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 Delta (x(n) - c(n)x(n-r)) + p(n)x(n-k) - q(n)x(n-l) = 0 where p(n), q(n), c(n) (n = 0, 1, 2,...) are real numbers with p(n) greater than or equal to 0, q(n) greater than or equal to 0, c(n) greater than or equal to 0, k, l, and r are integers with 0 less than or equal to l less than or equal to k - 1, r > 0, p(n) - q(n-k+l) greater than or equal to 0, and not identically sere. 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 Sigma(n=0)(infinity) (p(n) - q(n-k+l)) = infinity and improve the all known results in the literature. The existence theorems for the positive solutions are also obtained. (C) 2000 Elsevier Science Ltd. All rights reserved.