【作者】 倪华；

【导师】 林发兴；

【作者基本信息】 福州大学 ， 基础数学， 2006， 硕士

【摘要】 本文考虑非线性微分方程 dx/da =A( t ) x f (t , x) (*) 及其线性化方程 dx/dt=A( t )x (**) 的零解、概周期解的存在性、唯一性和稳定性。首先,利用指数型二分性和稳定性有关理论得到了系统(*)的零解稳定性可由其线性部分来决定,即(**)的零解的一致稳定性、一致渐进稳定性和不稳定性可蕴涵(*)的零解的一致稳定性、一致渐进稳定性和不稳定性,推广了李雅普诺夫的关于非线性微分方程零解的稳定性的相关定理,获得了一些便于应用的新的广泛的结论。接着,本文将这个结论应用到三类方程。(一)非线性微分方程; 对概周期非线性微分方程(*)进行了研究,利用指数型二分性、压缩映像原理以及稳定性有关理论得到了系统(*)的唯一的概周期解的存在性,并着重讨论了概周期解的稳定性,在一定的条件下,并分别得到了系统(*)存在着和系统(**)的零解相同稳定性的概周期解,并进一步将该结论推广到有界解、周期解的存在性、唯一性以及稳定性。(二)非线性时滞微分方程,对非线性时滞微分方程dx/dt= A( t ) x f (t , x (t r)) (***) 的概周期解进行了研究,得到了系统(***)在一定条件下存在着和非线性微分方程相似但又略有区别的结论。(三)Lienard方程; 研究了著名的Lienard方程的概周期解的存在性和稳定性和零解的稳定性的蕴涵关系。最后,对全文作了归纳、总结。更多还原

【Abstract】 This paper deals with the zero solutions of nonlinear differential equation dx/da =A( t ) x f (t , x) (*) and its linear equation dx/dt=A( t )x (**)Firstly, in the less stronger conditions , we obtain that the stability of zero solution of (*) is decided by the zero solution of (**) with the help of exponential dichotomy and theories of stability. That is to say , in the less stronger conditions, the uniform stability ,uniformly asymptotic stability and instability of zero solution of system (*) is decided by the uniform stability ,uniformly asymptotic stability and instability of zero solution of system (**) .This paper extends the existed conclusions of the corresponding articles and obtains the new results of wide applications. Next, we put the conclusions to three aspects. (一) nonlinear differential equation; we study the almost periodic solutions of system (*) , mainly by using exponential dichotomy, fixed pointed theories and theories of stability, we obtain the existence, uniqueness and stability of the system (*) , especially discuss the stability of the almost periodic solution. Under the given conditions, we obtain that the system (*) exists the same stability of the unique almost periodic solution as the stability of zero solution of the system (**) , we further extend the results to the bounded solution and periodic solution of the system (*) and obtain the existence ,uniqueness and stability of the bounded solution and periodic solution of the system (*) . (二) nonlinear differential equation with delay; the equation dx/dt= A( t ) x f (t , x (t r)) (***)Is considered and its almost periodic solutions are discussed , the existence ,uniqueness and stability of the system (***) are obtained, the results we obtained are mainly the same as the system (*) ,but have a little different from (*) ,the results are very interesting and well applied. (三)Lienard equation; we study the existence and stability of almost periodic solutions of famous Lienard equation. At last ,we draw a conclusion of the total paper.更多还原