Asymptotic behavior of nonoscillatory solutions of neutral difference equations

The authors consider the mth-order neutral difference equation Dm(y(n) + p(n)y(n − k) + q(n)f(y(σ(n))) = e(n), where m ≥ 1, {p(n)}, {q(n)}, {e(n)}, and {a1(n)}, {a2(n)}, …, {am−1(n)} are real sequences, ai(n) > 0 for i = 1,2,…, m−1, am(n) ≡ 1, D0z(n) = y(n)+p(n)y(n − k), Diz(n) = ai(n)ΔDi−1z(n) for i = 1,2, …, m, k is a positive integer, {σ(n)} → ∞ is a sequence of positive integers, and R → R is continuous with u f(u) > 0 for u ≠ 0. In the case where {q(n)} is allowed to oscillate, they obtain sufficient conditions for all bounded nonoscillatory solutions to converge to zero, and if {q(n)} is a nonnegative sequence, they establish sufficient conditions for all nonoscillatory solutions to converge to zero. Examples illustrating the results are included throughout the paper.