Stability problems of functional differential equations with abstract Volterra operator
type: Item
instance of: scholarly article
title: Stability problems of functional differential equations with abstract Volterra operator
zbMATH Open document ID: 0866.45005
author: Yizeng Li
published in: Journal of Integral Equations and Applications
publication date: 1997-07-06T00:00:00Z
review text: The functional-differential equations \[ \dot x(t)= (Vx)(t)+(Fx)(t),\;0<t_0<t<T\leq\infty, \;x(t)=\Phi(t),\;0\leq t\leq t_0,\;x(t_0)=x^0\in\mathbb{R}^n \] are studied with either an abstract Volterra operator \(V\) and \(F=0\), \(T=\infty\), or a linear continuous operator \(V\) and a nonlinear \(F\) (nonhomogeneous case) with a Nemytzki type operator \(F\) has a most useful example. Five different stability problems are treated: stability, uniform stability, asymptotic stability, uniformly asymptotic stability, exponential asymptotic stability. For a linear \(V\) and \(F=0\) the necessary and sufficient stability conditions are presented. For the second (nonhomogeneous) case ``some asymptotic behavior of the solution'' is discussed.
Mathematics Subject Classification ID: 45N05
Mathematics Subject Classification ID: 45G10
Mathematics Subject Classification ID: 45M10
zbMATH DE Number: 934746
zbMATH Keywords: Nemytzki operator
zbMATH Keywords: functional-differential equations
zbMATH Keywords: abstract Volterra operator
zbMATH Keywords: uniform stability
zbMATH Keywords: asymptotic stability
zbMATH Keywords: asymptotic behavior
reviewed by: Roland Duduchava
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1216/jiea/1181075916
OpenAlex ID: W2075893990
cites work: Volterra integral and differential equations
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DOI: 10.1216/JIEA/1181075916
description: scientific article; zbMATH DE number 934746