On basis property of eigenfunctions and associated functions for nonlocal boundary value problems
type: Item
instance of: scholarly article
title: On basis property of eigenfunctions and associated functions for nonlocal boundary value problems
zbMATH Open document ID: 0868.35079
DOI: 10.1007/BF02674617
author: I. V. Aliev
published in: Siberian Mathematical Journal
publication date: 1997-07-22T00:00:00Z
review text: The author considers the spectral problem \[ -D_x^2 u(x,y) + (-1)^m D_y^{2m} u(x,y) + \sum\limits_{j=0}^{m-1} a_j(x,y) D_xD_y^j u(x,y) + \] \[ +\sum\limits_{j=0}^{2m-1} b_j(x,y) D_y^j u(x,y) + q(y)u(x,y) = \lambda u(x,y) ,\;(x,y) \in Q=(0,1)\times \mathbb{R}, \eqno{(1)} \] \[ L_{\nu}u = \alpha_{\nu}u^{(m_{\nu})}_{x=0} + \beta_{\nu} u^{(m_{\nu})}_{x=1} + \alpha_{\nu 0} u{}_{x=0} + \beta_{\nu 0} u{}_{x=1} =0,\;\;\nu =1,2, \eqno{(2)} \] where \(\alpha_{\nu 0} = \beta_{\nu 0} =0 \) for \(m_{\nu} =0\). The problem is reduced to the following problem in the space \(L_2(0,1;H)\) (\(H\) is a Hilbert space; to problem (1), (2) there corresponds \(H=L_2(\mathbb{R})\)): \[ -u''(x) + A u(x) + A_1(x) u'(x) + A_2(x) u(x) = \lambda u(x),\;\;L_{\nu}u=0. \eqno{(3)} \] Here \(A\) is a positive operator whose resolvent set contains a set of the form \(\{\lambda:\;{}\lambda{} \geq r,\;{}{\operatorname{arg}}\lambda{} \geq \delta \}\) and such that \(A^{-1} \in \sigma_p\) for some \(p\). The operators \(A_1\) and \(A_2\) are assumed to be subordinate to the operator \(-\frac{d^2}{dx^2} + A\). Under natural conditions on the boundary operators \(L_{\nu}\) it is shown that the spectrum of problem (3) consists of isolated eigenvalues of finite algebraic multiplicity, and the eigenfunctions and the associated functions form a basis in \(L_2(0,1;H)\) for the Abel summation method of order \(\gamma \in ((2p+1)/2,\pi/2\delta)\).
Mathematics Subject Classification ID: 35P10
Mathematics Subject Classification ID: 35P05
zbMATH DE Number: 1037973
zbMATH Keywords: basis property
zbMATH Keywords: spectral problems
zbMATH Keywords: higher order equations
zbMATH Keywords: boundary value problems
MaRDI profile type: Publication
cites work: Completeness of the system of eigenfunctions and associated functions of a nonself-adjoint boundary value problem for a differential-operator equation of second order
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full work available at URL: https://doi.org/10.1007/bf02674617
OpenAlex ID: W2084852465
description: scientific article; zbMATH DE number 1037973