Homoclinic solutions of singular nonautonomous second-order differential equations
type: Item
instance of: scholarly article
title: Homoclinic solutions of singular nonautonomous second-order differential equations
zbMATH Open document ID: 1190.34028
DOI: 10.1155/2009/959636
author: Irena Rachůnková
author: Jan Tomeček
published in: Boundary Value Problems
publication date: 2009-11-06T00:00:00Z
review text: This article is concerned with the existence of solutions of the singular boundary value problem \[ (p(t)u')'=p(t)f(u),\;t\geq0;\quad u'(0)=0, \quad u(+\infty)=L\tag{1} \] Here \(p\) is \(C^1\), positive and increasing, \(p(0)=0\) and \(\lim_{t\to\infty}\frac{p'(t)}{p(t)}=0\). \(f\) is locally Lipschitz and there are \(\bar B<0<L\) such that \(f>0\) on \([\bar B,0)\), \(f<0\) on \((0, L)\), \(f(L)=0\) and, setting \(F(x)=-\int_0^xf(z)\,dz\) the equality \(F(\bar B)=F(L)\) holds. The authors prove that under these conditions, if in addition \(0<\liminf_{x\to-\infty}\frac{|x|}{f(x)}<\infty\), problem (1) has a solution with initial value \(u(0)<\bar B\). The arguments involve shooting and connectedness.
Mathematics Subject Classification ID: 34B40
Mathematics Subject Classification ID: 34B15
Mathematics Subject Classification ID: 34B16
zbMATH DE Number: 5626180
zbMATH Keywords: singular boundary value problem
zbMATH Keywords: homoclinic solution
Wikidata QID: Q59219989
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1155/2009/959636
OpenAlex ID: W2046710216
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description: scientific article; zbMATH DE number 5626180