# Fractional Langevin Equation Involving Two Fractional Orders: Existence and Uniqueness Revisited

^{1}

^{2}

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## Abstract

**:**

## 1. Introduction

## 2. Preliminaries and Auxiliary Results

**Definition**

**1.**

**Definition**

**2.**

**Definition**

**3**

**.**Let $\alpha ,\beta >0$, $\lambda \in \mathbb{R}$ and $x\in {L}^{1}[0,1]$. The Prabhakar integral can be written as

**Lemma**

**1**

**.**Let $\alpha ,\beta ,\gamma \ge 0$ and $x\in {L}^{1}[0,1]$. Then

**Lemma**

**2.**

**Proof.**

**Theorem**

**1.**

## 3. Existence and Uniqueness

**Theorem**

**2.**

**Hypothesis**

**1.**

**Hypothesis**

**2.**

**Hypothesis**

**3.**

**Proof.**

**Remark**

**1.**

**Theorem**

**3.**

**Hypothesis**

**4.**

**Hypothesis**

**5.**

**Proof.**

## 4. Illustrative Examples

**Example**

**1.**

**Example**

**2.**

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

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Fazli, H.; Sun, H.; Nieto, J.J.
Fractional Langevin Equation Involving Two Fractional Orders: Existence and Uniqueness Revisited. *Mathematics* **2020**, *8*, 743.
https://doi.org/10.3390/math8050743

**AMA Style**

Fazli H, Sun H, Nieto JJ.
Fractional Langevin Equation Involving Two Fractional Orders: Existence and Uniqueness Revisited. *Mathematics*. 2020; 8(5):743.
https://doi.org/10.3390/math8050743

**Chicago/Turabian Style**

Fazli, Hossein, HongGuang Sun, and Juan J. Nieto.
2020. "Fractional Langevin Equation Involving Two Fractional Orders: Existence and Uniqueness Revisited" *Mathematics* 8, no. 5: 743.
https://doi.org/10.3390/math8050743