Asymptotic and Oscillatory Behaviour of First-Order Neutral Difference Equations With Non-Monotone Advanced Arguments

This paper studies the oscillatory and asymptotic behaviour of a class of first-order nonlinear neutral difference equations with non-monotone advanced arguments of the form \(\begin{equation}\Delta\left[\Theta(\vartheta)-\sum\limits_{i=1}^r \rho_i(\vartheta) \Theta\left(\tau_i(\vartheta)\right)\right]-\sum\limits_{i=1}^k \phi_i(\vartheta) g_i\left(\Theta\left(\varphi_i(\vartheta)\right)\right)=0 ; \quad \vartheta \geq \vartheta_0 . \end{equation}\) (a) Using appropriate comparison methods, auxiliary sequences, and exponential inequalities, new sufficient conditions are established for the oscillation and convergence to zero of all solutions of the equation under investigation. Auxiliary lemmas are proved to examine the asymptotic characteristics of eventually positive and eventually negative solutions, and improved oscillation criteria are established in terms of the limit inferior and limit superior associated with equation (a). The obtained results improve and extend several known oscillation theorems in the literature by allowing multiple non-monotone advanced arguments and nonlinear neutral terms under weaker hypotheses.