关于一般Liénard系统解的唯一性问题的注记

研究dxdt=h(y)-F(x),dydt=-g(x)关于初值问题解的唯一性问题,给出了如下定理:定理A,设系统(2)仅有有限奇点,若F(x)和g(x)在R上连续,h(y)在R上具有连续导数且h′(y)>0,则系统(2)满足初始条件x(t0)=x0,y(t0)=y0的解唯一.其中M0(x0,y0)不为奇点.同时,当h(y)为严格下凸函数时,给出了类似的定理B.

A note on uniqueness of solution of generic Liénand to the initial value problem

In this paper,the problem on the uniqueness of solution to the initial value problem of the system dxdt=h(y)-F(x),dydt=-g(x)is discussed and the following theorems are obtained:Theorem A,let system(2) have at most finite singular points,if F(x) and g(x) are continuous in R,and h′(y)>0 is continuous,then the trajectory of (2) passing through arbitrary point M_0(x_0,y_0) is unique,where M_0 is not a singular point.Meanwhile,when h(y) is a strictly lower convex function,we also give the similar theorem B.