Oscillation Criteria for Third-Order Nonlinear Neutral Dynamic Equations with Mixed Deviating Arguments on Time Scales

: Under a couple of canonical and mixed canonical-noncanonical conditions, we investigate the oscillation and asymptotic behavior of solutions to a class of third-order nonlinear neutral dynamic equations with mixed deviating arguments on time scales. By means of the double Riccati transformation and the inequality technique, new oscillation criteria are established, which improve and generalize related results in the literature. Several examples are given to illustrate the main results.


Introduction
The theory of time scales provides a powerful tool for unifying and extending the knowledge about continuous and discrete systems, which has attracted the attention of many scholars in recent years, see the monographs [1,2] for the essentials about the subject. In particular, the research on the oscillation and asymptotic behavior of solutions to different types of differential equations and dynamic equations has been a topic of interest in the past two decades, see, for instance, Refs.  and the references cited therein.
Following this trend, in this paper, we are concerned with the oscillation and the asymptotic behavior of solutions to third-order nonlinear neutral functional dynamic equations with mixed deviating arguments of the form on a time scale T, where z(t) = x(t) + p(t)x(τ(t)).
Since we are interested in the oscillatory and asymptotic behavior of solutions, we assume that the given time scale T is unbounded from above. By a solution of (1), we understand a nontrivial function x ∈ C 1 rd ([T a , ∞) T , R) with T a ∈ [t 0 , ∞) T , which has the property and satisfies (1) on [T a , ∞) T .
We restrict our attention to only those solutions of (1) which exist on some half-line [T b , ∞) T and satisfy the condition sup{|x(t)| : t ∈ [T c , ∞) T } > 0 for any T c ∈ [T b , ∞) T . We tacitly assume that (1) admits such a solution.A solution of Equation (1) is said to be oscillatory, if it is neither eventually positive nor eventually negative. Otherwise, it is called non-oscillatory. Equation (1) is said to be oscillatory, if all its solutions are oscillatory. Otherwise, it is called non-oscillatory.
Equation (1) can be seen as a natural generalization of the half-linear differential equation, whose applications range over many science and technology areas (such as in the investigations of non-Newtonian fluid theory and properties of solutions to porous medium problems); see, e.g., the papers [7,33] for more details.
One of the most used classification rules for these results depends on which of the following conditions: is assumed to be satisfied. Conditions (2) are termed double canonical, conditions (3) and (4) are named mixed canonical-noncanonical and mixed noncanonical-canonical, respectively, and conditions (5) are termed double noncanonical. In fact, a majority of the related research (e.g., the results from references stated in (i)-(viii)) were given under the double canonical condition. In (ix), the authors investigated the equation under either the double canonical or the double noncanonical conditions. To the best of our knowledge, nothing is known regarding the oscillation of (1) when 0 ≤ p(t) ≤ p 0 < ∞, and mixed canonical-noncanonical conditions hold. The natural question now arises regarding whether it is possible to find new oscillation conditions which improve the results established in the above-mentioned papers and can be applied in the above case. One of our aims in this paper is to give an affirmative answer to this question and establish some sufficient conditions for oscillation by employing the Riccati substitution and the appropriate inequality technique.
The main results in this paper are organized into two parts in accordance with different assumptions on the coefficients a(t) and b(t). In Section 2.1, under the double canonical conditions (2), oscillation results for Equation (1) are established for δ(t) ≥ τ(t), δ(t) ≥ t and δ(t) ≥ σ(t), respectively. In Section 2.2, under the mixed canonical-noncanonical conditions (3), oscillation results for Equation (1) are given also for δ(t) ≥ τ(t), δ(t) ≥ t and δ(t) ≥ σ(t), respectively. In Section 3, we give three examples to illustrate the results reported in Sections 2.1 and 2.2.

