Improved Oscillation Results for Functional Nonlinear Dynamic Equations of Second Order

: In this paper, the functional dynamic equation of second order is studied on an arbitrary time scale under milder restrictions without the assumed conditions in the recent literature. The Nehari, Hille, and Ohriska type oscillation criteria of the equation are investigated. The presented results conﬁrm that the study of the equation in this formula is superior to other previous studies. Some examples are addressed to demonstrate the ﬁnding.


Introduction
In order to combine continuous and discrete analysis, the theory of dynamic equations on time scales was proposed by Stefan Hilger in [1]. There are different types of time scales applied in many applications (see [2]). The cases when the time scale T as an arbitrary closed subset is equal to the reals or to the integers represent the classical theories of differential and of difference equations. The theory of dynamic equations includes the classical theories for the differential equations and difference equations cases and other cases in between these classical cases. That is, we are eligible to consider the q-difference equations when T =q N 0 := {q k : k ∈ N 0 for q > 1} which has significant applications in quantum theory (see [3]) and different types of time scales like T =hN, T = N 2 and T = T n (the set of the harmonic numbers) can also be applied. For more details of time scales calculus, see [2,4,5]. The study of nonlinear dynamic equations is considered in this work because these equations arise in various real-world problems like the turbulent flow of a polytrophic gas in a porous medium, non-Newtonian fluid theory, and in the study of p−Laplace equations. Therefore, we are interested in the oscillatory behavior of the nonlinear functional dynamic equation of second order with deviating arguments a(ζ)ϕ γ z ∆ (ζ) ∆ + q(ζ)ϕ β (z(η(ζ))) = 0 (1) on an above-unbounded time scale T, where ϕ α (u) := |u| α sgnu, α > 0; a and q are positive rd-continuous functions on T such that and η : T → T is a rd-continuous function such that lim ζ→∞ η(ζ) = ∞.
By a solution of Equation (1) we mean a nontrivial real-valued function z ∈ C 1 rd [ζ z , ∞) T for some ζ z ≥ ζ 0 with ζ 0 ∈ T such that z ∆ , a(ζ)ϕ γ z ∆ (ζ) ∈ C 1 rd [ζ z , ∞) T and z(ζ) satisfies Equation (1) on [ζ z , ∞) T , where C rd is the space of right-dense continuous functions. It should be mentioned that in a particular case when T = R then σ(ζ) = ζ, µ(ζ) = 0, g ∆ (ζ) = g (ζ), b a g(ζ)∆ζ = b a g(ζ)dζ, and (1) turns as the nonlinear functional differential equation a(ζ)ϕ γ z (ζ) + q(ζ)ϕ β (z(η(ζ))) = 0. ( The oscillation properties of Equation (3) and special cases were investigated by Nehari [6], Fite [7], Hille [8], Wong [9], Erbe [10], and Ohriska [11] as follows: The oscillatory behavior of the linear differential equation of second order is investigated in Nehari [6] and showed that if then all solutions of (4) are oscillatory. Fite [7] proved that if then all solutions of Equation (4) are oscillatory. Hille [8] developed the condition (6) and illustrated that if lim inf then all solutions of Equation (4) are oscillatory. For the delay differential equation the Hille-type condition (7) is generalized by Wong [9], where η(ζ) ≥ γζ with 0 < γ < 1, and showed that if lim inf then all solutions of (8) are oscillatory. Erbe [10] enhanced the condition (9) and examined that if then all solutions of (8) are oscillatory where η(ζ) ≤ ζ. Ohriska [11] proved that, if lim sup then all solutions of (8) are oscillatory. When T = Z, then and (1) turns as the nonlinear functional difference equation The oscillation of Equation (12) when a(ζ) = 1, η(ζ) = ζ, and γ = β is the quotient of odd positive integers was elaborated by Thandapani et al. [12] in which q(ζ) is a positive sequence and showed that every solution of (12) is oscillatory, if We will examine that our results not only unite some of the known oscillation results for differential and difference equations but they also can be applied on other cases in which the oscillatory behavior of solutions for these equations on various types of time scales was not known. Note that, if T =hZ, h > 0, then and (1) turns as the nonlinear functional difference equation If where t 0 = q n 0 , and (1) turns as the second order q−nonlinear difference equation If and (1) turns as the second order nonlinear difference equation If T = {H n : n ∈ N 0 } where H n is the harmonic numbers defined by and (1) turns as the second order nonlinear harmonic difference equation For dynamic equations, Erbe et al. in [13,14] expanded the Hille and Nehari oscillation criteria to the half-linear delay dynamic equation of second order where γ is a quotient of odd positive integers, The authors showed that if either of the following conditions holds , then all solutions of (17) are oscillatory. We refer the reader to related results  and the references cited therein. A natural question now is: Do the oscillation criteria (5), (6), (7) and (11) for the differential equations of second order by Nehari, Fite, Hille and Ohriska extend to the nonlinear dynamic equation of second order (1) without the restrictive condition (18) in both cases η(ζ) ≤ ζ and η(ζ) ≥ ζ, and when β ≥ γ and β ≤ γ.
The aim of this paper is to propose an obvious answer to the above question. We will establish Nehari, Hille and Ohriska type oscillation criteria for (1) without imposing the restrictive condition (18), which generalize and improve the aforementioned results in the literature.

