A Gronwall inequality for a general Caputo fractional operator

Abstract

In this paper we present a new type of fractional operator, which is a generalization of the Caputo and Caputo--Hadamard fractional derivative operators. We study some properties of the operator, namely we prove that it is the inverse operation of a generalized fractional integral. A relation between this operator and a Riemann--Liouville type is established. We end with a fractional Gronwall inequality type, which is useful to compare solutions of fractional differential equations.

This is a preprint of a paper whose final and definite form is with "Mathematical Inequalities & Applications"