Asymptotic behavior results for nonlinear neutral delay difference equations

This paper is concerned with the nonlinear neutral delay difference equation equation(∗)Δ[x(n)-c(n)x(n-m)]+p(n)f(x(n-k))=0,n∈N(n0),where Δ is the forward difference operator defined by Δ x(n) = x(n + 1) − x(n), {c(n)} is a sequence of real numbers, {p(n)} is a positive sequence, f ∈ C(R, R), m and k are positive integers, n0 is a nonnegative integer and N(n0) = {n0, n0 + 1, n0 + 2, …}. Sufficient conditions are obtained under which every solution of equation (∗) is bounded and tends to a constant as n → ∞. Our results improve and extend some known results. One example is given to illustrate our results.