On the Existence of Smooth Solutions to a Hyperbolic Differential-Difference Equation

The paper studies the question of the existence of classical solutions to a two-dimensionalhyperbolic equation in a half-plane containing translation operators in lower derivatives actingwith respect to a spatial variable. All translations take arbitrary real values. A multiparameterfamily of infinitely smooth solutions to the specified equation is constructed in explicit form usingan operational scheme. A theorem is proved that the constructed solutions are classical if the realpart of the symbol of the differential-difference operator in the equation is positive.