Emden–Fowler-type neutral differential equations: oscillatory properties of solutions

In this paper, we study the oscillation of a class of fourth-order Emden–Fowler delay differential equations with neutral term. Using the Riccati transformation and comparison method, we establish several new oscillation conditions. These new conditions complement a number of results in the literature. We give examples to illustrate our main results.

The study of differential equations has been the object of many researchers over the last decades. Different approaches and various techniques are adopted to investigate the qualitative properties of their solutions. Recently, driven by their widespread applications, the investigation of fourth-order differential equations has drawn significant attention. The existence uniqueness, stability, and oscillation of solutions were the main features that attracted consideration [1][2][3].
In spite of the increasing interest in the study of second-order differential equations, the oscillation and nonoscillation of solutions for differential equations are still considered as an open area to investigate [4][5][6][7][8][9]. Equations with neutral terms are of particular significance as they arise in many applications including systems of control, electrodynamics, mixing liquids, neutron transportation, networks, and population models. In the qualitative analysis of such systems, indeed, the oscillatory behavior of solutions of equations, where the rate of the growth depends not only on the current and the past states but also on the rate of change in the past, play an important role [10][11][12][13][14]. In the light of this motivation and justification, different results have been reported regarding the asymptotic behavior of higher-order differential equations with neutral terms [15][16][17][18]. For relevant results on the application of oscillation theory, the reader can consult [19][20][21]. In the past 20 years, there have been a lot of research results on the oscillation of differential equations. As a matter of fact, Eq. (1) is a natural of the half-linear/Emden-Fowler differential equation (including the related differential equation), which arises in a variety of realworld problems such as the study of p-Laplace equations, non-Newtonian fluid theory, the turbulent flow of a polytrophic gas in a porous medium, and so on; see, for example, the papers [22,23] for more detail.
In this paper, we establish oscillatory properties of solutions of (1) and give some examples for applying the criteria.

Preliminaries
We first provide some notations which help us to easily display the results. Moreover, we present some auxiliary lemmas.
is of fixed sign and not identically zero on [t 0 , ∞) and that there exists t 1 ≥ t 0 such that u (j-1) (t)u (j) (t) ≤ 0 for all t ≥ t 1 . If lim t→∞ u(t) = 0, then for every μ ∈ (0, 1), there exists t μ ≥ t 1 such that For convenience, we impose the following hypothesis: (H1) x is an eventually positive solution of (1).

Theorem 3.1 Assume that
If then (1) is oscillatory.

Conclusion
In this paper, we consider the oscillation and asymptotic behavior of a class of fourthorder Emden-Fowler neutral differential equations. Using the Riccati transformation and comparison method, we establish new oscillation conditions for the solutions of fourthorder neutral differential equations. Our results unify and extend some known results for differential equations. In the future work, we will discuss the oscillatory behavior of these equations by using comparing technique with second-order equations.