Copyright q 2010 M. Liu and Z. Guo. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The existence of bounded nonoscillatory solutions of a higher-order nonlinear neutral delay di ff erence equation Δa kn · · · Δa 2 n Δa 1 n Δx n b n x n − d f n, x n − r 1 n , x n − r 2 n , . . . , x n − r sn 0, n ≥ n 0 , where n 0 ≥ 0, d > 0, k > 0, and s > 0 are integers, { a in } n ≥ n 0 i 1 , 2 , . . . , k and { b n } n ≥ n 0 are real sequences, s