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 Zhao2
1
2
Department of Mathematics, Sichuan University, Chengdu, Sichuan 610064, China
College of Business and Management, Sichuan University, Chengdu, Sichuan 610064, China
Correspondence should be addressed to Nan-jing Huang, nanjinghuang@hotmail.com
Received 4 April 2008; Accepted 9 June 2008
Recommended by Tomas Domı́nguez 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 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
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, Jung 5,
Luo 6, and Zhang 7 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, {Ft }t≥0 , P be a complete filtered probability space and let Wt denote a onedimensional standard Brownian motion defined on Ω, F, {Ft }t≥0 , P such that {Ft }t≥0 is the
natural filtration of Wt. Let at, bt, bt, ct, et, qt ∈ CR , R, and τt, δt ∈ CR , R with t − τt → ∞ and t − δt → ∞ as t → ∞. Here CS1 , S2 denotes the set of all continuous
functions φ : S1 → S2 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 τt r and there exists a constant α < 1 such that
t
bs rds t−r
t
t
bs re− s burdu
0
s
bu rdu ds ≤ α
2.2
s−r
∞
for all t ≥ 0 and 0 bsds ∞. Then, for every continuous initial function φ : −r, 0 → R, the
solution xt xt, 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
bj tx t − τj t
2.3
j
1
and obtained the following theorem.
Theorem B Zhang 7. Suppose that τj is differential, the inverse function gj t of t − τj t exists,
t
and there exists a constant α ∈ 0, 1 such that for t ≥ 0, lim inft→∞ 0 Qsds > −∞ and
n t
j
1
t−τj t
t
e
bj gj s ds t
− s Qudu 0
Qs
t
s
0
t
e− s Qudu bj sτj sds
s−τj s
bj gj v dv ds ≤ α,
2.4
where Qt nj
1 bj gj t. Then the zero solution of 2.3 is asymptotically stable if and only if
t
Qsds → ∞, as t → ∞.
0
Very recently, Luo 6 considered the following neutral stochastic differential equation:
d xt − qtx t − τt atxt btx t − τt dt ctxt etx 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
t a constant α ∈ 0, 1 and a
continuous function ht : 0, ∞ → R such that for t ≥ 0, lim inft→∞ 0 hsds > −∞ and
qt t
t
ashsds
t−τt
t
e− s hudu hs
t
e− s hudu a s−τs h s− τs 1−τ s bs−qshsds
0
s
0
t
au hudu ds t
s−τs
t
2
e−2 s hudu cs es ds
1/2
0
≤ α.
2.6
Then the zero solution of 2.5 is mean square asymptotically stable if and only if
t → ∞.
t
0
hsds → ∞, as
Now, we consider the generalization of 2.5:
d xt −
n
n
n
qj tx t − τj t bj tx t − τj t dt cj tx t − δj t dWt,
j
1
j
1
2.7
j
1
with the initial condition
xs φs
for s ∈ mt0 , t0 ,
2.8
where φ ∈ Cmt0 , t0 , R, bj t, cj t, qj t ∈ CR , R, τj t, δj t ∈ CR , R , t − τj t → ∞,
and t − δj t → ∞ as t → ∞ and for each t0 ≥ 0,
mj t0 min inf s − τj s, s ≥ t0 , inf s − δj s, s ≥ t0 ,
m t0 min mj t0 , 1 ≤ j ≤ n .
2.9
Note that 2.7 becomes 2.5 for n 2, τ1 t 0, τ2 t τt, b1 t at, b2 t bt, q1 t 0, q2 t qt, δ1 t 0, δ2 t δt, c1 t ct, and c2 t et. Thus, we know that 2.7
includes 2.1, 2.3, and 2.5 as special cases.
Our aim here is to generalize Theorems B and C to 2.7.
