Existence of solutions with turning points for nonlinear singularly perturbed boundary value problems.
type: Item
instance of: scholarly article
title: Existence of solutions with turning points for nonlinear singularly perturbed boundary value problems.
zbMATH Open document ID: 1036.34068
DOI: 10.1007/BF02676400
author: A. Yu. Kolesov
author: N. Kh. Rozov
published in: Mathematical Notes
publication date: 2000-11-16T00:00:00Z
review text: Consider the boundary value problem \[ \varepsilon u''=f(x,u,u'), \;0 < \varepsilon \ll 1, \quad g_j (u_{| x=0}, u_{| x=1}, u_{| x=0}', u_{| x=1}' ) =0,\;j=1,2, \tag \(*\) \] on the interval \(0 \leq x \leq 1\). Let \(u_0 \in C^\infty [0,1]\) be a solution to the degenerate equation \(f(x,u,u')=0\) such that there is a point \(x_0 \in (0,1)\) with the property that the function \(a (x):= f_v' (x,u_0 (x), u_0' (x))\) has a simple zero at \(x_0\). The authors derive six conditions guaranteeing that \((*)\) has a unique solution. The proof is based on the existence of a duck trajectory for some auxiliary system of differential equations.
Mathematics Subject Classification ID: 34E20
Mathematics Subject Classification ID: 34B15
zbMATH DE Number: 1531347
zbMATH Keywords: nonlinear boundary value problem
zbMATH Keywords: singular perturbation
zbMATH Keywords: solution with turning points
zbMATH Keywords: existence and uniqueness of solutions
Wikidata QID: Q59448686
reviewed by: Klaus R. Schneider
MaRDI profile type: Publication
cites work: Q5330830
cites work: Q4864744
full work available at URL: https://doi.org/10.1007/bf02676400
OpenAlex ID: W10194706
description: scientific article; zbMATH DE number 1531347