Asymptotic behavior results for nonlinear neutral delay difference equations

Abstract

This paper is concerned with the nonlinear neutral delay difference equation (∗) Δ [ x ( n ) - c ( n ) x ( n - m ) ] + p ( n ) f ( x ( n - k ) ) = 0 , n ∈ N ( n 0 ) , 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, n 0 is a nonnegative integer and N( n 0) = { n 0, n 0 + 1, n 0 + 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.