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Hindawi Publishing Corporation
Fixed Point Theory and Applications
Volume 2008, Article ID 407352, 11 pages
doi:10.1155/2008/407352
Research Article
Fixed Points and Stability in Neutral Stochastic
Differential Equations with Variable Delays
Meng Wu,
1
Nan-jing Huang,
1
and Chang-Wen Zhao
2
1
Department of Mathematics, Sichuan University, Chengdu, Sichuan 610064, China
2
College of Business and Management, Sichuan University, Chengdu, Sichuan 610064, China
Correspondence should be addressed to Nan-jing Huang,
Received 4 April 2008; Accepted 9 June 2008
Recommended by Tomas Dom
´
ınguez Benavides
We consider the mean square asymptotic stability of a generalized linear neutral stochastic
differential equation with variable delays by using the fixed point theory. An asymptotic mean
square stability theorem with a necessary and sufficient condition is proved, which improves and
generalizes some results due to Burton, Zhang and Luo. Two examples are also given to illustrate
our results.
Copyright q 2008 Meng Wu et a l. 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
Liapunov’s direct method has been successfully used to investigate stability properties of a
wide variety of differential equations. However, there are many difficulties encountered in
the study of stability by means of Liapunov’s direct method. Recently, Burton 1–4,Jung5,
Luo 6,andZhang7 studied the stability by using the fixed point theory which solved the
difficulties encountered in the study of stability by means of Liapunov’s direct method.
Up till now, the fixed point theory is almost used to deal with the stability for
deterministic differential equations, not for stochastic differential equations. Very recently, Luo
6 studied the mean square asymptotic stability for a class of linear scalar neutral stochastic
differential equations. For more details of the stability concerned with the stochastic differential
equations, we refer to 8, 9 and the references therein.
Motivated by previous papers, in this paper, we consider the mean square asymptotic
stability of a generalized linear neutral stochastic differential equation with variable delays by
using the fixed point theory. An asymptotic mean square stability theorem with a necessary
2 Fixed Point Theory and Applications
and sufficient condition is proved. Two examples is also given to illustrate our results. The
results presented in this paper improve and generalize the main results in 1, 6, 7.
2. Main results
Let Ω, F, {F
t
}
t≥0
,P be a complete filtered probability space and let Wt denote a one-
dimensional standard Brownian motion defined on Ω, F, {F
t
}
t≥0
,P such that {F
t
}
t≥0
is the
natural filtration of Wt.Letat,bt,
bt,ct,et,qt ∈ CR
,R, and τt,δt ∈ CR
,R
with t − τt →∞and t − δt →∞as t →∞.HereCS
1
,S
2
denotes the set of all continuous
functions φ : S
1
→S
2
with the supremum norm ·.
In 2003, Burton 1 studied the equation
x
t−btx
t − τt
2.1
and proved the following theorem.
Theorem A Burton 1. Suppose that τtr and there exists a constant α<1 such that
t
t−r
bs r
ds
t
0
bs r
e
−
t
s
burdu
s
s−r
bu r
du ds ≤ α 2.2
for all t ≥ 0 and
∞
0
bsds ∞. Then, for every continuous initial function φ : −r, 0 →R,the
solution xtxt, 0,φ of 2.1 is bounded and tends to zero as t →∞.
Recently, Zhang 7 studied the generalization of 2.1 as follows:
x
t−
n
j1
b
j
tx
t − τ
j
t
2.3
and obtained the following theorem.
Theorem B Zhang 7. Suppose that τ
j
is differential, the inverse function g
j
t of t − τ
j
t exists,
and there exists a constant α ∈ 0, 1 such that for t ≥ 0, lim inf
t→∞
t
0
Qsds > −∞ and
n
j1
t
t−τ
j
t
b
j
g
j
s
ds
t
0
e
−
t
s
Qudu
b
j
s
τ
j
s
ds
t
0
e
−
t
s
Qudu
Qs
s
s−τ
j
s
b
j
g
j
v
dv ds
≤ α,
2.4
where Qt
n
j1
b
j
g
j
t. Then the zero solution of 2.3 is asymptotically stable if and only if
t
0
Qsds →∞,ast →∞.
Very recently, Luo 6 considered the following neutral stochastic differential equation:
d
xt − qtx
t − τt
atxtbtx
t − τt
dt
ctxtetx
t − δt
dWt
2.5
and obtained the following theorem.
