An Oscillation Criterion of Nonlinear Differential Equations with Advanced Term

: The aim of the present paper is to provide oscillation conditions for fourth-order damped differential equations with advanced term. By using the Riccati technique, some new oscillation criteria, which ensure that every solution oscillates, are established. In fact, the obtained results extend, unify and correlate many of the existing results in the literature. Furthermore, two examples with speciﬁc parameter values are provided to conﬁrm our results.


Introduction
Fourth-order advanced differential equations have an enormous potential for applications in engineering, medicine, aviation and physics, etc. The oscillation of differential equations contributes to many applications in science and technology and self-excited oscillation phenomena which occur in bridges and in the oscillatory muscle movement model; see [1,2].

Definition 2.
When all the solutions of the equation in (1) are oscillatory, the equation is called oscillatory.

Definition 3.
If condition c(z) ≥ z hold, then the Equation (1) is called an advanced differential equation.
Park et al. [22] studied some oscillation properties of the solutions of differential equations with advanced term, by employing the comparison technique. Agarwal et al. [23,24] established the properties of oscillation for advanced equations using integral averaging technique.
Bazighifan et al. [25,26] considered fourth-order differential equations with advanced term where m is even and p > 1 .
The authors in [4], obtained some oscillation conditions for equation where m is even and p > 1 . Moreover, the authors used the comparison method to obtain oscillation conditions for this equation.
Other work has been done on similar equations with advanced term. Li et al. [3] investigated some oscillation criteria of equation The purpose of this paper is to establish new oscillation criteria for (1). The methods used in this paper simplify and extend some of the known results that are reported in the literature [4,26]. The authors in [4,26] used a comparison technique that differs from the one we used in this article. Moreover, the authors in [4,26] also studied the equation under the condition The organization of this article is as follows. After this introduction, in Section 2, we propose some preliminary lemmas that are used in the proof of our main theorems. In Section 3, we establish some oscillation criteria for (1) by Riccati technique; our results extend and correlate many of the existing results in the literature. Then, some examples are considered to check the efficiency of our main results.

Some Lemmas
These are some of the important Lemmas for all positive y, C > 0 and D be positive constant for z ≥ z 1 where z 1 ≥ z 0 is sufficiently large.

Oscillation Criteria
The motivation for this section is to create new oscillation criteria, established for (1) by the Riccati technique.
For ease of use, here are some notations.
Therefore, we see that Define we obtain B(z) < 0 also, from (11) and (12), we have From (12), we find Using (1), we obtain From (12), we see = Using Lemma 3, we find From (13)- (15), we obtain From (11), we see Using the latter inequality and (4), we see which implies that ζ (z)/ϑ(z) is nondecreasing. Thus, it follows from c(z) ≥ z that So, by (14) and (15), we see Multiplying (17) by ξ p−1 (z)G(z 0 , z) and integrating from z 1 to z, we get The new proposed criteria complement several results in the literature. We provide two examples with specific parameters to illustrate the applicability of our theorems. In future work, we will discuss the oscillatory behavior of these equations by using the integral averaging technique and under the condition