Some oscillation and nonoscillation criteria for neutral delay difference equations with positive and negative coefficients

In this study, a new companion transformation is used for the neutral delay difference equation Δ[x(n)−R(n)x(n−r)]+P(n)x(n−k)−Q(n)x(n−l)=0for n≥n0, where n∈Zn∈Z, R,P,QR,P,Q are nonnegative sequences and r,k,lr,k,l are positive integers. New criteria, which do not need the conditions equation(⋆ )∑i∞[P(i+k−l)−Q(i)]=∞ and/or equation(⋆⋆)R(n)+∑i=n−k+ln−1Q(i)≡1 for all sufficiently large nn, are introduced. All the recent results in the literature depend on either the condition (⋆) or the limitation (⋆⋆). We give illustrating examples of which neither oscillatory nor nonoscillatory behaviors are known by the results in the literature, and graphics of these examples are plotted by the mathematical programming language Mathematica 6. Even in the scalar case, our results still improve the literature. Moreover, some mistakes in the literature and their corrections are given.