Behavior of Non-Oscillatory Solutions of Fourth-Order Neutral Differential Equations

: In this paper, we deal with the asymptotics and oscillation of the solutions of fourth-order neutral differential equations of the form (cid:0) r ( t ) ( z (cid:48)(cid:48)(cid:48) ( t )) α (cid:1) (cid:48) + q ( t ) x α ( g ( t )) = 0, where z ( t ) : = x ( t ) + p ( t ) x ( δ ( t )) . By using a generalized Riccati transformation, we study asymptotic behavior and derive some new oscillation criteria. Our results extend and improve some well-known results which were published recently in the literature. Symmetry ideas are often invisible in these studies, but they help us decide the right way to study them, and to show us the correct direction for future developments. An example is given to illustrate the importance of our results.


Introduction
To date, the oscillatory behavior of the solutions to differential equations has been discussed in many papers. Among them, there are many papers about the oscillation of the solutions to functional differential equations. In a related field, the asymptotic behavior of the solutions to delay and neutral delay differential equations were discussed in many works, and there have been very fruitful achievements see .
By a solution of (1) we mean a function x ∈ C 3 ([t x , ∞)) , t x ≥ t 0 , which has the property r (t) (z (t)) α ∈ C 1 ([t x , ∞)) , and satisfies (1) on [t x , ∞). We consider only those solutions x of (1) which satisfy sup{|x (t)| : t ≥ T} > 0, for all T ≥ t x . A solution x of (1) is said to be non-oscillatory if it is positive or negative, ultimately; otherwise, it is said to be oscillatory.
Delay differential equations are often studied in one of two cases or (2) which it is said to be in canonical or noncanonical. For canonical, Moaaz et al. [21] proved that (1) is oscillatory if and lim inf where P n (t) In [25], the authors proved that (1) is oscillatory if the first-order differential equation Now, we state some lemmas that will be useful in establishing our main results: 18]). If the function x satisfies x (i) (t) > 0, i = 0, 1, ..., n, and x (n+1) (t) < 0, then Assume that x (n) (t) is of fixed sign and not identically zero on [t 0 , ∞) and that there exists a t 1 ≥ t 0 such that x (n−1) (t) x (n) (t) ≤ 0 for all t ≥ t 1 . If lim t→∞ x (t) = 0, then for every µ ∈ (0, 1) there exists t µ ≥ t 1 such that

Lemma 3 ([10]
). Let β be a ratio of two odd numbers, C > 0 and D are constants. Then In this work, we obtain some new oscillation criteria for (1). The paper is organized as follows. Firstly, we study the behavior of non-oscillatory solutions of (1) andwe obtain the sufficient conditions which guarantee that every non-oscillatory solution of (1) tends to zero. Secondly, we will use the Riccati transformation technique to give some conditions for the oscillation of (1). Finally, an example is provided to illustrate the main results.

The Behavior of Non-Oscillatory Solutions
In this section, we study the behavior of non-oscillatory solutions of (1) when p 0 ∈ (0, 1). We use an approach that leads to only three independent conditions, but we obtain sufficient conditions which guarantee that every non-oscillatory solution of (1) tends to zero.

Definition 1.
A solution x of (1) is said to be non-oscillatory if it is positive or negative; otherwise, it is said to be oscillatory.

Lemma 4.
Assume that x is an eventually positive solution of (1). Then, r (t) (z (t)) α is non-increasing. Moreover, we have the following cases: Lemma 5. Let x be a positive solution of (1) with property (S 1 ) or (S 2 ). Then the equation has a non-oscillatory solution.