Main Results
As usual, we assume that all functional inequalities hold for all t large enough. In the sequel, we adopt the following notation for a compact presentation of our results: We start by recalling an auxiliary result which will be useful in the proofs of our main results.
hold. If a positive function φ(t) ∈ C 1 rd (T, R) and t 1 ≥ t 0 exist, such that lim sup then every non-oscillatory solution x(t) of Equation (1) tends to zero as t → ∞.

Proof.
Assume that x(t) is a non-oscillatory solution of Equation (1). Without loss of generality, one can pick t 1 ∈ [t 0 , ∞) T such that x(t) > 0, x(τ(t)) > 0, x(δ(t)) > 0 for all t ∈ [t 1 , ∞) T . Then, the corresponding function z(t) satisfies one of two possible cases: Assume first that case (I) holds. By virtue of Equation (1) and condition (A 4 ), we get In addition, from the condition τ ∆ (t) ≥ τ 0 ≥ 0, the above inequality can be reduced to From (9) and (10), we have Using the first inequality in (6) and the definition of z(t), we get Substituting the above inequality into (11), we obtain Now, we define the function ω(t) by the generalized Riccati substitution Obviously, we see that ω(t) > 0 and, by the product rule and the quotient rule on time scales, By the Pötzsche chain rule (see [1] (Theorem 1.90)), when α ≥ 1, we have Substituting the above inequality into (15), we obtain In addition, by (17), (18) and the definition (14) of ω(t), we can obtain Furthermore, we define another function υ(t), by the generalized Riccati substitution It is clear that Similarly to the process of proving (19), we can easily get Combining (13), (19) and (21), we have Hence, Using which also implies that Thus, combining (24) and (27) yields Substituting (28) into (22), we conclude that Multiplying both sides of (29) by φ(t) and replacing t with s, integrating with respect to s from t 2 to t > t 2 , we have Using integration by parts, (30) yields Let It is easy to verify that Φ(ν) reaches its maximum value on [0, ∞) at see [32] (Lemma 2.4), and so Substituting the above inequality into (31), we have t t 2 that is, Taking the limsup on both sides of (34) as t → ∞ contradicts (8).
(2) and (7) hold. If a positive function φ(t) ∈ C 1 rd (T, R) and t 1 ≥ t 0 exist, such that lim sup then every non-oscillatory solution x(t) of Equation (1) tends to zero as t → ∞.
Proof. The proof is similar to that of Theorem 1 except that the second inequality in (6) is used instead of the first one to obtain (12). The details are omitted.
If we put φ(t) = C in Theorems 1 and 2, we obtain the following corollary.
holds, then every non-oscillatory solution x(t) of Equation (1) tends to zero as t → ∞.
In addition, the case when δ(t) ≥ σ(t) has not been considered in the literature (see, e.g., [23]), hence we will obtain new theorems in this case.

Proof. Assume x(t) is a non-oscillatory solution of Equation
Then, the corresponding z(t) satisfies case (I) or case (II) in the proof of Theorem 1.
Since we have that G(t 1 , t) = G 0 (t 1 , t)A(t, δ(t)), substituting (79) and (77) into (76), it can be concluded that Taking the limsup on both sides of the above inequality as t → ∞, which contradicts (62). The proof is complete.
If a positive function φ(t) ∈ C 1 rd (T, R) and t 1 ≥ t 0 exist, such that (41) holds, then every non-oscillatory solution x(t) of Equation (1) tends to zero as t → ∞.

Remark 2.
With an appropriate choice of the function φ(t), one can derive a number of oscillation criteria for Equation (1) using Theorems 7-12.

Examples
In this section, we give three examples to illustrate our main results in this paper.

Example 1. Consider a third-order neutral dynamic equation
Actually, there are many directions for future research: first, it is an interesting task to generalize the obtained results for odd-order dynamic equations with deviating arguments. Second, with regard to the results of this paper, there remains an open problem associated with removing all non-oscillatory solutions of (1). Third, our further plan is to take advantage of a recent approach [34] developed for third-order differential equations in order to state more effective results for dynamic equations on time scales.