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Fixed Point Theorems for Maps With Local and Pointwise Contraction Properties
- Source :
- Canadian Journal of Mathematics. 70:538-594
- Publication Year :
- 2018
- Publisher :
- Canadian Mathematical Society, 2018.
-
Abstract
- This paper constitutes a comprehensive study of ten classes of self-maps on metric spaces ⟨X, d⟩ 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 ⟨X, d⟩ 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 ℝ with f′ ≡ 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.
- Subjects :
- Pointwise
Discrete mathematics
Social connectedness
General Mathematics
010102 general mathematics
Periodic point
Fixed-point theorem
Fixed point
01 natural sciences
Metric space
Compact space
0103 physical sciences
010307 mathematical physics
Differentiable function
0101 mathematics
Mathematics
Subjects
Details
- ISSN :
- 14964279 and 0008414X
- Volume :
- 70
- Database :
- OpenAIRE
- Journal :
- Canadian Journal of Mathematics
- Accession number :
- edsair.doi...........c2e970ae7c1fc124e99b0ca4e7ac24d5