A note on solutions in \(L^1[0,1]\) to Hammerstein integral equations
type: Item
instance of: scholarly article
title: A note on solutions in \(L^1[0,1]\) to Hammerstein integral equations
zbMATH Open document ID: 0897.45008
author: Donal O'Regan
published in: Journal of Integral Equations and Applications
publication date: 1998-10-19T00:00:00Z
full work available at URL: contents.htm
review text: The author presents some existence principles for the Hammerstein integral equation \[ y(t)= g(t)+ \int_0^1 k(t,s) f(s,y(s))ds, \] where \(k:[0,1]\times [0,1]\to\mathbb{R}\) and \(f:[0,1]\times \mathbb{R}\to\mathbb{R}\). He considers solutions which lie in \(L^1[0,1]\) (for completeness he also discusses the case \(L^p[0,1]\), \(1<p<\infty\)). His analyses rely on the notion of measure of weak noncompactness and on the Schauder fixed point theorem. It is worth remarking here also that essentially the same reasoning would establish existence principles for Volterra and even Uryson integral equations.
Mathematics Subject Classification ID: 45G10
zbMATH DE Number: 1136060
zbMATH Keywords: Volterra integral equations
zbMATH Keywords: Hammerstein integral equation
zbMATH Keywords: measure of weak noncompactness
zbMATH Keywords: Schauder fixed point theorem
zbMATH Keywords: Uryson integral equations
reviewed by: Ulrich Kosel
MaRDI profile type: Publication
OpenAlex ID: W2071870541
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DOI: 10.1216/JIEA/1181076002
description: scientific article; zbMATH DE number 1136060