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# Fixed point theorems for maps with local and pointwise contraction properties

Published:2017-06-22

• Krzysztof Chris Ciesielski,
Department of Mathematics, West Virginia University, Morgantown, WV 26506-6310
• Jakub Jasinski,
Mathematics Department, University of Scranton, Scranton, PA 18510
 Format: LaTeX MathJax PDF

## Abstract

The paper constitutes a comprehensive study of ten classes of self-maps 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 self-maps 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 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 $\mathbb R$ 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, d-convex, geodesic, remetrization contraction mapping principle
 MSC Classifications: 54H25 - Fixed-point and coincidence theorems [See also 47H10, 55M20] 37C25 - Fixed points, periodic points, fixed-point index theory

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