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On the Picard-Lindelöf Argument and the Banach-Caccioppoli Contraction Mapping Principle

Alexander I. Bufetov, Ilya I. Zavolokin

TL;DR

This paper shows that a minor refinement of the Contraction Mapping Principle, implemented on a decreasing chain of metric subspaces, recovers the exact convergence rate for the Picard-Lindelöf iteration, achieving a factorial rate $O((\alpha L)^n/n!)$ rather than just a geometric rate. By constructing $H_j$ and $d_j$ and proving completeness and contraction bounds that scale with $j$, the authors derive a sharp bound on the distance between the $n$-th Picard iterate and the fixed point, $||y^n-y^\infty||\le e^{\alpha L}(\alpha L)^n/n!\,M$, in the real-time setting; they extend the same framework to complex time. The work unifies and extends the historical contraction-mapping approach to ODEs by providing an intrinsic multilevel metric structure that yields precise convergence rates, with potential implications for both theory and numerical approximation strategies in differential equations. The analysis clarifies how different metric choices at each level enable the factorial rate and demonstrates the generality of the chain-of-spaces method for existence-uniqueness proofs via fixed-point arguments.

Abstract

The aim of this note is to present the simple observation that a slight refinement of the Contraction Mapping Principle allows one to recover the precise convergence rate in the Picard-Lindelöf Theorem.

On the Picard-Lindelöf Argument and the Banach-Caccioppoli Contraction Mapping Principle

TL;DR

This paper shows that a minor refinement of the Contraction Mapping Principle, implemented on a decreasing chain of metric subspaces, recovers the exact convergence rate for the Picard-Lindelöf iteration, achieving a factorial rate rather than just a geometric rate. By constructing and and proving completeness and contraction bounds that scale with , the authors derive a sharp bound on the distance between the -th Picard iterate and the fixed point, , in the real-time setting; they extend the same framework to complex time. The work unifies and extends the historical contraction-mapping approach to ODEs by providing an intrinsic multilevel metric structure that yields precise convergence rates, with potential implications for both theory and numerical approximation strategies in differential equations. The analysis clarifies how different metric choices at each level enable the factorial rate and demonstrates the generality of the chain-of-spaces method for existence-uniqueness proofs via fixed-point arguments.

Abstract

The aim of this note is to present the simple observation that a slight refinement of the Contraction Mapping Principle allows one to recover the precise convergence rate in the Picard-Lindelöf Theorem.
Paper Structure (15 sections, 3 theorems, 107 equations)

This paper contains 15 sections, 3 theorems, 107 equations.

Table of Contents

  1. Introduction
  2. A contraction mapping on a chain of metric spaces
  3. A proof of the Picard-Lindelöf Theorem
  4. Preliminary remarks
  5. A chain of subspaces
  6. Completeness of the spaces $H_j$
  7. Nested images
  8. The rate of convergence
  9. Differential equations with complex time
  10. Preliminary remarks
  11. A chain of subspaces
  12. Completeness of the spaces $H_j$
  13. Nested images
  14. The rate of convergence
  15. Historical remarks

Key Result

Lemma 1

The map $P$ has a unique fixed point $x_\infty\in S$. Moreover, for all $x\in H$ and for all $n\geqslant j = 0, 1, \ldots$ the sequence $(x_m)_m$ defined by the formula: satisfies the relations where

Theorems & Definitions (16)

  • Remark 1
  • Lemma 1
  • Remark 2
  • Remark 3
  • Remark 4
  • proof
  • Remark 5
  • Remark 6
  • Theorem 1: Picard-Lindelöf
  • Remark 7
  • ...and 6 more