Counterexamples for abstract linear Volterra equations
type: Item
instance of: scholarly article
title: Counterexamples for abstract linear Volterra equations
zbMATH Open document ID: 0778.45013
author: W. Desch
author: Jan Pruess
published in: Journal of Integral Equations and Applications
publication date: 1994-01-03T00:00:00Z
review text: The main object of this paper is the study of the abstract Volterra equation \(u(t)=\int^ t_ 0a(t-s)Au(s)ds+f(t)\) where \(A\) is an unbounded linear opertor in a Banach space \(X\), \(f\) is an \(X\)-valued function and \(a\) a scalar valued function through the existence and properties of the resolvent operator \(t\mapsto S(t)\in{\mathcal L}(x)\) satisfying \(S(t)Ax=AS(t)x\), \(S(t)x=x+A\int^ t_ 0a(t-s)S(s)xds\). More precisely the authors show that (i) there exists an exponentially bounded \(a\) such that a resolvent operator exists but is not exponentially bounded (ii) there exists a negative self-adjoint unbounded operator \(A\) in a Hilbert space \(X\) and a kernel \(a\) such that the resolvent operator is analytic and \(1/|\hat a|\) grows faster than polynomially \((\hat a\) is the Laplace transform of \(a)\). Finally it is shown that there exists an analytic function \(F\) defined on the right half plane and satisfying \(| F(s)|\leq M/| s|\) on the right half plane, such that \(F\) is not the Laplace transform of an \(L^ \infty\) function on the positive half axis.
Mathematics Subject Classification ID: 45N05
Mathematics Subject Classification ID: 44A10
zbMATH DE Number: 205193
zbMATH Keywords: counterexamples
zbMATH Keywords: abstract linear Volterra equations
zbMATH Keywords: Banach space
zbMATH Keywords: resolvent operator
zbMATH Keywords: Hilbert space
zbMATH Keywords: Laplace transform
reviewed by: Gabriella Di Blasio
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1216/jiea/1181075726
OpenAlex ID: W2020039349
Wikidata QID: Q124986160
cites work: Q4049588
cites work: Q3872710
cites work: Q5621638
cites work: Q3719254
cites work: Q4692383
cites work: Linear integro-differential equations in Banach spaces
cites work: Q3697587
DOI: 10.1216/JIEA/1181075726
description: scientific article; zbMATH DE number 205193