Model Example for Study Smoothness of Solutions of Boundary-Value Problems for Differential-Difference Equations

The purpose of this article is to demonstrate with a simple example the complexities that arise when considering boundary-value problems for differential-difference equations. In this paper we discuss in detail smoothness issues of solutions to the Dirichlet problem for linear differential-difference equation with integer shifts. After analysis of this example we consider more general statement of the problem and certain results concerning the solvability and smoothness of solutions.