Criterion for unique solvability of a periodic boundary value problem for a functional-differential equation (Q1902832)

The author considers the boundary value problem for the functional differential equation \[ x^{(n)} (t) - \sum^{n - 1}_{i = 0} \int^b_a x^{(i)} (s) d_s r_i (t,s) = f(t),\;t \in [a,b], \quad x^{(i)} (b) - x^{(i)} (a) = \alpha_i,\;i = 0, \dots, n - 1, \] where the functions \(r_i (t,s)\) are measurable on \([a,b] \times [a,b]\). She obtains a necessary and sufficient condition for this problem to have a unique solution.