Quasi-intermediate value theorem and outflanking arc theorem for plane maps

Jiehua Mai; Enhui Shi; Kesong Yan; Fanping Zeng

doi:10.3934/dcds.2024162

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2025, Volume 45, Issue 7: 2215-2240. Doi: 10.3934/dcds.2024162

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Quasi-intermediate value theorem and outflanking arc theorem for plane maps

School of Mathematics and Quantitative Economics, Guangxi University of Finance and Economics, Nanning, Guangxi, 530003, China
2.
Institute of Mathematics, Shantou University, Shantou, Guangdong, 515063, China
3.
School of Mathematics and Sciences, Soochow University, Suzhou, Jiangsu 215006, China
4.
School of Mathematics and Statistics, Hainan Normal University, Haikou, Hainan, 571158, China

^*Corresponding author: Kesong Yan
^*Corresponding author: Kesong Yan

Received: May 2024

Revised: November 2024

Early access: December 6, 2024

Published online: December 6, 2024

Jiehua Mai and Fanping Zeng are supported by NNSF of China (Grant No. 12261006) and Project of Guangxi First Class Disciplines of Statistics (No. GJKY-2022-01); Enhui Shi is supported by NNSF of China (Grant No. 12271388); Kesong Yan is supported by NNSF of China (Grant No. 12171175).

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Abstract

For a disk $ D $ in the plane $ \mathbb R^2 $ and a plane map $ f $, we give several conditions on the restriction of $ f $ to the boundary $ \partial D $ of $ D $ which imply the existence of a fixed point of $ f $ in some specified domain in $ D $. These conditions are similar to those appeared in the intermediate value theorem for maps on the real line. As an application of this result, we establish a fixed point theorem for plane maps having an outflanking arc, which extends the famous theorem due to Brouwer: if $ f $ is an orientation-preserving homeomorphism on the plane and has a periodic point, then it has a fixed point.

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Mathematics Subject Classification: Primary: 37E30.

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