Let X be a uniformly convex Banach space which satisfies Opial’s condition or has a Fréchet differentiable norm, C a bounded closed convex subset of X, and
T
:
C
→
C
T:C \to C
an asymptotically nonexpansive mapping. It is then shown that the modified Mann and Ishikawa iteration processes defined by
x
n
+
1
=
t
n
T
n
x
n
+
(
1
−
t
n
)
x
n
{x_{n + 1}} = {t_n}{T^n}{x_n} + (1 - {t_n}){x_n}
and
x
n
+
1
=
t
n
T
n
(
s
n
T
n
x
n
+
(
1
−
s
n
)
x
n
)
+
(
1
−
t
n
)
x
n
{x_{n + 1}} = {t_n}{T^n}({s_n}{T^n}{x_n} + (1 - {s_n}){x_n}) + (1 - {t_n}){x_n}
, respectively, converge weakly to a fixed point of T.