Spectral properties of some regular boundary value problems for fourth order differential operators

In this paper we consider the problem where λ is a spectral parameter; p j ( x ) ∈ L 1 (0, 1), j = 0, 1, 2, are complex-valued functions; α s;l , s = 1, 2, 3, , are arbitrary complex constants; and σ = 0, 1. The boundary conditions of this problem are regular, but not strongly regular. Asymptotic fo...

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Bibliographic Details
Published in:Central European journal of mathematics Vol. 11; no. 1; pp. 94 - 111
Main Authors: Kerimov, Nazim B., Kaya, Ufuk
Format: Journal Article
Language:English
Published: Heidelberg SP Versita 01.01.2013
Versita
De Gruyter Brill Sp. z o.o., Paradigm Publishing Services
De Gruyter
Subjects:
34B05
34L10
34L20
Algebra
Asymptotic behavior of eigenvalues and eigenfunctions
Basis properties of the system of root functions
Boundary conditions
Boundary value problems
Complex constants
Differential equations
Eigenvalues
Eigenvectors
Fourth order eigenvalue problem
Geometry
Lie Groups
Mathematics
Mathematics and Statistics
Not strongly regular boundary conditions
Number Theory
Operators (mathematics)
Probability Theory and Stochastic Processes
Research Article
Spectra
Topological Groups
Asymptotic behavior of eigenvalues and eigenfunctions
34B05
Fourth order eigenvalue problem
Basis properties of the system of root functions
Not strongly regular boundary conditions
34L10
34L20
ISSN:1895-1074, 1644-3616, 2391-5455
Online Access:Get full text
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Summary:In this paper we consider the problem where λ is a spectral parameter; p j ( x ) ∈ L 1 (0, 1), j = 0, 1, 2, are complex-valued functions; α s;l , s = 1, 2, 3, , are arbitrary complex constants; and σ = 0, 1. The boundary conditions of this problem are regular, but not strongly regular. Asymptotic formulae for eigenvalues and eigenfunctions of the considered boundary value problem are established in the case α 3,2 + α 1,0 ≠ α 2,1 . It is proved that the system of root functions of this spectral problem forms a basis in the space L p (0, 1), 1 < p < ∞, when α 3,2 + α 1,0 ≠ α 2,1 , p j ( x ) ∈ W 1 j (0, 1), j = 1, 2, and p 0 ( x ) ∈ L 1 (0, 1); moreover, this basis is unconditional for p = 2.
Bibliography:ObjectType-Article-1
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content type line 14
ISSN:1895-1074
1644-3616
2391-5455
DOI:10.2478/s11533-012-0059-x