# A Note on the Periodic Solutions for a Class of Third Order Differential Equations

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## Abstract

**:**

## 1. Introduction and Statement of the Main Results

**Definition**

**1.**

**Theorem**

**1.**

## 2. Some Results on the Other Averaging Theory

**Theorem**

**2.**

- (i)
- $\mathcal{Z}=\left\{{\mathbf{z}}_{\alpha}=\left(\alpha ,{\beta}_{0}\left(\alpha \right)\right):\alpha \in Cl\left(V\right)\right\}\subset D$ and that for every ${\mathbf{z}}_{\alpha}\in \mathcal{Z}$ the solution $\mathbf{x}\left(t,{\mathbf{z}}_{\alpha}\right)$ of (5) is periodic of period T.
- (ii)
- for every ${\mathbf{z}}_{\alpha}\in \mathcal{Z}$ there exists a fundamental matrix ${M}_{{\mathbf{z}}_{\alpha}}\left(t\right)$ of (6) such that the matrix ${M}_{{\mathrm{z}}_{\alpha}}^{-1}\left(0\right)-{M}_{{\mathrm{z}}_{\alpha}}^{-1}\left(T\right)$ has in the upper right corner the $k\times (n-k)$ zero matrix, and in the lower right corner a $(n-k)\times (n-k)$ matrix ${\Delta}_{\alpha}$ with $det\left({\Delta}_{\alpha}\right)\ne 0.$

## 3. Proof of Theorem 1

**Remark**

**1.**

## 4. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

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**MDPI and ACS Style**

Diab, Z.; Guirao, J.L.G.; Vera, J.A.
A Note on the Periodic Solutions for a Class of Third Order Differential Equations. *Symmetry* **2021**, *13*, 31.
https://doi.org/10.3390/sym13010031

**AMA Style**

Diab Z, Guirao JLG, Vera JA.
A Note on the Periodic Solutions for a Class of Third Order Differential Equations. *Symmetry*. 2021; 13(1):31.
https://doi.org/10.3390/sym13010031

**Chicago/Turabian Style**

Diab, Zouhair, Juan L. G. Guirao, and Juan A. Vera.
2021. "A Note on the Periodic Solutions for a Class of Third Order Differential Equations" *Symmetry* 13, no. 1: 31.
https://doi.org/10.3390/sym13010031