In this paper the three-dimensional nonlinear difference system $$\begin{array}{cc}\hfill \Delta x_n& =a_{n}f\left(y_{n-l}\right),\hfill \\ \hfill \Delta y_n& =b_{n}g\left(z_{n-m}\right),\hfill \\ \hfill \Delta z_n& =\delta c_{n}h\left(x_{n-k}\right),\hfill \end{array}$$ is investigated. Sufficient conditions under which the system is oscillatory or almost oscillatory are presented.

We study the oscillatory behavior of the second-order quasi-linear retarded difference equation $$\Delta \left(p\left(n\right){\left(\Delta y\left(n\right)\right)}^{\alpha }\right)+\eta \left(n\right)y^{\beta }(n-k)=0$$ under the condition ${\sum }_{n=n_0}^{\infty }{p}^{-\frac{1}{\alpha }}\left(n\right)<\infty$ (i.e., the noncanonical form). Unlike most existing results, the oscillatory behavior of this equation is attained by transforming it into an equation in the canonical form. Examples are provided to show the importance of our main results.

Sufficient conditions for the oscillation of some nonlinear difference equations are established.

Some new criteria for the oscillation of third order nonlinear neutral difference equations of the form $$\Delta \left(a_n{\left({\Delta }^2(x_n+b_n{x}_{n-\delta })\right)}^{\alpha }\right)+q_n{x}_{n+1-\tau }^{\alpha }=0$$ and $$\Delta \left(a_n{\left({\Delta }^2(x_n-b_n{x}_{n-\delta })\right)}^{\alpha }\right)+q_n{x}_{n+1-\tau }^{\alpha }=0$$ are established. Some examples are presented to illustrate the main results.

We obtain some new sufficient conditions for the oscillation of the solutions of the second-order quasilinear difference equations with delay and advanced neutral terms. The results established in this paper are applicable to equations whose neutral coefficients are unbounded. Thus, the results obtained here are new and complement some known results reported in the literature. Examples are also given to illustrate the applicability and strength of the obtained conditions over the known ones.

Sufficient conditions are obtained for the third order nonlinear delay difference equation of the form $$\Delta \left(a_n\left(\Delta \left(b_n{\left(\Delta y_n\right)}^{\alpha }\right)\right)\right)+q_{n}f\left(y_{\sigma \left(n\right)}\right)=0$$ to have property $\left(\mathrm{A}\right)$ or to be oscillatory. These conditions improve and complement many known results reported in the literature. Examples are provided to illustrate the importance of the main results.

This paper is concerned with the oscillatory behavior of first-order nonlinear difference equations with variable deviating arguments. The corresponding difference equations of both retarded and advanced type are studied. Examples illustrating the results are also given.