Abstract: The critical point theory and the variational method were used to study the existence of the solution for the Caputo type fractional differential equation with the Sturm-Liouville boundary condition in Banach space. By defining the appropriate fractional derivative space, the existence of the solution to the boundary value problem of fractional differential equation was transformed into finding the critical point defined as the corresponding functional in a certain space, and a series of unbounded generalized solutions to the boundary value problem were obtained.

Key words: Sturm-Liouville boundary condition, critical point theory, variational method, discontinuous fractional derivative