An. Şt. Univ. Ovidius Constanţa
Vol. 20(1), 2012, 467–488
Stability of a Class of Nonlinear Neutral
Stochastic Differential Equations with Variable
Time Delays
Meng Wu, Nanjing Huang and Changwen Zhao
Abstract
In this paper, we study the mean square asymptotic stability of a
class of generalized nonlinear neutral stochastic differential equations
with variable time delays by using fixed point theory. An asymptotic
mean square stability theorem with a necessary and sufficient condition is proved which improves and generalizes some well-known results.
Finally, two examples are given to illustrate our results.
1
Introduction
It is well know that stochastic differential equation plays a very important
role in formulation and analysis in mechanical, electrical, control engineering,
neural network, economic and social sciences. Stochastic delay differential
equation, also known as stochastic functional differential equation, is a natural generalization of stochastic ordinary differential equation by allowing the
coefficients to depend on the past values. Recently, the studies of stochastic
differential equations have attracted the considerable attentions of scholars.
Many interesting results concerned with stochastic differential equations have
Key Words: fixed point, stability, neutral stochastic differential equations, variable
delays.
2010 Mathematics Subject Classification: Primary 34K50; Secondary 34K20.
Received: August, 2010.
Revised: May, 2011.
Accepted: February, 2012.
467
468
Meng Wu, Nanjing Huang and Changwen Zhao
been obtained over the last few years (see, for example, [13, 19] and the references therein).
Liapunov’s direct method has been successfully used to investigate stability problems in deterministic/stochastic differential equations and functional
differential equations for more than one hundred years. However, there are
many difficulties encountered in the study of stability by means of Liapunov’s
direct method. Recently, Burton and other scholars studied the stability for
deterministic systems by using fixed point theory which overcame the difficulties encountered in the study of stability by means of Liapunov’s direct
method (see, for example, [2, 3, 4, 5, 6, 7, 8, 9, 10, 14, 20, 23, 24] and the
references therein).
Very recently, many scholars have began to deal with the stability of
stochastic delay differential equations by using fixed point theory (see, for
example, [1, 2, 14, 15, 16, 17, 18, 21, 22]). More precisely, Appleby in [1] (also
see [2], pp.315-328) considered the almost sure stability for some classical
equations by splitting the stochastic differential equation into two equations,
one is a fixed stochastic problem and the other is a deterministic stability
problem with forcing function. Luo [14] studied the mean square asymptotic
stability for a class of linear scalar neutral stochastic differential equations by
means of fixed point theory. Furthermore, Luo [15, 16], Luo and Taniguchi [18]
used fixed point theory to study the exponential stability of mild solutions of
stochastic partial differential equations with bounded delays and with infinite
delays. Wu et al. [21, 22] applied fixed point theory to study the stability of a
general linear neutral stochastic differential equation and a half-linear neutral
stochastic differential equation with variable delays respectively. Luo [17] investigated the exponential stability for the classical stochastic Volterra-Levin
equations by using fixed point theory.
Motivated by the previous works mentioned above, in this paper, we study
the mean square asymptotic stability of a nonlinear neutral stochastic differential equation with variable delays by applying fixed point theory. An
asymptotic mean square stability theorem with a necessary and sufficient condition is proved. Two examples are given to illustrate our results. The results
presented in this paper improve and generalize the main results in [3], [14],
[23] and [24].
2
Main Results
Let (Ω, F, {Ft }t≥0 , P ) be a complete filtered probability space and W (t) 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 a(t), b(t), b̄(t), c(t), e(t),
STABILITY OF A CLASS OF NONLINEAR NEUTRAL STOCHASTIC
DIFFERENTIAL EQUATIONS WITH VARIABLE TIME DELAYS
469
q(t) ∈ C(R+ , R) and τ (t), δ(t) ∈ C(R+ , R+ ) with t − τ (t) → ∞ and t − δ(t) →
∞ as t → ∞. Here C(S1 , S2 ) denotes the set of all continuous function
φ : S1 → S2 with the supremum norm k · k.
Burton [3] and Zhang [23] studied the equation
x′ (t) = −b̄(t)x(t − τ (t))
(1)
and proved the following theorems.
Theorem A. (Burton [3]) Suppose that τ (t) = r and there exists a constant α < 1 such that
Z s
Z t
Z t
R
− st b̄(u+r)du
|b̄(u + r)|duds ≤ α
|b̄(s + r)|e
|b̄(s + r)|ds +
0
t−r
s−r
R∞
for all t ≥ 0 and 0 b̄(s)ds = ∞. Then, for every continuous initial function
φ : [−r, 0] → R, the solution x(t) = x(t, 0, φ) of (1) is bounded and tends to
zero as t → ∞.
Theorem B. (Zhang [23]) Suppose that τ is differentiable, the inverse
function g(t) of t − τ (t) exists, and there exists a constant α ∈ (0, 1) such that
Rt
for t ≥ 0, lim inf t→∞ 0 b̄(g(s))ds > −∞ and
Z t R
Z t
t
e− s b̄(g(u))du |b̄(s)||τ ′ (s)|ds
|b̄(g(s))|ds +
0
t−τ (t)
+
Z
t
e−
Rt
s
b̄(g(u))du
0
|b̄(g(s))|
Z
s
|b̄(g(v))|dvds ≤ α < 1.
(2)
s−τ (s)
Rt
Then the zero solution of (1) is asymptotically stable if and only if 0 b̄(g(s))ds
→ ∞, as t → ∞.
