Fixed Points and Stability in Neutral Stochastic Differential Equations with Variable Delays
© Meng Wu et al. 2008
Received: 4 April 2008
Accepted: 9 June 2008
Published: 16 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.
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 , Luo , and Zhang  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  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 .
and proved the following theorem.
Theorem A (Burton ).
for all and . Then, for every continuous initial function , the solution of (2.1) is bounded and tends to zero as .
and obtained the following theorem.
Theorem B (Zhang ).
where . Then the zero solution of (2.3) is asymptotically stable if and only if , as .
and obtained the following theorem.
Theorem C (Luo ).
Then the zero solution of (2.5) is mean square asymptotically stable if and only if as
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).
Suppose that is differential, and there exist continuous functions for and a constant such that for
Moreover, we set for and , as .
Therefore, is mean square continuous on .
Thus, , as . Similarly, we can show that , , as . Thus, as . This yields .
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 .
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.
for all , where satisfies .
which contradicts (2.29). Therefore, (2.11) is necessary for Theorem 2.1. This completes the proof.
Theorem 2.1 still holds if condition (ii) is satisfied for for some .
Theorem 2.1 improves Theorem C under different conditions.
where . Then the zero solution of (2.7) is mean square asymptotically stable if and only if as
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.
Then the zero solution of (3.1) is mean square asymptotically stable.
It easy to check that . Let . Then, and the zero solution of (3.1) is mean square asymptotically stable by Theorem 2.1.
Then the zero solution of (3.3) is asymptotically stable.
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.
Combining (3.6), (3.8), and (3.9), we see that the condition (2.4) of Theorem B does not hold with .
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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