Boundary value problems for hybrid differential equations with fractional order

This note is motivated by some papers treating the fractional hybrid differential equations involving Riemann-Liouville differential operators of order 0 < α< 1 . An existence theorem for this equation is proved under mixed Lipschitz and Caratheodory conditions. Some fundamental fractional differential inequalities which are utilized to prove the existence of extremal solutions are also established. Necessary tools are considered and the comparison principle is proved, which will be useful for further study of qualitative behavior of solutions.