Topological transversality and periodic solutions of neutral functional differential equations

In this paper, we study the existence of periodic solutions of neutral functional differential equations (NFDEs). A topological transversality theorem is used to obtain fixed points of certain nonlinear compact operators, which correspond to periodic solutions of the original differential equations. The method relies on a priori bounds on periodic solutions to a family of appropriately constructed NFDEs. A general existence theorem is proved and several illustrative examples are given where we use Liapunov-like functions in deriving such a priori bounds on periodic solutions. Due to the topological nature of the approach, the theorem applies as well to NFDEs of mixed type and NFDEs with state-dependent delay. Some comparisons between our results and the existing ones are also provided.