Bounds on Kuhfittig’s iteration schema in uniformly convex hyperbolic spaces

The purpose of this paper is to extract an explicit effective and uniform bound on the rate of asymptotic regularity of an iteration schema involving a finite family of nonexpansive mappings. The results presented in this paper contribute to the general project of proof mining as developed by the second author as well as generalize and improve various classical and corresponding quantitative results in the current literature. More precisely, we give a rate of asymptotic regularity of an iteration schema due to Kuhfittig for finitely many nonexpansive mappings in the context of uniformly convex hyperbolic spaces. The rate only depends on an upper bound on the distance between the starting point and some common fixed point, a lower bound 1 / N ≤ λ n ( 1 − λ n ) , the error ϵ > 0 and a modulus η of uniform convexity.