Spectral Properties of a Fourth-Order Differential Operator with Eigenvalue Parameter-Dependent Boundary Conditions

This paper is devoted to the study of the spectral properties of one eigenvalue problem for the fourth-order ordinary differential equations with a spectral parameter contained in two of the boundary conditions. This spectral problem arises when the Fourier method is applied to a boundary value problem for partial differential equations describing small bending vibrations of a homogeneous rod under the action of a longitudinal force in cross sections. The left end of the rod is either free or freely supported, and the inertial mass is concentrated on the right end. We find the arrangement of the eigenvalues on the real axis, determine the multiplicities of all eigenvalues, and study the asymptotic behavior of the eigenvalues and eigen functions of this problem. Moreover, sufficient conditions were found under which the system of root functions with two removed functions is a basis in the space L-p, 1 < p < infinity.