Oscillation criteria for a class of third-order Emden–Fowler delay dynamic equations with sublinear neutral terms on time scales

In this paper, we study the oscillation of a class of third-order Emden–Fowler delay dynamic equations with sublinear neutral terms on time scales. By using Riccati transformation and integral inequality, we establish several new theorems to ensure that each solution of the equation oscillates or asymptotically approaches zero, and the results in the literature are supplemented and extended. Examples are given to illustrate our main results.


Introduction
In this paper, we are concerned with the oscillation and asymptotic behavior of the thirdorder Emden-Fowler delay dynamic equation with sublinear neutral terms of the form b(t) a(t) x(t) + p(t)x α δ(t) γ + f t, x τ (t) = 0, t ∈ [t 0 , ∞] T (1.1) on a time scale T. Throughout this paper, the following assumptions are tacitly satisfied: (A 1 ) a(t), b(t), p(t) ∈ C 1 rd (T, R), and b (t) ≥ 0, 0 < p(t) ≤ p < 1, (1.2) (A 2 ) α, β, γ are quotients of odd positive integers, where β ≥ γ , 0 < α ≤ 1; (A 3 ) δ(t), τ (t) ∈ C 1 rd (T, R) such that τ (t) ≤ t, δ(t) ≤ t, lim t→+∞ τ (t) = lim t→+∞ δ(t) = ∞. (A 4 ) f ∈ C(T × R, R) is assumed that uf (t, u) > 0 for u = 0, t ∈ T, and there exists a function q(t) ∈ C 1 rd (T, R) such that f (t,u) u β ≥ q(t) for u = 0, t ∈ T. For the basic theory and notation of calculus on the time scale T, we can see, for instance, the monograph [1], which systematically gives the definition of delta (or nabla) differentiation, the basic algorithm, and the important properties, such as the following: Assume that f : T → R is a function, and let t ∈ T k . We define f (t) to be the number, provided it exists, with the property that, for any > 0, there exists a neighborhood U of t, U = (tδ, t + δ) ∩ T for some δ > 0 such that We call f (t) the delta or Hilger derivative of f at t. We say that f is delta or Hilger differentiable, shortly differentiable, in T k if f (t) exists for all t ∈ T k . The function f : T → R is said to be the delta derivative or Hilger derivative, shortly derivative, of f in T k , where σ (t) = inf{s ∈ T : s > t} is called the forward jump operator. The properties of the delta ( ) derivative and the knowledge of nabla (∇) derivatives can be found in the monograph [1], which is omitted here.
Since we are interested in oscillation and asymptotic behavior of solutions to equation, we assume that the time scale T is unbounded. In this paper, we only consider those solutions of Eq. (1.1) which satisfy sup{|x(t)| : t ∈ [T, ∞) T } > 0 and assume that such solutions exist. A solution of (1.1), which is nontrivial for all large t, is said to be oscillatory if it is neither eventually positive nor eventually negative. Otherwise, it is called nonoscillatory.
The theory of dynamic equations on time scales not only unifies the theory of differential equations and difference equations, but it also extends these classical cases to cases "in between", e.g., T = q N 0 , T = hZ, and other different time scales, which have wide application value in quantum theory, mathematical, theoretical, and chemical physics and ecology. Therefore, the study of the properties of dynamic equations on time scales has become a hot topic. In the past 20 years, there have been a lot of research results on the oscillation of dynamic equations. As a matter of fact, Eq. (1.1) is a natural generalization of the half-linear/Emden-Fowler dynamic equation (including related differential equation) which arises in a variety of real world problems such as in the study of p-Laplace equations, non-Newtonian fluid theory, the turbulent flow of a polytrophic gas in a porous medium, and so on; see, e.g., the papers [2][3][4][5] for more details. In this paper, our main purpose is to study the oscillation and asymptotic behavior of Eq. (1.1). In order to enlighten the research ideas, we briefly review the following research progress and important results of this problem.
Agarwal et al. [6] investigated the oscillation of a certain class of second-order differential equations with a sublinear neutral term where 0 < α < 1 is a ratio of odd positive integers. Dzurina et al. [3], Grace and Graef [7], and Tamilvanan et al. [8] established sufficient conditions for the oscillation of all solutions of a nonlinear differential equation where α and β are ratios of odd positive integers. Agarwal et al. [9] and Erbe et al. [10] established several Hille and Nehari type criteria for the third-order dynamic equation Agarwal et al. [11,12], Han et al. [13], and Li et al. [14] considered the oscillatory behavior of the third-order dynamic equation where γ > 0 is the ratio of positive odd integers, and proved the following results.
Yang [15] studied the oscillatory and asymptotic behavior of third-order nonlinear variable delay dynamic equations is the ratio of positive odd integers, and got the following results.
Grace [16] was concerned with new oscillation criteria for third-order nonlinear difference equations with a nonlinear non-positive neutral term of the form where α, γ , and β are the ratios of positive odd integers. Candan [17] was concerned with oscillation of the third-order nonlinear neutral dynamic equation where γ ≥ 1 is a ratio of odd positive integers. It would also be interesting to establish sufficient conditions for the oscillation and asymptotic behavior of solutions to Eq. (1.1) for the other ranges of the neutral coefficient p(t) such as (1.10). For this research issue, see, e.g., the papers [18,19] for more details.
It is not difficult to see that the results on the oscillation of delay dynamic (differential/difference) equations with sublinear neutral terms are mainly concentrated in the second order (see [3,[6][7][8] and the references cited therein), but the results of the third order are relatively lower [16,20]. Numerous researchers have studied the special case of Eq. (1.1), see, e.g., Ref . Inspired by the above papers, in this paper, by employing the Riccati transformation and some integral inequality, we present sufficient conditions to ensure that every solution of Eq. (1.1) is oscillatory or asymptotically converges to zero. Our results improve and generalize the results of the papers [13,15].

Preliminaries
Before proving the main theorems, we need some useful lemmas which will be used later.
In [1], p. 190, Taylor's monomials h n (t, s) ∞ n=0 are defined recursively by It follows from [1] that h 1 (t, s) = ts for any time-scale, but simple formulas in general do not hold for n ≥ 2.

3)
where z(t) is -differentiable and eventually positive or eventually negative.
Its proof is not difficult from the monograph [1, Theorem 2.57], and so it is omitted here.
It is obvious that Lemma 2.4 is an improvement of Lemma 2.3 in [31] about the constant γ > 0, which must be a ratio of two positive odd integers is no longer suitable for Eq. (1.1).