Spectra Of A Discrete Sturm-Liouville Problems With Eigenparameter-Dependent Boundary Conditions In Pontryagin Space

In the paper,we consider the second order discrete Sturm-Liouville problems-▽(p(t)△u(t))+q(t)y(t)=λm(t)y(t),t∈[1,T]Z,(a0λ+b0)y(0)=(c0λ+d0)△y(0),(a1λ+b1)y(T+1)=(c1λ+d1)▽y(T+1),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,p:[0,T]z→(0,+∞),q:[1,T]z→R,m:[1,T]z→(0,+∞),σ0=a0d0-b0c0<0,σ1=a1d1-b1c1<0.Different from the results of Gao and Ma[21],the operator corresponding to the problem here is not self-adjoint in the corresponding Hilbert space.According to the construction of the new Lagrange-type identites,the existence,simplicity and interlacing properties of real eigenvalues,nonreal eigenvalues and the oscillation properties of eigenfunctions are obtained in this paper.