# A new metastable convergence criterion and an application in the theory of uniformly convex Banach spaces

## TL;DR

This paper introduces a generalized convergence criterion called metastable rate of asymptotic decreasingness, applies it to fixed point theory in uniformly convex Banach spaces, and provides quantitative convergence rates for Picard iterates.

## Contribution

It develops a new metastable convergence criterion and applies it to derive explicit rates of convergence in fixed point theory within Banach spaces.

## Key findings

- Established a quantitative metastability rate for Picard iterates.
- Applied the criterion to fixed point convergence in uniformly convex Banach spaces.
- Provided explicit bounds for convergence in the context of Kirk and Sims' proof.

## Abstract

We study a convergence criterion which generalises the notion of being monotonically decreasing, and introduce a quantitative version of this criterion, a so called metastable rate of asymptotic decreasingness. We then present a concrete application in the fixed point theory of uniformly convex Banach spaces, in which we carry out a quantitative analysis of a convergence proof of Kirk and Sims. More precisely, we produce a rate of metastability (in the sense of Tao) for the Picard iterates of mappings T which satisfy a variant of the convergence criterion, and whose fixed point set has nonempty interior.