Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Minimal displacement of points under holomorphic mappings and fixed point properties for unions of convex sets

Authors: Tadeusz Kuczumow, Simeon Reich and Adam Stachura
Journal: Trans. Amer. Math. Soc. 343 (1994), 575-586
MSC: Primary 47H10; Secondary 32K05, 47H09
MathSciNet review: 1242784
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Abstract: Let D be an open convex bounded subset of a complex Banach space $ (X,\left\Vert \cdot \right\Vert)$, and let C be the union of a finite number of closed convex sets lying strictly inside D. Using the Kuratowski measure of noncompactness with respect to the Kobayashi distance in D, we first show that if $ f:D \to D$ is a holomorphic mapping which leaves C invariant, and if the Lefschetz number $ \lambda ({f_{\vert C}}) \ne 0$, then $ \inf \{ \left\Vert {x - f(x)} \right\Vert:x \in C\} = 0$. We then deduce several new fixed point theorems for holomorphic and nonexpansive mappings.

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Keywords: Fixed point, holomorphic mapping, Kobayashi distance, measure of noncompactness, minimal displacement, nonexpansive mapping
Article copyright: © Copyright 1994 American Mathematical Society