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An application of Ramsey's Theorem to the Banach Contraction Principle


Authors: James Merryfield, Bruce Rothschild and James D. Stein Jr.
Journal: Proc. Amer. Math. Soc. 130 (2002), 927-933
MSC (2000): Primary 05C55, 47H10
DOI: https://doi.org/10.1090/S0002-9939-01-06169-X
Published electronically: August 28, 2001
MathSciNet review: 1873763
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Abstract:

One of the most fundamental fixed-point theorems is Banach's Contraction Principle, of which the following conjecture is a generalization.


Generalized Banach Contraction Conjecture (GBCC). Let $T$ be a self-map of a complete metric space $(X,d)$, and let $0<M<1$. Let $J$ be a positive integer. Assume that for each pair $x,y\in X$, $\min\{d(T^kx, T^ky):1\le k\le J\}\le M\,d(x,y)$. Then $T$ has a fixed point.


Unlike Banach's original theorem (the case $J=1$), the above hypothesis does not compel $T$ to be continuous. In this paper we use Ramsey's Theorem from combinatorics to establish the GBCC for arbitrary $J$in the case when $T$ is assumed to be continuous, and also derive a result which enables us to prove the GBCC when $J=3$ without the assumption of continuity; it is known that the case $J=3$ includes instances where $T$ is not continuous.


References [Enhancements On Off] (What's this?)

  • 1. Ronald Graham, Bruce Rothschild, and Joel Spencer, Ramsey Theory, Wiley-Interscience, New York, 1980. MR 82b:05001
  • 2. Jacek R. Jachymski, Bernd Schroder, and James D. Stein, Jr., A connection between fixed-point theorems and tiling problems, J. Combin. Theory Ser. A 87 (1999), 273-286. MR 2000g:54075
  • 3. Jacek R. Jachymski and James D. Stein, Jr., A minimum condition and some related fixed-point theorems, J. Austral Math. Soc. Ser. A 66 (1999), 224-243. MR 2000a:54072

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Additional Information

James Merryfield
Affiliation: Long Beach Polytechnic High School, 1600 Atlantic Ave., Long Beach, California 90813
Address at time of publication: Department of Mathematics, University of California at Berkeley, Berkeley, California 94720
Email: kmerry@csulb.edu

Bruce Rothschild
Affiliation: Department of Mathematics, University of California at Los Angeles, 405 Hilgard Ave., Los Angeles, California 90024
Email: blr@math.ucla.edu

James D. Stein Jr.
Affiliation: Department of Mathematics, California State University at Long Beach, 1250 Bellflower Blvd., Long Beach, California 90840
Email: jimstein@csulb.edu

DOI: https://doi.org/10.1090/S0002-9939-01-06169-X
Keywords: Ramsey's Theorem, Banach Contraction Principle, fixed point
Received by editor(s): March 3, 2000
Received by editor(s) in revised form: May 10, 2000, July 14, 2000, and September 25, 2000
Published electronically: August 28, 2001
Communicated by: John R. Stembridge
Article copyright: © Copyright 2001 American Mathematical Society

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