Periodic boundary value problems for fractional implicit differential equations involving Hilfer fractional derivative

In this paper, a new class of the periodic boundary value problem for nonlinear implicit fractional differential equations involving Hilfer fractional derivative is considered in the weighted space of functions. We establish sufficient conditions for existence, uniqueness, Ulam-Hyers and Ulam-Hyers-Rassias stability of the given problem. The main results are based upon the technique of the Schaefer fixed point theorem, the Banach fixed point theorem, generalized Gronwall inequality, and with the help of some properties of Mittag-Leffler functions. An example is presented to illustrate our main results.