Oscillation Criteria for Even Order Nonlinear Neutral Differential Equations

In this paper, we consider the oscillation criteria for even order nonlinear neutral differential equations of the form " r(t)z (n−1) (t) " ′ + q(t)f (x(σ(t))) = 0, where z(t) = x(t) + p(t)x(τ (t)), n ≥ 2 is a even integer. The results are obtained both for the case R ∞ r −1 (t)dt = ∞, and in case R ∞ r −1 (t)dt < ∞. These criteria here derived extend and improve some known results in literatures. Some examples are given to illustrate our main results.

For the particular case when n = 2, (1.1) reduces to the following equations Han et al. [9] studied the oscillation criteria for the solutions of (1.4), where In 2011, Baculíková and Dzurina [13] studied the oscillatory behavior of the solutions of the second order neutral differential equations where Basing on the new comparison principles, the authors obtained some sufficient conditions for the oscillation of (1.5), which reduce the problem of the oscillation of the second order differential equations to the oscillation of a first order differential inequality.In this paper, Theorem 1 is quite general, since usual restrictions on the coefficients of (1.5), like τ (t) ≤ t, σ(t) ≤ τ (t), σ(t) ≤ t, 0 ≤ p(t) < 1, etc. are not assumed.Further, τ could be a delay or advanced argument, and σ could be a delay argument, hence the results obtained here improved and extended some known results in literature, such as [1,5,7].
Zhang et al. [26] studied the even-order nonlinear neutral functional differential equations where n is even, 0 ≤ p(t) < 1 and τ (t) ≤ t.The authors established a comparison theorem for (1.6) and the obtained results improved and generalized some known results.Using the Riccati transformation technique, Li et al. [25] obtained some new oscillation criteria for (1.6), when 0 ≤ p(t) ≤ p 0 < ∞.These oscillation criteria, at least in some sense, complemented and improved those of Zafer [20] and Zhang et al. [26].
In 2011, Zhang et al. [28] studied the oscillatory behavior of the following higher-order halflinear delay differential equation under the condition The authors obtained some sufficient conditions, which guarantee that every solution of (1.7) is oscillatory or tends to zero.Clearly, the above equations are special cases of (1.1).To the best of our knowledge, there are few results regarding the oscillation criteria for (1.1) under the condition (1.3).The purpose of this paper is to derive some oscillation theorems of (1.1).Our results obtained here improve and extend the main results of [9-11, 13, 20, 23, 25, 26].EJQTDE, 2012 No. 30, p. 2 In this section, we present some useful lemmas, which will be used in the proofs of our main results.
Lemma 2.2 [19] Let u be as in Lemma 2.1.Assume that u (n) (t) is not identically zero on any interval [t 0 , ∞), and there exists a t 1 ≥ t 0 such that u (n−1) (t)u (n) (t) ≤ 0 for all t ≥ t 1 .If lim t→∞ u(t) = 0, then for every λ, 0 < λ < 1, there exists T ≥ t 1 , such that for all t ≥ T, Lemma 2.3 Assume that (1.2) holds.Furthermore, assume that x is an eventually positive solution of (1.1).Then there exists t 1 ≥ t 0 , such that The proof is similar to that of Meng and Xu [24, Lemma 2.3], so is omitted.

