Logically Contractive Mappings: Fixed Points and Event-Indexed Rates

TL;DR

This paper introduces logically contractive mappings, extending fixed-point theory with event-indexed convergence rates, unifying various generalized contraction concepts, and providing explicit iteration bounds under certain conditions.

Contribution

It defines logically contractive mappings, proves a fixed-point theorem extending Banach's principle, and establishes convergence rates related to event sparsity and variable contraction factors.

Findings

Extended Banach's fixed-point theorem for logically contractive mappings.

Derived explicit convergence rates based on event gaps and contraction factors.

Unified several generalized contraction phenomena under a common framework.

Abstract

We introduce "logically contractive mappings" nonexpansive self-maps that contract along a subsequence of iterates and prove a fixed-point theorem that extends Banach's principle. We obtain event-indexed convergence rates and, under bounded gaps between events, explicit iteration-count rates. A worked example shows a nonexpansive map whose square is a strict contraction, and we clarify relations to Meir--Keeler and asymptotically nonexpansive mappings. We further generalize to variable-factor events and show that $\prod_k \lambda_k = 0$ (equivalently $\sum_k -\ln \lambda_k = \infty$) implies convergence. These results unify several generalized contraction phenomena and suggest new rate questions tied to event sparsity.