Boundary value problems for hybrid differential equations with fractional order

Abstract

This paper is motivated by some papers treating the fractional hybrid differential equations involving Caputo differential operators of order $0 < \alpha < 1$. An existence theorem for this equation is proved under mixed Lipschitz and Caratheodory conditions. Some fundamental fractional differential inequalities which are used 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 more study of qualitative behavior of solutions.