Continuous dependence on modeling for nonlinear ill-posed problems
type: Item
instance of: scholarly article
title: Continuous dependence on modeling for nonlinear ill-posed problems
zbMATH Open document ID: 1169.34042
author: Beth M. Campbell Hetrick
author: Rhonda J. Hughes
published in: Journal of Mathematical Analysis and Applications
publication date: 2008-11-06T00:00:00Z
review text: The authors study the nonlinear ill-posed Cauchy problem \[ \frac{du}{dt}=Au(t)+h(t,u(t)),\;\;u(0)=\varkappa, \] where \(A\) is a positive self-adjoint operator on a Hilbert space \(\mathcal{H}\) and \(h:[0,T)\times\mathcal{H}\to\mathcal{H}\) is a uniformly Lipschitz function with respect to both variables. They prove continuous dependence on modeling for this problem. If the solutions exist, they depend continuously on solutions to corresponding approximate well-posed problems \[ \frac{dv}{dt}=f(A)v(t)+h(t,v(t)),\;\;v(0)=\varkappa, \] where \(f\) is a real-valued function that approximates \(A\) in a suitable sense. The results are obtained by extending the solutions into the complex plane and introducing a related holomorphic function whose properties yield the Hölder continuous dependence.
Mathematics Subject Classification ID: 34G20
zbMATH DE Number: 5362167
zbMATH Keywords: abstract Cauchy problem
zbMATH Keywords: continuous depending on modeling
zbMATH Keywords: nonlinear ill-posed problem
reviewed by: Narcisa Apreutesei
MaRDI profile type: Publication
full work available at URL: j.jmaa.2008.08.052
OpenAlex ID: W2170651764
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DOI: 10.1016/J.JMAA.2008.08.052
description: scientific article; zbMATH DE number 5362167