  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.
