On the Dirichlet problem for differential-difference elliptic equations in a half-plane

The Dirichlet problem is considered in a half-plane (with continuous and bounded boundaryvalue function) for the model elliptic differential-difference equation
$$ u_{xx}+au_{xx}(x+h,y)+u_{yy}=0,\qquad|a|<1. $$
Its solvability is proved in the sense of generalized functions, the integral representation of the solution is constructed, and it is proved that everywhere but the boundary hyperplane this solution satisfies the equation in the classic sense as well.