Periodic solutions of delay equations with several fixed delays

Abstract

We present an existence result for periodic solutions of autonomous differential delay equations with several fixed delays \begin{eqnarray*} x'(t) = \sum_{i=1}^D f_i (x(t - d_i)), \end{eqnarray*} where the functions $f_i$ are close to two-valued step functions $h_i$. We give conditions under which existence of periodic solutions of the more tractable problem $$ y'(t) = \sum_{i=1}^D h_i (y(t - d_i)) $$ implies the existence of similar periodic solutions of the original problem. Our approach is to show that an analog of a Poincaré map has non-trivial fixed-point index.