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

AbstractIn this paper, we investigate the oscillation and nonoscillation of the neutral difference equation with variable coefficientsΔ(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∑n=0∞ (pn − qn−k+l) = ∞and improve the all known results in the literature. The existence theorems for the positive solutions are also obtained