
Fixed point theorems for maps with local and pointwise contraction properties
The paper constitutes a comprehensive study of ten classes of
selfmaps on metric spaces $\langle X,d\rangle$ with the local
and pointwise (a.k.a. local radial) contraction properties.
Each of those classes appeared previously in the literature in
the context of fixed point theorems.
We begin with presenting an overview
of these fixed point results, including concise self contained
sketches of their proofs.
Then, we proceed with a discussion of the relations among the
ten classes of selfmaps with domains $\langle X,d\rangle$
having various topological properties which often appear in the
theory of fixed point theorems: completeness, compactness, (path)
connectedness, rectifiable path connectedness, and $d$convexity.
The bulk of the results presented in this part consists of examples
of maps that show nonreversibility of the previously established
inclusions between theses classes.
Among these examples, the most striking is a differentiable autohomeomorphism
$f$ of a compact perfect subset $X$ of $\mathbb R$
with $f'\equiv 0$, which constitutes also a minimal dynamical
system.
We finish with discussing a few remaining open problems on weather
the maps with specific pointwise contraction
properties must have the fixed points.
Keywords:fixed point, periodic point, contractive map, locally contractive map, pointwise contractive map, radially contractive map, rectifiably path connected space, dconvex, geodesic, remetrization contraction mapping principle Categories:54H25, 37C25 