**1. Introduction**

Delay differential equations (DDE) are differential equations (DE) that take into account the effect of different times. Therefore, they are a better way to model natural phenomena in engineering and physical problems. It is easy to note the recent increase in research into the qualitative theory of DDEs. This is not only due to their practical importance, but also because they are rich in analytical problems and interesting open issues.

One of the most important branches of qualitative theory is oscillation theory, which studies the asymptotic and oscillatory behavior of solutions of DEs. Finding adequate conditions to guarantee that all DE solutions oscillate is one of the main objectives of oscillation theory. Ladas et al. [1] is one of the earliest monographs addressing oscillation theory, including the findings up until 1984. The main objective of this monograph is to investigate the deviating arguments on the oscillation of solutions; neutral delay equations are not discussed in this monograph. The monograph by Gyori and Ladas [2], which made significant contributions to the creation of linearized oscillation theory and the relationship between the oscillation of all solutions and the distribution of the roots of characteristic equations, is one of the key works in the theory of oscillation. Additional topics that are crucial to the theory of oscillation are covered in [3], including determining the conditions for the existence of solutions with particular asymptotic properties and calculating the separation between zeros of oscillatory solutions. The monographs [4–9] covered and summarized many of the results known in the literature up to the past ten years for further results, approaches, and references.

In addition to the theoretical importance and many interesting analytical problems, delay differential equations have many vital applications in engineering and physics, as

they appear when modeling many phenomena that are fundamentally time dependent. For example, we find that such equations appear in the modeling of electrical networks that contain lossless transmission lines (such as high-speed computers). Understanding the qualitative properties and behavior of equation solutions greatly helps in studying and developing the studied models.

It is easy to notice the research movement that aims to improve and develop the criteria for oscillation of solutions of DDEs, especially of the second-order, which is led by the Slovakian school; see, for example, the papers of Baculíková, Džurina and Jadlovská [10–13].

Mostly, we find that the study of the oscillation of solutions of DDEs of different orders adopts one of two approaches, either substituting Riccati or comparing with equations of lower orders, often first-order. In 1999, Koplatadze et al. [14], with a different approach than the traditional one, studied the asymptotic and oscillatory behavior of solutions to the DE

$$\frac{\mathbf{d}^n \mathbf{x}}{\mathbf{d} \mathbf{u}^n} + q \cdot [\mathbf{x} \circ \mathbf{g}] = 0,\tag{1}$$

where *u* > 0, *n* ≥ 2, and [*x* ◦ *g*](*u*) = *x*(*g*(*u*)). They considered the even- and odd-order of this equation. One of their results was to ensure that solutions of DDE (1) oscillate under the conditions *g*(*u*) ≤ *u*, *g* (*u*) ≥ 0, and

$$\begin{split} \limsup\_{u \to \infty} \left[ \mathcal{g}(u) \int\_{u}^{\infty} \mathcal{g}^{n-2}(\upsilon) q(\upsilon) \mathbf{d}\upsilon + \int\_{\mathcal{S}(u)}^{u} \mathcal{g}^{n-1}(\upsilon) q(\upsilon) \mathbf{d}\upsilon \right. \\ &+ \frac{1}{\mathcal{g}(u)} \int\_{0}^{\mathcal{S}(u)} \upsilon \mathcal{g}^{n-1}(\upsilon) q(\upsilon) \mathbf{d}\upsilon \right] > (n-1)!. \end{split}$$

Before and after that, many researchers also verified the oscillation of the higher order DDEs solutions in the canonical case by using traditional methods; see, for example, [15–20].

