Oscillation results of third-order nonlinear dynamic equations with damping on time scales

In this paper, we study the oscillatory criteria of solutions for third-order dynamic equations with damping and obtain some suﬃcient conditions by using the generalized Riccati transformation. We extend and improve some well-known existing results. We also provide an example for illustrating our main result.


Introduction
The study of oscillatory behavior of nonlinear damped differential equations has become a well-researched area because of the fact that such equations appear in many real-life problems; for example, see [5,8,10-12, 14,17,18,21-23,25-30] and the references cited therein.
Hilger [7] introduced the time scales theory which unified the representation of continuous analysis and discrete analysis; see also [2,3]. The past decade has witnessed the tremendous development of time scale theory in many fields. This theory has received a large amount of attention and studies. In general, we cannot obtain analytical solutions of higher order dynamic equations, so the oscillation and asymptotic behavior of solutions is what we often focus on. Very recently, there have been many researches regarding the oscillation criteria for solutions of dynamic equations on any time scales such as [4, 6, 9, 13-15, 15, 16, 18, 20-22, 24, 25, 28] and the references therein.
Aktaş et al. [1] gave some new oscillation results for the following difference equation: (c n (d n z n )) + p n z n+1 + q n f (z n−σ ) = 0, n ≥ n 0 , where n 0 ∈ N is a fixed integer and σ is a nonnegative integer. By a solution for dynamic equations on time scales, we mean a nontrivial real valued function satisfying the dynamic equation on any time scales T. If a solution of the dynamic equation is neither eventually negative nor eventually positive, it is called oscillatory. Or else, the solution is called non-oscillatory. If all of the solutions of dynamic equation are oscillatory, then we say that the dynamic equation is oscillatory.
Ş enel [21] studied the oscillation behavior of the following second-order dynamic equations: on a time scale T. Hassan [6] obtained some oscillation results for the equation: Afterwards, Erbe et al. [4] gave some new results for this problem. In [16], Qiu explored the following third-order damped dynamic equation: on any time scale T such that inf T = t 0 and sup T = ∞. * Corresponding author (f.serap.topal@ege.edu.tr).
The aim of the present study is to give an oscillation behavior for the following third-order nonlinear damped dynamic equation with delay term on any time scale T: (1) Throughout this study, we assume that the following assumptions hold: (C 2 ). g(t) ∈ C(T, T) and for all t ∈ T; and there exists a function q(t) ∈ C rd (T, (0, ∞)) such that for u, v and w with a same sign; satisfies whenever γ ∈ (0, 1); (C 6 ). γ is a quotient of odd positive integers.
The remaining part of this paper is arranged as follows. In Section 2, we present some lemmas that are required to prove the main result. In Section 3, we give the main oscillation result of the problem (1) and provide an example for illustrating it.

Preliminaries
In this section, we give some lemmas for establishing the oscillation result for the problem (1). Lemma 2.1. Suppose that the conditions (C 1 )-(C 6 ) (given in the previous section) hold. Also, suppose that there exists a sufficiently large Proof. Let t 1 ∈ [t 0 , ∞) T and x(t) be a solution of (1) satisfying x(t) > 0 for t ∈ [t 1 , ∞) T . Then, we have x(σ(t)) and x(g(t)) > 0. Using (1) and (C 3 ), we have we can say that r 2 (t)(x ∆ (t)) γ ) ∆ is eventually of one sign. We want to show that the inequality Then By using the fact that and hence .
Then either lim t→∞ x(t) = 0 or there exists a sufficiently large t 4 such that Proof. Using Lemma 2.1, we conclude that x ∆ (t) is eventually of one sign. Thus there exists a sufficiently large t 4 such that at least one of the inequalities x ∆ (t) < 0 and x ∆ (t) > 0 is satisfied on the interval [t 4 , ∞) T . We suppose that x ∆ (t) < 0, by x(t) is a positive solution of (1) on [t 0 , ∞) T , we get lim t→∞ x(t) = α ≥ 0 and lim t→∞ r 2 (t)(x ∆ (t)) γ = β ≤ 0. We assert that ∆s.
Thus, we get By integrating (5) from t to ∞, we get Again, by integrating both sides of (6) from t 5 to t, we have ∆s ∆τ ∆ξ.

q(s)∆s
and hence

Main result
In this section, we give a new oscillation criteria for the third-order nonlinear dynamic equation (1) with damping term by using the inequality technique and the generalized Riccati transformation. where Then, every solution of (1) is either oscillatory on [t 1 , ∞) T or satisfies lim t→∞ x(t) exists.