Mittag-Leffler Stability for Non-instantaneous Impulsive Generalized Proportional Caputo Fractional Differential Equations

Fractional calculus is a powerful tool in applied mathematics and is used to study problems in mathematical physics, finance, hydrology, biophysics, thermodynamics, control theory, statistical mechanics, astrophysics, cosmology and bioengineering. In studying stability for nonlinear fractional differential equations, there are several approaches in the literature, one of which is the Lyapunov approach. There are however several difficulties encountered when one applies the Lyapunov technique to fractional differential equations and one of the main difficulties is connected with the appropriate definition of derivatives of Lyapunov functions among differential equations of fractional order. In this paper fractional differential equations with non-instantaneous impulses and generalized proportional Caputo fractional derivative are studied. The case of changeable lower limit of the fractional derivative at any and time point of the non-instantaneuos impulse is considered. The Mitatg-Leffler stability with respect to the impulses is defined. Also, its partial case of exponential stability with respect to impulses is given. Sufficient conditions by the help with Lyapunov like functions are obtained. An example is given to illustrate our results.