On nonlinear pantograph fractional differential equations with Atangana-Baleanu-Caputo derivative

In this paper, we obtain sufficient conditions for the existence and uniqueness results of the pantograph fractional differential equations (FDEs) with nonlocal conditions involving Atangana-Baleanu-Caputo (ABC) derivative operator with fractional orders. Our approach is based on the reduction of FDEs to fractional integral equations and on some fixed point theorems such as Banach's contraction principle and the fixed point theorem of Krasnoselskii. Further, Gronwall's inequality in the frame of the Atangana-Baleanu fractional integral operator is applied to develop adequate results for different kinds of Ulam-Hyers stabilities. Lastly, the paper includes an example to substantiate the validity of the results.