Oscillation of solutions of a generalized Liénard equation

Abstract:

The generalized Liénard equation, $\ddot x + f(x,\dot x) + h(x) = 0$, with $xh(x) > 0$ and $yf(x,y) > 0$ for nonzero x and y, is considered here, subject to the additional condition that $|f(x,y)|$ is not greater than $k(x)|y{|^\alpha }$ where $\alpha$ is a positive number and $k(x)$ is a continuous function which is positive for nonzero x. In case $\alpha \geqq 2$, all solutions of this Liénard equation are oscillatory. In case $0 < \alpha < 2$, sufficient conditions are given which insure that all solutions are oscillatory.