Abstract and Applied AnalysisVolume 2011 (2011), Article ID 745451, 19 pagesdoi:10.1155/2011/745451

Research Article

Weak Convergence of the Projection Type Ishikawa Iteration Scheme for Two Asymptotically Nonexpansive Nonself-Mappings

Tanakit Thianwan

School of Science, University of Phayao, Phayao 56000, Thailand

Received 31 May 2011; Revised 26 August 2011; Accepted 30 August 2011

Academic Editor: Victor M. Perez Garcia

Copyright © 2011 Tanakit Thianwan. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

We study weak convergence of the projection type Ishikawa iteration scheme for two asymptotically nonexpansive nonself-mappings in a real uniformly convex Banach space which has a Fréchet differentiable norm or its dual has the Kadec-Klee property. Moreover, weak convergence of projection type Ishikawa iterates of two asymptotically nonexpansive nonself-mappings without any condition on the rate of convergence associated with the two maps in a uniformly convex Banach space is established. Weak convergence theorem without making use of any of the Opial's condition, Kadec-Klee property, or Fréchet differentiable norm is proved. Some results have been obtained which generalize and unify many important known results in recent literature.

1. Introduction and Preliminaries

Let be a nonempty closed convex subset of real normed linear space . Let be a mapping. A point is called a fixed point of if and only if . The set of all fixed points of a mapping is denoted by . A self-mapping is said to be nonexpansive if for all . A self-mapping is called asymptotically nonexpansive if there exists a sequence as such that for all and . A mapping is said to be uniformly -Lipschitzian if there exists a constant such that for all and .   is uniformly Hölder continuous if there exist positive constants and such that for all and .   is termed as uniformly equicontinuous if, for any , there exists such that whenever for all and or, equivalently, is uniformly equicontinuous if and only if whenever as .

It is easy to see that if is an asymptotically nonexpansive, then it is uniformly Lipschitzian with the uniform Lipschitz constant .

Remark 1.1. It is clear that asymptotically nonexpansiveness uniformly L-Lipschitz uniformly Hölder continuous uniformly equicontinuous.

However, their converse fail in the presence of the following example.

Example 1.2 (see [1]). Define by for all .

Fixed-point iteration process for nonexpansive self-mappings including Mann and Ishikawa iteration processes has been studied extensively by various authors [2–8]. For nonexpansive nonself-mappings, some authors (see [9–13]) have studied the strong and weak convergence theorems in Hilbert spaces or uniformly convex Banach spaces.

In [14], Tan and Xu introduced a modified Ishikawa iteration process: to approximate fixed points of nonexpansive self-mappings defined on nonempty closed convex bounded subsets of a uniformly convex Banach space . The mapping remains self-mapping of a nonempty closed convex subset of a uniformly convex Banach space. If, however, the domain of is a proper subset of (and this is the case in several applications) and maps into then, the sequence generated by (1.6) may not be well defined. More precisely, Tan and Xu [14] proved weak convergence of the sequences generated by (1.6) to some fixed point of in a uniformly convex Banach space which satisfies Opial's condition or has a Fréchet differentiable norm.

Note that each satisfies Opial's condition, while all do not have the property unless and the dual of reflexive Banach spaces with a Fréchet differentiable norm has the Kadec-Klee property. It is worth mentioning that there are uniformly convex Banach spaces, which have neither a Fréchet differentiable norm nor Opial property; however, their dual does have the Kadec-Klee property (see [15, 16]).

In 2005, Shahzad [11] extended Tan and Xu's result [14] to the case of nonexpansive nonself-mapping in a uniformly convex Banach space. He studied weak convergence of the modified Ishikawa type iteration process: in a uniformly convex Banach space whose dual has the Kadec-Klee property. The result applies not only to spaces with but also to other spaces which do not satisfy Opial's condition or have a Fréchet differentiable norm. Meanwhile, the results of [11] generalized the results of [14].

The class of asymptotically nonexpansive self-mappings is a natural generalization of the important class of nonexpansive mappings. Goebel and Kirk [17] proved that if is a nonempty closed convex and bounded subset of a real uniformly convex Banach space, then every asymptotically nonexpansive self-mapping has a fixed point.

In 1991, the modified Mann iteration which was introduced by Schu [18] generates a sequence in the following manner: where is a sequence in the interval and is an asymptotically nonexpansive mapping. To be more precise, Schu [18] obtained the following weak convergence result for an asymptotically nonexpansive mapping in a uniformly convex Banach space which satisfies Opial's condition.

Theorem 1.3 (see [18]). Let be a uniformly convex Banach space satisfying Opial’s condition, closed bounded and convex, and asymptotically nonexpansive with sequence for which and is bounded away. Let be a sequence generated in (1.8). Then, the sequence converges weakly to some fixed point of  .

Since then, Schu's iteration process has been widely used to approximate fixed points of asymptotically nonexpansive self-mappings in Hilbert space or Banach spaces (see [6, 14, 19, 20]).

In 1994, Tan and Xu [21] obtained the following results.

