Some Oscillatory Results of Nonlinear Neutral Differential Equation

In the last decades, functional differential equations have attracted the attention of many researchers; they were interested in the theory and its applications. The most common differential equations of functional type are advanced, neutral, and delay DEs. The theory of oscillatory DEs with retarded arguments has a paramount effect on the qualitative properties of DEs. It is essential to deduce conditions for oscillatory and non-oscillatory solutions. The objective of this paper is to obtain oscillatory conditions for differential equations with retarded arguments. So, the oscillatory behavior has been considered in the nonlinear differential equations (DEs) of neutral type with three delays. Some important conditions of all functions have been submitted. The sufficient conditions to secure the oscillatory property have been deduced. We dealt with special cases for delays to obtain some desired conditions for oscillation property. In fact, all new results and conditions innovate, and improved some oscillation properties that appeared in the literature. Some application examples with specific functions for important results have been illustrated and applied to all conditions. Two results with some different conditions have been obtained to get oscillatory behavior for DE. A new relationship between delays and other functions to get desired property has been formulated. Some application examples explained to ensure the importance of our results compared with other previous studies.