Existence, uniqueness, and controllability for Hilfer differential equations on times scales

We introduce a new version of ?-Hilfer fractional derivative, on an arbitrary time scale. The fundamental properties of the new operator are investigated, and in particular, we prove an integration by parts formula. Using the Laplace transform and the obtained integration by parts formula, we then propose a ?-Riemann-Liouville fractional integral on times scales. The applicability of the new operators is illustrated by considering a fractional initial value problem on an arbitrary time scale, for which we prove existence, uniqueness, and controlla-bility of solutions in a suitable Banach space. The obtained results are interesting and nontrivial even for the following particular choices: (i) of the time scale, (ii) of the order of differentiation, and/or (iii) function ?, opening new directions of investigation. Finally, we end the article with comments and future work.