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.

Some comparison theorems of Liapunov-Razumikhin type are provided for uniform (asymptotic) stability and uniform (ultimate) boundedness of solutions to neutral functional differential equations with infinite delay with respect to a given phase space pair. Examples are given to illustrate how the comparison theorems and stability and boundedness of solutions depend on the choice(s) of phase space(s) and are related to asymptotic behavior of solutions to some difference and integral equations.

An analog of the Hopf bifurcation theorem is proved for implicit neutral functional differential equations of the form F(xt, D′(xt, α), α) = 0. The proof is based on the method of S1-degree of convex-valued mappings. Examples illustrating the theorem are provided.

In this paper, we present some results on the existence of periodic solutions to Volterra integro-differential equations of neutral type. The main idea is to show the convergence of an equibounded sequence of periodic solutions of certain limiting equations which are of finite delay. This makes it possible to apply the existing Liapunov–Razumikhin technique for neutral equations with finite delay to obtain existence of periodic solutions of Volterra neutral integro-differential equations (of infinite delay). Some comparisons between our results and the existing ideas are also provided.