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

Abstract

We prove a theorem on the existence of solutions of the differential inclusion \(D_0^\alpha u(x) \in F(x,u(x)), u_{1 - \alpha } (0) = \gamma , \left( {u_{1 - \alpha } (x) = 1_0^{1 - \alpha } u(x)} \right),\) where \(\alpha \in (0,1), D_0^\alpha u(x) \left( {1_0^{1 - \alpha } u(x)} \right)\) is the Riemann-Liouville derivative (integral) of order α, and the multivalued mappingF(x, u) is upper semicontinuous inu