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

• Main Authors: Ziyatkhan S. Aliyev, Nazim B. Kerimov
• Format: Article
• Language: English
• Published: Hindawi Limited 2012-01-01
• Series: International Journal of Mathematics and Mathematical Sciences
• Online Access: http://dx.doi.org/10.1155/2012/456517

We consider the fourth-order spectral problem 𝑦(4)(𝑥)−(𝑞(𝑥)𝑦′(𝑥))=𝜆𝑦(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 𝐿𝑝(0,𝑙), 𝑝∈(1,∞), of the system of root functions of this problem.