥
I
k
z
∗
x
,μ
k
1
≥
I
k
1
,k∈ N,z
∗
x
z
1
, ,z
n
T
. 4.11
Theorem 4.3. Assume that H
1
–H
7
hold. Then S {φ ∈ PC
1
| φ
τ
≤ e
μ
1
} is a global
attracting set of 4.1.
Proof. Denote ˙xtyt.Letsgn· be the sign function. For x x
1
, ,x
n
T
, define
Sgnxdiag{sgnx
1
, ,sgnx
n
}.
Calculating the upper right derivative D
xt
along system 4.1.From4.1, H
2
and H
3
we have
D
xt
≤ ht
− D
1
U
xt
VG
xt
τ
W
yt
τ
J
,t∈
t
k−1
,t
k
,t≥ σ, k ∈ N.
4.12
8 Journal of Inequalities and Applications
On the other hand, from 4.1 and H
2
–H
4
, we have
0 ≤ ht
−
1
ht
yt
D
2
U
xt
V
xt
τ
W
yt
τ
J
≤ ht
− δ
yt
D
2
Uxt
V
xt
τ
W
yt
τ
J
,t∈ t
k−1
,t
k
,k∈ N.
4.13
Let
utxt,yt
T
∈ R
2n
, 4.14
then from 4.12–4.14 and H
6
, we have
KD
ut
≤ ht
P
ut
Q
ut
τ
L
,t∈ t
k−1
,t
k
,k∈ N. 4.15
By the conditions H
6
and the definition of M-matrix, we may choose a vector z
∗
z
1
, ,z
2n
T
∈ Ω
M
D such that
Dz
∗
−P Qz
∗
> 0. 4.16
By using continuity, we obtain that there must be a positive constant λ satisfying the
inequality 4.10.Letz
∗
x
z
1
, ,z
n
T
and z
∗
y
z
n1
, ,z
2n
T
, then z
∗
z
∗
x
,z
∗
y
T
. Since
z
∗
> 0, denote
d
1
min
1≤i≤2n
z
i
, 4.17
then dz
∗
≥ e
2n
1, ,1
T
∈ R
2n
. From the property of M-cone, we have, dz
∗
∈ Ω
M
D.
For the initial conditions xt
0
sϕs, s ∈ −τ, 0, where ϕ ∈ PC
1
and t
0
∈ R no
loss of generality, we assume t
0
≤ t
1
,andt ∈ t
0
− τ, t
0
, we can get
xt
≤
ϕt
τ
e
−λ
t
t
0
hsds
dz
∗
x
,
yt
≤
ϕ
t
τ
e
−λ
t
t
0
hsds
dz
∗
y
,
4.18
Then 4.18 yield
ut
≤ dz
∗
ϕ
1τ
e
−λ
t
t
0
hsds
, t
0
− τ ≤ t ≤ t
0
. 4.19
Let N
∗
{1, ,2n}, S {1, ,n} N and S
∗
{n 1, ,2n} N
∗
− S.Thus,all
conditions of Theorem 3.1 are satisfied. By Theorem 3.1, we have
ut
≤ dz
∗
ϕ
1τ
e
−λ
t
t
0
hsds
, t
0
≤ t<t
1
. 4.20
Journal of Inequalities and Applications 9
Suppose that for all m 1, ,k, the inequalities
ut
≤
m−1
j0
η
j
dz
∗
ϕ
1τ
e
−λ
t
t
0
hsds
m−1
j0
μ
j
, t
m−1
≤ t<t
m
,t≥ t
0
, 4.21
hold, where η
0
μ
0
1.
From 4.21, H
5
,andH
7
, we can get
x
t
k
≤
I
k
x
t
−
k
≤
k
j0
η
j
dz
∗
x
ϕt
1τ
e
−
t
k
t
0
hsds
k
j0
μ
j
1
.
