On the Generalization of Tempered-Hilfer Fractional Calculus in the Space of Pettis-Integrable Functions

We propose here a general framework covering a wide range of fractional operators for vector-valued functions. We indicate to what extent the case in which assumptions are expressed in terms of weak topology is symmetric to the case of norm topology. However, taking advantage of the differences between these cases, we emphasize the possibly less-restrictive growth conditions. In fact, we present a definition and a serious study of generalized Hilfer fractional derivatives. We propose a new version of calculus for generalized Hilfer fractional derivatives for vector-valued functions, which generalizes previously studied cases, including those for real functions. Note that generalized Hilfer fractional differential operators in terms of weak topology are studied here for the first time, so our results are new. Finally, as an application example, we study some n -point boundary value problems with just-introduced general fractional derivatives and with boundary integral conditions expressed in terms of fractional integrals of the same kind, extending all known cases of studies in weak topology.