The Strict Stability of Impulsive Differential Equations with a Caputo Fractional Derivative with Respect to Other Functions

The aim of this paper is to study a nonlinear system of impulsive fractional differential equations and Caputo fractional derivatives with respect to another function (CFF). The main characteristics of these fractional derivatives are two-fold: first, the lower limit of CFF equals the impulsive time of the considered interval; second, the applied function in CFF is changeable at each interval without impulses. An auxiliary system of two linear scalar impulsive fractional differential equations with CFF is considered, and strict stability in a couple is defined. The behavior of its solutions is illustrated with several examples. Also, we use appropriate Lyapunov functions to obtain sufficient conditions for the strict stability of the studied system. These sufficient conditions depend significantly on the type of impulsive function.