4.22
Since −
P Q
−1
L, we have
D
2
U
V
1
W
2
J δ
2
. 4.23
On the other hand, it follows from H
7
that
D
2
U
z
∗
x
Vz
∗
x
Wz
∗
y
e
λσ
<δz
∗
y
. 4.24
Then from 4.21–4.24, we have
y
t
k
≤
k
j0
η
j
dz
∗
y
ϕ
1τ
e
−λ
t
k
t
0
hsds
k
j0
μ
j
2
, 4.25
which together with 4.22 yields that
u
t
k
≤
k
j0
η
j
dz
∗
ϕ
1τ
e
−λ
t
k
t
0
hsds
k
j0
μ
j
. 4.26
Then, it follows from 4.21 and 4.26 that
u
t
≤
k
j0
η
j
dz
∗
ϕ
1τ
e
−λ
t
t
0
hsds
k
j0
μ
j
k
j0
η
j
dz
∗
ϕ
1τ
e
−λ
t
k
t
0
hsds
e
−λ
t
t
k
hsds
k
j0
μ
j
, ∀ t ∈
t
k
− τ, t
k
.
4.27
10 Journal of Inequalities and Applications
Using Theorem 3.1 again, we have
ut
≤
k
j0
η
j
dz
∗
ϕ
1τ
e
−λ
t
k
t
0
hsds
e
−λ
t
t
k
hsds
k
j0
μ
j
k
j0
η
j
dz
∗
ϕ
1τ
e
−λ
t
t
0
hsds
k
j0
μ
j
, t
k
≤ t<t
k1
.
4.28
By mathematical induction, we can conclude that
ut
≤
k
j0
η
j
dz
∗
ϕ
1τ
e
−λ
t
t
0
hsds
k
j0
μ
j
, t
k
≤ t<t
k1
,k∈ N. 4.29
Noticing that η
k
≤ e
ν
t
k
t
k−1
hsds
,byH
7
, we can use 4.29 to conclude that
ut
≤ dz
∗
ϕ
1τ
e
ν
t
t
0
hsds
e
−λ
t
t
0
hsds
e
μ
dz
∗
ϕ
1τ
e
−λ−ν
t
t
0
hsds
e
μ
, t
k−1
≤ t<t
k
,k∈ N.
4.30
This implies that the conclusion of the theorem holds.
By using Theorem 4.3 with d 0, we can obtain a positive invariant set of 4.1,and
the proof is similar to that of Theorem 4.3.
Theorem 4.4. Assume that H
1
–H
7
with
I
k
E
n
hold. Then S {φ ∈ PC
1
| φ
τ
≤
1
} is a
positive invariant set and also a global attracting set of 4.1.
Remark 4.5. Suppose that c
ij
≡ 0,i,j∈Nin H
5
,andht ≡ 1, then we can get Theorems 1
and 2 in 9.
Remark 4.6. If I
k
t, xt
−
x ∈ R
n
then 4.1 becomes the nonautonomous neutral neural
networks without impulses, we can get Theorem 4.1 in 22.
5. Illustrative Example
The following illustrative example will demonstrate the effectiveness of our results.
Example 5.1. Consider nonlinear impulsive neutral neural networks:
˙x
1
t−
7 cos
2
t
x
1
tsin t tan
x
1
t − τ
11
t
−
1
4
cos t
˙x
2
t − r
12
t
− 1.5 cos t,
˙x
2
t−
6 sin
2
t
x
2
t − 2 cos t
x
2
t − τ
22
t
1
4
sin t tan
˙x
1
t − r
21
t
2.5sint,
t
/
k,
5.1
Journal of Inequalities and Applications 11
with
x
1
t I
1
t, x
t
−
,
t k, k ∈ N,
x
2
t I
2
t, x
t
−
,
5.2
where τ
ij
t1/4| cosi jt|≤1/4
Δ
τ, r
ij
t1/4 − 1/8| sini jt|, i, j 1, 2,
I
k
t, xa
1k
tx
1
b
1k
tx
2
,a
2k
tx
1
b
2k
tx
2
T
,k∈ N.
