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Volume 2008, Article ID 407352, 11 pagesdoi:10.1155/2008/407352 Research Article Fixed Points and Stability in Neutral Stochastic Differential Equations with Variable Delays Meng Wu, 1 N

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

Received 4 April 2008; Accepted 9 June 2008

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.

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, Jung 5, Luo6, 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 to8,9 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

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, P such that {Ft}t≥0 is the

natural filtration of W t 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→ S2with the supremum norm·

In 2003, Burton1 studied the equation

t − τt 2.1 and proved the following theorem

Theorem A Burton 1

t

t −r

b s rds t

0

b s re−t

s b u rdus

s −r

b u rdu ds ≤ α 2.2

for all t ≥ 0 and ∞0b

solution x 2.1 is bounded and tends to zero as t → ∞.

Recently, Zhang7 studied the generalization of 2.1 as follows:

x

n



j

b j txt − τj t 2.3 and obtained the following theorem

Theorem B Zhang 7 Suppose that τj is differential, the inverse function g j t of t − τj t exists,

and there exists a constant α ∈ 0, 1 such that for t ≥ 0, lim inft→∞t

0Q sds > −∞ and n



j

t

t −τ j t

b j

g jsds t

0

e−s t Q udub j sτ

j sds

t

0

e−s t Q uduQ ss

s −τ j s

b j

g jvdv ds ≤ α, 2.4

where Q n j b j gj t Then the zero solution of 2.3 is asymptotically stable if and only if

t

0Q sds → ∞, as t → ∞.

Very recently, Luo6 considered the following neutral stochastic differential equation:

d

x t − qtxt − τt a txt btxt − τtdt c txt etxt − δtdW t

2.5 and obtained the following theorem

Theorem C Luo 6 Let τt be derivable Assume that there exists a constant α ∈ 0, 1 and a

continuous function h t : 0, ∞ → R such that for t ≥ 0, lim inft→∞t

0h sds > −∞ and

q t t

t −τt

a s hsds t

0

e−s t h udua

s −τs hs − τs1−τs bs−qshsds

t

0

e−s t h uduh ss

s −τs

a u hudu ds t

0

e−2s t h uduc s es2

ds

1/2

≤ α.

2.6

Then the zero solution of 2.5 is mean square asymptotically stable if and only ift

0h sds → ∞, as

t → ∞.

Now, we consider the generalization of2.5:

d



x t −n

j

q jtxt − τj t



n



j

b j txt − τjtdt n

j

c jtxt − δjtdW t, 2.7 with the initial condition

m t0, t0

where φ ∈ Cmt0, t0, R, bj t, cjt, qj t ∈ CR , R , τj t, δj t ∈ CR , R , t − τjt → ∞, and t − δjt → ∞ as t → ∞ and for each t0 ≥ 0,

m j t0

 inf

s − τj s, s ≥ t0



, inf

s − δj s, s ≥ t0



,

m

t0

m j



t0



, 1 ≤ j ≤ n. 2.9 Note that2.7 becomes 2.5 1 2 1 2 1

includes2.1, 2.3, and 2.5 as special cases

Our aim here is to generalize Theorems B and C to2.7

Theorem 2.1 Suppose that τ j is differential, and there exist continuous functions h jt : 0, ∞ → R

for j

i lim inft→∞t

0H sds > −∞,

ii

n



j

q j t n

j

t

t −τ j t

h j sds n

j

t

0

e−s t H uduh j

s − τjs1− τ

j s bj s− qjsHsds

n

j

t

0

e−s t H uduH ss

s −τ j s

h j udu ds 2⎛⎝t

0

e−2s t H udu

n

j

c js2

ds

⎞

⎠

1/2

≤ α < 1,

2.10

where H n j h jt.

Then the zero solution of 2.7 is mean square asymptotically stable if and only if

t

0

H sds −→ ∞ as t −→ ∞. 2.11

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

 sup

s ≥mt0 

ψ s, ω2

1/2

Moreover, we set ψ 0, t0 and E|ψt, ω|2→ 0, as t → ∞.