Main results
In this section, we state the main results which guarantee that every solution of (1.1) is oscillatory.
Remark 3.1 Recently, when studying the properties of the neutral differential equations, there are many further restrictions on the coefficients, such as τ (t) ≤ t, σ(t) ≤ τ (t), 0 ≤ p(t) < 1, etc.In Theorem 3.1 no such constraints are assumed, and therefore our results are of high generality.
Proof.Suppose, on the contrary, x is a nonoscillatory solution of (1.1).Without loss of generality, we may assume that there exists a constant t 1 ≥ t 0 , such that x(t) > 0, x(τ (t)) > 0 and x(σ(t)) > 0 for all t ≥ t 1 .Proceeding as in the proof of Theorem 3.1, we have (3.2).By Lemma 2.2 and (3.2), for every λ, 0 < λ < 1, we obtain for every t sufficiently large.Let u(t) = r(t)z (n−1) (t) > 0. Then for all t large enough, we have Therefore, ω is a positive solution of (3.8).Now, we consider the following two cases, depending on whether (3.4) But according to Lemma 2.4, (3.9) guarantees that (3.8) has no positive solution, which is a contradiction.Case (II): Using the definition of ω and (3.2), we obtain Thus From the above inequality, we obtain r(σ(s)) ds ≤ 0.
Hence from (3.11), we have (3.12) Taking the upper limit as t → ∞ in (3.12), we get lim sup t→∞ t σ(t) which is in contradiction with (3.13).This completes the proof. ) where Q is defined as in Theorem 3.2, then every solution of (1.1) is oscillatory.
EJQTDE, 2012 No. 30, p. 5 Proof.Suppose, on the contrary, x is a nonoscillatory solution of (1.1).Without loss of generality, we may assume that there exists a constant t 1 ≥ t 0 , such that x(t) > 0, x(τ (t)) > 0 and x(σ(t)) > 0 for all t ≥ t 1 .Proceeding as in the proof of Theorem 3.2, we have (3.6) Therefore, ω is a positive solution of (3.17 Thus The rest of the proof is similar to that of Theorem 3.2, leading to a contradiction to (3.15), so it can be omitted.This completes the proof.
Proof.Suppose, on the contrary, x is a nonoscillatory solution of (1.1).Without loss of generality, we may assume that there exists a constant t 1 ≥ t 0 , such that x(t) > 0, x(τ (t)) > 0 and x(σ(t)) > 0 for all t ≥ t 1 .Proceeding as in the proof of Theorem 3.1, we can see that r(t)z (n−1) (t) is a decreasing function.Consequently it is easy to conclude that there exist two possible cases of the sign of z (n−1) (t), that is, z (n−1) (t) is either eventually positive or eventually negative for t ≥ t 2 ≥ t 1 .
Case (I): z (n−1) (t) > 0, t ≥ t 2 .The proof of this case is similar to that of Theorem 3.3, so we omit the details.
where Q is defined as in Theorem 3.2, 0 < λ 0 < 1 is a constant and δ is defined as in Theorem 3.4, then every solution of (1.1) is oscillatory.
Proof.Suppose, on the contrary, x is a nonoscillatory solution of (1.1).Without loss of generality, we may assume that there exists a constant t 1 ≥ t 0 , such that x(t) > 0, x(τ (t)) > 0 and x(σ(t)) > 0 for all t ≥ t 1 .Proceeding as in the proof of Theorem 3.1, we can see that r(t)z (n−1) (t) is a decreasing function.Consequently it is easy to conclude that there exist two possible cases of the sign of z (n−1) (t), that is, z (n−1) (t) is either eventually positive or eventually negative for t ≥ t 2 ≥ t 1 .
Case (I): z (n−1) (t) > 0, t ≥ t 2 .The proof of this case is similar to that of Theorem 3.2, so we omit the details.

Examples
In this section, we will show the application of our main results.
Example 4.1 Consider the even order nonlinear neutral differential equations EJQTDE, 2012 No. 30, p. 9 Here respectively, which guarantees that every solution of (4.1) is oscillatory.Consequently, for all α > 0, we cover the oscillation criteria for (4.1) whether τ (t) = αt is delay or advanced argument.When n = 2, (4.1) becomes (E 5 ) in [13], and the conditions (4.2) and (4.3) reduce to the inequalities in Example 1 in [13].So our results contain the main results in [13].

Lemma 2 . 4 [ 18 ,
Theorem 2.1.1]Consider the oscillatory behavior of solutions of the following linear differential inequality y