Obviously, Theorem B improves Theorem A. Recently, Zhang [24] studied
the following half-linear equation
x′ (t) = −a(t)x(t) + b(t)g(x(t − τ (t)))
(3)
where g : R → R is continuous and obtained Theorem C.
Theorem C. (Zhang [24]) Suppose that τ (t) ≥ 0 such that for t ≥ 0,
t − τ (t) → ∞ as t → ∞, and there exists a constant L > 0, for |x|, |y| ≤ L,
Rt
|g(x) − g(y)| ≤ |x − y| and q(0) = 0. For t > 0, lim inf t→∞ 0 a(s)ds > −∞
and
Z t R
t
e− s a(u)du |b(s)|ds < 1.
(4)
sup
t≥0
0
Then the zero solution of (3) is asymptotically stable if and only if
∞, as t → ∞.
Rt
0
a(s)ds →
470
Meng Wu, Nanjing Huang and Changwen Zhao
Recently, Luo [14] considered a linear neutral stochastic differential equation
d[x(t) − q(t)x(t − τ (t))]
=
[a(t)x(t) + b(t)x(t − τ (t))]dt
+[c(t)x(t) + e(t)x(t − δ(t))]dW (t)
(5)
and obtained Theorem D.
Theorem D. (Luo [14]) Let τ (t) be differentiable. Assume that there
exists a constant α ∈ (0, 1) and a continuous function h(t) : [0, ∞) → R such
Rt
that for t ≥ 0, lim inf t→∞ 0 h(s)ds > −∞ and
Z s
Z t R
t
|a(u) + h(u)|duds
e− s h(u)du |h(s)|
|q(t)| +
s−τ (s)
0
t
+
Z
+
Z
e−
Rt
h(u)du
| a(s − τ (s)) + h(s − τ (s)) (1 − τ ′ (s))
0
Z t
|a(s) + h(s)|ds
+ b(s) − q(s)h(s)|ds +
s
t−τ (t)
t
e
−2
Rt
s
h(u)du
2
(|c(s)| + |e(s)|) ds
0
12
≤ α < 1.
Then the zero solution of (5) is mean square asymptotically stable if and only
Rt
if 0 h(s)ds → ∞, as t → ∞.
Very recently, Wu et al. [21, 22] generalized Theorems B, C, and D to
a general linear neutral stochastic differential equation and to a half-linear
neutral stochastic differential equation, respectively. In general, time delay
and system uncertainty are commonly encountered and are often sources of
instability (see [12]). Thus, it should be interesting to consider the nonlinear
stochastic differential equation and study the stability of nonlinear stochastic
differential equation with variable time delays.
In this paper, we consider a class of nonlinear neutral stochastic differential
equations,
d[x(t) − k t, x(t − τ (t) ] = [a(t)x(t) + f t, x(t), x(t − τ (t)) ]dt
+[c(t)x(t) + g t, x(t), x(t − δ(t)) ]dW (t) (6)
with the initial condition
x(s) = φ(s)
for s ∈ [m(0), 0],
where f, g : [0, ∞) × R × R → R and k : [0, ∞) × R → R are continuous,
φ ∈ C([m(0), 0], R), x : [m(0), ∞) × Ω → R and
m(0) = min inf{s − τ (s), s ≥ 0}, inf{s − δ(s), s ≥ 0} ≤ 0.
STABILITY OF A CLASS OF NONLINEAR NEUTRAL STOCHASTIC
DIFFERENTIAL EQUATIONS WITH VARIABLE TIME DELAYS
471
If f t, x(t), x(t −τ (t)) = b(t)x(t − τ (t)), g t, x(t), x(t − δ(t)) = e(t)x(t −
δ(t)), k t, x(t − τ (t) = q(t)x(t − τ (t)), then it is easy to see that (6) reduces
to (5). Thus, (6) includes (1), (3)and (5) as special cases. And so, the main
aim of this paper is to generalize Theorems B, C and D to apply to (6).
For any φ ∈ C([m(0), 0], R), we define kφk = sups∈[m(0),0] |φ(s)|. For each
λ > 0, we define C(λ) := {φ ∈ C([m(0), 0], R) : kφk ≤ λ}. Denote by F the
Banach space of all F-adapted processes ψ(t, ω) : [m(0), ∞) × Ω → R which
are almost surely continuous in t with norm
kψkF =
E
2 sup ψ(s, ω)
s≥m(0)
21
.
[r,t]
Further, we define F (λ) = {ψ ∈ F : kψkF ≤ λ} for each λ > 0. Let kψkF =
2 1
2 1
[0,∞)
E(sups∈[r,t] ψ(s, ω) ) 2 for r < t. Then kψkF
= E(sups≥0 ψ(s, ω) ) 2 .
