This paper investigated the properties and general Laplace transform of general fractional operators with fixed memory length. Subsequently, a class of general fractional differential equations with fixed memory length was systematically studied. First, the existence of solutions to this type of equations was established via integral methods. Then, the uniqueness of the solution was rigorously proven by applying the Banach fixed-point theorem.

This paper focuses on studying the existence and robust finite-time stability (FTS) for a more generalized class of nonlinear Caputo-Katugampola fractional differential equations with perturbation. The existence of mild solutions is proved by using the Sadovskii fixed-point theorem (SFPT). Then, sufficient conditions for robust FTS are obtained by applying the Jensen inequality and Hölder inequality. Finally, our theoretical results are supported by the research on the Lasota-Wazewska red blood cell (LWRBC) model.