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Generalizations of the Intermediate Value Theorem for Approximating Fixed Points and Zeros of Continuous Functions

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Numerical Computations: Theory and Algorithms (NUMTA 2019)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 11974))

Abstract

Generalizations of the traditional intermediate value theorem are presented. The obtained generalized theorems are particular useful for the existence of solutions of systems of nonlinear equations in several variables as well as for the existence of fixed points of continuous functions. Based on the corresponding criteria for the existence of a solution emanated by the intermediate value theorems, generalized bisection methods for approximating fixed points and zeros of continuous functions are given. These bisection methods require only algebraic signs of the function values and are of major importance for tackling problems with imprecise (not exactly known) information.

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Acknowledgment

The author would like to thank the anonymous reviewers for their helpful comments.

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Authors and Affiliations

  1. Department of Mathematics, University of Patras, 26110, Patras, Greece

    Michael N. Vrahatis

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  1. Michael N. Vrahatis
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Correspondence to Michael N. Vrahatis .

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  1. University of Calabria, Rende, Italy

    Yaroslav D. Sergeyev

  2. University of Calabria, Rende, Italy

    Dmitri E. Kvasov

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Vrahatis, M.N. (2020). Generalizations of the Intermediate Value Theorem for Approximating Fixed Points and Zeros of Continuous Functions. In: Sergeyev, Y., Kvasov, D. (eds) Numerical Computations: Theory and Algorithms. NUMTA 2019. Lecture Notes in Computer Science(), vol 11974. Springer, Cham. https://doi.org/10.1007/978-3-030-40616-5_17

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  • DOI: https://doi.org/10.1007/978-3-030-40616-5_17

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-030-40615-8

  • Online ISBN: 978-3-030-40616-5

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