Initial value problems for equations with a composition of fractional derivatives

We study the unique solvability of initial problems for linear equations in Banach spaces with a composition of two fractional derivatives and with a bounded operator on the right side. It is shown that the compositions of fractional derivatives of Riemann — Liouville and (or) Gerasimov — Caputo are derivatives of Dzhrbashyan — Nersesyan. With the help of the previously obtained general results on the initial problem for a linear equation with the Dzhrbashyan — Nersesyan fractional derivative, statements are formulated about the existence and uniqueness of a solution for initial problems to the equations under study with a composition of two fractional derivatives. The solutions are presented using the Mittag-Leffler functions. The general results are demonstrated by the example of an initial boundary value problem for an equation with polynomials with respect to the Laplace operator.