On the multiple completeness of eigenfunctions and adjoint functions for ordinary differential bundles with irregular boundary conditions

We consider an irregular problem of the form
\begin{gather} y^{(n)}+\lambda p_1y^{(n-1)}+\dots+\lambda^n p_ny=0, \label{1}\\ y^{(\varkappa_i)}(0)+\sum_{k=1}^{\varkappa_i}\alpha_{ik}y^{(\varkappa_i-k)}(0)=0, \quad i=\overline{1,l}, \label{2}\\ y^{(\varkappa_i)}(1)+\sum_{k=1}^{\varkappa_i}\beta_{ik}y^{(\varkappa_i-k)}(1)=0, \quad i=\overline{l+1,n};\quad l>n-l, \label{3} \end{gather}
The following theorem is proved. If all argument the roots of the characteristic equation of (1) are different then the system of eigenfunctions and adjoint function of the problem (1) to (3) is $n$-multiply complete in $L_2(0,1)$.