New Monotonic Properties for Solutions of a Class of Functional Differential Equations and Their Applications

: This paper delves into the enhancement of asymptotic and oscillatory behaviors in solutions to even-order neutral differential equations with multiple delays. The main objective is to establish improved inequalities to advance the understanding of oscillation theory for these equations. The paper’s approach is centered on improving the understanding of the intricate relationship between solutions and their corresponding functions. This is achieved by harnessing the modiﬁed monotonic properties of positive solutions, which provide valuable insights into oscillation behavior. Furthermore, leveraging the symmetry between positive and negative solutions, we derived criteria that ensure oscillation for all solutions, with a speciﬁc emphasis on excluding only positive solutions. To illustrate the signiﬁcance of our ﬁndings, we provide an illustrative example.

A solution to Equation ( 1) is termed oscillatory if it does not tend towards either eventual positivity or eventual negativity.Otherwise, it is classified as non-oscillatory.Equation ( 1) is considered oscillatory when all of its solutions exhibit oscillatory behavior.
A neutral DE is a specialized type of DE in which the rate of change of a function depends not only on its current state but also on past values, introducing a time delay.These equations find significant relevance in various fields such as biology, physics, and engineering, where systems exhibit delayed responses.Neutral DEs provide a crucial mathematical framework for modeling dynamic systems with memory and are instrumental in analyzing real-world phenomena with time-delay effects.Understanding their solutions and behavior is vital for accurately describing and predicting the dynamics of such systems, making them indispensable in scientific and engineering applications, see [1][2][3][4][5].
Oscillation solutions, which refer to the periodic behavior of solutions oscillating around a specific function or value, are commonly observed in physical systems, like mechanical systems, electrical circuits, and biological oscillators.The oscillation theorem is an essential result in the theory of DEs that describes the oscillatory nature of solutions.Its wide-ranging applications span various fields, such as physics, engineering, and economics.The implications of this theorem have a broad range of applications, spanning diverse fields.These include its relevance in analyzing oscillatory systems, like pendulums and vibrating strings, as well as its utility in examining population dynamics and the spread of infectious diseases.Furthermore, the oscillation theorem bears notable importance in the domains of control theory and signal processing.Here, it assumes a pivotal role in evaluating the stability and performance of feedback systems, as demonstrated in [6][7][8].
Even-order quasilinear DEs represent a significant class of mathematical models that find wide-ranging applications in science and engineering.These equations, often characterized by terms involving the function and its derivatives of the same order, offer a versatile framework for studying complex phenomena.Their utility extends to various fields, including physics, biology, and control theory.Specifically, they are employed to analyze dynamic systems with even-order dynamics, such as mechanical systems, electrical circuits, and heat transfer problems.This versatility makes the study of evenorder quasilinear DEs a vital endeavor, as it provides essential insights into the behavior of numerous real-world systems.
In the field of mathematical research, there has been a notable surge in interest in the investigation of delay DEs in unconventional contexts.This keen academic interest is apparent in the body of work such as [9][10][11].Similarly, refs.[12][13][14][15], have directed their efforts toward understanding neutral DEs.Furthermore, Moaaz et al. extended this analytical exploration to encompass even-order equations in their publications, such as [16,17].
Many investigations have explored the complex topic of oscillations in even-order neutral DEs.These investigations have proposed diverse methodologies aimed at establishing criteria for identifying oscillatory behavior in these equations.It is worth highlighting that this topic has received extensive attention, particularly in the canonical scenario denoted by the integral expression: as evidenced by the comprehensive body of prior scholarly works, including references such as [18][19][20].
We will now discuss several essential findings from previous research papers that have significantly advanced the study of even-order differential equations.
Lastly, the study carried out by Xing et al. [30] explored various oscillation theorems for the equation: Li and Rogovchenko [31] explored the asymptotic behavior of solutions to higher-order quasilinear neutral DEs, represented as follows with a particular focus on both even-and odd-order equations featuring diverse argument patterns, including alternating delayed and advanced characteristics.The exploration of asymptotic and oscillatory properties in neutral DEs (NDEs) relies on the intricate relationship between the solution χ and its corresponding function y.In the typical context of second-order equations, the standard association is often defined as: This expression is widely utilized.Conversely, in the case of positive, decreasing solutions within non-canonical settings, the relevant relationship takes the following form: This relationship has been demonstrated in previous studies, such as [32,33].
Moaaz et al. [34] developed new inequalities that improve the understanding of the asymptotic and oscillatory behaviors of solutions for fourth-order neutral DEs of the form r( ) z ( ) + q( )χ(σ( )) = 0, specifically in the noncanonical case.Lemma 1 ([34]).If χ represents an eventually positive solution of Equation ( 1), then eventually we have for any κ ≥ 0.
In this study, our primary objective is to build upon earlier work [34], which applied a similar approach to fourth-order equations.Our research is primarily motivated by the desire to extend this methodology, pushing the boundaries of our understanding by encompassing higher-order equations in our current investigation.This expansion marks a significant advancement in the scope of our research, opening up new avenues for exploration and discovery in the field.
Lemma 3 ([36]).Let f ∈ C m ([ 0 , ∞), R + ).Assume that f (m) ( ) has a fixed sign and is not identically zero on [ 0 , ∞) and that there exists 1 ≥ 0 , such that f (m−1) ( ) f (m) ( ) ≤ 0 for all Lemma 4 ([37]).Assuming χ > 0 is a solution of Equation ( 1), we have that r y (n−1) α is a decreasing function, and y fulfills one of the subsequent scenarios: Notation 1.The symbol Ω i is defined as the collection of all solutions that eventually become positivity, with their respective functions satisfying condition (C i ) for i = 1, 2, 3.

