Skip to main content
SpringerLink
Log in

Spectral properties of some regular boundary value problems for fourth order differential operators

  • Research Article
  • Published:
Central European Journal of Mathematics

Abstract

In this paper we consider the problem

$\begin{gathered} y^{iv} + p_2 (x)y'' + p_1 (x)y' + p_0 (x)y = \lambda y,0 < x < 1, \hfill \\ y^{(s)} (1) - ( - 1)^\sigma y^{(s)} (0) + \sum\limits_{l = 0}^{s - 1} {\alpha _{s,l} y^{(l)} (0) = 0,} s = 1,2,3, \hfill \\ y(1) - ( - 1)^\sigma y(0) = 0, \hfill \\ \end{gathered} $

where λ is a spectral parameter; p j (x) ∈ L 1(0, 1), j = 0, 1, 2, are complex-valued functions; α s;l , s = 1, 2, 3, \(l = \overline {0,s - 1} \), are arbitrary complex constants; and σ = 0, 1. The boundary conditions of this problem are regular, but not strongly regular. Asymptotic formulae for eigenvalues and eigenfunctions of the considered boundary value problem are established in the case α 3,2 + α 1,0 ≠ α 2,1. It is proved that the system of root functions of this spectral problem forms a basis in the space L p (0, 1), 1 < p < ∞, when α 3,2+α 1,0 ≠ α 2,1, p j (x) ∈ W j 1 (0, 1), j = 1, 2, and p 0(x) ∈ L 1(0, 1); moreover, this basis is unconditional for p = 2.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Bari N.K., A Treatise on Trigonometric Series. Vol. II, Macmillan, New York, 1964

    Google Scholar 

  2. Dernek N., Veliev O.A., On the Riesz basisness of the root functions of the nonself-adjoint Sturm-Liouville operator, Israel J. Math., 2005, 145, 113–123

    Article  MathSciNet  MATH  Google Scholar 

  3. Djakov P., Mityagin B.S., Instability zones of one-dimensional periodic Shrödinger and Dirac operators, Russian Math. Surveys, 2006, 61(4), 663–766

    Article  MathSciNet  MATH  Google Scholar 

  4. Djakov P., Mityagin B., Convergence of spectral decompositions of Hill operators with trigonometric polynomials as potentials, Doklady Math., 2011, 83(1), 5–7

    Article  MathSciNet  MATH  Google Scholar 

  5. Djakov P., Mityagin B., Convergence of spectral decompositions of Hill operators with trigonometric polynomial potentials, Math. Ann., 2011, 351(3), 509–540

    Article  MathSciNet  MATH  Google Scholar 

  6. Djakov P., Mityagin B., Criteria for existance of Riesz bases consisting of root functions of Hill and 1D Dirac operators, preprint available at http://arxiv.org/abs/1106.5774

  7. Dunford N., Schwartz J.T., Linear Operators. Part III, Wiley Classics Lib., John Wiley & Sons, New York, 1988

    MATH  Google Scholar 

  8. Gesztesy F., Tkachenko V., A Schauder and Riesz basis criterion for non-self-adjoint Schrödinger operators with periodic and antiperiodic boundary conditions, preprint available at http://arxiv.org/abs/1104.4846

  9. Gohberg I.C., Kreĭn M.G., Introduction to the Theory of Linear Nonselfadjoint Operators, Transl. Math. Monogr., 18, American Mathematical Society, Providence, 1969

    Google Scholar 

  10. Il’in V.A., Kritskov L.V., Properties of spectral expansions corresponding to non-self-adjoint differential operators, J. Math. Sci. (N.Y.), 2003, 116(5), 3489–3550

    Article  MathSciNet  MATH  Google Scholar 

  11. Ionkin N.I., The solution of a certain boundary value problem of the theory of heat conduction with a nonclassical boundary condition, Differ. Uravn., 1977, 13(2), 294–304 (in Russian)

    MathSciNet  MATH  Google Scholar 

  12. Kashin B.S., Saakyan A.A., Orthogonal Series, Transl. Math. Monogr., 75, American Mathematical Society, Providence, 1989

    Google Scholar 

  13. Kerimov N.B., Mamedov Kh.R., On the Riesz basis property of the root functions in certain regular boundary value problems, Math. Notes, 1998, 64(4), 483–487

    Article  MathSciNet  MATH  Google Scholar 

  14. Keselman G.M., On the unconditional convergence of eigenfunction expansions of certain differential operators, Izv. Vyssh. Uchebn. Zaved. Mat., 1964, 39(2), 82–93 (in Russian)

