Journal of Mathematical Sciences

, Volume 150, Issue 5, pp 2317–2325 | Cite as

On oscillation of eigenfunctions of a fourth-order problem with spectral parameters in the boundary conditions

Article

Abstract

In the paper, we study the problem on the number of zeros of eigenfunctions of the fourth-order boundary-value problem with spectral and physical parameters in the boundary conditions. We show that the number of zeros of the eigenfunctions corresponding to eigenvalues of positive type behaves in a usual way (it is equal to the serial number of an eigenvalue increased by 1), but, however, the number of zeros of the eigenfunction corresponding to an eigenvalue of negative type can be arbitrary. In the case of a sufficient smoothness of coefficients of the differential expression, we write the asymptotics in the physical parameter for such a number.

Keywords

Spectral Problem Negative Eigenvalue Positive Eigenvalue Positive Type Negative Type 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
This is a preview of subscription content, log in to check access.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    D. O. Banks and G. J. Kurowski, “A Prüfer transformation for the equation of the vibrating beam,” Trans. Amer. Math. Soc., 199, 203–222 (1974).MATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    J. Ben Amara, “Fourth-order spectral problem with eigenvalue in the boundary conditions,” in: V. Kadets, ed., Functional Analysis and Its Applications. Proc. Int. Conf. dedicated to the 110th anniversary of Stefan Banach, Lviv National University, Lviv, Ukraine, May 28–31, 2002, North-Holland Math. Stud., Vol. 197, Elsevier, Amsterdam (2004), pp. 49–58.CrossRefGoogle Scholar
  3. 3.
    J. Ben Amara and A. A. Shkalikov, “Sturm-Liouville problem with physical and spectral parameters in the boundary condition,” Mat. Zametki, 66, No. 2, 163–172 (1999).MathSciNetGoogle Scholar
  4. 4.
    J. Ben Amara and A. A. Vladimirov, “On a certain fourth-order problem with spectral and physical parameters in the boundary condition,” Izv. Ross. Akad. Nauk, Ser. Mat., 68, No. 4, 3–18 (2004).MathSciNetGoogle Scholar
  5. 5.
    A. V. Borovskii and Yu. V. Pokornyi, “Chebyshev-Haar systems in the theory of Kellogg discontinuous kernels,” Usp. Mat. Nauk, 49, No. 3, 3–42 (1994).Google Scholar
  6. 6.
    F. R. Gantmakher and M. G. Krein, Oscillatory Matrices and Kernels and Small Oscillations of Mechanical Systems [in Russian], Gostekhizdat, Moscow-Leningrad (1950).Google Scholar
  7. 7.
    C. Karlin, Total Positivity, Stanford Univ. Press (1968).Google Scholar
  8. 8.
    T. Kato, Perturbation Theory of Linear Operators [Russian translation], Mir, Moscow (1972).Google Scholar
  9. 9.
    W. Leighton and Z. Nehari, “On the oscillation of solutions of self-adjoint linear differential equations of the fourth order,” Trans. Amer. Math. Soc., 89, 325–377 (1958).CrossRefMathSciNetGoogle Scholar
  10. 10.
    A. Yu. Levin and G. D. Stepanov, “One-dimensional boundary-value problems with operators not reducing the number of sign alternations,” Sib. Mat. Zh., 17, No. 4, 813–830 (1976).MATHMathSciNetGoogle Scholar
  11. 11.
    M. A. Naimark, Linear Differential Operators [in Russian], Nauka, Moscow (1969).Google Scholar

Copyright information

© Springer Science+Business Media, Inc. 2008

Authors and Affiliations

  1. 1.University 7 November CarthageBizerte, Tunis
  2. 2.M. V.^Lomonosov Moscow State UniversityRussia

Personalised recommendations