In this paper fixed point theorems are established first for mappings
T
T
, mapping a closed bounded convex subset
K
K
of a reflexive Banach space into itself and satisfying
\[
|
|
T
x
−
T
y
|
|
≦
1
2
{
|
|
x
−
T
x
|
|
+
|
|
y
−
T
y
|
|
,
x
,
y
∈
K
,
||Tx - Ty|| \leqq \tfrac {1}{2}\{ ||x - Tx|| + ||y - Ty||,\quad x,y \in K,
\]
and then an analogous result is obtained for nonexpansive mappings giving rise to a question regarding the unification of these theorems.