Fractional calculus and fractional differential equations in nonreflexive Banach spaces

•We introduce the notion of pseudo-fractional derivatives.•We introduce the notion of fractional Pettis integrals.•We discuss functions in non-reflexive Banach spaces equipped with the weak topology.•We discuss fractional differential equations in an abstract setting.

In this paper we establish an existence result for the fractional differential equationDpαy(t)=f(t,y(t)),y(0)=y0,where Dpαy(·) is a fractional pseudo-derivative of a weakly absolutely continuous and pseudo-differentiable function y(·) : T → E, the function f (t, ·) : T × E → E is weakly–weakly sequentially continuous for every t ∈ T and f (·, y(·)) is Pettis integrable for every weakly absolutely continuous function y(·) : T → E, T is a bounded interval of real numbers and E is a nonreflexive Banach space.