Let
f
:
X
→
X
f:X \to X
be a continuous compact mapping of a metric space
(
X
,
d
)
(X,d)
into itself with the property that
x
,
y
∈
X
x,y \in X
and
x
≠
y
x \ne y
implies
d
(
f
(
x
)
,
f
(
y
)
)
>
1
2
[
d
(
x
,
f
(
x
)
)
+
d
(
y
,
f
(
y
)
)
]
d(f(x),f(y)) > \tfrac {1} {2}[d(x,f(x)) + d(y,f(y))]
. It is shown that under these conditions
f
f
has a unique fixed point and, moreover,
f
f
is a Banach contraction relative to a suitable remetrization of the space
X
X
. A similar result concerning condensing mappings is also obtained.