The authors consider the mth order nonlinear difference equations of the form Dmyn+qnf(yσ(n))=ei, where m1, n={0,1,2,}, ani>0 for i=1,2,,m1, anm1, D0yn=yn, Diyn=aniΔDi1yn, i=1,2,,m, σ(n) as n, and f: is continuous with uf(u)>0 for u0. They give sufficient conditions to ensure that all bounded nonoscillatory solutions tend to zero as n without assuming that n=01/ani=, i=1,2,,m1, {qn} is positive, or en0 as is often required. If {qn} is positive, they prove another such result for all nonoscillatory solutions.