Existence and uniqueness of limit cycles for generalized φ-Laplacian Liénard equations

The Liénard equation x″+f(x)x′+g(x)=0x″+f(x)x′+g(x)=0 appears as a model in many problems of science and engineering. Since the first half of the 20th century, many papers have appeared providing existence and uniqueness conditions for limit cycles of Liénard equations. In this paper we extend some of these results for the case of the generalized φ  -Laplacian Liénard equation, (φ(x′))′+f(x)ψ(x′)+g(x)=0(φ(x′))′+f(x)ψ(x′)+g(x)=0. This generalization appears when derivations of the equation different from the classical one are considered. In particular, the relativistic van der Pol equation, (x′/1−(x′/c)2)′+μ(x2−1)x′+x=0, has a unique periodic orbit when μ≠0μ≠0.