Let
E
E
be a
q
q
-uniformly smooth Banach space possessing a weakly sequentially continuous duality map (e.g.,
ℓ
p
,
1
>
p
>
∞
\ell _p, \ 1>p>\infty
). Let
T
T
be a Lipschitzian pseudocontractive selfmapping of a nonempty closed convex and bounded subset
K
K
of
E
E
and let
ω
∈
K
\omega \in K
be arbitrary. Then the iteration sequence
{
z
n
}
\{z_n\}
defined by
z
0
∈
K
,
z
n
+
1
=
(
1
−
μ
n
+
1
)
ω
+
μ
n
+
1
y
n
;
y
n
=
(
1
−
α
n
)
z
n
+
α
n
T
z
n
z_0\in K, \ \ z_{n+1}=(1-\mu _{n+ 1})\omega + \mu _{n+1}y_n; \ \ y_n = (1-\alpha _n)z_n+\alpha _nTz_n
, converges strongly to a fixed point of
T
T
, provided that
{
μ
n
}
\{\mu _n\}
and
{
α
n
}
\{\alpha _n\}
have certain properties. If
E
E
is a Hilbert space, then
{
z
n
}
\{z_n\}
converges strongly to the unique fixed point of
T
T
closest to
ω
\omega
.