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.
Similar content being viewed by others
References
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).
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.
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).
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).
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).
F. R. Gantmakher and M. G. Krein, Oscillatory Matrices and Kernels and Small Oscillations of Mechanical Systems [in Russian], Gostekhizdat, Moscow-Leningrad (1950).
C. Karlin, Total Positivity, Stanford Univ. Press (1968).
T. Kato, Perturbation Theory of Linear Operators [Russian translation], Mir, Moscow (1972).
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).
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).
M. A. Naimark, Linear Differential Operators [in Russian], Nauka, Moscow (1969).
Author information
Authors and Affiliations
Corresponding author
Additional information
__________
Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 12, No. 4, pp. 41–52, 2006.
Rights and permissions
About this article
Cite this article
Ben Amara, J., Vladimirov, A.A. On oscillation of eigenfunctions of a fourth-order problem with spectral parameters in the boundary conditions. J Math Sci 150, 2317–2325 (2008). https://doi.org/10.1007/s10958-008-0131-z
Issue Date:
DOI: https://doi.org/10.1007/s10958-008-0131-z