Theorem 2.1. Suppose that τj is differential, and there exist continuous functions hj t : 0, ∞ → R
for j 1 · · · n and a constant α ∈ 0, 1 such that for t ≥ 0
t
i lim inft→∞ 0 Hsds > −∞,
ii
n n t
qj t j
1
j
1
n t
n t t
hj sds e− s Hudu hj s − τj s 1 − τj s bj s− qj sHsds
t−τj t
j
1 0
t
s
e− s Hudu Hs
s−τj s
j
1 0
⎛
2 ⎞1/2
t n t
−2
Hudu
hj udu ds 2⎝ e s
cj s ds⎠ ≤ α < 1,
0
j
1
2.10
where Ht n
j
1
hj t.
4
Fixed Point Theory and Applications
Then the zero solution of 2.7 is mean square asymptotically stable if and only if
t
Hsds −→ ∞
as t −→ ∞.
2.11
0
Proof. For each t0 , denote by S the Banach space of all F-adapted processes ψt, ω : mt0 , ∞×
Ω → R which are almost surely continuous in t with norm
ψS E
2
sup ψs, ω
1/2
2.12
.
s≥mt0 Moreover, we set ψt, ω φt for t ∈ mt0 , t0 and E|ψt, ω|2 → 0, as t → ∞.
At first, we suppose that 2.11 holds. Define an operator P : S → S by P xt φt for
t ∈ mt0 , t0 and for t ≥ t0 ,
n
n t0
φt0 − qj t0 φ t0 − τj t0 −
P xt j
1
n
j
1
qj tx t − τj t j
1
t
j
1
t
e− s Hudu
t0
−
t−τj t
t0 −τj t0 hj sφsds e
−
t
t0 Hudu
hj sxsds
n
hj s − τj s 1 − τj s bj s − qj sHs x s − τj s ds
j
1
t
e
t
− s Hudu
Hs
t0
n t
s
n
j
1
t
e
t
− s Hudu
t0
n
s−τj s
hj uxudu ds
cj sx s − δj s
dWs :
j
1
5
Ii t.
i
1
2.13
Now, we show the mean square continuity of P on t0 , ∞. Let x ∈ S, T1 > 0, and let |r|
be sufficiently small. Then
5
2
2
EP x T1 r − P x T1 ≤ 5 EIi T1 r − Ii T1 .
2.14
i
1
It is easy to verify that
2
EIi T1 r − Ii T1 −→ 0,
as r −→ 0, i 1, 2, 3, 4.
2.15
Meng Wu et al.
5
It follows from the last term I5 in 2.13 that
n
T1 T1
T1 r
2
−
Hudu
E e− s Hudu e T 1
−1
cj sx s − δj s dWs
E I5 T1 r − I5 T1
t0
j
1
T1 r
e
−
T
s
1 r Hudu
T1
≤ 2E
T1
e
2
n
cj sx s − δj s dWs
j
1
T1 r
T
− T Hudu
−2 s 1 Hudu
1
e
2
n 2 −1
ds
cj s · x s − δj s
t0
j
1
2
T1 r n T r
−2 s 1 Hudu
cj s·x s − δj s ds −→ 0,
2E
e
T1
as r −→ 0.
j
1
2.16
Therefore, P is mean square continuous on t0 , ∞.
Next, we verify that P x ∈ S. Since E|xt| → 0, t − δj t → ∞ as t → ∞, for each > 0, there
exists a T1 > t0 such that s ≥ T1 implies E|xs|2 < and E|xs − δj s|2 < . Thus, for t ≥ T1 ,
the last term I5 in 2.13 satisfies
2
EI5 t
≤E
T1
e
t
−2 s Hudu
t0
≤E
n
cj sx s − δj s
s≥mt0 ds E
t
e
t
−2 s Hudu
n
T1
j
1
2
sup xs
2
T
1
e
t
−2 s Hudu
2
cj sx s − δj s
ds
j
1
2
2
t
n n t
−2
Hudu
cj s ds e s
cj s ds.
t0
j
1
T1
j
1
2.17
By condition ii and 2.11, there exists T2 > T1 such that t ≥ T2 implies
E|I5 t|2 < α.
