Noncanonical Neutral DDEs of Second-Order: New Sufﬁcient Conditions for Oscillation

: In this paper, new oscillation conditions for the 2nd-order noncanonical neutral differential equation ( a 0 ( t )(( u ( t ) + a 1 ( t ) u ( g 0 ( t ))) (cid:48) ) β ) (cid:48) + a 2 ( t ) u β ( g 1 ( t )) = 0, where t ≥ t 0 , are established. Using Riccati substitution and comparison with an equation of the ﬁrst-order, we obtain criteria that ensure the oscillation of the studied equation. Furthermore, we complement and improve the previous results in the literature. ,

A DDE is a single-variable differential equation, usually called time, in which the derivative of the solution at a certain time is given in terms of the values of the solution at earlier times. Moreover, if the highest-order derivative of the solution appears both with and without delay, then the DDE is called of the neutral type.
The neutral DDEs have many interesting applications in various branches of applied science, as these equations appear in the modeling of many technological phenomena; see [1][2][3][4]. The problem of studying the oscillatory and nonoscillatory properties of DDEs has been a very active area of research in the past few decades, and many well-known and interesting results can be found in Agarwal et al. [5] and Saker [6].
In this work, we study the oscillatory properties of solutions of second-order neutral DDE (1) in the noncanonical case, that is: where Although there are many works that have dealt with the study of the oscillation of this type of equation, in this work, we present a new approach that provides us with improved sufficient conditions for testing the oscillation of the studied equation. Contrary to the previous results, which studied the noncanonical case, our results test the oscillation of (1) when c 0 ≥ 1 along with c 0 < 1.
The paper is organized as follows: Section 2 is concerned with presenting a review of the relevant literature. In Section 3, Section 3.1, we infer some qualitative properties of the positive solutions of (1). In Section 3.2, we use the new properties to obtain improved oscillation conditions. Finally, in Section 4, we summarize the main conclusions extracted from our present work and discuss potential applications and future extensions of this study.

Literature Review
It is easy to note the continuing and growing interest in the study of the oscillatory behavior of DDEs, and improved results, methods, and approaches can be found in [7][8][9][10][11][12]. In more detail, contrary to most previous results, Baculíková [7,8] attained the oscillation of the second-order DDE (not neutral) in the noncanonical case (2) via only one condition. While using an improved approach, Chatzarakis et al. [9] studied the oscillatory behavior of the second-order noncanonical DDE with an advanced argument. On the other hand, Jadlovská et al. [10] and Moaaz et al. [11,12] studied the oscillatory behavior of higher-order equations.

Baculikova and Dzurina
Even though the establishment of the oscillation criteria for (1) in [13,14] and the insertion of the nonstandard Riccati substitution in [15,16] constitute significant progress in the subject of noncanonical neutral DDEs of second-order, the relationship between the corresponding function with delay and without delay is used in the traditional form and has not been improved, and none of these works took into account the case c 0 ≥ 1.
The main goal of our present work is create a better estimate of the ratio (v • g 1 )/v, which contributes to improving the oscillation criteria of (1). Moreover, our results take into account the case c 0 ≥ 1, along with c 0 < 1.

Main Results
We begin with the following notations: U + is the set of all eventually positive solutions of (1), and:

Auxiliary Lemmas
Below, we obtain some asymptotic properties of the positive solutions of (1). First, from the definition of η and the fundamental theorem of calculus, we obtain that Lemma 1. Assume that v ∈ U + and there exists a δ 0 ∈ (0, 1) such that: Then, v eventually satisfies: (C 1 ) v is decreasing and converges to zero; and v η is increasing, and: Proof. Let u ∈ U + . Then, we have that u(t), u(g 0 (t)), and u(g 1 (t)) are positive for t ≥ t 1 , for some t 1 ≥ t 0 . Therefore, it follows from (1) that: v(t) > 0 and V β (t) ≤ 0.
Using (1) and Lemma 1 in [17], we see that: and so: Integrating this inequality from t 1 to t and using the fact V β (t) ≤ 0, we find: (C 1 ) Assume the contrary, that v (t) > 0 for t ≥ t 1 . Thus, from (5), we have: This, from (3), implies: Letting t → ∞ and taking the fact that η(t) → 0 as t → ∞, we obtain V β (t) → −∞, which contradicts the positivity of V(t).
(C 2 ) Since V(t) is decreasing, we obtain: Then, (7), we obtain: Thus, from (4) and the fact V (t) ≤ 0, we obtain: and then: The proof is complete.

Oscillation Theorems
In the next theorem, by using the principle of comparison with an equation of the first-order, we obtain a new criterion for the oscillation of (1). Theorem 6. Assume that g 1 (t) ≤ g 0 (t) and there exists a δ 0 ∈ (0, 1) such that (3) holds. If the delay differential equation: is oscillatory, then every solution of (1) is oscillatory.
Proof. Assume the contrary, that (1) has a solution u ∈ U + . Then, we have that u(t), u(g 0 (t)), and u(g 1 (t)) are positive for t ≥ t 1 , for some t 1 ≥ t 0 . From Lemmas 1 and 2, we have that (C 1 )-(C 4 ) hold for t ≥ t 1 . Next, we define: From (C 1 ), w(t) > 0 for t ≥ t 1 . Thus, Thus, it follows from (C 3 ) that: Using (C 4 ), we obtain that: which with (10) gives: Now, we set: Then, W(t) ≤ c 0 w(g 0 (t)), and so, (11) becomes: which has a positive solution. In view of [20] (Theorem 1), (9) also has a positive solution, which is a contradiction. The proof is complete.
Proof. It follows from Theorem 2 in [21] that the condition (12) implies the oscillation of (9).
Next, by introducing two Riccati substitution, we obtain a new oscillation criterion for (1).

Conclusions
In this work, the oscillatory properties of the solutions of a class of second-order neutral DDEs were studied. Using the Riccati technique and comparison principles, we obtained new criteria that guarantee the oscillation of all solutions of the studied equation.
The new approach, taken in this work, relies on creating a better estimate of the ratio (v • g 1 )/v by establishing the new decreasing function v/η γ 0 δ 0 . This new estimate enables us to obtain new oscillation conditions that directly improve the previous related results. Moreover, our results considered the case where c 0 ≥ 1, which was not taken into account in the previous results.
An interesting issue is obtaining results that take into account all c 0 and do not adhere to the condition g 0 • g 1 = g 1 • g 0 . It is also interesting to extend our results to higher-order equations. It is also interesting to extend the results of this paper to study the oscillatory behavior of some concrete examples that may appear in physics, astronomy, medicine, hydrodynamics, etc.; as an example, see [22].