Intersection with the vertical isocline in the Liénard plane

In this paper, we study the problem of whether all trajectories of the Liénard system ẋ=y−F(x) and ẏ=−g(x) cross the vertical isocline, which is very important for the existence of periodic solutions and oscillation theory. We solve this problem completely in some sense. Our results include a proof for an old conjecture of Hara and Sugie.