On a certain property of a regular difference operator with variable coefficients

We consider a regular difference operator with variable coefficients in a bounded domain. It will be proved that this operator maps continuously and bijectively the Sobolev space of order k with the homogeneous Dirichlet boundary conditions to the subspace of Sobolev space of order k with nonlocal boundary conditions on the shifts of the boundary. This allows to apply the results obtained for nonlocal elliptic problems to the investigation of elliptic differential-difference equations. © 2019, © 2019 Informa UK Limited, trading as Taylor & Francis Group.