On fundamental solution of differential equation with time delay in Banach space
type: Item
instance of: scholarly article
title: On fundamental solution of differential equation with time delay in Banach space
zbMATH Open document ID: 0658.34056
published in: Proceedings of the Japan Academy. Series A
publication date: 1988-01-01T00:00:00Z
review text: This paper is concerned with the construction of the fundamental solution in the sense of \textit{S. Nakagiri}, J. Differ. Equations 41, 349-368 (1981; Zbl 0441.35068) to the linear differential equation with time delay in a Banach space: \[ du(t)/dt=Au(t)+A_ 1u(t-r)+\int^{0}_{- r}a(s)A_ 2u(t+s)ds \] of the type investigated by \textit{G. Di Blasio}, \textit{K. Kunisch} and \textit{E. Sinestrari} [J. Math. Anal. Appl. Vol. 102, 38-57 (1984; Zbl 0538.45007), Israel J. Math. 50, 231-263 (1985; Zbl 0594.45014), and \textit{E. Sinestrari}, Math. Z. 186, 223-246 (1984; Zbl 0521.35044)]. Here, A is the infinitesimal generator of an analytic semigroup such that D(A) is dense, \(A_ 1\) and \(A_ 2\) are operators such that \(D(A_ 1)\supset D(A)\), \(D(A_ 2)\supset D(A)\) and a(s) is a real valued Hölder continuous function in the interval [-h,0]. It is shown that the fundamental solution W(t) exists and satisfies \(\| dW(t)/dt\| \leq C_ n/(t-nr),\quad \| AW(t)\| \leq C_ n/(t-nr)\) in the interval \((nr,(n+1)r]\).
Mathematics Subject Classification ID: 34K05
Mathematics Subject Classification ID: 34G20
Mathematics Subject Classification ID: 35G10
Mathematics Subject Classification ID: 35K25
Mathematics Subject Classification ID: 45K05
zbMATH DE Number: 4075596
zbMATH Keywords: time delay
zbMATH Keywords: Banach space
author: Hiroki Tanabe
reviewed by: Hiroki Tanabe
MaRDI profile type: Publication
full work available at URL: pjaa.64.131
OpenAlex ID: W2085545747
cites work: Q3346708
cites work: \(L^ 2\)-regularity for parabolic partial integrodifferential equations with delay in the highest-order derivatives
cites work: Stability for abstract linear functional differential equations
cites work: On the fundamental solution of delay-differential equations in Banach spaces
cites work: Optimal control of linear retarded systems in Banach spaces
cites work: Pointwise completeness and degeneracy of functional differential equations in Banach spaces. I: General time delays
cites work: Structural properties of functional differential equations in Banach spaces
cites work: Q3810201
cites work: Controllability and observability of linear retarded systems in Banach spaces
cites work: Q3667394
cites work: On a class of retarded partial differential equations
Wikidata QID: Q115219941
DOI: 10.3792/PJAA.64.131
description: scientific article; zbMATH DE number 4075596