Fixed point theorems for maps with local and pointwise contraction properties


Krzysztof Chris Ciesielski and Jakub Jasinski

57 pages, Canad. J. Math., to appear.

The paper constitutes a comprehensive study of ten classes of self-maps on metric spaces (X,d) 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. It is the notion of pointwise contraction, that is the main motivation for this work. Several new results have been recently published in this context and there are still interesting open problems. Pointwise contraction is also most closely related to the notion of a derivative, while the theory of differential equation is one of the most prominent areas of the applications of fixed point theorems.

We begin with presenting a survey 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 self-maps with domains (X,d) 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 non-reversibility of the previously established inclusions between theses classes. Among these examples, the most striking is a differentiable auto-homeomorphism f of a compact perfect subset X of R with f'(x)=0 for all x in X, which constitutes also a minimal dynamical system. We finish with discussing a few remaining open problems on wether the maps with specific pointwise contraction properties must have the fixed points. All these problems are for compact and (path) connected action spaces (X,d).

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Last modified January 10, 2017.