Oscillatory behavior of second-order nonlinear neutral differential equations

We shall consider a class of second-order nonlinear neutral differential equations. Some new oscillation criteria are established by using the Riccati transformation technique. One example is given to show the applicability of the main results.

and x(t) is an eventually positive solution of Eq. (1). Then z(t) has the following two possible cases: Proof Since x(t) is an eventually positive solution of (1), there exists a t 1 ≥ t 0 such that hence r(t)(z (t)) α is decreasing function and of one sign, therefore z (t) is also of one sign, that is, there exists a t 2 ≥ t 1 such that, for t ≥ t 2 , z (t) > 0 or z (t) < 0. If z (t) > 0, we have (i) or (ii). Now, we prove that z (t) < 0 will not happen. If z (t) < 0, we have Integrating this inequality from t 2 to t, we have by condition (2), lim t→∞ z(t) = -∞. We will consider the following two cases.
We get Hence, z(t) satisfies one of the cases (i) and (ii).

Lemma 2.2 Assume that x(t) is a positive solution of Eq. (1) and z(t) satisfies case
Thus, we conclude that (1) and

Lemma 2.3 Assume that x(t) is an eventually positive solution of
Then the impossibility for z(t) satisfies case (ii) of Lemma 2.1.
Proof Assume that z(t) satisfies case (ii) of Lemma 2.1, we have That is, We deduce that From (1) and (H 4 ), we have We get Integrating this inequality from s to t, we conclude that That is, Integrating this inequality from τ -1 (σ (t)) to t, we get Since z(t) < 0, we have

Theorem 3.1 Assume that (2) and (3) be satisfied. If there exists a positive function ρ
Proof Assume that x(t) > 0. From Lemma 2.1, z(t) satisfies one of the cases (i) and (ii).
Case (i). Suppose that case (i) holds, from Lemma 2.2, we have That is, We get That is, From (1), we conclude that Then we have That is, where We define a function w(t) of the generalized Riccati transformation by Then
Example Consider the following equation: x(t)px(t -1)