Some Important Criteria for Oscillation of Non-Linear Differential Equations with Middle Term

: In this work, we present new oscillation conditions for the oscillation of the higher-order differential equations with the middle term. We obtain some oscillation criteria by a comparison method with ﬁrst-order equations. The obtained results extend and simplify known conditions in the literature. Furthermore, examining the validity of the proposed criteria is demonstrated via particular examples.


Introduction
Neutral equations contribute to many applications in physics, engineering, biology, non-Newtonian fluid theory, and the turbulent flow of a polytrophic gas in a porous medium. Also, oscillation of neutral equations contribute to many applications of problems dealing with vibrating masses attached to an elastic bar, see [1].
The main results are obtained under the following conditions: is an even natural number, ζ is not identically zero for large x.
Bazighifan et al. [2] examined the oscillation of higher-order delay differential equations with damping of the form This time, the authors used the Riccati technique. Zhang et al. in [3] considered a higher-order differential equation Bazighifan and Ramos [4] considered the oscillation of the even-order nonlinear differential equation with middle term of the form Liu et al. [5] investigated the higher-order differential equations where n is even and used integral averaging technique. The authors in [6,7] discussed oscillation criteria for the equations where is even and p > 1 is a real number, the authors used comparison method with first and second-order equations.
Li et al. [8] studied the oscillation of fourth-order neutral differential equations In [9,10], the authors considered the equation by using the Riccati method, they proved that this equation is oscillatory if where z(x) We can easily apply conditions (6) and (7) to the equation Hence, [10] improved the results in [9]. Thus, the main purpose of this article is to extend the results in [9,10,23]. An example is considered to illustrate the main results.
We mention some important lemmas:
Then, we have these cases:

Theorem 2. Assume that
and are oscillatory, where then (1) is oscillatory.
, then we have that z > 0 is a solution of the delay inequality It is clear (see [17] Theorem 1) that the Equation (14) also has a positive solution, this is a contradiction.
Let (I 2 ) hold, from Lemma 2, we obtain and then which with (18) yields Integrating (22) from x to ∞, we obtain Integrating this inequality − 3 times from x to ∞, we find which with (20) gives Thus, if we put ω(x) := (x), then we conclude that ω > 0 is a solution of It is clear (see [17] Theorem 1) that the equation (15) also has a positive solution, this is a contradiction. The proof is complete.

Applications
This section presents some interesting application which are addressed based on above hypothesis to show some interesting results in this paper.

Example 1. Let the equation
Thus, we find Hence, the condition becomes Therefore, by Corollary 1, every solution of (28) is oscillatory if ζ 0 > 25.5.

Conclusions
In this paper, we obtain sufficient criteria for oscillation of solutions of higher-order differential equation with middle term. We discussed the oscillation behavior of solutions for Equation (1). We obtain some oscillation criteria by comparison method with first order equations. Our results unify and improve some known results for differential equations with middle term. In future work, we will discuss the oscillatory behavior of these equations using integral averaging method and under condition For researchers interested in this field, and as part of our future research, there is a nice open problem which is finding new results in the following cases: