| In the long process of scientific research,many of the conclusions obtained in continuous systems are also applicable to discrete systems.In order to explore the nature of both the continuous systems and discrete systems,the concept of time scale arises at the historic moment.This new concept can be used to combine these two systems together,and avoid duplication of research,so the research of Sturm-Liouville(S-L)problem on time scales has become a new research hotspot in recent years.Some scholars at home and abroad have conducted in-depth research and obtained fruitful results.Simultaneously,since the spectral parameters often appear not only in differential equations but also in boundary conditions in practical problems,the operators determined by them will vary with different spectral parameters.In addition,when people study regular S-L problems,they also study singular S-L problems,one of which is a typical case with a singular point within the interval,that is,the S-L problem with a distributional potential function,and a series of research results have been obtained on these problems.Moreover,for the S-L problem with finite spectrum,including the S-L problem with eigenparameter-dependent boundary conditions,there are a lot of mature research results.However,the finite spectrum of S-L problem with eigenparameter-dependent boundary conditions on time scales has not been discussed yet.Therefore,the study of this kind of problem can give a more general and definite conclusion in theory and supplement the study of finite spectrum of S-L problems.Firstly,we consider S-L equation on time scale together with separated spectral parameter boundary conditions and coupled spectral parameter boundary conditions respectively.By partition the bounded time scale such that the coefficients of S-L equation satisfy certain conditions on the adjacent subintervals,the S-L problem with finite spectrum on the time scale is constructed,and the finite eigenvalue results are obtained.The conclusion is extended to the S-L problem with nonlinear polynomial eigenparameter-dependent boundary conditions,and some mathematical examples are given to illustrate the finite spectrum conclusion.Secondly,we consider the problem of the S-L equation with distribution potential function on time scales under the general spectral parameter boundary conditions,and also prove the conclusion of the finite eigenvalue of this kind of problem.Finally,The spectral analysis of a class of fourth order boundary value problems with self-adjoint boundary conditions on time scales.is investigated.And we extend the conclusion of finite spectrum on time scales to higher-order problems on time scales. |
