Forced oscillation of higher-order nonlinear neutral difference equations with positive and negative coef“cients

In this paper, we study the forced oscillation of the higher-order nonlinear difference equation of the form m x(n) – p(n)x(n – τ) + q 1 (n) α (n – σ 1) + q 2 (n) β (n – σ 2) = f (n), where m ≥ 1, τ , σ 1 and σ 2 are integers, 0 < α < 1 < β are constants, * (u) = |u| *–1 u, p(n), q 1 (n), q 2 (n) and f (n) are real sequences with p(n) > 0. By taking all possible values of τ , σ 1 and σ 2 into consideration, we establish some new oscillation criteria for the above equation in two cases: (i) q 1 = q 1 (n) ≤ 0, q 2 = q 2 (n) > 0; (ii) q 1 ≥ 0, q 2 < 0.