Abstract and Applied Analysis
Volume 2013 (2013), Article ID 361989, 10 pages
http://dx.doi.org/10.1155/2013/361989
Research Article

Inverse Problems for the Quadratic Pencil of the Sturm-Liouville Equations with Impulse

Rauf Kh. Amırov1 and A. Adiloglu Nabıev2

1Department of Mathematics, Faculty of Sciences, Cumhuriyet University, 58140 Sivas, Turkey
2Department of Secondary Science and Mathematics Education, Faculty of Education, Cumhuriyet University, 58140 Sivas, Turkey

Received 22 November 2012; Revised 2 March 2013; Accepted 4 March 2013

Academic Editor: Juan J. Nieto

Copyright © 2013 Rauf Kh. Amırov and A. Adiloglu Nabıev. 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

In this study some inverse problems for a boundary value problem generated with a quadratic pencil of Sturm-Liouville equations with impulse on a finite interval are considered. Some useful integral representations for the linearly independent solutions of a quadratic pencil of Sturm-Liouville equation have been derived and using these, important spectral properties of the boundary value problem are investigated; the asymptotic formulas for eigenvalues, eigenfunctions, and normalizing numbers are obtained. The uniqueness theorems for the inverse problems of reconstruction of the boundary value problem from the Weyl function, from the spectral data, and from two spectra are proved.