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This paper deals with the fractional Caputo--Fabrizio derivative and somebasic properties related. A computation of this fractional derivative to powerfunctions is given in terms of Mittag--Lefler functions. The inverse operatornamed the fractional Integral of Caputo--Fabrizio is also analyzed. The mainresult consists in the proof of existence and uniqueness of a global solutionto a nonlinear fractional differential equation, which has been solvedpreviously 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 theobtained results when $\al \nearrow 1$ (and the classical first derivative isrecovered) are analyzed throughout the paper.

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