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

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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Correspondence to Ziyatkhan S. Aliyev.

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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