On the uniqueness of limit cycles in a generalized Liénard system (Q2282112)

The paper is devoted to show that a result on uniqueness and hyperbolicity of the limit cycles for the Liénard system \[ x'=h(y)-F(x),\quad y'=-g(x), \] given by \textit{R. E. Kooij} and \textit{J. Sun} [J. Math. Anal. Appl. 208, No. 1, 260--276 (1997; Zbl 0872.34012)] fails, constructing an example under the hypotheses of that result with at least two limit cycles. The authors also propose some new hypotheses to be considered in order to get a uniqueness result. Their proof uses the classical tool of proving that the characteristic exponent of any limit cycle has a given sign, dividing the corresponding integral in several pieces.