Establishing New Criteria for Oscillation of Odd-Order Nonlinear Differential Equations

: By establishing new conditions for the non-existence of so-called Kneser solutions, we can generate sufﬁcient conditions to ensure that all solutions of odd-order equations are oscillatory. Our results improve and expand the previous results in the literature.


Introduction
In the 20th century, the extremely fast development of science led to applications in the fields of biology, population, chemistry, medicine dynamics, social sciences, genetic engineering, economics, and others. Many of these phenomena are modeled by delay differential equations. All these disciplines were promoted to a higher level and discoveries were made with the help of this kind of mathematical modeling.
The neutral differential equations are the differential equations in which the delayed argument occurs in the highest derivative of the state variable. The neutral equations appear in the modeling of the networks containing lossless transmission lines (as in high-speed computers where the lossless transmission lines are used to interconnect switching circuits); see [1].
It is known that determination of the signs of the derivatives of the solution is necessary and causes a significant effect before studying the oscillation of delay differential equations. The other essential thing is to establish relationships between derivatives of different orders, which may lead to additional restrictions during the study. In odd-order differential equations, in some cases, it is difficult to find relationships between derivatives of different orders, which in turn is central to the study of oscillatory behavior. Therefore, it can very easily be observed that differential equations of odd-order received less attention than differential equations with even-order. Additionally, most studies are concerned with finding sufficient conditions that guarantee that every non-oscillating solution tends to zero; see [4,[10][11][12][13][14][15][16][17][18][19][20].
In this paper, in Section 2, we offer some auxiliary lemmas that define the different cases of signs of derivatives and the relationships between derivatives of different orders. In Section 3, we establish a set of new criteria that ensure that there are no non-oscillating solutions in each case of derivatives separately. In Section 4, we establish new criteria for the oscillation of all solutions of the studied equation. Finally, in conclusion, we discuss the results and compare them to the related works.
In detail, we investigate the oscillatory properties of solutions to the odd-order neutral equation where n is an odd natural number. Moreover, we suppose that is not eventually zero on any half line I * for t * ≥ t 0 , and I s : Hypothesis 2 (H2). f ∈ C (R, R) and there exists a positive constant k such that f (x) ≥ kx α .
Next, we present the basic definitions.
is called the canonical operator.
Definition 2. Let x be a real-valued function defined for all t in a real interval I x , t x ≥ t 0 , and having a n th derivative for all t ∈ I x . The function x is called a solution of the differential equation (Equation (1)) on I if x is continuous; r z (n−1) α is continuously differentiable and x satisfies (1), for all t in I x .

Definition 3.
A nontrivial solution x of (1) is said to be oscillatory if it has arbitrary large zeros; that is, there exists a sequence of zeros {t n } ∞ n=0 (i.e., x (t n ) = 0) of x such that lim n→∞ t n = ∞. Otherwise, it is said to be non-oscillatory. Notation 1. The set of all eventually positive solutions of (1) is denoted by X + .
We restrict our discussion to those solutions x of (1) which satisfy sup {|x (t)| : t 1 ≤ t 0 } > 0 for every t 1 ∈ I x . All functional inequalities and properties, such as increasing, decreasing, positive, and so on, are assumed to hold eventually; that is, they are satisfied for all t large enough.

Preliminary Results
During this part of the paper, we provide auxiliary lemmas. These lemmas will be the cornerstone of the main results.

Lemma 4.
Suppose that x ∈ X + and z satisfies N. Then for ρ ≤ σ.
Proof. It follows from the monotonicity of r(z (n−1) ) (t) that Integrating (4) n − 3 times from ρ to σ, we get The proof is complete.

Proof.
Assume that x ∈ X + and z satisfies P. Then, there exists a t 1 ≥ t 0 such that Using Lemma 2 with F = z , we obtain for every λ ∈ (0, 1) , which with the fact that z (n) ≤ 0 gives Hence, from (5), (7) and (8), we get The proof is complete.
From (1), we get 1 Combining (1) and (13) and taking into account that τ (t) ≥ τ 0 , we obtain This implies that Using Lemma 1, we obtain From the definition of z, it is easy to conclude that Next, from (1), we get and 1 Using (15) and (16) and taking into account (9) and (11), we obtain By replacing (14) with (17), this part of proof is similar to that of the previous case and so we omit it.

Nonexistence Criteria of Non-Oscillatory Solutions
At the beginning of this section, we define the following classes: The set of all positive solutions of (1) whose corresponding function z satisfies P or N is denoted by X + P or X + N , respectively. Now, we create various criteria that ensure that there are no positive solutions of (1) whose corresponding function satisfies P.
then X + P is an empty class.
Proof. Assume the contrary that x ∈ X + P . Then, there exists a t 1 ≥ t 0 such that x (t) > 0, x (τ (t)) > 0 and x (g (t)) > 0 for t ∈ I 1 . Using Lemma 5 with δ (t) := 1, we arrive at By integrating the last inequality from t to ∞, we find This implies that From (19), we note that w (t) ≥ Θ (t). Thus, we have Taking into account (18) and (21), (20) becomes which contradicts the expected value of β > 1 and α > 0; therefore, the proof is complete.
m=0 be a sequence of continuous functions defined as follows: S 0 (t) = Θ (t) and By using the definition of {S m (t)} ∞ m=0 , we can infer more new criteria as follows: then X + P is an empty class, where Proof. Assume the contrary that x ∈ X + P . Then, there exists a t 1 ≥ t 0 such that x (t) > 0, x (τ (t)) > 0 and x (g (t)) > 0 for t ∈ I 1 . From Theorem 1, we have that (19) holds. By induction, using (19), it is easy to see that the sequence {S m (t)} ∞ m=0 is non-decreasing and w (t) ≥ S m (t). Thus the sequence {S m (t)} ∞ m=0 converges to S (t). By the Lebesgue monotone convergence theorem and letting m → ∞ in (22), we get and so Thus, we get that Integrating (24) from t 1 to t, we obtain However, letting t → ∞ and using (23), the above inequality yields ϕ (t) S (t) → −∞, which contradicts the fact that ϕ (t) S (t) is nonnegative. The proof is complete.

