The following theorem is proved: Suppose H is a complex Hilbert space, and
T
:
H
→
H
T:H \to H
is a monotonic, nonexpansive operator on H, and
f
∈
H
f \in H
. Define
S
:
H
→
H
S:H \to H
by
S
u
=
−
T
u
+
f
Su = - Tu + f
for all
u
∈
H
u \in H
. Suppose
0
⩽
t
n
⩽
1
0 \leqslant {t_n} \leqslant 1
for all
n
=
1
,
2
,
3
,
…
,
n = 1,2,3, \ldots ,
and
Σ
n
=
1
∞
t
n
(
1
−
t
n
)
\Sigma _{n = 1}^\infty \;{t_n}(1 - {t_n})
diverges. Then the iterative process
V
n
+
1
=
(
1
−
t
n
)
V
n
+
t
n
S
V
n
{V_{n + 1}} = (1 - {t_n}){V_n} + {t_n}S{V_n}
converges to the unique solution
u
=
p
u = p
of the equation
u
+
T
u
=
f
u + Tu = f
.