Use of Mixed Operator Method to a fractional Hadamard Dirichlet boundary value problem

The purpose of this paper is to deal with the following nonlinear Hadamard fractional boundary value problem HDα1+ u(t) + f(t, u(t), u(t)) + g(t, u(t)) = 0,1 < t < e, 1 < α ≤ 2,u(1) = u(e) = 0, where HDα 1+ is the Hadamard fractional derivative operator. Using the mixed monotone operator method, we prove an existence and uniqueness result for this mixed fractional Hadamard boundary value problem. As an application of this result, we give one example to establish an existence and uniqueness of a positive solution.