On the Uniqueness of Limit Cycles in a Generalized Liénard System

Kooij and Sun (J Math Anal Appl 208:260–276, 1997) proposed a theorem to guarantee the uniqueness of limit cycles for the generalized Liénard system dx/dt=h(y)-F(x),dy/dt=-g(x)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$dx/dt=h(y)-F(x),\ dy/dt=-g(x)$$\end{document}. We will give a counterexample to their theorem. Moreover, we shall give some sufficient conditions for the existence, uniqueness and hyperbolicity of limit cycles.