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

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.