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Spectral properties of the equation of a vibrating rod at both ends of which the masses are concentrated

Banach Journal of Mathematical Analysis volume 14, pages 585–606 (2020)Cite this article

Abstract

In this paper we consider a spectral problem for ordinary differential equation of fourth order with a spectral parameter in the boundary conditions. This problem arises when variables are separated in the dynamical boundary value problem describing bending vibrations of a homogeneous rod, in cross-sections of which the longitudinal force acts, the both ends of which are fixed elastically and on these ends the masses are concentrated. We investigate locations, multiplicities of eigenvalues, study the oscillation properties of eigenfunctions and establish sufficient conditions for the subsystems of root functions of this problem to form a basis in the space \(L_p,\,1< p < \infty \).

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Acknowledgements

We are very grateful to the referees for their valuable comments and suggestions.

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Authors and Affiliations

  1. Baku State University, Z. Khalilov str., 23, 1148, Baku, Azerbaijan

    Ziyatkhan S. Aliyev & Faiq M. Namazov

  2. Institute of Mathematics and Mechanics, Azerbaijan National Academy of Sciences, B. Vahabzadeh str., 9, 1141, Baku, Azerbaijan

    Ziyatkhan S. Aliyev

Authors
  1. Ziyatkhan S. Aliyev
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  2. Faiq M. Namazov
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Corresponding author

Correspondence to Ziyatkhan S. Aliyev.

Additional information

Communicated by Maria Alessandra Ragusa.

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Aliyev, Z.S., Namazov, F.M. Spectral properties of the equation of a vibrating rod at both ends of which the masses are concentrated. Banach J. Math. Anal. 14, 585–606 (2020). https://doi.org/10.1007/s43037-019-00009-1

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  • DOI: https://doi.org/10.1007/s43037-019-00009-1

Keywords

  • Ordinary differential equations of fourth order
  • Bending vibrations of a homogeneous rod
  • Pontryagin space
  • Basis property of eigenfunctions

Mathematics Subject Classification

  • 34B05
  • 34B08
  • 34B09
  • 34L10
  • 34L15
  • 47A75
  • 47B50
  • 74H45