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

Open Mathematics (2013)

- Volume: 11, Issue: 1, page 94-111
- ISSN: 2391-5455

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topNazim Kerimov, and Ufuk Kaya. "Spectral properties of some regular boundary value problems for fourth order differential operators." Open Mathematics 11.1 (2013): 94-111. <http://eudml.org/doc/269195>.

@article{NazimKerimov2013,

abstract = {In this paper we consider the problem $\{\begin\{array\}\{c\}y^\{iv\} + p_2 (x)y^\{\prime \prime \} + p_1 (x)y^\{\prime \} + 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\{array\}\} $ 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.},

author = {Nazim Kerimov, Ufuk Kaya},

journal = {Open Mathematics},

keywords = {Fourth order eigenvalue problem; Not strongly regular boundary conditions; Asymptotic behavior of eigenvalues and eigenfunctions; Basis properties of the system of root functions; fourth order eigenvalue problem; not strongly regular boundary conditions; asymptotic behavior of eigenvalues and eigenfunctions; basis properties of the system of root functions},

language = {eng},

number = {1},

pages = {94-111},

title = {Spectral properties of some regular boundary value problems for fourth order differential operators},

url = {http://eudml.org/doc/269195},

volume = {11},

year = {2013},

}

TY - JOUR

AU - Nazim Kerimov

AU - Ufuk Kaya

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

JO - Open Mathematics

PY - 2013

VL - 11

IS - 1

SP - 94

EP - 111

AB - In this paper we consider the problem ${\begin{array}{c}y^{iv} + p_2 (x)y^{\prime \prime } + p_1 (x)y^{\prime } + 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{array}} $ 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.

LA - eng

KW - Fourth order eigenvalue problem; Not strongly regular boundary conditions; Asymptotic behavior of eigenvalues and eigenfunctions; Basis properties of the system of root functions; fourth order eigenvalue problem; not strongly regular boundary conditions; asymptotic behavior of eigenvalues and eigenfunctions; basis properties of the system of root functions

UR - http://eudml.org/doc/269195

ER -

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