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

Abstract: 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.