Asymptotic and Oscillatory Properties of Noncanonical Delay Differential Equations

: In this work, by establishing new asymptotic properties of non-oscillatory solutions of the even-order delay differential equation, we obtain new criteria for oscillation. The new criteria provide better results when determining the values of coefﬁcients that correspond to oscillatory solutions. To explain the signiﬁcance of our results, we apply them to delay differential equation of Euler-type.

The interest in studying the qualitative properties of differential equations have been increasing in recent years due to several applications of such equations in different life sciences see [1][2][3].Works [4][5][6][7] contributed to the development of the oscillation theory of second-order DDEs, and works [8][9][10] to the development of the oscillation theory of neutral DDEs.
Even-order differential equations are frequently experienced in mathematical models of different biological, physical, and chemical phenomena.Applications include, for example, issues of flexibility, deformity of constructions, or soil settlement; see [11].
Our interest in this work is focused on DDEs of the even-order, which has attracted the attention of researchers, for a follow-up to developments in the study the oscillation of even-order DDEs in the canonical case, see for example [12][13][14].
Baculíková et al. [15] studied the oscillatory properties of the DDE in the canonical case and the non-canonical case In the non-canonical case (4), they proved that if the first-order DDE is oscillatory for some 1 ∈ (0, 1), then there are no solutions to (3) that belong to the following class 2) (s) > 0, and u (n−1) (s) < 0, eventually .
By Riccati substitution, Zhang et al. [16,17] studied Equation (3) when f (u) := u α where α is a quotient of odd positive integers, and created the criterion lim sup for some 2 ∈ (0, 1), to ensure that the class K is empty.As an extension and complement to the results in [17], Moaaz et al. [18] recently used a generalized Riccati substitution to prove that if there is a ρ ∈ for some 3 ∈ (0, 1), then the class K is empty.On the other hand, the study of oscillation of odd-order differential equations has received great interest in the last two years, see for example [19][20][21][22][23].The study of odd and even differential equations differs in that when studying odd differential equations, the different states of the derivatives of the positive solutions increase, which increases the restrictions imposed when testing the oscillation.Therefore, most of the works interested in studying the oscillation of delay differential equations focus only on one type, either even or odd differential equations.
In this paper, we derive new asymptotic properties of the solutions to Equation (1), which belong to class K.Then, we improve these properties by using approaches of an iterative nature.After that we get a new criterion that guarantees that there are no solutions in class K. Finally, we discuss the effect of this new criterion on the oscillatory properties of the solutions of (1).
The following lemmas are needed in the proofs of our main results.

Conclusions
A new criterion of oscillation of a class of even-order delay differential equations is established.The approach used is based on improving the asymptotic properties of the positive solutions of the studied equation.The new criterion inferred provides more sharp results compared to the related results in the literature.It is interesting to extend the results obtained on the neutral delay differential equations.