The Extension Of The Theories About Liapunov's Concerning The Stability Of Zero Solutions Of Nonlinear Differential Equations And Its Applications

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.