# Generalized Mittag–Leffler Stability of Hilfer Fractional Order Nonlinear Dynamic System

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Preliminaries

**Definition**

**1**

**.**The Hilfer fractional derivative of order α and type β for a function g is defined as

**Lemma**

**1**

**.**Let $0<\alpha <1$, then

**Lemma**

**2**

**.**Let $0<\alpha <1$, $0\le \beta \le 1$, and $\gamma =\alpha +\beta -\alpha \beta $, then

**Remark**

**1**

**.**The Laplace transform of Hilfer fractional derivative is

**Definition**

**2**

**.**The one-parameter and two-parameter Mittag–Leffler functions are defined by respectively

**Definition**

**3.**

**Definition**

**4.**

**Remark**

**2.**

**Definition**

**5**

**.**ω is called a K-class function, if $\omega (0)=0$, and $\omega :{\mathbb{R}}^{+}\to {\mathbb{R}}^{+}$ is strictly increasing.

**Lemma**

**3.**

**Proof.**

## 3. Main Theory

**Theorem**

**1.**

**Proof.**

**Theorem**

**2.**

**Proof.**

## 4. Examples

**Example**

**1.**

**Proof.**

**Example**

**2.**

**Proof.**

**Remark**

**4.**

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

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**MDPI and ACS Style**

Wang, G.; Qin, J.; Dong, H.; Guan, T. Generalized Mittag–Leffler Stability of Hilfer Fractional Order Nonlinear Dynamic System. *Mathematics* **2019**, *7*, 500.
https://doi.org/10.3390/math7060500

**AMA Style**

Wang G, Qin J, Dong H, Guan T. Generalized Mittag–Leffler Stability of Hilfer Fractional Order Nonlinear Dynamic System. *Mathematics*. 2019; 7(6):500.
https://doi.org/10.3390/math7060500

**Chicago/Turabian Style**

Wang, Guotao, Jianfang Qin, Huanhe Dong, and Tingting Guan. 2019. "Generalized Mittag–Leffler Stability of Hilfer Fractional Order Nonlinear Dynamic System" *Mathematics* 7, no. 6: 500.
https://doi.org/10.3390/math7060500