**Theorem 2.1**
*Let*
*E*
*be an arbitrary uniformly smooth real Banach space*, *D*
*be a nonempty closed convex subset of*
*E*, *and*
*be a generalized Lipschitz* Φ-*hemi*-*contractive mapping with*
. *Let*
,
,
,
*be four real sequences in*
*and satisfy the conditions* (i)
*as*
*and*
; (ii)
. *For some*
, *let*
,
*be any bounded sequences in*
*D*, *and*
*be an Ishikawa iterative sequence with errors defined by* (1.1). *Then* (1.1) *converges strongly to the unique fixed point*
*q*
*of*
*T*.

*Proof* Since

is a generalized Lipschitz Φ-hemi-contractive mapping, there exists a strictly increasing continuous function

with

such that

for any
and
.

Step 1. There exists
and
such that
(range of Φ). Indeed, if
as
, then
; if
with
, then for
, there exists a sequence
in *D* such that
as
with
. Furthermore, we obtain that
is bounded. Hence, there exists a natural number
such that
for
, then we redefine
and
.

Step 2. For any

,

is bounded. Set

, then from Definition 1.2(2), we obtain that

. Denote

,

. Since

*T* is generalized Lipschitz, so

*T* is bounded. We may define

. Next, we want to prove that

. If

, then

. Now, assume that it holds for some

*n*,

*i.e.*,

. We prove that

. Suppose it is not the case, then

. Since

*J* is uniformly continuous on a bounded subset of

*E*, then for

, there exists

such that

when

,

. Now, denote

Owing to

as

, without loss of generality, assume that

for any

. Since

, denote

. So, we have

Using Lemma 1.6 and the above formulas, we obtain

Substitute (2.11) into (2.10)

this is a contradiction. Thus,
, *i.e.*,
is a bounded sequence. So,
,
,
are all bounded sequences.

Step 3. We want to prove
as
. Set
.

By (2.10), (2.11), we have

where
,
and
as
.

Taking (2.14) into (2.13),

where
as
.

Set

, then

. If it is not the case, we assume that

. Let

, then

,

*i.e.*,

. Thus, from (2.15) it follows that

Let

,

,

. Then we get that

Applying Lemma 1.7, we get that

as

. This is a contradiction and so

. Therefore, there exists an infinite subsequence such that

as

. Since

, then

as

. In view of the strictly increasing continuity of Φ, we have

as

. Hence,

as

. Next, we want to prove

as

. Let

, there exists

such that

,

,

,

, for any

. First, we want to prove

. Suppose it is not the case, then

. Using (1.1), we may get the following estimates:

Since Φ is strictly increasing, then (2.19) leads to

. From (2.15), we have

which is a contradiction. Hence,
. Suppose that
holds. Repeating the above course, we can easily prove that
holds. Therefore, for any *m*, we obtain that
, which means
as
. This completes the proof. □

**Theorem 2.2**
*Let*
*E*
*be an arbitrary uniformly smooth real Banach space*,

*and*
*be a generalized Lipschitz* Φ-

*quasi*-

*accretive mapping with*
.

*Let*
,

,

,

*be four real sequences in*
*and satisfy the conditions* (i)

*as*
*and*
; (ii)

.

*For some*
,

*let*
,

*be any bounded sequences in*
*E*,

*and*
*be an Ishikawa iterative sequence with errors defined by*
*where*
*is defined by*
*for any*
. *Then*
*converges strongly to the unique solution of the equation*
(*or the unique fixed point of*
*S*).

*Proof* Since

*T* is a generalized Lipschitz and Φ-quasi-accretive mapping, it follows that

for all
,
. The rest of the proof is the same as that of Theorem 2.1. □

**Corollary 2.3**
*Let*
*E*
*be an arbitrary uniformly smooth real Banach space*, *D*
*be a nonempty closed convex subset of*
*E*, *and*
*be a generalized Lipschitz* Φ-*hemi*-*contractive mapping with*
. *Let*
,
*be two real sequences in*
*and satisfy the conditions* (i)
*as*
*and*
; (ii)
. *For some*
, *let*
*be any bounded sequence in*
*D*, *and*
*be the Mann iterative sequence with errors defined by* (1.2). *Then* (1.2) *converges strongly to the unique fixed point*
*q*
*of*
*T*.

**Corollary 2.4**
*Let*
*E*
*be an arbitrary uniformly smooth real Banach space*,

*and*
*be a generalized Lipschitz* Φ-

*quasi*-

*accretive mapping with*
.

*Let*
,

*be two real sequences in*
*and satisfy the conditions* (i)

*as*
*and*
; (ii)

.

*For some*
,

*let*
*be any bounded sequence in*
*E*,

*and*
*be the Mann iterative sequence with errors defined by*
*where*
*is defined by*
*for any*
. *Then*
*converges strongly to the unique solution of the equation*
(*or the unique fixed point of*
*S*).

**Remark 2.5** It is mentioned that in 2006, Chidume and Chidume [1] proved the approximative theorem for zeros of generalized Lipschitz generalized Φ-quasi-accretive operators. This result provided significant improvements of some recent important results. Their result is as follows.

**Theorem CC** ([[1], Theorem 3.1])

*Let*
*E*
*be a uniformly smooth real Banach space and*
*be a mapping with*
.

*Suppose*
*A*
*is a generalized Lipschitz* Φ-

*quasi*-

*accretive mapping*.

*Let*
,

*and*
*be real sequences in*
*satisfying the following conditions*: (i)

; (ii)

; (iii)

; (iv)

.

*Let*
*be generated iteratively from arbitrary*
*by*
*where*
*is defined by*
,
*and*
*is an arbitrary bounded sequence in*
*E*. *Then*, *there exists*
*such that if*
,
, *the sequence*
*converges strongly to the unique solution of the equation*
.

However, there exists a gap in the proof process of above Theorem CC. Here,
(
) does not hold in line 14 of Claim 2 on page 248, *i.e.*,
is a wrong case. For instance, set the iteration parameters:
, where
,
,
;
. Then
, but
. Therefore, the proof of above Theorem CC is not reasonable. Up to now, we do not know the validity of Theorem CC. This will be an open question left for the readers!