Abstract

We introduce a general iteration method for a finite family of generalized asymptotically quasi-nonexpansive mappings. The results presented in the paper extend and improve some recent results in the works by Shahzad and Udomene (2006); L. Qihou (2001); Khan et al. (2008).

1. Introduction and Preliminaries

Let be a nonempty subset of a real Banach space and a self-mapping of . The set of fixed points of is denoted by and we assume that . The mapping is said to be(i)contractive mapping if there exists a constant in such that , for all ;(ii)asymptotically nonexpansive mapping if there exists a sequence in with such that , for all and ;(iii)asymptotically quasi-nonexpansive if there exists a sequence in with such that , for all , and ;(iv)generalized asymptotically quasi-nonexpansive [1] if there exist two sequences , in with and such that where ;(v)uniformly -Lipschitzian if there exists a constant such that , for all and ;(vi) uniform -Lipschitz if there are constants and such that , for all and ;(vii)semicompact if for a sequence in with , there exists a subsequence of such that .

In (1), if for all , then becomes an asymptotically quasi-nonexpansive mapping; if and for all , then becomes a quasi-nonexpansive mapping. It is known that an asymptotically nonexpansive mapping is an asymptotically quasi-nonexpansive and a uniformly -Lipschitzian mapping is uniform -Lipschitz.

The mapping is said to be demiclosed at if for each sequence converging weakly to and converging strongly to , we have .

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

Let be a nonempty closed convex subset of a real Banach space and a finite family of asymptotically nonexpansive mappings of into itself. Suppose that , , and . Then we consider the following mapping of into itself: where (identity mapping). Such a mapping is called the modified -mapping generated by and (see [2, 3]).

In the sequel, we assume that .

In 2008, Khan et al. [4] introduced the following iteration process for a family of asymptotically quasi-nonexpansive mappings, for an arbitrary : where , , , and proved that the iterative sequence defined by (4) converges strongly to a common fixed point of the family of mappings if and only if , where . With the help of (3), we write (4) as

Recently, Chang et al. [5] introduced the following iteration process of asymptotically nonexpansive mappings in Banach space: where and is a fixed contractive mapping, and necessary and sufficient conditions are given for the iterative sequence to converge to the fixed points of .

For a family of mappings, it is quite significant to devise a general iteration scheme which extends the iteration processes (4) and (6), simultaneously. Thereby, to achieve this goal, we introduce a new iteration process for a family of mappings as follows.

Let be a nonempty closed convex subset of a real Banach space , a family of generalized asymptotically quasi-nonexpansive mappings, and a fixed contractive mapping with contractive coefficient . For a given , the iteration scheme is defined as follows: where and is the modified -mapping generated by , and for all positive integers .

The purpose of this paper is to study the convergence problem of the iterative sequences defined by (7). The obtained results extend the corresponding results in [48], and Lemma 11 partly improves the method of proof of Lemma  3.1 in [4].

In what follows, we need the following useful known lemmas.

Lemma 1 (see [9]). Let , , and be nonnegative real sequences satisfying the following condition: where and ; then exists.

Moreover, if in addition, , then .

Lemma 2 (see [4]). Let be a uniformly convex Banach space, for all , and let and be sequences in . Assume that , , and for some . Then .

2. Main Results

Lemma 3. Let be a nonempty closed convex subset of a real Banach space and an asymptotically quasi-nonexpansive self-mapping of with for all . Suppose . Then is a closed subset in .

Proof. Let be an arbitrary sequence of and as . Since is closed, we have . For any , there exists a natural number such that
Thus, we get
Since is arbitrary, it follows that ; that is, . Hence and is closed. This completes the proof.

Lemma 4. Let be a nonempty closed convex subset of a real Banach space . Let be generalized asymptotically quasi-nonexpansive self-mappings of with such that and for all . Suppose and for all . Let be the modified -mapping generated by and . Let the sequence be defined by (7) and assuming , then(1)there exist two sequences and in with such that (2)there exists a constant , such that where and .

Proof. Let , for all . Since for each , we can get . For all , it follows from (3) that
Assume that for some . Then
Thus, by induction, we have for all . Hence,
By (7) and (16), we obtain
Since , is bounded. Setting , we get that where and . This completes the proof of .
If , then and consequently, , . Thus, from part , we get for any positive integers , , where , . This completes the proof of .

Remark 5. Lemma 4 generalizes Lemma  2.1 in [4].

Theorem 6. Let be a nonempty closed convex subset of a real Banach space . Let be generalized asymptotically quasi-nonexpansive self-mappings of with such that and for all . Let for all and let be a modified -mapping generated by and . Suppose that is closed and . Starting from arbitrary , define the sequence by the recursion (7); then the sequence converges strongly to if and only if .

