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Keywords:
finite fractional variation; weak $\sigma $-additive fractional; derivative; fractional impulsive equation; Dirac measure; Cauchy formula
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.
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