On Implicit Time–Fractal–Fractional Differential Equation

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.

Keywords

• well-posedness
• exponential growth bound
• fractal–fractional operators
• Mittag–Leffler type kernel