On an ill-posed problem for the Laplace operator with nonlocal boundary condition

In this paper a nonlocal problem for the Poisson equation in a rectangular is considered. It is shown that this problem is ill-posed as well as the Cauchy problem for the Laplace equation. The method of spectral expansion via eigenfunctions of the nonlocal problem for equations with deviating argument allows us to establish a criterion of the strong solvability of the considered nonlocal problem. It is shown that the ill-posedness of the nonlocal problem is equivalent to the existence of an isolated point of the continuous spectrum for a nonself-adjoint operator with the deviating argument.