Existence of positive solutions of the Liénard differential equation (Q1260882)

Existence and qualitative properties are obtained for positive solutions \(x(t)\) of the Liénard-type differential equation (1) \(x''+f(x)x'+g(x)=0\), \(t\geq t_ 0\). In particular, conditions on \(f\), \(g\) are given for the existence of a positive solution such that \(0<x'(t)<1\), \(\lim_{t\to\infty}x(t)=+\infty\), and \(\lim_{t\to\infty}x'(t)<1\). Also, a positive decreasing solution is obtained with both \(x(t)\) and \(x'(t)\) decaying to zero at \(\infty\). Corollaries concern nonoscillatory properties and the nonexistence of periodic solutions.