Abstract
In this paper, we obtain some results of the existence and uniqueness of ageneralized solution for a singular fractional initial boundary value problem in the Caputo sense subject to Neumann and weighted integral conditions. We show that a priori estimate or energy inequality methods can be successfully applied to obtaining a priori estimates for the solution of initial fractional boundary problems as in the classical case. The obtained results will contribute in the development of the functional analysis method and enrich the existing nonextensive literature on the nonlocal fractional mixed problems in the Caputo sense.