Non-local boundary value problem for a system of ordinary differential equations with Riemann–Liouville derivatives

We study a nonlocal boundary value problem for a linear system of ordinary differential equations of fractional order with constant coefficients on the interval $[0,l]$. The fractional derivative of order $\alpha \in (0,1]$ is understood in the Riemann–Liouville sense. The boundary conditions connect the trace of the fractional integral of the desired vector function at the left end of the segment – at the $x=0$, with the trace of the vector function itself at the right end of the segment at the point $x=l$. The purpose of this work is to construct an explicit representation of the solution of this problem in terms of the Green's function. The structure of the solution to the boundary value problem is investigated, the corresponding Green's function is defined and constructed, and the representation of the solution is obtained. A theorem on the unique solvability of the boundary value problem under study is proved.