Functions of finite fractional variation and their applications to fractional impulsive equations

summary:We introduce a notion of a function of finite fractional variation and characterize such functions together with their weak $\sigma $-additive fractional derivatives. Next, we use these functions to study differential equations of fractional order, containing a $\sigma $-additive term---we prove existence and uniqueness of a solution as well as derive a Cauchy formula for the solution. We apply these results to impulsive equations, i.e. equations containing the Dirac measures