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 $$dx/dt=h(y)-F(x),\ dy/dt=-g(x)$$ . We will give a counterexample to their theorem. Moreover, we shall give some sufficient conditions for the existence, uniqueness and hyperbolicity of limit cycles.