On the solutions of Caputo-Hadamard Pettis-type fractional differential equations

Let E be a Banach space with the topological dual E *. The aim of this paper is two-fold. On the one hand, we prove some basic properties of Hadamard-type fractional integral operators. These results are related to earlier results about integral operators acting on different function spaces, but for the vector-valued case they are of independent interest. Note that we discuss it in a rather general setting. We study Hadamard-Pettis integral operators in both single and multivalued case. On the other hand, we apply these results to obtain the existence of solutions of the fractional-type problem dax(t)/ dta =. f (t, x(t)), a. (0, 1), t. [1, e], x(1) + bx(e) = h with certain constants., b, where h. E and f : [1, e] xE. E is Pettis integrable function such that, for every.. E *,. f lies in an appropriate Orlicz spaces. Here da dta stands the Caputo-Hadamard fractional differential operator.