Theorem 3.
If there exist some λ ∈ (0, 1) and S m (t) such that then X + P is an empty class.
Proof. Assume the contrary that x ∈ X + P . Using Lemma 2 and taking into account the fact that z (n−1) (t) is non-increasing, we find for λ ∈ (0, 1). Then, from definition of w (t) with δ (t) = 1, we have 1 which contradicts (25). The proof is complete.
then X + P is an empty class.
Next, by using comparison principles, we will create various criteria that ensure that there are no positive solutions of (1) whose corresponding function satisfies N.

Theorem 4. If the first-order advanced inequality
then X + N is an empty class.
Proof. Assume the contrary that x ∈ X + and z satisfy N. Then, there exists a t 1 ≥ t 0 such that x (t) > 0, x (τ (t)) > 0 and x (g (t)) > 0 for t ∈ I 1 . From Lemmas 4 and 6, we arrive at (2) and (10), respectively. Now from (2), we get which, by virtue of (10) yields that Now, set Using the fact that r (t) (z (n−1) (t)) is non-increasing, we obtain Using (31) in (29), we see that G is a positive solution of the inequality This a contradiction, and thus the proof is complete.

Theorem 5.
If there exists a function ϑ (t) ∈ C (I 0 , (0, ∞)) satisfying and the first-order delay equation is oscillatory, then X + N is an empty class.
By replacing (28) with (34) and proceeding as in proof of Theorem 4, we arrive at G (defined as in (30)) which is a positive solution of the inequality In view of ( [23], Theorem 1), we have that (33) also has a positive solution, a contradiction. Thus, the proof is complete.
Proof. Assume the contrary that x ∈ X + and z satisfy N. Then, there exists a t 1 ≥ t 0 such that x (t) > 0, x (τ (t)) > 0 and x (g (t)) > 0 for t ∈ I 1 . It is easy to notice that lim t→∞ z (j) = 0 for j = 1, 2, ..., n − 2 and lim t→∞ r (t) z (n−1) (t) α = 0. Hence, by integrating (1) from t to ∞, we obtain and hence Integrating the last inequality n − 2 times from t to ∞, we obtain Thus, we get From the definition of z, we have which with (1) yields Integrating the last inequality from t to θ (t), we arrive at Integrating the last inequality n − 2 times from t to θ (t), we get If we set then ω is a positive solution of the inequality ω (t) + B n−2 (t) ω τ θ n−1 (t) ≤ 0. In view of ([23], Theorem 1), we have that (37) also has a positive solution, a contradiction. The proof is complete.

Asymptotic and Oscillatory Properties
Theorem 7. Each non-oscillatory solution of (1) tends to zero if and one of the conditions (18) or (26) is fulfilled.
Proof. Let x be a non-oscillatory solution of (1). Without loss of generality, we assume that x ∈ X + . From Lemma 3, we have only two cases for z. Each of the conditions (18) or (26) contradicts that z fulfills P. Now, we suppose that z satisfies N. Since z (t) > 0 and z (t) < 0, we get that z → c as t → ∞, where c ≥ 0. Suppose that c > 0. Then, for every > 0, there exists a T ≥ t 0 such that c < z (t) < c + for all t > T. By set ε < (1 − p) (c/p), we get that where M = (c − p(c + )) / (c + ) > 0. Thus, integrating from t to ∞, we have By integrating from t to , we find Taking the limit of both sides as t → ∞ and using (39), we get that z (n−2) (t) → −∞ as t → ∞. But, z n−2 is a negative increasing function, this a contradiction. Therefore, lim t→∞ z(t) = 0, which implies that lim t→∞ x(t) = 0. The proof is complete.
In the following, based on the fact that there are only two cases for the corresponding function z, we infer new criteria for oscillation of all solutions of the Equation (1). In each of the following theorems, we refer to two conditions through which it is possible to exclude the existence of solutions in X + P or X + N . Thus, we rule out the existence of non-oscillatory solutions.

Conclusions
When studying the oscillatory behavior of solutions of differential equations with odd-order, it is customary to find conditions that ensure solutions are either oscillatory or tend to zero. Dzurina et al. [5] and Vidhyaa et al. [24] established criteria for the oscillation of all solutions of a third-order linear and half-linear neutral differential equation, respectively. As an extension and also an improvement of these results, we obtained new oscillation criteria for the odd-order non-linear neutral Equation (1).
If we consider the third order differential equation From Example 1 in [5], Equation (43) is oscillatory if q 0 > 120. However, by using our criterion (41), we get that (43) is oscillatory if q 0 > 111.11. Moreover, we consider the equation From Example 3 in [24], by choosing β = 4/3 Equation (44) is oscillatory if q 0 > 4. However, if we choose γ = 4/3, then our criterion (42) becomes q 0 > 2, and hence (44) is oscillatory. Thus, our results improve the results in [5,24]. In the future, we can try to study the advanced odd-order differential equations by the same approach.
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.