Two topics in nonlinear system analysis through fixed point theorems

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    Abstract

    This paper reviews two topics of nonlinear system analysis done in Japan. The first half of this paper concerns with nonlinear system analysis through the nondeterministic operator theory. The nondeterministic operator is a set-valued or fuzzy set valued operator introduced by K. Horiuchi. From 1975 Horiuchi has developed fixed point theorems for nondeterministic operators. Using such fixed point theorems, he developed a unique theory for nonlinear system analysis. Horiuchi's theory provides a fundamental view point for analysis of fluctuations in nonlinear systems. In this paper, it is pointed out that Horiuchi's theory can be viewed as an extension of the interval analysis. Next, Urabe's theory for nonlinear boundary value problems is discussed. From 1965 Urabe has developed a method of computer assisted existence proof for solutions of nonlinear boundary value problems. Urabe has presented a convergence theorem for a certain simplified Newton method. Urabe's theorem is essentially based on Banach's contraction mapping theorem. In this paper, reformulation of Urabe's theory using the interval analysis is presented. It is shown that sharp error estimation can be obtained by this reformulation. Both works discussed in this paper have been done independently with the interval analysis. This paper points out that they have deep relationship with the interval analysis. Moreover, it is also pointed out that these two works suggest future directions of the interval analysis.

    Original languageEnglish
    Pages (from-to)1144-1152
    Number of pages9
    JournalIEICE Transactions on Fundamentals of Electronics, Communications and Computer Sciences
    VolumeE77-A
    Issue number7
    Publication statusPublished - 1994 Jul

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    Interval Analysis
    Systems Analysis
    Nonlinear Analysis
    Fixed point theorem
    Nonlinear systems
    Nonlinear Systems
    Systems analysis
    Mathematical operators
    Boundary value problems
    Nonlinear Boundary Value Problems
    Reformulation
    Operator
    Newton-Raphson method
    Fuzzy sets
    Set theory
    Contraction Mapping
    Operator Theory
    Error analysis
    Error Estimation
    Stefan Banach

    ASJC Scopus subject areas

    • Hardware and Architecture
    • Information Systems
    • Electrical and Electronic Engineering

    Cite this

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    title = "Two topics in nonlinear system analysis through fixed point theorems",
    abstract = "This paper reviews two topics of nonlinear system analysis done in Japan. The first half of this paper concerns with nonlinear system analysis through the nondeterministic operator theory. The nondeterministic operator is a set-valued or fuzzy set valued operator introduced by K. Horiuchi. From 1975 Horiuchi has developed fixed point theorems for nondeterministic operators. Using such fixed point theorems, he developed a unique theory for nonlinear system analysis. Horiuchi's theory provides a fundamental view point for analysis of fluctuations in nonlinear systems. In this paper, it is pointed out that Horiuchi's theory can be viewed as an extension of the interval analysis. Next, Urabe's theory for nonlinear boundary value problems is discussed. From 1965 Urabe has developed a method of computer assisted existence proof for solutions of nonlinear boundary value problems. Urabe has presented a convergence theorem for a certain simplified Newton method. Urabe's theorem is essentially based on Banach's contraction mapping theorem. In this paper, reformulation of Urabe's theory using the interval analysis is presented. It is shown that sharp error estimation can be obtained by this reformulation. Both works discussed in this paper have been done independently with the interval analysis. This paper points out that they have deep relationship with the interval analysis. Moreover, it is also pointed out that these two works suggest future directions of the interval analysis.",
    author = "Shinichi Oishi",
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