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

Let $(X,\text{dist})$ be a complete metric space and let $C\subseteq X$ be a closed invariant set. We study fixed points of maps $T\text{\hspace{0.17em}⁣}:C\to C$ governed by a \emph{verifiable} contractive modulus. The modulus is encoded by a contractive gauge $\omega$ and a certified constant $\kappa ={\sup }_{0<r\le R}\omega (r)\mathrm{/}r<1$ on a computable working radius $R$. From this datum we derive explicit a priori bounds $\text{dist}(x_n,x^{\ast })\le \mathrm{\Phi }(n;\kappa ,{\delta }_0)$ for Picard iterates, a residual-to-error estimate, and a quantitative data dependence bound $\text{dist}(x^{\ast },y^{\ast })\le (1-\kappa {)}^{-1}{\sup }_{x\in C}\text{dist}(Tx,Sx)$. We further treat inexact evaluations $\text{dist}({\tilde{x}}_{n+1},T{\tilde{x}}_n)\le {\eta }_n$ 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.