The authors consider the m th order nonlinear difference equations of the form D m y n + q n f ( y σ ( n ) ) = e i , where m ≥ 1 , n ∈ ℕ = { 0 , 1 , 2 , … } , 0$"> a n i > 0 for i = 1 , 2 , … , m − 1 , a n m ≡ 1 , D 0 y n = y n , D i y n = a n i Δ D i − 1 y n , i = 1 , 2 , … , m , σ ( n ) → ∞ as n → ∞ , and f : ℝ → ℝ is continuous with 0$"> u f ( u ) > 0 for u ≠ 0 . They give sufficient conditions to ensure that all bounded nonoscillatory solutions tend to zero as n → ∞ without assuming that ∑ n = 0 ∞ 1 / a n i = ∞ , i = 1 , 2 , … , m − 1 , { q n } is positive, or e n ≡ 0 as is often required. If { q n } is positive, they prove another such result for all nonoscillatory solutions.