On the existence of solutions for a differential inclusion of fractional order with upper-semicontinuous right-hand side

We prove a theorem on the existence of solutions of the differential inclusion\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$D_0^\alpha u(x) \in F(x,u(x)), u_{1 - \alpha } (0) = \gamma , \left( {u_{1 - \alpha } (x) = 1_0^{1 - \alpha } u(x)} \right),$$ \end{document} where\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\alpha \in (0,1), D_0^\alpha u(x) \left( {1_0^{1 - \alpha } u(x)} \right)$$ \end{document} is the Riemann-Liouville derivative (integral) of order α, and the multivalued mappingF(x, u) is upper semicontinuous inu.