Oscillatory Behavior of Third-Order Quasi-Linear Neutral Differential Equations

: In this paper, we consider a class of quasilinear third-order differential equations with a delay argument. We establish some conditions of such certain third-order quasi-linear neutral differential equation as oscillatory or almost oscillatory. Those criteria improve, complement and simplify a number of existing results in the literature. Some examples are given to illustrate the importance of our results.

where α is the quotient of odd positive integers.
By a solution of (1), we mean a nontrivial function x ∈ C([T x , ∞), R) with T x ≥ t 0 , which satisfies the property r(y ) α ∈ C 1 ([T x , ∞), R), moreover, satisfies (1) on [T x , ∞). We only consider those solutions of (1) satisfying, on some half-line, [T x , ∞) and satisfying the condition sup{|x(t)| : T ≤ t < ∞} > 0 for any T ≥ T x . A solution of (1) is oscillatory if it has arbitrarily large zeros on [T x , ∞); otherwise, it is said to be nonoscillatory. The equation itself is termed oscillatory if all its solutions oscillate, and is said to be almost oscillatory if all its solutions are oscillatory or asymptotically convergent to zero.
The neutral differential equations have numerous applications in electrical engineering, chemical reactions analysis, and economics.
Such equations are essential tools to model and study the dynamics and stability properties of electrical power systems, as in the works of Milano et al. [1,2]. The asymptotic behavior of solutions of associated delay differential equations have been used to describe the behavior of solutions to third-order partial differential equations. Additionally, they are employed for the study of distributed networks containing lossless transmission lines; see [3,4] for more details.
Baculikova and Dzurina [25,26] and Grace et al. [27] considered the third-order nonlinear delay differential equation Saker and Dzurina [28] studied the third-order nonlinear delay differential equation and obtained several oscillation criteria, which guarantee that all non-oscillatory solutions of such Equation (2) tend towards zero. Ravi et al. [29] investigated a third-order delay differential equation and established some oscillation results that supplemented and improved the results in [27]. Sidorov and Trufanov [30] considered nonlinear operator equations with a functional perturbation of the argument of neutral type and reduced the problem to quasilinear operator equations with a functional perturbation of the argument. Moaaz, in [11], studied a third-order nonlinear delay differential (2) under the condition π(t) = ∞; he developed some results of previous references and established several sufficient conditions, which ensure that all solutions of (2) are oscillatory.
In previous papers, the authors used an integral averaging technique and a Riccati transformation to establish some sufficient conditions which ensure that any solution of ( 1) oscillates or converges to zero. The purpose of this paper is to improve and generalize these results and present some new sufficient conditions, which ensure that any solution of (1) oscillates or, for all its nonoscillatory solutions, tend towards zero as t → ∞.

Auxiliary Results
In this section, we state and prove the following lemmas, which will be useful in the proofs of the main results. Lemma 1 ([29]). Assume x(t) is nonoscillatory solution of (1). Then, y(t) > 0 and there are three possible cases of y(t): where B > 0, A and C are constants, α be a ratio of two odd positive numbers. Then, h attains its maximum value on R at u

Proof.
Assume that x is a positive solution of (1). From hypothesis (I 4 ), (1) becomes Since y (t) > 0, we find That is Combining (4) and (5), we have This completes the proof.
Using (4) with the above inequality, and taking into account (7), we have Thus, This completes the proof.
Proof. Since y is nondecreasing, this implies that Integrating from σ(t) to t 1 , we get Hence, for any ς ∈ (0, 1) and t ≥ t 2 , we see that The proof is complete.
then case N 1 is impossible.

Proof.
Assume that x(t) is a positive solution of (1) on [t 0 , ∞). Then, there exists t 1 ≥ t 0 such that x(τ(t)) > 0 and x(σ(t)) > 0 for all t ≥ t 1 . On the contrary, assume that y(t) satisfies case N 1 . Integrating (6) from t 2 to t and using (8), we get which is a contradiction.

