Regular variation and differential equations
type: Item
instance of: scholarly article
title: Regular variation and differential equations
zbMATH Open document ID: 0946.34001
author: Vojislav Marić
published in: Lecture Notes in Mathematics
publication date: 2000-05-02T00:00:00Z
review text: This little monograph deals with the precise asymptotic behavior of solutions to second-order differential equations of the form \(y''= f(x)\phi(y)\), both for the linear and for some nonlinear cases. By that as usual one means to find a continuous, positive function \(g\) such that \(y(x)/g(x)\to 1\) as \(x\to\infty\). Equations of the form \(y''+ g(x) y'+ h(x)y= 0\) and \(y'''- yy''+ \lambda(1+ y^{\prime 2})= 0\) are also studied. The novelty of the approach in this book consists of using the notion of regular variation of \textit{J. Karamata} [Mathematica 4, 38-53 (1930; JFM 56.0907.01)] and its extensions. A positive measurable function \(\rho\) defined on some neighbourhood \([a,\infty)\) of infinity is called regularly varying at infinity of index \(\alpha\) \((\alpha\in\mathbb{R})\) if for each \(\lambda> 0\), \(\lim_{x\to\infty} \rho(\lambda x)/\rho(x)= \lambda^\alpha\). The real number \(\alpha\) is called the index of regular variation. The book has two parts. Part 1 deals with the existence and asymptotic behavior of regular solutions for linear equations. Necessary and sufficient conditions for the solutions to belong to the Karamata class of functions are presented, here. Part 2 is devoted to nonlinear equations of Thomas-Fermi type and to an equation arising in boundary-layer theory. An Appendix summarizing basic properties of regularly varying and related functions, a list of references (67 titles) and a general index conclude the book. The book is a wonderful addition to the literature on asymptotic behavior of solutions to ordinary differential equations. The writing is concise, precise, elegant and enjoyable. These recommend it to graduate students and to all interested in this topic.
Mathematics Subject Classification ID: 34-02
Mathematics Subject Classification ID: 34C10
Mathematics Subject Classification ID: 34E05
Mathematics Subject Classification ID: 26A12
zbMATH DE Number: 1438390
zbMATH Keywords: asymptotic behavior
zbMATH Keywords: solutions
zbMATH Keywords: second-order differential equations
zbMATH Keywords: regular variation
zbMATH Keywords: JFM 56.0907.01
zbMATH Keywords: Karamata class of functions
zbMATH Keywords: nonlinear equations of Thomas-Fermi type
reviewed by: Radu Precup
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1007/bfb0103952
OpenAlex ID: W649677041
DOI: 10.1007/BFB0103952
description: scientific article; zbMATH DE number 1438390