On maximum principles in the class of oscillating functions
type: Item
instance of: scholarly article
title: On maximum principles in the class of oscillating functions
zbMATH Open document ID: 1081.34030
author: Agnieszka Kałamajska
author: Andrzej Stryjek
published in: Aequationes Mathematicae
publication date: 2005-06-06T00:00:00Z
review text: The paper discusses a maximum principle technique that is applicable to solutions of ordinary differential equations that might oscillate near the boundary, and thus covering an area left untouched by classical theory. The main result is Theorem 2.1: Assume that (i) \(\alpha <\beta \) are given numbers; (ii) \(a,b,c:(\alpha ,\beta )\rightarrow \mathbb{R}\) are given functions such that \(c\neq 0\) a.e., \(a/c\in W_{loc}^{1,1}(\alpha ,\beta )\), \(b/c\), \(h\in L_{\text{loc}}^1(\alpha ,\beta )\), where \(h=(a/c)^{\prime }-2(b/c)\); (iii) \(M\) is the set of functions \(u\) such that \(au^{\prime \prime }+bu^{\prime }+cu=0\) a.e. in \((\alpha ,\beta )\), \(u\in \) \(W_{\text{loc}}^{2,1}(\alpha ,\beta )\cap C[\alpha ,\beta ]\); (iv) the following conditions for \(s\in \{\alpha ,\beta \}\) are considered: \(c_1\)) \((a/c)(x)>0\) for \(x\) close to \(s\); \(c_2\)) \((a/c)(x)<0\) for \(x\) close to \(s\); \(c_3\)) lim\(_{x\rightarrow s}(a/c)(x)=0\) and \(M\subseteq \{u:\limsup_{y\rightarrow s}| u^{\prime }(y)| <+\infty \}\); \(c_4\)) \(\limsup_{x\rightarrow s}| a/c| (x)<+\infty \) and \(M\subseteq \{u:\lim_{y\rightarrow s}u^{\prime }(y)=0\}\). Then we have: (1) if \(h(x)\leq 0\) a.e. in \((\alpha ,\beta )\) and \(\alpha \) satisfies one of the conditions \(c_2),c_3),c_4)\), then the elements of \(M\) attain their maximum in \(\{\alpha ,\beta \}\). If additionally \(\beta \) satisfies one of the conditions \(c_1)\), \(c_3)\), \(c_4)\), then the elements of \(M\) attain their maximum in \(\alpha \). (2) if \(h(x)\geq 0\) a.e. in \((\alpha ,\beta )\) and \(\beta \) satisfies one of the conditions \(c_2)\), \(c_3)\), \(c_4)\), then the elements of \(M\) attain their maximum in \(\{\alpha ,\beta \}\). If additionally \(\alpha \) satisfies one of the conditions \(c_1)\), \(c_3)\), \(c_4)\), then the elements of \(M\) attain their maximum in \(\beta \). A result in the case when \(h\) changes its sign in \((\alpha ,\beta )\) is obtained in Corollary 2.1. The question of the dime…
Mathematics Subject Classification ID: 34C10
zbMATH DE Number: 2173760
zbMATH Keywords: maximum principles
zbMATH Keywords: hypergeometric functions
zbMATH Keywords: orthogonal polynomials
Wikidata QID: Q115181034
reviewed by: Octavian Mustafa
MaRDI profile type: Publication
full work available at URL: s00010-004-2761-7
OpenAlex ID: W1979904973
DOI: 10.1007/S00010-004-2761-7
description: scientific article; zbMATH DE number 2173760