Integrated resolvent operators
type: Item
instance of: scholarly article
title: Integrated resolvent operators
zbMATH Open document ID: 0846.45005
author: Hirokazu Oka
published in: Journal of Integral Equations and Applications
publication date: 1996-09-23T00:00:00Z
review text: Der Autor betrachtet die Integro-Differential-Gleichung \[ u'(t)=Au (t)+\int^t_0 B(t-s) u(s) ds+f(t), \quad t\in [ 0,T ], \quad u(0) =x. \tag{VE} \] Dabei ist \(A\) ein linearer abgeschlossener Operator mit (nicht notwendig dichtem) Definitionsbereich \(D(A)\) in einem Banachraum \(X\), der die Hille-Yoshida-Bedingung erfüllt. Es gibt reelle Konstanten \(M\) und \(\omega\), so daß \((\omega, \infty) \subset \rho (A)\) (die Resolventenmenge von \(A\)) und \(|(\lambda-A)|\leq M(\lambda-\omega)^{-n}\) für \(n=1, 2, \dots\) und \(\lambda> \omega\). \(\{B (t): t\geq 0\}\) ist eine Familie linearer Operatoren in \(X\) mit \(D(A) \subset D(B (t))\) für \(t\geq 0\) linearer beschränkter Operatoren von \(D(A)\) mit der Graphennorm nach \(X\), so daß die Funktion \(B(\cdot) x\) für \(x\in D(A)\) auf jedem Intervall \([ 0,T ]\) im starken Sinn von beschränkter Variation ist. Der Autor zeigt, daß es dann für (VE) ``a unique locally Lipschitz-continuous integrated resolvent operator \(\{ R(t): t\geq 0\}\)'' gibt, und daß dann (VE) für alle \(x\in X\) eine eindeutige ``Integral-Lösung'' und eine eindeutige ``schwache Lösung'' hat, Begriffe, die in der Arbeit definiert werden. Mit Hilfe dieser Ergebnisse wird für die beiden Cauchy-Probleme \[ u'(t)-Au(t)-Bu(t)=f(t), \quad t\in [0, T], \quad u(0)=x, \quad u'(t)=y \] und \[ u'(t)=A\Biggl( u(t)+\int^t_0 F(t-s) u(s) ds \Biggr)+Ku(t)+f(t), \quad t\in [0, T], \quad u(0)=x \] (unter weiteren Bedingungen an die auftretenden Größen) die Existenz klassischer Lösungen nachgewiesen.
reviewed by: Marianne Reichert
Mathematics Subject Classification ID: 45N05
Mathematics Subject Classification ID: 45J05
zbMATH DE Number: 826724
zbMATH Keywords: integrodifferential equation
zbMATH Keywords: Banach space
zbMATH Keywords: Hille-Yoshida condition
zbMATH Keywords: integral solution
zbMATH Keywords: weak solution
zbMATH Keywords: Cauchy problem
zbMATH Keywords: integrated resolvent operator
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1216/jiea/1181075869
OpenAlex ID: W2066352261
cites work: Vector-valued Laplace transforms and Cauchy problems
cites work: Q3830008
cites work: Shorter Notes: Strongly Continuous Semigroups, Weak Solutions, and the Variation of Constants Formula
cites work: Q3896020
cites work: Q3797548
cites work: Some considerations for linear integrodifferential equations
cites work: Q4039882
cites work: Integrated semigroups amd integrodifferential equations
cites work: On linear Volterra equations in Banach spaces
cites work: Q3336187
cites work: Q3244654
cites work: Integrated semigroups
cites work: Q4320583
cites work: Well-Posedness of Higher Order Abstract Cauchy Problems
cites work: Integrated semigroups and their application to complete second order Cauchy problems
cites work: A rate of convergence for the set compound estimation in a family of certain retracted distributions
cites work: Semigroups of linear operators and applications to partial differential equations
cites work: Q3756038
cites work: Semiflows generated by Lipschitz perturbations of non-densely defined operators
cites work: On linear integro-differential equations in a Banach space
DOI: 10.1216/JIEA/1181075869
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description: scientific article; zbMATH DE number 826724