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A Piecewise-Linear Fixed Point Theorem.

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Title: A Piecewise-Linear Fixed Point Theorem.
Authors: Simpson, David J.W. (AUTHOR) d.j.w.simpson@massey.ac.nz
Source: American Mathematical Monthly. Aug/Sep2025, Vol. 132 Issue 7, p695-699. 5p.
Subject Terms: Bifurcation theory, Fixed point theory, Stability theory, Dynamical systems, Mathematical functions, Continuous functions, Linear algebra
Abstract: We prove that if a continuous piecewise-smooth map on R n is comprised of two linear functions, has a bounded orbit, and satisfies a certain non-degeneracy condition, then it has a fixed point. The result has important consequences to the bifurcation theory of nonsmooth dynamical systems, yet the proof requires only elementary linear algebra. [ABSTRACT FROM AUTHOR]
Copyright of American Mathematical Monthly is the property of Taylor & Francis Ltd and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)
Database: Education Research Complete
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DbLabel: Education Research Complete
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AccessLevel: 6
PubType: Academic Journal
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  Data: A Piecewise-Linear Fixed Point Theorem.
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  Data: <searchLink fieldCode="JN" term="%22American+Mathematical+Monthly%22">American Mathematical Monthly</searchLink>. Aug/Sep2025, Vol. 132 Issue 7, p695-699. 5p.
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  Data: We prove that if a continuous piecewise-smooth map on R n is comprised of two linear functions, has a bounded orbit, and satisfies a certain non-degeneracy condition, then it has a fixed point. The result has important consequences to the bifurcation theory of nonsmooth dynamical systems, yet the proof requires only elementary linear algebra. [ABSTRACT FROM AUTHOR]
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  Data: <i>Copyright of American Mathematical Monthly is the property of Taylor & Francis Ltd and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract.</i> (Copyright applies to all Abstracts.)
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        Value: 10.1080/00029890.2025.2495534
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      – Code: eng
        Text: English
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      – SubjectFull: Fixed point theory
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      – SubjectFull: Stability theory
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      – SubjectFull: Dynamical systems
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      – SubjectFull: Mathematical functions
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      – SubjectFull: Continuous functions
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      – SubjectFull: Linear algebra
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      – TitleFull: A Piecewise-Linear Fixed Point Theorem.
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              Text: Aug/Sep2025
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