Existence and multiplicity for an O.D.E. with nonlinear conditions
type: Item
instance of: scholarly article
title: Existence and multiplicity for an O.D.E. with nonlinear conditions
zbMATH Open document ID: 0870.34025
author: O. Diekmann
published in: Differential Equations and Dynamical Systems
publication date: 1997-08-28T00:00:00Z
review text: The nonlinear problem \[ u''(x)+g(x,u)=0,\;u'(0)=-f(u(0)),\;u'(\pi)=f(u(\pi)) \] is considered, where \(f\) and \(g\) are continuous, \(f\) is increasing and \(g\) either satisfies a sign condition or is increasing with respect to \(u\). Under suitable additional assumptions on \(f\) and \(g\), which give among others that \(u=0\) is a solution, it is shown that the problem has multiple solutions, where a lower bound of the number of solutions is given. If it is not known that \(u=0\) is a solution, a different set of conditions on \(f\) and \(g\) is given such that the problem has at least one solution.
Mathematics Subject Classification ID: 34B15
zbMATH DE Number: 1011017
zbMATH Keywords: shooting method
zbMATH Keywords: variational method
zbMATH Keywords: lower bound of the number of solutions
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1011017