Dependence Of Eigenvalues Of Fourthorder Differential Boundary Value Problems With Transmission Conditions

The differential boundary value problems(or differential operators)consists of differential equation together with boundary conditions.So it has dependence on endpoints of the domain interval,coefficient functions and boundary conditions.Moreover,its eigenvalues will show different characteristics under different parameters.Investigation on the dependence of the eigenvalues can not only better understand the influence of different parameters on the eigenvalues,but also play a crucial role for the numerical calculation of the eigenvalues.In recent years,the dependence of the eigenvalues on the boundary value problems has become a hot topic for many scholars.On the other hand,the boundary value problems with transmission conditions have attracted a large number of researchers because of their wide application in physics and practices.And some practical problems need to be transformed into higher order boundary value problems in real world.In this paper,we combine them and discuss the dependence of eigenvalues of fourth-order differential boundary value problems with transmission conditions on the problem.Firstly,we investigate a class of fourth-order boundary value problems,where the boundary conditions are of a kind of separated conditions,and the transmission conditions are satisfied with an appropriate condition.It is obtained that the eigenvalues of the problem depend not only continuously but also smoothly on the parameters of boundary conditions ?,? and the matrices of transmission conditions C,D.Secondly,the dependence of eigenvalues of the classical fourth-order differential boundary value problems(without transmission conditions)is considered.Different from other literatures,we use the so called fundamental canonical forms of self-adjoint boundary conditions to obtain the results.Furthermore,we give that the eigenvalues are continuous and differentiable functions of boundary conditions when the conditions are separated,mixed and coupled self-adjoint boundary conditions respectively,the results are more general.Finally,on the basis of previous conclusions,we show that eigenvalues of fourth-order differential boundary value problems with transmission conditionsdepend not only continuously but also smoothly on the boundary conditions,transmission conditions and coefficient functions,where the boundary conditions are the fundamental canonical forms of self-adjoint boundary conditions and the transmission conditions are in the most general form,moreover we find the expressions for their derivatives.