Basis properties of root functions of a regular fourth order boundary value problem

Year 2020, Volume: 49 Issue: 1, 338 - 351, 06.02.2020

Ufuk Kaya , Esma Kara Kuzu

https://doi.org/10.15672/hujms.552213

Cited By: 1

Abstract

In this paper, we consider the following boundary value problem

\[ y^{(4)}+q(x) y=\lambda y,~\ \ \ 0<x<1, \]

\[ y^{\prime\prime\prime}\left(1\right)-\left(-1\right)^{\sigma}y^{\prime\prime\prime}\left(0\right)+\alpha y\left(0\right) =0, \]

\[ y^{(s)}(1) -( -1) ^{\sigma}y^{(s) }( 0) =0,\ \ \ s=\overline{0,2}, \]

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 $q( x) \in W_{1}^{1}( 0,1)$.

Keywords

fourth order eigenvalue problem, not strongly regular boundary conditions, asymptotic behavior of eigenvalues and eigenfunctions, basis properties of the system of root functions

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