Volume 2008, Article ID 407352,11pages doi:10.1155/2008/407352
Research Article
Fixed Points and Stability in Neutral Stochastic
Differential Equations with Variable Delays
Meng Wu,1Nan-jing Huang,1and Chang-Wen Zhao2
1Department of Mathematics, Sichuan University, Chengdu, Sichuan 610064, China
2College of Business and Management, Sichuan University, Chengdu, Sichuan 610064, China
Correspondence should be addressed to Nan-jing Huang,[email protected]
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.
Copyrightq2008 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, Burton1–4, Jung5, Luo6, and Zhang7studied 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
6studied 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 to8,9and the references therein.
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 in1,6,7.
2. Main results
Let Ω,F,{Ft}t≥0, P be a complete filtered probability space and let Wt denote a
one-dimensional standard Brownian motion defined on Ω,F,{Ft}t≥0, Psuch that {Ft}t≥0 is the
natural filtration ofWt. Letat, bt, bt, ct, et, qt∈CR , R,andτt, δt∈CR , R
with t−τt→ ∞andt−δt→ ∞ast→ ∞. HereCS1, S2denotes the set of all continuous
functionsφ:S1→S2with the supremum norm·.
In 2003, Burton1studied the equation
xt −btxt−τt 2.1
and proved the following theorem.
Theorem ABurton1. Suppose thatτt rand there exists a constantα <1such that
t
t−r
bs rds t
0
bs re−t sbu rdu
s
s−r
bu rdu ds≤α 2.2
for all t ≥ 0 and∞0bsds ∞. Then, for every continuous initial function φ : −r,0→R, the solutionxt xt,0, φof 2.1is bounded and tends to zero ast→ ∞.
Recently, Zhang7studied the generalization of2.1as follows:
xt −
n
j1
bjtx
t−τjt
2.3
and obtained the following theorem.
Theorem BZhang7. Suppose thatτj is differential, the inverse functiongjtoft−τjtexists, and there exists a constantα∈0,1such that fort≥0,lim inft→∞0tQsds >−∞and
n
j1
t
t−τjt
bj
gjsds
t
0
e−stQudub
jsτjsds
t
0
e−stQuduQs
s
s−τjs
bj
gjvdv ds ≤α,
2.4
where Qt nj1bjgjt. Then the zero solution of 2.3is asymptotically stable if and only if
t
0Qsds→ ∞, ast→ ∞.
Very recently, Luo6considered the following neutral stochastic differential equation:
dxt−qtxt−τtatxt btxt−τtdt ctxt etxt−δtdWt
2.5
Theorem C Luo 6. Let τtbe derivable. Assume that there exists a constant α ∈ 0,1and a continuous functionht:0,∞→Rsuch that fort≥0,lim inft→∞0thsds >−∞and
qt t
t−τt
as hsds t
0
e−sthuduas−τs hs−τs1−τs bs−qshsds
t
0
e−sthuduhs
s
s−τs
au hudu ds t
0
e−2sthuducs es2ds
1/2
≤α.
2.6
Then the zero solution of 2.5is mean square asymptotically stable if and only if0thsds→ ∞, as
t→ ∞.
Now, we consider the generalization of2.5:
d
xt−
n
j1
qjtx
t−τjt
n j1
bjtx
t−τjt
dt
n
j1
cjtx
t−δjt
dWt, 2.7
with the initial condition
xs φs for s∈mt0, t0
, 2.8
whereφ ∈ Cmt0, t0, R,bjt, cjt, qjt ∈CR , R,τjt, δjt ∈CR , R ,t−τjt→ ∞, andt−δjt→ ∞ast→ ∞and for eacht0 ≥0,
mjt0 min
infs−τjs, s≥t0
, infs−δjs, s≥t0
,
mt0
minmj
t0
, 1≤j ≤n. 2.9
Note that2.7becomes2.5forn 2,τ1t 0, τ2t τt,b1t at, b2t bt,q1t
0, q2t qt, δ1t 0, δ2t δt, c1t ct,andc2t et. Thus, we know that2.7
includes2.1,2.3, and2.5as special cases.
Our aim here is to generalize Theorems B and C to2.7.
Theorem 2.1. Suppose thatτj is differential, and there exist continuous functionshjt: 0,∞→R forj 1· · ·nand a constantα∈0,1such that fort≥0
ilim inft→∞0tHsds >−∞,
ii
n
j1
qjt n
j1
t
t−τjt
hjsds n
j1
t
0
e−stHuduh j
s−τjs
1−τjs bjs−qjsHsds
n
j1
t
0
e−stHuduHs
s
s−τjs
hjudu ds 2⎛⎝t
0
e−2stHudu
n
j1
cjs2
ds ⎞ ⎠
1/2
≤α <1,
2.10
Then the zero solution of 2.7is mean square asymptotically stable if and only if
t
0
Hsds−→ ∞ ast−→ ∞. 2.11
Proof. For eacht0, denote bySthe Banach space of allF-adapted processesψt, ω:mt0,∞×
Ω→Rwhich are almost surely continuous intwith norm
ψS
E
sup s≥mt0
ψs, ω2
1/2
. 2.12
Moreover, we setψt, ω φtfort∈mt0, t0andE|ψt, ω|2→0, ast→ ∞.
