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A new eigenvalue problem for the difference operator with nonlocal conditions

Vilnius University Press

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Abstract

In the paper, the spectrum structure of one-dimensional differential operator with nonlocal conditions and of the difference operator, corresponding to it, has been exhaustively investigated. It has been proved that the eigenvalue problem of difference operator is not equivalent to that of matrix eigenvalue problem Au = λu, but it is equivalent to the generalized eigenvalue problem Au = λBu with a degenerate matrix B. Also, it has been proved that there are such critical values of nonlocal condition parameters under which the spectrum of both the differential and difference operator are continuous. It has been established that the number of eigenvalues of difference problem depends on the values of these parameters. The condition has been found under which the spectrum of a difference problem is an empty set. An elementary example, illustrating theoretical expression, is presented.

Most cited references19

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Arnoldi and Jacobi-Davidson methods for generalized eigenvalue problems $Ax=\lambda Bx$ with singular $B$

(2007)
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Stepwise Stability for the Heat Equation with a Nonlocal Constraint

(1995)
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• Record: found

On the spectral stability in subspaces for difference schemes with nonlocal boundary conditions

(2013)
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Author and article information

Journal
Nonlinear Analysis: Modelling and Control
Vilnius University Press
1392-5113
April 23 2019
April 23 2019
: 24
: 3
: 462-484
Affiliations
[1 ]Faculty of Mathematics and Informatics, Vilnius University
[2 ]Vilnius Gediminas Technical University
[3 ]Department of Applied Mathematics, Kaunas University of Technology
[4 ]Institute of Data Science and Digital Technologies, Vilnius University
Article
10.15388/NA.2019.3.9