Lemma 8.
Let y(t) be a positive increasing solution of (1). If then y satisfies case N 2 for t ≥ t 1 and (a) y(t) ≥ ty (t) and y(t)/t is decreasing, and lim t→∞ y(t)/t = y = 0, Since y is increasing, y satisfies either case N 1 or N 2 . In view of π(t) < ∞ and (11), we see that (9) hold. By Lemma 7, y(t) satisfies case N 2 .
On the other hand, it follows from y (t) is decreasing, such that there exists a constant λ ≥ 0 such that lim t−→∞ y (t) = λ ≥ 0. We claim that λ = 0. As the proof of Lemma 7 we have (10). Take into account r(t)(y (t)) α ≤ 0 and y (t) < 0, we have It follows that Integrating from t 2 to t, we obtain In view of (11), this contradicts the positivity of y (t). Thus λ = 0. By "Hospital's rule", we see that lim t→∞ y(t) t = 0 and lim t→∞ y (t) = 0. Thus, Therefore, for t ≥ t 2 . Hence, by the monotonicity of y (t), one can obtain that Now, it is easy to see that Thus, The proof is complete.

Main Results
then possible positive solution of (1) satisfies case N 3 .
Since y is increasing, then it follows that In (6), we obtain Integrating (16) from t 2 to t and using (14), we obtain First, let y(t) satisfies case N 1 . We note that (t) > 0. Using the fact π(t) < ∞ together with (13) yields that t t 0Ô (s)ds contradicts the positivity of (t). If y(t) satisfies case N 2 , using (17) in (18) becomes that is Integrating from t 2 to t, we have we obtain a cotradiction with the positivity of y (t). The proof of the theorem is complete.
then, a possible positive solution to (1) satisfies case N 3 .
Combining the above inequality along with (6), we get Integrating from t 2 to t and using (17), we have Using the fact that y (t) < 0, we see that In view of ( [13], Theorem 1), however, the associated delay Equation (21) has a positive solution, which is a contradiction. The proof is complete. Remark 1. Theorem 2 does not require the existence of auxiliary functions such as ( [27], Theorem3), which uses the same principles (compared with first-order delay equations).

Theorem 3. Assume that (11) hold. If
then, the possible positive solution to (1) satisfies case N 3 .

Proof.
Suppose that y satisfies case N 1 or N 2 . We see that (9) holds due to π(t) < ∞ (this mean that lim t→∞ π(t) = 0) and condition (22). Hence, by Lemma 8, y(t) satisfies case N 2 in addition to properties (a) and (b) in Lemma 8. As in the proof of Theorem 2 with the fact r(t)(y (t)) α is nonincreasing, and from (20), we obtain −r(t) y (t) This contradicts (22). The proof is complete.

Proof.
Assume that x(t) is a positive solution of (1) on [t 0 , ∞). Then, there exists t 1 ≥ t 0 such that x(τ(t)) > 0 and x(σ(t)) > 0 for all t ≥ t 1 . Suppose that y satisfies case N 1 or N 2 . By Lemma 8, y(t) satisfies case N 2 and the properties (a) and (b). Define the function w(t) by .

Theorem 5.
Assume that (11) holds. If there is a nondecreasing function then, the possible positive solution to (1) satisfies case N 3 .

Proof.
Assume that x(t) is a positive solution of (1) on [t 0 , ∞); then, there exists t 1 ≥ t 0 such that x(τ(t)) > 0 and x(σ(t)) > 0 for all t ≥ t 1 . Suppose that y satisfies case N 1 or N 2 . By Lemma 8, y(t) satisfies case N 2 and the properties (a) and (b). Define the function w(t) by From Lemma 8, it is easy to see that That is, w(t) > 0 and Using (16) and the fact y (t) is decreasing, we have.
In view of (b) in Lemma 8, we find Using Lemma 2, we obtain It is clear that In (33), we obtain Integrating the above inequality from t 2 to t yields From (30), we are led to The proof is complete.
. Let x(t) be a positive solution to (1) and y(t), satisfying case N 3 .

Oscillation
In the following Theorem, we combine Theorems 2-5 with Theorem (37) to obtain new criteria for oscillation of (1) Theorem 12. If all assumptions of Theorem 1 or 2 or 3 or 4 or 5 and (37) hold, then (1) is oscillatory.

Remark 2.
Compared to the existing results of [25,26], oscillation of (1) is attained by easier conditions.

Conclusions
In this paper, we introduced a simplified theorem for near oscillation; furthermore, we established oscillation criteria for (1). Using comparison theorems and the Riccati technique, we established criteria to check the oscillation under fewer restrictions, and compared this with some results published in the literature. Our results are an extension of and complement to existing results in some previous studies, such as [15,27,29].
The establishment of criteria for the oscillation of Equation (1)

Conflicts of Interest:
There are no competing interests.