2
2.18
2
Thus, E|I5 t| → 0, as t → ∞. Similarly, we can show that E|Ii t| → 0, i 1, 2, 3, 4, as t → ∞.
Thus, E|P xt|2 → 0 as t → ∞. This yields P x ∈ S.
Now we show that P : S → S is a contraction mapping. From ii, we can choose ε > 0
such that α2 ε < 1. Thus, for each t0 ≥ 0, we can find a constant L > 0 such that
n n t t
s
1
− s Hudu hj udu ds
qj t Hs
e
1
L
s−τj s
j
1
j
1 t0
2
n t t
− s Hudu hj sds
hj s−τj s1−τj s bj s− qj sHs ds
e
n t
j
1
t
41 L
t0
t−τj t
t
e−2 s Hudu
j
1
t0
2
n cj s ds ≤ α2 ε < 1.
j
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 Lp -inequality see
10 that
2
e sup pxs − pys
s≥mt0 n s
n
e sup qj s x s − τj s − y s − τj s s≥t0
j
1
s
t0
j
1
s−τj s
hj v xv − yv dv
n
e− v hudu
hj v − τj v 1 − τj v bj v − qj vhv
s
j
1
× x v − τj v − y v − τj v dv
s n v
s
hj u xu − yu du dv
− e− v hudu hv
t0
s
t0
j
1
v−τj v
2
n
− v hudu
e
cj v x v − δj v − y v − δj v
dwv
s
j
1
n 1
qj s·x s − τj s − y s − τj s e sup
≤ 1
l
s≥t0
j
1
n s
hj v·xv − yvdv
j
1
s
s−τj s
s
e− v hudu
t0
n hj v − τj v 1 − τ v bj v − qj vhv
j
j
1
· x v − τj v − y v − τj v dv
v
2
s n
s
− v hudu
hj u · xu − yu du dv
e
hv
t0
j
1
v−τj v
2 s n s
cj v·x v − δj v − y v − δj v dv
41 l sup e e− v hudu
s≥t0
t0
j
1
2
≤ e sup xs − ys
s≥mt0 n n s s
v
1
− v hudu hj udu ds
qj s hv
· sup
e
1
l
s≥t0
v−τj v
j
1
j
1 t0
s
n
hj vdv
j
1 s−τj s
n s
s
e− v hudu
j
1 t0
× hj v − τj v 1 − τj v bj v − qj vhvdv
s
41 l
t0
s
e−2 v hudu
2
n 2
cj v dv ≤ α2 ε e sup xs − ys2 .
j
1
s≥mt0 2.20
Meng Wu et al.
7
Therefore, P is contraction mapping with contraction constant α2 ε. By the contraction
mapping principle, P has a fixed point x ∈ S, which is a solution of 2.7 with xs φs
on mt0 , t0 and E|xt|2 → 0 as t → ∞.
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 > 0 and δ < satisfying the
following condition:
t0
4δK 2 1 Le2 0
Hudu
α2 ε < ,
2.21
t
where K supt≥0 {e− 0 Hsds }. If xt xt, t0 , φ is a solution of 2.7 with φ2 < δ, then
xt P xt defined in 2.13. We assume that E|xt|2 < for all t ≥ t0 . Notice that E|xt|2 φ2 < for t ∈ mt0 , t0 . If there exists t∗ > t0 such that E|xt∗ |2 and E|xt|2 < for
t ∈ mt0 , t∗ , then 2.13 and 2.19 imply that
2
n n t0
t∗
∗ 2
2
hj sds e−2 t0 Hudu
Ex t ≤ 1 Lφ 1 qj t0 j
1 t0 −τj t0 j
1
n n t∗
1
qj t∗ hj sds
1
L
j
1
j
1 t∗ −τj t∗ t∗
e
−
t∗
s
Hudu
s
n
j
1 s−τj s
t0
t∗
t∗
e− s
Hudu
t0
t∗
e
t∗
−2 s Hudu
hj udu Hsds
2
n hj s − τj s 1 − τ s bj s − qj sHsds
j
j
1
2
n cj s ds
t0
j
1
n n t0
≤ 1 Lδ 1 qj t0 j
1 t0 −τj t0 j
1
hj sds
2
e
t∗
−2 t Hudu
0
α2 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 {tn } with
t
tn → ∞ as n → ∞ such that limn→∞ 0n Hudu β, where β ∈ R. Then, we can choose a constant
tn
J > 0 satisfying 0 Hudu ∈ −J, J for all n ≥ 1. Denote
n s
hj s − τj s 1 − τ s bj s − qj sHs Hs
hj udu
2.23
ωs j
s−τj s
j
1
for all s ≥ 0. From ii, we have
tn
0
tn
e− s
Hudu
ωsds ≤ α,
2.24
8
Fixed Point Theory and Applications
which implies
tn
s
tn
e 0 Hudu ωsds ≤ αe 0
Hudu
≤ eJ .
