On a class of singular boundary value problems with singular perturbation
type: Item
instance of: scholarly article
title: On a class of singular boundary value problems with singular perturbation
zbMATH Open document ID: 1241.34027
author: Feng Xie
published in: Journal of Differential Equations
publication date: 2012-01-16T00:00:00Z
review text: This paper is devoted to singular boundary value problems with singular perturbation of the form \[ \varepsilon y''-{{1}\over{{x-\lambda}}}f(x,y)y'+g(x,y)=0,\;x\in(0,\lambda)\cup(\lambda,1), \] \[ y(0)=A,\;y(\lambda)=\mu,\;y(1)=B, \] where \(\varepsilon\) is a small positive parameter, the constants \(A, B, \mu\) and \(\lambda\in[0,1]\) are such that if \(\lambda=0\) or \(\lambda=1\), then \(\mu=A\) or \(\mu=B\), respectively, and the functions \(f(x,y)\) and \(g(x,y)\) are \(C^1\)-smooth on \([0,1]\times \mathbb R\) and there is a positive constant \(\sigma_0\) such that \(f(x,y)\geq\sigma_0>0\) for \((x,y)\in[0,1]\times \mathbb R\). Using the lower and upper solutions technique and the Schauder fixed point theorem, the author studies the existence and uniqueness of solutions to the considered problem as well as their asymptotic behavior with respect to \(\varepsilon.\) Moreover, it is assumed in addition that the problem \[ f(x,y)y'=(x-\lambda)g(x,y),\;y(\lambda)=\mu, \] has a solution \(y=\phi(x)\) in \(C^2[0,1]\).
reviewed by: Petio S. Kelevedjiev
Mathematics Subject Classification ID: 34B16
Mathematics Subject Classification ID: 34E15
Mathematics Subject Classification ID: 34E05
Mathematics Subject Classification ID: 34B08
Mathematics Subject Classification ID: 47N20
zbMATH DE Number: 5998055
zbMATH Keywords: singular boundary value problem
zbMATH Keywords: singular perturbations
zbMATH Keywords: lower and upper solutions
zbMATH Keywords: existence and uniqueness
zbMATH Keywords: asymptotic estimate
MaRDI profile type: Publication
full work available at URL: j.jde.2011.10.003
OpenAlex ID: W2009238143
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DOI: 10.1016/J.JDE.2011.10.003
description: scientific article; zbMATH DE number 5998055