On the geometry and fixed point free mappings in hyperbolic spaces

In the paper, we focus on two problems in geodesic hyperbolic spaces. First, we will consider the behavior of Picard iterative sequences of fixed point free nonexpansive mappings in spaces of constant curvature. Second, motivated by Hotchkiss (Proc Am Math Soc 125:1903–1912, 1997) we will show under which additional assumptions the Gromov boundary of a hyperbolic space equipped with the topology defined by the visual metric coincides with the geodesic boundary equipped with the cone topology.