Remark on methods of estimating the number of solutions of nonlinear boundary value problems for ordinary differential equations
type: Item
instance of: scholarly article
title: Remark on methods of estimating the number of solutions of nonlinear boundary value problems for ordinary differential equations
zbMATH Open document ID: 0853.34027
DOI: 10.1007/BF02304558
published in: Mathematical Notes
publication date: 1997-01-06T00:00:00Z
review text: The author extends the results by Petrov about the number of solutions of the Dirichlet problem (1) \(x'' = g(t,x) + f(t,x,x')\), \(x(0) = 0\), \(x (\pi) = 0\), to the case of a jumping nonlinearity \(g\) and \(f\) sublinear in \(x\), \(x'\). Assuming that \(\xi\) is a solution of (1), he proves that the number of solutions to (1) is not less than \(|m_+ - k |+ |k - m_- |\), where \(k(m_+\) or \(m_-)\) is the number of zeros in \((0, \pi)\) of a solution (which does not vanish in \(\pi)\) of the initial value problem \(-y'' = g_x (t, \xi) y + f_x (t, \xi, \xi') y + f_{x'} (t, \xi, \xi') y'\), \(y(0) = 0\), \(y'(0) = 1\), \((- z'' = g_+ (t) z^+ - g_- (t) z^-\), \(z(0) = 0\), \(z'(0) = 1\) or \(z'(0) = - 1)\).
Mathematics Subject Classification ID: 34B15
zbMATH DE Number: 898894
zbMATH Keywords: Dirichlet problem
zbMATH Keywords: number of solutions
Wikidata QID: Q57554721
author: Felix Sadyrbaev
reviewed by: Irena Rachůnková
MaRDI profile type: Publication
cites work: Q4838092
cites work: Q4013457
cites work: Number of solutions of boundary value problems
cites work: Multiplicity and eigenvalue intersecting nonlinearities
cites work: On a conjecture related to the number of solutions of a nonlinear Dirichlet problem
cites work: Q3488423
cites work: Q3041524
cites work: Q5848416
description: scientific article; zbMATH DE number 898894