The authors consider the third order neutral delay difference equation with positive and negative coefficients ∆(a n ∆(b n ∆(x n + px n−m))) + p n f (xn−k) − q n (x n−l) = 0, n ≥ n 0 , and give some new sufficient conditions for the existence of nonoscillatory solutions. Banach's fixed point theorem plays a major role in the proofs. Examples are provided to illustrate their main results.