A criterion of the nonexistence of periodic solutions to a generalized Liénard system and its application (Q1288277)

For the generalized Liénard system: \[ \dot x=P\bigl(Q(y)-F(x) \bigr),\quad \dot y=-g(x),\tag{1} \] where \(P,Q,F\), \(g:\mathbb{R} \to\mathbb{R}\) are continuous, \(F(0)=0\), \(xg(x)>0\) for \(x\neq 0\); \(P(y)\) is strictly monotone increasing, \(P(\pm\infty)= \pm\infty\), \(yP(y)>0\) for \(y\neq 0\); \(Q(y)\) satisfies the same conditions; then \(O(0,0)\) is the only critical point of (1). When \(P(y) \equiv y\), \(Q(y)\equiv y\), (1) is reduced to a Liénard system. Under some other conditions, the authors prove that (1) has no nontrivial period solution, which generalizes and extends some known results of \textit{J. Sugie} and \textit{T. Yoneyama} [Math. Proc. Camb. Philos. Soc. 113, No. 2, 413-422 (1993; Zbl 0778.34030)]; \textit{Z. Zheng} [Nonlinear Anal., Theory Methods Appl. 16, No. 2, 101-110 (1991; Zbl 0760.34037)], \textit{J. Zhou} [Nonlinear Anal., Theory Methods Appl. 27, No. 12, 1463-1470 (1996; Zbl 0864.34031)], \textit{L. Huang} [Ann. Differ. Equations 10, No. 1, 16-23 (1994; Zbl 0798.34050)].