An application of diophantine approximations to the theory of differential equations
type: Item
instance of: scholarly article
title: An application of diophantine approximations to the theory of differential equations
zbMATH Open document ID: 0651.10024
DOI: 10.1007/BF01830945
published in: Aequationes Mathematicae
publication date: 1988-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/137265
review text: Mittels einiger Resultate aus der Gleichverteilungstheorie beweist der Autor: Sei \(\psi: \mathbb N\to \mathbb R_+\) monoton mit \(\psi(n)\to \infty\) bei \(n\to\infty\). Dann gibt es über \(\mathbb R\) linear unabhängige Funktionen \(u_1,\ldots,u_s\) und eine unendliche Teilmenge \(Q\) von \(\mathbb N\), so daß für \(j=1,\ldots,s\) die folgenden beiden Bedingungen erfüllt sind. (1) \(u_j\) genügt der Differentialgleichung \(P_j(u_j(t),u'_j(t), u''_j(t))=0,\) wobei \(P_j\) ein Polynom in \(u_j\), \(u'_j\), \(u''_j\) ist. (2) Die Ungleichung \(| u_j(q)| \geq \psi(q)\) gilt für jedes \(q\in Q\). Weiter wird ein ähnliches Resultat über Funktionen mehrerer Variablen, die einem System partieller Differentialgleichungen zweiter Ordnung genügen, angegeben. \{Die Note schließt an eine Arbeit von \textit{N. M. Basu}, \textit{S. N. Bose} und \textit{T. Vijayaraghavan} an [J. Lond. Math. Soc. 12, 250--252 (1937; Zbl 0017.25004, JFM 63.0419.02)].\}
reviewed by: Peter Bundschuh
Mathematics Subject Classification ID: 11J71
Mathematics Subject Classification ID: 34A34
Mathematics Subject Classification ID: 34A45
Mathematics Subject Classification ID: 35G99
zbMATH DE Number: 4061333
zbMATH Keywords: uniform distribution
zbMATH Keywords: solutions of ordinary differential equations of the second order
zbMATH Keywords: system of partial differential equations of the second order
MaRDI profile type: Publication
cites work: Q4192121
OpenAlex ID: W2032663269
author: Edmund Hlawka
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description: scientific article; zbMATH DE number 4061333