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An application of Ramsey’s Theorem to the Banach Contraction Principle. (English) Zbl 1001.47042
The following conjecture generalizes the Banach contraction principle: 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 $$x,y\in X$$, $$\min\{d(T^kx, T^ky):1 \leq k\leq J\}\leq Md(x,y)$$. Then $$T$$ has a fixed point.
Banach’s original theorem is simply the case $$J=1$$, in which $$T$$ is uniformly continuous. If $$T$$ is uniformly continuous, then GBCC is true for arbitrary $$J$$ Theorem 2 from J. R. Jachymski and J. D. Stein, jun., J. Aust. Math. Soc. Ser. A 66, No. 2, 224-243 (1999; Zbl 0931.47042)]. In [J. R. Jachymski, B. SchrĂ¶der and J. D. Stein, jun., J. Comb. Theory, Ser. A 87, No. 2, 273-286 (1999; Zbl 0983.54047)] it is proved that if $$J=2$$ the GBCC is true without any additional assumption of $$T$$ and if $$J=3$$ and $$T$$ is continuous the GBCC is true. It is shown that case $$J=3$$ includes examples where $$T$$ is discontinuous.
In the present paper the authors show that Ramsey’s theorem (Ramsey’s Theorem: Let $$S$$ be an infinite set, $$n$$ a positive integer. Assume that every subset of $$S$$ of cardinality $$n$$ has been given one of a finite number of colors. Then there exists an infinite subset $$T$$ of $$S$$ such that $$T$$ is monochromatic; i.e. every subset of $$T$$ of cardinality $$n$$ has the same color.) can be used to prove the GBCC for arbitrary $$J$$ under the assumption that $$T$$ is continuous. In the last part of this paper it is proved that if $$T$$ satisfies the GBCC condition for $$J=3$$, then $$T$$ has a fixed point.
Reviewer: V.Popa (Bacau)

##### MSC:
 47H10 Fixed-point theorems 05D10 Ramsey theory
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##### References:
 [1] Ronald L. Graham, Bruce L. Rothschild, and Joel H. Spencer, Ramsey theory, John Wiley & Sons, Inc., New York, 1980. Wiley-Interscience Series in Discrete Mathematics; A Wiley-Interscience Publication. · Zbl 0455.05002 [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), no. 2, 273 – 286. · Zbl 0983.54047 [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), no. 2, 224 – 243. · Zbl 0931.47042
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