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 a connection between these problems and nonlocal mixed problems for strongly elliptic differential equations. We prove the uniqueness and the smoothness of their solutions.