On functions without pseudo derivatives having fractional pseudo derivatives

Abstract

A classical question of the fractional calculus of real-valued functions— the existence of a continuous nowhere differentiable function which has a continuous Riemann-Louville fractional derivative of any order less than one—was studied in detail by B. Ross et al. [16]). This paper deals with said question within the framework of abstract functions. Indeed, we give examples of weakly continuous vector-valued functions which fail to be pseudo differentiable but have weakly continuous fractional-pseudo derivatives of some critical order less than one. Based on these examples, it can be seen that even if an initial value problem of fractional order has a weakly continuous right-hand side, the equivalence between differential and integral form of the problem can be lost.