New Asymptotic Properties of Positive Solutions of Delay Differential Equations and Their Application

: In this study, new asymptotic properties of positive solutions of the even-order delay differential equation with the noncanonical operator are established. The new properties are of an iterative nature, which allows it to be applied several times. Moreover, we use these properties to obtain new criteria for the oscillation of the solutions of the studied equation using the principles of comparison.


Introduction
Our interest in this work revolves around the study of the asymptotic behavior of positive solutions of the delay differential equation (DDE): under the following hypotheses: Hypothesis 1 (H1). n ≥ 4 is an even natural number; Hypothesis 2 (H2). a and q are continuous real functions on I 0 := [t 0 , ∞), a(t) > 0, a (t) ≥ 0, q(t) ≥ 0, and: Hypothesis 3 (H3). g is a continuous nondecreasing real function on I 0 , g(t) ≤ t, and lim t→∞ g(t) = ∞.
DDEs are a type of functional differential equation that takes into account the effect of past time. Therefore, DDEs are a better way to describe natural phenomena and timerelated problems. For example, the oscillation of contacts of electromagnetic switches could be described by the oscillation of solutions of the second-order DDE (see [1]), and in mathematical ecology, by DDE, Israelsson and Johnsson [2] introduced a model for geotropic circumnutations of Helianthus annuus.
Recently, a research movement has been active that deals with the qualitative properties of solutions such as these equations, especially their oscillatory behavior. Bac-ulíková [3,4], Džrina and Jadlovská [5], and Chatzarakis et al. [6] developed approaches and techniques for studying oscillatory behavior in order to improve the oscillation criteria of second-order delay/advanced differential equations. Bohner et al. [7], Grace et al. [8], and Moaaz et al. [9,10] also extended this evolution to DDEs of the neutral type. On the other hand, Džurina et al. [11,12] and Moaaz et al. [13] dealt by different methods with the asymptotic properties of the solutions of DDEs of the odd-order.
For even-order DDEs, Moaaz et al. [14] and Park et al. [15] were interested in studying the oscillation of the even-order DDE: (or some of its special cases) where γ, β are the ratio of odd natural numbers. They only focused on studying the oscillation of (2) in the canonical case, that is, For the canonical DDE of the neutral type, see [16]. In the noncanonical case, that is, Zhang et al. [17] studied the qualitative properties of (2). They obtained conditions that ensured that all nonoscillatory solutions of Equation (2) tend to zero. In [18], Zhang et al. established criteria for the oscillation of all solutions of (2) by using Riccati substitution. By establishing comparison theorems that compare the nth-order equation with one or a couple of first-order delay differential equations, Baculíková et al. [19] studied the oscillatory properties of the DDE: (a(t)(ψ (n−1) (t)) γ ) + q(t) f (ψ(g(t))) = 0, where f is nondecreasing and − f (−xψ) ≥ f (xψ) ≥ f (x) f (ψ), for xψ > 0. Moreover, by introducing a generalized Riccati substitution, Moaaz and Muhib [20] extended the technique used in [21] to study the oscillation of (2). In this study, we first obtain new asymptotic properties of the positive solutions of DDE (1). Then, we improve these asymptotic properties by using an iterative technique. Finally, we use these new properties to study the oscillatory behavior of the solutions of (1). Our results in this paper extend and complement the results in [17][18][19]. (1), the delay appears only in the solution ψ, but not in its derivatives, which makes it quite a special high-order DDE.

Applications in Oscillation Theory
In the following, we use our results in the previous section to obtain the criteria of the oscillation for the solutions of (1). A solution u of (1) is called nonoscillatory if it is eventually positive or eventually negative; otherwise, it is called oscillatory.

Remark 3. Consider the DDE:
Note that, the condition (23), with m = 0, reduces to 0.185 log 5 > 0.815 e , which is not satisfied, and thus, the oscillatory behavior of (24) cannot be verified. However, using the iterative nature of (23), we find that: and δ 1 = 0.26208. Now, the condition (23) with m = 1 reduces to 0.185 log 5 > 0.73792 e , which is satisfied. Hence, every solution of (24) is oscillatory. Remark 4. By using comparison principles, Baculíková et al. [19] studied the oscillatory properties of DDE (3). In order to rule out the existence of positive solutions in class (4), they assumed that there is a ξ ∈ C 1 ([t 0 , ∞)) with ξ(t) > t, ξ (t) ≥ 0 and ξ n−2 (σ(t)) < t such that the DDE: z (t) + 1 a(t) t t 0 q(η)dη J n−2 (g(t))(z(ξ n−2 (g(t)))) = 0 is oscillatory, where: for i = 1, 2, ..., n − 3. Since there is no general rule as to how to choose ξ and ξ i satisfying the imposed conditions, our results in this paper improve the results in [19], as our results do not require unknown functions.

Conclusions
In this work, new results of studying the oscillatory behavior of a class of even-order DDEs were presented. In the noncanonical case, using the principles of comparison, we obtained new criteria that guarantee the oscillation of all solutions of the studied equation. By comparing with previous results in the literature that used the same approach, we found that our results are easy to apply and do not require unknown functions. Moreover, the new criteria have an iterative nature. An interesting problem is to extend our results to even-order DDEs of the neutral type.