Oscillation of higher order neutral functional difference equations with positive and negative coefficients

Sufficient conditions are obtained so that every solution of the neutral functional difference equation $$ \Delta ^m (y_n - p_n y_{\tau (n)} ) + q_n G(y_{\sigma (n)} ) - u_n H(y_{\alpha (n)} ) = f_n , $$ oscillates or tends to zero or ±∞ as n → ∞, where Δ is the forward difference operator given by Δxn = xn+1 − xn, pn, qn, un, fn are infinite sequences of real numbers with qn > 0, un ≥ 0, G,H ∈ C(ℝ,ℝ) and m ≥ 2 is any positive integer. Various ranges of {pn} are considered. The results hold for G(u) ≡ u, and fn ≡ 0. This paper corrects, improves and generalizes some recent results.