Quantitative fixed-point theorems with verifiable hypotheses: rates and stability

Abstract

Let $(X,\dist)$ be a complete metric space and let $C\subseteq X$ be a closed invariant set. We study fixed points of maps $T:C\to C$ governed by a \emph{verifiable} contractive modulus. The modulus is encoded by a contractive gauge $\ddot{I}‰$ and a certified constant $\hat{I}º={\sup }_{0<r\le R}\ddot{I}‰(r)/r<1$ on a computable working radius $R$. From this datum we derive explicit a priori bounds $\dist(x_n,x^\ast)\le Φ(n;κ,δ_0)$ for Picard iterates, a residual-to-error estimate, and a quantitative data dependence bound $\dist(x^\ast,y^\ast)\le (1-κ)^{-1}\sup_{x\in C}\dist(Tx,Sx)$. We further treat inexact evaluations $\dist(\tilde x_{n+1},T\tilde x_n)\le η_n$ and obtain certified resilience bounds with the same stability factor. The framework applies to Hammerstein--Volterra integral equations and to boundary value problems via Green operators, where kernel bounds yield certified convergence rates.