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.

The aim of this work is to study oscillation properties for a scalar linear difference equation of mixed type $$\Delta x\left(n\right)+\sum _{k=-p}^q{a}_k\left(n\right)x(n+k)=0,n>n_0,$$ where $\Delta x\left(n\right)=x(n+1)-x\left(n\right)$ is the difference operator and $\left\{a_k\left(n\right)\right\}$ are sequences of real numbers for $k=-p,...,q$, and $p>0$, $q\ge 0$. We obtain sufficient conditions for the existence of oscillatory and nonoscillatory solutions. Some asymptotic properties are introduced.

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.