The uniqueness of a solution to the inverse Cauchy problem for a fractional differential equation in a Banach space

We consider linear fractional differential operator equations involving Caputo derivative. The goal of this paper is to establish conditions of the unique solvability of the inverse Cauchy problem for these equations. We use properties of the Mittag-Leffler function and the calculus of sectorial operators in a Banach space. For equations with operators in a general form we obtain sufficient conditions for the unique solvability, and for equations with densely defined sectorial operators we obtain necessary and sufficient unique solvability conditions.