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

100%
Kerimov N., Kaya U.
Open Mathematics
|
2013
|
tom 11
|
nr 1
94-111
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.
2
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Basis properties of a fourth order differential operator with spectral parameter in the boundary condition

100%
Aliyev Z.
Open Mathematics
|
2010
|
tom 8
|
nr 2
378-388
EN
We consider a fourth order eigenvalue problem containing a spectral parameter both in the equation and in the boundary condition. The oscillation properties of eigenfunctions are studied and asymptotic formulae for eigenvalues and eigenfunctions are deduced. The basis properties in L p(0; l); p ∈ (1;∞); of the system of eigenfunctions are investigated.
3
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Some global results for nonlinear fourth order eigenvalue problems

88%
Aliyev Z.
Open Mathematics
|
2014
|
tom 12
|
nr 12
1811-1828
EN
In this paper, we consider the nonlinear fourth order eigenvalue problem. We show the existence of family of unbounded continua of nontrivial solutions bifurcating from the line of trivial solutions. These global continua have properties similar to those found in Rabinowitz and Berestycki well-known global bifurcation theorems.
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