Existence and continuous dependence of mild solutions to semilinear functional differential equations in Banach spaces
type: Item
instance of: scholarly article
title: Existence and continuous dependence of mild solutions to semilinear functional differential equations in Banach spaces
zbMATH Open document ID: 0964.34068
author: Jong Son Shin
author: Toshiki Naito
published in: Tôhoku Mathematical Journal. Second Series
publication date: 2000-10-26T00:00:00Z
review text: The authors consider the initial value problem for the semilinear functional-differential equation with infinite delay \[ \frac {d} {dt} u(t) =Au(t)+F(t,u_t), \qquad \sigma < t\leq \sigma +a, \] \[ u(t)=\varphi (t-\sigma), \qquad t\in (-\infty,\sigma), \] where \(A\) is the infinitesimal generator of a \(C_0\)-semigroup on a Banach space \(E\), \(F\) is a continuous mapping defined on a suitably chosen memory function space \(B\), \(\varphi\in B\). They prove some existence and continuous dependence on the data results under the additional assumption that either the semigroup generated by \(A\) is compact, or \(F\) satisfies a certain condition expressed in terms of the measure of compactness.
Mathematics Subject Classification ID: 34K30
Mathematics Subject Classification ID: 34G20
Mathematics Subject Classification ID: 45K05
zbMATH DE Number: 1445642
zbMATH Keywords: infinite delay equation
zbMATH Keywords: compact semigroup
zbMATH Keywords: measure of noncompactness
zbMATH Keywords: existence
zbMATH Keywords: continuous dependence
reviewed by: Ioan I. Vrabie
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2748/tmj/1178224719
OpenAlex ID: W2036663410
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DOI: 10.2748/TMJ/1178224719
description: scientific article; zbMATH DE number 1445642