Theorem 2.1. Suppose that τ is differentiable, and there exist continuous
functions h(t) : [0, ∞) → R, l(t), m(t), n(t) : [0, ∞) → R+ and constants
L > 0, α ∈ (0, 1) such that
Rt
(i) lim inf t→∞ 0 h(s)ds > −∞;
(ii) for any t ≥ 0,
n(t) +
+
Z
Z
Rt
t
e
−
0
t
s
Rt
h(u)du
h(u)du
|h(s)|
Z
s
|a(u) + h(u)|duds
s−τ (s)
|(a(s − τ (s)) + h(s − τ (s)))(1 − τ ′ (s))|
Z t
|a(s) + h(s)|ds
+ l(s) + m(s) + |h(s)|n(s) ds +
e−
s
0
t−τ (t)
+2
Z
t
e
−2
Rt
s
h(u)du
2
(|c(s)| + l(s) + m(s)) ds
0
12
≤ α < 1;
(iii) for any t ≥ 0,
|f (t, x, y) − f (t, x̄, ȳ)|
and
_
|g(t, x, y) − g(t, x̄, ȳ)| ≤ l(t)|x − x̄| + m(t)|y − ȳ|
|k(t, x) − k(t, x̄)| ≤ n(t)|x − x̄|
for all x, x̄, y, ȳ ∈ F (L) with f (t, 0, 0) = g(t, 0, 0) = k(t, 0) = 0.
472
Meng Wu, Nanjing Huang and Changwen Zhao
Then the zero solution of (6) is mean square asymptotically stable if and only
if
Z t
h(s)ds → ∞ as t → ∞.
(7)
0
Proof. At first, we suppose that (7) holds.
Choose δ > 0, δ < L such that
Rt
2δK + αL ≤ L, where K = supt≥0 {e− 0 h(s)ds }. Let φ ∈ C(δ) and set
S = x : [m(0), ∞) × Ω → R x(t, ω) = φ(t) for t ∈ [m(0), 0],
x(t, ω) ∈ F (L) for t ≥ 0, E|x(t, ω)|2 → 0 as t → ∞ .
Then it is easy to check that S is a closed subset of F . From the definitions
of k · k and k · kF , for any x ∈ S and t > 0,
[0,∞)
kxkF = max{kφk, kxkF
} ≤ L.
(8)
Define an operator P : S → S by (P x)(t) = φ(t) for t ∈ [m(0), 0] and for
t ≥ 0,
=
(P x)(t)
Z
φ(0) − k 0, φ(−τ (0)) −
+ k t, x(t − τ (t)) +
Z
t
e−
Rt
h(u)du
h(u)du
Z
0
−τ (0)
t
R
t
(a(s) + h(s))φ(s)ds e− 0 h(s)ds
(a(s) + h(s))x(s)ds
t−τ (t)
(a(s − τ (s)) + h(s − τ (s)))(1 − τ ′ (s))x(s − τ (s))
+ f s, x(s), x(s − τ (s)) − h(s)k s, x(s − τ (s)) ds
Z s
Z t R
t
e− s h(u)du h(s)
a(u) + h(u) x(u)du ds
−
+
s
0
s−τ (s)
0
+
:=
Z
5
X
t
e−
0
Rt
s
c(s)x(s) + g s, x(s), x(s − δ(s))
dW (s)
Ii (t).
(9)
i=1
Now, we show the mean square continuity of P on [0, ∞). Let x ∈ S, T1 > 0
and |r| be sufficiently small. Then
E|(P x)(T1 + r) − (P x)(T1 )|2 ≤ 5
5
X
i=1
E|Ii (T1 + r) − Ii (T1 )|2 .
STABILITY OF A CLASS OF NONLINEAR NEUTRAL STOCHASTIC
DIFFERENTIAL EQUATIONS WITH VARIABLE TIME DELAYS
473
It is easy to verify that
E|Ii (T1 + r) − Ii (T1 )|2 → 0,
as r → 0,
i = 1, 2, 3, 4.
From the last term I5 in (9), we have
E|I5 (T1 + r) − I5 (T1 )|2
Z T1 R
R T1 +r
T1
e− s h(u)du e− T1 h(u)du − 1
E 0
· c(s)x(s) + g s, x(s), x(s − δ(s)) dW (s)
2
Z T1 +r R
− sT1 +r h(u)du
c(s)x(s) + g s, x(s), x(s − δ(s)) dW (s)
e
+
=
T1
T1
≤
2E
Z
e−2
0
RT
s
1
h(u)du −
e
R T1 +r
T1
h(u)du
2
− 1
2
· c(s)x(s) + g s, x(s), x(s − δ(s)) ds
Z T1 +r
R T +r
2
1
h(u)du e−2 s
+ 2E
c(s)x(s) + g s, x(s), x(s − δ(s)) ds
T1
→
0, as r → 0.
Therefore, P is mean square continuous on [0, ∞).
Next, we verify that kP xkF ≤ L. As φ ∈ C(δ) and x ∈ S,
[0,∞)
kP xkF
=
1
1
5
2 2
X
2 2
E sup P x(s)
= E sup
Ii (s)
s≥0
≤
5 X
i=1
s≥0
2 E sup Ii (s)
s≥0
12
i=1
.
