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

Nazim Kerimov; Ufuk Kaya

Open Mathematics (2013)

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

Abstract

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In this paper we consider the problem y i v + p 2 ( x ) y ' ' + p 1 ( x ) y ' + p 0 ( x ) y = λ y , 0 < x < 1 , y ( s ) ( 1 ) - ( - 1 ) σ y ( s ) ( 0 ) + ∑ l = 0 s - 1 α s , l y ( l ) ( 0 ) = 0 , s = 1 , 2 , 3 , y ( 1 ) - ( - 1 ) σ y ( 0 ) = 0 , 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 = 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.

How to cite

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Nazim 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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