Theory of fractional hybrid differential equations with linear perturbations of second type

In this paper, we develop the theory of fractional hybrid differential equations with linear perturbations of second type involving Riemann-Liouville differential operators of order 0<q<1 . An existence theorem for fractional hybrid differential equations is proved under the φ -Lipschitz condition. 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 which will be useful for further study of qualitative behavior of solutions is proved. MSC: 34A40, 34A12, 34A99.