Asymptotic behavior of solutions of first order nonlinear delay difference equations

We obtain sufficient conditions which ensure that all solutions of the first order non-linear delay difference equation Δyn + F(n,yn−k) = 0, n N(n0) tend to zero as n → ∞. In (E), N(n0) = n0, n0 + 1, n0 + 2,…, n0 is a fixed nonnegative integer, k is a nonnegative integer, F : N(n0) × R → R is a real-valued function, and Δyn = yn + 1 − yn.