OSCILLATION OF SECOND ORDER NONLINEAR DYNAMIC EQUATIONS WITH A NONLINEAR NEUTRAL TERM ON TIME SCALES ∗

In this article, we consider the oscillation of second order nonlinear dynamic equations with a nonlinear neutral term on time scales. Some new sufficient conditions which insure that any solution of the equation oscillates are established by means of an inequality technique and Riccati transformation. This paper improves and generalizes some known results. Several illustrative examples are given throughout.


Introduction
The theory of the calculus on time scales, which has recently received a lot of attention, was introduced by Stefan Hilger in his PhD thesis in 1988 [5] in order to unify continuous and discrete analysis.In recent years, there has been an increasing interest in obtaining sufficient conditions for oscillatory behavior of different classes of dynamic equations on time scales, we refer the reader to [2,7,8] and the references cited therein.
The neutral functional differential equation arises in the design of high-speed computer lossless transmission lines.It also finds wide applications in certain hightech fields, such as control, communication, mechanical engineering, biomedicine, physics, mechanics, economics and so on.Also the neutral dynamic equation having a nonlinearity in the neutral term arises in many applications.
S. H. Saker [9] studied the oscillation of the second-order nonlinear neutral delay dynamic equation on time scales and presented some necessary and sufficient conditions for oscillation.
Motivated by the above articles, in this article, we consider the following nonlinear dynamic equation where I = [t 0 , ∞) T and 0 < α ≤ 1 is the ratio of two odd positive integers, and σ(t) is a jump operator on time scales, σ(t) ≥ t.Assume that the following conditions are satisfied: A function x is called a solution of (1.1), and Let T be a time scale with sup T = ∞.We restrict our attention to those solutions of (1.1) which exist on I = [t 0 , ∞) T and satisfy the condition As usual, a solution x of (1.1) is said to be oscillatory, if it is neither eventually positive nor eventually negative.Otherwise, it is called nonoscillatory.The equation (1.1) is called oscillatory, if all its solutions are oscillatory.Otherwise, it is called nonoscillatory.
Throughout this paper we assume that And we investigate the oscillatory behavior of (1.1) under the condition (1.2).By using a Riccati transformation and an inequality technique, we present some new sufficient conditions which ensure that any solution of (1.1) oscillates.Specifically, we study neutral dynamic equation and the constants α and β are independent with each other.

Preliminaries
In this section, we will present some necessary background.Without loss of generality, we can only deal with the positive solutions of equation (1.1) since the proof of the other case is similar.For the sake of simplicity, we define ∆s.
Next, we state and prove the following lemmas.
Lemma 2.2.Suppose that (2.1) of Lemma 2.1 holds.If x(t) is an eventually positive solution of equation (1.1), then there exists a T ≥ t 1 such that where Proof.By the definition of z(t) and (H1), we get From (2.1), we see that z(t) is positive and increasing, since A(t) is positive, decreasing and tends to zero as t → ∞, there exists a T ≥ t 1 such that By (2.5) and (2.6), we have This completes the proof.

Oscillation Results
We are now in a position to state and prove our main results in this paper.Theorem 3.1.Assume that (1.2) holds, β ≥ 1 and max{p 1 (r(t)), p 2 (r(t))} < 1, t ∈ I, where p 1 (t) and p 2 (t) are defined as in Lemma 2.1 and Lemma 2.2.Suppose that the condition r(σ(t)) = σ(r(t)) holds.If there exists a positive, nondecreasing and ∆-differentiable function b(t) such that lim sup or lim sup where K > 0 and M > 0 are any positive constants, then every solution of (1.1) is oscillatory.
Proof.By contradiction, suppose that (1.1) is nonoscillatory and x(t) is a nonoscillatory solution for (1.1).Without loss of generalization, assume that x(t) is eventually positive, and there exists a t 1 ≥ t 0 such that x(t) > 0, x(m(t)) > 0 and x(r(t)) > 0 for any t ≥ t 1 .By the definition of z(t), we have where T is chosen so that (2.1) and (2.2) of Lemma 2.1 hold for all t ≥ T .We shall show that in each case we are led to a contradiction.
(3.8) Then by completing the square in (3.8), we get (3.9) Integrating (3.9) from T to t, we obtain Case (2): Suppose that (2.2) of Lemma 2.1 holds, by (2.3) and (2.7), we obtain Define the following Riccati transformation: Then u(t) ≤ 0, t ≥ T .By (2.9), we have So by −a(t)z ∆ (t) > 0 and (3.11), we obtain where L = −a(T )z ∆ (T ).Differentiating (3.11), we get By the corollary of the Keller chain rule [4] and β ≥ 1 we have By (2.10) and −a(t)z ∆ (t) is positive and increasing, there exists a constant K > 0 such that Using (3.17) in (3.16), we obtain Multiplying (3.18) by A β (t) and integrating the resulting inequality from T to t, we have By using (3.13), we get This contradicts (3.2) and the proof is complete.Here T = R + , and a(t It is easy to verify that all conditions of Theorem 3.  where K > 0 is any positive constant, then every solution of (1.1) is oscillatory.
Proof.Proceeding as in the proof of Theorem 3.1, we see that two cases of Lemma 2.1 hold.