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Fixed Points and Stability in Neutral Stochastic Differential Equations with Variable Delays
Fixed Point Theory and Applications volume 2008, Article number: 407352 (2008)
Abstract
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.
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 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 be a complete filtered probability space and let
denote a one-dimensional standard Brownian motion defined on
such that
is the natural filtration of
. Let
and
with
and
as
. Here
denotes the set of all continuous functions
with the supremum norm
.
In 2003, Burton [1] studied the equation
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F407352/MediaObjects/13663_2008_Article_1081_Equ1_HTML.gif)
and proved the following theorem.
Theorem A (Burton [1]).
Suppose that and there exists a constant
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F407352/MediaObjects/13663_2008_Article_1081_Equ2_HTML.gif)
for all and
. Then, for every continuous initial function
, the solution
of (2.1) is bounded and tends to zero as
.
Recently, Zhang [7] studied the generalization of (2.1) as follows:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F407352/MediaObjects/13663_2008_Article_1081_Equ3_HTML.gif)
and obtained the following theorem.
Theorem B (Zhang [7]).
Suppose that is differential, the inverse function
of
exists, and there exists a constant
such that for
,
and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F407352/MediaObjects/13663_2008_Article_1081_Equ4_HTML.gif)
where . Then the zero solution of (2.3) is asymptotically stable if and only if
, as
.
Very recently, Luo [6] considered the following neutral stochastic differential equation:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F407352/MediaObjects/13663_2008_Article_1081_Equ5_HTML.gif)
and obtained the following theorem.
Theorem C (Luo [6]).
Let be derivable. Assume that there exists a constant
and a continuous function
such that for
,
and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F407352/MediaObjects/13663_2008_Article_1081_Equ6_HTML.gif)
Then the zero solution of (2.5) is mean square asymptotically stable if and only if as
Now, we consider the generalization of (2.5):
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F407352/MediaObjects/13663_2008_Article_1081_Equ7_HTML.gif)
with the initial condition
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F407352/MediaObjects/13663_2008_Article_1081_Equ8_HTML.gif)
where ,
,
,
and
as
and for each
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F407352/MediaObjects/13663_2008_Article_1081_Equ9_HTML.gif)
Note that (2.7) becomes (2.5) for ,
,
,
,
,
and
. 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 is differential, and there exist continuous functions
for
and a constant
such that for
-
(i)
,
-
(ii)
(2.10)
where .
Then the zero solution of (2.7) is mean square asymptotically stable if and only if
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F407352/MediaObjects/13663_2008_Article_1081_Equ11_HTML.gif)
Proof.
For each , denote by
the Banach space of all
-adapted processes
which are almost surely continuous in
with norm
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F407352/MediaObjects/13663_2008_Article_1081_Equ12_HTML.gif)
Moreover, we set for
and
, as
.
At first, we suppose that (2.11) holds. Define an operator by
for
and for
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F407352/MediaObjects/13663_2008_Article_1081_Equ13_HTML.gif)
Now, we show the mean square continuity of on
. Let
,
and let
be sufficiently small. Then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F407352/MediaObjects/13663_2008_Article_1081_Equ14_HTML.gif)
It is easy to verify that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F407352/MediaObjects/13663_2008_Article_1081_Equ15_HTML.gif)
It follows from the last term in (2.13) that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F407352/MediaObjects/13663_2008_Article_1081_Equ16_HTML.gif)
Therefore, is mean square continuous on
.
Next, we verify that . Since
,
as
, for each
, there exists a
such that
implies
and
. Thus, for
, the last term
in (2.13) satisfies
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F407352/MediaObjects/13663_2008_Article_1081_Equ17_HTML.gif)
By condition (ii) and (2.11), there exists such that
implies
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F407352/MediaObjects/13663_2008_Article_1081_Equ18_HTML.gif)
Thus, , as
. Similarly, we can show that
,
, as
. Thus,
as
. This yields
.
Now we show that is a contraction mapping. From (ii), we can choose
such that
. Thus, for each
, we can find a constant
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F407352/MediaObjects/13663_2008_Article_1081_Equ19_HTML.gif)
For any , it follows from (2.13), conditions (i) and (ii), and Doob's
-inequality (see [10]) that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F407352/MediaObjects/13663_2008_Article_1081_Equ20_HTML.gif)
Therefore, is contraction mapping with contraction constant
. By the contraction mapping principle,
has a fixed point
, which is a solution of (2.7) with
on
and
as
.
