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

On the solvability of the mixed problem for a fractional-order equation  with a time-delay argument and pseudodifferential operators with non-local  boundary conditions in Sobolev classes