Periodic solutions of second order equations via rotation numbers
type: Item
instance of: scholarly article
title: Periodic solutions of second order equations via rotation numbers
zbMATH Open document ID: 1410.34125
author: Pedro J. Torres
author: Dingbian Qian
author: Pei-Yu Wang
published in: Journal of Differential Equations
publication date: 2019-02-08T00:00:00Z
review text: 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.
reviewed by: Guglielmo Feltrin
Mathematics Subject Classification ID: 34C25
Mathematics Subject Classification ID: 37E45
zbMATH DE Number: 7018384
zbMATH Keywords: periodic solution
zbMATH Keywords: second order equation
zbMATH Keywords: Poincaré-Birkhoff theorem
zbMATH Keywords: superlinear growth
zbMATH Keywords: rotation number
MaRDI profile type: Publication
full work available at URL: j.jde.2018.10.010
OpenAlex ID: W2897733703
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Wikidata QID: Q129092125
DOI: 10.1016/J.JDE.2018.10.010
description: scientific article; zbMATH DE number 7018384