Meng Wu et al. 3
Theorem C Luo 6. Let τt be derivable. Assume that there exists a constant α ∈ 0, 1 and a
continuous function ht : 0, ∞ →
R such that for t ≥ 0, lim inf
t→∞
t
0
hsds > −∞ and
qt
t
t−τt
ashs
ds
t
0
e
−
t
s
hudu
a
s−τs
h
s−τs
1−τ
s
bs−qshs
ds
t
0
e
−
t
s
hudu
hs
s
s−τs
auhu
du ds
t
0
e
−2
t
s
hudu
cs
es
2
ds
1/2
≤ α.
2.6
Then the zero solution of 2.5 is mean square asymptotically stable if and only if
t
0
hsds →∞, as
t →∞.
Now, we consider the generalization of 2.5:
d
xt −
n
j1
q
j
tx
t − τ
j
t
n
j1
b
j
tx
t − τ
j
t
dt
n
j1
c
j
tx
t − δ
j
t
dWt, 2.7
with the initial condition
xsφs for s ∈
mt
0
,t
0
, 2.8
where φ ∈ Cmt
0
,t
0
,R, b
j
t,c
j
t,q
j
t ∈ CR
,R, τ
j
t,δ
j
t ∈ CR
,R
, t − τ
j
t →∞,
and t − δ
j
t →∞as t →∞and for each t
0
≥ 0,
m
j
t
0
min
inf
s − τ
j
s,s≥ t
0
, inf
s − δ
j
s,s≥ t
0
,
m
t
0
min
m
j
t
0
, 1 ≤ j ≤ n
.
2.9
Note that 2.7 becomes 2.5 for n 2, τ
1
t0,τ
2
tτt, b
1
tat,b
2
tbt, q
1
t
0,q
2
tqt, δ
1
t0,δ
2
tδt, c
1
tct, and c
2
tet. Thus, we know that 2.7
includes 2.1, 2.3,and2.5 as special cases.
Our aim here is to generalize Theorems B and C to 2.7.
Theorem 2.1. Suppose that τ
j
is differential, and there exist continuous functions h
j
t : 0, ∞ →R
for j 1 ···n and a constant α ∈ 0, 1 such that for t ≥ 0
i lim inf
t→∞
t
0
Hsds > −∞,
ii
n
j1
q
j
t
n
j1
t
t−τ
j
t
h
j
s
ds
n
j1
t
0
e
−
t
s
Hudu
h
j
s − τ
j
s
1 − τ
j
s
b
j
s−q
j
sHs
ds
n
j1
t
0
e
−
t
s
Hudu
Hs
s
s−τ
j
s
h
j
u
du ds 2
⎛
⎝
t
0
e
−2
t
s
Hudu
n
j1
c
j
s
2
ds
⎞
⎠
1/2
≤ α<1,
2.10
where Ht
n
j1
h
j
t.
4 Fixed Point Theory and Applications
Then the zero solution of 2.7 is mean square asymptotically stable if and only if
t
0
Hsds −→ ∞ as t −→ ∞. 2.11
Proof. For each t
0
,denotebyS the Banach space of all F-adapted processes ψt, ω : mt
0
, ∞×
Ω →
R which are almost surely continuous in t with norm
ψ
S
E
sup
s≥mt
0
ψs, ω
2
1/2
. 2.12
Moreover, we set ψt, ωφt for t ∈ mt
0
,t
0
and E|ψt, ω|
2
→0, as t →∞.
At first, we suppose that 2.11 holds. Define an operator P : S →S by Pxtφt for
t ∈ mt
0
,t
0
and for t ≥ t
0
,
Pxt
φt
0
−
n
j1
q
j
t
0
φ
t
0
− τ
j
t
0
−
n
j1
t
0
t
0
−τ
j
t
0
h
j
sφsds
e
−
t
t
0
Hudu
n
j1
q
j
tx
t − τ
j
t
n
j1
t
t−τ
j
t
h
j
sxsds
t
t
0
e
−
t
s
Hudu
n
j1
h
j
s − τ
j
s
1 − τ
j
s
b
j
s − q
j
sHs
x
s − τ
j
s
ds
−
t
t
0
e
−
t
s
Hudu
Hs
n
j1
s
s−τ
j
s
h
j
uxudu
ds
t
t
0
e
−
t
s
Hudu
n
j1
c
j
sx
s − δ
j
s
dWs :
5
i1
I
i
t.