At first, we suppose that2.11holds. Define an operatorP :S→SbyP xt φtfor
t∈mt0, t0and fort≥t0,
P xt
φt0−
n
j1
qjt0φ
t0−τjt0
−n j1
t0
t0−τjt0
hjsφsds
e−
t t0Hudu
n
j1
qjtx
t−τjt
n
j1
t
t−τjthjsxsds
t
t0
e−stHudu n
j1
hj
s−τjs
1−τjs bjs−qjsHs
xs−τjs
ds
−
t
t0
e−stHuduHs
n
j1
s
s−τjshjuxudu
ds
t
t0
e−stHudu
n
j1
cjsx
s−δjs
dWs:
5
i1
Iit.
2.13
Now, we show the mean square continuity ofP ont0,∞. Letx∈S,T1 >0,and let|r|
be sufficiently small. Then
EP xT1 r
−P xT12 ≤5 5
i1
EIi
T1 r
−Ii
T12. 2.14
It is easy to verify that
EIi
T1 r
−Ii
It follows from the last termI5in2.13that
EI5
T1 r
−I5
T12E
T1
t0
e−sT1Hudu
e−
T1r
T1 Hudu−1
n
j1
cjsx
s−δjs
dWs
T1 r
T1
e−sT1 rHudu n
j1
cjsx
s−δjs
dWs
2
≤2E T1
t0
e−2sT1Hudu
e−
T1 r
T1 Hudu−1
2n
j1
cjs·x
s−δjs
2
ds
2E T1 r
T1
e−2sT1rHudu
n
j1
cjs·x
s−δjs
2
ds−→0, as r−→0.
2.16 Therefore,P is mean square continuous ont0,∞.
Next, we verify thatP x∈S. SinceE|xt| →0,t−δjt→ ∞ast→ ∞, for each >0, there exists aT1 > t0such thats≥ T1impliesE|xs|2 < andE|xs−δjs|2 < . Thus, fort≥ T1,
the last termI5in2.13satisfies
EI5t2
≤E T1
t0
e−2stHudu
n
j1
cjsx
s−δjs
2
ds E
t
T1
e−2stHudu
n
j1
cjsx
s−δjs
2
ds
≤E
sup s≥mt0
xs2T1
t0
e−2stHudu
n
j1
cjs2
ds
t
T1
e−2stHudu
n
j1
cjs2
ds.
2.17 By conditioniiand2.11, there existsT2> T1such thatt≥T2implies
E|I5t|2< α. 2.18
Thus,E|I5t|2→0, as t→ ∞. Similarly, we can show that E|Iit|2→0, i 1,2,3,4, as t→ ∞. Thus,E|P xt|2→0 ast→ ∞. This yieldsP x∈S.
Now we show thatP : S→Sis a contraction mapping. Fromii, we can chooseε > 0 such thatα2 ε <1. Thus, for eacht
0≥0, we can find a constantL >0 such that
1 1
L
n
j1
qjt n
j1
t
t0
e−stHuduHs
s
s−τjs
hjudu ds
n
j1
t
t−τjt
hjsds n
j1
t
t0
e−stHuduh
js−τjs1−τjs bjs−qjsHsds
2
41 L
t
t0
e−2stHudu
n
j1
cjs2ds≤α2 ε <1.
For anyx, y ∈ S, it follows from2.13, conditionsiandii, and Doob’sLp-inequalitysee 10that
e sup
s≥mt0
pxs−pys2
esup s≥t0
n
j1
qjs
xs−τjs
−ys−τjs
n
j1
s
s−τjshjv
xv−yvdv
s
t0
e−vshudu n
j1
hj
v−τjv
1−τjv bjv−qjvhv
×xv−τjv
−yv−τjv
dv
−
s
t0
e−vshuduhv
n
j1
v
v−τjv
hju
xu−yudu
dv
s
t0
e−vshudu
n
j1
cjv
xv−δjv
−yv−δjv
dwv
2 ≤ 1 1 l esup
s≥t0
n
j1
qjs·x
s−τjs
−ys−τjs
n
j1
s
s−τjs
hjv·xv−yvdv
s
t0
e−vshudu n
j1
hj
v−τjv
1−τjv bjv−qjvhv
·xv−τjv
−yv−τjvdv
s
t0
e−vshuduhv
n
j1
v
v−τjv
hju·xu−yududv2
41 lsup s≥t0
e
s
t0
e−vshudu
n
j1
cjv·x
v−δjv
−yv−δjv
2
dv
≤e sup s≥mt0
xs−ys2
·sup s≥t0
1 1
l
n
j1
qjs n
j1
s
t0
e−vshuduhv
v
v−τjv
hjudu ds
n
j1
s
s−τjs
hjvdv
n
j1
s
t0
e−vshudu
×hj
v−τjv
1−τjv bjv−qjvhvdv
2
41 l
s
t0
e−2vshudu
n
j1
cjv2
dv
≤α2 εesup s≥mt0
xs−ys2
.
