New Comparison Theorems to Investigate the Asymptotic Behavior of Even-Order Neutral Differential Equations

: Based on a comparison with ﬁrst-order equations, we obtain new criteria for investigating the asymptotic behavior of a class of differential equations with neutral arguments. In this work, we consider the non-canonical case for an even-order equation. We concentrate on the requirements for excluding positive solutions, as the method used considers the symmetry between the positive and negative solutions of the studied equation. The results obtained do not require some restrictions that were necessary to apply previous relevant results in the literature.


Introduction
Differential equations (DEs) have been widely used in both pure and applied mathematics since they were first introduced in the middle of the 17th century.New connections between the various branches of mathematics, beneficial interactions with practical domains, and reformulations of fundamental problems and theories in diverse sciences have all led to a vast variety of new models and issues.
DEs can be used to simulate almost any physical, technological, or biological activity, including astronomical motion, the construction of bridges, and interactions between neurons.Most models that represent real phenomena and applications cannot have closedform solutions.The available options in this case include finding approximate solutions or studying the qualitative properties of the solutions of these models, which include stability, symmetry, oscillation, periodicity, and others.
A type of functional differential equation known as "neutral differential equations (NDEs)" occurs when the highest derivative of the unknown function appears on the solution both with and without delay.NDEs are used to simulate a wide range of phenomena in many applied sciences, see [1].
The various differential models that have been proposed in various applied sciences have served as a great source of inspiration for research into the qualitative theory of DEs.According to this approach, the oscillation theory of DEs has made huge strides in recent decades, see [2][3][4][5].Numerous authors have examined the oscillation of even-order differential equations and various methods for developing oscillatory criteria for these equations [6][7][8][9].
This study aims to create new conditions to investigate the asymptotic behavior of the even-order NDE a(s)w where s ≥ s 0 , n ≥ 4 are even integers, w(s For a solution to (1), we select a function x ∈ C([s x , ∞)), s x ≥ s 0 , which has the properties w ∈ C (n−1) ([s x , ∞)) and aw (n−1) ∈ C 1 ([s x , ∞)), and x satisfies (1) on [s x , ∞).We consider only those solutions to Equation (1) that will not vanish eventually.If a solution x of ( 1) is eventually positive or negative, then it is said to be non-oscillatory; otherwise, it is said to be oscillatory.
The oscillation theory, which has lately seen major growth and development, covers the study of oscillation for delay, neutral, mixed, and damping ordinary, fractional, and partial DEs.Second-order delay DEs have received the majority of attention in the literature, notably in the non-canonical case, see, for example, [10][11][12][13][14][15][16].Recently, Bohner et al. [17] presented improved criteria for testing the oscillation of solutions of non-canonical secondorder advanced differential equations.
In the non-canonical case, even-order delay DEs have gained more attention than neutral equations, see, for example, [6,7,18,19].
Li and Rogovchenko [20] considered the NDE where α and γ are ratios of odd positive integers.They obtained the oscillation criteria for Equation (2) by using comparison techniques and assuming three unknown functions that satisfy certain conditions.Moreover, the results in [20] required the following restrictions: Recently, Moaaz et al. [21] studied the asymptotic behavior of solutions to the NDE (1).

Main Results
For any eventually positive solution x to Equation (1), we find that the corresponding function w has one of the following cases, based on Lemma 1.1 in [22]: Case 1 w, w , and w (n−1) are positive and w (n) is nonpositive; Case 2 w, w , and w (n−2) are positive and w (n−1) is negative; For ease, the symbol S i indicates the category of eventually positive solutions whose corresponding function satisfies case (i) for i = 1, 2, 3.Moreover, we define Lemma 4 (Lemma 1 in [24]).Suppose that x is an eventually positive solution of (1).Then, eventually, for any integer m ≥ 0.
Proof.Assume that x ∈ S 1 ∪ S 2 .Then, assume that there is an integer ∈ [1, n − 1] such that w ( +1) is the first nonpositive derivative of w.Using Lemma 1 with ψ = w and m = , we obtain w(s) ≥ s w (s) for all ∈ (0, 1).Thus, d ds Using this property with the fact that w (s) > 0, we have that Hence, it follows from Lemma 4 that The proof is now complete.
Lemma 6. Suppose that x is an eventually positive solution of (1), where Then, x ∈ S 3 .

Criteria for Convergence of Non-Oscillatory Solutions to Zero
Theorem 2. Suppose that ( 6) and ( 7) hold.If then all solutions of Equation ( 1) oscillate or converge to zero.
In the following theorem, we prove that the nonoscillatory solutions of Equation ( 1) converge to zero without using an additional condition such as condition (14) in Theorem 2.

Theorem 3. Suppose that
and lim inf where Then, all solutions of Equation ( 1) oscillate or converge to zero.
Proof.Assume the contrary, i.e., that x is an eventually positive solution of (1) and lim s→∞ x(s) = 0. From the fact that w is of fixed sign, we have that w is increasing or decreasing.Assume that w is increasing.Since φ n−2 (s) ≤ 0, we get that φ n−2 (δ(s)) ≥ φ n−2 (s) and Assume that w is decreasing.Then, (−1) k w (k) are positive for all k = 0, 1, 2, . . ., n − 1.
Using the fact that a(s)w (n−1) (s) ≤ 0, we obtain Then, w (n−2) /φ 0 ≥ 0, and so By repeating this procedure, we arrive at (w/φ n−2 ) ≥ Using this property, we get that (20) holds.Therefore, Equation (1) becomes Now, we classify the positive solutions of Equation ( 1) into the following only two categories: (C1) w and w (n−1) are positive and w (n) is non-positive; (C2) w and w (n−2) are positive and w (n−1) is negative.
By following the same approach as in Lemma 6 and using inequality (21) instead of (8), we get the required result.
The proof is now complete.
The proof is now complete.
Proof.Assume the contrary, i.e., that x is an eventually positive solution of (1).From Lemma 6, we have x ∈ S 3 .Using Lemma 8, we get (25).Integrating (25) from s 1 to s, we arrive at It follows from Lemma 7 that w (s) ≤ a(s)w (n−1) (s) φ n−3 (s), and so Therefore, w is a positive solution of (27).It follows from [25] (Theorem 1), that the equation also has a positive solution.Although, Theorem 2 in [26] asserts that condition (26) ensures the oscillation of Equation (28), which is a contradiction.The proof is now complete.

Conclusions
In this work, the asymptotic behavior of solutions to even-order neutral differential equations in the non-canonical case is studied.We obtained a new relationship between the solution and its corresponding function.We then used this new relationship to derive criteria that ensure that all non-oscillatory solutions converge to zero.The new criteria do not require additional restrictions to delay functions (as in (3)).Furthermore, Theorem 3 improves Theorem 1, as it does not require verification of the extra condition (4).