Oscillatory Behavior of Third-Order Nonlinear Difference Equations with a Nonlinear-Nonpositive Neutral Term

We shall present some new oscillation criteria for third-order nonlinear difference equations with a nonlinear-nonpositive neutral term of the form: $$\begin{aligned} \Delta \left( \left( a(t)\left( \Delta ^{2}\left( x(t)-p(t)x^{\alpha }(t-k) \right) \right) ^{\gamma } \right) +q(t)x^{\beta }(t-m+1)=0,\right. \end{aligned}$$ with positive coefficients via comparison with first-order equations whose oscillatory behavior are known, or via comparison with second-order inequalities with solutions having certain properties. The obtained results are new, improve and correlate many of the known oscillation criteria appeared in the literature even for the case of Eq. (1.1) with $$\hbox {p (t)} = 0$$ . Examples are given to illustrate the main results.