Theory of hybrid differential equations on time scales

Abstract

‎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.