Existence and stability results for fractional integro-differential equation with Hilfer fractional derivative

In this paper, we discuss the existence, uniqueness of solutions and Ulam-Hyers stability for the Hilfer fractional integro-differential equation with nonlocal Erdélyi-Kober fractional condition. First, the equivalence of this class of problem and a nonlinear Volterra integral equation is established. Next, the existence and uniqueness results are obtained by using a variety of fixed point theorems, such as Banach's and Krasnoselskii's fixed point theorems. Further, due to the Gronwall inequality, we obtain Ulam-Hyers stability of the considered problem. Finally, two examples are given to illustrate our theory results.