Open Access

Fixed Points and Stability in Neutral Stochastic Differential Equations with Variable Delays

Fixed Point Theory and Applications20082008:407352

https://doi.org/10.1155/2008/407352

Received: 4 April 2008

Accepted: 9 June 2008

Published: 16 June 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 [14], 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
(2.1)

and proved the following theorem.

Theorem A (Burton [1]).

Suppose that and there exists a constant such that
(2.2)

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:
(2.3)

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
(2.4)

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:
(2.5)

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
(2.6)

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):
(2.7)
with the initial condition
(2.8)
where , , , and as and for each ,
(2.9)

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

  1. (i)

    ,

     
  2. (ii)
    (2.10)
     

where .

Then the zero solution of (2.7) is mean square asymptotically stable if and only if
(2.11)

Proof.

For each , denote by the Banach space of all -adapted processes which are almost surely continuous in with norm
(2.12)

Moreover, we set for and , as .

At first, we suppose that (2.11) holds. Define an operator by for and for ,
(2.13)
Now, we show the mean square continuity of on . Let , and let be sufficiently small. Then
(2.14)
It is easy to verify that
(2.15)
It follows from the last term in (2.13) that
(2.16)

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
(2.17)
By condition (ii) and (2.11), there exists such that implies
(2.18)

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
(2.19)
For any , it follows from (2.13), conditions (i) and (ii), and Doob's -inequality (see [10]) that
(2.20)

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:
(2.21)
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
(2.22)

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
(2.23)
for all . From (ii), we have
(2.24)
which implies
(2.25)
Therefore, the sequence has a convergent subsequence. Without loss of generality, we can assume that
(2.26)
for some . Let be an integer such that
(2.27)

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
(2.28)
It follows from (2.13) and (2.28) with that for ,
(2.29)
If the zero solution of (2.7) is mean square asymptotic stable, then as . Since , as and condition (ii) and (2.11) hold,
(2.30)

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
(2.31)

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:
(3.1)

Then the zero solution of (3.1) is mean square asymptotically stable.

Proof.

Choosing and in Theorem 2.1, we have
(3.2)

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:
(3.3)

Then the zero solution of (3.3) is asymptotically stable.

Proof.

Choosing in Theorem 2.1, we have and
(3.4)
Notice that and
(3.5)

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 ,
(3.6)
Notice that
(3.7)
It follows from (3.7) that
(3.8)
From (3.6), we obtain
(3.9)

Combining (3.6), (3.8), and (3.9), we see that the condition (2.4) of Theorem B does not hold with .

Declarations

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).

Authors’ Affiliations

(1)
Department of Mathematics, Sichuan University
(2)
College of Business and Management, Sichuan University

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Copyright

© Meng Wu et al. 2008

This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.