Norm continuity and stability for a functional differential equation in Hilbert space
type: Item
instance of: scholarly article
title: Norm continuity and stability for a functional differential equation in Hilbert space
zbMATH Open document ID: 1013.34076
DOI: 10.1016/S0022-247X(02)00075-6
published in: Journal of Mathematical Analysis and Applications
publication date: 2002-09-08T00:00:00Z
review text: The author considers stability properties for the functional-differential equation \[ u'(t)= Au(t)+ \int^0_{-h} a(r) Au(t+ r) dr,\quad t> 0,\tag{1} \] \[ u(0)= \phi^0,\quad u(r)= \phi^1(r),\quad r\in [-h,0). \] He proves that when \(a(\cdot)\) is a measurable square integrable complex function then the solution semigroup is norm continuous for \(g> h\). The norm continuity of the solution semigroup is used to consider the stability of the solution to (1) in the case when \(a(\cdot)\) is a real function, more precisely, the author proves that the solution is asymptotically stable.
Mathematics Subject Classification ID: 34K30
Mathematics Subject Classification ID: 34K20
zbMATH DE Number: 1797508
zbMATH Keywords: solution semigroup
zbMATH Keywords: norm continuity
zbMATH Keywords: asymptotic stability
author: Miklavž Mastinšek
reviewed by: V. Petrov
MaRDI profile type: Publication
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Wikidata QID: Q115340228
full work available at URL: s0022-247x(02)00075-6
OpenAlex ID: W2071556410
description: scientific article; zbMATH DE number 1797508