Let
E
E
be a uniformly smooth real Banach space and let
A
:
E
→
E
A: E \rightarrow E
be a mapping with
N
(
A
)
≠
∅
N(A)\neq \emptyset
. Suppose
A
A
is a generalized Lipschitz generalized
Φ
\Phi
-quasi-accretive mapping. Let
{
a
n
}
,
{
b
n
}
,
\{a_{n}\}, \{b_{n}\},
and
{
c
n
}
\{c_{n}\}
be real sequences in [0,1] satisfying the following conditions: (i)
a
n
+
b
n
+
c
n
=
1
a_{n} + b_{n} + c_{n} = 1
; (ii)
∑
(
b
n
+
c
n
)
=
∞
\sum (b_{n} + c_{n} ) = \infty
; (iii)
∑
c
n
>
∞
\sum c_{n} > \infty
; (iv)
lim
b
n
=
0.
\lim b_{n} = 0.
Let
{
x
n
}
\{x_{n}\}
be generated iteratively from arbitrary
x
0
∈
E
x_{0}\in E
by
\[
x
n
+
1
=
a
n
x
n
+
b
n
S
x
n
+
c
n
u
n
,
n
≥
0
,
x_{n+1} = a_{n}x_{n} + b_{n}Sx_{n} + c_{n}u_{n}, n\geq 0,
\]
where
S
:
E
→
E
S: E\rightarrow E
is defined by
S
x
:=
x
−
A
x
∀
x
∈
E
Sx:=x-Ax ~\forall x\in E
and
{
u
n
}
\{u_{n}\}
is an arbitrary bounded sequence in
E
E
. Then, there exists
γ
0
∈
ℜ
\gamma _{0}\in \Re
such that if
b
n
+
c
n
≤
γ
0
∀
n
≥
0
,
b_{n} + c_{n} \leq \gamma _{0} ~\forall ~ n\geq 0,
the sequence
{
x
n
}
\{x_{n}\}
converges strongly to the unique solution of the equation
A
u
=
0
Au = 0
. A related result deals with approximation of the unique fixed point of a generalized Lipschitz and generalized
ϕ
\phi
-hemi-contractive mapping.