Well-posedness of differential-operator problems. II: The Cauchy problem for complete second-order equations in Banach spaces
type: Item
instance of: scholarly article
title: Well-posedness of differential-operator problems. II: The Cauchy problem for complete second-order equations in Banach spaces
zbMATH Open document ID: 0922.34056
DOI: 10.1007/BF02365213
author: A. Filinkov
author: I. V. Melnikova
published in: Journal of Mathematical Sciences (New York)
publication date: 1999-05-27T00:00:00Z
review text: [For Part I, see the review Zbl 0922.34055 above. ] Consider the Cauchy problem for the complete second-order equation \[ u'' = B u' + Au,\quad t \geq 0,\qquad u(0)=u_0,\quad u'(0) = u_1, \] where \(A\) and \(B\) are closed linear operators on a Banach space \(X\). The authors introduce the notion of the \(\omega\)-well-posedness of the above Cauchy problem and construct a theory of \(M,N\)-functions for commuting operators \(A,B\). Necessary and sufficient conditions for the \(\omega\)-well-posedness of the above problem are given in terms of conditions on the resolvent of the operator pair \(A,B\), which is \[ R(\lambda^2) = (\lambda^2-\lambda B - A)^{-1}. \]
Mathematics Subject Classification ID: 34K10
Mathematics Subject Classification ID: 34K30
Mathematics Subject Classification ID: 46F20
zbMATH DE Number: 1292311
zbMATH Keywords: \(\omega\)-well-posed problem
zbMATH Keywords: uniformly well-posed problem
zbMATH Keywords: the resolvent of an operator pair
zbMATH Keywords: \(M,N\)-function family
zbMATH Keywords: \(\omega\)-closed operator pair
reviewed by: Narcisa Apreutesei
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf02365213
OpenAlex ID: W1974365627
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description: scientific article; zbMATH DE number 1292311