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Spectral properties of a beam equation with eigenvalue parameter occurring linearly in the boundary conditions

Part of: General theory of linear operators Ordinary differential operators Qualitative theory Dynamical problems Boundary value problems

Published online by Cambridge University Press:  30 July 2021

Ziyatkhan S. Aliyev Open the ORCID record for Ziyatkhan S. Aliyev [Opens in a new window]  and
Gunay T. Mamedova Open the ORCID record for Gunay T. Mamedova [Opens in a new window]
Ziyatkhan S. Aliyev
Affiliation:
Baku State University, Baku AZ1148, Azerbaijan Institute of Mathematics and Mechanics NAS of Azerbaijan, Baku AZ1141, Azerbaijan National Aviation Academy of Azerbaijan, Baku AZ1045, Azerbaijan (z_aliyev@mail.ru)
Gunay T. Mamedova
Affiliation:
Ganja State University, Ganja AZ2001, Azerbaijan (gunaymamedova614@gmail.com)
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Abstract

In this paper, we consider an eigenvalue problem for ordinary differential equations of fourth order with a spectral parameter in the boundary conditions. The location of eigenvalues on real axis, the structure of root subspaces and the oscillation properties of eigenfunctions of this problem are investigated, and asymptotic formulas for the eigenvalues and eigenfunctions are found. Next, by the use of these properties, we establish sufficient conditions for subsystems of root functions of the considered problem to form a basis in the space $L_p,1 < p < \infty$.

Keywords

Differential operatorEigenvalueEigenfunctionBasis property of root functions

MSC classification

Primary: 34B05: Linear boundary value problems
Secondary: 34B08: Parameter dependent boundary value problems 34C10: Oscillation theory, zeros, disconjugacy and comparison theory 34L10: Eigenfunctions, eigenfunction expansions, completeness of eigenfunctions 47A75: Eigenvalue problems 74H45: Vibrations
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 152 , Issue 3 , June 2022 , pp. 780 - 801
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Copyright
Copyright © The Author(s), 2021. Published by Cambridge University Press on behalf of The Royal Society of Edinburgh

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