In the non-canonical case, the oscillation conditions for solutions of the DDE

$$\frac{\mathbf{d}}{\mathbf{d}u}\left(a \cdot \left(\frac{\mathbf{d}^{n-1}x}{\mathbf{d}u^{n-1}}\right)^{a}\right) + q \cdot \left[f \circ x \circ g\right] = 0\tag{2}$$

were established by Baculíková et al. [21] and Moaaz et al. [22] by using the comparison technique. Using the Riccati substitution, Zhang et al. [23] and Moaaz and Muhib [24] studied oscillation of the DDE

$$\frac{\mathbf{d}}{\mathbf{d}u} \left( a \cdot \left( \frac{\mathbf{d}^{n-1} \mathbf{x}}{\mathbf{d}u^{n-1}} \right)^{\alpha} \right) + q \cdot \left[ \mathbf{x}^{\emptyset} \circ \mathbf{g} \right] = 0.$$

The results of the second-order equation were most recently expanded to the even-order equations in the non-canonical case by Moaaz et al. [25]. In order to develop iteratively new oscillation criteria, they developed an approach that involved obtaining new monotonic properties for positive decreasing solutions.

For the neutral equations, which the higher derivative appears on the solution with and without delay, Li and Rogovchenko [26] related oscillation of solutions of the DDE

$$\frac{\mathbf{d}}{\mathbf{d}u}\left(a\cdot\left(\frac{\mathbf{d}^{n-1}}{\mathbf{d}u^{n-1}}(\mathbf{x}+p\cdot[\mathbf{x}\circ h])\right)^{a}\right)+q\cdot\left[\mathbf{x}^{\mathcal{G}}\circ\mathbf{g}\right]=0\tag{3}$$

to three equations of the first-order by using comparison techniques. In [27], a criterion for ruling out the existence of so-called Kneser solutions to DDE (3) was developed. The results in [27] are more accurate and effective than those in [26] since they do not rely on unknown functions. Very recently, Elabbasy et al. [28] studied the asymptotic behavior of the solutions of the DDE (3). Let us review the following theorem, which gives the conditions ensuring that non-oscillatory solutions of (3) tend to zero.

On the other hand, for neutral equations of the second order, the study of oscillation of these equations has been developed with many improved techniques; see, for example, [29–31].

In this article, we consider the neutral DDE of the form

$$\frac{\mathbf{d}}{\mathbf{d}u} \left( a \cdot \frac{\mathbf{d}^{n-1}}{\mathbf{d}u^{n-1}} (\mathbf{x} + p \cdot [\mathbf{x} \circ h]) \right) + q \cdot [F \circ \mathbf{x} \circ \mathbf{g}] = \mathbf{0},\tag{4}$$

where *u* ≥ *u*0, *n* ≥ 4 is an even natural number. We also suppose the following:

(H1) *a*, *p*, *q* ∈ *C*([*u*0, ∞), [0, ∞)), *a*(*u*) > 0, *p*(*u*) ≤ E0(*u*)/E0(*h*(*u*)) and E*n*−2(*u*0) < ∞, where

$$\mathcal{E}\_0(u) := \int\_u^\infty a^{-1}(v) \mathrm{d}v,\text{ and } \mathcal{E}\_k(u) := \int\_u^\infty \mathcal{E}\_{k-1}(v) \mathrm{d}v.$$

for *k* = 1, 2, . . . , *n* − 2.


First, we define the corresponding function of solution *x* as *ω* := *x* + *p* · [*x* ◦ *h*]. By a solution of (4), we mean a function *<sup>x</sup>* <sup>∈</sup> *<sup>C</sup>n*−1([*ux*, <sup>∞</sup>), <sup>R</sup>), for *ux* <sup>≥</sup> *<sup>u</sup>*0, which *<sup>a</sup>* · *<sup>ω</sup>*(*n*−1) <sup>∈</sup> *<sup>C</sup>*1([*ux*, <sup>∞</sup>), <sup>R</sup>) and *<sup>x</sup>* satisfies (4) for all *ux* <sup>≥</sup> *<sup>u</sup>*0. We consider only those solutions of (4) that do not eventually vanish. A solution of DDE (4) is called oscillatory if it has arbitrarily large zeros; otherwise, it is said to be nonoscillatory. DDE (4) is called oscillatory if all its solutions are oscillatory.

The aim of the study is to provide new conditions for determining the oscillation parameters of all solutions of Equation (4) in the non-canonical case. We also aim to develop the oscillation theorems of higher-order neutral delay differential equations by deriving new oscillation parameters characterized by an iterative nature. The method employed is an extension of the method Koplatadze et al. [14] and, later, by Baculková [10].
