Generalized Dirichlet problem for an ordinary delay differential equation with Dzhrbashyan - Nersesyan derivative

In recent decades, interest in the study of differential equations involving fractional derivatives has noticeably increased. This interest is due to the fact that the number of fields of science in which equations containing fractional derivatives are used varies from biology and medicine to management theory, engineering, finance, as well as optics, physics and so on. In this paper, the generalized Dirichlet problem is investigated for a linear ordinary delay differential equation with Dzhrbashyan - Nersesyan fractional differentiation operator. A condition for unique solvability is obtained. The existence and uniqueness theorem to the solution is proved. The solution of the problem is written out in terms of the special function $W_\nu(t)$, which is defined in terms of the generalized Mittag - Leffler function (Prabhakar function).