2.13
Now, we show the mean square continuity of P on t
0
, ∞.Letx ∈ S, T
1
> 0, and let |r|
be sufficiently small. Then
E
Px
T
1
r
− Px
T
1
2
≤ 5
5
i1
E
I
i
T
1
r
− I
i
T
1
2
. 2.14
It is easy to verify that
E
I
i
T
1
r
− I
i
T
1
2
−→ 0, as r −→ 0,i 1, 2, 3, 4. 2.15
Meng Wu et al. 5
It follows from the last term I
5
in 2.13 that
E
I
5
T
1
r
− I
5
T
1
2
E
T
1
t
0
e
−
T
1
s
Hudu
e
−
T
1
r
T
1
Hudu
− 1
n
j1
c
j
sx
s − δ
j
s
dWs
T
1
r
T
1
e
−
T
1
r
s
Hudu
n
j1
c
j
sx
s − δ
j
s
dWs
2
≤ 2E
T
1
t
0
e
−2
T
1
s
Hudu
e
−
T
1
r
T
1
Hudu
− 1
2
n
j1
c
j
s
·
x
s − δ
j
s
2
ds
2E
T
1
r
T
1
e
−2
T
1
r
s
Hudu
n
j1
c
j
s
·
x
s − δ
j
s
2
ds −→ 0, as r −→ 0.
2.16
Therefore, P is mean square continuous on t
0
, ∞.
Next, we verify that Px ∈ S. Since E|xt|→0, t −δ
j
t →∞as t →∞,foreach>0, there
exists a T
1
>t
0
such that s ≥ T
1
implies E|xs|
2
<and E|xs − δ
j
s|
2
<. Thus, for t ≥ T
1
,
the last term I
5
in 2.13 satisfies
E
I
5
t
2
≤ E
T
1
t
0
e
−2
t
s
Hudu
n
j1
c
j
sx
s − δ
j
s
2
ds E
t
T
1
e
−2
t
s
Hudu
n
j1
c
j
sx
s − δ
j
s
2
ds
≤ E
sup
s≥mt
0
xs
2
T
1
t
0
e
−2
t
s
Hudu
n
j1
c
j
s
2
ds
t
T
1
e
−2
t
s
Hudu
n
j1
c
j
s
2
ds.
2.17
By condition ii and 2.11, there exists T
2
>T
1
such that t ≥ T
2
implies
E|I
5
t|
2
< α. 2.18
Thus, E|I
5
t|
2
→0, as t →∞. Similarly, we can show that E|I
i
t|
2
→0, i 1, 2, 3, 4, as t →∞.
Thus, E|Pxt|
2
→0ast →∞. This yields Px ∈ S.
Now we show that P : S →S is a contraction mapping. From ii, we can choose ε>0
such that α
2
ε<1. Thus, for each t
0
≥ 0, we can find a constant L>0 such that
1
1
L
n
j1
q
j
t
n
j1
t
t
0
e
−
t
s
Hudu
Hs
s
s−τ
j
s
h
j
u
du ds
n
j1
t
t−τ
j
t
h
j
s
ds
n
j1
t
t
0
e
−
t
s
Hudu
h
j
s−τ
j
s1−τ
j
s b
j
s−q
j
sHs
ds
2
41 L
t
t
0
e
−2
t
s
Hudu
n
j1
c
j
s
2
ds ≤ α
2
ε<1.
