On the homoclinic orbits of the generalized Liénard equations

In this work we study the existence of homoclinic orbits of the planar system of Liénard type ẋ=1a(x)[h(y)−F(x)],ẏ=−a(x)g(x), where a(x)>0a(x)>0, for every x∈Rx∈R, and hh is strictly increasing, but it is not assumed that h(±∞)=±∞h(±∞)=±∞, h(y)≤myh(y)≤my, or h(y)≥myh(y)≥my. We present sufficient and necessary conditions for this system to have a positive orbit which starts at a point on the curve h(y)=F(x)h(y)=F(x) and approaches the origin without intersecting the xx-axis. The conditions obtained are very sharp. Our results extend the results presented by Hara and Yoneyama for this system with a(x)=1a(x)=1, and h(y)=yh(y)=y, and improve the results presented by Sugie.