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Search: MSC category 54H25 ( Fixed-point and coincidence theorems [See also 47H10, 55M20] )

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1. CJM Online first

Ciesielski, Krzysztof Chris; Jasinski, Jakub
 Fixed point theorems for maps with local and pointwise contraction properties 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 principleCategories:54H25, 37C25

2. CJM 2011 (vol 63 pp. 533)

Espínola, Rafa; Fernández-León, Aurora
 On Best Proximity Points in Metric and Banach Spaces In this paper we study the existence and uniqueness of best proximity points of cyclic contractions as well as the convergence of iterates to such proximity points. We take two different approaches, each one leading to different results that complete, if not improve, other similar results in the theory. Results in this paper stand for Banach spaces, geodesic metric spaces and metric spaces. We also include an appendix on CAT$(0)$ spaces where we study the particular behavior of these spaces regarding the problems we are concerned with. Categories:54H25, 47H09
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