Oscillation and nonoscillation of neutral difference equations with positive and negative coefficients

In this paper, we investigate the oscillation and nonoscillation of the neutral difference equation with variable coefficients Δ(xn − cncn−r) + pnxn−k − qnxn−l = 0, where pn, qn, cn (n = 0,1,2,…) are real numbers with pn ≥ 0, qn ≥ 0, cn ≥ 0, k, l, and r are integers with 0 ≤ l ≤ k − 1, r> 0, pn − qn−k+l ≥ 0, and not identically zero. Several new sufficient conditions for the oscillation of all solutions of equation (*) are established, some of them are “sharp”. Our results do not need the usual hypothesis and improve the all known results in the literature. The existence theorems for the positive solutions are also obtained.