Logically Contractive Mappings: Fixed Points and Event-Indexed Rates

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.