Equivalence of functional-differential equations of neutral type and abstract Cauchy problems
type: Item
instance of: scholarly article
title: Equivalence of functional-differential equations of neutral type and abstract Cauchy problems
zbMATH Open document ID: 0583.34050
DOI: 10.1007/BF01298925
author: Kangpei Zhang
author: Franz Kappel
published in: Monatshefte für Mathematik
publication date: 1986-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/178268
review text: The authors prove equivalence of neutral functional differential equations \((1)\quad (d/dt)Dx_ t=Lx_ t+f(t)\) and abstract Cauchy problems in appropriate state spaces. If \(D\in {\mathcal B}(C;{\mathbb{R}}^ n)\) and \(L\in {\mathcal B}(W^{1,p};{\mathbb{R}}^ n)\) then (1) is equivalent to the abstract Cauchy problem \((2)\quad \dot u(t)={\mathcal A}u(t)+(f(t),0)\) in \({\mathbb{R}}^ n\times L^ p\), where \({\mathcal A}\) is the infinitesimal generator of the \(C_ 0\)-semigroup on \({\mathbb{R}}^ n\times L^ p\) generated by the generalized solutions of (1). D need not to be atomic at zero. If \(D\in {\mathcal B}(L^ p;{\mathbb{R}}^ n)\) then the abstract Cauchy problem (2) is in the state space \(L^ p\) instead of \({\mathbb{R}}^ n\times L^ p\).
Mathematics Subject Classification ID: 34K05
Mathematics Subject Classification ID: 34K30
Mathematics Subject Classification ID: 47D03
zbMATH DE Number: 3933552
zbMATH Keywords: first order differential equation
zbMATH Keywords: neutral functional differential equations
zbMATH Keywords: abstract Cauchy problems
zbMATH Keywords: state spaces
MaRDI profile type: Publication
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OpenAlex ID: W2065223890
description: scientific article; zbMATH DE number 3933552