A new result for fractional differential equation with nonlocal initial value using Caputo-Fabrizio derivative

Abstract

In this paper, using Caputo-Fabrizio fractional derivative, we obtain some new results for the existence and uniqueness of solutions of differential equation with nonlocal initial value about fractional order.  These results are applied with the help of Arzela-Ascoli theorem and Schauder fixed point.
Also based on some α-contractive maps for such problems, some new unique theorem has been introduced and proved. Finally, some illustrative example is considered to show the effectiveness of the results.