A fixed point operator for a nonlinear boundary value problem
type: Item
instance of: scholarly article
title: A fixed point operator for a nonlinear boundary value problem
zbMATH Open document ID: 1007.34015
published in: Journal of Mathematical Analysis and Applications
publication date: 2002-07-04T00:00:00Z
review text: The authors study the solutions to the following boundary value problem \[ u''+g(t,u,u')=0,\qquad u'(0)=-f(u(0)),\quad u'(\pi)=f(u(\pi)), \tag{1} \] with \(g:[0,\pi]\times\mathbb{R}^2\to\mathbb{R}\) and \(f:\mathbb{R}\to\mathbb{R}\) continuous. Two existence and localization results are provided: one finite-dimensional based on the Brouwer degree and one infinite-dimensional, valid for \(f\) invertible, based on Schauder's theorem. Furthermore, when \(f\) is invertible, by embedding of (1) into the family of problems \[ u''+\lambda g(t,u,u')=0, \qquad u(0)=f^{-1}(-u'(0)),\quad u(\pi)=f^{-1}(u'(\pi)), \tag{\(1_\lambda\)} \] the authors give an iterative construction of a solution to (1). Namely, for \(\varepsilon>0\) small enough, taking a solution \(u_0\) to (\(1_{\lambda_0}\)) as ``seed'', they define a sequence converging in \(C^1([0,\pi])\) to a solution to (\(1_{\lambda_0+\varepsilon}\)). Then, since (\(1_0\)) is solvable, they show, under suitable assumptions, the existence of a sequence \(0=\lambda_0<\lambda_1<\ldots <\lambda_n=1\) with the property that a solution to (\(1_{\lambda_i}\)) can be constructed recursively.
reviewed by: Marco Spadini
Mathematics Subject Classification ID: 34B15
Mathematics Subject Classification ID: 47H10
Mathematics Subject Classification ID: 47N20
zbMATH DE Number: 1763700
author: Pablo Amster
author: Maria C. Mariani
MaRDI profile type: Publication
full work available at URL: jmaa.2001.7722
OpenAlex ID: W1999233172
cites work: Q4319450
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DOI: 10.1006/JMAA.2001.7722
description: scientific article; zbMATH DE number 1763700