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
T
be a self-map of a complete metric space
(
X
,
d
)
(X,d)
, and let
0
>
M
>
1
0>M>1
. Let
J
J
be a positive integer. Assume that for each pair
x
,
y
∈
X
x,y\in X
,
min
{
d
(
T
k
x
,
T
k
y
)
:
1
≤
k
≤
J
}
≤
M
d
(
x
,
y
)
\min \{d(T^kx, T^ky):1\le k\le J\}\le M\,d(x,y)
. Then
T
T
has a fixed point. Unlike Banach’s original theorem (the case
J
=
1
J=1
), the above hypothesis does not compel
T
T
to be continuous. In this paper we use Ramsey’s Theorem from combinatorics to establish the GBCC for arbitrary
J
J
in the case when
T
T
is assumed to be continuous, and also derive a result which enables us to prove the GBCC when
J
=
3
J=3
without the assumption of continuity; it is known that the case
J
=
3
J=3
includes instances where
T
T
is not continuous.