On equations determining Green's matrix of a general boundary value problem for a functional-differential equation (Q1902767)

The authors consider a functional differential equation of the type \[ {d\over dt} \int^b_a Q(t, \tau) \dot x(\tau)d\tau+ A(t)x(a)= f(t), \quad t\in [a, b],\tag{1} \] \[ \int^b_a \phi(\tau) \dot x(\tau) d\tau+ \psi x(a)= \alpha.\tag{2} \] They obtain necessary and sufficient conditions for the existence of a Green's matrix \(G(t, s)\) such that the solutions of (1), (2) can be represented in the form \[ x(t)= X(t) \alpha+ \int^b_a G(t, s) f(s),\quad t\in [a, b]. \]