Existence and uniqueness of positive solutions for first-order nonlinear Liouville–Caputo fractional differential equations

We study the existence and uniqueness of positive solutions of the first-order nonlinear Liouville–Caputo fractional differential equation CDαxt-g(t,x(t))=ft,xt,0<t≤1,x0=x0>g0,x0>0,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \left\{ \begin{array}{l} ^{C}D^{\alpha }\left( x\left( t\right) -g( t,x( t) ) \right) =f\left( t,x\left( t\right) \right) ,\quad 0<t\le 1, \\ x\left( 0\right) =x_{0}>g\left( 0,x_{0}\right) >0, \end{array} \right. \end{aligned}$$\end{document}where 0<α≤1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$0<\alpha \le 1$$\end{document}. In the process we convert the given fractional differential equation into an equivalent integral equation. Then we construct appropriate mappings and employ the Krasnoselskii fixed point theorem and the method of upper and lower solutions to show the existence of a positive solution of this equation. We also use the Banach fixed point theorem to show the existence of a unique positive solution. Finally, an example is given to illustrate our results.