Lemma 3.1.
Let be a uniformly convex Banach space with the convex modulus of power type , a nonempty closed convex subset of satisfying , and hemicontractive mappings with . Let , , , and be the sequences in (II) and
where is the constant in Remark 2.4. Then,
(1) exists for all ,
(2) exists,

(3)
if is continuous, then , for all .
Proof. (1) Let . By the boundedness assumption on , there exists a constant , for any , such that . From the definition of hemicontractive mappings, we have
Using Lemmas 2.3, 2.5, and (3.2), we obtain
Hence,
It follows from (II) and Lemma 2.5 that
By the condition , we may assume that
Therefore,
Substituting (3.7) into (3.4), we get
Assumptions (i) and (ii) imply that there exists a positive integer such that for every ,
Hence, for all ,
where
From (3.9) and conditions (i) and (ii), it follows that
By Lemma 2.6, we see that exists and the sequence is bounded.
(2) It is easy to verify that exists.
(3) By the boundedness of , there exists a constant such that , for all . From (3.10), we get, for ,
which implies
Thus,
It implies that
Therefore, by (3.7), we have
Using (II), we obtain
By a combination with the continuity of (, we get
It is clear that for each , there exists such that . Consequently,
This completes the proof.
Theorem 3.2.
Let the assumptions of Lemma 3.1 hold, and let be continuous. Then, converges strongly to a common fixed point of if and only if .
Proof.
The necessity is obvious.
Now, we prove the sufficiency. Since , it follows from Lemma 3.1 that .
For any , we have
Hence, we get
So, is a Cauchy sequence in . By the closedness of , we get that the sequence converges strongly to . Let a sequence , for some , be such that converges strongly to . By the continuity of , we obtain
Therefore, . This implies that is closed. Therefore, is closed. By , we get . This completes the proof.
Theorem 3.3.
Let the assumptions of Lemma 3.1 hold. Let be continuous and satisfy condition. Then, converges strongly to a common fixed point of .
Proof.
Since satisfies condition, and for each , it follows from the existence of that . Applying the similar arguments as in the proof of Theorem 3.2, we conclude that converges strongly to a common fixed point of . This completes the proof.
As a direct consequence of Theorem 3.3, we get the following result.
Corollary 3.4 (see [12, Theorem 3]).
Let H be a real Hilbert space, a nonempty closed convex subset of satisfying , and continuous hemicontractive mapping which satisfies condition (A). Let be a real sequence in with . For any , the sequence is defined by
Then, converges strongly to a fixed point of .
Proof.
Employing the similar proof method of Lemma 3.1, we obtain by (3.10)
This implies
By , we have . Equation (3.7) implies that . Since satisfies condition (A) and the limit exists, we get . The rest of the proof follows now directly from Theorem 3.2. This completes the proof.
Remark 3.5.
Theorems 3.2 and 3.3 extend [12, Theorem 3] essentially since the following hold.

(i)
Hilbert spaces are extended to uniformly convex Banach spaces.

(ii)
The requirement of compactness on domain on [12, Theorem 3] is dropped.

(iii)
A single mapping is replaced by a family of mappings.

(iv)
The Manntype implicit iteration is replaced by the generalized Mann iteration. So the restrictions of with for some are relaxed to . The error term is also considered in the iteration (II).
Moreover, if , then is well defined by (II). Hence, Theorems 3.2 and 3.3 are also answers to the question proposed by Qing [13].
Theorem 3.6.
Let and be as the assumptions of Lemma 3.1. Let be strictly pseudocontractive mappings with being nonempty. Let , , , , and be the sequences in (II) and
where is the constant in Remark 2.4. Then,
(1) converges strongly to a common fixed point of if and only if .

(2)
If satisfies condition () , then converges strongly to a common fixed point of .
Remark 3.7.
Theorem 3.6 extends the corresponding result [6, Theorem 3.1].