Third-Order Neutral Delay Differential Equations: New Iterative Criteria for Oscillation

This study is aimed at developing new criteria of the iterative nature to test the oscillation of neutral delay differential equations of third order. First, we obtain a new criterion for the nonexistence of the so-called Kneser solutions, using an iterative technique. Further, we use several methods to obtain different criteria, so that a larger area of the models can be covered. The examples provided strongly support the importance of the new results.

If the solution x is either ultimately positive or ultimately negative, then x is called nonoscillatory; otherwise, it is called an oscillatory solution. The equation itself is termed oscillatory if all its solutions oscillate. Solutions x whose corresponding function z satisfies zðlÞz′ðlÞ < 0 are called Kneser solutions. We denote the class of all Kneser solutions of (1) with the symbol X K . Otherwise, X N denote to the class of all positive solutions of (1) whose z satisfies zðlÞz′ðlÞ > 0.
Delay differential equations as a subclass of functional differential equations take into account the dependence on the system's past history, which results in predicting the future in a more reliable and efficient way. Neutral delay differential equations arise in various phenomena including problems concerning electric networks containing lossless transmission lines (as in high-speed computers where such lines are used to interconnect switching circuits), in the study of vibrating masses attached to an elastic bar or in the solution of variational problems with time delays, or in the theory of automatic control and in neuromechanical systems in which inertia plays a major role, see [1][2][3][4][5][6].
At the beginning of any study of the oscillatory properties of solutions of differential equations, it is easy to notice the importance of classifying signs of derivatives of nonoscillatory solutions. For the positive solutions, based on the canonical condition Ð ∞ l 0 r −1/α ðsÞds ⟶ ∞ as l ⟶ ∞, it follows from ( [21], Lemma 1) that ðr 1 z ′ Þ ′ > 0 and there are two possible cases for z ′ ðlÞ: either z ′ ðlÞ > 0 or z ′ ðlÞ < 0. By creating conditions that ensure X N and X K are empty sets, we can directly set the criteria for oscillation. There are numerous results interested in finding conditions that ensure class X N is empty, which include Hille and Nehari types and Philos type. Baculikova and Dzurina [7] established a condition of Hille and Nehari type and proved that if where p 0 < 1 and r ′ ðlÞ > 0, then X N = ∅. By comparison principles, Baculikova and Dzurina [8] proved that if the first-order DDE is oscillatory, then X N = ∅. We can easily notice that the delay argument τðlÞ has been neglected in (2) and (3). Otherwise, by using the Riccati transformation, Thandapani and Li [16] guaranteed that class X N is empty if where σ′ðlÞ > 0, τ′ðlÞ ≥ τ 0 > 0,QðlÞ ≔ min fqðlÞ, qðτðlÞÞg, and ρ ∈ Cð½l 0 ,∞Þ, ð0,∞ÞÞ. All previous results focused on the class X N only and proved that every solution that belongs to X K tends to zero. On the other hand, by establishing conditions for the nonexistence of Kneser solutions (X K = ∅), Dzurina et al. [12] attained the oscillation of all solutions of (1) in the linear case α = 1. They proved that if (4) and hold, then equation (1) is oscillatory, where ρ ∈ Cð½l 0 ,∞Þ, ð0 ,∞ÞÞ satisfying σ < ρ < τ. One purpose of this study is to further complement and improve the well-known results reported in the literature. In Section 2, by using an iterative technique, we get analogous iterative estimates for Kneser solutions of (1). These iterative estimates enable us to establish new criteria that ensure the nonexistence of Kneser solutions. Further, criteria of an iterative nature help check the oscillation, even when the other criteria fail to apply. In Section 3, we use the Riccati transformation method and comparison principles to obtain different criteria which guarantee that X N = ∅. Examples illustrating the new results are also given.
For the sake of ease and assistance in presenting the main results, we provide the following notations and lemmas: Lemma 1 (see [7], Lemma 1). All eventually positive solutions x of (1) have the following properties: ðPÞz and z ″ are positive, z ′ is of fixed sign, and rðlÞ ðz″ðlÞÞ α is nonincreasing, for l large enough.

Main Results 1: Iterative Technique
Lemma 3. Assume that x belongs to X K and there is a function ρ ∈ CðΤ l 0 , ð0, ∞ÞÞ with the property Proof. Suppose x belongs to X K . Thus, there is a l 1 ≥ l 0 satisfying xðlÞ, and xðσðlÞÞ and xðτðlÞÞ are positive for l ≥ l 1 . As a direct result of Lemma 1, x achieves property ðPÞ.
2 Journal of Function Spaces Using induction, we will prove the iterative relationship (8).

Journal of Function Spaces
Integrating (24) twice over ½u, vÞ, we get This completes the proof.
Proof. Suppose x belongs to X K . As a direct result of Lemma 2, we get that (8) holds. By following the same approach as in proof of Lemma 2, we get the relationships from (14) to (21). Now, assume H is defined as in (18). From Lemma 1, we have that z ″ ðlÞ > 0, and hence, HðlÞ > 0, for l ∈ Τ l 1 . Then, the delay inequality (21) has a positive solution. From Theorem 1 in [22], the associated equation of (21) is has also a positive solution. However, it is well known from ( [23], Theorem 2) that (34) implies oscillation of (27), a contradiction. This completes the proof.
Proof. Assume that x belongs to X N on ½l 1 , ∞Þ. Then, it follows from Lemma 1 that there is a l 1 ≥ l 0 , such that zðlÞ, z ′ ðlÞ, and z ″ ðlÞ are positive and z ″ is nonpositive, for all l ≥ l 1 . It is easy to conclude that which, with the fact that z″ ≤ 0, gives Now, we define From (39), we get Thus, we have that F is an increasing function with Fðl 1 Þ = 0, and so, FðlÞ > 0 for all l ≥ l 1 . Therefore, from the definition of F, we get the required directly.