Asymptotic Behavior of Solutions of Even-Order Differential Equations with Several Delays

: The higher-order delay differential equations are used in the describing of many natural phenomena. This work investigates the asymptotic properties of the class of even-order differential equations with several delays. Our main concern revolves around how to simplify and improve the oscillation parameters of the studied equation. For this, we use an improved approach to obtain new properties of the positive solutions of these equations.

Delay differential equations as one of the branches of functional differential equations appear when modeling several phenomena in different branches of science, see Hale [1], Arino et al. [2], and Rihan [3]. In mathematical models of basic and applied sciences phenomena, even-order differential equations are frequently encountered. Elasticity difficulties, structural deformation, and soil settling are examples of applications; see [4].
The study of second-order DDEs and their properties has always been a subject of continuous interest by researchers. For more information about the oscillation neutral DDEs of second-order. In [5], Bohner et al. investigated the oscillatory properties of the class of second-order DDE of neutral type. They improved and simplified the results of Agarwal et al. [6] and Han et al. [7]. The results in [8][9][10][11][12], recently, also contributed to the development of the study of qualitative behavior of DDEs of second-order.
Although higher-order equations are important, higher-order DDEs have not received as much attention as in the case of second-order DDEs. Since 2011, a research movement focused on the study the asymptotic behavior of DDEs of even-order in the noncanonical case (2). Zhang et al. [13][14][15] established criteria to ensure the oscillation of solutions of a class of DDEs of even-order. Using a different approach, Baculikova et al. [16] studied the asymptotic behavior of even-order DDEs in the canonical and noncanonical cases. For more interesting, very recently, results about oscillation of higher-order DDEs, see [17][18][19].
In this paper, we obtain the oscillatory properties of the even-order DDE with several delays (1). We extend Bohner's results in [5] to higher order equations in order to improve and simplify previous results in the literature.

Lemma 1.
[20] (Lemma 1.1) Let v ∈ C m (I 0 , (0, ∞)) and v (n) be eventually of one sign for all large k. Then, there is an integer a ∈ [0, m], with m + a even for v (m) ≥ 0, or m + a odd for v (m) ≤ 0 such that eventually.
First, we assume that Case (3) of Lemma 3 holds on which is since γ is a ratio of two odd integers. If we divide (6) by β 1/γ and integrating from k to , we get Integrating (7) (m − 2) times from k to ∞, we obtain and From (1) and (9), we have Integrating (10) from k 1 to k, we obtain Integrating (11) from k 1 to k, we get At k → ∞, we get a contradiction with (4). Now, let Case (1) of Lemma 3 holds on I 1 . Also, we find from (4) and (2) that Integrating (1) from k 2 to k, we get which in view of (12) contradicts to the positivity of s (m−1) as k → ∞. The proof is complete.