Hadamard-type fractional calculus in Banach spaces

In this pages, we present the definitions and some properties of the Hadamard-type fractional Pettis integrals (and corresponding fractional derivatives) for the functions that take values in Banach space. Further, we show that a well known properties of the Hadamard-type fractional calculus over the space of real-valued functions also hold in Banach spaces. Some emphasizes examples are demonstrated. Meanwhile, we construct an example of a function that has no pseudo derivative everywhere, but has a Hadamard-type fractional derivative. As far as we know, the topic of this paper was never investigated before, and so is new.