Let K be a subset of a Banach space X. A mapping
F
:
K
→
K
F:K \to K
is said to be asymptotically nonexpansive if there exists a sequence
{
k
i
}
\{ {k_i}\}
of real numbers with
k
i
→
1
{k_i} \to 1
as
i
→
∞
i \to \infty
such that
‖
F
i
x
−
F
i
y
‖
≦
k
i
‖
x
−
y
‖
,
x
,
y
∈
K
\left \| {{F^i}x - {F^i}y} \right \| \leqq {k_i}\left \| {x - y} \right \|,x,y \in K
. It is proved that if K is a non-empty, closed, convex, and bounded subset of a uniformly convex Banach space, and if
F
:
K
→
K
F:K \to K
is asymptotically nonexpansive, then F has a fixed point. This result generalizes a fixed point theorem for nonexpansive mappings proved independently by F. E. Browder, D. Göhde, and W. A. Kirk.