Oscillation Conditions for Solutions of First Order Piecewise Constant Neutral Differential Equations

The research seeks to examine a new form of equations as well as possible criteria for oscillation and non-oscillation for all solutions of first-order neutral differential equations with the piecewise constant, whether these oscillatory solutions are convergent or non-convergent, by finding sufficient conditions that guarantee the oscillation of all solutions of these equations. According to NDEPC, these equations contain three delays and three coefficients, which requires that part of the coefficients in this equation be piecewise continuous. In witch A function that assumes constant values over predetermined intervals is known as a piecewise constant argument (PCA). The biggest integer function [ω], which rounds down any real number ω to the closest integer, is a typical example. The oscillation criteria for this type of equation necessitate the acquisition of some essential and sufficient conditions to ensure the achievement of this objective. The biggest integer function represents the piecewise constant. All solutions to this type of problem must be oscillatory, and necessary and sufficient conditions have been devised to assure this. Based on these conditions, one can deduce that the technique used for producing results relied on classifying coefficients into two independent types.