Abstract Volterra equations of the second kind
type: Item
instance of: scholarly article
title: Abstract Volterra equations of the second kind
zbMATH Open document ID: 0917.45011
author: Martin Väth
published in: Journal of Integral Equations and Applications
publication date: 1999-07-27T00:00:00Z
full work available at URL: CONT10-3.html
review text: The author studies the equation \(x = Vx+f\) where \(V\) is a nonlinear Volterra operator. Given a family \(P_i\) of projections indexed by a linearly ordered set an operator \(V\) is said to be a \(0\)-Volterra operator provided \(P_ix=P_iy\) implies that \(P_iVx= P_iVy\), and \(V\) is said to be a Volterra operator provided one in addition has \(V(P_ix + (I-P_i)y) +V((I-P_i)x + P_iy) = Vx + Vy\). The author then studies the existence of local solutions and the connection between local and global solutions and shows in a quite general setting that compact linear Volterra operators have spectral radius zero. A number of additional results is established as well and this supports the claim that this more general definition of a Volterra operator gives useful new results.
reviewed by: Gustaf Gripenberg
Mathematics Subject Classification ID: 45N05
Mathematics Subject Classification ID: 45G10
Mathematics Subject Classification ID: 47J05
zbMATH DE Number: 1240967
zbMATH Keywords: nonlinear Volterra equation
zbMATH Keywords: abstract Volterra operator
zbMATH Keywords: local solutions
zbMATH Keywords: global solutions
zbMATH Keywords: compact linear Volterra operators
MaRDI profile type: Publication
OpenAlex ID: W1985409287
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DOI: 10.1216/JIEA/1181074233
description: scientific article; zbMATH DE number 1240967