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 2.7with xs φs
onmt0, t0andE|xt|2→0 ast→ ∞.
To obtain the mean square asymptotic stability, we need to show that the zero solution of 2.7is mean square stable. Let > 0 be given and chooseδ > 0 and δ < satisfying the following condition:
4δK21 Le20t0Hudu α2 ε < , 2.21
where K supt≥0{e−0tHsds}. If xt xt, t0, φis a solution of 2.7 with φ2 < δ, then
xt P xtdefined in2.13. We assume thatE|xt|2< for allt≥t0. Notice thatE|xt|2
φ2 < for t ∈ mt
0, t0. If there existst∗ > t0 such thatE|xt∗|2 andE|xt|2 < for
t∈mt0, t∗, then2.13and2.19imply that
Ext∗2≤1 Lφ2
1 n
j1
qj t0
n
j1
t0
t0−τjt0
hjsds2
e−2 t∗
t0Hudu
1 1
L
n
j1
qj
t∗
n
j1
t∗
t∗−τjt∗
hjsds
t∗
t0
e−t
∗ sHudu
n
j1
s
s−τjs
hjuduHsds
t∗
t0
e−t
∗ sHudu
n
j1
hj
s−τjs
1−τjs bjs−qjsHsds
2
t∗
t0
e−2t
∗ sHudu
n
j1
cjs2ds
≤1 Lδ
1 n
j1
qj t0
n
j1
t0
t0−τjt0
hjsds2e−2tt0∗Hudu α2 ε < ,
2.22
which contradicts the definition oft∗. Thus, the zero solution of2.7is stable. It follows that the zero solution of2.7is mean square asymptotically stable if2.11holds.
Conversely, we suppose that 2.11 fails. From i, there exists a sequence {tn} with
tn→ ∞asn→ ∞such that limn→∞
tn
0 Huduβ, whereβ∈R. Then, we can choose a constant
J >0 satisfyingtn
0 Hudu∈−J, Jfor alln≥1. Denote
ωs
n
j1
hj
s−τjs
1−τjs bjs−qjsHs Hs
s
s−τjs
hjudu 2.23
for alls≥0. Fromii, we have
tn
0
which implies
tn
0
e0sHuduωsds≤αe
tn
0 Hudu≤eJ. 2.25
Therefore, the sequence {tn
0 e
s
0Huduωsds} has a convergent subsequence. Without loss of generality, we can assume that
lim n→∞
tn
0
e0sHuduωsdsγ 2.26
for someγ >0. Letkbe an integer such that
tn
tk
e0sHuduωsds < δ0
8K 2.27
for alln≥k, whereδ0>0 satisfies 8δ0K2e2J α2 ε<1.
Now we consider the solutionxt xt, tk, φof2.7withφtk2 δ0andφs2 <
δ0fors < tk. By the similar method in2.22, we haveE|xt|2<1 fort≥tk. We may chooseφ so that
Gtk:φtk− n
j1
qjtkφ
tk−τjtk
−n j1
tk
tk−τjtk
hjsφsds≥ 1
2δ0. 2.28
It follows from2.13and2.28withxt P xtthat forn≥k,
Extn− n
j1
qjtnx
tn−τj
tn
−n j1
tn
tn−τjtn
hjsxsds
2
≥G2tke−2
tn
tkHudu−2Gtke−
tn
tkHudu
tn
tk
e−stnHuduωsds
≥ δ0
2 e
−2tktnHuduδ0
2 −2K
tn
tk
e0sHuduωsds
≥ δ02
8 e
−2J >0.
2.29
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− τjtn→ ∞, tn − δjtn→ ∞ as n→ ∞ and condition ii and2.11hold,
Extn− n
j1
qjtnx
tn−τj
tn
−n j1
tn
tn−τjtn
hjsxsds
2
−→0, as n−→ ∞, 2.30
which contradicts 2.29. Therefore, 2.11 is necessary for Theorem 2.1. This completes the proof.
