Existence of multiple positive solutions for singular boundary value problems of nonlinear fractional differential equations

In this paper, we consider the properties of the Green’s function for the nonlinear fractional differential equation boundary value problem D0+qu(t)=f(t,u(t)), t∈J:=[0,1], u(0)=u′(1)=0, where 1<q≤2 is a real number, and D0+q is the standard Riemann-Liouville differentiation. As an application of the Green’s function, we give some multiple positive solutions for singular boundary value problems, and we also give the uniqueness of solution for a singular problem by means of the Leray-Schauder nonlinear alternative, a fixed-point theorem on cones, and a mixed monotone method.