New Criteria for Oscillation of Half-Linear Differential Equations with p -Laplacian-Like Operators

: In this paper, new oscillatory properties for fourth-order delay differential equations with p -Laplacian-like operators are established, using the Riccati transformation and comparison method. Moreover, our results are an extension and complement to previous results in the literature. We provide some examples to examine the applicability of our results. using Riccati and comparison techniques under (3) and (4). These results contribute to adding some important conditions that were previously studied in the subject of oscillation of differential equations with neutral terms. To prove our main results, we give some examples.


Introduction
Delay differential equations arise in a variety of phenomena, including mixing liquids, economics problems, biology, medicine, physics, engineering and automatic control problems, as well as vibrational motion in flight and to explain human self-balancing; see [1,2].
Li et al. [8] considered the oscillation for the delay equation where φ i (ı) ≤ ı, and they used the Riccati technique to find oscillation conditions for this equation. Park et al. [26] studied the asymptotic properties of the solutions of the delay equation where φ(ı) ≤ ı, κ is even, and they used the integral average technique to obtain some oscillation results for this equation under the condition Zhang et al. [9] discussed the equation The purpose of this paper is to continue the authors' work [13,14]. Many researchers have used the comparison method to find oscillation conditions for this equation.
The authors in [8,9,26] used the integral average and comparison techniques that differ from the approach used in this article. Their approach is based on using the comparison technique to reduce Equations (1) and (2) into a first-order equation, while our article is based on using the Riccati technique to reduce Equations (1) and (2) into a first-order inequality to find more effective oscillation conditions for Equations (1) and (2).
Motivated by the reasons mentioned above, in this paper, we extend the results using Riccati and comparison techniques under (3) and (4). These results contribute to adding some important conditions that were previously studied in the subject of oscillation of differential equations with neutral terms. To prove our main results, we give some examples.

Lemma 3.
In [17], let κ be a ratio of two odd numbers, V > 0 and U, that are constants. Then,

Lemma 4. Let
Proof. The proof is obvious and therefore is omitted.

This implies that
which contradicts (11). The proof is complete.

Conclusions
In this work, we study the asymptotic and oscillatory properties of solutions of the fourth-order delay differential equations with p-Laplacian-like operators. Using the Riccati transformation, we obtained new criteria that guarantee the oscillation of all solutions of the studied equations. In future work, we will study oscillatory properties of Equation (1) under the condition An interesting problem is to extend our results to even-order damped differential equations with p-Laplacian-like operators η(ı) w (κ−1) (ı) p−1 + β(ı) w (ı) p 1 −2 w (ı) + ϑ(ı) f (w(φ(ı))) = 0.