#### Abstract

We introduce a general iteration scheme for a finite family of generalized asymptotically quasi-nonexpansive mappings in Banach spaces. The new iterative scheme includes the multistep Noor iterations with errors, modified Mann and Ishikawa iterations, three-step iterative scheme of Xu and Noor, and Khan and Takahashi scheme as special cases. Our results generalize and improve the recent ones announced by Khan et al. (2008), H. Fukhar-ud-din and S. H. Khan (2007), J. U. Jeong and S. H. Kim (2006), and many others.

#### 1. Introduction

Let be a subset of real Banach space . Let be a self-mapping of and let denote the fixed points set of , that is . Recall that a mapping is said to be *asymptotically nonexpansive* on if there exists a sequence in with such that for each ,

If for all , then is known as a *nonexpansive mapping*. is called *generalized asymptotically quasi-nonexpansive* [1] if there exist sequences in with such that

for all and all . If for all , then is known as an asymptotically quasi-nonexpansive mapping. *is called asymptotically nonexpansive mapping in the intermediate sense* [2] provided that is uniformly continuous and

*T* is said to be *uniform Lipschitz* [3] if there are constants and such that

for all . A mapping is called *semicompact* if any bounded sequence in with , there exists a subsequence of such that converges strongly to some in .

*Remark 1.1. *Let be asymptotically nonexpansive mapping in the intermediate sense. Put , .

If , we obtain that for all and all . Since , therefore is a generalized asymptotically quasi-nonexpansive mapping.

Recall that a mapping with is said to satisfy *condition* (I) [4] if there is a nondecreasing function with and for all such that for all , where .

Fixed-point iteration processes for asymptotically quasi-nonexpansive mapping in Banach spaces including Mann and Ishikawa iterations processes have been studied extensively by many authors; see [3, 5–11]. Many of them are used widely to study the approximate solutions of the certain problems. In 1974, Senter and Dotson [4] studied the convergence of the Mann iteration scheme defined by ,

in a uniformly convex Banach space, where is a sequence satisfying and is a nonexpansive (or a quasi-nonexpansive) mapping. They established a relation between *condition* (I) and *demicompactness*.

Recall that a mapping is demicompact if for every bounded sequence in such that converges, there exists a subsequence say of that converges strongly to some in . Every compact and semicompact mapping is demicompact. They actually showed that *condition* (I) is weaker than demicompactness for a nonexpansive mapping defined on bounded set.

Xu and Noor [12], in 2002, introduced a three-step iterative scheme as follows:

where are appropriate sequences in . The theory of three-step iterative scheme is very rich, and this scheme, in the context of one or more mappings, has been extensively studied (e.g., see Khan et al. [6], Plubtieng and Wangkeeree [7], Fukhar-ud-din and Khan [5], Petrot [13], and Suantai [14]). It has been shown in [15] that three-step method performs better than two-step and one-step methods for solving variational inequalities.

In 2001, Khan and Takahashi [16] have approximated common fixed points of two asymptotically nonexpansive mappings by the modified Ishikawa iteration. Jeong and Kim [17] have approximated common fixed points of two asymptotically nonexpansive mappings. Plubtieng et al. [18], in 2006, modified Noor iterations with errors and have approximated common fixed points of three asymptotically nonexpansive mappings. Shahzad and Udomene [10] established convergence theorems for the modified Ishikawa iteration process of to asymptotically quasi-nonexpansive mappings to a common fixed point of the mappings. Plubtieng and Wangkeeree [7], in 2006, established strong convergence theorems of the modified multistep Noor iterations with errors for an asymptotically quasi-nonexpansive mapping and asymptotically nonexpansive mapping in the intermediate sense.

Very recently, Khan et al. [6], in 2008, established convergence theorems for the modified multistep Noor iterations process of finite family of asymptotically quasi-nonexpansive mappings to a common fixed point of the mappings. For rerated results with errors terms, we refer to [5–7, 17–21]. Inspired and motivated by these facts, we introduce a new iteration process for a finite family of of generalized asymptotically quasi-nonexpansive mappings as follows.

Let be mappings and For a given and a fixed ( denote the set of all positive integers), compute the iterative sequences and by

where and are bounded sequences in with , and are appropriate real sequences in such that for all and all . Our iteration includes and extends the Mann iteration (1.5), three-step iteration by Xu and Noor (1.6), the multistep Noor iterations with errors by Plubtieng and Wangkeeree [7], and the iteration defined by Khan et al. [6] simultaneously.

The purpose of this paper is to establish several strong convergence theorems of the iterative scheme (1.7) for a finite family of generalized asymptotically quasi-nonexpansive mappings when one mapping satisfies a condition which is weaker than demicompactness and we also weak convergence theorem for a finite family of generalized asymptotically quasi-nonexpansive mappings in a uniformly convex Banach space satisfying Opial's property. Our results generalize and improve the corresponding ones announced by Khan et al. [6], Fukhar-ud-din and Khan [5], and many others.

