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**Attraction, stability and robustness for stochastic functional differential equations with infinite delay.**
*(English)*
Zbl 1241.60028

The authors established a LaSalle theorem for a stochastic functional differential equation with infinite delay. The main result is given in Theorem 1, which contains existence and uniqueness of the global solution, and attraction and boundedness of this solution. They assume some Lipschitz condition on the coefficient functions more general than the linear growth assumption. The proof is based on the standard truncation technique [X. Mao, Exponential stability of stochastic differential equations. New York: Marcel Dekker (1994; Zbl 0806.60044), Theorem 3.2.2]. This result extends some other results given by X. Mao [Stochastic differential equations and their applications. Chichester: Horwood Publishing (1997; Zbl 0892.60057)], Y. Ren and N. Xia [Appl. Math. Comput. 214, No. 2, 457–461 (2009; Zbl 1221.34222)], R.Z. Khas’minskiĭ [Stochastic stability of differential equations. Rockville, Maryland, USA: Sijthoff & Noordhoff (1980; Zbl 0441.60060)], X. Mao and M. J. Rasias [Stochastic Anal. Appl. 23, No. 5, 1045–1069 (2005; Zbl 1082.60055)], Y. Shen, Q. Luo and X. Mao [J. Math. Anal. Appl. 318, No. 1, 134–154 (2006; Zbl 1090.60059)].

Other very important results are given in Theorem 2 and Theorem 3 under a slightly modified assumption than the above mentioned Lipschitz condition. As an interesting application of these results, a scalar stochastic integro-differential equation with infinite delay is considered.

Other very important results are given in Theorem 2 and Theorem 3 under a slightly modified assumption than the above mentioned Lipschitz condition. As an interesting application of these results, a scalar stochastic integro-differential equation with infinite delay is considered.

Reviewer: Romeo Negrea (Timisoara)

### MSC:

60H10 | Stochastic ordinary differential equations (aspects of stochastic analysis) |

34F05 | Ordinary differential equations and systems with randomness |

34K20 | Stability theory of functional-differential equations |

### Keywords:

attraction; stability; robustness; stochastic functional differential equations; infinite delay### Citations:

Zbl 0806.60044; Zbl 0892.60057; Zbl 1221.34222; Zbl 0441.60060; Zbl 1082.60055; Zbl 1090.60059
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\textit{F. Wu} and \textit{S. Hu}, Automatica 47, No. 10, 2224--2232 (2011; Zbl 1241.60028)

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### References:

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