Oscillation Criteria of (1) when β ≥ γ
In the subsequent results, we will use the subsequent notations Furthermore, l > 0 is assuming in the next results. First, we derive Nehari type to the nonlinear dynamic equation of second order (1).
Hence z ∆ (ζ) > 0, otherwise, it leads to a contradiction. Define Using the product and quotient rules, we reach From (1) and the definition of w(ζ), we have Therefore, In both cases and from the definition of φ(ζ) we have that and so Then by using the Pötzsche chain rule ([2], Theorem 1.90), we get that and if γ ≥ 1, then Multiplying both sides of (24) by A γ+1 (ζ) and integrating from By integration by parts, we have Using the Pötzsche chain rule, we arrive Hence and where a * is defined by a * := lim inf ζ→∞ A γ (ζ)w (σ(ζ)) ≤ 1.

Dividing both sides by A(ζ), we have
Taking the lim sup of both sides as ζ → ∞ and using (2), we get Since k and ε > 0 are arbitrary constants, we achieve the demanded inequality lim inf From (30) and (33), we obtain lim inf which contradicts the condition (20) if 0 < γ ≤ 1.
(II) When γ ≥ 1. Using the product rule, we have Again by the Pötzsche chain rule we obtain and so

Dividing both sides by A(ζ), we have
Taking the lim sup of both sides as ζ → ∞ and by (2), we obtain Since k, ε > 0 are arbitrary constants, we reach the demanded inequality lim inf From (30) and (34), we get lim inf which is in contrast to the condition (20) if γ ≥ 1. The proof is accomplished.

Dividing both sides by A(ζ), we have
Taking the lim sup of both sides as ζ → ∞ and by (2), we obtain Since k, ε > 0 are arbitrary, we get the required inequality From (30) and (38), we obtain lim inf which is in contrast to the condition (35). The proof is accomplished.
Then every solution of Equation (1) is oscillatory.
From Theorem 3, we assume without loss of generality that Otherwise, we have that (40) holds due to φ(ζ) ≤ 1, which implies that Equation (1) is oscillatory by Theorem 3. The next theorem is generalized Hille type to the second order nonlinear dynamic Equation (1).
Then every solutions of Equation (1) is oscillatory.

Remark 1.
We could refer to the recent results due to [13,14] and others do not apply to Equations (39) and (45). Proof. Assume z (t) is a nonoscillatory solution of Equation (1) on [ζ 0 , ∞) T . Thus, without loss of generality, let z(ζ) > 0 and z(η(ζ)) > 0 on [ζ 0 , ∞) T . Integrating both sides of the dynamic Equation (1) from As shown in the proof of Theorem 1, there exists and From (47) and (48), we obtain Since z ∆ (ζ) > 0, we get that From (49) and (50), we get Taking v → ∞, we have Since k > 0 is arbitrary, we have which gives us the contradiction lim sup The proof of Theorem 5 is accomplished.

Conclusions
(1) In this paper, several Nehari, Hille and Ohriska type oscillation criterion have been given.

Conflicts of Interest:
The authors declare that they have no competing interests. There are not any non-financial