The positive properties of Green’s function for three point boundary value problems of nonlinear fractional differential equations and its applications

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 ) ) + e ( t ) = 0 , 0 < t < 1 , u ( 0 ) = 0 , D 0 + β u ( 1 ) = a D 0 + β u ( ξ ) where 1 < α ⩽ 2 , 0 < β ⩽ 1 , 0 ⩽ a ⩽ 1 , 0 < ξ < 1 , α - β - 1 ⩾ 0 , D 0 + α is the standard Riemann–Liouville derivative. Here our nonlinearity f may be singular at u = 0. As applications of Green’s function, we give some multiple positive solutions for singular boundary value problems by means of Schauder fixed-point theorem.