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An application of Ramsey's Theorem to the Banach Contraction Principle |
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| Authors: | James Merryfield Bruce Rothschild James D Stein Jr |
| Institution: | Long Beach Polytechnic High School, 1600 Atlantic Ave., Long Beach, California 90813 ; Department of Mathematics, University of California at Los Angeles, 405 Hilgard Ave., Los Angeles, California 90024 ; Department of Mathematics, California State University at Long Beach, 1250 Bellflower Blvd., Long Beach, California 90840 |
| 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 Unlike Banach's original theorem (the case |
| Keywords: | Ramsey's Theorem Banach Contraction Principle fixed point |
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be a self-map of a complete metric space 
, and let 
. Let 
be a positive integer. Assume that for each pair 
, 
. Then 
has a fixed point. 
), the above hypothesis does not compel 
to be continuous. In this paper we use Ramsey's Theorem from combinatorics to establish the GBCC for arbitrary 
in the case when 
is assumed to be continuous, and also derive a result which enables us to prove the GBCC when 
without the assumption of continuity; it is known that the case 
includes instances where 
is not continuous.