Stability by Lyapunov functions of Caputo fractional differential equations with non-instantaneous impulses

The stability of the zero solution of a nonlinear Caputo fractional differential equation with noninstantaneous impulses is studied using Lyapunov like functions. The novelty of this paper is based on the new definition of the derivative of a Lyapunov like function along the given noninstantaneous impulsive fractional differential equations. On one side this definition is a natural generalization of Caputo fractional Dini derivative of a function and on the other side it allows us the assumption for Lyapunov functions to be weakened to continuity. By appropriate examples it is shown the natural relationship between the defined derivative of Lyapunov functions and Caputo derivative. Several sufficient conditions for uniform stability and asymptotic uniform stability of the zero solution, based on the new definition of the derivative of Lyapunov functions are established. Some examples are given to illustrate the results.