On the asymptotic behavior of second order nonlinear differential systems of Li´enard type

In this paper we consider the generalized Li´enard system x˙ = φ (z − F(x)), z˙ =−g(x), where φ is strictly increasing and φ (±∞) = ±∞. We present some sufficient and necessary conditions for this system to have a positive and a negative semiorbit which starts at a point on the curve z = F(x) and approaches the origin without intersecting the x-axes which are very important in the theory of oscillation and global asymptotic stability of the solutions of this system.