Intersection with the vertical isocline in the generalized Liénard equations

We consider the generalized Liénard systemdxdt=1a(x)[h(y)−F(x)],dydt=−a(x)g(x), where a, F, g, and h are continuous functions on R and a(x)>0a(x)>0, for x∈Rx∈R. Under the assumptions that the origin is a unique equilibrium, we study the problem whether all trajectories of this system intersect the vertical isocline h(y)=F(x)h(y)=F(x), which is very important in the global asymptotic stability of the origin, oscillation theory, and existence of periodic solutions. Under quite general assumptions we obtain sufficient and necessary conditions which are very sharp. Our results extend the results of Villari and Zanolin, and Hara and Sugie for this system with h(y)=yh(y)=y, and a(x)=1a(x)=1 and improve the results presented by Sugie et al. and Gyllenberg and Ping.