Analysis of Cauchy problems for variable-order derivatives with Mittag-Leffler kernel

In this paper, the Cauchy problem for variable-order fractional differential equations incorporating the Mittag-Leffler kernel is explored. The variable-order derivative is modeled as a bounded function that adapts to the underlying dynamics of the system. The existence of a solution by utilizing a fixed-point theorem along with an iterative series that converges to the precise solution is established. The uniqueness of the solution is guaranteed by enforcing conditions like generalized Lipschitz continuity and linear growth conditions. This study contributes to the broader understanding of fractional calculus and its applications in complex systems where classical models are insufficient.