Nonlinear second-order periodic boundary value problems
type: Item
instance of: scholarly article
title: Nonlinear second-order periodic boundary value problems
zbMATH Open document ID: 0678.34022
DOI: 10.1016/0022-247X(88)90383-6
published in: Journal of Mathematical Analysis and Applications
publication date: 1988-01-01T00:00:00Z
review text: For the nonlinear boundary value problem (BVP) with periodic boundary conditions \(u''+f(t,u)=0\), \(u(0)=u(2\pi)\), \(u'(0)=u'(2\pi)\), the existence of lower and upper solutions \(\alpha\) and \(\beta\) with \(\alpha\leq \beta\) on [0,2\(\pi\) ] guarantees the existence of a solution u(t) of the BVP with \(\alpha\leq u\leq \beta\). In this paper, the case of reversed lower and upper solutions \(\alpha\) and \(\beta\) with \(\alpha\geq \beta\) is considered. Examples are given to show that the existence of reversed upper and lower solutions will not imply the existence of a solution of the BVP in between, and indeed, the BVP may not have a solution. Conditions are given which imply the existence of solutions of the BVP when reversed lower and upper solutions exist.
Mathematics Subject Classification ID: 34B15
Mathematics Subject Classification ID: 34A30
zbMATH DE Number: 4111083
zbMATH Keywords: lower solutions
zbMATH Keywords: periodic boundary conditions
zbMATH Keywords: Examples
zbMATH Keywords: upper solutions
author: Juan J. Nieto
reviewed by: Louis J. Grimm
MaRDI profile type: Publication
Wikidata QID: Q60199494
cites work: Le correspondant topologique de l'unicité dans la théorie des équations différentielles
cites work: Periodic boundary value problems for systems of second order differential equations
cites work: An introduction to nonlinear boundary value problems
cites work: Linear monotone method for nonlinear boundary value problems in Banach spaces
cites work: A monotone method for quasilinear boundary value problems
cites work: Periodic solutions of nonlinear boundary value problems
cites work: Existence of periodic solutions of nonlinear boundary value problems and the method of upper and lower solutions
cites work: Sufficient conditions for existence of solutions of nonlinear boundary value problems at resonance
cites work: Q3807603
cites work: Periodic boundary value problems of first and second order differential equations
cites work: Existence of periodic solutions of semilinear parabolic equations and the method of upper and lower solutions
cites work: Q3216921
cites work: Existence and monotone method for periodic solutions of first-order differential equations
cites work: Remarks on first and second order periodic boundary value problems
cites work: Monotone method for second order periodic boundary value problems
cites work: Existence of periodic solutions for first order differential equations
cites work: On the structure of the solution set for first order differential equations
cites work: Q3819589
cites work: Funnel sections
cites work: The shape of a cross-section of the solution funnel of an ordinary differential equation
cites work: Q5628581
cites work: Q3338582
cites work: On the existence of periodic quasi-solutions for first order systems
full work available at URL: 0022-247x(88)90383-6
OpenAlex ID: W2009909776
description: scientific article; zbMATH DE number 4111083