Our interest in this article is to develop oscillation conditions for solutions of higher order differential equations and to extend recent results in the literature to differential equations of several delays. We obtain new asymptotic properties of a class from the positive solutions of an even higher order neutral delay differential equation. Then we use these properties to create more effective criteria for studying oscillation. Finally, we present some special cases of the studied equation and apply the new results to them.

The main objective of our paper is to investigate the oscillatory properties of solutions of differential equations of neutral type and in the noncanonical case. We follow an approach that simplifies and extends the related previous results. Our results are an extension and reflection of developments in the study of second-order equations. We also derive criteria for improving conditions that exclude the decreasing positive solutions of the considered equation.

In this work, we investigate the oscillatory properties of the neutral differential equation (r(l)[(s(l)+p(l)s(g(l)))′]v)′+∑i=1nqi(l)sv(hi(l))=0, where s≥s0. We first present new monotonic properties for the solutions of this equation, and these properties are characterized by an iterative nature. Using these new properties, we obtain new oscillation conditions that guarantee that all solutions are oscillate. Our results are a complement and extension to the relevant results in the literature. We test the significance of the results by applying them to special cases of the studied equation.

In this paper, we obtain new monotonic properties for positive solutions of even-order delay differential equations in the non-canonical case. Using these properties, we establish a new oscillation criterion for solutions by comparison with an equation of the first order. The approach adopted is based on the use of symmetry between positive and negative solutions.