Oscillation criteria for damped second-order equations with time delays based on the parameter-Riccati method

This work presents a comprehensive oscillation theory for a class of second-order nonlinear neutral differential equations with a variable damping term and variable delay functions. The primary contribution consists of a set of new oscillation criteria whose validity depends on the interplay between the key parameters $\alpha$ and $\beta$. To address the limitations of existing approaches, a novel framework is introduced based on the construction of parameterized auxiliary equations and the use of a generalized Riccati transformation. This approach reduces the problem to the analysis of an associated integro-differential inequality, which is then analyzed via refined inequality techniques and iterative integration. The main results reveal that oscillatory behavior is dictated by a threshold condition comparing $\alpha$ with the rate of change of the delay, quantified by $\beta$. These results are shown to be both sharp and easily applicable, as illustrated by examples that also highlight their improvement over existing results.