Cauchy Problem for Hyperbolic Equations with Nonlocal Potential-Type Terms: Series Expansions of Solutions

We consider the Cauchy problem for equations containing sums of the two-dimensional d’Alembertian and translation operators with respect to the spatial independent variable. In the case where the boundary-value functions belong to the spaces $ C^{2}(-\infty,+\infty) $ and $ C^{1}(-\infty,+\infty) $ , respectively, classical solutions are explicitly represented by function series consisting of iterated means of translated solutions to the Cauchy problem for the wave equation (with the same initial-value functions). The constructed series converge absolutely and uniformly in each finite-width band.