# On the Oscillatory Behavior of a Class of Fourth-Order Nonlinear Differential Equation

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## Abstract

**:**

## 1. Introduction

## 2. Auxiliary Lemmas

**Notation**

**1.**

**Lemma**

**1.**

**Lemma**

**2.**

**Lemma**

**3.**

**Lemma**

**4.**

**Lemma**

**5.**

- (
**i**_{1}) - If x satisfies $\left({S}_{1}\right)$, then$${\omega}_{1}^{\prime}\left(t\right)+Q\left(t\right)+{R}_{1}\left(t\right){\omega}_{1}^{1+1/\alpha}\left(t\right)\le 0;$$
- (
**i**_{2}) - If x satisfies $\left({S}_{2}\right)$, then$${\omega}_{2}^{\prime}\left(t\right)+{\omega}_{2}^{2}\left(t\right)+{B}^{\beta -\alpha}\tilde{R}\left(t\right)\le 0.$$

**Proof.**

## 3. Oscillation Criteria

**Theorem**

**1.**

**Proof.**

**Definition**

**1.**

**Theorem**

**2.**

**Proof.**

**Theorem**

**3.**

**Proof.**

**Corollary**

**1.**

**Proof.**

**Example**

**1.**

**Remark**

**1.**

**Remark**

**2.**

## 4. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**MDPI and ACS Style**

Moaaz, O.; Kumam, P.; Bazighifan, O.
On the Oscillatory Behavior of a Class of Fourth-Order Nonlinear Differential Equation. *Symmetry* **2020**, *12*, 524.
https://doi.org/10.3390/sym12040524

**AMA Style**

Moaaz O, Kumam P, Bazighifan O.
On the Oscillatory Behavior of a Class of Fourth-Order Nonlinear Differential Equation. *Symmetry*. 2020; 12(4):524.
https://doi.org/10.3390/sym12040524

**Chicago/Turabian Style**

Moaaz, Osama, Poom Kumam, and Omar Bazighifan.
2020. "On the Oscillatory Behavior of a Class of Fourth-Order Nonlinear Differential Equation" *Symmetry* 12, no. 4: 524.
https://doi.org/10.3390/sym12040524