Mediterranean Journal of Mathematics

, Volume 11, Issue 2, pp 447–462 | Cite as

Resolvent Operator and Self-Adjointness of Sturm–Liouville Operators with a Finite Number of Transmission Conditions



In this paper, we consider a Sturm–Liouville operator with eigenparameter-dependent boundary conditions and transmission conditions at a finite number of interior points. We introduce a Hilbert space formulation such that the problem under consideration can be interpreted as an eigenvalue problem for a suitable self-adjoint linear operator. We construct Green function of the problem and resolvent operator. We establish the self-adjointness of the discontinuous Sturm–Liouville operator.

Mathematical Subject Classification (1991)

34B24 34B27 47E05 


Sturm–Liouville problems eigenparameter-dependent boundary conditions transmission conditions self-adjoint resolvent operator green function 


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  1. 1.
    Binding P.A., Browne P.J., Watson B.A.: Sturm–Liouville problems with boundary conditions rationally dependent on the eigenparameter, II. J. Comput. Appl. Math. 148(1), 147–168 (2002)CrossRefMATHMathSciNetGoogle Scholar
  2. 2.
    Akdoğan Z., Demirci M., Mukhtarov O.Sh.: Discontinuous Sturm–Liouville problems with eigenparameter-dependent boundary and transmissions conditions. Acta Appl. Math. 86(3), 329–344 (2005)CrossRefMATHMathSciNetGoogle Scholar
  3. 3.
    Coddington, E., Levinson, N.:Theory of Ordinary Differential Equations. McGraw Hill, New York (1995)Google Scholar
  4. 4.
    Dehghani Tazehkand, I., Jodayree Akbarfam, A.: On inverse Sturm–Liouville problems with spectral parameter linearly contained in the boundary conditions. ISRN Math. Anal. 2011, Article ID 754718 (2011)Google Scholar
  5. 5.
    Fulton C.T.: Two-point boundary value problems with eigenvalue parameter contained in the boundary conditions. Proc. R. Soc. Edinburgh A 77, 293–308 (1977)CrossRefMATHMathSciNetGoogle Scholar
  6. 6.
    Kadakal M., Mukhtarov O.Sh.: Sturm–Liouville problems with discontinuities at two points. Comput. Math. Appl. 54, 1367–1379 (2007)CrossRefMATHMathSciNetGoogle Scholar
  7. 7.
    Likov, A.V., Mikhailov, Yu.A.:The Theory of Heat and Mass Transfer. Gosenergoizdat, Moscow (1963, Russian)Google Scholar
  8. 8.
    Mukhtarov O.Sh., Yakubov S.: Problems for differential equations with transmission conditions. Appl. Anal. 81, 1033–1064 (2002)CrossRefMATHMathSciNetGoogle Scholar
  9. 9.
    Naimark, N.: Linear Differential Operators, Parts I and II. Frederick Ungar Publishing Co., New York (1968)Google Scholar
  10. 10.
    Shkalikov, A.A.: Boundary value problems for ordinary differential equations with a parameter in boundary conditions. Trudy Sem. Petrovsk. 9:190–229 (1983) (Russian)Google Scholar
  11. 11.
    Tunç E., Mukhtarov O.Sh.: Resolvent operator and self-adjointness of boundary avlue problems with transmission conditions. Int. J. Math. Game Theory Algebra 14(2), 89–99 (2004)MATHMathSciNetGoogle Scholar
  12. 12.
    Zettl, A.: Sturm–Liouville theory. In: Mathematical Surveys and Monographs, vol. 121. American Mathematical Society, Providence (2005)Google Scholar

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© Springer Basel 2013

Authors and Affiliations

  1. 1.Department of MathematicsPayame Noor UniversityAharIran
  2. 2.Faculty of Mathematical SciencesUniversity of TabrizTabrizIran

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