The Differentiability Of Eigenvalues ​​of Several Types Of Differential Operators

It is well known that the research of eigenvalues of spectral problems,whether in theory and in application,are of great importance.In this paper,we introduce the eigenvalues of regular spectral problems with self-adjoint boundary conditions.Firstly,we investigate the eigenvalues of a regular fourth-order spectral problem with self-adjoint boundary conditions.The continuous dependence on the problem of eigenvalues and normalized eigenfunctions is researched.The derivative formulas of eigenvalues with respect to a given parameter: an endpoint,a boundary condition,a coefficient or the weight function are obtained,which are both theoretical and computational significance.As a consequence of differentiability results,we have the monotone properties of eigenvalues with respect to the coefficients and the weight function.Then we study a regular 2nth-order spectral problem with self-adjoint boundary conditions.Eigenvalues and normalized eigenfunctions depend continuously on the problem is proved.The eigenvalues are differentiable and the derivative formulas of eigenvalues with respect to a given parameter: an endpoint,a boundary condition,a coefficient or the weight function,are found.This paper contains three chapters.The first chapter: an introduction.The second chapter: dependence of eigenvalues of fourth-order spectral problems.The third chapter:dependence of eigenvalues of 2nth-order spectral problems.