The existence of positive solutions of singular fractional boundary value problems

We discuss the existence of positive solutions for the singular fractional boundary value problem Dαu+f(t,u,u′,Dμu)=0Dαu+f(t,u,u′,Dμu)=0, u(0)=0u(0)=0, u′(0)=u′(1)=0u′(0)=u′(1)=0, where 2<α<32<α<3, 0<μ<10<μ<1. Here DαDα is the standard Riemann–Liouville fractional derivative of order αα, ff is a Carathéodory function and f(t,x,y,z)f(t,x,y,z) is singular at the value 0 of its arguments x,y,zx,y,z.