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

- Meng Wu
^{1}, - Nan-jing Huang
^{1}Email author and - Chang-Wen Zhao
^{2}

**2008**:407352

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

© Meng Wu et al. 2008

**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 [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 .

and proved the following theorem.

Theorem A (Burton [1]).

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 [7]).

where . Then the zero solution of (2.3) is asymptotically stable if and only if , as .

and obtained the following theorem.

Theorem C (Luo [6]).

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).

Theorem 2.1.

Suppose that is differential, and there exist continuous functions for and a constant such that for

Proof.

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.

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.

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.

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

Proof.

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.

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

Proof.

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 .

## 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

## References

- Burton TA: Stability by fixed point theory or Liapunov theory: a comparison.
*Fixed Point Theory*2003, 4(1):15-32.MATHMathSciNetGoogle Scholar - Burton TA: Liapunov functionals, fixed points, and stability by Krasnoselskii's theorem.
*Nonlinear Studies*2002, 9(2):181-190.MATHMathSciNetGoogle Scholar - Burton TA: Fixed points and stability of a nonconvolution equation.
*Proceedings of the American Mathematical Society*2004, 132(12):3679-3687. 10.1090/S0002-9939-04-07497-0MATHMathSciNetView ArticleGoogle Scholar - Burton TA, Furumochi T: Fixed points and problems in stability theory for ordinary and functional differential equations.
*Dynamic Systems and Applications*2001, 10(1):89-116.MATHMathSciNetGoogle Scholar - Jung S-M: A fixed point approach to the stability of a Volterra integral equation.
*Fixed Point Theory and Applications*2007, 2007:-9.Google Scholar - Luo J: Fixed points and stability of neutral stochastic delay differential equations.
*Journal of Mathematical Analysis and Applications*2007, 334(1):431-440. 10.1016/j.jmaa.2006.12.058MATHMathSciNetView ArticleGoogle Scholar - Zhang B: Fixed points and stability in differential equations with variable delays.
*Nonlinear Analysis*2005, 63(5–7):e233-e242.MATHView ArticleGoogle Scholar - Kolmanovskii VB, Shaikhet LE: Matrix Riccati equations and stability of stochastic linear systems with nonincreasing delays.
*Functional Differential Equations*1997, 4(3-4):279-293.MATHMathSciNetGoogle Scholar - Liu K:
*Stability of Infinite Dimensional Stochastic Differential Equation with Applications, Chapman & Hall/CRC Monographs and Surveys in Pure and Applied Mathematics*.*Volume 135*. Chapman & Hall/CRC, Boca Raton, Fla, USA; 2006:xii+298.Google Scholar - Karatzas I, Shreve SE:
*Brownian Motion and Stochastic Calculus, Graduate Texts in Mathematics*.*Volume 113*. 2nd edition. Springer, New York, NY, USA; 1991:xxiv+470.Google Scholar

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