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Spectral asymptotics and basis properties of fourth order differential operators with regular boundary conditions
Author(s) -
Kerimov Nazim B.,
Kaya Ufuk
Publication year - 2013
Publication title -
mathematical methods in the applied sciences
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.719
H-Index - 65
eISSN - 1099-1476
pISSN - 0170-4214
DOI - 10.1002/mma.2827
Subject(s) - mathematics , eigenfunction , eigenvalues and eigenvectors , boundary value problem , basis (linear algebra) , mathematical analysis , differential operator , order (exchange) , function (biology) , space (punctuation) , pure mathematics , geometry , linguistics , physics , philosophy , finance , quantum mechanics , evolutionary biology , economics , biology
In this paper, we consider the problemy ıv + q ( x ) y = λy ,0 < x < 1 ,y( s )( 1 ) −( − 1 )σy( s )( 0 ) + α sy( s − 1 )( 0 ) = 0 ,s = 1 , 2 , 3 ,y ( 1 ) −( − 1 )σ y ( 0 ) = 0 ,where λ is a spectral parameter; q ( x ) ∈ L 1 (0,1) is complex‐valued function; α s , s = 1,2,3, are arbitrary complex constants that satisfy α 2 = α 1 + α 3 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. It is proved that all the eigenvalues, except for finite number, are simple and the system of root functions of this spectral problem forms a basis in the space L p (0,1), 1 < p < ∞ , whenα 1α 3(α 1 2 + α 1α 3 + α 3 2) ≠ 0 ; moreover, this basis is unconditional for p = 2. We note that the considered problem was previously investigated in the condition of α 2 ≠ α 1 + α 3 . Copyright © 2013 John Wiley & Sons, Ltd.