At first, we suppose that2.11

t ∈ mt0, t0 and for t ≥ t0,



φ t0 −n

j

q jt0φt0− τj t0−n

j

t0

t0−τ j t0 h j sφsds



e−

t t0 H udu

n

j

q jtxt − τj t n

j

t

t −τ j t h j sxsds

t

t0

e−s t H udun

j



h j



s − τjs1− τ

j s bjs − qjsHsx

s − τjsds

−

t

t0

e−s t H udu H s

n

j

s

s −τ j s h j uxudu



ds

t

t0

e−s t H udu

n

j

c j sxs − δjs



dW

5



i

I it.

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

E PxT1 r− PxT12 ≤ 55

i

EI i

T1 r− IiT12

. 2.14

It is easy to verify that

EI i

T1 r− IiT12 2.15

It follows from the last term I5in2.13 that

EI5

T1 r− I5



T12 





T1

t0

e−s T1 H udu

e−

T1 r

T1 H udu− 1n

j

c j sxs − δjsdW s

T1 r

T1

e−s T1 r H udun

j

c jsxs − δj sdW s





2

≤ 2E

T1

t0

e−2s T1 H udu

e−

T1 r

T1 H udu− 12

n

j

c j s·xs − δjs2ds

2E

T1 r

T1

e−2s T1 r H udu

n

j

c j s·xs − δj s2ds −→ 0, as r −→ 0.

2.16

Therefore, P is mean square continuous on t0,∞

Next, we verify that P x ∈ S Since E|xt| → 0, t − δjt → ∞ as t → ∞, for each  > 0, there exists a T1 > t0such that s ≥ T1implies E|xs|2 <  and E |xs − δjs|2 <  Thus, for t ≥ T1,

the last term I5in2.13 satisfies

EI5t2

≤ E

T1

t0

e−2s t H udu

 n



j

c jsxs − δj s

2

ds E

t

T1

e−2s t H udu

 n



j

c j sxs − δjs

2

ds

≤ E



sup

s ≥mt0 

x s2T1

t0

e−2s t H udu

 n



j

c j s2

ds 

t

T1

e−2s t H udu

 n



j

c js2

ds.

2.17

By conditionii and 2.11, there exists T2> T1such that t ≥ T2implies

Thus, E|I5t|2→ 0, as t → ∞ Similarly, we can show that E|Iit|2

Thus, E|Pxt|2→ 0 as t → ∞ This yields Px ∈ 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

1 1

L

n

j

q jt n

j

t

t0

e−s t H uduH ss

s −τ j s

h judu ds

n

j

t

t −τ j t

h jsds n

j

t

t0

e−s t H udu hjs−τj s1−τ

j s bjs− qjsHsds2

41 L

t

t0

e−2s t H udu

n

j

c j s2ds ≤ α2 ε < 1.

2.19

For any x, y ∈ S, it follows from 2.13, conditions i and ii, and Doob’s L p-inequalitysee

10 that

e sup

s ≥mt0 pxs − pys2

s ≥t0



n

j

q j sx

s − τj s− ys − τj s n

j

s

s −τ j s h j vx v − yvdv

s

t0

e−v s h udun

j



h j



v − τjv1− τ

j v bj v − qjvhv

×x

v − τjv− yv − τj vdv

−

s

t0

e−v s h udu h v n

j

v

v −τ j v h jux u − yudu



dv

s

t0

e−v s h udu n

j

c jvx

v − δj v− yv − δj vdw v

2

≤ 1 1

l



e sup

s ≥t0

n

j

q j s·xs − τj s− ys − τjs

n

j

s

s −τ j s

h j v·xv − yvdv

s

t0

e−v s h udun

j

h j

v − τjv1− τ

j v bjv − qjvhv

·x

v − τj v− yv − τj vdv

s

t0

e−v s h udu h v

n

j

v

v −τ j v

h j u·xu − yududv

2

41 l sup

s ≥t0

e

s

t0

e−v s h udu n

j

c j v·xv − δjv

− yv − δjv2

dv

≤ e sup

s ≥mt0 

x s − ys2

· sup

s ≥t0

1 1

l

n



j

q j s n

j

s

t0

e−v s h uduh vv

v −τ j v

h j udu ds

n

j

s

s −τ j s

h j vdv

n

j

s

t0

e−v s h udu

×h j

v − τjv1− τ

j v bj v − qjvhvdv2

41 l

s

t0

e−2v s h udu n

j

c jv2

dv

≤α2 εe sup

s ≥mt0 

x s − ys2

.

