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

• Autorzy: Nazim Kerimov , Ufuk Kaya
• Języki publikacji: EN

Abstrakty

EN

In this paper we consider the problem $\begin{gathered} y^{iv} + p_2 (x)y'' + p_1 (x)y' + p_0 (x)y = \lambda y,0 < x < 1, \hfill \\ y^{(s)} (1) - ( - 1)^\sigma y^{(s)} (0) + \sum\limits_{l = 0}^{s - 1} {\alpha _{s,l} y^{(l)} (0) = 0,} s = 1,2,3, \hfill \\ y(1) - ( - 1)^\sigma y(0) = 0, \hfill \\ \end{gathered} $ 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, $l = \overline {0,s - 1} $, 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 1j(0, 1), j = 1, 2, and p 0(x) ∈ L 1(0, 1); moreover, this basis is unconditional for p = 2.

Słowa kluczowe

EN

Fourth order eigenvalue problem; Not strongly regular boundary conditions; Asymptotic behavior of eigenvalues and eigenfunctions; Basis properties of the system of root functions

Kategorie tematyczne

• 34L10: Eigenfunctions, eigenfunction expansions, completeness of eigenfunctions
• 34B05: Linear boundary value problems
• 34L20: Asymptotic distribution of eigenvalues, asymptotic theory of eigenfunctions

• Wydawca: De Gruyter Open
• Czasopismo: Open Mathematics
• Rocznik: 2013
• Tom: 11
• Numer: 1
• Strony: 94-111

Daty

wydano

2013-01-01

online

2012-10-24

Twórcy

autor

Nazim Kerimov

• Mersin University

autor

Ufuk Kaya

• Mersin University

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Identyfikatory

DOI

10.2478/s11533-012-0059-x