On the existence and stability of periodic solutions for pendulum-like equations with friction and \(\phi \)-Laplacian
type: Item
instance of: scholarly article
title: On the existence and stability of periodic solutions for pendulum-like equations with friction and \(\phi \)-Laplacian
zbMATH Open document ID: 1275.34059
author: Pedro J. Torres
author: J. Ángel Cid
published in: Discrete and Continuous Dynamical Systems
publication date: 2013-03-19T00:00:00Z
review text: The authors consider the differential equation \[ (\phi (x'))'+cx'+g(x)=e(t)+s \] and discuss the existence and stability properties of periodic solutions. The function \(e\) is periodic and has zero average, and the numbers \(c\geq 0\) and \(s\in \mathbb{R}\) act as parameters. The assumptions on the function \(\phi\) are typical of the so-called \(\phi\)-Laplacian operators, and they include the examples \(\phi (z)=\frac{z}{\sqrt{1\pm z^2}}\) corresponding to the mean curvature and relativistic operators. With respect to the nonlinearity \(g\), the main focus is on the function \(g(x)=k\sin x\) leading to some variants of the forced pendulum equation. In the study of stability properties, the authors observe that the linearized equation can be reduced to an equation of Hill's type via a change of the independent variable.
reviewed by: Rafael Ortega
Mathematics Subject Classification ID: 34C25
Mathematics Subject Classification ID: 34C15
Mathematics Subject Classification ID: 34D20
zbMATH DE Number: 6146678
zbMATH Keywords: periodic solution
zbMATH Keywords: asymptotic stability
zbMATH Keywords: \(\phi\)-Laplacian
zbMATH Keywords: pendulum
MaRDI profile type: Publication
full work available at URL: dcds.2013.33.141
OpenAlex ID: W2321541387
DOI: 10.3934/DCDS.2013.33.141
description: scientific article; zbMATH DE number 6146678