Theory of hybrid differential equations on time scales

In this paper, we develop the theory of hybrid differential equations on time scales. An existence theorem for hybrid differential equations on time scales is given under Lipschitz conditions. Some fundamental fractional differential inequalities are also established which are utilized to prove the existence of extremal solutions. Necessary tools are considered and the comparison principle is proved which will be useful for further study of qualitative behavior of solutions. Our results in this paper extend and improve some known results.