  In this work, a mixed problem for a time-fractional equation with a delayed
argument and pseudodifferential operators related to Laplace operators with
non-local boundary conditions in Sobolev classes is studied. The solutions to
the initial-boundary problem are constructed as a series of the eigenfunctions
of a multidimensional spectral problem. The eigenvalues of the spectral problem
are found, and the corresponding system of eigenfunctions is constructed. It is
shown that this system of eigenfunctions is complete and forms a Riesz basis in
the Sobolev subspaces. Based on the completeness of the system of
eigenfunctions, the uniqueness theorem for the solution of the problem is
proven. The existence of a regular solution to the posed initial-boundary
problem in the Sobolev subspaces is also demonstrated.
