This paper investigates the oscillatory behavior of second-order differential equations featuring a mixed neutral term along with a $p$-Laplace differential operator. The analysis employs Riccati transformations and includes a comparative study with first-order equations, which facilitates the development of oscillation criteria. The findings culminate in a significant theorem that addresses the oscillation properties of these equations. In addition, two illustrative examples are presented to demonstrate the practical application of the established criteria.

In this paper, we obtain some oscillatory properties for the noncanonical second-order differential equation with mixed neutral terms. We established our results first by transforming the equation into canonical type and then by using the Riccati technique to get new oscillatory properties for the considered equation. We obtained these results to extend and simplify existing criteria in the literature. We discussed some examples to illustrate the effectiveness of our main results.

This paper studies the oscillatory behavior of a class of second-order neutral delay differential equations (NDDEs). Using comparison principles and Riccati transformation techniques, we derived new sufficient conditions for oscillation. A recursive framework was introduced to enhance the monotonic properties of positive solutions, leading to sharper criteria. Auxiliary inequalities, exponential estimates, and integral averaging methods support the analysis. Illustrative examples demonstrate the applicability and improvements over existing results, positioning our findings within a broader context relevant to engineering and biological models.