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 Mat. Zametki, 2017, Volume 101, Issue 5, Pages 768–778 (Mi mz11468)

A Regular Differential Operator with Perturbed Boundary Condition

M. A. Sadybekova, N. S. Imanbaevab

a Institute of Mathematics and Mathematical Modeling, Ministry of Education and Science, Republic of Kazakhstan
b South Kazakhstan State Pedagogical institute

Abstract: The operator $\mathcal{L}_{0}$ generated by a linear ordinary differential expression of $n$th order and regular boundary conditions of general form is considered on a closed interval. The characteristic determinant of the spectral problem for the operator $\mathcal{L}_{1}$, where $\mathcal{L}_{1}$ is an operator with the integral perturbation of one of its boundary conditions, is constructed, assuming that the unperturbed operator $\mathcal{L}_{0}$ possesses a system of eigenfunctions and associated functions generating an unconditional basis in $L_{2}(0,1)$. Using the obtained formula, we derive conclusions about the stability or instability of the unconditional basis properties of the system of eigenfunctions and associated functions of the problem under an integral perturbation of the boundary condition. The Samarskii–Ionkin problem with integral perturbation of its boundary condition is used as an example of the application of the formula. \renewcommand{\qed}

Keywords: basis, regular boundary condition, eigenvalue, root function, spectral problem, integral perturbation of the boundary condition, characteristic determinant.

 Funding Agency Grant Number Ministry of Education and Science of the Republic of Kazakhstan 0825/ÃÔ4 This work was supported by the Committee of Science of the Ministry of Education and Science of the Republic of Kazakhstan under grant 0825/GF4.

Author to whom correspondence should be addressed

DOI: https://doi.org/10.4213/mzm11468

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English version:
Mathematical Notes, 2017, 101:5, 878–887

Bibliographic databases:

UDC: 517.927
PACS: 02.30.Jr, 02.30.Tb
Revised: 20.11.2016

Citation: M. A. Sadybekov, N. S. Imanbaev, “A Regular Differential Operator with Perturbed Boundary Condition”, Mat. Zametki, 101:5 (2017), 768–778; Math. Notes, 101:5 (2017), 878–887

Citation in format AMSBIB
\Bibitem{SadIma17} \by M.~A.~Sadybekov, N.~S.~Imanbaev \paper A Regular Differential Operator with Perturbed Boundary Condition \jour Mat. Zametki \yr 2017 \vol 101 \issue 5 \pages 768--778 \mathnet{http://mi.mathnet.ru/mz11468} \crossref{https://doi.org/10.4213/mzm11468} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=3646481} \elib{https://elibrary.ru/item.asp?id=29106617} \transl \jour Math. Notes \yr 2017 \vol 101 \issue 5 \pages 878--887 \crossref{https://doi.org/10.1134/S0001434617050133} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000404236900013} \scopus{https://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-85021285330} 

• http://mi.mathnet.ru/eng/mz11468
• https://doi.org/10.4213/mzm11468
• http://mi.mathnet.ru/eng/mz/v101/i5/p768

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