An index for periodic orbits of functional differential equations (Q1114070)

Let \(r>0\) and call \(C:=C([-r,0],{\mathbb{R}}^ n)\). Let f: \(C\to {\mathbb{R}}^ n\) be a \(C^ 2\)-mapping which maps bounded sets in C into bounded sets in \({\mathbb{R}}^ n\). \textit{S. N. Chow} and \textit{J. Mallet-Paret} [J. Differ. Equations 39, 66-85 (1978; Zbl 0349.34027)] sketched a proof for the existence of an index for periodic orbits of the local semiflow generated by the functional differential equation \(y'(t)=f(y(t+\cdot)).\) We show that one can obtain such an index by a direct generalization of ideas due to \textit{F. B. Fuller} [Am. J. Math.89,137-148 (1967; Zbl 0152.402)] to an infinite-dimensional setting. We avoid the complicated bifurcation argument by Chow and Mallet-Paret by applying the results of our earlier article [J. Math. Anal. Appl. 129, 517-532 (1988)]. Moreover, a substantial part of our construction works for general semiflows on arbitrary ANR spaces without requirements on differentiability.