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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
Published electronically: August 28, 2001
MathSciNet review: 1873763
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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.

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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

Bruce Rothschild
Affiliation: Department of Mathematics, University of California at Los Angeles, 405 Hilgard Ave., Los Angeles, California 90024

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

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