Sufﬁcient Conditions for Oscillation of Fourth-Order Neutral Differential Equations with Distributed Deviating Arguments

: Some new sufﬁcient conditions are established for the oscillation of fourth order neutral differential equations with continuously distributed delay of the form (cid:0) r ( t ) ( N (cid:48)(cid:48)(cid:48) x ( t )) α (cid:1) (cid:48) + (cid:82) ba q ( t , ϑ ) x β ( δ ( t , ϑ )) d ϑ = 0, where t ≥ t 0 and N x ( t ) : = x ( t ) + p ( t ) x ( ϕ ( t )) . An example is provided to show the importance of these results.


Introduction
The theory of differential equations is an adequate mathematical apparatus for the simulation of processes and phenomena observed in biotechnology, neural networks, physics etc, see [1]. One area of active research in recent times is to study the sufficient criterion for oscillation of delay differential equations, see .

Definition 2.
A solution of (1) is called oscillatory if it has arbitrarily large zeros on [t x , ∞), and otherwise is called to be nonoscillatory. (1) is called oscillatory if every its solutions are oscillatory.
Hence, [22] improved the results in [24]. Thus, the motivation in studying this paper is complement results in [9] and improve results [22,24].
By using the Riccati transformations, we establish a new oscillation criterion for a class of fourth-order neutral differential equations (1). An example is provided to illustrate the main results.

Some Auxiliary Lemmas
We shall employ the following lemmas Assume that x (n) (t) is of fixed sign and not identically zero on [t 0 , ∞) and there exists a t 1 ≥ t 0 such that

Lemma 4 ([9]).
Assume that x is an eventually positive solution of (1). Then, there exist two possible cases:

Notation 1.
We consider the following notations:

Main Results
In this part, we will discuss some oscillation criteria for Equation (1).

Lemma 5.
Assume that x is an eventually positive solution of (1) and Then Proof. Let 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, (7) holds. This completes the proof. (6) holds. If there exist positive functions θ, for some µ 1 ∈ (0, 1) and every M 1 , M 2 > 0, then (1) is oscillatory.
Proof. Let x be a non-oscillatory solution of (1) on [t 0 , ∞). Without loss of generality, we can assume that x is eventually positive. It follows from Lemma 4 that there exist two possible cases (S 1 ) and (S 2 ). Let (S 1 ) holds. From Lemma 2, we obtain N x (t) ≥ 1 3 tN x (t) and hence the function From (7) and (10), we get that From (1) and (11), we obtain Since δ (t, ξ) is nondecreasing with respect tos, we get δ (t, ϑ) ≥ δ (t, a) for ξ ∈ (a, b) and so Next, we define a function ω by Differentiating and using (12), we obtain Recalling that r (t) (N x (t)) α is decreasing, we get

This yields
It follows from Lemma 1 that for all µ 1 ∈ (0, 1). Thus, by (13)-(15), we get Hence, Since N x (t) > 0, there exist a t 2 ≥ t 1 and a constant M > 0 such that for all t ≥ t 2 . Using the inequality This implies that which contradicts (8).
In the case where (S 2 ) satisfies, by using Lemma 2, we find that and hence t −1 N x (t) ≤ 0. Therefore, From (7) and (18), we have which with (1) gives Integrating this inequality from t to , we obtain From (17), we get that Letting → ∞ in (19) and using (20), we obtain Integrating this inequality again from t to ∞, we get for all µ 2 ∈ (0, 1).

Now, we define
Then w (t) > 0 for t ≥ t 1 . By differentiating w and using (21), we find Thus, we obtain and so Then, we get which contradicts (9). This completes the proof.
Using (see ( [15] Theorem 1)), we see (24) also has a positive solution, a contradiction. Suppose that Case (S 2 ) holds. From Theorem 1, we get that (21) holds. Since σ (t) ≤ δ (t) and N x (t) > 0, we have that Using Lemma 2, we get that From (18) and (29), we obtain Now, we choose φ (t) := N x (t), thus, we find that φ is a positive solution of Using (see ( [15] Theorem 1)), we see (25) also has a positive solution, a contradiction. The proof is complete. Example 1. Consider the differential equation Thus, by using Theorem 1, then Equation (31) is oscillatory.

Remark 2.
One can easily see that the results obtained in [24] cannot be applied to conditions in Theorem 1, so our results are new.

Conclusions
In this work, our method is based on using the Riccati transformations to get some oscillation criteria of (1). There are numerous results concerning the oscillation criteria of fourth order equations, which include various forms of criteria as Hille/Nehari, Philos, etc. This allows us to obtain also various criteria for the oscillation of (1). Further, we can try to get some oscillation criteria of (1) if N x (t) := x (t) − p (t) x (ϕ (t)) in the future work.

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:
The authors received no direct funding for this work.

Acknowledgments:
The authors thank the reviewers for for their useful comments, which led to the improvement of the content of the paper.

Conflicts of Interest:
There are no competing interests between the authors.