2.25
0
t s
Therefore, the sequence { 0n e 0 Hudu ωsds} has a convergent subsequence. Without loss of
generality, we can assume that
tn
lim
n→∞ 0
s
e 0 Hudu ωsds γ
2.26
for some γ > 0. Let k be an integer such that
tn
s
e 0 Hudu ωsds <
tk
δ0
8K
2.27
for all n ≥ k, where δ0 > 0 satisfies 8δ0 K 2 e2J α2 ε < 1.
Now we consider the solution xt xt, tk , φ of 2.7 with φtk 2 δ0 and φs2 <
δ0 for s < tk . By the similar method in 2.22, we have E|xt|2 < 1 for t ≥ tk . We may choose φ
so that
Gtk :
φtk −
n
n
qj tk φ tk − τj tk −
j
1
j
1
tk
1
hj sφsds ≥ δ0 .
2
tk −τj tk 2.28
It follows from 2.13 and 2.28 with xt P xt that for n ≥ k,
2
n
n tn
Extn −
qj tn x tn − τj tn −
hj sxsds
j
1
j
1 tn −τj tn ≥ G tk e
2
t
−2 t n Hudu
k
− 2Gtk e
−
tn
tk
Hudu
tn
tn
e− s
Hudu
ωsds
2.29
tk
tn s
δ2
δ0 −2ttn Hudu δ0
− 2K e 0 Hudu ωsds ≥ 0 e−2J > 0.
≥ e k
2
2
8
tk
If the zero solution of 2.7 is mean square asymptotic stable, then E|xt|2 E|xt, tk , φ|2 → 0 as t → 0. Since tn − τj tn → ∞, tn − δj tn → ∞ as n → ∞ and condition ii
and 2.11 hold,
2
n
n tn
Extn −
qj tn x tn − τj tn −
hj sxsds −→ 0,
j
1
j
1 tn −τj tn 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 ≥ ta for some ta ∈ R .
Meng Wu et al.
9
Remark 2.3. Theorem 2.1 improves Theorem C under different conditions.
Corollary 2.4. Suppose that τj is differential, the inverse function gj t of t − τj t exists, and there
t
exists a constant α ∈ 0, 1 such that for t ≥ 0, lim inft→∞ 0 Qsds > −∞ and
n n t
n t t
qj t bj gj s ds e− s Qudu bj sτj s − qj sQsds
j
1
j
1
t−τj t
n t s
t
− s Qudu Qs
e
s−τj s
j
1 0
j
1
0
⎛
2 ⎞1/2
t n t
bj gj u du ds2⎝ e−2 s Qudu
cj s ds⎠ ≤ α < 1,
0
j
1
2.31
n
where Qt j
1 − bj gj t. Then the zero solution of 2.7 is mean square asymptotically stable if
t
and only if 0 Qsds → ∞ as t → ∞.
Remark 2.5. When hj t −bj gj t for j 1 · · · n, Theorem 2.1 reduces to Corollary 2.4. On the
other hand, we choose qj t ≡ cj t ≡ 0 and bj ≡ −bj for j 1 · · · n, then Corollary 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:
xt
xt−sin t
xt−t/2 3sin t4
xt − t/2
t
dWt.