(10)
By condition (iii), we have
|f (t, x, y)|
_
|g(t, x, y)| ≤ l(t)|x| + m(t)|y|
and
|k(t, x)| ≤ n(t)|x|
(11)
for all x, y ∈ F (L). It follows from (9), (10), (11), condition (ii) and Doob’s
474
Meng Wu, Nanjing Huang and Changwen Zhao
Lp -inequality (see [11]) that
[0,∞)
kP xkF
≤
sup{e−
Rs
0
h(v)dv
s≥0
Z
+

0
} · |φ(0)| + n(0) · |φ(−τ (0))|
|a(v) + h(v)| · |φ(v)|dv
−τ (0)
+ E sup n(s) · |x(s − τ (s))| +
s≥0
Z
+ E sup
s≥0
s
e−
Rs
v
h(u)du
Z
s
s−τ (s)
!2  21
|a(v) + h(v)| · |x(v)|dv 
|(a(v − τ (v)) + h(v − τ (v)))(1 − τ ′ (v))|
0
2 12
·|x(v − τ (v))| + |f (v, x(v), x(v − δ(v)))| + |h(v)||k(v, x(v − τ (v)))| dv
1
Z v
Z s R
2 2
s
|a(u) + h(u)| · |x(u)|du dv
e− v h(u)du |h(v)|
+ E sup
s≥0
Z
+2 sup E
s≥0
≤
v−τ (v)
0
s
e−2
Rs
v
h(u)du
0
δK 1 + n(0) +
(
Z
0
|a(v) + h(v)|dv
−τ (0)
+kxkF · sup n(s) +
s≥0
2 12
|c(v)| · |x(v)| + |g(v, x(v), x(v − δ(v)))| dv
Z
s
|a(v) + h(v)|dv +
s−τ (s)
Z
s
e−
Rs
v
h(u)du
0
′
· |(a(v − τ (v)) + h(v − τ (v)))(1 − τ (v))| + l(v) + m(v) + |h(v)|n(v) dv
Z v
Z s R
s
e− v h(u)du |h(v)|
|a(u) + h(u)|du dv
+
0
+2
≤
Z
v−τ (v)
s
e
−2
Rs
v
h(u)du
2
|c(v)| + l(v) + m(v) dv
0
2δK + αL ≤ L.
21 )
Further, from (8), we have
[0,∞)
kP xkF = max{kφk, kP xkF
} ≤ L.
Thirdly, we verify that E|(P x)(t)|2 → 0 as t → ∞. Since E|x(t)|2 → 0,
t − δ(t) → ∞ as t → ∞, for each ǫ > 0, there exists a T1 > 0 such that s ≥ T1
implies E|x(s)|2 < ǫ and E|x(s − δ(s))|2 < ǫ. By condition (ii), for t ≥ T1 ,
STABILITY OF A CLASS OF NONLINEAR NEUTRAL STOCHASTIC
DIFFERENTIAL EQUATIONS WITH VARIABLE TIME DELAYS
475
the last term I5 in (9) satisfies
2
E I5 (t)
Z T1
2
Rt
e−2 s h(u)du |c(s)| · |x(s)| + l(s)|x(s)| + m(s)|x(s − δ(s))| ds
≤ E
0
Z t
2
Rt
e−2 s h(u)du |c(s)| · |x(s)| + l(s)|x(s)| + m(s)|x(s − δ(s))| ds
+E
T1
≤
≤
≤
[m(0),T1 ] 2
kxkF
kxk2F · e
L2 α 2 e
−2
−2
Rt
T1
Rt
T1
Z
T1
e−2
Rt
s
h(u)du
0
h(u)du
h(u)du
Z
T1
e−2
RT
1
s
|c(s)| + l(s) + m(s)
h(u)du
0
+ αǫ.
2
ds + αǫ
|c(s)| + l(s) + m(s)
−2
Rt
2
ds + αǫ
h(u)du
T1
< ǫ for t ≥ T2 .
From (7), there exists T2 > T1 such that L2 α2 e
2
2
Thus, for t ≥ T2 , E I5 (t) < ǫ + αǫ. This proves that E I5 (t) → 0, as
2
t → ∞. Similarly, we can show that E Ii (t) → 0, i = 1, 2, 3, 4, as t → ∞.
Thus, E|(P x)(t)|2 → 0 as t → ∞. Hence P x ∈ S.
Now we show that P : S → S is a contraction mapping. For any x, y ∈ S,
we have
=
kP x − P ykF
2 1
E sup (P x)(s) − (P y)(s) 2
s≥m(0)
=
E sup k s, x(s − τ (s)) − k s, y(s − τ (s))
s≥0
Z s R
Z s
− vs h(u)du
(a(v − τ (v))
e
(a(v) + h(v))(x(v) − y(v))dv +
+
0
s−τ (s)
′
+ h(v − τ (v)))(1 − τ (v))(x(v − τ (v)) − y(v − τ (v)))
+ f (v, x(v), x(v − τ (v))) − f (v, y(v), y(v − τ (v)))
dv
− h(v) k v, x(v − τ (v)) − k v, y(v − τ (v))
Z v
Z s R
− vs h(u)du
(a(u) + h(u))(x(u) − y(u))du dv
e
h(v)
−
v−τ (v)
0
+
Z
s
e−
Rs
v
h(u)du
c(v)(x(v) − y(v)) + g(v, x(v), x(v − τ (v)))
0
2 21
− g(v, y(v), y(v − τ (v))) dW (v)
476
≤
Meng Wu, Nanjing Huang and Changwen Zhao
(
kx − ykF · sup n(s) +
s≥0
+
Z
s
e
−
Rs
v
h(u)du
|h(v)|
Z
s
|a(v) + h(v)|dv
s−τ (s)
Z v
|a(u) + h(u)|dudv +
v−τ (v)
0
Z
s
e−
Rs
v
h(u)du
0
· |(a(v − τ (v)) + h(v − τ (v)))(1 − τ ′ (v))| + l(v) + m(v) + |h(v)|n(v) dv
Z s
21 )
R
−2 vs h(u)du
2
e
+2
(|c(v)| + l(v) + m(v)) dv
0
≤
αkx − ykF .
Therefore, P : S → S is contraction mapping and so P has a fixed point x ∈ S,
which is a solution of (6) with x(s) = φ(s) on [m(0), 0] and E|x(t)|2 → 0 as
t → ∞.
