Fredholm’s boundary-value problems for differential systems with a single delay

Conditions are derived for the existence of solutions of linear Fredholm’s boundary-value problems for systems of ordinary differential equations with constant coefficients and a single delay. Utilizing a delayed matrix exponential and a method of pseudo-inverse by Moore–Penrose matrices led to an explicit and analytical form of a criterion for the existence of solutions in a relevant space and, moreover, to the construction of a family of linearly independent solutions of such problems in a general case with the number of boundary conditions (defined by a linear vector functional) not coinciding with the number of unknowns of a differential system with a single delay.