\(C^{k}\) invariant manifolds for maps of Banach spaces (Q1604223)

By using some fixed point theorems for self maps on closed subsets of a Banach space of functions, the author gives a unified proof for the existence, under appropriate gap conditions, of the standard manifolds for a class of \(C^{k, \delta}\) self maps, \(0 \leq \delta \leq 1\), of a Banach space. These results extends to the case \(\delta=0\) his own results recently obtained in \textit{M. S. ElBialy} [J. Math. Anal. Appl. 232, 229-258 (1999; Zbl 0938.47041)].