Dependence of eigenvalues of fourth-order Sturm-Liouville problems on canonical boundary conditions

In this paper, we discuss the dependence of eigenvalues of fourth-order SturmLiouville (SL) problems on the fundamental canonical forms of separated, coupled and mixed self-adjoint boundary conditions. The eigenvalues depend not only continuously but smoothly on the boundary conditions and boundary points. Simultaneously, the derivative expression of the eigenvalues with respect to all the parameters in the boundary conditions are given, and thus its monotonicity with respect to some parameters is derived.