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 L p , 1 < p < ∞.

Spectral properties of a beam equation with eigenvalue parameter occurring linearly in the boundary conditions

“…As in [8, 33], we will seek the solution of (1.1), (1.2), (1.4), (1.5) in the form where , are some constants.…”

“…These properties of the root functions of eigenvalue problems for ordinary differential equations of the fourth order, one of the boundary conditions of which depends on the spectral parameter, were studied in detail in [2,12,13,22,23,29,30,32,33,39]. In the case when two of the boundary conditions contain a spectral parameter, these problems were studied in [5,6,9,10,32], and when three of the boundary conditions contain a spectral parameter, they were studied in [8]. The problems studied in these works describe bending vibrations of a rod, the left end of which is either fixed or at this end a load is concentrated or a tracking force acts, and at the right end an inertial load is concentrated (the tracking force can also act at this end) (see.…”

“…Oscillatory properties of eigenfunctions and basis properties in various functional spaces of root functions of Sturm-Liouville problems with a spectral parameter in the boundary conditions were investigated in [1, 3, 4, 11, 15–18, 21, 24, 26–28, 31, 34, 36–38, 40]. These properties of the root functions of eigenvalue problems for ordinary differential equations of the fourth order, one of the boundary conditions of which depends on the spectral parameter, were studied in detail in [2, 12, 13, 22, 23, 29, 30, 32, 33, 39]. In the case when two of the boundary conditions contain a spectral parameter, these problems were studied in [5, 6, 9, 10, 32], and when three of the boundary conditions contain a spectral parameter, they were studied in [8].…”