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Fractional Calculus and Applied Analysis

Editor-in-Chief: Kiryakova, Virginia


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A survey of Lyapunov functions, stability and impulsive Caputo fractional differential equations

Ravi Agarwal / Snezhana Hristova / Donal O’Regan
Published Online: 2016-05-04 | DOI: https://doi.org/10.1515/fca-2016-0017

Abstract

We present an overview of the literature on solutions to impulsive Caputo fractional differential equations. Lyapunov direct method is used to obtain sufficient conditions for stability properties of the zero solution of nonlinear impulsive fractional differential equations. One of the main problems in the application of Lyapunov functions to fractional differential equations is an appropriate definition of its derivative among the differential equation of fractional order. A brief overview of those used in the literature is given, and we discuss their advantages and disadvantages. One type of derivative, the so called Caputo fractional Dini derivative, is generalized to impulsive fractional differential equations. We apply it to study stability and uniform stability. Some examples are given to illustrate the results.

Key Words and Phrases: stability; Caputo derivative; Lyapunov functions; impulses; fractional differential equations

MSC 2010: Primary 34A34; Secondary 34A08; 34D20

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About the article

Received: 2015-07-20

Revised: 2016-01-25

Published Online: 2016-05-04

Published in Print: 2016-04-01


Citation Information: Fractional Calculus and Applied Analysis, Volume 19, Issue 2, Pages 290–318, ISSN (Online) 1314-2224, ISSN (Print) 1311-0454, DOI: https://doi.org/10.1515/fca-2016-0017.

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