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 fractionalpseudo 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.

Key words: Fractional calculus, Pettis integrals, pseudo derivative.