Global Classical Solutions of Some Two-Dimensional Hyperbolic Differential-Difference Equations

For a two-dimensional hyperbolic equation with a differential-difference operator acting with respect to the spatial variable, we construct a one-parameter family of global solutions. We prove that these solutions are classical for all values of the real parameter provided that the real part of the symbol of the difference operator occurring in the equation is positive. We present a class of equations for which this condition is satisfied.