On multiple completeness of the root functions of ordinary differential polynomial pencil with constant coefficients

In the space of square integrable functions on a finite segment we consider a class of polynomial pencils of $n$th-order ordinary differential operators with constant coefficients and two-point boundary-value conditions (at the edges of the segment). We suppose that roots of the characteristic equation of pencils of this class are simple and nonzero. We establish sufficient conditions for $m$-multiple completeness ($1\le m\le n$) of the system of root functions of pencils from this class in the space of square integrable functions on this segment.