Integral equations in measure spaces
type: Item
instance of: scholarly article
title: Integral equations in measure spaces
zbMATH Open document ID: 1031.45008
author: László Horváth
published in: Integral Equations and Operator Theory
publication date: 2004-03-08T00:00:00Z
review text: Let \((X,A,\mu_i)\) be measure spaces and let \(s: X\to A\) be a function such that (i) \(x\not\in s(x)\) for \(x\in X\), (ii) if \(y\in s(x)\) then \(s(y)\subset s(x)\), (iii) the set \(\{(x_1, x_2)\in X^2: x_2\in s(x_1)\}\) is \(\mu_i\times \mu_j\) measurable for \(i,j= 1,2,\dots, n\). The following integral equation is considered \[ y(x)= f(x)+ \sum^n_{i=1} g_i(x) \int_{s(x)} h_i\circ yd\mu_i,\quad\text{for }x\in X,\tag{1} \] where \(f\) and \(g_i\) \((i= 1,2,\dots,n)\) are real-valued functions on \(X\) and \(h_i\) is a function from a subset \(I_i\) of \(\mathbb{R}\) into \(\mathbb{R}\) for \(i= 1,2,\dots, n\). A number of necessary and sufficient conditions is given which together with the Lipschitz condition imply the existence and uniqueness of solution of the equation (1). Some applications are also derived.
Mathematics Subject Classification ID: 45N05
Mathematics Subject Classification ID: 45G10
zbMATH DE Number: 1925517
zbMATH Keywords: measure spaces
zbMATH Keywords: integral equation
zbMATH Keywords: existence
zbMATH Keywords: uniqueness
reviewed by: Józef Banaś
MaRDI profile type: Publication
full work available at URL: s000200300001
OpenAlex ID: W41152610
DOI: 10.1007/S000200300001
description: scientific article; zbMATH DE number 1925517