  We construct stable periodic solutions for a simple form nonlinear delay
differential equation (DDE) with a periodic coefficient. The equation involves
one underlying nonlinearity with the multiplicative periodic coefficient. The
well-known idea of reduction to interval maps is used in the case under
consideration, when both the defining nonlinearity and the periodic coefficient
are piece-wise constant functions. The stable periodic dynamics persist under a
smoothing procedure in a small neighborhood of the discontinuity set. This work
continues the research in recent paper [7] on stable periodic solutions of
differential delay equations with periodic coefficients.
