Toward a Sturm-Liouville theory for an equation with generalized coefficients
type: Item
instance of: scholarly article
title: Toward a Sturm-Liouville theory for an equation with generalized coefficients
zbMATH Open document ID: 1078.34012
author: S. A. Shabrov
author: Yu. V. Pokornij
published in: Journal of Mathematical Sciences (New York)
publication date: 2005-06-07T00:00:00Z
review text: Consider the generalized equation \[ -(p u')' + Q' u = \lambda R' u + F' \] over \([0, 1]\), with \(p, Q, R, F\) having bounded variation in \([0, 1]\). Let \(\mu (x)\) be an increasing function such that \(Q, R, F\) are \(\mu\)-absolutely continuous (an example of such a \(\mu\) is given). Let \(E_\mu\) be the set of functions \(u(x)\) such that \(u(x)\) is absolutely continuous on \([0, 1]\), \(u'(x)\) has bounded variation in \([0, 1]\) as well as \(p(x) u'(x)\) is \(\mu\)-absolutely continuous. A ``solution'' of the above equation is defined to be a function in \(E_\mu\) that satisfies the equality almost everywhere (with respect to the \(\mu\)-measure).\ The authors establish some well-known classical results to the above mentioned generalized settings which include properties concerning the existence, uniqueness, continuity, Sturm-type oscillation of the solution of the initial value problems as well as the properties concerning Green's function and the eigenpairs of the Dirichlet boundary value problems.
Mathematics Subject Classification ID: 34B24
zbMATH DE Number: 2174270
zbMATH Keywords: Sturm-Liouville theory
zbMATH Keywords: generalized coefficients
zbMATH Keywords: integral theory
MaRDI profile type: Publication
DOI: 10.1023/B:JOTH.0000012755.40752.A1
description: scientific article; zbMATH DE number 2174270