Solution branching in linear differential equations
type: Item
instance of: scholarly article
title: Solution branching in linear differential equations
zbMATH Open document ID: 0770.34006
author: Grant Gustafson
author: Jerry R. Ridenhour
published in: Journal of Differential Equations
publication date: 1993-05-16T00:00:00Z
review text: This paper deals with the \(n\)th order differential equation (1) \(L(y)\equiv y^{(n)}+p_{n-1}(t) y^{(n-1)}+\dots+p_ 0(t)y=0\), where \(p_ i\) are assumed to be continuous and real valued on \(\mathbb{R}\), together with the boundary conditions (2) \(y^{(r)}(a_ j)=0\), \(r=0,\dots,i_ j-1\), \(j=1,\dots,k\), where \(a_ 1,\dots,a_ k\) are distinct points and \(\sum_{j=1}^ k i_ j=n\). The principal motivating result of this paper is due to \textit{P. Hartman} [Rend. Circ. Mat. Palermo, II. Ser. 7, 123-142 (1958; Zbl 0085.045)]: If (1) has a nontrivial solution \(y\) with \(n\) zeros counting multiplicities on an open interval \(I\), then there exists a solution \(z\) with \(n\) distinct zeros on \(I\). Hartman's proof does not establish any relation between the solution \(z\) and origin solution \(y\). That is, \(z\) is not necessarily close (in the sense of continuous functions) to \(y\) and the distinct zeros of \(z\) are not necessarily close to the zeros of \(y\). The authors explain why the solutions \(y\) and \(z\) in Hartman's theorem are unrelated, in general, and give assumptions under which we may write \(z=y+\sum_{i=1}^ m c_ i w_ i\), where constants \(c_ 1,\dots,c_ m\) are near zero, \(w_ 1,\dots,w_ m\) are solutions of \(L(y)=0\), \(\| w_ i\|=1\), and zeros of \(z\) are simple and close to zeros of \(y\). Key assumptions are 1- dimensionality of the solution space of BVP (1), (2), an exactness condition on the zeros of \(y\), and a restriction on multiple zero counts.
Mathematics Subject Classification ID: 34A30
Mathematics Subject Classification ID: 34M99
Mathematics Subject Classification ID: 34C10
zbMATH DE Number: 166904
zbMATH Keywords: \(\varepsilon\)-simple branching
zbMATH Keywords: extremal solution
zbMATH Keywords: \(n\)th order differential equation
zbMATH Keywords: zeros
reviewed by: Svatoslav Staněk
MaRDI profile type: Publication
full work available at URL: jdeq.1993.1016
OpenAlex ID: W1965863841
DOI: 10.1006/JDEQ.1993.1016
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description: scientific article; zbMATH DE number 166904