Eigenvalue comparisons for second order difference equations with Neumann boundary conditions

The boundary value problem for second order difference equationΔ(ri-1Δyi-1)-biyi+λaiyi=0,1⩽i⩽ny0-τy1=yn+1-δyn=0with δ,τ∈[0,1]δ,τ∈[0,1] and τ+δ≠2τ+δ≠2 was recently discussed in Ji and Yang (2007) [J. Ji, B. Yang, Eigenvalue comparisons for a class of boundary value problems of second order difference equations, Linear Algebra Appl. 420 (1) (2007) 218–227]. In this paper we extend our earlier results to the second order difference equations with Neumann boundary conditions (the case of τ=δ=1τ=δ=1). As in Ji and Yang (2007) mentioned above, we will also focus on the structure of its eigenvalues and comparisons of all eigenvalues as the coefficients {ai},{bi}{ai},{bi}, and {ri}{ri} change.