Dependence Of Eigenvalues Of Some Boundary Value Problems On Time Scales

Stefan Hilger,a German mathematician,first put forward the concept of time scales in his doctoral dissertation.Because time scales is widely used in population dynamics model,biology,medical science and other fields,more and more scholars pay attention to the study of the theory of time scales.The so-called time scales(measure chain)refer to any non-empty closed subset of the real number field R,which can combine continuous systems with discrete systems organically.In recent years,people have made a series of important research achievements in the theory of time scales and it's applications,which makes it develop rapidly.Nowadays,many scholars have made a series of researches on the properties of solutions of dynamic equations on time scales from many aspects,and obtained a lot of classical theories.Therefore,it is of great value to further study the theory of time scales.At the same time,studying the dependence of eigenvalues of the boundary value problem itself plays an important role in understanding the change trend and behavior of the eigenvalues in theory,so as to carry out the numerical calculation of the eigenvalues.Therefore,many scholars at home and abroad have carried out a lot of research on the dependence of eigenvalues of Sturm-Liouville problems,and achieved rich results.In addition,the boundary value problems with eigenparameter dependent boundary conditions are widely used in practical problems,such as heat conduction and vibration of loaded string.Since the eigenparameter not only appears in the differential equation but also appears in the boundary conditions,the change of eigenparameter will cause the change of both the boundary conditions and differential equations.So the boundary value problems with eigenparameter dependent boundary conditions have been concerned by many scholars,a lot of achievements have been made in the studies of Sturm-Liouville problems with eigenparameter dependent boundary conditions.At present,there are few studies on the dependence of eigenvalues of boundary value problems on time scales,especially the boundary value problems with eigenparameter dependent boundary conditions on time scales.According to the above situation,this paper studies the eigenvalue dependence of boundary value problems in three forms on time scales.Firstly,the dependence of eigenvalues of the second order Sturm-Liouville problems is considered on a special time scale(two-interval).It is proved that the eigenvalues depend not only continuously but also differentiable with respect to the coefficient functions and the boundary condition parameters,and the differential expressions are given.Secondly,the dependence of eigenvalues of two-interval third-order boundary value problems is considered.Finally,the dependence of eigenvalues of Sturm-Liouville problems on time scales with eigenparameter dependent boundary conditions is discussed.It is proved that the eigenvalues are continuously and differentiable with respect to the coefficient functions and the boundary condition parameters,here for boundary condition parameters,the differential expressions of each of them and as a parameter matrix are given,respectively.