To obtain the mean square asymptotic stability, we need to show that the
zero solution of (6) is mean square stable. From (ii), we can choose ∆ > 0
such that α2 + ∆ < 1. Thus, we can find a constant N > 0 such that
Z s
Z t R
1
− st h(u)du
|a(u) + h(u)|duds
e
|h(s)|
(1 + ) n(t) +
N
s−τ (s)
0
Z t R
Z t
t
e− s h(u)du
|a(s) + h(s)|ds +
+
0
t−τ (t)
· |(a(s − τ (s)) + h(s − τ (s)))(1 − τ ′ (s))| + l(s) + m(s) + |h(s)|n(s) ds
+ 4(1 + N )
Z
t
e−2
Rt
s
h(u)du
(|c(s)| + l(s) + m(s))2 ds ≤ α2 + ∆ < 1.
2
(12)
0
Let ǫ > 0 and ǫ < L be given and choose δ0 > 0 and δ0 < ǫ satisfying the
following condition
4(1 + N )δ02 K 2 + (α2 + ∆)ǫ < ǫ,
where N is defined in (12). If x(t) = x(t, 0, φ) is a solution of (6) with kφk < δ0 ,
then x(t) = (P x)(t) which is defined in (9). We claim that E|x(t)|2 < ǫ for
all t ≥ 0. Notice that x(t) = φ(t) for t ∈ [m(0), 0], so E|x(t)|2 , kφ(t)k2 < ǫ for
t ∈ [m(0), 0]. If there exists t∗ > 0 such that E|x(t∗ )|2 = ǫ and E|x(t)|2 < ǫ
STABILITY OF A CLASS OF NONLINEAR NEUTRAL STOCHASTIC
DIFFERENTIAL EQUATIONS WITH VARIABLE TIME DELAYS
477
for t ∈ [m(0), t∗ ), then (9) and (12) imply that
E|x(t∗ )|2
≤
Z
(1 + N )kφk2 1 + n(0) +
0
|a(s) + h(s)|ds
−τ (0)
2
e−2
R t∗
0
h(u)du
Z t∗ R ∗
Z s
t
1 +ǫ(1 + ) |q(t∗ )| +
|a(u) + h(u)|du |h(s)|ds
e− s h(u)du
N
s−τ (s)
0
Z t∗
Z t∗ R ∗
t
+
|a(s) + h(s)|ds +
e− s h(u)du
t∗ −τ (t∗ )
0
2
· |(a(s − τ (s)) + h(s − τ (s)))(1 − τ ′ (s))| + l(s) + m(s) + |h(s)|n(s) ds
Z t∗
R t∗
e−2 s h(u)du (|c(s)| + l(s) + m(s))2 ds
+ǫ
0
≤
(1 + N )δ02 1 + n(0) +
<
ǫ,
Z
0
−τ (0)
R t∗
2
|a(s) + h(s)|ds e−2 0 h(u)du + (α2 + ∆)ǫ
(13)
which contradicts the definition of t∗ . Thus, the zero solution of (6) is mean
square asymptotically stable if (7) holds.
Conversely, we suppose that (7) fails. From condition (i), there exists a
Rt
sequence {tn } with tn → ∞ as n → ∞ such that lim 0 n h(u)du = ζ for some
n→∞
Rt
ζ ∈ R. Then, we can choose a constant J > 0 satisfying 0 n h(u)du ∈ [−J, J]
for all n ≥ 1. Denote
ω(s)
:=
|(a(s − τ (s)) + h(s − τ (s)))(1 − τ ′ (s))|
Z s
+ l(s) + m(s) + |h(s)|n(s) + |h(s)|
|a(u) + h(u)|du,
s−τ (s)
for all s ≥ 0. From condition (ii), we have
Z
tn
e−
R tn
s
h(u)du
ω(s)ds ≤ α,
0
which implies
Z
tn
e
Rs
0
h(u)du
ω(s)ds ≤ αe
R tn
0
h(u)du
≤ eJ .
0
Rt Rs
Therefore, the sequence { 0 n e 0 h(u)du ω(s)ds} has a convergent subsequence.
478
Meng Wu, Nanjing Huang and Changwen Zhao
Without loss of generality, we can assume that lim
n→∞
γ for some γ > 0. Let m be an integer such that
Z
tn
e
Rs
0
h(u)du
ω(s)ds <
tm
R tn
0
e
Rs
0
h(u)du
ω(s)ds =
δ1
8K
(14)
for all n ≥ m, where 0 < δ1 < 1 satisfies 8δ12 K 2 e2J + (α2 + ∆) < 1.
Now, we consider the solution x(t) = x(t, tm , φ) of (6) with kφ(tm )k = δ1
and kφ(t)k < δ1 for t < tm . By the similar method in (13), we have E|x(t)|2 <
1 for t ≥ tm . We may choose φ so that
G(tm )
:=
φ(tm ) − k tm , φ(tm − τ (tm ))
Z tm
δ1
(a(s) + h(s))φ(s)ds ≥ .
