A new eigenvalue problem for the difference operator with nonlocal conditions
Articles
Mifodijus Sapagovas
Vilnius University
https://orcid.org/0000-0002-7139-3468
Regimantas Čiupaila
Vilnius Gediminas Technical University
Kristina Jakubėlienė
Kaunas University of Technology
Stasys Rutkauskas
Vilnius University
Published 2019-04-23
https://doi.org/10.15388/NA.2019.3.9
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Keywords

eigenvalue problem
nonlocal condition
difference operator

How to Cite

Sapagovas M., Čiupaila R., Jakubėlienė K. and Rutkauskas S. (2019) “A new eigenvalue problem for the difference operator with nonlocal conditions”, Nonlinear Analysis: Modelling and Control, 24(3), pp. 462-484. doi: 10.15388/NA.2019.3.9.

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.

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