On the Karlsson-Nussbaum conjecture for resolvents of nonexpansive mappings

Let $D\subset {\mathbb{R}}^n$ be a bounded convex domain and $F:D\to D$ a $1$-Lipschitz mapping with respect to the Hilbert metric $d$ on $D$ satisfying condition $d(sx+(1-s)y,sz+(1-s)w)\le \max \{d(x,z),d(y,w)\}$. 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 $\mathrm{\partial }D\mathrm{.}$ As a consequence, we show a Wolff-Denjoy type theorem for resolvents of nonexpansive mappings acting on an ellipsoid $D$.