Abstract
Let
E
be a real Banach space, and let
A
:
D
(
A
)
⊆
E
→
E
be a Lipschitz,
ψ
-expansive and accretive mapping such that
c
o
¯
(
D
(
A
)
)
⊆
∩
λ
>
0
R
(
I
+
λ
A
)
. Suppose that there exists
x
0
∈
D
(
A
)
, where one of the following holds: (i) There exists
R
>
0
such that
ψ
(
R
)
>
2
‖
A
(
x
0
)
‖
; or (ii) There exists a bounded neighborhood
U
of
x
0
such that
t
(
x
−
x
0
)
∉
A
x
for
x
∈
∂
U
∩
D
(
A
)
and
t
<
0
. An iterative sequence
{
x
n
}
is constructed to converge strongly to a zero of
A
. Related results deal with the strong convergence of this iteration process to fixed points of
ψ
-expansive and pseudocontractive mappings in real Banach spaces. The convergence results established in this paper are new for this more general class of
ψ
-expansive and accretive or pseudocontractive mappings.