Linear Volterra equations and integrated solution families
type: Item
instance of: scholarly article
title: Linear Volterra equations and integrated solution families
zbMATH Open document ID: 0862.45017
DOI: 10.1007/BF02574144
author: Hirokazu Oka
published in: Semigroup Forum
publication date: 1996-12-15T00:00:00Z
full work available at URL: https://eudml.org/doc/135492
review text: Consider the linear Volterra equation \[ u(t)=f(t)+A\int^t_0 a(t-s)u(s)ds\quad\text{for }t\geq 0.\tag{\(*\)} \] Here \(A\) is an unbounded closed operator in a Banach space \(X\), \(u\) and \(f\) take their values in \(X\), and \(a\) is a scalar function. Often the operator \(A\) is alternatively moved inside the integral. There is a variation of constants formula \[ u(t)= {d\over dt}\int^t_0 S(t-s)f(s)ds, \] where \(S\) is the solution family (the resolvent operator), characterized by the fact that \(AS(t)=S(t)A\) on the domain of \(A\) for all \(t\geq 0\), and \(S(t)x=x+A\int^t_0 a(t-s)S(s)x ds\) for all \(x\in X\) and \(t\geq 0\). To get a more general variation of constants formula one can use an \(n\)-times integrated solution family \(S_n\) instead (which can be thought of as the \(n\)th integral of a solution family), and write the variation of constants formula in the form \[ u(t)={d^{n+1}\over dt^{n+1}}\int^t_0 S(t-s)f(s)ds, \] which can then be interpreted in a weaker or stronger sense. This paper studies the existence and uniqueness of \(n\)-times integrated solution families and their relations to the existence of more or less smooth solutions of \((*)\). As two special cases one gets \(n\)-times integrated semigroups (take \(a(t)=1\)) or cosine families (take \(a(t)=t\)). A number of perturbation results are given where \(A\) is replaced by either \(A+B\) or \(CA\). The adjoint problem is studied as well.
Mathematics Subject Classification ID: 45N05
Mathematics Subject Classification ID: 45D05
zbMATH DE Number: 943669
zbMATH Keywords: integrated solution families
zbMATH Keywords: linear Volterra equation
zbMATH Keywords: unbounded closed operator
zbMATH Keywords: Banach space
zbMATH Keywords: resolvent operator
zbMATH Keywords: variation of constants formula
zbMATH Keywords: semigroups
zbMATH Keywords: perturbation results
reviewed by: Olof Staffans
MaRDI profile type: Publication
cites work: Vector-valued Laplace transforms and Cauchy problems
cites work: Q3830008
cites work: Linear integro-differential equations in Banach spaces
cites work: Q3797548
cites work: Non autonomous evolution operators of hyperbolic type
cites work: Counterexamples for abstract linear Volterra equations
cites work: Q3479064
cites work: Q3254057
cites work: Integrated semigroups
cites work: On an extension of the Trotter-Kato theorem for resolvent families of operators
cites work: On Volterra Equations Associated with a Linear Operator
cites work: Q4320583
cites work: Integrated resolvent operators
cites work: Evolutionary Integral Equations and Applications
cites work: Positive perturbations of linear Volterra equations and sine functions of operators
cites work: Multiplicative Perturbations of Linear Volterra Equations
cites work: Q3746374
OpenAlex ID: W2016284340
description: scientific article; zbMATH DE number 943669