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 0 < ? < 1 . These results are applied with the help of Arzela-Ascoli theorem and Schauder fixed point. Also based on some ?-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.