Oscillatory Behavior of Higher-Order DEs 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 techniques for establishing new oscillation conditions for all solutions of higher-order differential equations will be presented. The first method to employ the Riccati transformations, which differ from those described in some published works. The second method employs comparison principles with first-order delay differential equations, from which it is straightforward to deduce oscillation for all studied equation solutions. In addition to improving, extending, and significantly simplifying the previously established criteria, the newly proposed criteria have the potential to serve as a benchmark for the theory of delay differential equations of higher order, which is still in its infancy of development. We were able to determine three fundamental theorems regarding the oscillation of this equation. Some examples will be provided to illustrate the findings.