Ill-posed problems, \(C_{0}\)-semigroups and the Showalter regularization
type: Item
instance of: scholarly article
title: Ill-posed problems, \(C_{0}\)-semigroups and the Showalter regularization
zbMATH Open document ID: 1077.47013
author: Claus Müller
author: Eberhard Schock
published in: Journal of Mathematical Analysis and Applications
publication date: 2005-03-08T00:00:00Z
review text: The authors study linear ill-posed equations \(Ax=y\) in Banach space \(X\), where \(-A\) generates a bounded \(C_0\)-semigroup. The Showalter method known for bounded operators \(A\) is here generalized to unbounded operators. Consider the abstract Cauchy problem \(u^{\prime}(t)=-Au(t)+y\), \(t \geq 0\), \(u(0)=0\), \(u \in C^1([0,\infty),X)\). Let \(T: [0,\infty) \to {\mathcal L}(X)\) be a semigroup generated by \(-A\). The unknown solution \(x\) is approximated by the elements \(\int_0^t T(s)y ds\) and \(t^{-1} \int_0^t u(s)ds\) \((t \to \infty)\). Rate of convergence estimates have the form \(O(t^{-\mu})\), provided that \(x \in \text{ Im}A^{\mu}\), \(\mu \in (0,1)\).
reviewed by: Mikhail Yu. Kokurin
Mathematics Subject Classification ID: 47A52
Mathematics Subject Classification ID: 47A50
Mathematics Subject Classification ID: 47D06
zbMATH DE Number: 2141777
zbMATH Keywords: ill-posed problems
zbMATH Keywords: Showalter regularization
zbMATH Keywords: abstract Cauchy problems
MaRDI profile type: Publication
full work available at URL: j.jmaa.2004.06.028
OpenAlex ID: W2048968536
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DOI: 10.1016/J.JMAA.2004.06.028
description: scientific article; zbMATH DE number 2141777