Proof. We will only prove the sufficiency; the necessity is obvious. From Lemma 4, we have for all and all . Therefore,
As , so . By Lemma 1 and , we get that . Next, we prove that is a Cauchy sequence. From Lemma 4, we have
Hence, for all integers and all ,
Taking infimum over in (23) gives
Now, since and , given , there exists an integer such that for all , and . So for all integers , , we obtain from (24) that
Hence, is a Cauchy sequence in . Since is complete, there exists such that . We now show that . Since and as , for each , there exists an integer such that, and for all . In particular, we have ; that is, there exists a such that ; hence
Since is a closed subset of , we obtain . This completes the proof.

Remark 7. Theorem 6 generalizes and extends Theorem 2.2 of Khan et al. [4], Theorem 3.1 of Ghosh and Debnath [8], Theorem 3.2 of Shahzad and Udomene [6], and Theorem of Qihou [7] together with its Corollaries and .

Asymptotically nonexpansive mappings and asymptotically quasi-nonexpansive mappings are all generalized asymptotically quasi-nonexpansive, by Theorem 6 and Lemma 3, so we have

Corollary 8. Let be a nonempty closed convex subset of a real Banach space . Let be asymptotically quasi-nonexpansive self-mappings of with such that for all . Let for all and let be a modified -mapping generated by and . Suppose and . Starting from arbitrary , define the sequence by the recursion (7). Then the sequence converges strongly to if and only if .

Corollary 9. Let be a nonempty closed convex subset of a real Banach space . Let be asymptotically nonexpansive self-mappings of with such that for all . Let for all and let be a modified -mapping generated by and . Suppose and . Starting from arbitrary , define the sequence by the recursion (7). Then the sequence converges strongly to if and only if .

Corollary 10. Let be a nonempty closed convex subset of a real Banach space . Let be generalized asymptotically quasi-nonexpansive self-mappings of with such that and for all . Let for all and let be a modified -mapping generated by and . Suppose that is closed and . Starting from arbitrary , define the sequence by the recursion (7). Then the sequence converges strongly to if and only if there exists a subsequence of which converges to .

3. Results in Uniformly Convex Banach Spaces

Lemma 11. Let be a nonempty closed convex subset of a uniformly convex Banach space . Let be uniform Lipschitz and generalized asymptotically quasi-nonexpansive self-mappings of with such that and for all . Let for some and let be a modified -mapping generated by and . Suppose and . Starting from arbitrary , define the sequence by the recursion (7). Then for each .

Proof. Let and , for all . By Lemma 1 and Lemma 4, it follows that exists for all . Assume that
From (15) and (27) we obtain that
From (7), we have therefore,
From (28) and (30) we can obtain that
Suppose that for some . Since so we obtain that
From (28) and (33), we have that
Thus, by induction, we have for each . That is, for each . From (28), we obtain for each . By Lemma 2, we get
If , from (38), we have
If , then we have
Hence,
Note that therefore, we have
Now, we observe that
By (41) and (43), we have for . This completes the proof.

Theorem 12. Let be a nonempty closed convex subset of a uniformly convex Banach space . Let be uniform Lipschitz and generalized asymptotically quasi-nonexpansive self-mappings of with such that and for all . Let for some and let be a modified -mapping generated by and . Suppose , and there exists one member in which is semicompact for some positive integer . Starting from arbitrary , define the sequence by the recursion (7). Then converges strongly to some common fixed point of the family .

Proof. By Lemma 11, we have for each . Without loss of generality, we may assume that is semicompact for some ; then we have
Since is semicompact, then there exists a subsequence of such that . Hence, we have for each . This implies that . By Corollary 10, converges strongly to some common fixed point of the family .

Theorem 13. Let be a nonempty closed convex subset of a uniformly convex Banach space . Let be uniform Lipschitz and generalized asymptotically quasi-nonexpansive self-mappings of with such that and for all . Let for some and let be a modified -mapping generated by and . Suppose , and each , , is demiclosed at . If satisfies Opial’s condition, then the sequence defined by (7) converges weakly to a common fixed point of the family .

Proof. From the proof of Lemma 11, we know that is a bounded sequence in . Since is uniformly convex, it must be reflexive. Therefore, there exists a subsequence in converging weakly to . By Lemma 11, and is demiclosed at for , so we obtain . That is, . Suppose that there exists another subsequence of converging weakly to . As above, we can prove . By (27) we know that and exist. Assume . Then by the Opial’s condition, we have which is a contradiction. Hence . This implies that converges weakly to a common fixed point of the family .

Remark 14. Lemma 11, Theorem 12, and Theorem 13 extend Lemma 3.1, Theorem 3.3, and Theorem 3.2 of Khan et al. [4], respectively.

Conflict of Interests

The author declares that there is no conflict of interests.