This paper deals with the structure of eigenvalues of the boundary value problem for the second order difference equation
under the assumption that the ’s can be negative at some , . Especially, the authors focus on the comparison of all eigenvalues as the coefficients , , and the parameter change. The results extend the authors earlier results to a more general setting, allowing some ’s to be negative [J. Ji, B. Yang, J. Math. Anal. Appl. 320, No. 2, 964–972 (2006; Zbl 1111.39012)].