Let
E
E
be a real Banach space with a uniformly convex dual, and let
K
K
be a nonempty closed convex and bounded subset of
E
E
. Let
T
:
K
→
K
T:K \to K
be a continuous strongly pseudocontractive mapping of
K
K
into itself. Let
{
c
n
}
n
=
1
∞
\{ {c_n}\} _{n = 1}^\infty
be a real sequence satisfying: (i)
0
>
c
n
>
1
0 > {c_n} > 1
for all
n
⩾
1
n \geqslant 1
; (ii)
∑
n
=
1
∞
c
n
=
∞
\sum \nolimits _{n = 1}^\infty {{c_n} = \infty }
; and (iii)
∑
n
=
1
∞
c
n
b
(
c
n
)
>
∞
\sum \nolimits _{n = 1}^\infty {{c_n}b({c_n}) > \infty }
, where
b
:
[
0
,
∞
)
→
[
0
,
∞
)
b:[0,\infty ) \to [0,\infty )
is some continuous nondecreasing function satisfying
b
(
0
)
=
0
,
b
(
c
t
)
⩽
c
b
(
t
)
b(0) = 0,\,b(ct) \leqslant cb(t)
for all
c
⩾
1
c \geqslant 1
. Then the sequence
{
x
n
}
n
=
1
∞
\{ {x_n}\} _{n = 1}^\infty
generated by
x
1
∈
K
{x_1} \in K
,
\[
x
n
+
1
=
(
1
−
c
n
)
x
n
+
c
n
T
x
n
,
n
⩾
1
,
{x_{n + 1}} = (1 - {c_n}){x_n} + {c_n}T{x_n},\qquad n \geqslant 1,
\]
converges strongly to the unique fixed point of
T
T
. A related result deals with the Ishikawa iteration scheme when
T
T
is Lipschitzian and strongly pseudocontractive.