On the fixed point property for nonexpansive mappings in hyperbolic geodesic spaces

Motivated by the well-known results concerning the complex Hilbert ball with the hyperbolic metric and metric trees, we give a characterization of the convex and closed subsets of Busemann and hyperbolic geodesic spaces in terms of the fixed point property for nonexpansive and firmly nonexpansive mappings. Furthermore, motivated by Goebel and Reich's results concerning the complex Hilbert ball with the hyperbolic metric, we describe how the fixed point free mappings behave in a much more general class of spaces.