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Firmly nonexpansive mappings in classes of geodesic spaces. (English) Zbl 1524.47056

Summary: Firmly nonexpansive mappings play an important role in metric fixed point theory and optimization due to their correspondence with maximal monotone operators. In this paper we do a thorough study of fixed point theory and the asymptotic behaviour of Picard iterates of these mappings in different classes of geodesic spaces, such as (uniformly convex) \( W\)-hyperbolic spaces, Busemann spaces and CAT(0) spaces. Furthermore, we apply methods of proof mining to obtain effective rates of asymptotic regularity for the Picard iterations.

MSC:

47H09 Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc.
47J26 Fixed-point iterations
03F10 Functionals in proof theory
54H25 Fixed-point and coincidence theorems (topological aspects)
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