Regularly varying solutions of half-linear functional differential equations with retarded arguments
type: Item
instance of: scholarly article
title: Regularly varying solutions of half-linear functional differential equations with retarded arguments
zbMATH Open document ID: 1199.34413
author: B. E. Eshmatov
published in: Acta Mathematica Hungarica
publication date: 2008-10-21T00:00:00Z
review text: If \(f:\mathbb R^+\to\mathbb R^+\), then \(f\) is said to be regularly varying of index \(\rho\) if \(\lim_{t\to\infty}\frac{f(\lambda t)}{f(t)}=\lambda^\rho\) for any \(\lambda >0\). In case \(f\) is regularly varying with index \(0\), we say that \(f\) is slowly varying. The authors considers the equation \[ (| x'(t)| ^\alpha\text{sgn}\, x'(t))'=g(t)| x(g(t))| ^\alpha\text{sgn}\, x(g(t)), \] where \(\alpha >0\), \(q(t)>0\) and continuous, \(g(t)>0\) increasing and continuous such that \(g(t)<t\), \(\lim_{t\to \infty} g(t)=\infty\). Under the assumption \(\limsup_{t\to \infty}\frac{t}{g(t)}<\infty\), it is shown that the equation has a slowly varying solution and a regular solution of index 1 if and only if \(\lim_{t\to\infty}t^\alpha\) \(\int^\infty_t q(s)\,ds=0\).
Mathematics Subject Classification ID: 34K25
Mathematics Subject Classification ID: 34K05
zbMATH DE Number: 5355148
zbMATH Keywords: half-linear functional differential equations
zbMATH Keywords: Karamata functions
zbMATH Keywords: asymptotic behavior
zbMATH Keywords: related functional
zbMATH Keywords: slowly varying functions
reviewed by: Stig-Olof Londen
MaRDI profile type: Publication
full work available at URL: s10474-007-7101-7
OpenAlex ID: W1983776222
cites work: Q4726487
cites work: On an extension of the Fubini theorem and its applications in ODEs
cites work: Remarks on the existence of regularly varying solutions for second order linear differential equations
cites work: Nonoscillation theory for second order half-linear differential equations in the framework of regular variation
cites work: On oscillation of half-linear functional differential equations with deviating arguments
cites work: On a class of functional differential equations having slowly varying solutions
cites work: Regular variation and differential equations
cites work: Q3209431
DOI: 10.1007/S10474-007-7101-7
description: scientific article; zbMATH DE number 5355148