Geodesic
On oscillation of eigenfunctions of a fourth-order problem with spectral parameter in boundary condition
Fundamentalʹnaâ i prikladnaâ matematika, Tome 12 (2006) no. 4, pp. 41-52 
J. Ben Amara; A. A. Vladimirov. On oscillation of eigenfunctions of a fourth-order problem with spectral parameter in boundary condition. Fundamentalʹnaâ i prikladnaâ matematika, Tome 12 (2006) no. 4, pp. 41-52. http://geodesic.mathdoc.fr/item/FPM_2006_12_4_a2/
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Voir la notice de l'article provenant de la source Math-Net.Ru

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.

[1] Ben Amara Zh., Vladimirov A. A., “Ob odnoi zadache chetvertogo poryadka so spektralnym i fizicheskim parametrami v granichnom uslovii”, Izv. RAN. Ser. mat., 68:4 (2004), 3–18 | MR

[2] Ben Amara Zh., Shkalikov A. A., “Zadacha Shturma–Liuvillya s fizicheskim i spektralnym parametrami v granichnom uslovii”, Mat. zametki, 66:2 (1999), 163–172 | MR | Zbl

[3] Borovskikh A. V., Pokornyi Yu. V., “Sistemy Chebysheva–Khaara v teorii razryvnykh yader Kelloga”, Uspekhi mat. nauk, 49:3 (1994), 3–42 | MR | Zbl

[4] Gantmakher F. R., Krein M. G., Ostsillyatsionnye matritsy i yadra i malye kolebaniya mekhanicheskikh sistem, Gostekhizdat, M., L., 1950

[5] Kato T., Teoriya vozmuschenii lineinykh operatorov, mir, M., 1972 | MR | Zbl

[6] Levin A. Yu., Stepanov G. D., “Odnomernye kraevye zadachi s operatorami, ne ponizhayuschimi chisla peremen znaka”, Sib. mat. zhurn., 17:3 (1976), 606–625 | MR

[7] Naimark M. A., Lineinye differentsialnye operatory, Nauka, M., 1969 | MR

[8] Banks D. O., Kurowski G. J., “A Prüfer transformation for the equation of the vibrating beam”, Trans. Amer. Math. Soc., 199 (1974), 203–222 | DOI | MR | Zbl

[9] Ben Amara J., “Fourth order spectral problem with eigenvalue in the boundary conditions”, Functional Analysis and Its Applications (Proc. of the Int. Conf., dedicated to the 110th anniversary of Stefan Banach, Lviv National University, Lviv, Ukraine, May 28–31), 197, ed. V. Kadets, Elsevier, Amsterdam, 2002, 49–58 | MR

[10] Karlin C., Total Positivity, Stanford Univ. Press, 1968

[11] Leighton W., Nehari Z., “On the oscillation of solutions of self-adjoint linear differential equations of the fourth order”, Trans. Amer. Math. Soc., 89 (1958), 325–377 | DOI | MR