New Monotonic Properties of the Class of Positive Solutions of Even-Order Neutral Differential Equations

: In this study, new asymptotic properties of positive solutions of the even-order neutral 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. Using these properties, we obtain new criteria to exclude a class from the positive solutions of the studied equation, using the comparison principles.


Introduction
Differential equations (DE) are crucial for understanding real-life problems and phenomena, or at the very least for knowing the characteristics of the solutions to the equations resulting from modeling these phenomena. However, DEs, such as the ones presented, that are utilized to address real-world issues may not be explicitly solvable, i.e., may not have closed-form solutions. Only equations with simple forms accept the solutions supplied by explicit formulae. In recent decades, different models of DEs have been established in various fields, which have led to stimulate research in the qualitative theory of DEs. Qualitative properties of differential equations have received a lot of attention, such as existence, oscillation, periodicity, boundedness, stability; see for example [1,2].
Neutral differential equations (NDE) are a type of functional differential equation in which the highest derivative of the unknown function appears with and without delay. The qualitative analysis of such equations has a lot of practical use in addition to its theoretical value. This is due to the fact that NDEs appear in a variety of situations, such as problems involving lossless transmission lines in electric networks (as in high-speed computers, where such lines are used to interconnect switching circuits), the study of vibrating masses attached to an elastic bar, and the solution of variational problems with time delays; see Hale [2].
The essence of oscillation theory is to establish conditions for the existence of oscillatory (non-oscillatory) solutions and/or convergence to zero, studying the laws of distribution of the zeros, obtaining lower limits for the separation between successive zeros, and considering the number of zeros of each given span, as well as looking at the relationship between the oscillatory properties of solutions and corresponding oscillatory processes in a system. The oscillation theory has become a significant numerical mathematical tool for many disciplines and high technologies. The subject of finding oscillation criteria for certain functional DEs has been a highly active study area in recent decades, and the monographs by Agarwal et al. [3,4] and Győri and Ladas [5] contain many references and descriptions of known results.
Let us denote the composition of two functions f and g by g • f , that is, where n ≥ 4 is an even natural number, a, p, τ and ζ in By a proper solution of (1), we mean a real-valued function In this paper, we study the asymptotic and oscillatory behavior of solutions of (1) in the non-canonical case, that is Jacob Robert Emden (1862-1940), a Swiss astrophysicist, and Sir Ralph Howard Fowler (1889Fowler ( -1944, an English astronomer, are the namesakes of the famous Emden-Fowler equation. Fowler investigated the equation to explain many fluid mechanics phenomena [6]. Since then, there has been a surge of interest in generalizing this equation and using it to explain a variety of physical processes [7,8]. Equation (1) is a generalization of the Emden-Fowler equation in the higher-order and the neutral case.
Studying the qualitative behavior of solutions to differential equations is of great importance, especially in the case of an inability to find a solution to differential equations. On the other hand, numerical studies are important in understanding, analyzing and interpreting different phenomena (see, for example, [9,10]).
In 2011, Zhang et al. [11] presented conditions that ensure the convergence of nonoscillatory solutions to zero of the equation where α and β are ratios of odd positive integers. Zhang et al. [12] provided criteria for oscillation of all solutions of (2). Using the comparison technique, Baculíková [13] investigated the oscillation of the solutions of the equation where , for xy > 0. Moaaz and Muhib [14] studied the oscillation of (2) and presented improved results in [12,13].
On the other hand, the study of the oscillatory behavior of solutions of second-order delay differential equations was recently developed. To track this development, see [15][16][17][18][19]. Baculíková [15] established the monotonic properties of nonoscillatory solutions of the linear equation in the delay and advanced cases. He provided criteria for oscillation, which improved the results in [16]. For the NDE Bohner et al. [18] and Moaaz et al. [19] verified the oscillatory behavior of this equation in the non-canonical case.
On the other hand, the study of the asymptotic behavior of delay differential equations in the non-canonical case differs greatly from the canonical case. The possibilities of signs of derivatives of positive solutions are more in the non-canonical case, and this opens the way for the use of different approaches and methods to exclude positive solutions. Anis and Moaaz [20] presented oscillation criteria for the equation and Moaaz et al. [21] verified the oscillatory behavior of (4) in the canonical case.
The main objective of this study is to find the new monotonic properties of a class of positive solutions of (1) in the non-canonical case. Then, we improve these properties by establishing them in an iterative nature. By using these properties, we can obtain an iterative criterion that ensures that there are no solutions in the class of the positive solutions under study. The results in this paper extend the approach used in [15] for the higher order as well as the neutral equations. Finally, we test the effect of this improvement on a special case of (1).

Main Results
Naturally, the qualitative study of the solutions of the NDDs begins with the classification of the signs of the derivatives of the function Assume that x is a positive solution to Equation (1). Since lim k→∞ τ(k) = ∞ and lim k→∞ ζ(k) = ∞, there is a k 1 > k 0 such that x • τ and x • ζ are positive for all k ≥ k 1 . Thus, υ(k) > 0 and a(k)υ (n−1) (k) ≤ 0. Taking into account Lemma 2.2.3 in [3], the following are the possible cases, eventually: Here, we define the class as the set of all positive solutions of (1) with υ satisfying P2. Further, we define the functions B m and Q by
Using Lemma 1 with r = n − 1 and g = υ, we obtain (r 1,2 ) for all µ 0 ∈ (0, 1). Next, Since υ (n−2) is a positive decreasing function, we conclude that υ (n−2) converges to a non-negative constant, and this with (6) gives This also confirms the positivity of the numerator of the derivative of υ (n−2) /B 0 , or otherwise, d dk This completes the proof.

Lemma 3.
Assuming that x belongs to and (c 1 ) there are δ ∈ (0, 1) and Proof. First of all, since x belongs to , we can say that (r 1,1 ) − (r 1,5 ) in Lemma 2 are satisfied for all k ≥ k 1 , with k 1 large enough. Now, since υ (n−2) is a positive decreasing function, we conclude that υ (n−2) converges to a non-negative constant, let us say l.
If y 0 ≤ 1/2, we can improve the properties in Lemma 3, as stated in the following result.
Next, if y 1 ≤ 1/2, then we define As in the proof of the case for m = 1, we can prove (r 3,1 ), (r 3,2 ) and (r 3,3 ) for m = 2, and so on. The proof is complete.

Theorem 1.
Assume that (c 1 ) and (c 2 ) hold. If there exists a positive integer m such that y m > 1/2 for some µ 0 ∈ (0, 1), then the class is empty, where y m is defined as in Lemma 4.
Proof. Assume the contrary, that x belongs to . From Lemma 4, we have that the functions υ (n−2) /B y m 0 and υ (n−2) /B 1−y m 0 are decreasing and increasing for k ≥ k 1 , respectively. Then, y m ≤ 1/2, which is a contradiction. The proof is complete.
then the class is empty, where y m is defined as in Lemma 4.
Note that in this paper, we have obtained a new criterion without requiring the existence of the unknown functions and without requiring the condition in (24).

Conclusions
In the non-canonical case, new monotonic properties of the positive solutions of a class of even-order neutral differential equations were obtained. Using these properties, we have presented some criteria to guarantee that = ∅. The new criteria are iterative in nature, which allows us to apply them more than once. The examples and figures show the importance of the new properties. It is interesting to extend the technique used in this work to advanced differential equations.