On the basic theory of some generalized and fractional derivatives

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.