Existence and uniqueness analysis of solutions for Hilfer fractional spectral problems with applications

In this work, we define Sturm-Liouville problems through Hilfer fractional derivative operator using its different types. First, we obtain an integral representation of solutions in eight different forms including Riemann-Liouville, Caputo, and ordinary form. These different forms are stemmed from the type of Hilfer fractional derivative. Then, we analyze the existence and uniqueness of solutions for the problem by the Banach fixed point theorem. Moreover, we investigate the behavior of solutions and analyze eigenvalues and eigenfunctions for three-point Hilfer fractional boundary value problem under different values of alpha and types of beta.