Solvability of a nonhomogeneous boundary value problem for a differential system with measures
type: Item
instance of: scholarly article
title: Solvability of a nonhomogeneous boundary value problem for a differential system with measures
zbMATH Open document ID: 1070.34024
DOI: 10.1023/A:1026017600916
author: V. V. Mazurenko
author: R. M. Tatsiĭ
published in: Differential Equations
publication date: 2005-03-15T00:00:00Z
review text: The authors consider nonhomogeneous linear boundary value problems of the form \[ JY'=(B'+\lambda A')Y+F' \] on an interval \([a,b]\) with certain inhomogeneous boundary conditions in \(a\) and \(b\). Here, \(A\) and \(B\) are Hermitian \(n\times n\)-matrix functions, \(J\) is a constant \(n\times n\)-matrix satisfying \(J^\ast=-J\), \(J^\ast J= E\), and \(F\) is an \(n\times 1\)-vector function. The functions \(A\), \(B\) and the inhomogeneity \(F\) are only assumed to be right continuous and locally of bounded variation over \([a,b]\) so that their (generalized) derivatives appearing in the differential equation are Stieltjes measures. The main result of this paper is a solvability condition for the above boundary value problem and a formula for the solution \(Y\) in the case that the complex parameter \(\lambda\) is an eigenvalue of the corresponding homogeneous boundary eigenvalue problem (with \(F'=0\)); the simpler case where \(\lambda\) is an eigenvalue of the latter has been considered in the previous work of the references. To this end, the problem is reduced to an equivalent loaded integral equation of Fredholm-Stieltjes type with matrix kernel.
reviewed by: Christiane Tretter
Mathematics Subject Classification ID: 34B05
zbMATH DE Number: 2146350
zbMATH Keywords: nonhomogeneous boundary value problem
zbMATH Keywords: solvability condition
zbMATH Keywords: distributional coefficients
MaRDI profile type: Publication
full work available at URL: a:1026017600916
OpenAlex ID: W7140074
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description: scientific article; zbMATH DE number 2146350