Bounds for periodic solutions of vector second-order nonlinear differential equations
type: Item
instance of: scholarly article
title: Bounds for periodic solutions of vector second-order nonlinear differential equations
zbMATH Open document ID: 1121.34049
author: M. A. Pinsky
author: A. A. Zevin
published in: Journal of Mathematical Analysis and Applications
publication date: 2007-05-23T00:00:00Z
review text: Consider the system of coupled second-order nonlinear differential equations \[ \ddot x+ f(x, \dot x,t)= 0,\tag{1} \] where \(x= (x_1,\dots, x_n)^T\in \mathbb{R}^n\), \(f\in C(\mathbb{R}^n\times \mathbb{R}^n\times \mathbb{R}, \mathbb{R}^n)\) and \(T\)-periodic in \(t\). Assume that (1) has a \(T\)-periodic solution \(x(t)\not\equiv\) constant. Let for \(1\leq i\leq n\) \[ A_i:= {1\over 2} \Biggl(\max_{t\in [0,T]} x_i(t)- \min_{t\in [0,T]} x_i(t)\Biggr),\quad V_i:= \max_{t\in [0,T]} |\dot x_i(t)|. \] The authors derive upper bounds for \(A_i\) and \(V_i\), \(1\leq i\leq n\), and a lower bound for the period \(T\) in the cases when (1) is autonomous or parametrically excited, that is \(f(0,0,t)\equiv 0\). To this end, the authors consider the auxiliary boundary value problem \[ \ddot x+ g(x,\dot x_0, t)= 0,\quad x(0)= \dot x(T/4)= 0,\tag{2} \] where the function \(g\) is associated with \(f\). They prove that the positive solutions of (2) provide the required bound.
reviewed by: Klaus R. Schneider
Mathematics Subject Classification ID: 34C25
zbMATH DE Number: 5156432
zbMATH Keywords: nonlinear equations
zbMATH Keywords: periodic solutions
zbMATH Keywords: bounds for amplitudes and periods
MaRDI profile type: Publication
full work available at URL: j.jmaa.2006.10.009
OpenAlex ID: W2084756559
cites work: Q3710830
cites work: Q5848315
cites work: Q5737099
cites work: Periods of Periodic Solutions and the Lipschitz Constant
cites work: A symmetry theorem for variational problems
cites work: On minimal periods of functional-differential equations and difference inclusions
cites work: A general symmetry principle and some implications
cites work: Q4513270
cites work: Minimal periods of semilinear evolution equations with Lipschitz nonlinearity
cites work: Sur les périodes des solutions de l'équation différentielle x'' + g(x) = 0
DOI: 10.1016/J.JMAA.2006.10.009
description: scientific article; zbMATH DE number 5156432