Nonlinear Hybrid Second-Order Neutral Delay Differential Equation with Mixed Sign Terms:Oscillation via Symmetric Transform

This paper investigates the oscillatory behavior of solutions to a class of second-order nonlinear neutral delay differential equations with both positive and negative terms.To facilitate the analysis, the equation is transformed into an equivalent binomial form using a positive solution of an auxiliary second-order ordinary differential equation. The comparison method and integral averaging technique combined with the arithmetic-geometry mean inequality are employed to establish new sufficient conditions for the oscillation of all solutions. The obtained results extend and improve several existing criteria in the literature. Finally, illustrative examples are provided to demonstrate the effectiveness, novelty, and applicability of the proposed oscillation criteria.