The existence of multiple positive solutions for boundary value problems of nonlinear fractional differential equations

In this paper, we study the existence on multiple positive solutions for the nonlinear fractional differential equation boundary value problem. D 0 + α u ( t ) + f ( t , u ( t ) ) = 0 , 0 < t < 1 , u ( 0 ) = u ′ ( 0 ) = u ′ ( 1 ) = 0 , 2 < α ⩽ 3 is a real number, D 0 + α is the Riemann–Liouville fractional derivative. By the properties of the Green function, the lower and upper solution method and fixed point theorem, some new existence criteria for singular and nonsingular fractional differential equation boundary value problem are established. As applications, examples are presented to illustrate the main results.