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Eigenvalue comparisons for second order difference equations with Neumann boundary conditions. (English) Zbl 1131.39015

The boundary value problem (BVP) for the second order difference equation

Δ(r i-1 Δy i-1 )-b i y i +λa i y i =0,1in,y 0 -τy 1 =y n+1 -δy n =0

with δ,τ[0,1] and τ+δ2 was recently discussed by the authors [see J. Ji and B. Yang, Linear Algebra Appl. 420, 218–227 (2007; Zbl 1113.39021)]. In this paper, they extend their earlier results to the special case τ=δ=1. The difficulty lies in the singularity of some matrix. A new approach is developed to handle the case. As in their earlier paper, they obtain the complete structure of the eigenvalues of the BVP

Δ(r i-1 Δy i-1 )-b i y i +λa i y i =0,1in,y 0 -y 1 =y n+1 -y n =0

and describe the monotonic behavior of all eigenvalues as the coefficients {a i } i=1 n ,{b i } i=1 n and {r i } i=1 n-1 change.

39A12Discrete version of topics in analysis
39A10Additive difference equations
34L05General spectral theory for OD operators