# Oscillatory Properties of Solutions of Even-Order Differential Equations

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^{2}

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^{†}

## Abstract

**:**

## 1. Introduction

**Lemma**

**1**

**Lemma**

**2**

**Lemma**

**3**

## 2. Oscillation Criteria

**Theorem**

**1.**

**Proof.**

**Corollary**

**1.**

## 3. Kamenev-Type Criteria

**Definition**

**1.**

- (i
_{1}) - $H\left(\right)open="("\; close=")">t,t$and$H\left(\right)open="("\; close=")">t,s\in {D}_{0}$for$t\ge {t}_{0},$
- (i
_{2}) - $H$has a nonpositive continuous partial derivative$\partial H/\partial s$on${D}_{0}$with respect to the second variable, and there exist functions$h\in C\left(\right)open="("\; close=")">{D}_{0},\mathbb{R}$and$\delta \in {C}^{1}\left(\right)open="("\; close=")">\left(\right)open="["\; close=")">{t}_{0},\infty $such that

**Theorem**

**2.**

**Proof.**

**Corollary**

**2.**

**Example**

**1.**

## 4. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**MDPI and ACS Style**

Elabbasy, E.M.; El-Nabulsi, R.A.; Moaaz, O.; Bazighifan, O.
Oscillatory Properties of Solutions of Even-Order Differential Equations. *Symmetry* **2020**, *12*, 212.
https://doi.org/10.3390/sym12020212

**AMA Style**

Elabbasy EM, El-Nabulsi RA, Moaaz O, Bazighifan O.
Oscillatory Properties of Solutions of Even-Order Differential Equations. *Symmetry*. 2020; 12(2):212.
https://doi.org/10.3390/sym12020212

**Chicago/Turabian Style**

Elabbasy, Elmetwally M., Rami Ahmad El-Nabulsi, Osama Moaaz, and Omar Bazighifan.
2020. "Oscillatory Properties of Solutions of Even-Order Differential Equations" *Symmetry* 12, no. 2: 212.
https://doi.org/10.3390/sym12020212