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February 2022 On the Banach mapping theorem and a related conjecture
Ming-Chia Li
Rocky Mountain J. Math. 52(1): 183-187 (February 2022). DOI: 10.1216/rmj.2022.52.183

Abstract

We extend, using an elementary method, results of Banach (1924), Fan (1952), and Sanders (1961), which concern a finite collection {fi:AiAi+1}i=1n of mappings with An+1=A1 which is decomposable as fi(Bi)=Ai+1Bi+1, where BiAi for all i and Bn+1=B1. Our theorem determines when such a collection is decomposable. We also show that such a set B1 is unique up to an addition of a certain set, which was conjectured by Sanders.

Citation

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Ming-Chia Li. "On the Banach mapping theorem and a related conjecture." Rocky Mountain J. Math. 52 (1) 183 - 187, February 2022. https://doi.org/10.1216/rmj.2022.52.183

Information

Received: 1 February 2021; Revised: 18 May 2021; Accepted: 23 May 2021; Published: February 2022
First available in Project Euclid: 19 April 2022

Digital Object Identifier: 10.1216/rmj.2022.52.183

Subjects:
Primary: 03E30 , 37B35 , 37C25 , 47H10

Keywords: Banach mapping theorem , fixed point , invariant set , mapping decomposition

Rights: Copyright © 2022 Rocky Mountain Mathematics Consortium

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Vol.52 • No. 1 • February 2022
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