Finite Spectrum Of Boundary Value Problems With Boundary And Transmission Conditions Dependent On The Spectral Parameter

The discontinuous boundary value problems are widely used in mathematics,mechanics,physics and other branches of natural sciences,and a lot of research results have been obtained on this kind of problems.Carlson has considered a class of nonuniform vibrating string problems,which can be described as a class of Sturm-Liouville(S-L)equations with appropriate conditions.It is proved that this class of problems can be transformed into S-L problems with transmission conditions dependent on spectral parameter in the sense that they have the same eigenvalues.Since then,many scholars have studied S-L problems with eigenparameter-dependent transmission conditions.On the other hand,boundary value problems with eigenparameter-dependent boundary conditions have attracted the attention of the majority of scholars.It appears in actual problems,especially some physical problems,such as heat conduction problems and vibrating string problems and so on.In addition,the time scale is a theory developed in the 1980s,which can organically unify both of the discrete and continuous systems.It is widely used in population dynamics model,biology,medical science and other actual problems.In recent years,many scholars have studied the boundary value problems with finite spectrum.However,the corresponding finite spectrum results of boundary value problems with eigenparameter-dependent transmission conditions have not been concluded.Moreover,finite spectrum theory of boundary value problems on time scales needs further improvement.Therefore,this paper mainly studies the finite spectrum of boundary value problems with transmission conditions dependent on the spectral parameter and boundary value problems with eigenparameter-dependent boundary conditions on time scales.Firstly,the finite spectrum of the second-order S-L problem with transmission conditions dependent on the spectral parameter is studied.By analyzing the characteristic function,it is proved that the S-L problem consists of a finite number of eigenvalues.Moreover,these eigenvalues can be located anywhere in the complex plane.Secondly,the finite spectrum problem is extended to the fourth-order case,and the finite spectrum of the fourth-order boundary value problem with boundary and transmission conditions dependent on the spectral parameter is studied.Finally,we study the finite spectrum of the third-order boundary value problem with eigenparameterdependent boundary conditions on time scales.By partitioning the time scale such that the coefficients of the equation satisfying certain conditions,we construct a kind of third-order boundary value problems with three types of eigenparameter-dependent boundary conditions on time scale which have a finite number of eigenvalues.