Existence, Uniqueness and Stability of Slowly Oscillating Periodic Solutions for Delay Differential Equations with Non-negativity Constraints

Deterministic dynamical system models with delayed feedback and state constraints arise in a variety of applications in science and engineering. Under certain conditions oscillatory behavior has been observed and it is of interest to know when there are periodic solutions. Here we consider one-dimensional delay differential equations with non-negativity constraints as prototypes for such models. We obtain sufficient conditions for the existence of slowly oscillating periodic solutions of such equations when the delay/lag interval is long and the dynamics depend only on the current and lagged state. Under further assumptions, including possibly longer delay /lag intervals and restricting the dynamics to depend only on the lagged state, we prove uniqueness and a strong form of asymptotic stability for such solution