2.20

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

onmt0, 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:

4δK21 Le2 t0

0 H udu α2 ε < , 2.21

where K t≥0{e−t

0H sds

0, φ is a solution of 2.7 with φ2 < δ, then

x 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 2 <  for

t ∈ mt0, t∗, then 2.13 and 2.19 imply that

Ex

t∗2≤ 1 Lφ2



1 n

j

q j

t0 n j

t0

t0−τ j t0 

h jsds2

e−2

t∗

t0 H udu

 1 1

L

n

j

q j

t∗ n j

t∗

t∗−τ j t∗ 

h j sds

t∗

t0

e−s t∗ H udu

n

j

s

s −τ j s

h j uduH sds

t∗

t0

e−s t∗ H udun

j

h j

s − τjs1− τ

j s bj s − qj sHsds2



t∗

t0

e−2s t∗ H udu

n

j

c j s2ds

≤ 1 Lδ



1 n

j

q j

t0 n j

t0

t0−τ j t0 

h j sds2e−2t∗

t0 H udu α2 ε < ,

2.22

which contradicts the definition of t∗ Thus, the zero solution of2.7 is stable It follows that the zero solution of2.7 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 n → ∞ as n → ∞ such that limn→∞t n

0 H

J > 0 satisfyingt n

0 H udu ∈ −J, J for all n ≥ 1 Denote

ω

n



j

h j

s − τj s1− τ

j s bjs − qjsHs Hss

s −τ j s

h j udu 2.23

for all s≥ 0 From ii, we have

t n

0

e−s tn H udu ω sds ≤ α, 2.24

which implies

t n

0

e0s H udu ω sds ≤ αe0tn H udu ≤ e J 2.25

Therefore, the sequence {t n

0 e0s H udu ω sds} has a convergent subsequence Without loss of

generality, we can assume that

lim

n→∞

t n

0

for some γ > 0 Let k be an integer such that

t n

t k

e0s H udu ω sds < δ0

for all n ≥ k, where δ0> 0 satisfies 8δ0K2e 2J α2 ε < 1.

Now we consider the solution x k , φ of 2.7 with φtk2

0andφs2 <

δ0for s < tk By the similar method in2.22, we have E|xt|2< 1 for t ≥ tk We may choose φ

so that

G tk k −n

j

q j tkφt k − τj tk−n

j

t k

t k −τ j t kh j sφsds ≥ 1

2δ0. 2.28

It follows from2.13 and 2.28

E



x tn −

n



j

q jtnxt n − τjt n



−n

j

t n

t n −τ j t nh jsxsds





2

≥ G2tk e−2tk tn H udu − 2Gtke−tk tn H udut n

t k

e−s tn H udu ω sds

≥ δ0

2 e

−2tn

tk H udu δ0

2 − 2K

t n

t k

e0s H udu ω sds



≥ 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 − τj tn → ∞, tn − δjtn → ∞ as n → ∞ and condition ii

and2.11 hold,

E



x tn −

n



j

q j tnxt n − τjt n



−n

j

t n

t n −τ j t nh 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 2.1still holds if conditionii is satisfied for t ≥ ta for some ta∈ R

Remark 2.3. Theorem 2.1improves Theorem C under different conditions.

