Abstract
•Establishing the Gronwall inequality for Atangana–Baleanu fractional integrals.•Discussing the existence and uniqueness criteria for ordinary differential equations containing Atangana–Baleanu fractional derivatives.•Discussing the Ulam stability for ordinary differential equations containing Atangana–Baleanu fractional derivatives.
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.