三类不动点与一类随机动力系统的稳定性

Three kinds of fixed points and stability of a class of stochastic dynamic systems

不动点理论已被成功地应用于随机动力系统零解稳定性的研究,但Krasnoselskii不动点方法使用的较少.本文在采用Banach和Schauder不动点方法研究的基础上进一步采用Krasnoselskii不动点方法研究了一类随机动力系统零解的指数均方稳定性,得出了使得该系统零解指数均方稳定的充分条件.通过实例与现有文献结论的比较表明,相比于Banach和Schauder等不动点方法,Krasnoselskii不动点方法的应用更加灵活和简便.本文的结论在一定程度上改进和拓展了相关文献的结果,完善了不动点理论在研究随机动力系统零解稳定性上的应用.

The fixed point theory has been successfully applied onto the study of zero solution stability for stochastic dynamic systems; however, the Krasnoselskii fixed point is relatively less used. In this paper, on the basis of the study for Banach and Schauder fixed point methods, we furtherly use the Krasnoselskii fixed point method to study the mean square exponential stability of zero solution in a class of stochastic dynamic systems. At the same time, we have achieved sufficient condition to make the zero solution exponential mean square of this system stable. Through the comparison of the detailed example and the conclusion in the existing articles, and comparing with Banach and Schauder fixed point methods,the Krasnoselskii fixed point is more flexible and more convenient. The conclusion in this paper has improved and extended the result of relative articles to some extent, as well as has perfected the application of fixed point methods on studying stability of zero solution in stochastic dynamical system.