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In this paper, we firstly establish the existence and uniqueness of solutions of the operator equation $A(x,x)+ B(x,x)+C(x)+e = x$ A ( x , x ) + B ( x , x ) + C ( x ) + e = x , where A and B are two mixed monotone operators, C is a decreasing operator, and $e\in P$ e ∈ P with $\theta \leq e \leq h$ θ ≤ e ≤ h . Then, using our abstract theorem, we prove a class of fractional boundary value problems with the derivative term to have a unique solution and construct the corresponding iterative sequences to approximate the unique solution.

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