Duality for the left and right fractional derivatives

TL;DR

This paper establishes a duality relationship between left and right fractional derivatives, enabling the transfer of results between them and facilitating advances in fractional calculus applications.

Contribution

It proves a general duality principle for fractional derivatives, independent of operator type, and demonstrates its utility through new integration by parts and minimizer existence results.

Findings

Duality between left and right fractional derivatives proven.

New fractional integration by parts formula for right Caputo derivative.

Existence of minimizers in fractional variational problems shown.

Abstract

We prove duality between the left and right fractional derivatives, independently on the type of fractional operator. Main result asserts that the right derivative of a function is the dual of the left derivative of the dual function or, equivalently, the left derivative of a function is the dual of the right derivative of the dual function. Such duality between left and right fractional operators is useful to obtain results for the left operators from analogous results on the right operators and vice versa. We illustrate the usefulness of our duality theory by proving a fractional integration by parts formula for the right Caputo derivative and by proving a Tonelli-type theorem that ensures the existence of minimizer for fractional variational problems with right fractional operators.