The parameters of conditions H
3
–H
9
are as follows:
htδ 1,
D
1
diag{7, 6},
D
2
diag{8, 7},σ
1
4
,
J1.5, 2.5
T
,
U
00
00
,
V
10
02
,
W
1
4
01
10
,
K
⎛
⎜
⎜
⎝
1000
0100
0000
0000
⎞
⎟
⎟
⎠
,
P
−
D
1
U 0
D
2
U −δE
2
⎛
⎜
⎜
⎝
−70 0 0
0 −60 0
80−10
070−1
⎞
⎟
⎟
⎠
,
Q
V
W
V
W
⎛
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎝
100
1
4
02
1
4
0
100
1
4
02
1
4
0
⎞
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎠
,
D −
P Q
⎛
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎝
600−
1
4
04−
1
4
0
−90 1 −
1
4
0 −9 −
1
4
1
⎞
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎠
.
5.3
It is easy to prove that
D is an M-matrix and
Ω
M
D
z
1
,z
2
,z
3
,z
4
T
> 0 | 9z
2
1
4
z
3
<z
4
< 24z
1
, 9z
1
1
4
z
4
<z
3
< 16z
2
. 5.4
Let z
∗
1, 1, 15, 20
T
, then z
∗
∈ Ω
M
D and z
∗
x
1, 1
T
.Letλ 0.1 which satisfies the
inequality
λ
K P Qe
λσ
z
∗
< 0. 5.5
12 Journal of Inequalities and Applications
Now, we discuss the asymptotical behavior of the system 5.1 as follows.
i If a
1k
tb
2k
t0,b
1k
t1/2e
1/5
2k
1 sin t,a
2k
t1/2e
1/5
2k
1 − cos t,
then
I
k
e
1/5
2k.
01
10
. 5.6
Thus η
k
μ
k
e
1/5
2k
≥ 1, ln η
k
e
1/5
2k
≤ 0.04, ν 0.04 <λ,andμ 1/24. Clearly,
all conditions of Theorem 4.3 are satisfied, by Theorem 4.3, S {φ ∈ PC
1
| φ
τ
≤
e
1/24
1
≈ e
1/24
1.196, 1.746
T
} is a global attracting set of 5.1.
ii If a
1k
tcos t, b
2k
tsin t, b
1k
ta
2k
t0, then
I
k
E
2
.ByTheorem 4.4,
S {φ ∈ PC
1
| φ
τ
≤
1
≈ 1.196, 1.746
T
} is a positive invariant set of 5.1.
Acknowledgments
This work was supported by the National Natural Science Foundation of China under Grant
no. 10671133 and the Scientific Research Fund of Sichuan Provincial Education Department
08ZA044.
References
1 W. Walter, Differential and Integral Inequalities, Springer, New York, NY, USA, 1970.
2 V. Lakshmikantham and S. Leela, Differential and Integral Inequalities: Theory and Applications. Vol. I:
Ordinary Differential Equations, vol. 55 of Mathematics in Science and Engineering, Academic Press, New
York, NY, USA, 1969.
3 V. Lakshmikantham and S. Leela, Differential and Integral Inequalities: Theory and Applications. Vol. II:
Functional, Partial, Abstract, and Complex Differential Equations, vol. 55 of Mathematics in Science and
Engineering, Academic Press, New York, NY, USA, 1969.
4 D. Y. Xu, “Integro-differential equations and delay integral inequalities,” Tohoku Mathematical Journal,
vol. 44, no. 3, pp. 365–378, 1992.
5 L. Wang and D. Y. Xu, “Global exponential stability of Hopfield reaction-diffusion neural networks
with time-varying delays,” Science in China. Series F, vol. 46, no. 6, pp. 466–474, 2003.
