Abstract
Suppose
K is a nonempty closed convex nonexpansive retract of a real uniformly convex Banach space
E with
P as a nonexpansive retraction. Let
T
:
K
→
E
be a nonexpansive non-self map with
F
(
T
)
:=
{
x
∈
K
:
Tx
=
x
}
≠
∅
. Suppose
{
x
n
}
is generated iteratively by
x
1
∈
K
,
x
n
+
1
=
P
(
(
1
-
α
n
)
x
n
+
α
n
TP
[
(
1
-
β
n
)
x
n
+
β
n
Tx
n
]
)
,
n
⩾
1
,
where
{
α
n
}
and
{
β
n
}
are real sequences in
[
ε
,
1
-
ε
]
for some
ε
∈
(
0
,
1
)
. (1) If the dual
E
*
of
E has the Kadec–Klee property, then weak convergence of
{
x
n
}
to some
x
*
∈
F
(
T
)
is proved; (2) If
T satisfies condition
(
A
)
, then strong convergence of
{
x
n
}
to some
x
*
∈
F
(
T
)
is obtained.