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.

在本文中，我们为受电弓分数阶微分方程（FDE）的存在和唯一性结果（具有非局部条件）提供了充分的条件，该条件涉及分数阶Atangana–Baleanu–Caputo（ABC）导数算子。我们的方法基于将FDE简化为分数积分方程，并且基于一些不动点定理，例如Banach的收缩原理和Krasnoselskii的不动点定理。此外，在Atangana–Baleanu分数积分算子的框架中，Gronwall的不等式被用于为各种Ulam–Hyers稳定性得出适当的结果。最后，本文包括一个实例来证实结果的有效性。