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Journal of Function Spaces
Volume 2014, Article ID 606815, 7 pages
http://dx.doi.org/10.1155/2014/606815
Research Article

New Type of Sturm-Liouville Problems in Associated Hilbert Spaces

1Department of Mathematics, Faculty of Science, Gaziosmanpaşa University, 60250 Tokat, Turkey
2Institute of Mathematics and Mechanics, Azerbaijan National Academy of Sciences, Baku, Azerbaijan

Received 6 November 2013; Accepted 2 January 2014; Published 16 April 2014

Academic Editor: Qingying Bu

Copyright © 2014 O. Sh. Mukhtarov and K. Aydemir. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We introduce a new type of discontinuous Sturm-Liouville problems, involving an abstract linear operator in equation. By suggesting own approaches we define some new Hilbert spaces to establish such properties as isomorphism, coerciveness, and maximal decreasing of resolvent operator with respect to spectral parameter. Then we find sufficient conditions for discreteness of the spectrum and examine asymptotic behaviour of eigenvalues. Obtained results are new even for continuous case, that is, without transmission conditions.