Existence and Boundary Behavior of Positive Solutions for a Semilinear Fractional Differential Equation

We consider the following semilinear fractional initial value problem Dαu(x)=a(x)uσ(x),x∈(0,1)andlimx⟶0+x1-αu(x)=0,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$D^{\alpha} u(x) = a(x)u^{\sigma} (x), x\in (0, 1) \quad {\rm and} \quad \lim\limits_{x \longrightarrow0^{+}}x^{1 - \alpha} u(x) = 0,$$\end{document}where 0<α<1,σ<1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${0 < \alpha < 1, \sigma < 1}$$\end{document} and a is a positive measurable function on (0, 1). We establish the existence and the uniqueness of a positive solution in the space of weighted continuous functions. We also give the boundary behavior of such solution.