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

Kuczumow, Tadeusz; Reich, Simeon; Stachura, Adam

doi:10.1090/S0002-9947-1994-1242784-0

Minimal displacement of points under holomorphic mappings and fixed point properties for unions of convex sets
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by Tadeusz Kuczumow, Simeon Reich and Adam Stachura

Trans. Amer. Math. Soc. 343 (1994), 575-586

DOI: https://doi.org/10.1090/S0002-9947-1994-1242784-0

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Abstract:

Let D be an open convex bounded subset of a complex Banach space $(X,\left \| \cdot \right \|)$, 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 $\lambda ({f_{|C}}) \ne 0$, then $\inf \{ \left \| {x - f(x)} \right \|:x \in C\} = 0$. We then deduce several new fixed point theorems for holomorphic and nonexpansive mappings.

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