Parametric Boundary Conditions Sturm-lioville,

This paper is mainly concerned with Sturm-Liouville problems with eigenparameter in the boundary conditions.In the first part of this paper, we give some basic concepts of regular Sturm-Liouville problems and some Sturm-Liouville problems with eigenparameter in the boundary conditions and the definition and properties of the Krein space. Using the Krein space, we describe a class of Sturm-Liouville problems with eigenparameter in two boundary conditions and prove that it can generate a self-adjoint operator in Krein space with only point spectrum.In the second part, Sturm-Liouville problems with a descreasing affine function of eigenparameter in the first boundary condition and an arbitrary rational function in the second one is considered. Asymptotic estimates of eigenvalues and eigenfunctions are given and oscillation results are developed.In the third part, we discuss a simple class of Sturm-Liouville problems with Dirichlet boundary condition in the first boundary condition and an affine function in the second one. we give the eigenvalue ratios of the above problem with potential function q≥0 or q≤0. Also we prove that all the eigenvalues are positive when potential function q≥0 and the constant term of the affine function b≤0.