A sum operator method for the existence and uniqueness of positive solutions to Riemann–Liouville fractional differential equation boundary value problems

In this paper, we are concerned with the existence and uniqueness of positive solutions for the following fractional boundary value problems given by - D 0 + α u ( t ) = f ( t , u ( t ) ) + g ( t , u ( t ) ) , 0 < t < 1 , 3 < α ⩽ 4 , where D 0 + α is the standard Riemann–Liouville fractional derivative, subject either to the boundary conditions u ( 0 ) = u ′ ( 0 ) = u ″ ( 0 ) = u ″ ( 1 ) = 0 or u ( 0 ) = u ′ ( 0 ) = u ″ ( 0 ) = 0 , u ″ ( 1 ) = β u ″ ( η ) for η , β η α - 3 ∈ ( 0 , 1 ) . Our analysis relies on a fixed point theorem of a sum operator. Our results can not only guarantee the existence of a unique positive solution, but also be applied to construct an iterative scheme for approximating it. Two examples are given to illustrate the main results.