Sturm theory for the equation of vibrating beam

In this paper we develop an extension of the classical Sturm theory [C. Sturm, Sur une classe d'equations à derivée partielle, J. Math. Pures Appl. 1 (1836) 373–444], to study the oscillation properties for the eigenfunctions of some fourth-order two point boundary value problems on the interval [0,1]. We are mainly interested in the case when these problems have negative eigenvalues induced by the sign of the parameters in the boundary conditions. In particular, we give an asymptotic estimate of the number of zeros in (0,1) of the first eigenfunction in terms of the variation of parameters in the boundary conditions.