关于二阶非线性常微分方程的振动问题

利用与文相似的方法研究二阶非线性常微分方程(A)(r(t)x′)′ f(t,x)=0的振动问题,得到主要结果定理1,并作为对特殊情形的应用导出了二阶微分方程(B)x″ q(t)x′ p(t)f(x)=0的一切解均振动的充分条件(推论1).同时指出,由文中定理2也可导出两个关于方程(B)为振动的相仿而又不同的充分条件(推论2及3).文中的推论2.1及2.2包括在本文的推论3之中.本文所讨论的方程(A)比文中研究的二阶方程更为一般.

ON THE OSCILLATION OF SECOND ORDER NONLINEAR DIFFERENTIAL EQUATIONS

The question of second order nonlinear differential equation (A) (r(t) x') ' f ( t, x) = 0 is examined by the similar method used in paper one,and as a main result, theorem 1 is hereby obtained, and is particularly applied to the second order differential equation (B) x" q(t)x' p (t)f(x) = 0. The sufficient condition ( corollary 1 )which is thus obtained determines that all solutions of equation ( B ) are oscillatory. At the same time, two similar yet different conditions ( corllary 2 and 3 ) which are deduced from theordm 2 in paper one determine that all solutions of equation ( B ) are oscillatory. Corollaries 2.1 and 2.2 are included in corollary 3 It is noticeable that equation ( A ) here ismore general than the second order differential equation in paper one.