An application of Ramsey’s Theorem to the Banach Contraction Principle.

*(English)*Zbl 1001.47042The 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.

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)

##### Keywords:

Ramsey’s theorem; Banach contraction; principle; fixed point; generalized Banach contraction conjecture
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\textit{J. Merryfield} et al., Proc. Am. Math. Soc. 130, No. 4, 927--933 (2002; Zbl 1001.47042)

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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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