Structural operators for linear delay-differential equations in Hilbert space
type: Item
instance of: scholarly article
title: Structural operators for linear delay-differential equations in Hilbert space
zbMATH Open document ID: 0668.45013
published in: Proceedings of the Japan Academy. Series A
publication date: 1988-01-01T00:00:00Z
review text: Let \[ (1)\quad du(t)/dt=A_ 0u(t)+A_ 1u(t-h)+\int^{0}_{- h}a(s)A_ 2u(t+s)ds \] be a linear delay-differential equation in a Hilbert space H. Let V be another Hilbert space such that \(V\subset H\subset V^*\). \(A_ 0\) is a linear operator associated with a strongly elliptic sesquilinear form defined on \(V\times V\). \(A_ i\), \(i=1,2\), are bounded linear operators from V to \(V^*\) such that \(A_ iA_ 0^{- 1}\), \(i=1,2\), map H into itself. If we apply the result of \textit{G. Di Blasio}, \textit{K. Kunisch} and \textit{E. Sinestrari} [J. Math. Anal. Appl. 102, 38-57 (1984; Zbl 0538.45007)] to (1) considering it as an equation in \(V^*\), then the associated solution semigroup S(t) is a \(C_ 0\)- semigroup in the space \(Z=H\times L^ 2(-h,0;V)\). Similarly, the solution semigroup \(S_ T(t)\) is defined for the equation (1) with \(A_ i\), \(i=0,1,2\), replaced by their adjoints. It is shown that \(FS(t)=S^*_ T(t)F\) and \(S^*(t)F^*=F^*S_ T(t)\) holds for \(t>0\), where F: \(V\to V^*\) is the structural operator for (1).
Mathematics Subject Classification ID: 45N05
Mathematics Subject Classification ID: 34K30
Mathematics Subject Classification ID: 45K05
zbMATH DE Number: 4094142
zbMATH Keywords: structural operator
zbMATH Keywords: linear delay-differential equation
zbMATH Keywords: Hilbert space
zbMATH Keywords: sesquilinear form
zbMATH Keywords: solution semigroup
zbMATH Keywords: adjoints
author: Hiroki Tanabe
reviewed by: Hiroki Tanabe
MaRDI profile type: Publication
full work available at URL: pjaa.64.265
OpenAlex ID: W2014375195
cites work: Stability for abstract linear functional differential equations
cites work: Structural properties of functional differential equations in Banach spaces
cites work: On fundamental solution of differential equation with time delay in Banach space
DOI: 10.3792/PJAA.64.265
description: scientific article; zbMATH DE number 4094142