Existence of Solutions of a Boundary Value Problem fora Diffusion Equation with Piecewise Constant Arguments

We study a boundary value problem for a diffusion equation with piecewise constant arguments. By separation of variables, the analysis of this boundary value problem is reduced to that of the existence conditions for solutions of initial value problems for a differential equation with piecewise constant arguments. Conditions for the existence of infinitely many solutions or the emptiness of the solution set are established for the differential equation in question, and explicit formulas for these solutions are obtained. Several examples are given to illustrate the results.