−
2
tm −τ (tm )
(15)
It follows from (9), (14) and (15) with x(t) = (P x)(t) that for n ≥ m,
≥
2
(a(s) + h(s))x(s)ds
tn −τ (tn )
Z tn R
R tn
R tn
tn
G2 (tm )e−2 tm h(u)du − 2G(tm )e− tm h(u)du
e− s h(u)du ω(s)ds
tn
Z
E x(tn ) − k tn , x(tn − τ (tn )) −
R tn
h(u)du
R tm
tm
tn
Z
Rs
≥
G(tm )e−2
≥
Z tn R
s
δ1 −2 Rttn h(u)du δ1
δ2
h(u)du
0
m
e
ω(s)ds ≥ 1 e−2J > 0. (16)
e
− 2K
2
2
8
tm
tm
G(tm ) − 2e−
0
h(u)du
e
0
h(u)du
ω(s)ds
tm
If the zero solution of (6) is mean square asymptotic stable, then E|x(t)|2 =
E|x(t, tm , φ)|2 → 0 as t → ∞. Since tn − τ (tn ) → ∞, tn − δ(tn ) → ∞ as
n → ∞ and conditions (ii) and (iii) hold, then
E x(tn ) − k tn , x(tn − τ (tn )) −
Z
tn
tn −τ (tn )
2
(a(s) + h(s))x(s)ds → 0,
as n → ∞ which contradicts (16). Thus, (7) is necessary for Theorem 2.1.
This completes the proof.
Remark 2.1. Theorem 2.1 is still true if condition (ii) is satisfied for t ≥ ta
with some ta ∈ R+ .
STABILITY OF A CLASS OF NONLINEAR NEUTRAL STOCHASTIC
DIFFERENTIAL EQUATIONS WITH VARIABLE TIME DELAYS
479
Remark 2.2. The method in this paper can be extended to the following nonlinear neutral stochastic differential equation with several variable delays:
n
X
ki t, x(t − τi (t)
d x(t) −
i=1
=
n
X
a(t)x(t) +
fi t, x(t), x(t − τi (t))
i=1
dt
m
X
gj t, x(t), x(t − δj (t)) dW (t).
+ c(t)x(t) +
j=1
Choosing h(t) ≡ −a(t) in Theorem 2.1, we have the following result.
Corollary 2.1. Suppose that τ is differential, and there exist continuous functions l(t), m(t), n(t) : [0, ∞) → R+ and constants L > 0, α ∈ (0, 1) such that
Rt
(i′ ) lim inf t→∞ 0 −a(s)ds > −∞;
(ii′ ) for any t ≥ 0,
n(t) +
Z
t R
t
e
s
a(u)du
0
+2
Z
t
e
2
Rt
s
l(s) + m(s) + |a(s)|n(s) ds
a(u)du
2
(|c(s)| + l(s) + m(s)) ds
0
12
≤ α < 1;
(iii′ ) for any t ≥ 0,
|f (t, x, y) − f (t, x̄, ȳ)|
and
_
|g(t, x, y) − g(t, x̄, ȳ)| ≤ l(t)|x − x̄| + m(t)|y − ȳ|
|k(t, x) − k(t, x̄)| ≤ n(t)|x − x̄|
for all x, x̄, y, ȳ ∈ F (L) with f (t, 0, 0) = g(t, 0, 0) = k(t, 0) = 0.
Then the zero solution of (6) is mean square asymptotically stable if and only
Rt
if 0 a(s)ds → ∞ as t → ∞.
Now, we consider a special case of nonlinear neutral stochastic differential
equation (6) that
d[x(t) − k t, x(t − τ (t) ]
= [a(t)x(t) + b(t)x(t − τ (t)) + f t, x(t), x(t − τ (t)) ]dt
+ [c(t)x(t) + e(t)x(t − δ(t)) + g t, x(t), x(t − δ(t)) ]dW (t), (17)
480
Meng Wu, Nanjing Huang and Changwen Zhao
Note that (17) reduces to (5)
when f t, x(t), x(t − τ (t)) ≡ g t, x(t), x(t −
δ(t)) ≡ 0 and k t, x(t − τ (t) = q(t)x(t − τ (t)). Then, we have the following
results.
Theorem 2.2. Suppose that τ is differentiable, and there exist continuous
functions h(t) : [0, ∞) → R, l(t), m(t), n(t) : [0, ∞) → R+ and constants
L > 0, α ∈ (0, 1) such that
Rt
(i◦ ) lim inf t→∞ 0 h(s)ds > −∞;
(ii◦ ) for any t ≥ 0,
n(t) +
+
Z
Z
t
e−
Rt
s
h(u)du
0
t
|h(s)|
Z
s
|a(u) + h(u)|duds
s−τ (s)
Z t
e−
|a(s) + h(s)|ds +
Rt
s
h(u)du
l(s) + m(s) + |h(s)|n(s)
0
t−τ (t)
+ |(a(s − τ (s)) + h(s − τ (s)))(1 − τ ′ (s)) + b(s)| ds
Z t
12
R
−2 st h(u)du
2
e
+2
(|c(s)| + |e(s)| + l(s) + m(s)) ds
≤ α < 1;
0
(iii◦ ) for any t ≥ 0,
|f (t, x, y) − f (t, x̄, ȳ)|
and
_
|g(t, x, y) − g(t, x̄, ȳ)| ≤ l(t)|x − x̄| + m(t)|y − ȳ|
|k(t, x) − k(t, x̄)| ≤ n(t)|x − x̄|
for all x, x̄, y, ȳ ∈ F (L) with f (t, 0, 0) = g(t, 0, 0) = k(t, 0) = 0.
Then the zero solution of (17) is mean square asymptotically stable if and only
Rt
if 0 h(s)ds → ∞ as t → ∞.
The proof is analogous to that of Theorem 2.1 and so we omit it here.
