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

Chandrasekhar Gokavarapu, Srinivasulu Ch, D V N S Sriram Murthy, Rajeev Muthu

TL;DR

This work develops a verifiable contractive framework for fixed-point problems on a closed invariant set, parameterized by a computable contractive gauge $\omega(\cdot;\theta)$ and a working radius $R$, yielding a certified contraction $\kappa<1$ and explicit rates. It derives actionable a priori bounds $\mathrm{d}(x_n, x^\ast) \le \Phi(n;\text{data})$, a residual-to-error conversion, and a sharp data-dependence bound $\mathrm{d}(x^\ast, y^\ast) \le C_{\mathrm{stab}}\, \|T-S\|$ with $C_{\mathrm{stab}}=\frac{1}{1-\kappa}$, all robust to inexact evaluations. The framework is instantiated for Hammerstein–Volterra integral equations and Green-operator-based boundary-value problems, where kernel/Lipschitz data yield explicit convergence rates and stability constants. By exporting computable certificates for rate and stability, the paper provides practical tools for rigorous numerical analysis and sensitivity assessment in applied fixed-point computations. This approach bridges classical contraction theory with verifiable, data-driven guarantees suitable for numerical verification and error control in applications.

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 $ω$ and a certified constant $κ=\sup_{0<r\le R}ω(r)/r<1$ 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.

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

TL;DR

This work develops a verifiable contractive framework for fixed-point problems on a closed invariant set, parameterized by a computable contractive gauge and a working radius , yielding a certified contraction and explicit rates. It derives actionable a priori bounds , a residual-to-error conversion, and a sharp data-dependence bound with , all robust to inexact evaluations. The framework is instantiated for Hammerstein–Volterra integral equations and Green-operator-based boundary-value problems, where kernel/Lipschitz data yield explicit convergence rates and stability constants. By exporting computable certificates for rate and stability, the paper provides practical tools for rigorous numerical analysis and sensitivity assessment in applied fixed-point computations. This approach bridges classical contraction theory with verifiable, data-driven guarantees suitable for numerical verification and error control in applications.

Abstract

Let be a complete metric space and let be a closed invariant set. We study fixed points of maps governed by a \emph{verifiable} contractive modulus. The modulus is encoded by a contractive gauge and a certified constant on a computable working radius . From this datum we derive explicit a priori bounds for Picard iterates, a residual-to-error estimate, and a quantitative data dependence bound . We further treat inexact evaluations 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.
Paper Structure (38 sections, 41 theorems, 113 equations)

This paper contains 38 sections, 41 theorems, 113 equations.

Table of Contents

  1. Introduction
  2. What is proved
  3. Verifiable models
  4. Plan of the paper
  5. Main contributions
  6. Preliminaries and standing hypotheses
  7. Metric and normed settings
  8. Operator norms and verifiable constants
  9. Iterative schemes used in the paper
  10. A verifiable contractive framework
  11. Contractive conditions with checkable parameters
  12. Admissible data and a diagnostic checklist
  13. Main quantitative fixed-point theorem
  14. Existence and uniqueness under the framework
  15. Explicit rate statement
  16. ...and 23 more sections

Key Result

Proposition 2.5

Let $(X,\mathrm{d})$ be complete and let $T\colon X\to X$ satisfy $\mathrm{Lip}(T)\le L<1$. Fix $x_0\in X$ and define $x_{n+1}:=Tx_n$. Then there exists a unique $x^\ast\in\mathrm{Fix}(T)$ and, for all $n\in\mathbb{N}$,

Theorems & Definitions (115)

  • Definition 2.1: Continuous-function model space
  • Definition 2.2: Kernel operators
  • Definition 2.3: Lipschitz constant
  • Definition 2.4: Uniform perturbation size
  • Proposition 2.5: A priori error bound for a contraction
  • Proposition 2.6: Data dependence for a common contraction modulus
  • Lemma 2.7: Verifiable Lipschitz constants for Hammerstein/Volterra maps
  • Definition 2.8: Residual
  • Definition 2.9: Picard iteration
  • Definition 2.10: Krasnosel'skii--Mann iteration
  • ...and 105 more