Twist periodic solutions of repulsive singular equations
type: Item
instance of: scholarly article
title: Twist periodic solutions of repulsive singular equations
zbMATH Open document ID: 1058.34052
author: Pedro J. Torres
author: Meirong Zhang
published in: Nonlinear Analysis. Theory, Methods \& Applications. Series A: Theory and Methods
publication date: 2004-09-22T00:00:00Z
review text: The paper deals with the periodic problem \[ x^{\prime\prime}+g(t,x)=0, \quad x(0)=x(2\pi),\quad x^{\prime}(0)=x^\prime(2\pi), \] where \(g(t,x)\) is \(2\pi\)-periodic in \(t\) and \(C^{0,4}\) in \((t,x)\). A motivation to study this problem is due to particular equations of this type: the Lazer-Solimini equation describing the motion of a charged particle in an electric field and the Brillouin equation governing a focusing system for an electron beam immersed in a periodic magnetic field. A solution is said to be of twist type if the first coefficient of Birkhoff, also called the twist coefficient, is nonzero. If strong resonance is avoided, a solution of twist type is Lyapunov-stable. In the paper, by using a compact expression of the twist coefficient, a new stability criterion is derived for a (positive) periodic solution to be of twist type. On this base an estimation on the region of parameters is obtained, where the equations have twist periodic solutions. An application to the motivating equations is given.
reviewed by: Ivan Ginchev
Mathematics Subject Classification ID: 34C25
Mathematics Subject Classification ID: 34C60
Mathematics Subject Classification ID: 34D20
zbMATH DE Number: 2101854
zbMATH Keywords: twist periodic solutions
zbMATH Keywords: Lyapunov stability
zbMATH Keywords: singular equations
MaRDI profile type: Publication
full work available at URL: j.na.2003.10.005
OpenAlex ID: W2152784128
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DOI: 10.1016/J.NA.2003.10.005
description: scientific article; zbMATH DE number 2101854