Remark 2.3. Theorem 2.2 improves Theorem D under different conditions.
Let h(t) ≡ −b(p(t)) in Theorem 2.2. Then we have the following corollary.
Corollary 2.2. Suppose that τ is differential, the inverse function p(t) of
t − τ (t) exists, and there exist continuous functions l(t), m(t), n(t) : [0, ∞) →
R+ and constants L > 0, α ∈ (0, 1) such that
Rt
(i∗ ) lim inf t→∞ 0 −b(p(s))ds > −∞;
STABILITY OF A CLASS OF NONLINEAR NEUTRAL STOCHASTIC
DIFFERENTIAL EQUATIONS WITH VARIABLE TIME DELAYS
481
(ii∗ ) for any t ≥ 0,
n(t) +
+
Z
t
Z
t R
t
e
s
b(p(u))du
|b(p(s))|
0
Z
|a(s) − b(p(s))|ds +
s
|a(u) − b(p(u))|duds
s−τ (s)
Z t R
e
0
t−τ (t)
t
s
b(p(u))du
· l(s) + m(s)
+ |b(p(s))|n(s) + |(a(s − τ (s)) − b(s))(1 − τ ′ (s)) + b(s)| ds
+2
Z
t
e
2
Rt
s
b(p(u))du
2
(|c(s)| + |e(s)| + l(s) + m(s)) ds
0
12
≤ α < 1;
(iii∗ ) for any t ≥ 0,
|f (t, x, y) − f (t, x̄, ȳ)|
and
_
|g(t, x, y) − g(t, x̄, ȳ)| ≤ l(t)|x − x̄| + m(t)|y − ȳ|
|k(t, x) − k(t, x̄)| ≤ n(t)|x − x̄|
for all x, x̄, y, ȳ ∈ F (L) with f (t, 0, 0) = g(t, 0, 0) = k(t, 0) = 0.
Then the zero solution of (17) is mean square asymptotically stable if and only
Rt
if 0 b(p(s))ds → ∞ as t → ∞.
Remark 2.4. When k(t, x(t − τ (t))) ≡ f (t, x(t), x(t − τ (t))) ≡ g(t, x(t), x(t −
δ(t))) ≡ a(t) ≡ c(t) ≡ e(t) ≡ 0 and b(t) ≡ −b̄(t), we know that Corollary 2.2
still holds if the condition (ii∗ ) is replaced by (2). Therefore, Corollary 2.2 is
a generalization of Theorem B.
Now, we consider another special case of nonlinear neutral stochastic differential equation (6) that
dx(t) = −a(t)x(t) + f t, x(t), x(t − τ (t)) dt,
(18)
Note that (18) reduces to (3) when f t, x(t), x(t − τ (t)) ≡ b(t)g x(t − τ (t)) .
Then, we have the following result.
Theorem 2.3. Suppose that τ is differential, and there exist continuous functions h(t) : [0, ∞) → R, l(t), m(t) : [0, ∞) → R+ and constants L > 0,
α ∈ (0, 1) such that
(i⋆ ) lim inf t→∞
Rt
0
h(s)ds > −∞;
482
Meng Wu, Nanjing Huang and Changwen Zhao
(ii⋆ ) for any t ≥ 0,
Z
t
e
−
Rt
s
h(u)du
0
+
Z
t
|h(s)|
Z
s
|h(u) − a(u)|duds
s−τ (s)
|h(s) − a(s)|ds +
t−τ (t)
Z
t
e−
Rt
s
h(u)du
l(s) + m(s)
0
+ |(h(s − τ (s)) − a(s − τ (s)))(1 − τ ′ (s))| ds ≤ α < 1;
(iii⋆ ) for any t ≥ 0,
|f (t, x, y) − f (t, x̄, ȳ)| ≤ l(t)|x − x̄| + m(t)|y − ȳ|
for all x, x̄, y, ȳ ∈ F (L) with f (t, 0, 0) = 0.
Then the zero solution of (18) is asymptotically stable if and only if
∞ as t → ∞.
Rt
0
h(s)ds →
The proof is analogous to that of Theorem 2.1 and so we omit it here.
Remark 2.5. When f t, x(t), x(t−τ (t)) ≡ b(t)g x(t−τ (t)) , choosing h(t) ≡
a(t), l(t) ≡ 0 and m(t) ≡ |b(t)|, Corollary 2.3 reduces to Theorem C.
3
Two Examples
In this section, we give two examples to illustrate the applications of our
main results.
Example 3.1. Consider the following nonlinear neutral stochastic delay differential equation
=
1
3t d x(t) − x2
8
4
e−2t
3t 3t − 2x(t) +
sin x(t) + x( ) · cos x(t) − x( ) dt
8
4
4
e−t
t t 1
cos x(t) + x( ) · sin x(t) − x( ) dW (t). (19)
+ x(t) +
9
7
2
2
Then the zero solution of (19) is mean square asymptotically stable.
STABILITY OF A CLASS OF NONLINEAR NEUTRAL STOCHASTIC
DIFFERENTIAL EQUATIONS WITH VARIABLE TIME DELAYS
483
Proof. Let
f (t, x(t), x(t − τ (t))
:=
g(t, x(t), x(t − δ(t))
:=
e−2t
t t sin x(t) + x(t − ) · cos x(t) − x(t − ) ,
8
4
4
t t e−t
cos x(t) + x(t − ) · sin x(t) − x(t − ) .