    MathSciNet  Google Scholar 

  15. Kiraç A.A., Riesz basis property of the root functions of non-selfadjoint operators with regular boundary conditions, Int. J. Math. Anal. (Ruse), 2009, 3(21–24), 1101–1109

    MathSciNet  MATH  Google Scholar 

  16. Makin A.S., On a class of boundary value problems for the Sturm-Liouville operator, Differ. Uravn., 1999, 35(8), 1058–1066 (in Russian)

    MathSciNet  Google Scholar 

  17. Makin A.S., On spectral decompositions corresponding to non-self-adjoint Sturm-Liouville operators, Dokl. Math., 2006, 73(1), 15–18

    Article  MathSciNet  MATH  Google Scholar 

  18. Makin A.S., Convergence of expansions in the root functions of periodic boundary value problems, Dokl. Math., 2006, 73(1), 71–76

    Article  MATH  Google Scholar 

  19. Makin A.S., On the basis property of systems of root functions of regular boundary value problems for the Sturm- Liouville operator, Differ. Equ., 2006, 42(12) 1717–1728

    Article  MathSciNet  MATH  Google Scholar 

  20. Makin A.S., Characterization of the spectrum of regular boundary value problems for the Sturm-Liouville operator, Differ. Equ., 2008, 44(3), 341–348

    Article  MathSciNet  MATH  Google Scholar 

  21. Makin A.S., Asymptotics of the spectrum of the Sturm-Liouville operator with regular boundary conditions, Differ. Equ., 2008, 44(5), 645–658

    Article  MathSciNet  MATH  Google Scholar 

  22. Mamedov Kh.R., On the basis property in L p(0, 1) of the root functions of a class non self adjoint Sturm-Liouville operators, Eur. J. Pure Appl. Math., 2010, 3(5), 831–838

    MathSciNet  MATH  Google Scholar 

  23. Mamedov Kh.R., Menken H., On the basisness in L 2(0, 1) of the root functions in not strongly regular boundary value problems, Eur. J. Pure Appl. Math., 2008, 1(2), 51–60

    MathSciNet  MATH  Google Scholar 

  24. Menken H., Accurate asymptotic formulas for eigenvalues and eigenfunctions of a boundary-value problem of fourth order, Bound. Value Probl., 2010, #720235

  25. Menken H., Mamedov Kh.R., Basis property in L p(0, 1) of the root functions corresponding to a boundary-value problem, J. Appl. Funct. Anal., 2010, 5(4), 351–356

    MathSciNet  MATH  Google Scholar 

  26. Mikhailov V.P., On Riesz bases in L 2(0, 1), Dokl. Akad. Nauk SSSR, 1962, 144(5), 981–984 (in Russian)

    MathSciNet  Google Scholar 

  27. Naimark M.A., Linear Differential Operators, 2nd ed., Nauka, Moskow, 1969 (in Russian)

    Google Scholar 

  28. Shkalikov A.A., Basis property of eigenfunctions of ordinary differential operators with integral boundary conditions, Vestnik Moskov. Univ. Ser. I Mat. Mekh., 1982, 6, 12–21

    MathSciNet  Google Scholar 

  29. Shkalikov A.A., Veliev O.A., On the Riesz basis property of eigen- and associated functions of periodic and antiperiodic Sturm-Liouville problems, Math. Notes, 2009, 85(5-6), 647–660

    Article  MathSciNet  MATH  Google Scholar 

  30. Veliev O.A., On the nonself-adjoint ordinary differential operators with periodic boundary conditions, Israel J. Math., 2010, 176, 195–207

    Article  MathSciNet  MATH  Google Scholar 

  31. Veliev O.A., Asymptotic analysis of non-self-adjoint Hill operators, preprint available at http://arxiv.org/abs/1107.2552

  32. Veliev O.A., Duman M.T., The spectral expansion for a nonself-adjoint Hill operator with a locally integrable potential, J. Math. Anal. Appl., 2002, 265(1), 76–90

    Article  MathSciNet  MATH  Google Scholar 

  33. Zygmund A., Trigonometric Series. II, 2nd ed., Cambridge University Press, New York, 1959

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Faculty of Science and Letters, Mersin University, Mersin, 33343, Turkey

    Nazim B. Kerimov & Ufuk Kaya

Authors
  1. Nazim B. Kerimov
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Ufuk Kaya
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Nazim B. Kerimov.

About this article

Cite this article

Kerimov, N.B., Kaya, U. Spectral properties of some regular boundary value problems for fourth order differential operators. centr.eur.j.math. 11, 94–111 (2013). https://doi.org/10.2478/s11533-012-0059-x

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.2478/s11533-012-0059-x

MSC

Keywords

Search

Navigation