#### Abstract

The purpose of this paper is to give a general and short principle for proving some convergence results of certain types of iterative sequences. A small gap in the paper by Imnang and Suantai (2009) is discussed and corrected. Finally, we prove that the generalized asymptotically quasi-nonexpansive mappings in the sense of Lan (2006) are nothing but asymptotically quasi-nonexpansive. Hence several results concerning these mappings become a special case of the known ones.

#### 1. Introduction

Let be a nonempty subset of a Banach space . A mapping is said to be

(i)*asymptotically nonexpansive*[1] if there exists a sequence in such that and

for all and ;(ii)

*asymptotically quasi-nonexpansive*[2] if and there exists a sequence in such that and

for all , and ;(iii)

*generalized asymptotically nonexpansive*if there exist sequences in such that and

for all and ;(iv)

*generalized asymptotically quasi-nonexpansive*[3] if and there exist sequences in such that and

for all , and .

Many researchers have paid their attention on the approximation of a fixed point of a single mapping or a common fixed point of a family of mappings. One effective way is to use a sequence generated by an appropriate iteration. In this paper, we propose a general and short principle for proving some convergence results of certain types of iterative sequences. We also discuss and correct a small gap in the recent paper by Imnang and Suantai [4]. In the last section, we give a remark on the generalized asymptotically quasi-nonexpansive mapping in the sense of Lan [5].

Let be a finite family of self-mappings of a closed convex subset of . The sequence is generated from , and

where are bounded sequences in , and , and are sequences in such that for all and .

#### 2. Main Results

##### 2.1. Sequences of Monotone Types (1) and (2)

*Definition 2.1. *Let be a sequence in a metric space and a subset of . We say that is of (i)*monotone type (1) with respect to * [6] if there exist sequences and of nonnegative real numbers such that , andfor all and ; (ii)*monotone type (2) with respect to * if for each there exist sequences and of nonnegative real numbers such that , andfor all .

Proposition 2.2. *If is of monotone type (1) with respect to , then it is of monotone type (2) with respect to .*

Lemma 2.3 ([7, Lemma ]). * Let , , and be sequences of nonnegative real numbers such that
**
If and then exists.*

Theorem 2.4. * Let be a complete metric space, , and a sequence in . Then one has the following assertions. *(a)*If is of monotone type (2) with respect to , then exists for all . *(b)*If is of monotone type (1) with respect to , then exists. *(c)*If is of monotone type (1) with respect to and , then for some satisfying . In particular, if is closed, then .*

*Proof. *(a)It is easy to see that the result follows from (2.2) and Lemma 2.3.(b) Note that and are independent of . Taking infimum over all in (2.1) gives
Again, by Lemma 2.3, we get that exists.(c) It follows from (b) and that
To show that is a Cauchy sequence, let . Since , we may assume without loss of generality that there is a sequence in such that for all . As is bounded, we put . From (2.1), we have
where . Consequently,
Notice that . So there exists such that . Then for all , we have
Hence, is a Cauchy sequence in . By the completeness of , we assume that for some . Since
we obtain . This completes the proof.

##### 2.2. A Correction of Recent Results of Imnang and Suantai

The following observation is an auxiliary result.

Proposition 2.5. *Let be a nonempty subset of a Banach space , and let be generalized asymptotically quasi-nonexpansive mappings with . Then there exist sequences in such that and
**
for all , and .*

From now on, we assume that generalized asymptotically quasi-nonexpansive mappings are equipped with the sequences in as mentioned in the preceding proposition.

Theorem 2.6. *Let be a nonempty closed convex subset of a Banach space , and a finite family of generalized asymptotically quasi-nonexpansive self-mappings of with the sequence such that and . Assume that is closed, and is the sequence in defined by (1.5) such that for each . Then the sequence converges strongly to a common fixed point of the family of mappings if and only if .*

*Remark 2.7. *There is a small gap in [4, Theorem ]. More precisely, the sequence generated by (1.5) is shown in [4, Theorem ] to be of monotone type (2) with respect to , that is, where each is a nonnegative real number depending on . Then the expression cannot warrant.

*Remark 2.8. *The same gap also appears in [8,Lemma ] and [6, Theorem ].