2.19
6 Fixed Point Theory and Applications
For any x, y ∈ S, it follows from 2.13, conditions i and ii, and Doob’s L
p
-inequality see
10 that
e sup
s≥mt
0
pxs − pys
2
e sup
s≥t
0
n
j1
q
j
s
x
s − τ
j
s
− y
s − τ
j
s
n
j1
s
s−τ
j
s
h
j
v
xv − yv
dv
s
t
0
e
−
s
v
hudu
n
j1
h
j
v − τ
j
v
1 − τ
j
v
b
j
v − q
j
vhv
×
x
v − τ
j
v
− y
v − τ
j
v
dv
−
s
t
0
e
−
s
v
hudu
hv
n
j1
v
v−τ
j
v
h
j
u
xu − yu
du
dv
s
t
0
e
−
s
v
hudu
n
j1
c
j
v
x
v − δ
j
v
− y
v − δ
j
v
dwv
2
≤
1
1
l
e sup
s≥t
0
n
j1
q
j
s
·
x
s − τ
j
s
− y
s − τ
j
s
n
j1
s
s−τ
j
s
h
j
v
·
xv − yv
dv
s
t
0
e
−
s
v
hudu
n
j1
h
j
v − τ
j
v
1 − τ
j
v
b
j
v − q
j
vhv
·
x
v − τ
j
v
− y
v − τ
j
v
dv
s
t
0
e
−
s
v
hudu
hv
n
j1
v
v−τ
j
v
h
j
u
·
xu − yu
du
dv
2
41 l sup
s≥t
0
e
s
t
0
e
−
s
v
hudu
n
j1
c
j
v
·
x
v − δ
j
v
− y
v − δ
j
v
2
dv
≤ e sup
s≥mt
0
xs − ys
2
·sup
s≥t
0
1
1
l
n
j1
q
j
s
n
j1
s
t
0
e
−
s
v
hudu
hv
v
v−τ
j
v
h
j
u
du ds
n
j1
s
s−τ
j
s
h
j
v
dv
n
j1
s
t
0
e
−
s
v
hudu
×
h
j
v − τ
j
v
1 − τ
j
v
b
j
v − q
j
vhv
dv
2
41 l
s
t
0
e
−2
s
v
hudu
n
j1
c
j
v
2
dv
≤
α
2
ε
e sup
s≥mt
0
xs − ys
2
.
2.20
Meng Wu et al. 7
Therefore, P is contraction mapping with contraction constant α
2
ε. By the contraction
mapping principle, P has a fixed point x ∈ S, which is a solution of 2.7 with xsφs
on mt
0
,t
0
and E|xt|
2
→0ast →∞.
To obtain the mean square asymptotic stability, we need to show that the zero solution
of 2.7 is mean square stable. Let >0 be given and choose δ>0andδ<satisfying the
following condition:
4δK
2
1 Le
2
t
0
0
Hudu
α
2
ε
<, 2.21
where K sup
t≥0
{e
−
t
0
Hsds
}.Ifxtxt, t
0
,φ is a solution of 2.7 with φ
2
<δ,then
xtPxt defined in 2.13. We assume that E|xt|
2
<for all t ≥ t
0
. Notice that E|xt|
2
φ
2
<for t ∈ mt
0
,t
0
.Ifthereexistst
∗
>t
0
such that E|xt
∗
|
2
and E|xt|
2
<for
t ∈ mt
0
,t
∗
,then2.13 and 2.19 imply that
E
x
t
∗
2
≤ 1 Lφ
2
1
n
j1
q
j
t
0
n
j1
t
0
t
0
−τ
j
t
0
h
j
s
ds
2
e
−2
t
∗
t
0
Hudu
1
1
L
n
j1
q
j
t
∗
n
j1
t
∗
t
∗
−τ
j
t
∗
h
j
s
ds
t
∗
t
0
e
−
t
∗
s
Hudu
n
j1
s
s−τ
j
s
h
j
u
du
Hs
ds
t
∗
t
0
e
−
t
∗
s
Hudu
n
j1
h
j
s − τ
j
s
1 − τ
j
s
b
j
s − q
j
sHs
ds
2
t
∗
t
0
e
−2
t
∗
s
Hudu
n
j1
c
j
s
2
ds
≤ 1 Lδ
1
n
j1
q
j
t
0
n
j1
t
0
t
0
−τ
j
t
0
h
j
s
ds
2
e
−2
t
∗
t
0
Hudu
α
2
ε
<,
2.22
which contradicts the definition of t
∗
. Thus, the zero solution of 2.7 is stable. It follows that
the zero solution of 2.7 is mean square asymptotically stable if 2.11 holds.
Conversely, we suppose that 2.11 fails. From i, there exists a sequence {t
n
} with
t
n
→∞as n →∞such that lim
n→∞
t
n
0
Hudu β,whereβ ∈ R. Then, we can choose a constant
J>0 satisfying
t
n
0
Hudu ∈ −J, J for all n ≥ 1. Denote
ωs
n
j1
h
j
s − τ
j
s
1 − τ
j
s
b
j
s − q
j
sHs
Hs
s
s−τ
j
s
h
j
u
du 2.23
for all s ≥ 0. From ii,wehave
t
n
0
e
−
t
n
s
Hudu
ωsds ≤ α, 2.24
8 Fixed Point Theory and Applications
which implies
t
n
0
e
s
0
Hudu
ωsds ≤ αe
t
n
0
Hudu
≤ e
J
. 2.25
Therefore, the sequence {
t
n
0
e
s
0
Hudu
ωsds} has a convergent subsequence. Without loss of
generality, we can assume that
lim
n→∞
t
n
0
e
s
0
Hudu
ωsds γ 2.26
for some γ>0. Let k be an integer such that
t
n
t
k
e
s
0
Hudu
ωsds <
δ
0
8K
2.27
for all n ≥ k,whereδ
0
> 0satisfies8δ
0
K
2
e
2J
α
2
ε < 1.
