Existence and uniqueness of positive solutions for a class of nonlinear fractional differential equations with mixed-type boundary value conditions

In this article, we study a class of nonlinear fractional differential equations with mixed-type boundary conditions. The fractional derivatives are involved in the nonlinear term and the boundary conditions. By using the properties of the Green function, the fixed point index theory and the Banach contraction mapping principle based on some available operators, we obtain the existence of positive solutions and a unique positive solution of the problem. Finally, two examples are given to demonstrate the validity of our main results.