Variational aspects of one-dimensional fourth-order problems with eigenvalue parameter in the boundary conditions

We study a general type of eigenvalue problems for one-dimensional fourth-order operators. The case where the spectral parameter linearly appears in the boundary conditions is discussed. It is well-known that the Galerkin methods depend on the variational formulations of the given boundary problem. The conditions, when the variational bilinear forms are symmetric and the eigenfunctions belong to an appropriate Hilbert space, are presented. Our investigation has a direct application to the eigenoscillations of the mechanical systems. The effect of the theoretical results are illustrated by some examples.