Some recent homotopy results given by B. Sims, H. K. Xu
and G. X. Z. Yuan
[in: Proceedings of the 6th international conference on nonlinear functional analysis and applications (Masan, Korea, September 1–5, 2000), 93–104 (2001; Zbl 1073.47519
)] are here extended for set-valued mappings in the CAT(0) setting. Also, using an ultrapower approach, several fixed point theorems for nonexpansive mappings in Banach spaces are given.