A fractional dimension-space theorem for Roll's and mean value theorems

Abstract

In this paper, we briefly review some accomplished research in the mean value and Roll's theorems of the fractional calculus. Then, we present a mathematical space so-called Fractional Dimension Space (FDS). Using this space, a Roll's and mean value transformation is made, which transmits the classical Roll's and mean value theorems into the FDS'. The goal is finding a fractional order $0\leq \alpha<1$ by one of the well-known methods such as Riemann-Liouville, Caputo, Gr$\ddot{u}$nwald-Letnikov, Hadamard, and Weyl fractional derivatives satisfying the classical mean value and Roll's theorems. Finally, we give a theorem proving that there exists an $0\leq \alpha<1$ in such space through at least one of the aforementioned methods of differentiation. The two applications of the FDS are mentioned in this paper as well.