70 Years of asymptotic fixed point theory

Abstract

In this expository article we aim to illustrate asymptotic fixed point theory from the viewpoints of its history, the main motivation from applications, the two basic techniques for proofs and a survey on the major results. The bibliography should cover most aspects but is, for sure, not complete. We confine ourselves to results from topological fixed point theory, i.e., metric fixed point theory is not at all touched.

Dold sequences, periodic points, and dynamics

“…For the case of n prime, the direct analogue of Fermat's little theorem, similar results were shown earlier by Steinlein [118] and by Zabreȋko and Krasnosel'skiȋ [134]. Using the Leray-Schauder degree to extend the concepts to infinite dimensions, the Dold relations have also been shown for suitable maps on Banach spaces by Steinlein [119].…”

“…In the terminology we have adopted (which of course is not that used by Dold) we have the following result. The recent survey by Steinlein [119] with an emphasis on topology and the earlier survey of Nussbaum [97] with an emphasis on non-linear functional analysis, contain much of the interesting history of proofs that (ind(f n )) and (L(f n )) are Dold sequences, described in the more general context of the Leray-Schauder degree. 6.1.1.…”

Generating sequences of Lefschetz numbers of iterates

“…Lemma 10 in [5]). In fact, this statement was known in the literature much earlier and appeared in different contexts (see [22,23]). …”

Generalized Dold Sequences on Partially-Ordered Sets

“…Then in our model (cf. Remark 23) the condition (41) implies that each fixed point x 0 has exactly one target type within the set {n 1 , . .…”