This paper deals with the following higher order neutral delay differential equation:

where n is a positive integer,      for  and  By employing the contraction mapping principle, we dicuss several existence results on nonoscillatory solutions for the above equation, establish a few Mann type iterative approximation algorithms for these nonoscillatory solutions and construct several error estimates between the approximate solutions and the nonoscillatory solutions. These results presented, in this paper, extend, improve and unify many known results due to Cheng and Annie [3], Graef et al. [6], Kulenović and Hadžiomerspahić [8, 9], Liu et al. [10], Zhang and Yu [14], Zhang [16] and Zhou and Zhang [18] and others.