Oscillation theorems for second order nonlinear differential equations.

Abstract:

The oscillatory and nonoscillatory behavior of the nonlinear second order differential equation $(1)\;x'' + p(t)f(x) = 0$ is related to that of ${(2)_\lambda }\;x'' + \lambda p(t)x = 0,\;\lambda > 0$. Under certain conditions on $p(t)$ and $f(x)$ it is shown that all solutions of $(1)$ are oscillatory if ${(2)_\lambda }$ is oscillatory for all $\lambda > 0$. In contrast to most of the literature on this subject, no sign or integrability conditions on $p(t)$ are explicitly assumed.