Existence of positive solutions for a class of fractional differential equations with the derivative term via a new fixed point theorem

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\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$A(x,x)+ B(x,x)+C(x)+e = x$\end{document}, where A and B are two mixed monotone operators, C is a decreasing operator, and e∈P\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$e\in P$\end{document} with θ≤e≤h\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\theta \leq e \leq h$\end{document}. 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.