On a boundary value problem with a spectral parameter in the boundary conditions
type: Item
instance of: scholarly article
title: On a boundary value problem with a spectral parameter in the boundary conditions
zbMATH Open document ID: 0942.34067
published in: Siberian Mathematical Journal
publication date: 2000-04-16T00:00:00Z
review text: The authors study the boundary value problem with a spectral parameter in the equation and the boundary conditions: \[ -u'' + q(x)u = \lambda^2u,\quad 0<x<1, \] \[ (\alpha_0 + \alpha_1\lambda + \alpha_2\lambda^2)u(0) + u'(0) = 0,\quad (\beta_0 + \beta_1\lambda + \beta_2\lambda^2)u(1) + u'(1) = 0. \] Here, \(\lambda\) is a spectral parameter, \(q(x)\) is a nonnegative continuous function on the interval \([0,1]\), and \(\alpha_i\) and \(\beta_i\) are real constants. The authors study properties of the eigenvalues and eigenfunctions of the boundary value problem. The main result is a theorem on the number of zeros of eigenfunctions under their natural enumeration and a theorem on asymptotic formulas for eigenvalues and eigenfunctions. It was Shkalikov who drew the authors' attention to the nontriviality of this problem in the case when the boundary conditions involve \(\lambda\) and \(\lambda^2\).
reviewed by: Vladimir N. Grebenev
Mathematics Subject Classification ID: 34L05
Mathematics Subject Classification ID: 34B24
Mathematics Subject Classification ID: 34L20
Mathematics Subject Classification ID: 34C10
zbMATH DE Number: 1432313
zbMATH Keywords: boundary value problem for an ordinary differential equation
zbMATH Keywords: spectral theory
zbMATH Keywords: spectral parameter in the boundary conditions
zbMATH Keywords: zeros of an eigenfunction
zbMATH Keywords: asymptotic formulas for eigenvalues and eigenfunctions
author: Nazim B. Kerimov
author: Kh. R. Mamedov
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1432313