On Hadamard fractional operator and three-point fractional boundary value problem in integral-form Hölder spaces

The paper aims to put forward and discuss the proper assumptions for the existence, in addition to the uniqueness, of the solutions to the non-local fractional boundary value problem containing a Hadamard-type fractional operator in integral-form Hölder Banach space $J_{\alpha ,\beta }$, which has much better properties than both classical Hölder spaces $C^{\alpha }$ and the space of continuous solutions $C$. Based on these results, we investigate and prove some essential properties of the Hadamard fractional operators, such as the boundedness, acting, and continuity in the studied spaces. Our analysis is based on Darbo's fixed point principle and the measure of noncompactness with fractional calculus. The results are confirmed with a numerical example.