Half-plane differential-difference elliptic problems with general-kind nonlocal potentials

In the half-plane, the Dirichlet problem is considered for elliptic differential-difference equations with nonlocal general-kind potentials, which are linear combinations of translations of the desired function, not bounded by commensurability conditions. We find a condition for the symbol of the corresponding differential-difference operator, providing the classical solvability of the specified problem for each continuous and bounded boundary-value function. The representation of the specified classical solution by a Poisson-type integral is constructed. © 2020 Informa UK Limited, trading as Taylor & Francis Group.