Basis properties of the system of eigenfunctions of a fourth order eigenvalue problem with spectral and physical parameters in the boundary conditions

In this paper we consider the eigenvalue problem for fourth order ordinary differential equation that describes the bending vibrations of a homogeneous rod, the both ends of which are fixed elastically and on these ends the tracking forces act. We investigate the spectral properties of this problem, including we study the basis properties of subsystems of root functions in the space Lp, 1 < p < ∞.