About Conformable Derivatives in Banach Spaces

TL;DR

This paper explores conformable derivatives in Banach spaces, establishing a key equivalence between conformable and first-order derivatives at a point, thus advancing the theoretical understanding of fractional calculus in infinite-dimensional spaces.

Contribution

It introduces a connection between different conformable derivatives and characterizes the existence of conformable derivatives via classical derivatives in Banach spaces.

Findings

Conformable derivatives in Banach spaces are equivalent to first-order derivatives at a point.

A function has a conformable derivative at a point if and only if it has a first-order derivative there.

The paper clarifies the behavior of conformable derivatives in infinite-dimensional settings.

Abstract

In the paper we discuss conformable derivative behavior in arbitrary Banach spaces and clear the connection between two conformable derivatives of different order. As a consequence we obtain the important result that an abstract function has a conformable derivative at a point (which does not coincide with the lower terminal of the conformable derivative) if and only if it has a first order derivative at the same point.