In this paper, some new results are obtained for the even order neutral delay difference equation ΔanΔm-1xn+pnxn-kα+qnxn-lβ=0, where m≥2 is an even integer, which ensure that all solutions of the studied equation are oscillatory. Our results extend, include, and correct some of the existing results. Examples are provided to illustrate the importance of the main results.

By using comparison theorems, the authors investigate the oscillatory and asymptotic behavior of solutions to the third-order quasi-linear neutral difference equation $$\begin{array}{} \displaystyle \Delta\left(a(n)\left(\Delta^2 z(n) \right)^\alpha \right)+q(n)x^\beta(\sigma(n))=0, \end{array}$$ where z(n) = x(n) + px(n − k). Under less restrictive conditions on the coefficient functions and on the delay argument σ(n) than currently found in the literature, their criteria improve a number of known related results. The results are illustrated with examples.

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In this paper, we present some new oscillation criteria for nonlinear neutral difference equations of the form ?(b(n)?(a(n)?z(n))) + q(n)x?(?(n)) = 0 where z(n) = x(n) + p(n)x(?(n)),? > 0, b(n) > 0, a(n) > 0, q(n) ? 0 and p(n) > 1. By summation averaging technique, we establish new criteria for the oscillation of all solutions of the studied difference equation above. We present four examples to show the strength of the new obtained results.

In this article, we introduce the oscillation of all solutions of third-order half-linear neutral difference equation(OSTOHLDE) ∆ (g(n)(∆(h(n)∆z(n))) α) + f (n)y α (n + 1) = 0, where z(n) = y(n) + e(n)y(n − k) and α is a ratio of odd positive integers(PI). Our results are new and complement to the existing ones.