Abstract
An implicit time-fractal-fractional differential equation involving the Atangana's fractal-fractional derivative in the sense of Caputo with the Mittag-Leffler law type kernel is studied. Using the Banach fixed point theorem, the well-posedness of the solution is proved. We show that the solution exhibits an exponential growth bound, and, consequently, the long-time (asymptotic) property of the solution. We also give examples to illustrate our problem.