Properties of Asymptotic and Monotonic Behaviors
We establish asymptotic and monotonic properties for the solutions of the neutral DE (1), in this section.
Proof.Assume that χ ∈ Ω 2 leads to a contradiction.According to Lemma 6, the functions y (n−2) /π β 0 0 and y (n−2) /π 1−β 0 0 are decreasing and increasing for ≥ 1 , respectively.As a result, we can conclude that This conclusion contradicts the initial assumption.Hence, the proof is considered complete.
Theorem 2. Let us assume that conditions (4) and (10) are satisfied.Suppose there exists a positive integer value m, such that the following inequality holds then Ω 2 = ∅, where α ≤ 1.
Proof.Let us assume the opposite scenario, where χ ∈ Ω 2 .According to the information provided in Lemma 7, it follows that both (Y 3,1 ) and (Y 3,2 ) hold.
After integrating the final inequality from to ∞, the result is expressed as or equivalently y (n−3) ≤ r 1/α y (n−1) π 1 .
Proof.Similar to the argument presented in the proof of Lemma 9, we obtain the Equation (17).Considering (Y 7,1,m ), we deduce that which with (17) yields Theorem 3. Let's assume that conditions ( 18) and ( 19) hold, and that there exists a positive integer m such that Under these conditions, we can conclude that Ω 3 = ∅, where α ≤ 1.
Based on (Y 4,2 ) at i = n − 2, we find that w ≥ 0 for ≥ 2 .Additionally, using (27), we obtain Hence, from the definition of w, we can deduce that Thus, Using (Y 5,2 ) and (Y 4,2 ) at i = n − 2, we deduce that Using (Y 4,1 ) in Lemma 8, we observe that y( )/π n−2 ( ) is increasing, then y( σ( )) which, from Equation (30), gives Therefore, the positive solution w to the differential inequality can be deduced from Equation (31).Notably, according to the findings in Theorem 2.1.1 in [8], the condition expressed in Equation (29) ensures that Equation (31).This logical contradiction serves as conclusive evidence for proving the Theorem.Theorem 4.Under the assumption that Equations ( 18) and ( 19) are satisfied, we consider a positive integer m, such that If the above inequality holds, then Ω 3 = ∅.
Proof.To demonstrate this, we utilize the relationship χ > p 2 ( ; κ)y, with respect to Equation (1), employing the identical proof technique used in the previous theorem.
Proof.If we consider the contrary scenario where χ ∈ Ω 1 , it becomes clear from the information provided by (C 1 ), that lim →∞ y( ) = 0.
Therefore, it can be deduced from Lemma 3 that, for any ∈ (0, 1), eventually.Using Equation (34) in Equation ( 1), we see that Consider the function θ = r y (n−1) α .By observing the last inequality, it becomes clear that θ( ) serves as a positive solution to the delay differential inequality, expressed as: Therefore, the positive solution θ to the differential inequality can be deduced from Equation (35).Notably, according to the findings in Theorem 2.1.1 in [8], the condition expressed in Equation (33) ensures that Equation (35) this logical contradiction serves as conclusive evidence for proving the Theorem.

Criteria for Oscillation
This section extends the groundwork laid in the preceding sections to introduce fresh criteria for confirming the oscillatory nature of all solutions within Equation (1).To be more precise, we have pinpointed particular conditions that conclusively exclude the existence of positive solutions in all three scenarios, denoted as (C 1 ), (C 2 ), and (C 3 ).By amalgamating these conditions, as expounded in the subsequent theorems, we can establish robust criteria for ascertaining oscillation.

Conclusions
In this study, we delved into the investigation of the oscillatory behavior and monotonic properties of even-order quasilinear neutral differential equations.Our main focus was on a specific type of such equations.Through our research, we were able to establish improved relationships that connect the solution and its corresponding function for two out of the three categories of positive solutions in the equation under study.By leveraging these newly derived relationships, we were able to develop criteria to ascertain that categories Ω 2 and Ω 3 contained no positive solutions.A significant contribution of this work was the introduction of novel criteria to assess the oscillation of Equation (1).These criteria provide a valuable tool for analyzing the oscillatory nature of the equation.Looking ahead, it would be intriguing to extend our findings to explore the behavior of non-linear odd-order neutral DEs, opening up exciting possibilities for future research directions.