7
2
2
Since | sin x| ≤ |x| for x ∈ R, the we get
|f (t, x(t), x(t − τ (t)) − f (t, y(t), y(t − τ (t))|
e−2t t e−2t
t
|x(t) − y(t)| +
≤
x(t − ) − y(t − ),
8
8
4
4
|g(t, x(t), x(t − δ(t)) − g(t, y(t), y(t − δ(t))|
t
e−t t e−t
|x(t) − y(t)| +
≤
x(t − ) − y(t − ).
7
7
2
2
As |x2 | ≤ |x| when |x| ≤ 1, we can choose L = 1/2, l(t) = m(t) = e−t /7 and
n(t) = 1/8 such that the condition (iii) of Theorem 2.1 holds. Moreover, it
is easy to verify that t − τ (t) = t − t/4 → ∞ and t − δ(t) = t − t/2 → ∞ as
t → ∞.
Choosing h(t) = 2 in Theorem 2.1, we have
Z t R
t
e− s h(u)du |(a(s − τ (s)) + h(s − τ (s)))(1 − τ ′ (s))| + l(s) + m(s)
0
Z t
2e−s
1
e−2(t−s)
+|h(s)|n(s) ds =
+
ds ≤ 0.18,
7
4
0
12
Z t
Rt
e−2 s h(u)du (|c(s)| + l(s) + m(s))2 ds
2
0
and
Z t
t−τ (t)
2e−s
Z t
1 12
≤ 0.68,
e−2(t−s)
=2
ds
+
7
9
0
|a(s) + h(s)|ds =
Z
t
e
0
−
Rt
s
h(u)du
|h(s)|
Z
s
|a(u) + h(u)|duds = 0.
s−τ (s)
Rt
It easy to check that 0 h(s)ds → ∞ as t → ∞. Let α = 1/8+0.18+0.68. Then
α = 0.985 < 1 and so the zero solution of (19) is mean square asymptotically
stable.
Example 3.2. Consider the following delay differential equation
1
1
′
−t
−t
x (t) = − x(t) + 2e sin
.
x t−e
5
10
(20)
484
Meng Wu, Nanjing Huang and Changwen Zhao
Then the zero solution of (20) is asymptotically stable.
Proof. Let
f (t, x(t), x(t − τ (t)) := 2e−t sin
x
Since | sin 10
|≤
1
10 |x|
1
x t − e−t .
10
for x ∈ R, we have
|f (t, x(t), x(t − τ (t)) − f (t, y(t), y(t − τ (t))| ≤
e−t
|x(t − e−t ) − y(t − e−t )|.
5
Therefore, we can choose l(t) ≡ 0, m(t) = e−t /5 and L for any positive
constant such that the condition (iii⋆ ) of Theorem 2.3 holds. Moreover, it is
easy to verify that t − τ (t) = t − e−t → ∞ as t → ∞. Choosing h(t) ≡ 0.3 in
Theorem 2.3, we have
t
Z
|h(s) − a(s)|ds =
t−τ (t)
Z t R
− st h(u)du
e
|h(s)|
0
Z
s
Z
t
(0.3 − 0.2)ds → 0.1e−t ≤ 0.1,
t−e−t
|h(u) − a(u)|duds ≤
s−τ (s)
Z
t
0
0.03
ds ≤ 0.1,
e0.3(t−s)
and
Z
=
Z
t
e−
Rt
s
h(u)du
0
t
|(h(s − τ (s)) − a(s − τ (s)))(1 − τ ′ (s))| + l(s) + m(s) ds
e−0.3(t−s) [(0.3 − 0.2)(1 + e−s ) +
0
e−s
]ds ≤ 0.36.
5
It is easy to see that all conditions of Theorem 2.3 hold for α = 0.1 + 0.1 +
0.36 = 0.56 < 1. Thus, Theorem 2.3 implies that the zero solution of (20) is
asymptotically stable.
However, Theorem C can not be used to verify that the zero solution of
y
x
− sin 10
| ≤ |x − y|
(20) is asymptotically stable. In fact, noticing that | sin 10
−t
for all x, y ∈ R, b(t) ≡ 2e , a(t) ≡ 1/5 and
Z
t
e−
0
Rt
s
a(u)du
|b(s)|ds =
Z
t
e−0.2(t−s) · 2e−s ds < 1.33.
0
Obviously, the condition (4) of Theorem C does not hold with α = 1.33 >
1.
STABILITY OF A CLASS OF NONLINEAR NEUTRAL STOCHASTIC
DIFFERENTIAL EQUATIONS WITH VARIABLE TIME DELAYS
485
Acknowledgement
This work was supported by the National Natural Science Foundation of China
(71101099, 11171237) and the Fundamental Research Funds for the Central
Universities (2009SCU11096).
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Meng Wu,
College of Business Administration,
Sichuan University,
Chengdu, Sichuan 610064, China.
Email: wumeng@scu.edu.cn, shancherish@hotmail.com
STABILITY OF A CLASS OF NONLINEAR NEUTRAL STOCHASTIC
DIFFERENTIAL EQUATIONS WITH VARIABLE TIME DELAYS
Nanjing Huang,
Department of Mathematics,
Sichuan University,
Chengdu, Sichuan 610064, China.
Email: njhuang@scu.edu.cn
Changwen Zhao,
College of Business Administration,
Sichuan University,
Chengdu, Sichuan 610064, China.
Email: cwzhao@scu.edu.cn
487
488
Meng Wu, Nanjing Huang and Changwen Zhao