New Results on the Oscillation of Solutions of Third-Order Differential Equations with Multiple Delays

: This study aims to examine the oscillatory behavior of third-order differential equations involving various delays within the context of functional differential equations of the neutral type. The oscillation criteria for the solutions of our equation have been obtained in this study to extend and supplement existing ﬁndings in the literature. In this study, a technique that relies on repeatedly improving monotonic properties was used in order to exclude positive solutions to the studied equation. Negative solutions are excluded based on the symmetry between the positive and negative solutions. Our results are important because they become sharper when applied to a Euler-type equation as compared to previous studies of the same equation. The signiﬁcance of the ﬁndings was illustrated through the application of these ﬁndings to speciﬁc cases of the investigated equation.


Introduction
Delay differential equations (DDEs), a subclass of functional differential equations, take into account the system's reliance on the past to produce predictions for the future that are more precise and effective.One of the most important roles that the concept of delay in systems is thought to play is modeling the length of time needed to complete certain unseen activities.The predator-prey model demonstrates a delay when the birth rate of the predator takes into account both present and past numbers of predators and prey.With the rapid development of communication technologies, transmitting measured signals to a remote control center has become much simpler.However, the biggest obstacle for engineers is the time that it takes for the signal to reach the controller after a measurement has been taken.In order to minimize the possibility of experimental instability and potential harm, this lag must be considered during the planning phase.Modeling such phenomena, as well as others, requires the use of DDEs (see [1][2][3][4][5]).
Neutral delay differential equations (NDDEs) are encountered in several kinds of phenomena, such as electric transmission line problems, which are utilized for interconnecting switching circuits in high-speed computers, the study of vibrating masses connected to elastic bars, the solution of variational problems involving time delays or in the theory of automatic control, and neuro-mechanical systems where inertia is a significant factor (see [6][7][8][9][10]).The reader is directed to consult the references [11][12][13][14][15] for comprehensive insights into the methodologies, techniques, and findings relating to the investigation of oscillatory behavior in third-order NDDEs.Furthermore, the aforementioned studies [16][17][18][19][20] primarily center their attention on the examination of DDEs with odd orders.
This study focuses on third-order NDDEs expressed in linear form with several delays where ≥ 0 , z( ) := y( ) + p( )y(σ( )), and Ω is a positive natural number.We suppose throughout this paper that the following hypotheses are fulfilled: for each i = 1, 2, . . ., Ω.
If a solution y is neither eventually positive nor eventually negative, then it is said to be oscillatory.Otherwise, it is said to be non-oscillatory.The equation itself is termed oscillatory if all of its solutions oscillate.
The previous studies on the oscillatory characteristics of neutral differential equations with odd orders primarily concentrated on establishing a suitable criterion for verifying whether the solutions exhibit oscillatory behavior or approach zero, as referenced in [21][22][23][24].In the following, we provide some background details regarding the study of various classes of neutral differential equations.
In 2010, Baculíková and Džurina [11,25] investigated the asymptotic properties of the third-order NDDE κ( ) z ( ) They obtained conditions that test the convergence of all non-oscillatory solutions to zero.In [11], they used comparisons with first-order equations, while in [25], they obtained Hille and Nehari criteria.Thandapani and Li [26] found some fulfilling conditions that confirm that every solution of (3) either converges to zero or is oscillatory by using the Riccati transformation.In [27], Baculíková and Džurina examined the oscillation of the NDDE κ( )(z( )) + q( )y(τ( )) = 0.
They obtained results based on the comparison theorems, which allowed them to reduce the problem of the oscillation in a third-order equation coupled to a first-order equation.
Their results tested the convergence of all non-oscillatory solutions to zero.Their results are also sharp when applied to the Euler-type DDE, and they improved all previous results with regard to the criterion that tests the convergence of all non-oscillatory solutions to zero.
Our paper investigates the oscillatory properties of a third-order NDDE with multiple delays.The main motivation of this study is to extend the results of [28] to equations with multiple delays with respect to the convergence of non-oscillatory solutions to zero.Moreover, we create standards that guarantee the oscillation of all solutions of the studied equation by establishing a standard that excludes so-called Kneser solutions.Applying our results to a particular case of the considered equation supported the findings.

Preliminary Results
For convenience, we define the following: Lemma 1. Ref. [28] Lemma 1-suppose that there is a constant l > 0 such that Then, M eventually for all ∈ (0, 1).
To proceed with proving our results we need to define the following limits:

Main Results
In this section, we provide sufficient conditions to ensure the oscillation of all solutions of the studied equation.For the following results, we assume that λ * , β * , k * ∈ (0, ∞).Lemma 2. Suppose that y is a positive solution of (1).Then, J 3 z( ) ≤ 0, and there are only two categories: Proof.The proof is straightforward; hence, we omit the details.Notation 1.By x ∈ 1 we mean that solution x with corresponding function has class (1) properties, while by x ∈ 2 we mean that the solution x with corresponding function has class (2) properties.

Class (1)
In this section, we present some characteristics of solutions that belong to class 1 .We also obtain criteria that rule out the existence of solutions with class 1 properties.
Proof.Assume that y ∈ 1 .We will employ an induction argument on n.For n = 0, the conclusion directly follows from Lemma 4 with ε 0 = k/k * .Next, assuming that (C n1 )-(C n3 ) hold when n ≥ 1 for ≥ n ≥ 1 , we need to demonstrate that these conditions also hold for n + 1.
Based on Lemma 4, the proof is exactly similar to the proof of Lemma 5 in [28]; therefore, it was omitted.
From the previous results and taking into account (18), the sequence {β n } has the limit where Theorem 1.If (20) does not possess a root on (0, 1), then 1 = ∅.
In the following theorem, we establish certain conditions that guarantee the absence of Kneser solutions, which are solutions whose corresponding function satisfies the properties in class (2).In the following, we need the conditions τ i (σ( )) = σ(τ i ( )), and σ ( ) is oscillatory, then 2 = ∅, where Proof.Assume that y ∈ 2 .This implies that z > 0, J 1 z < 0, and J 2 z > 0.
By using the latter inequality in (27), we obtain Since z is decreasing, then On the other hand, it follows from the monotonicity of J 2 z that Integrating ( 29) from to ς, we have Thus, we have which, by virtue of ( 28), yields that From the fact that J 2 z is non-increasing, we have Using (32) in (31), we see that w is a positive solution of the differential inequality In view of [31] Theorem 1, we have that (24) also has a positive solution, a contradiction.Thus, the proof is complete.

Oscillatory Theorems and Examples
We obtain the criteria in the following theorems by directly combining the results in the previous two subsections.Assuming that the solution is positive means that it belongs to one of two categories: 2 or 2 .Therefore, when it is confirmed that categories Theorem 3. Suppose β * > 1, and (22) holds.Then, every solution of (1) either converges to zero or is oscillatory.Theorem 4. Suppose ( 21) and ( 22) hold.Then, every solution of (1) either converges to zero or is oscillatory.Theorem 5. Suppose β * > 1, and (33) holds.Then, every solution of (1) is oscillatory.Theorem 6. Suppose ( 21) and (33) hold.Then, every solution of (1) is oscillatory.
The following example demonstrates the significance of the results obtained.
Then, we can compute the value of β * as follows: For β * ≥ 1, we have Thus, the assumption of Theorem 3 is satisfied, and then, every solution of (34) either converges to zero or is oscillatory.