Solutions with exponential character to a linear functional equation and roots of its characteristic equation
type: Item
instance of: scholarly article
title: Solutions with exponential character to a linear functional equation and roots of its characteristic equation
zbMATH Open document ID: 1109.39024
published in: Journal of Mathematical Analysis and Applications
publication date: 2007-01-09T00:00:00Z
review text: Continuing his work [Exponential behaviour of solutions of a linear functional equation. Aequationes Math., to appear] the author offers, among others, the following result: Let \(\lambda\) be a real number, \((S,\Sigma,\sigma)\) a measure space, \(\sigma\) being a finite measure, and \(M: S\to\mathbb{R}\) be a \(\Sigma\)-measurable bounded function, for which the \(\sigma\)-measure of the set where \(M\neq 0\) is positive. If, for some solution \(\phi\) of (1)\(\;\phi(x)=\int_S\phi(x+M(s))\sigma(ds),\;\) there exists a finite limit (2)\(\;\lim_{x\to\infty}\;\phi(x)e^{-\lambda x}\;/x\neq 0,\) then (3)\(\;\int_S e^{\lambda M(s)}\sigma(ds)=1.\;\) If, on the other hand, for some regular solution \(\phi\) of (1), a finite limit (2) exists, then (3) is satisfied, \(\int_S M(s)e^{\lambda M(s)}\sigma(ds)\leq 0,\;\sigma(M<0)>0\) and \(\sigma(M>0)>0\) hold.
reviewed by: János Aczél
Mathematics Subject Classification ID: 39B22
zbMATH DE Number: 5083282
zbMATH Keywords: linear integral functional equations and inequalities
zbMATH Keywords: characteristic roots
author: Dariusz Sokołowski
MaRDI profile type: Publication
full work available at URL: j.jmaa.2006.04.019
OpenAlex ID: W2052191717
cites work: A convolution inequality
cites work: Solutions with constant sign at infinity of a linear functional equation of infinite order
cites work: Exponential behaviour of solutions of a linear functional equation
DOI: 10.1016/J.JMAA.2006.04.019
description: scientific article; zbMATH DE number 5083282