When all trajectories in the Liénard plane cross the vertical isocline?

Abstract

In this paper we study the problem whether all trajectories of the system$\dot \chi$=y−F(x) and$\dot y$=−g(x) cross the vertical isocline which is very important for the existence of periodic solutions and oscillation theory. The problem has not been solved for the critical case:

$$\mathop {\lim }\limits_{x \to \infty } F(x) = - \infty and \mathop {\lim }\limits_{x \to \infty } \frac{1}{{F(x)}}\int_b^x {\frac{{g(\xi )}}{{F(\xi )}}d\xi = \frac{1}{4}} .$$

We concentrate our attention on this point and given an answer.