Some new oscillation criteria of fourth-order quasi-linear differential equations with neutral term

In this article, we are interested in studying the asymptotic behavior of fourth-order neutral differential equations. Despite the growing interest in studying the oscillatory behavior of delay differential equations of second-order, fourth-order equations have received less attention. We get more than one criterion to check the oscillation by the generalized Riccati method and the integral average technique. Our results are an extension and complement to some results published in the literature. Examples are given to prove the significance of new theorems.

Xing et al. [33] presented criteria for oscillation of the equation where 0 ≤ỹ(x) <ỹ 0 < ∞ and ω(x) := min{ω(θ -1 (x)),ω(θ -1 (θ(x)))}. Bazighifan et al. [18], Li and Rogovchenko [25], and Zhang et al. [26,28] presented oscillation results for fourth-order equation and they used the Riccati technique. Zhang et al. [36] established oscillation criteria for the equation and under the condition By using the Riccati transformation technique, Chatzarakis et al. [19] established asymptotic behavior for the neutral equation In this work, a new oscillation condition is created for fourth-order differential equations with a canonical operator. We use the Riccati technique and the integral averaging technique to prove our results.
Here are the notations used for our study:

Oscillation criteria
We next present the lemmas needed for the proof of the original results.
is of a fixed sign and not identically zero on [x 0 , ∞) and that there exists where Y > 0 and X are constants.
Let (N 2 ) hold. Integrating (7) from x to u, we find From Lemma 2.1, we obtain and hence For (17), letting u → ∞ and using (18), we get Integrating (19) from x to ∞, we find From the definition of A(x), we see that A(x) > 0 for x ≥ x 1 , and using (16) and (20), we find Since δ (x) > 0, there exist x 2 ≥ x 1 and A 2 > 0 such that Thus, we obtain Thus, (10) holds. The proof of the theorem is completed. Now, we present some Philos-type oscillation criteria for (1).

Conclusion
In this work, we proved some new oscillation theorems for (1). New oscillation results are established that complement related contributions to the subject. We used the Riccati technique and the integral averages technique to get some new results to oscillation of equation (1) under the condition ∞ x 0 1 z 1/r 1 (s) ds = ∞. We may say that, in future work, we will study this type of equation under the condition ∞ x 0 1 z 1/r 1 (s) ds < ∞.
Also we will try to introduce some important oscillation criteria of differential equations of fourth-order and under