On a generalization of the Osgood condition
type: Item
instance of: scholarly article
title: On a generalization of the Osgood condition
zbMATH Open document ID: 0969.45007
DOI: 10.1155/S102558340000028X
published in: Journal of Inequalities and Applications
publication date: 2001-10-02T00:00:00Z
full work available at URL: https://eudml.org/doc/121245
review text: The authors prove as main result a generalization of the Osgood uniqueness condition for a nonlinear Volterra integral equation of the second kind. Namely, it concerns a nonlinear equation of the Abel type. The main theorem says in which cases a nonlinear Volterra equation of this kind has trivial and nontrivial solutions for different values of \(\alpha\geq 1\) in the Abel kernel \((x-s)^{\alpha-1}\). In particular, for \(\alpha=1\) we obtain the classical Osgood condition. The existence of trivial or nontrivial solutions depends upon the behavior of the function, which gives the nonlinearity of the equation. As it is mentioned the case \(\alpha\in(1,2)\) is still open to conclude that the Osgood type condition is necessary for the existence of nontrivial solutions of a nonlinear Volterra equation of this type.
Mathematics Subject Classification ID: 45G10
zbMATH DE Number: 1559355
zbMATH Keywords: nonlinear Abel integral equation
zbMATH Keywords: nonlinear Volterra integral equation
zbMATH Keywords: Osgood condition
zbMATH Keywords: nontrivial solutions
author: Wojciech Mydlarczyk
author: Wojciech Okrasiński
reviewed by: S. Yakubovich
MaRDI profile type: Publication
OpenAlex ID: W1964915781
description: scientific article; zbMATH DE number 1559355