A Fixed Point Theorem, Intermediate Value Theorem, and Nested Interval Property

Z. Wu

doi:10.1007/s10476-018-0612-3

A Fixed Point Theorem, Intermediate Value Theorem, and Nested Interval Property

Z. Wu^*

^*Corresponding author for this work

Department of Applied Mathematics

Research output: Contribution to journal › Article › peer-review

2 Citations (Scopus)

Abstract

For a continuous function f : [a, b] → R, we prove that f has a fixed point if and only if the intervals [a₀, b₀]:= [a, b] and [a_n, b_n]:= [a_n−1, b_n−1] ∩ f([a_n−1, b_n−1]) (n = 1, 2, · · ·) are all nonempty. More equivalent statements for the existence of fixed points of f have also been obtained and used to derive the intermediate value theorem and the nested interval property.

Original language	English
Pages (from-to)	443-447
Number of pages	5
Journal	Analysis Mathematica
Volume	45
Issue number	2
DOIs	https://doi.org/10.1007/s10476-018-0612-3
Publication status	Published - 1 Jun 2019

Keywords

fixed point
intermediate value theorem
nested interval property

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10.1007/s10476-018-0612-3

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AB - For a continuous function f : [a, b] → R, we prove that f has a fixed point if and only if the intervals [a0, b0]:= [a, b] and [an, bn]:= [an−1, bn−1] ∩ f([an−1, bn−1]) (n = 1, 2, · · ·) are all nonempty. More equivalent statements for the existence of fixed points of f have also been obtained and used to derive the intermediate value theorem and the nested interval property.

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KW - nested interval property

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A Fixed Point Theorem, Intermediate Value Theorem, and Nested Interval Property

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Keywords

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