On the Oscillation of Solutions of Differential Equations with Neutral Term

: In this work, new criteria for the oscillatory behavior of even-order delay differential equations with neutral term are established by comparison technique, Riccati transformation and integral averaging method. The presented results essentially extend and simplify known conditions in the literature. To prove the validity of our results, we give some examples.


Introduction
Neutral/delay differential equations are used in a variety of problems in economics, biology, medicine, engineering and physics, including lossless transmission lines, vibration of bridges, as well as vibrational motion in flight, and as the Euler equation in some variational problems, see [1][2][3].
Nowadays, there is an ongoing interest in obtaining several sufficient conditions for the oscillatory properties of the solutions of different kinds of differential equations, especially their the oscillation and asymptotic, see Agarwal et al. [4] and Saker [5].
In [26], Zhang et al. studied the equation γ −1/α (s)ds < ∞ and they used comparison and Riccati techniques. In case γ(t) = 1 and α = 1, the authors in [27,28] studied the oscillatory properties for equation where r is an even and under the condition 0 ≤ h(t) < 1.
Agarwal et al. [31] studied the oscillation conditions of the equation where α > 1. The authors used comparison method to find this conditions. Elabbasy et al. [32] were interested in discussing the oscillatory properties of the equation and r is an even positive integer. Based on the above results of previous scholars, in this work, we are concerned with the following differential equations with neutral term of the form where j ≥ 1, and Throughout this work, we suppose the following hypotheses: .., j; r and p are positive integers, r is even, r ≥ 2, p > 1.

Definition 2.
A solution of (2) is said to be non-oscillatory if it is positive or negative, ultimately; otherwise, it is said to be oscillatory.
The motivation for this article is to continue the previous works [33]. The authors in [34] used the comparison technique that differs from the one we used in this article. Our approach is based on using integral averaging method and the Riccati technique to reduce the main equation into a first-order inequality to obtain more effective oscillation conditions for Equation (2). Therefore, in order to highlight the novelty of the results that we obtained in this work, we presented a comparison between the previous results and our main results, represented in the Example 2.
Motivated by these reasons mentioned above, in this paper, we extend the results using integral averaging method and Riccati transformation under These results contribute to adding some important conditions that were previously studied in the subject of oscillation of differential equations with neutral term. To prove our main results, we give some examples.

Oscillation Results
Now, we mention some important lemmas.
for all large t.

Lemma 4.
Assume that x(t) is a positive solution of Equation (2). Then Proof. Suppose that x(t) is a positive solution of Equation (2). Then, we can assume that x(t) > 0, x(β(t)) > 0 and x(w(t)) > 0 for t ≥ t 1 . Hence, we deduce z(t) > 0 and Which means that γ(t)z (r−1) (t) is decreasing and z (r−1) (t) is eventually of one sign. We see that z (r−1) (t) > 0. Otherwise, if there exists a t 2 ≥ t 1 such that z (r−1) (t) < 0 for t ≥ t 2 , and Integrating (7) from t 2 to t we find So, we get Letting t → ∞, we have lim t→∞ z (r−2) (t) = −∞, which contradicts the fact that z(t) is a positive solution by Lemma 1. Hence, we obtain z (r−1) (t) ≥ 0 for t ≥ t 1 .
By using Theorem 2.1.1 in [35], we get the following corollary.
for ε ∈ (0, 1), then (2) is oscillatory, where Proof. Assume on the contrary that (2) has a nonoscillatory, say positive solution x. From Lemma 2 with x = z , there exists a > 0 and w i (t) ≤ t such that Defining we have From (12), we obtain By using (14), we have Using the inequality Integrating (16) from t 1 to t we find which contradicts (13). Theorem 2 is proved.

Definition 3.
Let A function G ∈ C(D, R) is said to belong to the function class ψ, written by G ∈ ψ, if (i) G(t, s) > 0 on D 0 and G(t, s) = 0 for t ≥ t 0 with (t, s) / ∈ D 0 ; (ii) G(t, s) has a continuous and nonpositive partial derivative ∂G/∂s on D 0 and g ∈ C(D 0 , R) such that ∂G(t, s) ∂s = −g(t, s) G(t, s).
Thus, we see that ∞ 1 If we set G(t, s) = (t − s) 2 , g(t, s) = 2 and = 1, then So, it can be easily verified that Using Theorem 3, Equation (19) is oscillatory.

Remark 1.
The results of [33] cannot solve (19) because of γ(t) = t. Thus, our results extend and complement upon the results of previous papers on this topic.

Conclusions
In this work, a large amount of attention has been focused on the oscillation problem of Equation (2). By Riccati transformation, comparison technique and integral averages method, we establish some new oscillation conditions. These results contribute to adding some important criteria that were previously studied in the literature. For future consideration, it will be of a great importance to study the oscillation of γ(t) z (r−1) (t) p−2 z (r−1) (t) + a(t)ϕ(x(β(t))) = 0, under the assumption that where z(t) = |x(t)| p−2 x(t) + h(t)x(β(t)) and p > 1 is a constant.