On the basic theory of some generalized and fractional derivatives

TL;DR

This paper advances the theory of generalized derivatives, establishing fundamental calculus theorems and principles, and explores the existence of fractional derivatives that do not exist everywhere, extending prior conformable fractional derivative work.

Contribution

It develops a comprehensive basic theory of generalized derivatives, including key calculus theorems and existence results, broadening the understanding of fractional derivatives.

Findings

Formulated versions of maximum principle, Rolle's theorem, and Mean value theorem for generalized derivatives.

Proved an existence and uniqueness theorem for a generalized Riccati equation.

Demonstrated that for each ter 1, there exists a fractional derivative and function where the derivative fails to exist everywhere.

Abstract

We continue the development of the basic theory of generalized derivatives as introduced in \cite{JPA} and give some of their applications. In particular, we formulate versions of a weak maximum principle, Rolle's theorem, the Mean value theorem, the Fundamental theorem of Calculus, Integration by parts, along with an existence and uniqueness theorem for a generalized Riccati equation, each of which includes, as corollaries, the corresponding version for conformable fractional derivatives considered by \cite{kat}, \cite{kha} among many others. Finally, we show that for each $\alpha > 1$ there is a fractional derivative and a corresponding function whose fractional derivative fails to exist everywhere on the real line.