Abstract
In this paper, we study Lipschitz stability of Caputo fractional differential equations with non-instantaneous impulses and state dependent delays. The study is based on Lyapunov functions and the Razumikhin technique. Our equations in particular include constant delays, time variable delay, distributed delay, etc. We consider the case of impulses that start abruptly at some points and their actions continue on given finite intervals. The study of Lipschitz stability by Lyapunov functions requires appropriate derivatives among fractional differential equations. A brief overview of different types of derivative known in the literature is given. Some sufficient conditions for uniform Lipschitz stability and uniform global Lipschitz stability are obtained by an application of several types of derivatives of Lyapunov functions. Examples are given to illustrate the results.
Highlights
Many papers in the literature study stability of solutions of differential equations via Lyapunov functions
We study the Lipschitz stability for a nonlinear system of non-instantaneous impulsive fractional differential equations with state dependent delay (NIFrDDE)
We study the case of changeable lower limit of the Caputo fractional derivative
Summary
Many papers in the literature study stability of solutions of differential equations via Lyapunov functions. We study the Lipschitz stability for a nonlinear system of non-instantaneous impulsive fractional differential equations with state dependent delay (NIFrDDE). Lipschitz stability; state dependent delays (note a special case is time varying delays); and models with non-instantaneous impulses. The lower limit of the fractional derivative is one and the same on the whole interval of study and at each point of jump we consider a boundary value problem defined by the impulsive function. We use the second approach to study Lipschitz stability properties of nonlinear non-instantaneous impulsive delay differential equations. A brief overview in the literature of different types of derivatives of Lyapunov functions among the studied fractional differential equation is given.
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