Abstract
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.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
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)
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)
Coddington, E., Levinson, N.:Theory of Ordinary Differential Equations. McGraw Hill, New York (1995)
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)
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)
Kadakal M., Mukhtarov O.Sh.: Sturm–Liouville problems with discontinuities at two points. Comput. Math. Appl. 54, 1367–1379 (2007)
Likov, A.V., Mikhailov, Yu.A.:The Theory of Heat and Mass Transfer. Gosenergoizdat, Moscow (1963, Russian)
Mukhtarov O.Sh., Yakubov S.: Problems for differential equations with transmission conditions. Appl. Anal. 81, 1033–1064 (2002)
Naimark, N.: Linear Differential Operators, Parts I and II. Frederick Ungar Publishing Co., New York (1968)
Shkalikov, A.A.: Boundary value problems for ordinary differential equations with a parameter in boundary conditions. Trudy Sem. Petrovsk. 9:190–229 (1983) (Russian)
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)
Zettl, A.: Sturm–Liouville theory. In: Mathematical Surveys and Monographs, vol. 121. American Mathematical Society, Providence (2005)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Dehghani, I., Akbarfam, A.J. Resolvent Operator and Self-Adjointness of Sturm–Liouville Operators with a Finite Number of Transmission Conditions. Mediterr. J. Math. 11, 447–462 (2014). https://doi.org/10.1007/s00009-013-0338-1
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00009-013-0338-1