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

## TL;DR

This paper introduces a new version of the $ ext{ extpsi}$-Hilfer fractional derivative on arbitrary time scales, investigates its properties, and applies it to prove existence, uniqueness, and controllability of solutions for fractional initial value problems.

## Contribution

It develops a novel $ ext{ extpsi}$-Hilfer fractional derivative on time scales and demonstrates its usefulness in analyzing fractional differential equations.

## Key findings

- Established properties of the new fractional derivative.
- Proved existence and uniqueness of solutions for fractional initial value problems.
- Demonstrated controllability of solutions in a Banach space.

## Abstract

We introduce a new version of $\psi$-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 $\psi$-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 controllability of solutions in a suitable Banach space. The obtained results are interesting and nontrivial even for particular choices: (i) of the time scale; (ii) of the order of differentiation; and/or (iii) function $\psi$; opening new directions of investigation.