Positive solutions for Dirichlet problems of singular nonlinear fractional differential equations

In this paper, we investigate the existence of positive solutions for the singular fractional boundary value problem: D α u ( t ) + f ( t , u ( t ) , D μ u ( t ) ) = 0 , u ( 0 ) = u ( 1 ) = 0 , where 1 < α < 2 , 0 < μ ⩽ α − 1 , D α is the standard Riemann–Liouville fractional derivative, f is a positive Carathéodory function and f ( t , x , y ) is singular at x = 0 . By means of a fixed point theorem on a cone, the existence of positive solutions is obtained. The proofs are based on regularization and sequential techniques.