Asymptotic forms of positive solutions of third-order Emden-Fowler equations
type: Item
instance of: scholarly article
title: Asymptotic forms of positive solutions of third-order Emden-Fowler equations
zbMATH Open document ID: 1043.34052
DOI: 10.1016/S0022-247X(02)00065-3
author: Ken-Ichi Kamo
author: Hiroyuki Usami
published in: Journal of Mathematical Analysis and Applications
publication date: 2002-11-28T00:00:00Z
review text: The study of qualitative behavior of third-order ordinary differential equations has been of interest from the seventies. In this setting, the authors consider equations of Emden-Fowler type, that is, \[ u'''(t)=p(t) | u(t)|^{\lambda-1}u(t), \] where \(\lambda>1\) and \(p(t)\) behaves like a positive \(C^{1}\) function near \(+\infty\) (that is \(lim_{t\rightarrow\infty} \frac{p(t)}{t^{\sigma}}\) for some \(\sigma\in\mathbb{R}\)). The main interest of such equations is to understand the asymptotic behavior of their solutions near \(+\infty\) and if it is possible to give their form. In fact \(u\in C^{3}\) is a solution of the former equation if \(u\) does not vanishes and satisfies the equation near \(+\infty\). \textit{I. T. Kiguradze} and \textit{T. A. Chanturiya} [Asymptotic properties of solutions of nonautonomous ordinary differential equations, Dordrecht: Kluwer Academic Publisher, (1992; Zbl 0782.34002)] gave five forms of asymptotic behavior (asymptotically constant solutions (AC), intermediate order solutions (I), asymptotically linear solutions (AL), asymptotically quadratic solutions (AQ) and asymptotically superquadratic solutions (ASQ)) and characterized them in terms of some integrals of \textit{p(t)} in the four first cases. Using these characterizations, the authors obtain the following results: The differential equation has: (1) an AC-solution if and only if \(\sigma+3<0\),(2) an AL-solution if and only if \(\sigma+\lambda+2<0\),(3) an AQ-solution if and only if \(\sigma+2\lambda+1<0\) and (4) an ASQ-solution if \(\sigma+2\lambda+1<0.\) Putting \(k=-\frac{\sigma+3}{\lambda-1}\), the authors improve also some results from \textit{P. A. Ohme} [Ann. Mat. Pura Appl. (4) 104, 43--65 (1975; Zbl 0307.34046)]: (a) If \(k>2\), then (1) has one of the following asymptotic forms: \(c_{1}\), \(c_{2}t\), \(c_{3}t^{2}\) or \(at^{k}\), where \(a^{\lambda-1}=k(k-1)(k-2)\) (\(c_{1}, c_{2}\) and \(c_{3}\) are positive constants). (b) If \(k=2\) the forms are: \(c_{1}…
reviewed by: Francisco Balibrea
Mathematics Subject Classification ID: 34D05
Mathematics Subject Classification ID: 34A34
zbMATH DE Number: 1836755
zbMATH Keywords: positive solutions
zbMATH Keywords: third order Emden-Fowler equations
zbMATH Keywords: asymptotic forms
MaRDI profile type: Publication
cites work: Asymptotic behavior of the solutions of the third order nonlinear differential equations \(u'''\pm t^\sigma u^n=0\)
cites work: Q5288929
cites work: Q5330830
cites work: On the Nature of the Spectrum of Singular Second Order Linear Differential Equations
full work available at URL: s0022-247x(02)00065-3
OpenAlex ID: W2081438898
description: scientific article; zbMATH DE number 1836755