Periodic solutions of some differential equations with nonlinear damping
type: Item
instance of: scholarly article
title: Periodic solutions of some differential equations with nonlinear damping
zbMATH Open document ID: 1023.34039
author: Pedro J. Torres
published in: Differential and Integral Equations
publication date: 2003-11-02T00:00:00Z
review text: Here, the author studies the existence and uniqueness of periodic solutions to the following equation \[ x'' + F(x, x') = p(t), \] where \(p\) is periodic in \(t\). Under some reasonable assumptions, he proves the existence of periodic solutions using the upper-and-lower solutions method. He gets a series of results when the function \(F\) satisfies one of the following conditions (1) \(F: \mathbb{R}^2\to \mathbb{R}\) is continuous in \(x\) and \(x^\prime\) (nonsingular case); (2) \(F: (0,+\infty)\times \mathbb{R}\to \mathbb{R}\) and \(\lim_{x\to 0^+}F(x,0) = +\infty\) (case of one singularity); (3) \(F: (0,1)\times \mathbb{R}\to \mathbb{R}\) and \(\lim_{x\to 0^+}F(x,0) = +\infty\), \(\lim_{x\to 1^-}F(x,0) = -\infty\) (the case of two attractive singularities). He also investigates the case of dry friction \(x'' + \mu\text{sgn}(x') + g(x) = p(t)\).
reviewed by: Bin Liu
Mathematics Subject Classification ID: 34C25
zbMATH DE Number: 1836160
zbMATH Keywords: periodic solutions
zbMATH Keywords: upper-and-lower solution methods
zbMATH Keywords: singularities
zbMATH Keywords: dry friction
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1836160