The third mixed boundary-value problem for strongly elliptic differential-difference equations in a bounded domain

We consider strongly elliptic differential-difference equations with mixed boundary conditions in a bounded domain. There are homogeneous Dirichlet conditions on a part of the boundary, and boundary conditions of the third kind on the other part of the boundary. We establish the connection between these problems and nonlocal mixed problems for strongly elliptic differential equations. We prove the uniqueness and the smoothness of their solutions.