Let
K
K
be a compact convex subset of a real Hilbert space,
H
H
;
T
:
K
→
K
T:K\rightarrow K
a continuous pseudocontractive map. Let
{
a
n
}
,
{
b
n
}
,
{
c
n
}
,
{
a
n
′
}
,
{
b
n
′
}
\{a_{n}\}, \{b_{n}\}, \{c_{n}\}, \{a_{n}^{’}\}, \{b_{n}^{’}\}
and
{
c
n
′
}
\{c_{n}^{’}\}
be real sequences in [0,1] satisfying appropriate conditions. For arbitrary
x
1
∈
K
,
x_{1}\in K,
define the sequence
{
x
n
}
n
=
1
∞
\{x_{n}\}_{n=1}^{\infty }
iteratively by
x
n
+
1
=
a
n
x
n
+
b
n
T
y
n
+
c
n
u
n
;
y
n
=
a
n
′
x
n
+
b
n
′
T
x
n
+
c
n
′
v
n
,
n
≥
1
,
x_{n+1} = a_{n}x_{n} + b_{n}Ty_{n} + c_{n}u_{n}; y_{n} = a_{n}^{’}x_{n} + b_{n}^{’}Tx_{n} + c_{n}^{’}v_{n}, n\geq 1,
where
{
u
n
}
,
{
v
n
}
\{u_{n}\}, \{v_{n}\}
are arbitrary sequences in
K
K
. Then,
{
x
n
}
n
=
1
∞
\{x_{n}\}_{n=1}^{\infty }
converges strongly to a fixed point of
T
T
. A related result deals with the convergence of
{
x
n
}
n
=
1
∞
\{x_{n}\}_{n=1}^{\infty }
to a fixed point of
T
T
when
T
T
is Lipschitz and pseudocontractive. Our theorems also hold for the slightly more general class of continuous hemicontractive nonlinear maps.