Positive properties of Green's function for three-point boundary value problems of nonlinear fractional differential equations and its applications

In this article, we consider the properties of Green's function for the nonlinear fractional differential equation boundary value problem D ₀₊ ^αu(t)+f(t,u(t)=0, 0 < t < 1, u(0)=0, u(1) = βu(η), where 1 < α < 2, 0 < β, η < 1, D ₀₊ ^α is the standard Riemann-Liouville derivative. Here our nonlinearity f may be singular at u = 0. As an application of Green's function, we establish some multiple positive solutions for singular positone and semipositone boundary value problems by means of the Leray-Schauder nonlinear alternative, a fixed-point theorem on cones, and also we give uniqueness of a solution for a singular problem by a mixed monotone method.