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.
Acknowledgements
The research was partially supported by the Fund NPD, Plovdiv University, No. MU15-FMIIT-008.
Please cite to this paper as published in: Fract. Calc. Appl. Anal., Vol. 19, No 2 (2016), pp. 290–318, DOI: 10.1515/fca-2016-0017
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© 2016 Diogenes Co., Sofia
Articles in the same Issue
- Frontmatter
- Editorial
- FCAA related news, events and books (FCAA-Volume 19-2-2016)
- Survey Paper
- A survey of Lyapunov functions, stability and impulsive Caputo fractional differential equations
- Survey Paper
- A free fractional viscous oscillator as a forced standard damped vibration
- Research Paper
- On a Legendre Tau method for fractional boundary value problems with a Caputo derivative
- Research Paper
- Nonlinear Dirichlet problem with non local regional diffusion
- Research Paper
- Generalized fraction evolution equations with fractional Gross Laplacian
- Research Paper
- A stochastic solution with Gaussian stationary increments of the symmetric space-time fractional diffusion equation
- Research Paper
- Convolutional approach to fractional calculus for distributions of several variables
- Research Paper
- Existence theorems for semi-linear Caputo fractional differential equations with nonlocal discrete and integral boundary conditions
- Research Paper
- Nonlinear Riemann-Liouville fractional differential equations with nonlocal Erdelyi-Kober fractional integral conditions
- Research Paper
- Spatial dispersion of elastic waves in a bar characterized by tempered nonlocal elasticity
- Research Paper
- Applications of the fractional Sturm-Liouville problem to the space-time fractional diffusion in a finite domain
- Short Paper
- Measurement of para-xylene diffusivity in zeolites and analyzing desorption curves using the Mittag-Leffler function
- Short Paper
- Monotonicity of functions and sign changes of their Caputo derivatives
- Short Note
- On the Volterra μ-functions and the M-Wright functions as kernels and eigenfunctions of Volterra type integral operators
Articles in the same Issue
- Frontmatter
- Editorial
- FCAA related news, events and books (FCAA-Volume 19-2-2016)
- Survey Paper
- A survey of Lyapunov functions, stability and impulsive Caputo fractional differential equations
- Survey Paper
- A free fractional viscous oscillator as a forced standard damped vibration
- Research Paper
- On a Legendre Tau method for fractional boundary value problems with a Caputo derivative
- Research Paper
- Nonlinear Dirichlet problem with non local regional diffusion
- Research Paper
- Generalized fraction evolution equations with fractional Gross Laplacian
- Research Paper
- A stochastic solution with Gaussian stationary increments of the symmetric space-time fractional diffusion equation
- Research Paper
- Convolutional approach to fractional calculus for distributions of several variables
- Research Paper
- Existence theorems for semi-linear Caputo fractional differential equations with nonlocal discrete and integral boundary conditions
- Research Paper
- Nonlinear Riemann-Liouville fractional differential equations with nonlocal Erdelyi-Kober fractional integral conditions
- Research Paper
- Spatial dispersion of elastic waves in a bar characterized by tempered nonlocal elasticity
- Research Paper
- Applications of the fractional Sturm-Liouville problem to the space-time fractional diffusion in a finite domain
- Short Paper
- Measurement of para-xylene diffusivity in zeolites and analyzing desorption curves using the Mittag-Leffler function
- Short Paper
- Monotonicity of functions and sign changes of their Caputo derivatives
- Short Note
- On the Volterra μ-functions and the M-Wright functions as kernels and eigenfunctions of Volterra type integral operators