#### 2. Preliminaries

In the sequel, the following lemmas are needed to prove our main results.

A mapping with domain and range in X is said to be at if whenever is a sequence in such that converges weakly to and converging strongly to , we have .

A Banach space is said to satisfy *Opial's property* if for each in and each sequence weakly convergent to , the following condition holds for :

It is well known that all Hilbert spaces and spaces have Opial's property while spaces have not. A family of self-mappings of with is said to satisfy the following conditions.

(1)*Condition*[22]. If there is a nondecreasing function with and for all such that for all , where .(2)

*Condition*[22]. If there is a nondecreasing function with and for all such that for all .(3)

*Condition*[22]. If there is a nondecreasing function with and for all such that for all and for at least one .

Note that and are equivalent, condition reduces to condition (I) when all but one of 's are identities, and in addition, it also condition .

It is well known that every continuous and demicompact mapping must satisfy condition (I) (see [4]). Since every completely continuous is continuous and demicompact so that it satisfies condition (I). Thus we will use condition instead of the demicompactness and complete continuity of a family .

Lemma 2.1 (see [8, Lemma ]). *Let , and be sequences of nonnegative real numbers satisfying the inequality
**
If and , then*(i)* exists;*(ii)* whenever .*

Lemma 2.2 (see [7, Lemma ]). *Let be a uniformly convex Banach space, , , real numbers , and let be a real sequence number which satisfies*(i)* and for some ;*(ii)* and ;*(iii)*. Then .*

Lemma 2.3 (see [14, Lemma ]). *Let be a Banach space which satisfies Opial's property and let be a sequence in . Let be such that and exist. If and are subsequences of which converge weakly to and , respectively, then .*

#### 3. Convergence Theorems in Banach Spaces

Our first result is the strong convergence theorems of the iterative scheme (1.7) for a finite family of generalized asymptotically quasi-nonexpansive mappings in a Banach space. In order to prove our main results, the following lemma is needed.

Lemma 3.1. *Let be a Banach space and a nonempty closed and convex subset of , and a finite family of generalized asymptotically quasi-nonexpansive self-mappings of with the sequences such that and for all . Assume that and for each . For a given , let the sequences and be defined by (1.7). Then*(a)*there exist sequences and in such that , , and , for all and all ;*(b)* exists for all ;*(c)*there exist constant and in such that and for all and .*

*Proof. *(a) Let , and for all .

Since and , for all , therefore and . For each we note that
where . Since is bounded, and , we obtain that . It follows from (3.1) that
where . Since , are bounded, , , and , it follows that . Moreover, we see that
where . Since , are bounded, , , and , it follows that . By continuing the above method, there are nonnegative real sequences in such that and
This completes the proof of (a).

(b) From part (a), for the case , we have
It follows from Lemma 2.1(i) that exists, for all .

(c) If , then and so, , for Thus, from (3.5), it follows that
where and .

Theorem 3.2. *Let be a Banach space and a nonempty closed and convex subset of and a finite family of generalized asymptotically quasi-nonexpansive self-mappings of with the sequences such that and for all . Assume that is closed and for each . Then the iterative sequence defined by (1.7) converges strongly to a common fixed point of the family of mappings if and only if *

*Proof. *We prove only the sufficiency because the necessity is obvious. From (3.5), we have .

Hence, we have
Since , it follows that . Since and , it follows from Lemma 2.1(ii) that . Next, we prove that is a Cauchy sequence. From Lemma 3.1(c), we have
Since and , therefore for , there exists such that
Therefore, there exists in such that
From (3.8) to (3.10), for all and , we have
This shows that is a Cauchy sequence, hence . It remains to show that . Notice that
Since , we obtain that .

The following corollary follows from Theorem 3.2.

Corollary 3.3. *Let be a Banach space and a nonempty closed and convex subset of and a finite family of generalized asymptotically quasi-nonexpansive self-mappings of with the sequences such that and for all . Assume that is closed and for each . Then the iterative sequence , defined by (1.7), converges strongly to a point if and only if there exists a subsequence of converging to .*

Since an asymptotically quasi-nonexpansive mapping is generalized asymptotically quasi-nonexpansive mapping, so we have the following result.

Corollary 3.4. *Let be a Banach space and a nonempty closed and convex subset of and a finite family of asymptotically quasi-nonexpansive self-mappings of with the sequences such that for all . Assume that and for each . Then the iterative sequence , defined by (1.7), converges strongly to a common fixed point of the family of mappings if and only if *

*Remark 3.5. *Theorem 3.2 generalizes and extends Theorem of Khan et al. [6], for a finite family of asymptotically quasi-nonexpansive mappings, Theorem of Fukhar-ud-din and Khan [5], and Theorem of Shahzad and Udomene [10] for two asymptotically quasi-nonexpanaive mappings to the more general class of generalized asymptotically quasi-nonexpansive mappings.

