Stable periodic solutions in scalar periodic differential delay equations

A class of nonlinear simple form differential delay equations with a $T$-periodic coefficient and constant $\tau >0$ is considered. It shown that for an arbitrary value the period $T>4\tau -d_0$, some $d_0>0$, there equation in such it possesses asymptotically stable $T$-period solution. The periodic solutions are constructed explicitly piecewise nonlinearities coefficients involved, by reduction problem to one-dimensional maps. their stability properties persist when “smoothed” at discontinuity points.