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

In this paper, we consider the following boundary value problem \[ y^{insert ignore into journalissuearticles values(4);}+qinsert ignore into journalissuearticles values(x); y=\lambda y,~\ \ \ 0<x<1, \] \[ y^{\prime\prime\prime}\leftinsert ignore into journalissuearticles values(1\right);-\leftinsert ignore into journalissuearticles values(-1\right);^{\sigma}y^{\prime\prime\prime}\leftinsert ignore into journalissuearticles values(0\right);+\alpha y\leftinsert ignore into journalissuearticles values(0\right); =0, \] \[ y^{insert ignore into journalissuearticles values(s);}insert ignore into journalissuearticles values(1); -insert ignore into journalissuearticles values( -1); ^{\sigma}y^{insert ignore into journalissuearticles values(s); }insert ignore into journalissuearticles values( 0); =0,\ \ \ s=\overline{0,2}, \] where $\lambda $ is a spectral parameter, $qinsert ignore into journalissuearticles values( x);\in L_{1}insert ignore into journalissuearticles values(0,1);$ 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 $L_{2}insert ignore into journalissuearticles values( 0,1);$. Furthermore, we show that the system of root functions forms a basis in the space $L_{p}insert ignore into journalissuearticles values( 0,1);$, $1<p<\infty$ $insert ignore into journalissuearticles values(p\neq 2);$, under the conditions $\alpha\ne 0$ and $qinsert ignore into journalissuearticles values( x); \in W_{1}^{1}insert ignore into journalissuearticles values( 0,1);$.