Now we consider the solution xtxt, t
k
,φ of 2.7 with φt
k
2
δ
0
and φs
2
<
δ
0
for s<t
k
. By the similar method in 2.22,wehaveE|xt|
2
< 1fort ≥ t
k
. We may choose φ
so that
Gt
k
: φt
k
−
n
j1
q
j
t
k
φ
t
k
− τ
j
t
k
−
n
j1
t
k
t
k
−τ
j
t
k
h
j
sφsds ≥
1
2
δ
0
. 2.28
It follows from 2.13 and 2.28 with xtPxt that for n ≥ k,
E
xt
n
−
n
j1
q
j
t
n
x
t
n
− τ
j
t
n
−
n
j1
t
n
t
n
−τ
j
t
n
h
j
sxsds
2
≥ G
2
t
k
e
−2
t
n
t
k
Hudu
− 2Gt
k
e
−
t
n
t
k
Hudu
t
n
t
k
e
−
t
n
s
Hudu
ωsds
≥
δ
0
2
e
−2
t
n
t
k
Hudu
δ
0
2
− 2K
t
n
t
k
e
s
0
Hudu
ωsds
≥
δ
2
0
8
e
−2J
> 0.
2.29
If the zero solution of 2.7 is mean square asymptotic stable, then E|xt|
2
E|xt, t
k
,φ|
2
→0ast →0. Since t
n
− τ
j
t
n
→∞, t
n
− δ
j
t
n
→∞ as n →∞ and condition ii
and 2.11 hold,
E
xt
n
−
n
j1
q
j
t
n
x
t
n
− τ
j
t
n
−
n
j1
t
n
t
n
−τ
j
t
n
h
j
sxsds
2
−→ 0, as n −→ ∞, 2.30
which contradicts 2.29. Therefore, 2.11 is necessary for Theorem 2.1. This completes the
proof.
Remark 2.2. Theorem 2.1 still holds if condition ii is satisfied for t ≥ t
a
for some t
a
∈ R
.
Meng Wu et al. 9
Remark 2.3. Theorem 2.1 improves Theorem C under different conditions.
Corollary 2.4. Suppose that τ
j
is differential, the inverse function g
j
t of t − τ
j
t exists, and there
exists a constant α ∈ 0, 1 such that for t ≥ 0, lim inf
t→∞
t
0
Qsds > −∞ and
n
j1
q
j
t
n
j1
t
t−τ
j
t
b
j
g
j
s
ds
n
j1
t
0
e
−
t
s
Qudu
b
j
sτ
j
s − q
j
sQs
ds
n
j1
t
0
e
−
t
s
Qudu
Qs
s
s−τ
j
s
b
j
g
j
u
du ds2
⎛
⎝
t
0
e
−2
t
s
Qudu
n
j1
c
j
s
2
ds
⎞
⎠
1/2
≤ α<1,
2.31
where Qt
n
j1
− b
j
g
j
t. Then the zero solution of 2.7 is mean square asymptotically stable if
and only if
t
0
Qsds →∞as t →∞.
Remark 2.5. When h
j
t−b
j
g
j
t for j 1 ···n, Theorem 2.1 reduces to Corollary 2.4.Onthe
other hand, we choose q
j
t ≡ c
j
t ≡ 0andb
j
≡−b
j
for j 1 ···n,thenCorollary 2.4 reduces
to Theorem B.
3. Two examples
In this section, we give two examples to illustrate applications of Theorem 2.1 and
Corollary 2.4.
Example 3.1. Consider the following linear neutral stochastic delay differential equation:
d
xt−
xt − t/2
1000
−
xt−t/2
1616t
−
3sin t4
4848t
x
t −
t
4
dt
xt
24
√
3 4t
−
xt−sin t
12
√
34t
dWt.
3.1
Then the zero solution of 3.1 is mean square asymptotically stable.
