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Nonexpansive set-valued mappings in metric and Banach spaces. (English) Zbl 1120.47043
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.

##### MSC:
 47H09 Contraction-type mappings, nonexpansive mappings, $$A$$-proper mappings, etc. 47H04 Set-valued operators 47H10 Fixed-point theorems 54H25 Fixed-point and coincidence theorems (topological aspects) 54C60 Set-valued maps in general topology
##### Keywords:
fixed point; nonexpansive mapping; Banach space