On differentiability of solutions of fractional differential equations with respect to initial data

In this paper, we deal with a Cauchy problem for a nonlinear fractional differential equation with the Caputo derivative of order $\alpha \in (0, 1)$. As initial data, we consider a pair consisting of an initial point, which does not necessarily coincide with the inferior limit of the fractional derivative, and a function that determines the values of a solution on the interval from this inferior limit to the initial point. We study differentiability properties of the functional associating initial data with the endpoint of the corresponding solution of the Cauchy problem. Stimulated by recent results on the dynamic programming principle and Hamilton--Jacobi--Bellman equations for fractional optimal control problems, we examine so-called fractional coinvariant derivatives of order $\alpha$ of this functional. We prove that these derivatives exist and give formulas for their calculation.