Geodesic
On the Karlsson–Nussbaum conjecture for resolvents of nonexpansive mappings
Aleksandra Huczek1 ; Andrzej Wiśnicki1
1Pedagogical University of Krakow, Department of Mathematics
Annales Fennici Mathematici, Tome 48 (2023) no. 1, pp. 153-161

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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 $D$.
DOI : 10.54330/afm.126009
Keywords: Karlsson–Nussbaum conjecture, Wolff–Denjoy theorem, geodesic space, Hilbert's projective metric, resolvent, nonexpansive mapping

Aleksandra Huczek  1   ; Andrzej Wiśnicki  1

1 Pedagogical University of Krakow, Department of Mathematics 
Aleksandra Huczek; Andrzej Wiśnicki. On the Karlsson–Nussbaum conjecture for resolvents of nonexpansive mappings. Annales Fennici Mathematici, Tome 48 (2023) no. 1, pp. 153-161. doi: 10.54330/afm.126009
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