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 journalArticlepeer-review

    3 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

    All Science Journal Classification (ASJC) codes

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

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