arXiv subjects
Publications and source records attributed to Aleksandra Huczek.
We show a Wolff-Denjoy type theorem in the case of a one-parameter continuous semigroups of nonexpansive mappings in which there is a compact mapping. Using the notion of attractor we are also able to prove some specific properties directly related to the Karlsson-Nussbaum conjecture.
Let $D\subset \mathbb{R}^{n}$ be a bounded convex domain and $F:D\rightarrow D$ 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 $\{R_{λ}\}_{λ>0}$ of resolvents of $F$ is a subset of $\partial D.$ As a consequence, we show a Wolff-Denjoy type theorem for resolvents of nonexpansive mappings acting on an ellipsoid $D$.
We show a Wolff-Denjoy type theorem in complete geodesic spaces in the spirit of Beardon's framework that unifies several results in this area. In particular, it applies to strictly convex bounded domains in $\mathbb{R}^{n}$ or $\mathbb{C}^{n}$ with respect to a large class of metrics including Hilbert's and Kobayashi's metrics. The results are generalized to $1$-Lipschitz compact mappings in infinite-dimesional Banach spaces.