On the solution of an integral-functional equation with a parameter
type: Item
instance of: scholarly article
title: On the solution of an integral-functional equation with a parameter
zbMATH Open document ID: 0897.45004
DOI: 10.4171/ZAA/814
published in: Zeitschrift für Analysis und ihre Anwendungen
publication date: 1998-07-28T00:00:00Z
review text: This paper studies the Fredholm integral equation \[ \phi(t) = (L\phi)(t) = b \int_{at - a + 1}^{at} \phi(\tau) d\tau, \tag{\(*\)} \] where \(a > 1\) and \(b = a/(a-1)\) (for \(a \leq 1\) no nonzero solutions of this equation exist). This equation can also be written in differential form \( \phi'(t) = ab[\phi(at) - \phi(at - a + 1)] \). It is shown that this equation has a solution supported on \([0,1]\), which is unique under the normalization \(\int_0^1 \phi(\tau) d\tau = 1\). This solution is \(C^\infty\) smooth, symmetric with respect to the point \(t = \frac 12\), nonnegative, nondecreasing in \([0,\frac 12]\), nonincreasing in \([\frac 12,1]\), and constant in \([1/a,1-1/a]\) if \(a > 2\). Several other properties of this solution are also established about its Laplace transform, convergence of successive approximations, and its Fourier expansion. Furthermore, a complete description is given of the spectrum and eigenfunctions of the compact operator \(L\) in \((*)\).
Mathematics Subject Classification ID: 45D05
Mathematics Subject Classification ID: 39B22
Mathematics Subject Classification ID: 42A16
Mathematics Subject Classification ID: 45C05
Mathematics Subject Classification ID: 45L05
zbMATH DE Number: 1129195
zbMATH Keywords: Fredholm integral equation
zbMATH Keywords: Laplace transform
zbMATH Keywords: convergence
zbMATH Keywords: successive approximations
zbMATH Keywords: Fourier expansion
zbMATH Keywords: spectrum
zbMATH Keywords: eigenfunctions
zbMATH Keywords: compact operator
zbMATH Keywords: integral functional equation
author: Lothar Berg
author: Manfred Krüppel
reviewed by: Olof Staffans
MaRDI profile type: Publication
cites work: Q4358539
cites work: Q3706042
Recommended article: On solutions of a certain functional-integral equation
Recommended article: Q3312694
Recommended article: Q3568009
Recommended article: Q3344687
Recommended article: Q3149081
Recommended article: Q4918735
Recommended article: Q3645904
Recommended article: Q3095724
Recommended article: Q5843992
Recommended article: Q3759338
description: scientific article; zbMATH DE number 1129195