Rates of convergence for asymptotically weakly contractive mappings in normed spaces

TL;DR

This paper introduces a generalized framework for analyzing the convergence rates of iterative algorithms approximating fixpoints of asymptotically weakly contractive mappings in normed spaces, using proof-theoretic methods.

Contribution

It defines a new notion of asymptotically ψ-weakly contractive mappings with explicit moduli and provides unified convergence theorems with quantitative rates, advancing the proof mining approach.

Findings

Established explicit convergence rates based on the new contractivity modulus

Unified various known results under a generalized theoretical framework

Applied proof-theoretic methods to derive quantitative convergence bounds

Abstract

We study Krasnoselskii-Mann style iterative algorithms for approximating fixpoints of asymptotically weakly contractive mappings, with a focus on providing generalised convergence proofs along with explicit rates of convergence. More specifically, we define a new notion of being asymptotically $\psi$-weakly contractive with modulus, and present a series of abstract convergence theorems which both generalise and unify known results from the literature. Rates of convergence are formulated in terms of our modulus of contractivity, in conjunction with other moduli and functions which form quantitative analogues of additional assumptions that are required in each case. Our approach makes use of ideas from proof theory, in particular our emphasis on abstraction and on formulating our main results in a quantitative manner. As such, the paper can be seen as a contribution to the proof mining program.