Multiple positive solutions for the boundary value problem of a nonlinear fractional differential equation

Abstract

In this paper, we consider the properties of Green’s function for the nonlinear fractional differential equation boundary value problem D 0 + α u ( t ) = f ( t , u ( t ) ) , 0 < t < 1 , u ( 0 ) = u ( 1 ) = u ′ ( 0 ) = u ′ ( 1 ) = 0 , where 3 < α ≤ 4 is a real number, and D 0 + α is the standard Riemann–Liouville differentiation. As an application of Green’s function, we give some multiple positive solutions for singular and nonsingular 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.