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Basis properties of root functions of a regular fourth order boundary value problem. (English) Zbl 1488.34474

Summary: In this paper, we consider the following boundary value problem \[ \begin{aligned} &y^{(4)}+q(x) y=\lambda y, \quad 0<x<1, \\ &y^{\prime\prime\prime}\left(1\right)-\left(-1\right)^{\sigma}y^{\prime\prime\prime}\left(0\right)+\alpha y\left(0\right) =0, \\ &y^{(s)}(1) -( -1)^{\sigma}y^{(s)}(0) =0, \quad s=\overline{0,2}, \end{aligned} \] where \(\lambda\) is a spectral parameter, \(q(x)\in L_1(0,1)\) is complex-valued function and \(\sigma =0,1\). 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. When \(\alpha\ne 0\), we proved that all the eigenvalues, except for finite number, are simple and the system of root functions of this spectral problem forms a Riesz basis in the space \(L_2(0,1)\). Furthermore, we show that the system of root functions forms a basis in the space \(L_p(0,1), 1<p<\infty (p\neq 2)\), under the conditions \(\alpha\ne 0\) and \(q( x) \in W_1^1(0,1)\).

MSC:

34L10 Eigenfunctions, eigenfunction expansions, completeness of eigenfunctions of ordinary differential operators
34B05 Linear boundary value problems for ordinary differential equations
34L20 Asymptotic distribution of eigenvalues, asymptotic theory of eigenfunctions for ordinary differential operators
34B09 Boundary eigenvalue problems for ordinary differential equations
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