Basic results on distributed order fractional hybrid differential equations with linear perturbations

In this article, we develop the distributed order fractional hy- brid differential equations (DOFHDEs) with linear perturbations involving the fractional Riemann-Liouville derivative of order 0 < q < 1 with respect to a nonnegative density function. Furthermore, an existence theorem for the fractional hybrid differential equations of distributed order is proved under the mixed ?-Lipschitz and Caratheodory conditions. Some basic fractional differential inequalities of distributed order are utilized to prove the existence of extremal solutions and comparison principle.