An application of Ramsey's Theorem to the Banach Contraction Principle (Q2781236)

The following conjecture generalizes the Banach contraction principle: Generalized Banach Contraction Conjecture (GBCC).NEWLINENEWLINENEWLINELet \(T\) 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.NEWLINENEWLINENEWLINEBanach'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 \textit{J. R. Jachymski} and \textit{J. D. Stein}, jun., J. Aust. Math. Soc. Ser. A 66, No. 2, 224-243 (1999; Zbl 0931.47042)]. In [\textit{J. R. Jachymski}, \textit{B. Schröder} and \textit{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.NEWLINENEWLINENEWLINEIn 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.