The Dependence On Boundary For Some Classes Of Differential Operators With Transmission Conditions

In recent years,more and more mathematical researchers focus on the problems of the dependence of eigenvalues and eigenfunctions.These kinds of problems play an important role in the theory of differential operators.In this paper,we investigate the dependence of eigenvalues for the fourth-order(even order case)and thirdorder(odd order case)differential operator with interface conditions,and the regular fifth-order(odd order case)differential operator with boundary endpoints.Firstly,under a kind of transmission conditions and special separated boundary conditions,we investigate a class of fourth-order differential operator and prove that the eigenvalues depend not only continuously but also smoothly on the problem.In particular,we give the differential expressions of eigenvalues with respect to boundary condition parameters and boundary endpoints.Secondly,we investigate the dependence of the eigenvalues of the third-order differential operator with transmission conditions and give the differential expressions of the eigenvalue on the endpoints.Finally,the dependence of eigenvalues on the boundary of fifth-order differential equations with three types of boundary conditions is studied,and the differential expression of eigenvalues on the boundary endpoint is obtained by the Lagrange's identity and normalized eigenfunction.