Proof. Choosing h
1
t1/8 16t and h
2
t7/48 64t in Theorem 2.1,wehave
Ht
1
8 16t
7
48 64t
,
11
48 64t
≤ Ht ≤
13
48 64t
,
2
j1
t
t−τ
j
t
h
j
s
ds
t
t/2
1
8 16s
ds
t
3t/4
7
48 64s
ds −→ 0.07479, as t −→ ∞,
2
j1
t
0
e
−
t
s
Hudu
Hs
s
s−τ
j
s
h
j
u
du ds ≤
t
0
e
−
t
s
11/4864udu
13
48 64s
·0.07479 ds ≤ 0.08839,
2
⎛
⎝
t
0
e
−2
t
s
Hudu
2
j1
c
j
s
2
ds
⎞
⎠
1/2
≤ 2
t
0
e
−
t
s
11/2432udu
1
824 32s
ds
1/2
≤ 0.21320,
2
j1
t
0
e
−
t
s
Hudu
h
j
s − τ
j
s1 − τ
j
s b
j
s − q
j
sHs
ds
≤
t
0
e
−
t
s
11/4864udu
0.013
48 64s
17
144 192s
ds ≤
0.013
11
17
33
0.51634.
3.2
10 Fixed Point Theory and Applications
It easy to check that
∞
0
Hsds ∞.Letα 0.001 0.07479 0.08839 0.21320 0.51634.
Then, α 0.89372 < 1andthezerosolutionof3.1 is mean square asymptotically stable by
Theorem 2.1.
Example 3.2. Consider the following delay differential equation:
x
t−
1
6 4t
x
t −
t
3
−
1
12 4t
x
t −
2
3
t
. 3.3
Then the zero solution of 3.3 is asymptotically stable.
Proof. Choosing h
1
th
2
t1/4 4t in Theorem 2.1,wehaveHt1/2 2t and
2
j1
t
t−τ
j
t
h
j
s
ds
t
2/3t
1
4 4s
ds
t
t/3
1
4 4s
ds −→
1
2
ln 3 −
1
4
ln 2 0.37602, as t −→ ∞,
2
j1
t
0
e
−
t
s
Hudu
Hs
s
s−τ
j
s
h
j
u
du ds ≤
t
0
e
−
t
s
1/22udu
1
2 2s
·0.37602 ds ≤ 0.37602.
3.4
Notice that q
j
tc
j
t ≡ 0and
2
j1
h
j
s − τ
j
s
1 − τ
j
s
b
j
s − q
j
sHs
3
12 8s
·
2
3
−
1
6 4s
3
12 4s
·
1
3
−
1
12 4s
0.
3.5
It is easy to see that all the conditions of Theorem 2.1 hold for α 0.376020.37602 0.75204 <
1. Thus, Theorem 2.1 implies that the zero solution of 3.3 is asymptotically stable.
However, Theorem B cannot be used to verify that the zero solution of 3.3 is
asymptotically stable. In fact,
b
1
t1/6 4t, b
2
t1/12 4t, b
1
g
1
t 1/6 6t,
b
2
g
2
t 1/12 12t,and|Qt| 1/4 4t.Ast →∞,
2
j1
t
t−τ
j
t
b
j
g
j
s
ds ≤
t
2/3t
1
6 6s
ds
t
t/3
1
12 12s
ds −→
1
4
ln 3 −
1
6
ln 2 0.15913.
3.6
Notice that
2
j1
b
j
sτ
j
s − q
j
sQs
1
18 12s
1
18 6s
≤
1
4 4s
. 3.7
It follows from 3.7 that
2
j1
t
0
e
−
t
s
Qudu
b
j
sτ
j
s − q
j
sQs
ds ≤
t
0
e
−
t
s
1/44udu
1
4 4s
ds ≤ 1. 3.8
Meng Wu et al. 11
From 3.6,weobtain
2
j1
t
0
e
−
t
s
Qudu
Qs
s
s−τ
j
s
b
j
g
j
u
du ds ≤
t
0
e
−
t
s
1/44udu
1
4 4s
·0.15913 ds ≤ 0.15913.
3.9
Combining 3.6, 3.8,and3.9, we see that the condition 2.4 of Theorem B does not hold
with α 1.31825.
Acknowledgement
This work was supported by the National Natural Science Foundation of China 10671135
and Specialized Research Fund for the Doctoral Program of Higher Education 20060610005.
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