In this paper, we discuss the conditions of existence and uniqueness of solutions to a certain class of ordinary differential equations involving Atangana–Baleanu fractional derivative . Benefiting from the Gronwall inequality in the frame of Riemann–Liouville fractional integral, we establish a Gronwall inequality in the frame of Atangana–Baleanu fractional integral. Then, we study the stability of such equations in the sense of Ulam.