Multiplicity of periodic solutions to Duffing's equations with Lipschitzian condition (Q1591539)

The author proves the existence and multiplicity of periodic solutions to Duffing's equation \(x''+g(x)=p(t)\) when \(g(x)\) satisfies the conditions: (i) \(\lim_{|x|\to+\infty}sgn(x)g(x)=+\infty\), (ii) \(g(x)\) is Lipschitzian and the time mapping \(\tau(e)\) related to equation \(x''+g(x)=0\) satisfies the weaker oscillating property, namely, \(\limsup_{e\to+\infty}\sqrt{e}\tau (e)=+\infty\), and \(\liminf_{e\to+\infty}\sqrt{e}\tau (e)=-\infty\). This weaker oscillating condition was first introduced in \textit{D. Hao} and \textit{S. Ma} [J. Differ. Equations 133, No.~1, 98-116 (1997; Zbl 0877.34036)]. The proof is performed by estimating the time of solutions \((x(t), x'(t))\) going to one turn around the origin in the \((x, x')\)-plane and by using the generalized Poincaré-Birkhoff twist theorem.