Periodic solutions of second order equations via rotation numbers
The paper deals with periodic solutions of the planar system \[ x'=-y, \quad y'=f(t,x), \] where \(f: \mathbb{R}\times\mathbb{R}\to\mathbb{R}\) is continuous, \(2\pi\)-periodic with respect to the first variable, locally Lipschitz-continuous with respect to the second variable, and such that \(f(t,0)\equiv 0\). Inspired by the so-called admissible spiral method introduced by Fonda and Sfecci and generalizing the classical results by Jacobowitz and Hartman, the authors exploit a rotation number approach in order to obtain multiplicity results. More precisely, the analysis of the gap between the rotation number of related linear systems at zero and at infinity allows them to apply the Poincaré-Birkhoff theorem, without needing global existence of the Cauchy problems or sign conditions on \(f\). As an application they consider a partially superlinear second-order equation.
scientific article; zbMATH DE number 7018384
scientific article