Theory of hybrid differential equations on time scales

In this paperý, ýwe develop the theory of hybrid differentialý ýequations on time scalesý. ýAn existence theorem for hybridý ýdifferential equations on time scales is given under Lipschitz ýconditionsý. ýSome fundamental fractional differential inequalitiesý ýare also established which are utilized to prove the existence ofý ýextremal solutionsý. ýNecessary tools are considered and theý ýcomparison principle is proved which will be useful for furtherý ýstudy of qualitative behavior of solutionsý. ýOur results in this ýpaper extend and improve some known results.