Nonexpansive iterations in uniformly convex $W$-hyperbolic spaces

TL;DR

This paper introduces a new class of uniformly convex hyperbolic spaces suitable for analyzing nonexpansive iterations and provides effective convergence rates for Ishikawa iterations using proof mining techniques.

Contribution

It defines UCW-hyperbolic spaces, generalizing known spaces, and derives new effective convergence rates for iterative methods in these spaces.

Findings

UCW-hyperbolic spaces generalize uniformly convex normed and CAT(0) spaces.

Effective asymptotic regularity rates are obtained for Ishikawa iterations.

Results are new even for uniformly convex Banach spaces.

Abstract

We propose the class of uniformly convex $W$-hyperbolic spaces with monotone modulus of uniform convexity ($UCW$-hyperbolic spaces for short) as an appropriate setting for the study of nonexpansive iterations. $UCW$-hyperbolic spaces are a natural generalization both of uniformly convex normed spaces and CAT(0)-spaces. Furthermore, we apply proof mining techniques to get effective rates of asymptotic regularity for Ishikawa iterations of nonexpansive self-mappings of closed convex subsets in $UCW$-hyperbolic spaces. These effective results are new even for uniformly convex Banach spaces.