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.
Similar content being viewed by others
References
Fulton C. T., “Two-point boundary value problems with eigenvalue parameter contained in the boundary conditions,” Proc. Roy. Soc. Edinburgh, 77A, 293–308 (1977).
Titchmarsh E. C., Eigenfunctions Expansion Associated with Second Order Differential Equations. I, Oxford Univ. Press, London (1962).
Walter J., “Regular eigenvalue problems with eigenvalue parameter in the boundary conditions,” Math. Z., 133, 301–312 (1973).
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).
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).
Schneider A., “A note on eigenvalue problems with eigenvalue parameter in the boundary conditions,” Math. Z., 136, 163–167 (1974).
Shkalikov A. A., “Boundary value problems for ordinary differential equations with a parameter in boundary conditions,” Trudy Sem. Petrovsk., 9, 190–229 (1983).
Yakubov S., Completeness of Root Functions of Regular Differential Operators, Longman, Scientific Technical, New York (1994).
Yakubov S. and Yakubov Y., “An Abel basis of root functions of regular boundary value problems,” Math. Nachr., 197, 157–187 (1999).
Yakubov S. and Yakubov Y., Differential-Operator Equations. Ordinary and Partial Differential Equations, Chapman and Hall/CRC, Boca Raton (2000).
Tikhonov A. N. and Samarskii A. A., Equations of Mathematical Physics, Pergamon, Oxford; New York (1963).
Titeux I. and Yakubov Ya. S., Application of Abstract Differential Equations to Some Mechanical Problems, Kluwer Academic Publishers, Dordrecht; Boston; London (2003).
Mukhtarov O. Sh. and Demir H., “Coerciveness of the discontinuous initial-boundary value problem for parabolic equations,” Israel J. Math., 114, 239–252 (1999).
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).
Mukhtarov O. Sh. and Yakubov S., “Problems for ordinary differential equations with transmission conditions,” Appl. Anal., 81, 1033–1064 (2002).
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).
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).
Author information
Authors and Affiliations
Additional information
Original Russian Text Copyright © 2005 Mukhtarov O. Sh. and Kadakal M.
__________
Translated from Sibirskii Matematicheskii Zhurnal, Vol. 46, No. 4, pp. 860–875, July–August, 2005.
Rights and permissions
About this article
Cite this article
Mukhtarov, O.S., Kadakal, M. Some Spectral Properties of One Sturm-Liouville Type Problem with Discontinuous Weight. Sib Math J 46, 681–694 (2005). https://doi.org/10.1007/s11202-005-0069-z
Received:
Issue Date:
DOI: https://doi.org/10.1007/s11202-005-0069-z