Siberian Mathematical Journal

, Volume 46, Issue 4, pp 681–694 | Cite as

Some Spectral Properties of One Sturm-Liouville Type Problem with Discontinuous Weight

  • O. Sh. Mukhtarov
  • M. Kadakal
Article

Abstract

We consider a discontinuous weight Sturm-Liouville equation together with eigenparameter dependent boundary conditions and two supplementary transmission conditions at the point of discontinuity. We extend and generalize some approaches and results of the classic regular Sturm-Liouville problems to the similar problems with discontinuities. In particular, we introduce a special Hilbert space formulation in such a way that the problem under consideration can be interpreted as an eigenvalue problem for a suitable selfadjoint operator, construct the Green’s function and resolvent operator, and derive asymptotic formulas for eigenvalues and normalized eigenfunctions.

Keywords

discontinuous Sturm-Liouville problems transmission conditions eigenvalues eigenfunctions Green’s function resolvent 

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References

  1. 1.
    Fulton C. T., “Two-point boundary value problems with eigenvalue parameter contained in the boundary conditions,” Proc. Roy. Soc. Edinburgh, 77A, 293–308 (1977).Google Scholar
  2. 2.
    Titchmarsh E. C., Eigenfunctions Expansion Associated with Second Order Differential Equations. I, Oxford Univ. Press, London (1962).Google Scholar
  3. 3.
    Walter J., “Regular eigenvalue problems with eigenvalue parameter in the boundary conditions,” Math. Z., 133, 301–312 (1973).CrossRefGoogle Scholar
  4. 4.
    Birkhoff G. D., “On the asymptotic character of the solution of the certain linear differential equations containing parameter,” Trans. Amer. Soc., 9, 219–231 (1908).Google Scholar
  5. 5.
    Hinton D. B., “An expansion theorem for an eigenvalue problem with eigenvalue parameter in the boundary condition,” Quart. J. Math. Oxford, 30, 33–42 (1979).Google Scholar
  6. 6.
    Schneider A., “A note on eigenvalue problems with eigenvalue parameter in the boundary conditions,” Math. Z., 136, 163–167 (1974).CrossRefGoogle Scholar
  7. 7.
    Shkalikov A. A., “Boundary value problems for ordinary differential equations with a parameter in boundary conditions,” Trudy Sem. Petrovsk., 9, 190–229 (1983).Google Scholar
  8. 8.
    Yakubov S., Completeness of Root Functions of Regular Differential Operators, Longman, Scientific Technical, New York (1994).Google Scholar
  9. 9.
    Yakubov S. and Yakubov Y., “An Abel basis of root functions of regular boundary value problems,” Math. Nachr., 197, 157–187 (1999).Google Scholar
  10. 10.
    Yakubov S. and Yakubov Y., Differential-Operator Equations. Ordinary and Partial Differential Equations, Chapman and Hall/CRC, Boca Raton (2000).Google Scholar
  11. 11.
    Tikhonov A. N. and Samarskii A. A., Equations of Mathematical Physics, Pergamon, Oxford; New York (1963).Google Scholar
  12. 12.
    Titeux I. and Yakubov Ya. S., Application of Abstract Differential Equations to Some Mechanical Problems, Kluwer Academic Publishers, Dordrecht; Boston; London (2003).Google Scholar
  13. 13.
    Mukhtarov O. Sh. and Demir H., “Coerciveness of the discontinuous initial-boundary value problem for parabolic equations,” Israel J. Math., 114, 239–252 (1999).Google Scholar
  14. 14.
    Mukhtarov O. Sh., Kandemir M., and Kuruoglu N., “Distribution of eigenvalues for the discontinuous boundary-value problem with functional-many-point conditions,” Israel J. Math., 129, 143–156 (2002).Google Scholar
  15. 15.
    Mukhtarov O. Sh. and Yakubov S., “Problems for ordinary differential equations with transmission conditions,” Appl. Anal., 81, 1033–1064 (2002).CrossRefGoogle Scholar
  16. 16.
    Tunc E. and Mukhtarov O. Sh. “Fundamental solutions and eigenvalues of one boundary-value problem with transmission conditions,” Appl. Math. Comput., 157, 347–355 (2004).CrossRefGoogle Scholar
  17. 17.
    Kadakal M., Muhtarov F. S., and Mukhtarov O. Sh., “The Green’s function of one discontinuous boundary value problem with transmission conditions,” Bull. Pure Appl. Sci., 21E, No.2, 357–369 (2002).Google Scholar

Copyright information

© Springer Science+Business Media, Inc. 2005

Authors and Affiliations

  • O. Sh. Mukhtarov
    • 1
  • M. Kadakal
    • 2
  1. 1.Gaziosmanpasa UniversitesiTokatTurkey
  2. 2.Ondokuz Mayis UniversitesiSamsunTurkey

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