Basis properties of a fourth order differential operator with spectral parameter in the boundary condition

Ziyatkhan Aliyev

Open Mathematics (2010)

  • Volume: 8, Issue: 2, page 378-388
  • ISSN: 2391-5455

Abstract

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We consider a fourth order eigenvalue problem containing a spectral parameter both in the equation and in the boundary condition. The oscillation properties of eigenfunctions are studied and asymptotic formulae for eigenvalues and eigenfunctions are deduced. The basis properties in L p(0; l); p ∈ (1;∞); of the system of eigenfunctions are investigated.

How to cite

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Ziyatkhan Aliyev. "Basis properties of a fourth order differential operator with spectral parameter in the boundary condition." Open Mathematics 8.2 (2010): 378-388. <http://eudml.org/doc/269549>.

@article{ZiyatkhanAliyev2010,
abstract = {We consider a fourth order eigenvalue problem containing a spectral parameter both in the equation and in the boundary condition. The oscillation properties of eigenfunctions are studied and asymptotic formulae for eigenvalues and eigenfunctions are deduced. The basis properties in L p(0; l); p ∈ (1;∞); of the system of eigenfunctions are investigated.},
author = {Ziyatkhan Aliyev},
journal = {Open Mathematics},
keywords = {Fourth order eigenvalue problem; Spectral parameter in the boudary condition; Oscillation properties of eigenfunctions; Basis properties of the system of eigenfunctions; fourth order eigenvalue problem; spectral parameter in the boundary condition; oscillation properties of eigenfunctions; basis properties of the system of eigenfunctions},
language = {eng},
number = {2},
pages = {378-388},
title = {Basis properties of a fourth order differential operator with spectral parameter in the boundary condition},
url = {http://eudml.org/doc/269549},
volume = {8},
year = {2010},
}

TY - JOUR
AU - Ziyatkhan Aliyev
TI - Basis properties of a fourth order differential operator with spectral parameter in the boundary condition
JO - Open Mathematics
PY - 2010
VL - 8
IS - 2
SP - 378
EP - 388
AB - We consider a fourth order eigenvalue problem containing a spectral parameter both in the equation and in the boundary condition. The oscillation properties of eigenfunctions are studied and asymptotic formulae for eigenvalues and eigenfunctions are deduced. The basis properties in L p(0; l); p ∈ (1;∞); of the system of eigenfunctions are investigated.
LA - eng
KW - Fourth order eigenvalue problem; Spectral parameter in the boudary condition; Oscillation properties of eigenfunctions; Basis properties of the system of eigenfunctions; fourth order eigenvalue problem; spectral parameter in the boundary condition; oscillation properties of eigenfunctions; basis properties of the system of eigenfunctions
UR - http://eudml.org/doc/269549
ER -

