Eigenvalues Of Discrete Sturm-Liouville Operators With Several Special Boundary Conditions

Discrete Sturm-Liouville operator is one of the representative of operators,it also provides an important mathematical tool for solving the problems of mathematical physics,etc.Many scholars have studied the problem,some articles about this problem has been published.The properties of eigenvalues and eigenfunctions are the hot topics in the discrete Sturm-Liouville operator.Inspired by relative reference,in this paper,we will study the spectrum of discrete Sturm-Liouville operators with two special boundary conditions.First,when the weight function is right-definite,we study the eigenvalue problems of discrete Sturm-Liouville with nonlinear parameters in the boundary conditions;second is the eigenvalue problems of discrete Sturm-Liouville with indefinite weights,under coupled boundary conditions.The main contents are as follows:In the first chapter,the research background and the main contents of this paper are introduced.In the second chapter,the eigenvalue problems of left indefinite discrete SturmLiouville with nonlinear parameters in the boundary conditions is studied.Using the pure analysis,the number of eigenvalues,the existence of eigenvalues and oscillation properties of corresponding eigenfunction are given.In the third chapter,under coupled boundary conditions,the spectrum of discrete Sturm-Liouville with indefinite weight are studied,by the properties of characteristic determinant and Sylvester Inertia Theorem,the properties of eigenvalue are given.When the parameters of the coupled boundary conditions satisfy some conditions,the number of sign changes of the eigenfunction is given by using pure analysis method.