New Criteria on Oscillatory and Asymptotic Behavior of Third-Order Nonlinear Dynamic Equations with Nonlinear Neutral Terms

In the paper, we provide sufficient conditions for the oscillatory and asymptotic behavior of a new type of third-order nonlinear dynamic equations with mixed nonlinear neutral terms. Our theorems not only improve and extend existing theorems in the literature but also provide a new approach as far as the nonlinear neutral terms are concerned. The main results are illustrated by some particular examples.

We define the solution x of Equation (1) as a continuous function on [T x , ∞) which satisfies Equation (1) on [T x , ∞), T x ≥ ς 0 . We only consider those solutions x of Equation (1) satisfying sup{|x(ς)| : ς ≥ T} > 0 for all T ≥ T x .
A solution x of Equation (1) is said to be oscillatory if there exists a sequence {ξ n } such that x(ξ n ) = 0 with lim n→∞ ξ n = 0, and otherwise it is non-oscillatory. If all solutions of Equation (1) are oscillatory, then it is said to be oscillatory.
The oscillatory behavior of dynamic equations on time scales has become a very popular subject for many researchers, and thus it has been widely developed. For recent investigations regarding the systematic treatments of oscillations of solutions for secondorder dynamic equations, we refer to [1][2][3][4] and the references cited therein. On the other hand, it has been realized that the oscillations of nonlinear third-order neutral equations contribute to many disciplines, including mechanical oscillation, earthquake structures, clinical applications, frequency measurements and harmonic oscillators that involve symmetrical properties; see, for instance, the pioneering monographs of [5,6]. Inspired by these extensive applications, many authors have paid more attention to studying the oscillatory behavior of third-order difference and differential equations. We review some relevant results for the sake of completeness.
In [7], the authors studied asymptotic properties of the third-order neutral differential equation of the form where a, q, p are positive functions, γ > 0 is a quotient of odd positive integers and τ(ς) ≤ ς, δ(ς) ≤ ς. Sufficient conditions are established which ensure that all nonoscillatory solutions of Equation (4) converge to zero. Very recently in [8], the following third-order nonlinear neutral differential equation was considered.
where z(ς) = x(ς) + p(ς)x(δ(ς)) and γ is a ratio of odd positive integers. New oscillation criteria have been introduced under the two cases For more significant results, the reader can consult the papers [6,[9][10][11][12][13]. After exploring the above-mentioned literature and to the best of authors' knowledge, there have been no results published with regard to the oscillation and asymptotic behavior of third-order, nonlinear neutral differential equations as far as the nonlinear neutral terms are concerned. In this paper, we recover this case and obtain some sufficient conditions which assure that Equation (1) is either oscillatory or any of its solutions converge to zero. Evidently, it is shown that the existing literature does not guarantee such behavior for the solutions of Equation (1). Several examples are presented to validate and support the proposed suppositions.
One should observe that finding h n for n ≥ 2 is not an easy task in general. For a particular time scale such as T = R or T = Z, we can easily find the functions h n . Indeed, we have h n (µ, s) = (µ − s) n n! (µ, s ∈ R) and h n (µ, s) = (µ − s)n n! (µ, s ∈ Z), where We present the main results of this paper in four parts. (1) When ω = 1 and β ≤ 1

Equation
The following result deals with the oscillation and asymptotic behavior of (1) with a sub-linear neutral term. If for ς 1 ∈ [ς 0 , ∞), then Equation (1) is oscillatory or every solution of it converges to zero.
for ς ≥ ς 2 and for some positive constant b. Integrating this inequality from ς 2 to ς and using condition (3) we obtain which is a contradiction, hence we have y ∆∆ (ς) > 0 for ς ≥ ς 2 .
Case I. (2) implies that Since y(ς) is non-decreasing, we have for some positive constant c * > 0 such that y ≥ c * . This implies that there exists a constant c ∈ (0, 1) such that x(ς) ≥ cy(ς).
Thus, we have Since a(ς) y ∆∆ (ς) α is a non-increasing function, we conclude that y ∆∆ (ς) > 0 and y ∆ (ς) > 0 for ς ≥ ς 2 . It is clear to see that Integration of both sides of the inequality above from ς 1 to ς gives Using the last inequality, (11) turns out to be where W(ς) = a(ς) y ∆∆ (ς) α . Now, integration of both sides of inequality (12) from τ(ς) to ς gives Through multiplying both sides of the resulting inequality by W −γ/α (τ(ς)), we obtain By taking the limit supremum of both sides of (13) as ς → ∞, we get which contradicts with condition (8) of the theorem. Case II. By condition (7), it is easy to see that any solution converges to zero. This completes the proof.
We present the following illustrative example. Example 1. Let T = R and consider the neutral functional differential equation: where It is easy to check that the conditions of Theorem 1 are satisfied, and hence every solution of Equation (14) is either oscillatory or converges to zero. (1) When ω = 1 and β ≥ 1

Equation
The following result is related to the oscillatory and asymptotic behavior of (1) with a super-linear neutral term.
Here we have a(n) = n 3 , p(n) = n −3 , α = 3 and τ(n) = δ(n) = n/2. It can be verified that the conditions of Theorem 2 are satisfied. Thus we conclude that Equation (16) is either oscillatory or every solution of it converges to zero.
Using (9) and (10) turns out to be The rest of the proof is left to the reader, since it is similar to that of the above case.
We have the following example.

Equation (1) When p(ς) = 0
In this subsection, we obtain a new oscillation criterion for the equation
By following the analogous steps as in the proof of Theorem 1 for case (I), we get a contradiction.
Integrating the last inequality, we get Using the inequality above in Equation (21), we have The rest of the proof is omitted since it is similar to that of Theorem 1.

Equation (1) When
In this context, we have the following result.
Case (IV): In this case we have z ∆∆ (ς) < 0 and so z ∆ (ς) ≥ 0. It is easy to see that Using the last inequality, (28) turns out that where Z(ς) = z ∆ (ς) > 0. We see that for v ≥ u ≥ ς 1 . Setting u = h * (ς) and v = ξ(ς) we have Using (29) and (30), we get Due to the similarity to the above cases, the rest of the proof is left to the reader, and hence is omitted.

Conclusions
In this paper, we discussed the oscillatory behavior of a new type of third-order nonlinear dynamic equations with mixed nonlinear neutral terms. Particular emphasis was paid to the consideration of nonlinear neutral terms in the main equation, which has not been considered before. The proof of the main results was given based on the cases β ≤ 1 and β ≥ 1. It was demonstrated that the equations considered in the examples cannot be commented on by the results obtained in the literature [6][7][8][9][10][11][12][13]. Thus, the results of this paper complement and generalize somehow the existing results in the literature.
We leave this problem for further consideration in the future. Funding: J. Alzabut is thankful to Prince Sultan University for funding this work.
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Data Availability Statement: Not applicable.