Homoclinic orbits in generalized Liénard systems

This paper is devoted to the investigation on the existence of homoclinic orbits of the planar system of Liénard type x˙=h(y)−F(x), y˙=−g(x). Here h(y) is strictly increasing, but is not imposed h(±∞)=±∞. Sufficient conditions are given for a positive orbit of the system starting at a point on the curve h(y)=F(x) to approach the origin without intersecting the x-axis. The obtained theorems include previous results as special cases. Our results are applied to a concrete system and their sharpness are improved.