Global Solution to a Nonlinear Fractional Differential Equation for the Caputo-Fabrizio Derivative

This paper deals with the fractional Caputo–Fabrizio derivative and some basic properties related. A computation of this fractional derivative to power functions is given in terms of Mittag–Lefler functions. The inverse operator named the fractional Integral of Caputo–Fabrizio is also analyzed. The main result consists in the proof of existence and uniqueness of a global solution to a nonlinear fractional differential equation, which has been solved previously for short times by Lozada and Nieto (Progr. Fract. Differ. Appl., 1(2):87–92, 2015). The effects of memory as well as the convergence of the obtained results when $\text{al}\nearrow 1$ (and the classical first derivative is recovered) are analyzed throughout the paper.