Leibniz type rule: ψ-Hilfer fractional operator

In this paper, we present a Leibniz type rule for the psi-Hilfer (psi-H) fractional derivative operator in two forms, one written in terms of the psi-Riemann-Liouville (psi-RL) fractional derivative operator and the other in terms of the psi-H fractional derivative operator. Direct consequences of this new formulation of a Leibniz type rule are the possibility of writing recurrence relations involving solutions of fractional differential equations and of investigating the existence, uniqueness and Ulam-Hyers stabilities of mild solutions of fractional differential equations involving psi-H fractional operator. We present some specific cases of Leibniz type rule for the psi-H fractional derivative operator which emerge from different choices of parameter beta and the function psi. (C) 2019 Elsevier B.V. All rights reserved.