A theorem for numerical verification on local uniqueness of solutions to fixed-point equations

Nobito Yamamoto, Mitsuhiro T. Nakao, Yoshitaka Watanabe

Research output: Contribution to journalArticle

2 Citations (Scopus)

Abstract

We give a theoretical result with respect to numerical verification of existence and local uniqueness of solutions to fixed-point equations which are supposed to have Fréchet differentiable operators. The theorem is based on Banach's fixed-point theorem and gives sufficient conditions in order that a given set of functions includes a unique solution to the fixed-point equation. The conditions are formulated to apply readily to numerical verification methods. We already derived such a theorem in [11], which is suitable to Nakao's methods on numerical verification for PDEs. The present theorem has a more general form and one may apply it to many kinds of differential equations and integral equations which can be transformed into fixed-point equations.

Original languageEnglish
Pages (from-to)1190-1204
Number of pages15
JournalNumerical Functional Analysis and Optimization
Volume32
Issue number11
DOIs
Publication statusPublished - 2011
Externally publishedYes

Keywords

  • Computer-assisted proof
  • Fixed-point equation
  • Local uniqueness
  • Numerical verification
  • Self-validated computing

ASJC Scopus subject areas

  • Analysis
  • Control and Optimization
  • Signal Processing
  • Computer Science Applications

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