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Fixed Points and Stability in Neutral Stochastic Differential Equations with Variable Delays
Fixed Point Theory and Applications volumeÂ 2008, ArticleÂ number:Â 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 [1â€“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 onedimensional 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
and proved the following theorem.
Theorem A (Burton [1]).
Suppose that and there exists a constant such that
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:
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
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:
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
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):
with the initial condition
where , , , and as and for each ,
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
Proof.
For each , denote by the Banach space of all adapted processes which are almost surely continuous in with norm
Moreover, we set for and , as .
At first, we suppose that (2.11) holds. Define an operator by for and for ,
Now, we show the mean square continuity of on . Let , and let be sufficiently small. Then
It is easy to verify that
It follows from the last term in (2.13) that
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
By condition (ii) and (2.11), there exists such that implies
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
For any , it follows from (2.13), conditions (i) and (ii), and Doob's inequality (see [10]) that
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:
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
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
for all . From (ii), we have
which implies
Therefore, the sequence has a convergent subsequence. Without loss of generality, we can assume that
for some . Let be an integer such that
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
It follows from (2.13) and (2.28) with that for ,
If the zero solution of (2.7) is mean square asymptotic stable, then as . Since , as and condition (ii) and (2.11) hold,
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
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:
Then the zero solution of (3.1) is mean square asymptotically stable.
Proof.
Choosing and in Theorem 2.1, we have
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:
Then the zero solution of (3.3) is asymptotically stable.
Proof.
Choosing in Theorem 2.1, we have and
Notice that and
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 ,
Notice that
It follows from (3.7) that
From (3.6), we obtain
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