*Proof of Theorem 2.6. *Necessity is obvious. Conversely, we show first that is of monotone type (2) with respect to . Let . We have that
where . Notice that and is bounded. Then . It follows from (2.12) that
where . Notice that and is bounded. Then . By continuing this process, there is a sequence of nonnegative real numbers such that and
Then is of monotone type (2) with respect to . By Theorem 2.4(a), we get that exists and is bounded. Next, we show that is of monotone type (1) with respect to . It follows from (2.11) that
where . Notice that , are bounded and . Then and is independent of . Again, by continuing this process, we obtain a sequence of nonnegative real numbers such that it is independent of , and
for all and . Then is of monotone type (1) with respect to . Hence the result follows from (2.16) and Theorem 2.4(c). This completes the proof.

*Remark 2.9. *Theorem 2.4 is a correction of [4,Theorem ]. In fact, the closedness of is not assumed there (this defect is now corrected *after* the submission of this article). Moreover, it is shown in the following example that the fixed point set of a generalized asymptotically nonexpansive mapping is not necessarily closed even in a Hilbert space.

*Example 2.10 (A generalized asymptotically nonexpansive mapping whose fixed point set is not closed). *
Let be a mapping defined by
Then is generalized asymptotically nonexpansive.

*Proof. *Notice that is not closed. We prove that
for all and . The inequality above holds trivially if or . Then it suffices to consider the following cases.*Case 1 (). * Then
*Case 2 ( and ). * Then
*Case 3 ( and ). * Then
*Case 4 ( and ). * Then
Hence, (2.18) holds. This completes the proof.

*Remark 2.11. *For which is defined in Example 2.10 and , we define
where and . It is not hard to show that and . Hence [4, Theorems and ] do not hold even for a single mapping if the closedness of the fixed point set is not assumed.

We present a sufficient condition guaranteeing the closedness of the fixed point set of a generalized asymptotically quasi-nonexpansive mapping.

Theorem 2.12. *Let be a nonempty subset of a Banach space and a generalized asymptotically quasi-nonexpansive mapping. If is closed, then is closed.*

*Proof. *Let be a sequence in such that . Since is a generalized asymptotically quasi-nonexpansive mapping with the sequence , we have
Then , and so . Hence, by the closedness of , . This completes the proof.

*Remark 2.13. *It is also worth mentioning that the uniform Lipschitz condition of mappings in [4, Theorems and ] implies the closedness of their graphs.

The following result shows that the closedness of can be dropped if is asymptotically quasi-nonexpansive.

Theorem 2.14. *Let be a nonempty subset of a Banach space , and an asymptotically quasi-nonexpansive mapping. Then is closed.*

*Proof. *Suppose that is an asymptotically quasi-nonexpansive mapping with the sequence . Let be a sequence in such that . We have
Then . This completes the proof.

*Remark 2.15. *Not every generalized asymptotically quasi-nonexpansive mapping is asymptotically quasi-nonexpansive. In fact, the mapping in Example 2.10 is not asymptotically quasi-nonexpansive since is not closed.

#### 3. Remark on Lan’s Generalized Asymptotically Quasi-Nonexpansive Mappings

The following mapping introduced by Lan [5] also bears the name generalized asymptotically quasi-nonexpansive mappings. We recall his definition here.

*Definition 3.1 (see [5, Definition ()]). *Let be a subset of a Banach space . A mapping is called *generalized asymptotically quasi-nonexpansive in the sense of Lan* if there exists two sequences and such that and
for all , , and .

Lan [5] and many authors (e.g., [8–11]) have investigated convergence theorems for such mappings without awareness that Lan's mappings are not new ones.

Proposition 3.2. *If is generalized asymptotically quasi-nonexpansive in the sense of Lan, then it is asymptotically quasi-nonexpansive.*

*Proof. *By Lan's definition, there exist two sequences and such that and
for all , , and . Consequently,
This implies
It is also clear that and this completes the proof.

#### Acknowledgments

The research of the first and second authors is partially supported by the Centre of Excellence in Mathematics, the Commission on Higher Education of Thailand. The third author was supported by the Human Resource Development in Science Project.