Nonlocal boundary value problem for a linear ordinary delay differential equation with Gerasimov–Caputo derivative

In this paper, for a linear ordinary delay differential equation with constant coefficients and with the Gerasimov–Caputo derivative, a solution to the nonlocal boundary value problem with conditions, connecting the value of the unknown function at the end of the interval with the values at interior points, is constructed. The solution to the problem is obtained in explicit form. The condition for the unique solvability of the problem is written out. A representation of Green’s function is obtained. Green’s function is defined in terms of the special function $W_\nu(t)$, which, in turn, is defined using the generalized Mittag-Leffler functions. The properties of the Green’s function are proven. The solution to the problem is formulated in terms of Green’s function. The existence and uniqueness theorem to the problem under study is formulated and proved. The proofs of the lemma and theorem are given using methods of the theory of fractional calculus, the theory of special functions, Green’s function method, and the theory of integral equations.