Non-oscillation domains of differential equations with two parameters
type: Item
instance of: scholarly article
title: Non-oscillation domains of differential equations with two parameters
zbMATH Open document ID: 0657.34035
author: Angelo Bernado Mingarelli
author: S. Gotskalk Halvorsen
published in: Lecture Notes in Mathematics
publication date: 1993-06-05T00:00:00Z
review text: The second order differential equation (1) \(y''+(-\alpha +\beta B(x))y=0\) where \(\alpha\), \(\beta\) are real parameters and x varies over some real interval, arises in many branches of pure and applied mathematics. Hill's equation occurs if B(x) is periodic, and a particularly important case is Mathieu's equation in which \(B(x)=\cos 2x\). Apart from direct applications, these equations, which exhibit parametric excitation, also arise as variational equations from the perturbation of periodic solutions in the theory of nonlinear oscillators. There has been renewed interest in the properties of such equations in the chaotic output of certain forced oscillators. Equation (1) is said to be disconjugate on (-\(\infty,\}\) if and only if every one of its nontrivial solutions has at most one zero in (- \(\infty,\infty)\) It is said to be non-oscillatory on (-\(\infty,\infty)\) if and only if every one of its nontrivial solutions has at most a finite number of zeros in (-\(\infty,\infty)\). The collection of all \((\alpha,\beta)\in R^ 2\) for which (1) is disconjugate (non- oscillatory) on (-\(\infty,\infty)\) will be called the disconjugacy (non- oscillatory) domain of (1). Suppose that these domains are denoted by D and N respectively. \textit{R. A. Moore} [J. Analyse Math. 5, 183-196 (1957; Zbl 0077.087)] showed that \(D=N\) for (1) and that N is a closed, convex and unbounded subset of the \(\alpha\) \(\beta\)-plane. This monograph starts by presenting a review of the properties of the disconjugacy and non-oscillation domains for the more general equation (2) \(y''+(-\alpha A(x)+\beta B(x))y=0,\) on the closed half-line [0,\(\infty)\). The requirements of A and B are the minimal ones of A,B:[0,\(\infty)\to R\) such that each is Lebesgue integrable on every compact subset of [0,\(\infty)\). It is shown that D is always closed and convex. In general \(D\subset N\) and N is also convex though not always closed. In chapter 2 various special properties of D and N are di…
Mathematics Subject Classification ID: 34C10
Mathematics Subject Classification ID: 34A30
Mathematics Subject Classification ID: 34C15
Mathematics Subject Classification ID: 34-02
Mathematics Subject Classification ID: 45D05
Mathematics Subject Classification ID: 39A10
Mathematics Subject Classification ID: 39A12
zbMATH DE Number: 193255
zbMATH Keywords: second order differential equation
zbMATH Keywords: Hill's equation
zbMATH Keywords: Mathieu's equation
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1007/bfb0080637
OpenAlex ID: W106482001
DOI: 10.1007/BFB0080637
description: scientific article; zbMATH DE number 193255