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In the paper, new comparison theorems for the half-linear difference equation Δ(RkΦ(Δzk)) + CkΦ(zk+1) = 0, Φ(u) = |u|p-2 u, p > 1,are derived. We show that if a solution of this equation has a generalized zero on the discrete interval [a, b], then the same holds for a solution of its majorant. The main tool used in the paper is the variational technique which relates nonexistence of a solution with a generalized zero with nonegativity of the p-degree functional defined on the suitable class of admissible functions.

Summation comparison theorems for half-linear second order difference equations on finite interval