Corollary 2.4 Suppose that τ j is differential, the inverse function g jt of t − τjt exists, and there

exists a constant α ∈ 0, 1 such that for t ≥ 0, lim inft→∞t

0Q sds > −∞ and n



j

q jt n

j

t

t −τ j t

b j

g jsds n

j

t

0

e−s t Q udub j sτ

j s − qj sQsds

n

j

t

0

e−s t Q uduQ ss

s −τ j s

b j

g judu ds 2⎛⎝t

0

e−2s t Q udu

n

j

c j s2ds⎞⎠

1/2

≤ α < 1,

2.31

where Q n j − bj gj t Then the zero solution of 2.7 is mean square asymptotically stable if

and only ift

0Q sds → ∞ as t → ∞.

Remark 2.5 When h j jgj Theorem 2.1reduces toCorollary 2.4 On the

other hand, we choose qj t ≡ cj t ≡ 0 and bj ≡ −bj for j Corollary 2.4reduces

to Theorem B

3 Two examples

In this section, we give two examples to illustrate applications of Theorem 2.1 and

Example 3.1 Consider the following linear neutral stochastic delay differential equation:

d x t− x t − t/2

1000



−x t−t/2

16 16t −

3sin t 4

48 48tx t−

t

4



dt x t

24√

3 4t−

x t−sin t

12√

3 4t



dW t.

3.1 Then the zero solution of3.1 is mean square asymptotically stable

8 16t

7

48 64t ,

11

48 64t ≤ Ht ≤

13

48 64t ,

2



j

t

t −τ j t

h jsds t

t/2

1

8 16s ds

t

3t/4

7

48 64s ds −→ 0.07479, as t −→ ∞,

2



j

t

0

e−t s H uduH ss

s −τ j s

h j udu ds≤t

0

e−s t 11/48 64udu 13

48 64s · 0.07479 ds ≤ 0.08839,

2

⎛

⎝t

0

e−2s t H udu



2



j

c j s2ds⎞⎠

1/2

≤ 2 t

0

e−s t 11/24 32udu 1

824 32sds

1/2

≤ 0.21320,

2



j

t

0

e−s t H udu hj s − τj s1 − τ

j s bjs − qjsHsds

≤

t

0

e−s t 11/48 64udu 0.013

48 64s

17

144 192s



ds≤ 0.013

11 17 33

3.2

It easy to check that ∞

0 H

Then, α 3.1 is mean square asymptotically stable by

Example 3.2 Consider the following delay differential equation:

6 4t x t−

t

3



− 1

12 4t x t−

2

3t



Then the zero solution of3.3 is asymptotically stable

Proof Choosing h1 2 Theorem 2.1, we have H

2



j

t

t −τ j t

h j sds t

2/3t

1

4 4s ds

t

t/3

1

4 4s ds−→

1

2ln 3− 1

4ln 2

2



j

t

0

e−s t H uduH ss

s −τ j s

h judu ds≤t

0

e−s t 1/2 2udu 1

2 2s · 0.37602 ds ≤ 0.37602.

3.4

Notice that qj j t ≡ 0 and

2



j

h j

s − τjs1− τ

j s bj s − qj sHs



123 8s·2

3 − 1

6 4s



 123 4s·1

3 − 1

12 4s

It is easy to see that all the conditions ofTheorem 2.1hold for α

1 Thus,Theorem 2.1implies that the zero solution of3.3 is asymptotically stable

However, Theorem B cannot be used to verify that the zero solution of 3.3 is

asymptotically stable In fact, b1 2 1g1

b2g2

2



j

t

t −τ j t

b j

g j sds≤t

2/3t

1

6 6s ds

t

t/3

1

12 12s ds−→

1

4ln 3−1

6ln 2

3.6 Notice that

2



j

b jsτ

j s − qj sQs

18 12s

1

18 6s ≤

1

4 4s . 3.7

It follows from3.7 that

2



j

t

0

e−s t Q udub jsτ

j s − qjsQsds≤t

0

e−s t 1/4 4udu 1

4 4s ds ≤ 1. 3.8

2.5 and obtained the following theorem

Theorem C Luo 6... class="text_page_counter">Trang 6

For any x, y ∈ S, it follows from 2.13, conditions i and ii, and Doob’s L p-inequalitysee

10...

2.20