Abstract
Let
E be a real Banach space with a uniformly Gâteaux differentiable norm and which possesses uniform normal structure,
K a nonempty bounded closed convex subset of
E,
T
:
K
⟶
K
an asymptotically nonexpansive mapping with sequence
{
k
n
}
n
⊂
[
1
,
∞
)
. Let
{
t
n
}
⊂
(
0
,
1
)
be such that
t
n
→
1
as
n
→
∞
and
f be a contraction on
K. Under suitable conditions on the sequence
{
t
n
}
, we show the existence of a sequence
{
x
n
}
n
satisfying the relation
x
n
=
(
1
-
t
n
k
n
)
f
(
x
n
)
+
t
n
k
n
T
n
x
n
, and prove that
{
x
n
}
n
converges strongly to the fixed point of
T, which solves some variational inequality, provided
∥
x
n
-
Tx
n
∥
→
0
as
n
→
∞
. As an application, we prove that the iterative process defined by
z
0
∈
K
,
z
n
+
1
≔
(
1
-
t
n
k
n
)
f
(
z
n
)
+
t
n
k
n
T
n
z
n
,
n
∈
N
, converges strongly to the same fixed point of
T.