Rocky Mountain Journal of Mathematics

Eigenvalues and eigenfunctions of discontinuous two-point boundary value problems with an eigenparameter in the boundary condition

S. C̣aliṣkan, A. Bayramov, Z. Oer, and S. Uslu

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Abstract

In this paper we get asymptotic formulas for eigenvalues and eigenfunctions of discontinuous two-point boundary value problems with the eigenparameter in the boundary conditions with transmission conditions at the two points of discontinuity.

When our problem is continuous the obtained results coincide with the corresponding results in [{\bf3}].

Article information

Source
Rocky Mountain J. Math., Volume 43, Number 6 (2013), 1871-1892.

Dates
First available in Project Euclid: 25 February 2014

Permanent link to this document
https://projecteuclid.org/euclid.rmjm/1393336660

Digital Object Identifier
doi:10.1216/RMJ-2013-43-6-1871

Mathematical Reviews number (MathSciNet)
MR3178447

Zentralblatt MATH identifier
1319.34150

Subjects
Primary: 34L10: Eigenfunctions, eigenfunction expansions, completeness of eigenfunctions 47E05: Ordinary differential operators [See also 34Bxx, 34Lxx] (should also be assigned at least one other classification number in section 47)

Keywords
Two-point boundary value problems transmission condition asymptotic of eigenvalues and eigenfunctions discontinuous boundaryvalue problems

Citation

C̣aliṣkan, S.; Bayramov, A.; Oer, Z.; Uslu, S. Eigenvalues and eigenfunctions of discontinuous two-point boundary value problems with an eigenparameter in the boundary condition. Rocky Mountain J. Math. 43 (2013), no. 6, 1871--1892. doi:10.1216/RMJ-2013-43-6-1871. https://projecteuclid.org/euclid.rmjm/1393336660


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References

  • N. Alt\in\i\ds\ik, M. Kadakal and O.Sh. Mukhtakov, Eigenvalues and eigenfunctions of discontinous Sturm-Liouville problems with eigenparameter dependent boundary conditions, Acta Math. Hungar. 102 (2004), 159-175.
  • A.L. Collatz, Eigenwertaufgaben mit technischen anwendungen, Leipzig, 1963.
  • C.T. Fulton, Two-point boundary value problems with eigenvalues parametercontained in the boundary conditions, Proc. Roy. Soc. Edin. 77 (1977), 293-308.
  • D.B. Hinton, An expansion theorem for eigenvalue problem with eigenvalue parameter in the boundary conditions, Quart. J. Math. Oxford 30 (1979), 33-42.
  • N.B. Kerimov and Kh. Mamedov, On a boundary value problem with a spectral parameter in the boundary conditions, Sibir. Math. Zh. 2 (1999), 325-335. Siberian Math. J. 40 (1999), 281–290 (in English).
  • M.N. Naimark, Linear differential operators, Ungar, New York, 1967.
  • A.A. Shkalikov, Boundary value problems for ordinary differential equations with a parameter in boundary conditions, Trudy Sem. Imeny I.G. Petrow. 9 (1983), 190-229.
  • E.C. Titchmarsh, Eigenfunctions expansion associated with second order differential equations, Oxford University Press, London, 1962.
  • A.N. Tychonov and A.A. Samarski, Partial differential equations of mathematical physics, Pergamon, London, 1964.
  • J. Walter, Regular eigenvalue problems with eigenvalue parameter in the boundary conditions, Math. Z. 133 (1973), 301-312.
  • S. Yakubov and Y. Yakubov, Differential operator equations. Ordinary and partial differential equations, Chapman and Hall/CRC, Boca Raton, 2000.
  • A. Zygmund, Trigonometric series I (2nd edition), Pergamon, London, 1959. \noindentstyle