Upper and lower solutions for first-order discontinuous ordinary differential equations
type: Item
instance of: scholarly article
title: Upper and lower solutions for first-order discontinuous ordinary differential equations
zbMATH Open document ID: 0962.34008
published in: Journal of Mathematical Analysis and Applications
publication date: 2000-12-14T00:00:00Z
review text: The paper contains a significant contribution to the classical method of upper and lower solutions for a nonlinear first-order differential equation \(x'(t)=f(t,x(t)), t\in I=[0,1]\), with \(f:I\times \mathbb{R}\to \mathbb{R}\). The author generalizes the previous results in several directions: 1. Function \(f\) is allowed to be discontinuous, satisfying certain conditions introduced by \textit{E. R. Hassan} and \textit{W. Rzymowski} [Nonlinear. Anal., Theory Methods Appl. 37A, No. 8, 997-1017 (1999; Zbl 0949.34005)], which are weaker than the so-called Carathéodory conditions. 2. Classical continuous upper and lower solutions are replaced by functions of bounded variation. 3. Discontinuous and nonlinear boundary conditions with functional dependence are considered. The main result establishes the existence of the minimal and the maximal solution to a general boundary value problem between a lower and an upper solution. Two examples are included which show that the generalization achieved allows to prove the existence of the extremal solutions to some problems not included in previous settings, and also to improve the approximation to such solutions, because the upper and lower solutions are taken from a wider class of functions.
reviewed by: Eduardo Liz
Mathematics Subject Classification ID: 34B15
Mathematics Subject Classification ID: 34A36
Mathematics Subject Classification ID: 34A12
zbMATH DE Number: 1449218
zbMATH Keywords: discontinuous differential equations
zbMATH Keywords: upper and lower solutions
zbMATH Keywords: initial value problem
zbMATH Keywords: boundary value problem
author: Rodrigo L. Pouso
MaRDI profile type: Publication
full work available at URL: jmaa.2000.6717
OpenAlex ID: W2018867836
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DOI: 10.1006/JMAA.2000.6717
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description: scientific article; zbMATH DE number 1449218