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

TL;DR

This paper develops explicit quantitative fixed-point theorems in complete metric spaces with verifiable hypotheses, providing convergence rates, stability bounds, and resilience estimates applicable to integral equations and boundary value problems.

Contribution

It introduces a framework for fixed-point analysis using verifiable contractive moduli, enabling explicit bounds and stability estimates for iterative solutions.

Findings

Derived explicit a priori bounds for Picard iterates.

Established residual-to-error estimates for fixed points.

Provided certified resilience bounds under inexact evaluations.

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 $T\colon C\to C$ governed by a \emph{verifiable} contractive modulus. The modulus is encoded by a contractive gauge $\omega$ and a certified constant $\kappa=\sup_{0<r\le R}\omega(r)/r<1$ on a computable working radius $R$. From this datum we derive explicit a priori bounds $\dist(x_n,x^\ast)\le \Phi(n;\kappa,\delta_0)$ for Picard iterates, a residual-to-error estimate, and a quantitative data dependence bound $\dist(x^\ast,y^\ast)\le (1-\kappa)^{-1}\sup_{x\in C}\dist(Tx,Sx)$. We further treat inexact evaluations $\dist(\tilde x_{n+1},T\tilde x_n)\le \eta_n$ 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.