An improved approach for studying oscillation of generalized Emden–Fowler neutral differential equation

The purpose of this work is to study the oscillation criteria for generalized Emden–Fowler neutral differential equation. We establish new oscillation criteria using both the technique of comparison with first order delay equations and the technique of Riccati transformation. Our new criteria are interesting as they improve, simplify, and complement some results that have been published recently in the literature. Moreover, we present an illustrating example.


Introduction
In aeromechanical systems, where they have a significant role, in the theory of automatic control, in study of vibrating masses attached to an elastic bar (as the Euler equation), in the networks that have lossless transmission lines (as is the case in high-speed computers), and other applications, delay or neutral differential equations can be seen in the modeling of the mentioned phenomena, see [1,2,5,15]. As a result of these applications, research groups including us still study the differential equations with delay. The theory of oscillation of delay differential equations comes at the forefront of topics that have received the attention of researchers in recent times, see . In the last decade, there has been a research movement to improve and develop the oscillation criteria of solutions of second order differential equations with delay (see [9,10]), neutral (see [3,7,13]) and advanced (see [4,13]).
For canonical form (if η(t 0 ) = ∞), there have been some studies that consider the oscillation and nonoscillation criteria of solutions of (1.1), see for example [19,24].
For noncanonical form (if η(t 0 ) < ∞), Liu et al. [18] got necessary and appropriate conditions that ensure all solutions of (1.1) can be oscillatory, or they can tend to zero, following the conditions lim t→∞ p(t) = C, p (t) ≥ 0 and τ (t) ≥ 0.
(1.2) Furthermore, Saker [23] developed the results of [18] in the sense that they established the conditions that assure all the solutions of Eq. (1.1) are oscillatory. The results of both [23] as well as [18] follow an approach that does lead to two conditions, and they are requested (1.2).
Wu et al. [28] established some criteria of oscillation for the neutral equation This work aims at developing the oscillation theory of second order quasi-linear equations with delay argument. The use of the technique of comparison with first order delay equations and the technique of Riccati transformation helps us to get two various conditions, ensuring oscillation of (1.1) without requiring (1.2). In this paper, in the first two theorems, we simplify results in [18,23,28] and obtain new criteria for ensuring oscillation of (1.1) without checking the additional conditions. Our criteria complement and extend the results in [7,8]. In [ In our paper, Theorems 2.5 and 2.6 substantially improve Theorem 2.2 in [7, Theorem 2.2], when W ≤ 1. The next lemma collects two useful inequalities that can be found in [29]. (1.5) and

Main results
In this section, we shall establish new oscillation criteria for (1.1). Let us define where t 1 ∈ [t 0 , ∞) and a 1 , a 2 are any positive constants.

Lemma 2.1
Assume that u is an eventually nonincreasing positive solution of (1.1). Then Proof Let υ be an eventually positive solution of (1.1) and υ (t) < 0. Then we have the following cases: In the case where α = β, it is easy to see that υ β-α (t) = 1. Let α > β. Since υ(t) is a nonincreasing positive function, there exists M 1 > 0 such that υ(t) ≤ M 1 , which implies that In the case α < β, by using the decreasing property of r(υ ) α , we obtain Integrating the last inequality from t to ∞, we get Thus, we include that The proof of the lemma is complete.

Lemma 2.3 Let u be a positive solution of
for t ≥ t 2 . Integrating two times this inequality from t 2 to t, we get, after the first integration, This implies that lim t→∞ υ(t) = -∞, which contradicts υ > 0. The proof of this lemma is complete.
Proof Let u be a positive solution of (1.1) on [t 0 , ∞) (assume the converse). Then we suppose that there exists Since π(t 0 ) < ∞ and (2.7), we have that t t 1 Q(ν)π β (σ (ν)) dν must be unbounded. Thus, and from the fact π (t) < 0, it is easy to see that (2.1) holds. Hence, from Lemma 2.2, we have that υ (t) < 0 and (2.2) holds. Since it follows that d dt In view of the definition of υ, we deduce Consequently, (2.2) becomes From the monotonicity property of r(t)(υ (t)) α , we have which in view of (2.8) implies Integrating (2.11) from t 1 to t, we obtain Integrating (2.12) from t 1 to t and using (2.7), we get which in view of (2.7) contradicts the positivity of υ(t). The proof of the theorem is complete.

Theorem 2.3
Assume that (2.1) holds. If the first order delay differential equation is oscillatory, then (1.1) is oscillatory.
Proof To the contrary, we suppose that u is a positive solution of (1.1) on [t 0 , ∞). Then there exists t 1 ≥ t 0 such that u(τ (t)) > 0 and u(σ (t)) > 0 for all t ≥ t 1 . Using (2.1) and Lemma 2.2, we get that υ < 0 on [t 1 , ∞). As in the proof of Theorem 2.2, we get (2.14) holds. From (2.14), it is clear that υ is a positive solution of the first order differential inequality In view of [25, Lemma 1], we see that the first-order delay differential equation (2.16) has a positive solution, a contradiction. Then the proof is complete.
is necessary for the validity of (2.1). Therefore, the proof is complete.  [12] and [17]. On the other hand, we see that (2.18) is necessary for the validity of (2.1). Therefore, the proof is complete.
For the simplicity, we define the following notations: Proof Suppose against the assumption of theorem that equation (1.1) has a nonoscillatory solution u on [t 0 , ∞). Without loss of generality, we may assume that u(t) > 0 and u(σ (t)) > 0 for t ≥ t 1 ≥ t 0 . Let Combining (2.22) and (2.23), and using inequality (2.9), we get Integrating (2.24) from t to ∞, we get It follows that and so By virtue of definition of m, we see that this contradicts the positivity of υ. Assume next that the case m ≤ α holds. Proceeding as in the proof of Theorem 2.2, we get (2.14). Thus, by (2.25), we get Proof Assume that u is a positive solution of (1.1) on [T, ∞). By Lemma 2.3, u(t) satisfies (H) and (2.6). Proceeding as in the proof of Theorem 2.2, we have (2.14) holds. Now, we see that d dt . (2.27) In view of (2.14), we note that Therefore, (2.27) becomes d dt Then the proof is complete.
In view of (2.8), we obtain It follows that Taking the lim sup on both sides, we obtain a contradiction. Then the proof is complete.
Taking lim sup on both sides, we obtain Therefore, Then the proof is complete.
In the next theorems, by using a generalized Riccati substitution, we establish new oscillation criteria of (1.1).
Proof To the contrary, we suppose that u is a positive solution of (1.1) on [t 0 , ∞). Thus, there exists t 1 ≥ t 0 such that u(τ (t)) > 0 and u(σ (t)) > 0 for all t ≥ t 1 . Then we get that υ has one sign eventually. Now, we let υ (t) < 0 for t ≥ t 1 . As in the proof of Theorem 2.1, we get (2.9) holds. Define the function ω(t) by .