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Fixed Point Theorems for Maps With Local and Pointwise Contraction Properties

Published online by Cambridge University Press:  20 November 2018

Krzysztof Chris Ciesielski  and
Jakub Jasinski
Krzysztof Chris Ciesielski
Affiliation:
Department of Mathematics, West Virginia University, Morgantown, WV 26506-6310 and Department of Radiology, MIPG, University of Pennsylvania, Philadelphia, PA 19104-6021 email: kcies@math.wvu.edu
Jakub Jasinski
Affiliation:
Mathematics Department, University of Scranton, Scranton, PA 18510 email: jakub.jasinski@scranton.edu
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Abstract

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This paper constitutes a comprehensive study of ten classes of self-maps on metric spaces $\langle X\,,\,d\rangle $ with the pointwise (i.e., local radial) and local contraction properties. Each of these classes appeared previously in the literature in the context of fixed point theorems.

We begin with 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 that 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 these 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 by discussing a few remaining open problems on whether the maps with specific pointwise contraction properties must have the fixed points.

Keywords

54H25, 37C25fixed pointperiodic pointcontractive maplocally contractive mappointwise contractive mapradially contractive maprectifiable-path connected spaced-convexgeodesicremetrizationcontraction mapping principle
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 70 , Issue 3 , 01 June 2018 , pp. 538 - 594
Copyright
Copyright © Canadian Mathematical Society 2018

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