Properties of the set of positive solutions to Dirichlet boundary value problems with time singularities
type: Item
instance of: scholarly article
title: Properties of the set of positive solutions to Dirichlet boundary value problems with time singularities
zbMATH Open document ID: 1271.34027
published in: Central European Journal of Mathematics
publication date: 2013-01-30T00:00:00Z
review text: The authors study the structure and properties of the set \(\mathcal {S}\) of all positive solutions to the singular Dirichlet boundary value problem \[ u''(t)+\frac{a}{t}u'(t)-\frac{a}{t^2}u(t)=f(t,u(t),u'(t)),\;\;u(0)=u(T)=0 \] with \(a\in (-\infty,-1)\). They obtain that, for each \(c\geq0\), the set \(\mathcal {S}_{c}=\{u\in \mathcal {S}, u'(T)=-c\}\) is nonempty and compact and \(\mathcal {S}=\bigcup_{c\geq 0} \mathcal {S}_{c}\). They also give an order in \(\mathcal {S}\) and show that the difference of any two solutions in \(\mathcal {S}_{c}\) keeps its sign on \([0,T]\). As an application, the equation \[ v''(t)+k \frac{v'(t)}{t}=\psi(t)+g(t,v(t))\quad (k\in (1,\infty)) \] is considered.
reviewed by: Yulian An
Mathematics Subject Classification ID: 34B16
Mathematics Subject Classification ID: 34B18
zbMATH DE Number: 6132918
zbMATH Keywords: singular ordinary differential equation
zbMATH Keywords: positive solutions
zbMATH Keywords: blow up solutions
zbMATH Keywords: uniqueness
author: Irena Rachůnková
author: Svatoslav Staněk
MaRDI profile type: Publication
full work available at URL: s11533-012-0047-1
OpenAlex ID: W2049285190
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DOI: 10.2478/S11533-012-0047-1
description: scientific article; zbMATH DE number 6132918