Spectra Of Discrete Sturm-Liouville Problems With Squared Eigenparameter-Dependent Boundary Conditions

In the paper,we consider the two problems as follows:1.In Chapter 1,we consider the spectra a discrete right-definite Sturm-Liouville problems with two squared eigenparameter-dependent boundary conditions:-▽(p(t)Δ(t))+q(t)y(t)=λr(t)y(t),t ∈[1,T]z,A(λ)y(0)+B(λ)p(0)Δy(0)=0,C(λ)y(T+1)+D(λ)p(T)▽y(T+1)=0,where Δ is the forward difference operator satisfying Δy(t)=y(t+1)-y(t),▽is the backward difference operator satisfying ▽y(t)=y(t)-y(t-1),λ is the spectrum parameter.By constructing some new Lagrange-type identities and two fundamental functions,we obtain not only the existence,the simplicity and the interlacing properties of the real eigenvalues,but also the oscillation properties,orthogonality of the eigenfunctions and the expansion theorem.Finally,we also give a computation scheme for computing eigenvalues and eigenfunctions of specific eigenvalue problems.2.In chapter 2,we consider the spectra a discrete left-definite Sturm-Liouville problems with squared eigenparameter-dependent boundary conditions:-▽(p(t)Δy(t))+q(t)y(t)=λr(t)y(t),t∈[1,T]z,boy(0)=doΔy(0),y(T+1)=-a1λ2\▽y(T+1),By constructing two new fundamental functions,we obtain the interlacing properties of the real eigenvalues.At last,we also give a specific eigenvalue problem example.