On the Classical and Generalized Solutions of Boundary-Value Problems for Difference-Differential Equations with Variable Coefficients

The first boundary-value problem for second-order difference-differential equations with variable coefficients on a finite interval $(0,d)$ is considered. The following question is studied: Under what conditions will the boundary-value problem for a difference-differential equation have a classical solution for an arbitrary continuous right-hand side? It is proved that a necessary and sufficient condition for the existence of a classical solution is that certain coefficients of the difference operators on the orbits generated by the shifts be equal to zero.