On the Karlsson–Nussbaum conjecture for resolvents of nonexpansive mappings

Authors

  • Aleksandra Huczek Pedagogical University of Krakow, Department of Mathematics
  • Andrzej Wiśnicki Pedagogical University of Krakow, Department of Mathematics

Keywords:

Karlsson–Nussbaum conjecture, Wolff–Denjoy theorem, geodesic space, Hilbert's projective metric, resolvent, nonexpansive mapping

Abstract

Let \(D\subset \mathbb{R}^{n}\) be a bounded convex domain and \(F\colon D\rightarrow D\) a 1-Lipschitz mapping with respect to the Hilbert metric \(d\) on \(D\) satisfying condition \(d(sx+(1-s)y,sz+(1-s)w)\leq \max \{d(x,z),d(y,w)\}\). We show that if \(F\) does not have fixed points, then the convex hull of the accumulation points (in the norm topology) of the family \(\{R_{\lambda}\}_{\lambda >0}\) of resolvents of \(F\) is a subset of \(\partial D\). As aconsequence, we show a Wolff-Denjoy type theorem for resolvents of nonexpansive mappings acting on an ellipsoid \(D\).
Section
Articles

Published

2023-01-15

How to Cite

Huczek, A., & Wiśnicki, A. (2023). On the Karlsson–Nussbaum conjecture for resolvents of nonexpansive mappings. Annales Fennici Mathematici, 48(1), 153–161. https://doi.org/10.54330/afm.126009