Fredholm boundary-value problems for linear delay systems defined by pairwise permutable matrices

摘要

The paper deals with a Fredholm boundary value problem for a linear delay system with several delays defined by pairwise permutable constant matrices. The initial value condition is given on a finite interval and the boundary condition is given by a linear vector functional. A sufficient condition for the existence of solutions of this type of boundary value problem is proved. Moreover, a family of linearly independent solutions in an explicit general analytic form is constructed under the assumption that the number of boundary conditions (defined by a dimension of linear vector functional) do not coincide with the number of unknowns of the system of the delay differential equations. The proof of this result is based on a representation of solutions by using so-called multi-delayed matrix exponential and a method of a pseudo-inverse matrix of the Moore-Penrose type.