Subharmonic solutions for nonlinear second order equations in presence of lower and upper solutions
type: Item
instance of: scholarly article
title: Subharmonic solutions for nonlinear second order equations in presence of lower and upper solutions
zbMATH Open document ID: 1275.34058
author: Alberto Boscaggin
author: Fabio Zanolin
published in: Discrete and Continuous Dynamical Systems
publication date: 2013-03-19T00:00:00Z
review text: This paper deals with the second-order equation \[ u''+f(t,u)=0, \] where \(f\) is \(T\)-periodic in \(t\). The authors prove the existence of infinitely many sub-harmonic solutions, and they also obtain precise information on the minimal periods and the number of zeros of these solutions. The main assumptions are the existence of ordered upper and lower solutions, \(\beta (t)>\alpha (t)\), together with a \(T\)-periodic solution \(u^* (t)\) lying between \(\alpha\) and \(\beta\) and satisfying the condition \[ \int_0^T \frac{\partial f}{\partial u} (t,u^* (t))\,dt >0. \] This elegant result can be applied to many known equations, in particular to the forced pendulum equation.
reviewed by: Rafael Ortega
Mathematics Subject Classification ID: 34C25
Mathematics Subject Classification ID: 34C15
Mathematics Subject Classification ID: 37J10
Mathematics Subject Classification ID: 37J45
zbMATH DE Number: 6146675
zbMATH Keywords: periodic solution
zbMATH Keywords: sub-harmonic solution
zbMATH Keywords: Poincaré-Birkhoff
zbMATH Keywords: lower and upper solution
MaRDI profile type: Publication
full work available at URL: dcds.2013.33.89
OpenAlex ID: W2252765792
DOI: 10.3934/DCDS.2013.33.89
description: scientific article; zbMATH DE number 6146675