Eigenvalue comparisons for second order difference equations with periodic and antiperiodic boundary conditions

We consider a class of boundary value problems of the second order difference equation $$\Delta(r_{i-1}\Delta y_{i-1})-b_{i}y_{i}+\lambda a_{i}y_{i}=0,\quad 1\le i\le n,\ y_{0}=\alpha y_{n},\ y_{n+1}=\alpha y_{1}.$$ The class of problems considered includes those with antiperiodic, Dirichlet, and periodic boundary conditions. We focus on the structure of eigenvalues of this class of problems and comparisons of all eigenvalues as the coefficients {a _i } _i=1 ⁿ ,{b _i } _i=1 ⁿ , and {r _i } _i=0 ⁿ change.