6 Y. M. Huang, D. Y. Xu, and Z. G. Yang, “Dissipativity and periodic attractor for non-autonomous
neural networks with time-varying delays,” Neurocomputing, vol. 70, no. 16–18, pp. 2953–2958, 2007.
7 D. Y. Xu and Z. Yang, “Impulsive delay differential inequality and stability of neural networks,”
Journal of Mathematical Analysis and Applications, vol. 305, no. 1, pp. 107–120, 2005.
8 D. Y. Xu, W. Zhu, and S. Long, “Global exponential stability of impulsive integro-differential
equation,” Nonlinear Analysis: Theory, Methods & Applications, vol. 64, no. 12, pp. 2805–2816, 2006.
9 D. Y. Xu and Z. Yang, “Attracting and invariant sets for a class of impulsive functional differential
equations,” Journal of Mathematical Analysis and Applications , vol. 329, no. 2, pp. 1036–1044, 2007.
10 D. Y. Xu, Z. Yang, and Z. Yang, “Exponential stability of nonlinear impulsive neutral differential
equations with delays,” Nonlinear Analysis: Theory, Methods & Applications, vol. 67, no. 5, pp. 1426–
1439, 2007.
11 R. D. Driver,
Ordinary and Delay Differential Equations, vol. 20 of Applied Mathematical Sciences,Springer,
New York, NY, USA, 1977.
12 K. Gopalsamy, Stability and Oscillations in Delay Differential Equations of Population Dynamics, vol. 74 of
Mathematics and Its Applications, Kluwer Academic Publishers, Dordrecht, The Netherlands, 1992.
13 A. Halanay, Differential Equations: Stability, Oscillations, Time Lags, Academic Press, New York, NY,
USA, 1966.
14 T. Amemiya, “Delay-independent stabilization of linear systems,” International Journal of Control, vol.
37, no. 5, pp. 1071–1079, 1983.
Journal of Inequalities and Applications 13
15 E. Liz and S. Trofimchuk, “Existence and stability of almost periodic solutions for quasilinear delay
systems and the Halanay inequality,” Journal of Mathematical Analysis and Applications, vol. 248, no. 2,
pp. 625–644, 2000.
16 A. Ivanov, E. Liz, and S. Trofimchuk, “Halanay inequality, Yorke 3/2 stability criterion, and
differential equations with maxima,” Tohoku Mathematical Journal, vol. 54, no. 2, pp. 277–295, 2002.
17 H. Tian, “The exponential asymptotic stability of singularly perturbed delay differential equations
with a bounded lag,” Journal of Mathematical Analysis and Applications, vol. 270, no. 1, pp. 143–149,
2002.
18 K. N. Lu, D. Y. Xu, and Z. C. Yang, “Global attraction and stability for Cohen-Grossberg neural
networks with delays,” Neural Networks, vol. 19, no. 10, pp. 1538–1549, 2006.
19 D. Y. Xu, S. Li, X. Zhou, and Z. Pu, “Invariant set and stable region of a class of partial differential
equations with time delays,” Nonlinear Analysis: Real World Applications, vol. 2, no. 2, pp. 161–169,
2001.
20 D. Y. Xu and H Y. Zhao, “Invariant and attracting sets of Hopfield neural networks with delay,”
International Journal of Systems Science, vol. 32, no. 7, pp. 863–866, 2001.
21 H. Zhao, “Invariant set and attractor of nonautonomous functional differential systems,” Journal of
Mathematical Analysis and Applications, vol. 282, no. 2, pp. 437–443, 2003.
22 Q. Guo, X. Wang, and Z. Ma, “Dissipativity of non-autonomous neutral neural networks with time-
varying delays,” Far East Journal of Mathematical Sciences, vol. 29, no. 1, pp. 89–100, 2008.
23 A. Berman and R. J. Plemmons, Nonnegative Matrices in the Mathematical Sciences, Computer Science
and Applied Mathematic, Academic Press, New York, NY, USA, 1979.