−
−
dt
d xt−
−
x t−
√
√
1000
1616t
4848t
4
24 3 4t 12 34t
3.1
Then the zero solution of 3.1 is mean square asymptotically stable.
Proof. Choosing h1 t 1/8 16t and h2 t 7/48 64t in Theorem 2.1, we have
1
7
11
13
Ht ,
≤ Ht ≤
,
8 16t 48 64t
48 64t
48 64t
t
t
t
2
1
7
hj sds ds ds −→ 0.07479, as t −→ ∞,
8
16s
48
64s
t/2
3t/4
j
1 t−τj t
t 2 t t
s
t
hj udu ds ≤ e− s 11/4864udu 13
e− s Hudu Hs
· 0.07479 ds ≤ 0.08839,
48 64s
s−τj s
0
j
1 0
⎛
2 ⎞1/2
t t 1/2
2 t
t
1
−2 s Hudu
− s 11/2432udu
⎠
⎝
cj s
e
ds
≤2
e
≤ 0.21320,
2
ds
824 32s
0
0
j
1
2 t t
e− s Hudu hj s − τj s1 − τj s bj s − qj sHsds
j
1 0
≤
t
e
0
t
− s 11/4864udu
0.013
17
0.013 17
ds ≤
0.51634.
48 64s 144 192s
11
33
3.2
10
Fixed Point Theory and Applications
∞
It easy to check that 0 Hsds ∞. Let α 0.001 0.07479 0.08839 0.21320 0.51634.
Then, α 0.89372 < 1 and the zero solution of 3.1 is mean square asymptotically stable by
Theorem 2.1.
Example 3.2. Consider the following delay differential equation:
1
1
t
2
−
x t −
x t−
x t− t .
6 4t
3
12 4t
3
3.3
Then the zero solution of 3.3 is asymptotically stable.
Proof. Choosing h1 t h2 t 1/4 4t in Theorem 2.1, we have Ht 1/2 2t and
2 t
j
1
t−τj t
2 t
j
1
e
hj sds t
1
ds 4
4s
2/3t
s
Hs
t
− s Hudu s−τj s
0
t
1
1
1
ds −→ ln 3 − ln 2 0.37602,
4
4s
2
4
t/3
hj udu ds ≤
t
t
e− s 1/22udu
0
as t −→ ∞,
1
· 0.37602 ds ≤ 0.37602.
2 2s
3.4
Notice that qj t cj t ≡ 0 and
2 hj s − τj s 1 − τ s bj s − qj sHs
j
j
1
3
2
1 3
1
1 0.
· −
·
−
12 8s 3 6 4s 12 4s 3 12 4s 3.5
It is easy to see that all the conditions of Theorem 2.1 hold for α 0.37602 0.37602 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, b1 t 1/6 4t, b2 t 1/12 4t, b1 g1 t 1/6 6t,
b2 g2 t 1/12 12t, and |Qt| 1/4 4t. As t → ∞,
2 t
j
1
t−τj t
bj gj s ds ≤
t
1
ds 6
6s
2/3t
t
1
1
1
ds −→ ln 3 − ln 2 0.15913.
12
12s
4
6
t/3
3.6
Notice that
2 bj sτ s − qj sQs j
j
1
1
1
1
≤
.
18 12s 18 6s 4 4s
3.7
It follows from 3.7 that
2 t
j
1
0
e
t
− s Qudu bj sτj s
− qj sQsds ≤
t
0
t
e− s 1/44udu
1
ds ≤ 1.
4 4s
3.8
Meng Wu et al.
11
From 3.6, we obtain
2 t
j
1
t
s
e− s Qudu Qs
0
s−τj s
bj gj u du ds ≤
t
0
t
e− s 1/44udu
1
· 0.15913 ds ≤ 0.15913.
4 4s
3.9
Combining 3.6, 3.8, and 3.9, 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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