Oscillation theorems for higher order dynamic equations with superlinear neutral term

: In this paper, several oscillation criteria for a class of higher order dynamic equations with superlinear neutral term are established. The proposed results provide a uniﬁed platform that adequately covers both discrete and continuous equations and further su ﬃ ciently comments on oscillatory behavior of more general class of equations than the ones reported in the literature. We conclude the paper by demonstrating illustrative examples.


Assume that
and we also assume that lim µ→∞ A(µ, µ 0 ) < ∞. (1.2) Define the time scale interval to be [µ 0 , ∞) T := [µ 0 , ∞) T. Since we are interested in the oscillatory behavior of solutions of (1.1), we assume that sup T = ∞. Recall that a solution of (1.1) is a nontrivial real-valued function x satisfying Eq (1.1) for µ ≥ µ 0 . We exclude from our consideration all solutions vanishing in some neighborhood of infinity. A solution x of (1.1) is called oscillatory if it is neither eventually positive nor eventually negative, otherwise it is called nonoscillatory. The theory of dynamic equations has been introduced to unify difference and differential equations; see, for instance [1]. Meanwhile, different theoretical aspects of this theory have been discussed in the last years. Particularly, the oscillation of dynamic equations has been the target of many researchers who succeeded in reporting relevant results. Exploring the literature, however, one can observe that most of the obtained results for the oscillation of dynamic equations have been carried out using the comparison, integral averaging, and Riccati transformation techniques [2][3][4][5][6][7][8][9][10][11][12]. Based on authors' observation, never the less, there are no known results regarding the oscillation of higher order dynamic equations with nonlinear (superlinear/sublinear) neutral term and advanced argument. The reader can consult relevant results for difference and differential equations in [13,14], and the references cited therein. More precisely, the existing literature does not provide any criteria for the oscillation of Eq (1.1). Motivated by this inspiration, we consider new sufficient conditions that ensure that all solutions of (1.1) are either oscillatory or converge to zero. Eq (1.1) is commonly used in a variety of applied problems. We state, in particular, the use of Eq (1.1) in the study of non-Newtonian fluid theory and the turbulent flow of a polytrophic gas in a porous medium; see the papers [15,16] for further details.

Main results
As our results will be based on Taylor monomials, we give the following definition. It follows that h 1 (µ, s) = µ − s on any time scale.
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 where µ n := µ(µ + 1) . . . (µ + n − 1).

Lemma 2.2. (Kneser's Theorem) [1, Theorem 5]
Let sup T = ∞, n ∈ N and x ∈ C n rd (T, R + ). Suppose that x n (µ) 0 is either nonnegative or nonpositive on T. Then there exists m ∈ [0, n) Z such that (−1) n−m x n (µ) ≥ 0 holds for all sufficiently large µ. Moreover, both of the following conditions hold: In what follows, we provide the following lemma which plays a crucial role in the sequel.
Moreover, suppose that Kneser's Theorem holds with m ∈ [0, n) Z and x n (µ) ≤ 0 on T. Then, there exists a sufficiently large µ 1 ∈ T such that If m = n − 1 in the above inequality, then upon integration it becomes For convenience, we let and then every solution of Eq (1.1) is either oscillatory or converges to zero.
Proceeding as in the above case, we obtain (2.8) and by Lemma 2.3 with m = n − 2, we see that Integrating this inequality from µ 3 to µ, we have where Q 2 is defined in (2.3). It follows that where Z(µ) = y n−2 (µ). The remaining part of the proof is similar to that of the above case and hence is omitted. Define By virtue of the above theorem, we conclude the following immediate consequence.
The following example is illustrative.
Remark 2.13. The oscillatory behavior of Eq (2.25) cannot be addressed via results existing in the literature.

Conclusions
Unlike previous techniques, a new approach is employed to establish easily verifiable sufficient conditions for the oscillation of higher order dynamic equations with superlinear neutral term and advanced argument. The obtained results improve and complement some previous theorems in the literature. Examples are provided to support theoretical findings. The results in this paper are presented in an essentially new form and of a high degree of generality. For future consideration, it will be of great importance to study the oscillation of Eq (1.1) when β > α.