Functional Differential Equations with Several Delays: Oscillatory Behavior

: In this work, we study the asymptotic behavior of even-order delay functional differential equation. As an extension of the recent development in the study of oscillation, we obtain improved and simpliﬁed criteria that test the oscillation of solutions of the studied equation. We adopt an approach that improves the relationships between the solution with and without delay. The symmetry between the positive and negative solutions also plays a key role in simplifying the presentation of the main results. Finally, we attach an example to illustrate the results and compare them together with the previous results in the literature.


Introduction
Differential equations (DE) are essential for comprehending real-world issues and phenomena, or at the very least for understanding the properties of the equations that come from modeling these occurrences. However, DEs like the ones shown here that are used to solve real-world problems may not be explicitly solvable, that is, they may not have closed-form solutions. Only equations with basic forms accept explicit formulae for solutions. Different models of DEs have been produced in numerous domains in recent decades, which has sparked interest in qualitative theory of DEs study.
The highly rapid progress of research in the 20th century led to applications in biology, population, chemistry, medicine dynamics, social sciences, genetic engineering, economy, and other domains. All of these areas advanced, and new discoveries were made because of this type of mathematical modeling, see [1,2].
Oscillation theory is one of the branches of qualitative theory that studies the qualitative properties of solutions of differential equations, such as stability, symmetry, oscillation, and others, without finding solutions. The solutions of the studied equation are classified into three disjointed classes, which are positive eventually, negative eventually, and oscillatory solutions. The studied equation is characterized by the property of symmetry between the positive and negative solutions, which means that if x is a solution to the equation, then −x is also a solution. Therefore, the conditions that exclude positive solutions also exclude negative solutions, and thus they are conditions that guarantee the oscillation of the studied equation.
In this paper, we consider the delay differential equation (DDE) in the noncanonical case, that is, Throughout this study, we assume that r is a quotient of odd positive integers, is a positive integers and κ ≥ 4 is an even natural number, a, h i ∈ C 1 (I 0 , R + ), a (s) ≥ 0, By a solution of (1), we denote to a nontrivial real valued function υ in for some s * ≥ s 0 , which has the property a υ (κ−1) r in C 1 ([s * , ∞)) and satisfies (1) on [s * , ∞). We take into account these solutions υ of (1) such that sup{|υ(u)| : u ≥ s u } > 0 for every s u in [s * , ∞). A solution υ of (1) is said to be nonoscillatory if it is eventually positive or eventually negative; otherwise, it is said to be oscillatory .
There has recently been a surge of interest in developing sufficient conditions for oscillatory or non-oscillatory behavior of many types of functional differential equations. For the existence of solutions to DDEs, see [3,4].
The study of second-order differential equations and their properties have long been the subject of constant interest by researchers. We refer to [5][6][7][8][9] for more information, approaches, and methods on the oscillation of second-order neutral DDEs. Additionally, many techniques and methods have recently been developed to study the asymptotic behavior of higher-order DDEs, see for example [10][11][12][13][14].
Fore some related works, Zhang et al. [15] discussed the asymptotic properties of DDE where r = β and obtained some oscillation criteria. Baculikova et al. [16] proved that the oscillation of first-order DDE y (s) + q(s) f δτ κ−1 (s) (κ − 1)!a 1 r (τ(s)) f y 1 r (τ(s)) = 0 guarantees the oscillation of even-order DDE a(s) υ (κ−1) (s) r + q(s) f (υ(τ(s))) = 0 (4) in the case where and obtained some comparison theorems for non-canonical case. Xing et al. [17] presented some theorems for oscillation of DDE of neutral type a(s) (υ(s) + p(t)υ(τ(t))) (κ−1) r + q(s)υ r (σ(s)) = 0, where r ≤ 1 is the quotient of odd positive integers. Recently, in an interesting work, Moaaz and Muhib [13] extended and improved the results in [15,16,18] and obtained unusual conditions for testing the oscillation of the solutions of (4) where f (υ) := υ β and β is a quotient of odd positive integers. We begin in this paper with a simplification of the oscillation conditions of (1) as an extension of the approach used in [19]. We then improve the oscillation criteria by finding new relationships and improved inequalities. We suppose the Riccati transformations in the general form and we obtain new conditions for oscillation.
First, we assume that (3) holds on I 1 . Since a(s) υ (κ−1) (s) which is If we divide (8) by a 1/r and integrating the resulting inequality from s to , we obtain Letting → ∞, we obtain Integrating (9) from s to ∞, we obtain integrating the above inequality from s to ∞ a total of (κ − 4) times, we find Integrating (10) from s to ∞, implies that From (1) and (11), we have Integrating (12) from s 1 to s, we obtain Integrating (13) from s 1 to s, we obtain At s → ∞, we obtain a contradiction with (6). Now, let Case (1) holds on I 1 . Now, from (2) and (6), we obtain that Integrating (1) from s 2 to s, we find which, from (14), contradicts to the fact that υ (κ−1) > 0. The proof is complete.
Proof. Using Lemma 1, we have that one of the cases (1)-(3) holds. First, we suppose that (3) holds on I 1 . Then, Integrating (16) from s to ∞, we obtain Integrating the above inequality from s to ∞ a total of (κ − 4) times, we find Integrating (17) from s to ∞, we arrive at Integrating (1) from s 1 to s, we find Since σ(s) ≤ s, we have From (18) and (20), we arrive at Dividing (21) by a(s)(υ (κ−1) (s)) r and taking the limsup, we obtain lim sup which is a contradiction with (15). Next, let Case (1) holds on I 1 . It follows from (15) and δ κ−2 (s) < ∞ that (14) holds. Now, we continue as in the proof of Theorem 1. The proof is complete.
Proof. Assume the contrary that (1) has a positive solution υ(s) and satisfies Case (2).
Proof. The proof is quite similar to the proof of Theorem 4, so it has been omitted.

Conclusions
In this paper, we aim to establish new oscillation criteria for DDE (1). We start by excluding two possible cases of positive solutions using one condition, as in Theorems 1 and 2. The development of the study of the oscillatory behavior of solutions of DDEs depends on the development of the inequalities and relationships used in the study. Proceeding from this, we used the condition (28) to obtain a new monotonical property of υ (κ−2) , which in turn contributes to finding a better estimate of the ratio υ (κ−2) • σ i /υ (κ−2) . This estimate plays a role in improving the criterion that ensures that no solutions meet case (2) in Lemma 1. By combining the conditions for excluding all possible cases of positive solutions, we obtain a new oscillation criterion.