Functions of bounded fractional differential variation: A new Banach algebra

Prajapati and Kachhia [10] introduced the concept of functions of bounded fractional differential variation and proposed an open problem whether the class of functions of bounded fractional differential variation on a closed bounded interval is a Banach algebra. We have given a solution to this problem. For a sequence of functions, we have discussed a condition for the convergence of sequence of Caputo fractional derivatives to the Caputo fractional derivative of the limit function. We have proved an additive property of the total fractional differential variation with respect to closed intervals and given an equivalent condition for vanishing of the total fractional differential variation of a function.