References

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  1. [1] Aliyev Z.S., On the defect basisity of the system of root functions of a fourth order spectral problem with spectral parameter in boundary condition, News of Baku University, Series of Phys.-Math. Sci., 2008, 4, 8–16 
  2. [2] Banks D., Kurowski G., A Prüfer transformation for the equation of a vibrating beam subject to axial forces, J. Differential Equations, 1977, 24, 57–74 http://dx.doi.org/10.1016/0022-0396(77)90170-X Zbl0313.73051
  3. [3] Ben Amara J., Fourth-order spectral problem with eigenvalue in boundary conditions, In: Kadets V., Zelasko W. (Eds), Proc. Int. Conf. Funct. Anal. and its Applications dedicated to the 110 anniversary of S. Banach (28–31 may 2002 Lviv Ukraine), North-Holland Math. Stud., Elsevier, 2004, 197, 49–58 Zbl1118.47321
  4. [4] Ben Amara J., Sturm theory for the equation of vibrating beam, J. Math. Anal. Appl., 2009, 349, 1–9 http://dx.doi.org/10.1016/j.jmaa.2008.07.055 Zbl1163.34330
  5. [5] Ben Amara J., Oscillation properties for the equation of vibrating beam with irreqular boundary conditions, J. Math. Anal. Appl., 2009, 360, 7–13 http://dx.doi.org/10.1016/j.jmaa.2009.05.042 Zbl1179.34033
  6. [6] Ben Amara J., Vladimirov A.A., On oscillation of eigenfunctions of a fourth-order problem with spectral parameter in boundary condition, J. Math. Sciences, 2008, 150(5), 2317–2325 http://dx.doi.org/10.1007/s10958-008-0131-z Zbl1151.34338
  7. [7] Fulton C.T., Two-point boundary-value problems with eigenvalue parameter contained in the boundary conditions, Proc. Roy.Soc. Edinburgh. Sect. A, 1977, 77, 293–308 Zbl0376.34008
  8. [8] Kapustin N.Yu., On a spectral problem arising in a mathematical model of torsional vibrations of a rod with pulleys at the ends, Differential Equations, 2005, 41(10), 1490–1492 http://dx.doi.org/10.1007/s10625-005-0303-2 Zbl05255987
  9. [9] Kapustin N.Yu., Moiseev E.I., The basis property in L p of the system of eigenfunctions corresponding to two problems with a spectral parameter in the boundary condition, Differential Equations, 2000, 36(10), 1498–1501 http://dx.doi.org/10.1007/BF02757389 Zbl1005.34075
  10. [10] Kashin B.S., Saakyan A.A., Orthogonal series, Amer. Math. Soc., Providence, Rhode Island, 1989 Zbl0668.42011
  11. [11] Kerimov N.B., Aliyev Z.S., On oscillation properties of the eigenfunctions of a fourth-order differential operator, Trans. Natl. Acad. Sci. Azerb. Ser. Phys.-Techn. Math. Sci., 2005, 25(4), 63–76 
  12. [12] Kerimov N.B., Aliyev Z.S., On the basis property of the system of eigenfunctions of a spectral problem with spectral parameter in the boundary condition, Differential Equations, 2007, 43(7), 905–915 http://dx.doi.org/10.1134/S0012266107070038 Zbl1189.34161
  13. [13] Kerimov N.B., Mirzoev V.S., On the basis properties of a spectral problem with a spectral parameter in a boundary condition, Siberian Math. J., 2003, 44(5), 813–816 http://dx.doi.org/10.1023/A:1025932618953 
  14. [14] Meleshko S.V., Pokornyi Yu.V., On a vibrational boundary value problem, Differentsialniye Uravneniya, 1987, 23(8), 1466–1467 (in Russian) Zbl0642.34016
  15. [15] Moiseev E.I., Kapustin N.Yu., On the singulatities of the root space of one spectral problem with a spectral parameter in the boundary condition, Doklady Mathematics, 2000, 66(1), 14–18 
  16. [16] Naimark M.A., Linear differential operators, Ungar, New York, 1967 
  17. [17] Roseau M., Vibrations in mechanical systems. Analytical methods and applications, Springer-Verlag, Berlin, 1987 
  18. [18] Russakovskii E.M., Operator treatment of boundary problems with spectral parameter entering via polynomials in the boundary conditions, Funct. Anal. Appl., 1975, 9, 358–359 http://dx.doi.org/10.1007/BF01075895 
  19. [19] Shkalikov A.A., Boundary-value problems for ordinary differential equations with a parameter in the boundary conditions, J. Soviet Math., 1986, 33, 1311–1342 http://dx.doi.org/10.1007/BF01084754 Zbl0609.34019
  20. [20] Timosenko S.P., Strength and vibrations of structural members, Nauka, Moscow, 1975 (in Russian) 
  21. [21] Tretter C., Boundary eigenvalue problems for differential equations N q = ρPη with λ-polynomial boundary conditions, J.Differential Equations, 2001, 170, 408–471 http://dx.doi.org/10.1006/jdeq.2000.3829 
  22. [22] Walter J., Regular eigenvalue problems with eigenparameter in the boundary conditions, Math. Zeitschrift, 1973, 133(4), 301–312 http://dx.doi.org/10.1007/BF01177870 Zbl0246.47058

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