Abstract
In this paper we consider a spectral problem for ordinary differential equations of fourth order with spectral parameter in the boundary conditions. This problem describes the bending vibrations of a homogeneous rod, in cross-sections of which the longitudinal force acts, on the right end of which a mass is concentrated and on the left end a tracking force acts. We investigate the location of eigenvalues on the real axis, we study the structure of root spaces and oscillation properties of eigenfunctions and we obtain sufficient conditions for the subsystems of root functions of this problem to form a basis in $$L_p,\, 1< p < \infty $$ .
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