Oscillatory Behavior of Higher-Order Differential Equations with Delay Terms

The aim of this research is to study the oscillatory properties of higher -order delay half linear differential equations with non-canonical operators. Two methods for establishing some new conditions for the oscillation of all solutions of higher-order differential equations will be presented. The first method to use the Riccati transformations which differ from those reported in some literature. The second method employs comparison principles with first-order delay differential equations which can easily deduce oscillation of all solutions of studied equations. Not only do the newly proposed criteria improve, extend, and greatly simplify the previously established criteria, but they also have the potential to act as a reference point for the theory of delay differential equations of higher order, which is still in its early stages of development. We were able to determine three fundamental theorems regarding the oscillation of this equation. There will be some examples provided to illustrate the findings.