Leibniz type rule: ψ-Hilfer fractional operator

Abstract

In this paper, we present a Leibniz type rule for the ψ-Hilfer (ψ-H) fractional derivative operator in two forms, one written in terms of the ψ-Riemann-Liouville (ψ-RL) fractional derivative operator and the other in terms of the ψ-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 ψ-H fractional operator. We present some specific cases of Leibniz type rule for the ψ-H fractional derivative operator which emerge from different choices of parameter β and the function ψ.