Representation of solutions to the Euler type differential equation of fractional order using the fractional analogue of the Green's function

A solution to the nonhomogeneous Euler-type differential equation with Riemann — Liouville fractional derivatives on the half-axis $(0;+\infty)$ in the class ${ I}_{0+}^{\alpha}\left({ L}_{1}(0;+\infty)\right)$ of functions represented by the fractional integral of the order of $\alpha$ with a density from ${ L}_{1}(0;+\infty)$ in terms of the fractional analogue of the Green's function is given by using the direct and inverse Mellin transforms. Fractional analogues of the Green's function are constructed in the case when all roots of the characteristic polynomial are different, and also in the case when there are multiple roots among the roots of the characteristic polynomial. Theorems of solvability of the nonhomogeneous fractional differential equations of Euler-type on the half-axis $(0;+\infty)$ are formulated and proved. Special cases and examples are considered.