International Journal of Mathematics and Mathematical Sciences
Volume 2012 (2012), Article ID 456517, 28 pages
http://dx.doi.org/10.1155/2012/456517
Research Article

Spectral Properties of the Differential Operators of the Fourth-Order with Eigenvalue Parameter Dependent Boundary Condition

Ziyatkhan S. Aliyev1 and Nazim B. Kerimov2

1Department of Mathematics, Baku State University, Baku AZ 1148, Azerbaijan
2Department of Mathematics, Mersin University, 33343 Mersin, Turkey

Received 15 August 2011; Accepted 12 November 2011

Academic Editor: Amin Boumenir

Copyright © 2012 Ziyatkhan S. Aliyev and Nazim B. Kerimov. 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 consider the fourth-order spectral problem 𝑦 ( 4 ) ( 𝑥 ) − ( 𝑞 ( 𝑥 ) 𝑦 ′ ( 𝑥 ) ) = 𝜆 𝑦 ( 𝑥 ) , 𝑥 ∈ ( 0 , 𝑙 ) with spectral parameter in the boundary condition. We associate this problem with a selfadjoint operator in Hilbert or Pontryagin space. Using this operator-theoretic formulation and analytic methods, we investigate locations (in complex plane) and multiplicities of the eigenvalues, the oscillation properties of the eigenfunctions, the basis properties in 𝐿 𝑝 ( 0 , 𝑙 ) , 𝑝 ∈ ( 1 , ∞ ) , of the system of root functions of this problem.