Remark 2.3. Theorem 2.1improves Theorem C under different conditions.
Corollary 2.4. Suppose thatτj is differential, the inverse function gjtof t−τjtexists, and there exists a constantα∈0,1such that fort≥0,lim inft→∞0tQsds >−∞and
n
j1
qjt n
j1
t
t−τjt
bj
gjsds
n
j1
t
0
e−stQudubjsτ
js−qjsQsds
n
j1
t
0
e−stQuduQs
s
s−τjs
bj
gjudu ds 2
⎛
⎝t
0
e−2stQudu
n
j1
cjs2ds⎞⎠
1/2
≤α <1,
2.31
whereQt nj1−bjgjt. Then the zero solution of 2.7is mean square asymptotically stable if and only ift0Qsds→ ∞ast→ ∞.
Remark 2.5. Whenhjt −bjgjtforj1· · ·n,Theorem 2.1reduces toCorollary 2.4. On the other hand, we chooseqjt≡ cjt≡ 0 andbj ≡ −bj forj 1· · ·n, thenCorollary 2.4reduces 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
16 16t −
3sint 4 48 48tx
t− t
4
dt
xt
24√3 4t−
xt−sint
12√3 4t
dWt.
3.1 Then the zero solution of3.1is mean square asymptotically stable.
Proof. Choosingh1t 1/8 16tandh2t 7/48 64tinTheorem 2.1, we have
Ht 1
8 16t
7 48 64t,
11
48 64t ≤Ht≤
13 48 64t,
2
j1
t
t−τjt
hjsdst
t/2
1 8 16sds
t
3t/4
7
48 64sds−→0.07479, as t−→ ∞,
2
j1
t
0
e−tsHuduHs
s
s−τjs
hjudu ds≤t
0
e−st11/48 64udu 13
48 64s·0.07479ds≤0.08839,
2
⎛
⎝t
0
e−2stHudu
2
j1
cjs
2
ds ⎞ ⎠
1/2
≤2
t
0
e−st11/24 32udu 1
824 32sds 1/2
≤0.21320,
2
j1
t
0
e−stHuduh
js−τjs1−τjs bjs−qjsHsds
≤
t
0
e−st11/48 64udu
0.013 48 64s
17 144 192s
ds≤ 0.013
11 17
33 0.51634.
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 of3.1is mean square asymptotically stable by
Theorem 2.1.
Example 3.2. Consider the following delay differential equation:
xt − 1
6 4tx
t− t
3
− 1
12 4tx
t−2
3t
. 3.3
Then the zero solution of3.3is asymptotically stable.
Proof. Choosingh1t h2t 1/4 4tinTheorem 2.1, we haveHt 1/2 2tand
2
j1
t
t−τjt
hjsdst 2/3t
1 4 4sds
t
t/3
1
4 4sds−→
1 2ln 3−
1
4ln 20.37602, ast−→ ∞,
2
j1
t
0
e−stHuduHs
s
s−τjs
hjudu ds≤t
0
e−st1/2 2udu 1
2 2s·0.37602ds≤0.37602.
3.4
Notice thatqjt cjt≡0 and
2
j1
hj
s−τjs
1−τjs bjs−qjsHs
3
12 8s·
2 3 −
1 6 4s
1234s·1
3 − 1 12 4s
0.
3.5
It is easy to see that all the conditions ofTheorem 2.1hold forα0.37602 0.376020.75204<
1. Thus,Theorem 2.1implies that the zero solution of3.3is asymptotically stable.
However, Theorem B cannot be used to verify that the zero solution of 3.3 is asymptotically stable. In fact, b1t 1/6 4t, b2t 1/12 4t, b1g1t 1/6 6t,
b2g2t 1/12 12t, and|Qt|1/4 4t. Ast→ ∞,
2
j1
t
t−τjt
bj
gjsds≤
t
2/3t 1 6 6sds
t
t/3
1
12 12sds−→
1 4ln 3−
1
6ln 20.15913.
3.6
Notice that
2
j1
bjsτ
js−qjsQs 1 18 12s
1 18 6s ≤
1
4 4s. 3.7
It follows from3.7that
2
j1
t
0
e−stQudubjsτ
js−qjsQsds≤
t
0
e−st1/4 4udu 1
From3.6, we obtain
2
j1
t
0
e−stQuduQs
s
s−τjs
bj
gjudu ds≤
t
0
e−st1/4 4udu 1
4 4s·0.15913ds≤0.15913.
3.9
Combining3.6,3.8, and3.9, we see that the condition2.4of Theorem B does not hold withα1.31825.
Acknowledgement
This work was supported by the National Natural Science Foundation of China 10671135 and Specialized Research Fund for the Doctoral Program of Higher Education20060610005.
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