Oscillation and basis properties for the equation of vibrating rod at one end of which an inertial mass is concentrated

Abstract

In this paper, we consider an eigenvalue 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, the left end of which is fixed and on the right end an inertial mass is concentrated. We study the oscillation properties of the eigenfunctions and their derivatives, and we use these properties to establish sufficient conditions for the subsystems of eigenfunctions of this problem to form a basis in the space Lp, 1 < p < ∞.