Naimark Problem for a Fractional Ordinary Differential Equation

For a fractional ordinary differential equation, we consider a problem where the boundary conditions are given in the form of linear functionals. This permits covering a fairly broad class of linear local and nonlocal conditions. The fractional derivative is understood in the sense of Gerasimov–Caputo. A necessary and sufficient condition for the unique solvability of the problem is obtained. A representation of the solution via special functions is found. A theorem on the existence and uniqueness of the solution is proved.