Asymptotic decomposition of boundary value problems for differential equations in Banach spaces
type: Item
instance of: scholarly article
title: Asymptotic decomposition of boundary value problems for differential equations in Banach spaces
zbMATH Open document ID: 0828.34047
DOI: 10.1007/BF01210426
published in: Mathematical Notes
publication date: 1994-11-16T00:00:00Z
review text: The author considers the boundary-value problem \[ \frac{du}{dt} = kA(t)u+ F,\;(t\in \mathbb{R}),\qquad \lim_{t\to\pm\infty} V_\pm u(t)= 0, \tag{P} \] where \(k\) is a major asymptotic parameter, \(u, F: \mathbb{R}\to B\) (\(B\) a Banach space), \(A(t): D(A(t))\subset B\to B\) are closed linear operators which are stabilized as \(t\to \pm\infty\). He assumes that \(B= B_1(t)+ B_2(t)\), where \(B_i(t)\) are invariant under \(A(t)\) and the spectra of \(A(t)/B_i(t)\), \(i= 1,2\), are uniformly separated for \(t\in \mathbb{R}\) and \(V_\pm\) are linear bounded operators for which \(B_i(\pm \infty)\) are invariant, \(i= 1,2\). Under some additional assumptions, the author shows that the asymptotic behaviour of the solutions of (P) as \(k\to \infty\) can be described by means of the asymptotic behaviour of some auxiliary problems in \(B_i(t)\), \(i= 1,2\), even though the system cannot be decoupled.
Mathematics Subject Classification ID: 34G10
Mathematics Subject Classification ID: 34D05
zbMATH DE Number: 687526
zbMATH Keywords: asymptotic decomposition
zbMATH Keywords: boundary-value problem
zbMATH Keywords: Banach space
zbMATH Keywords: closed linear operators
zbMATH Keywords: asymptotic behaviour
author: S. L. Ehdel'shtejn
reviewed by: Ioan I. Vrabie
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 687526