Fractional calculus and fractional differential equations in nonreflexive Banach spaces

In this paper we establish an existence result for the fractional differential equation D p α y ( t ) = f ( t , y ( t ) ) , y ( 0 ) = y 0 , where D p α y ( · ) 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.