First-order singular and discontinuous differential equations
type: Item
instance of: scholarly article
title: First-order singular and discontinuous differential equations
zbMATH Open document ID: 1190.34007
DOI: 10.1155/2009/507671
author: Daniel C. Biles
author: Rodrigo L. Pouso
published in: Boundary Value Problems
publication date: 2009-11-06T00:00:00Z
full work available at URL: https://eudml.org/doc/45612
review text: The authors present sufficient conditions for the existence of extremal solutions (the greatest and the smallest Carathéodory solutions) between some given lower and upper solutions for the initial value problem \[ x'=f(t,x),\,\,\,t\in I=[t_0,t_0+L],\,\,\,\,x(t_0)=x_0, \] where the function \(f\) is bounded by an integrable function on the sector delimited by the graphs of the lower and upper solutions, and also verifies some additional assumptions. Then they show that the existence of a pair of well-ordered piecewise continuous lower and upper solutions implies the existence of a better pair of continuous lower and upper solutions. Two existence results for weak solutions in the case \(f\) does not have a strong bound (the singular case) are also proved. Some considerations for singular quasimonotone systems and several examples which illustrate the obtained results are finally addressed.
reviewed by: Rodica Luca
Mathematics Subject Classification ID: 34A36
Mathematics Subject Classification ID: 34A12
zbMATH DE Number: 5626158
zbMATH Keywords: initial value problem
zbMATH Keywords: upper solution
zbMATH Keywords: lower solution
zbMATH Keywords: Caratheodory solution
zbMATH Keywords: extremal solution
zbMATH Keywords: discontinuous nonlinearities
zbMATH Keywords: singular nonlinearities
zbMATH Keywords: admissible non-quasi-semicontinuity curve
Wikidata QID: Q59249356
MaRDI profile type: Publication
OpenAlex ID: W2140028977
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description: scientific article; zbMATH DE number 5626158