Oscillation and Convergence of Solutions of Second Order Neutral Differential Equations with Periodic Coefficients

This research aims to study the behavior of solutions of second-order neutral differential equations with periodic coefficients. Some necessary and sufficient conditions have been obtained that classify all solutions of these equations into three categories: either oscillatory, non-oscillatory, and convergent to zero, or non-oscillatory and divergent. The extent to which periodic coefficients influence the occurrence of oscillation, convergence, or divergence for each solution has been explained. The equation under consideration contained a variable delay and a constant delays in which the coefficients are periodic. Not much previous research has discussed the oscillation of solutions of second-order neutral equations with periodic coefficients. In each case, some illustrative examples have been provided that illustrate the ease of achieving the conditions for the obtained results.