To obtain the mean square asymptotic stability, we need to show that the zero solution of (2.7) is mean square stable. Let be given and choose
and
satisfying the following condition:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F407352/MediaObjects/13663_2008_Article_1081_Equ21_HTML.gif)
where . If
is a solution of (2.7) with
, then
defined in (2.13). We assume that
for all
. Notice that
for
. If there exists
such that
and
for
, then (2.13) and (2.19) imply that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F407352/MediaObjects/13663_2008_Article_1081_Equ22_HTML.gif)
which contradicts the definition of . 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 with
as
such that
, where
. Then, we can choose a constant
satisfying
for all
. Denote
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F407352/MediaObjects/13663_2008_Article_1081_Equ23_HTML.gif)
for all . From (ii), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F407352/MediaObjects/13663_2008_Article_1081_Equ24_HTML.gif)
which implies
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F407352/MediaObjects/13663_2008_Article_1081_Equ25_HTML.gif)
Therefore, the sequence has a convergent subsequence. Without loss of generality, we can assume that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F407352/MediaObjects/13663_2008_Article_1081_Equ26_HTML.gif)
for some . Let
be an integer such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F407352/MediaObjects/13663_2008_Article_1081_Equ27_HTML.gif)
for all , where
satisfies
.
Now we consider the solution of (2.7) with
and
for
. By the similar method in (2.22), we have
for
. We may choose
so that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F407352/MediaObjects/13663_2008_Article_1081_Equ28_HTML.gif)
It follows from (2.13) and (2.28) with that for
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F407352/MediaObjects/13663_2008_Article_1081_Equ29_HTML.gif)
If the zero solution of (2.7) is mean square asymptotic stable, then as
. Since
,
as
and condition (ii) and (2.11) hold,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F407352/MediaObjects/13663_2008_Article_1081_Equ30_HTML.gif)
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 for some
.
Remark 2.3.
Theorem 2.1 improves Theorem C under different conditions.
Corollary 2.4.
Suppose that is differential, the inverse function
of
exists, and there exists a constant
such that for
,
and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F407352/MediaObjects/13663_2008_Article_1081_Equ31_HTML.gif)
where . Then the zero solution of (2.7) is mean square asymptotically stable if and only if
as
Remark 2.5.
When for
, Theorem 2.1 reduces to Corollary 2.4. On the other hand, we choose
and
for
, 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:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F407352/MediaObjects/13663_2008_Article_1081_Equ32_HTML.gif)
Then the zero solution of (3.1) is mean square asymptotically stable.
Proof.
Choosing and
in Theorem 2.1, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F407352/MediaObjects/13663_2008_Article_1081_Equ33_HTML.gif)
It easy to check that . Let
. Then,
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:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F407352/MediaObjects/13663_2008_Article_1081_Equ34_HTML.gif)
Then the zero solution of (3.3) is asymptotically stable.
Proof.
Choosing in Theorem 2.1, we have
and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F407352/MediaObjects/13663_2008_Article_1081_Equ35_HTML.gif)
Notice that and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F407352/MediaObjects/13663_2008_Article_1081_Equ36_HTML.gif)
It is easy to see that all the conditions of Theorem 2.1 hold for . 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, ,
,
,
, and
. As
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F407352/MediaObjects/13663_2008_Article_1081_Equ37_HTML.gif)
Notice that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F407352/MediaObjects/13663_2008_Article_1081_Equ38_HTML.gif)
It follows from (3.7) that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F407352/MediaObjects/13663_2008_Article_1081_Equ39_HTML.gif)
From (3.6), we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F407352/MediaObjects/13663_2008_Article_1081_Equ40_HTML.gif)
Combining (3.6), (3.8), and (3.9), we see that the condition (2.4) of Theorem B does not hold with .
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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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Wu, M., Huang, Nj. & Zhao, CW. Fixed Points and Stability in Neutral Stochastic Differential Equations with Variable Delays. Fixed Point Theory Appl 2008, 407352 (2008). https://doi.org/10.1155/2008/407352
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DOI: https://doi.org/10.1155/2008/407352