Oscillation of a class of nonlinear neutral difference equations of higher order

Sufficient conditions are obtained under which all solutions of Δm y(n) + p(n)y(n −s) + q(n)G y(n − k) = f (n), n 0, (∗) oscillate. One of these conditions states the existence of a real-valued function F(n) on {0, 1, 2, . . .} such that F(n) changes sign and ΔmF(n) = f (n). Unlike earlier works, it is not assumed that F(n)→0 as n→∞. As the results for (∗) do not apply to homogeneous equations Δm y(n) + p(n)y(n −s) + q(n)G y(n − k) = 0, n 0, (∗∗) the behaviour of its solutions is studied separately. In most of our results for (∗∗), G is sublinear. In our discussion m 2 is even or odd and p(n) lies in different ranges. Existence of a positive bounded solution of (∗) is established where the bound of oscillation of F(n) depends on the range of p(n).  2003 Published by Elsevier Inc.