An application of continuous logic to fixed point theory

TL;DR

This paper develops a continuous logic framework that handles discontinuous functions and applies ultraproducts to establish uniform convergence results in functional analysis.

Contribution

It introduces a novel continuous logic approach with a linear structure concept to extend ultraproduct methods to discontinuous functions.

Findings

Achieved uniform metastable convergence for specific functional analysis sequences.

Extended ultraproduct techniques to discontinuous functions.

Provided a general framework for applying continuous logic to fixed point theory.

Abstract

In aiming to apply to a broader class of examples the Avigad-Iovino "ultraproducts and metastability" approach to obtaining uniformity for convergence of sequences, we construct a framework using continuous logic that in particular is able to handle discontinuous functions in its domain of discourse. This setup weakens the usual continuity requirements for functions, but compensates for the loss of control by introducing a notion of "linear structure" that captures in a quite general way the situation of having geodesics between every pair of points, and has as a special case the vector space structure of Banach spaces. We use this to apply the Avigad-Iovino method to specific convergence results from functional analysis involving iterations of discontinuous functions, and so obtain uniform metastable convergence in those results.