Theorem 3.6. *Let be a Banach space and a nonempty closed and convex subset of and a finite family of generalized asymptotically quasi-nonexpansive self-mappings of with the sequences such that and for all . Suppose that is closed. Let and be the sequence defined by (1.7). If , for all and satisfies condition , then converges strongly to a common fixed point of the family of mappings.*

*Proof. *From for all and satisfying condition , there is a nondecreasing function with and for all such that for some , it follows that . From Theorem 3.2, we obtain that converges strongly to a common fixed point of the family of mappings.

#### 4. Convergence Theorems in Uniformly Convex Banach Spaces

In this section, we establish weak and strong convergence theorems of the iterative scheme (1.7) for a finite family of generalized asymptotically quasi-nonexpansive and uniform Lipschitz mappings in a uniformly convex Banach space. In order to prove our main results, we need the following lemma.

Lemma 4.1. *Let be a nonempty closed and convex subset of a uniformly convex Banach space and a finite family of uniform Lipschitz and generalized asymptotically quasi-nonexpansive self-mappings of with the sequences such that and for all . Assume that and for all . For a given let and be the sequences defined by (1.7) with , for all and all and for some . Then *(i)* for all ,*(ii)* for all ,*(iii)* for all . *

*Proof. *Let , and for all .

(i) From Lemma 3.1(b), we have that exists for all . Suppose that

From (3.4) and (4.1), we get that
For each and , we have
By using (4.3) and (1.7), for each , we have
Since , for all and all , we have that for all and all ,
where and . Since and , it follows that
From (4.2) and (4.6), we have
That is, for each , we have
Since
it follows that
From (4.8) to (4.11), we can conclude from Lemma 2.2 that

(ii) It follows from part (i) in the case that For , we obtain from part (i) that
Therefore,
Since
it follows from (i) and (4.14) that

(iii) Since and
for all , (iii) is directly obtained by (i).

Theorem 4.2. *Let be a nonempty closed and convex subset of a uniformly convex Banach space satisfying the Opial's property, and a finite family of uniform Lipschitz and generalized asymptotically quasi-nonexpansive self-mappings of with the sequences such that and for all . Assume that and for all . For a given let be the sequence defined by (1.7) with , for all and all and for some . If , , is demiclosed at , then converges weakly to a common fixed point of the family .*

*Proof. *By Lemma 4.1(ii), we have , for all Since is uniformly convex and is bounded, without loss of generality we may assume that weakly as for some . Since , , is demiclosed at , we have . Suppose that there are subsequences and of that converge weakly to and , respectively. Again, as above, we can prove that . By Lemma 3.1(b), and exist. It follows from Lemma 2.3 that . Therefore converges weakly to a common fixed point of .

Theorem 4.3. *Under the hypotheses of Lemma 4.1, assume that the family satisfies condition . Then and converge strongly to a common fixed point of the family of mappings for all .*

*Proof. *From (3.5), we have
Therefore,
Since , it follows that . Since , we obtain from Lemma 2.1(i) that exists. By Lemma 4.1(ii), we have for all . Since satisfies condition , there is a nondecreasing function with and for all such that for some , it follows that . By Theorem 3.2, we can conclude that converges strongly to a common fixed point of the family . From Lemma 4.1(iii), we have for all , and we obtain that for all .

*Remark 4.4. *The family of generalized asymptotically quasi-nonexpansive mappings in Theorem 4.2 and 4.3 can be replaced by a family of asymptotically quasi-nonexpansive mappings. Lemma 3.1 and 4.1 generalize and improve [6, Lemma ], [19, Lemmas and ], [18, Lemma ], [7, Lemma ], and [17, Lemma ] to a finite family of uniform Lipschitz and generalized asymptotically quasi-nonexpansive mappings. Theorem 4.2 generalizes and improves [6, Theorems and ], [18, Theorem ], [17, Theorem ], and [21, Theorem ] to the more general class of a finite family of uniform Lipschitz and generalized asymptotically quasi-nonexpansive mappings. Theorem 4.3 generalizes and improves [6, Theorem ], [7, Theorem ], [5, Theorem ], [18, Theorem ], [19, Theorem ], [17, Theorem ], [10, Theorem ], [20, Theorem ] and [21, Theorem ] by using condition instead of condition or semicompactness or completely continuous or compactness to the more general class of a finite family of uniform Lipschitz and generalized asymptotically quasi-nonexpansive mappings.

*Remark 4.5. *From Remark 1.1, Theorems 3.2 to 4.3 hold true for a finite family of asymptotically nonexpansive mappings in the intermediate sense.

#### Acknowledgments

The authors would like to thank the Commission on Higher Education, the Thailand Research Fund, the Thaksin University, and the Graduate School of Chiang Mai University, Thailand for their financial support.