Nonoscillation of Mathieu's equation whose coefficient is a finite Fourier series approximating a square wave
type: Item
instance of: scholarly article
title: Nonoscillation of Mathieu's equation whose coefficient is a finite Fourier series approximating a square wave
zbMATH Open document ID: 1401.34035
author: Jitsuro Sugie
published in: Monatshefte für Mathematik
publication date: 2018-08-20T00:00:00Z
review text: In this paper, a parametric non-oscillation region is given for the Mathieu-type equation \[ x''+(-\alpha+\beta c(t))x=0,\qquad\alpha,\beta\in{\mathbb R},\tag{1} \] where \(c(t)\) is periodic with mean value zero, i.e., \[ \int_0^\omega c(t){\mathrm d}t=0\qquad(\omega\quad\text{is the period of}\quad c(t)). \] If \(\alpha=0\) and \(\beta\neq0\), then all non-trivial solutions (1.1) are oscillatory. It follows from Sturm's comparison theorem that if \(\alpha<0\) and \(\beta\neq0\), then all non-trivial solutions (1.1) are oscillatory. It is clear that {\parindent=0.7cm \begin{itemize}\item[{\(\bullet\)}] if \(\alpha<0\) and \(\beta=0\), then all non-trivial solutions (1.1) are oscillatory. \item[{\(\bullet\)}] if \(\alpha=\beta=0\), then all non-trivial solutions (1.1) are non-oscillatory. \end{itemize}} Thus, a necessary condition for all non-trivial solutions of (1.1) to be non-oscillatory is that \(\alpha\geq0\). Oscillation problem about a kind of Meissner's equation (for which \(c(t)\) is a piecewise constant function) is also discussed. The obtained result is proved by using Sturm's comparison theorem and phase plane analysis of the second-order equation \[ y''+a(t)y'+b(t)y=0, \] where \(a,b\in\mathrm C[0,\infty)\). Parametric non-oscillation region about \((\alpha,\beta)\) and some solution orbits are drawn to help to understand the result.
reviewed by: A. Özbekler
Mathematics Subject Classification ID: 34C10
Mathematics Subject Classification ID: 34A36
zbMATH DE Number: 6920375
zbMATH Keywords: Lyapunov type inequality
zbMATH Keywords: mixed nonlinear
zbMATH Keywords: sub-linear
zbMATH Keywords: super-linear
MaRDI profile type: Publication
full work available at URL: s00605-017-1049-7
OpenAlex ID: W2605453498
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DOI: 10.1007/S00605-017-1049-7
description: scientific article; zbMATH DE number 6920375