Proof.
Suppose the x is a positive solution of (1) with property (S 1 ) or (S 2 ). Then, we have that Thus, from Lemma 2, we obtain From definition of z, we see that (1) gives Hence, from (7), if we set w := r (z ) α > 0, then the differential inequality From [4] (Corollary 1), we have that (6) also has a positive solution, and this completes the proof.
Lemma 6. Let x be a positive solution of (1) with property (S 3 ). Then the equation has a non-oscillatory solution.
Proof. Suppose the x is a positive solution of (1) with property (S 3 ). Using Lemma 2, we obtain As in the proof of Lemma 6, we can obtain that (8). Next, if we set G := r (z /z ) α < 0, then we get Hence, from the fact that z < 0 and (10), we find Therefore, there exists a function G ∈ C 1 ([t 0 , ∞) , R) such that (11) holds. It follow from [1] that (9) has a non-oscillatory solution, and this completes the proof.
Theorem 1. Assume that the differential equations (6) and (9) are oscillatory. Then every non-oscillatory solution of (1) tends to zero if Proof. Assume the contrary that x is a positive solution of (1) with property lim t→∞ x (t) = 0. From Lemma 4, we have cases (S 1 ) − (S 4 ). Using Lemmas 5 and 6 with the fact that the differential Equations (6) and (9) are oscillatory, we conclude that x satisfies case (S 4 ). Then, since z is a positive decreasing function, we get that lim t→∞ z (t) = c ≥ 0. Suppose the contrary that c > 0. Thus, for all ε > 0 and t enough large, we have c ≤ z(t) < c + ε. Choosing ε < (1 − p 0 ) (c/p 0 ), we obtain where L = (c − p 0 (c + ε)) / (c + ε) > 0. Hence, from (1), we have Integrating this inequality from t 1 to t, we get Letting t → ∞ and taking into account (12), we get that lim t→∞ z (t) = −∞. This contradicts the fact that z (t) > 0. Therefore, c = 0; moreover the fact x (t) ≤ z (t) implies lim t→∞ x (t) = 0, a contradiction. This completes the proof.

Lemma 7.
Assume that x is an eventually positive solution of (1). If z is an increasing and p (t) for any odd positive integer n, where Proof. From the definition of z (t), we obtain for t ≥ t 2 , where t 2 ≥ t 0 sufficiently large, and any odd positive integer n. Since δ 2k+1 (t) ≤ δ 2k (t), we find z δ i (t) ≤ z (t) , for i = 0, 1, ..., n, which with (18) gives The proof is complete.

New Oscillation Criteria
For convenience, we denote: Also, we define the Riccati substitutions (19) and At studying the asymptotic behavior of positive solutions, there are three Cases (S 1 ) − (S 4 ). We recall an existing criterion for Cases (S 1 ) and (S 2 ) in the following lemma: 21]). Assume that x be an eventually positive solution of (1). If (4) and (5) hold, then z is neither satisfied (S 1 ) nor (S 2 ).

Lemma 9.
Assume that x be an eventually positive solution of (1) and Then Proof. Assume that x be an eventually positive solution of (1) on [t 0 , ∞). From the definition of z (t), we see that Repeating the same process, we obtain which yields Thus, (22) holds. This completes the proof.

Lemma 11.
Assume that x be an eventually positive solution of (1) and (S 3 ) holds. If we have the function ω ∈ C 1 [t, ∞) defined as (19), then for all t > t 1 , where t 1 large enough.

Remark 1.
The results of this paper can be extended to the more general equation of the form r (t) z (n−1) (t) α + q (t) x β (g (t)) = 0.
The statement and the formulation of the results are left to the interested reader.

Remark 2.
One can easily see that the results obtained in [25] cannot be applied to Theorem 2, so our results are new.

Conclusions
This paper is concerned with oscillatory behavior of a class of fourth-order delay differential equations. Using a Riccati transformation, a new asymptotic criterion for (1) is presented. In future work, we will aim to present a new comparison theorem that compares the higher-order Equation (1) with first-order equations. There are numerous results concerning the oscillation criteria of first order equations, which include various forms of criteria such as Hille/Nehari, Philos, etc. This allows us to obtain various criteria for the oscillation of (1). Further, we can try to get some oscillation criteria of (1) if z (t) := x (t) − p (t) x (δ (t)).

Author Contributions:
The authors claim to have contributed equally and significantly in this paper. All authors have read and agreed to the published version of the manuscript.
Funding: This research received no external funding.