# Averaging Methods for Second-Order Differential Equations and Their Application for Impact Systems

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## Abstract

**:**

## 1. Introduction

## 2. Preliminaries

- $u\xb7v$—the inner product of vectors $u,v\in {\mathbb{R}}^{N}$;
- $\left|u\right|$—the Euclidean norm of $u\in {\mathbb{R}}^{N}$;
- $\mathrm{dist}(A,B):=inf\left\{\right|a-b|\mid a\in A,\phantom{\rule{0.166667em}{0ex}}b\in B\}$—the distance of nonempty sets $A,B\subset {\mathbb{R}}^{N}$;
- $\dot{x}\left(t\right)$ ($\ddot{x}\left(t\right)$)—the first (second) derivative of function x at time t;
- $\dot{x}{\left(t\right)}^{+}$ ($\dot{x}{\left(t\right)}^{-}$)—the right (left) derivative of function x at time t.

**Definition**

**1.**

## 3. Results and Proofs

**Theorem**

**1.**

**Proof.**

**Remark**

**1.**

**Lemma**

**1.**

**Proof.**

**Remark**

**2.**

**Theorem**

**2.**

**Remark**

**3.**

**Proof.**

**Remark**

**4.**

## 4. Example

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

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Fečkan, M.; Pačuta, J.
Averaging Methods for Second-Order Differential Equations and Their Application for Impact Systems. *Mathematics* **2020**, *8*, 916.
https://doi.org/10.3390/math8060916

**AMA Style**

Fečkan M, Pačuta J.
Averaging Methods for Second-Order Differential Equations and Their Application for Impact Systems. *Mathematics*. 2020; 8(6):916.
https://doi.org/10.3390/math8060916

**Chicago/Turabian Style**

Fečkan, Michal, and Július Pačuta.
2020. "Averaging Methods for Second-Order Differential Equations and Their Application for Impact Systems" *Mathematics* 8, no. 6: 916.
https://doi.org/10.3390/math8060916