Existence and uniqueness of solution for Sturm–Liouville fractional differential equation with multi-point boundary condition via Caputo derivative

We investigate the existence and uniqueness of a solution for a Sturm–Liouville fractional differential equation with a multi-point boundary condition via the Caputo derivative; existence and uniqueness results for the given problem are obtained via the Banach fixed point theorem. Also we study its continuous dependence on coefficients of the nonlocal condition. We discuss our results for more general boundary conditions, we present the existence of solutions under nonlocal integral conditions and also extend our results to an ordinary Sturm–Liouville problem. Two examples illustrating the main results are also presented.