Theorem 1.4 (see [21]). Let be a uniformly convex Banach space whose norm is Fréchet differentiable, a nonempty closed and convex subset of , and an asymptotically nonexpansive mapping with a sequence such that such that is nonempty. Let be sequence generated in (1.8), where is a real sequence bounded away from 0 and 1. Then, the sequence converges weakly to some point in .

In 2001, Khan and Takahashi [22] constructed and studied the following Ishikawa iteration process: where  ,  are asymptotically nonexpansive self-mappings on with  (rate of convergence) and , .

Note that the rate of convergence condition, namely, has remained in extensive use to prove both weak and strong convergence theorems to approximate fixed points of asymptotically nonexpansive maps. The conditions like Opial's condition, Kadec-Klee property, or Fréchet differentiable norm have remained key to prove weak convergence theorems.

In 2010, Khan and Fukhar-Ud-Din [23] established weak convergence of Ishikawa iterates of two asymptotically nonexpansive self-mappings without any condition on the rate of convergence associated with the two mappings. They got that the following new weak convergence theorem does not require any of Opial’s condition, Kadec-Klee property or Fréchet differentiable norm.

Theorem 1.5 (see [23]). Let be a nonempty bounded closed convex subset of a uniformly convex Banach space . Let be asymptotically nonexpansive maps with sequences such that , , respectively. Let the sequence be as in (1.9) with . for some . If , then converges weakly to a common fixed point of and .

The concept of asymptotically nonexpansive nonself-mappings was introduced by Chidume et al. [24] in 2003 as the generalization of asymptotically nonexpansive self-mappings. The asymptotically nonexpansive nonself-mapping is defined as follows.

Definition 1.6 (see [24]). Let be a nonempty subset of a real normed linear space . Let be a nonexpansive retraction of onto . A nonself-mapping is called asymptotically nonexpansive if there exists a sequence , as such that for all and .   is said to be uniformly L-Lipschitzian if there exists a constant such that for all and .

By studying the following iteration process: Chidume et al. [24] got the following weak convergence theorem for asymptotically nonexpansive nonself-mapping.

Theorem 1.7 (see [24]). Let be a real uniformly convex Banach space which has a Fréchet differentiable norm and a nonempty closed convex subset of . Let be an asymptotically nonexpansive map with sequence such that and . Let be such that , for all and some . From an arbitrary , define the sequence by (1.12). Then, converges weakly to some fixed point of .

If is a self-mapping, then becomes the identity mapping so that (1.10) and (1.11) reduce to (1.1) and (1.2), respectively. Equation (1.12) reduces to (1.8).

In 2006, Wang [25] generalizes the iteration process (1.12) as follows: , where are asymptotically nonexpansive nonself-mappings and , are real sequences in . He studied the strong and weak convergence of the iterative scheme (1.13) under proper conditions. Meanwhile, the results of [25] generalized the results of [24].

Recently, an iterative scheme which is called the projection type Ishikawa iteration for two asymptotically nonexpansive nonself-mappings was defined and constructed by Thianwan [26]. It is given as follows: where and are appropriate real sequences in .

In [26], Thianwan gave the following weak convergence theorem.

Theorem 1.8. Let be a uniformly convex Banach space which satisfies Opial's condition and a nonempty closed convex nonexpansive retract of with as a nonexpansive retraction. Let be two asymptotically nonexpansive nonself-mappings of with sequences such that , , respectively, and . Suppose that and are real sequences in for some . Let and be the sequences defined by (1.14). Then, and converge weakly to a common fixed point of and .

The iterative schemes (1.14) and (1.13) are independent: neither reduces to the other. If and for all , then (1.14) reduces to (1.12). It also can be reduces to Schu process (1.8).

Inspired and motivated by the recent works, we prove some new weak convergence theorems of the sequences generated by the projection type Ishikawa iteration scheme (1.14) for two asymptotically nonexpansive nonself-mappings in uniformly convex Banach spaces.

Now, we recall some well-known concepts and results.

Let be a Banach space with dimension . The modulus of is the function defined by Banach space is uniformly convex if and only if for all . It is known that a uniformly convex Banach space is reflexive and strictly convex.

Recall that a Banach space is said to satisfy Opial's condition [27] if weakly as and implying that The norm of is said to be Fréchet differentiable if for each with the limit exists and is attained uniformly for , with . In the case of Fréchet differentiable norm, it has been obtained in [21] that for all , in , where is the normalized duality map from to defined by

is the duality pairing between and and is an increasing function defined on such that .

A subset of is said to be retract if there exists continuous mapping such that for all . Every closed convex subset of a uniformly convex Banach space is a retract. A mapping is said to be a retraction if . If a mapping is a retraction, then for every , range of . A set is optimal if each point outside can be moved to be closer to all points of . It is well known (see [28]) that(1)if is a separable, strictly convex, smooth, reflexive Banach space, and if is an optimal set with interior, then is a nonexpansive retract of ;(2)a subset of , with , is a nonexpansive retract if and only if it is optimal.

Note that every nonexpansive retract is optimal. In strictly convex Banach spaces, optimal sets are closed and convex. Moreover, every closed convex subset of a Hilbert space is optimal and also a nonexpansive retract.

Recall that weak convergence is defined in terms of bounded linear functionals on as follows.

A sequence in a normed space is said to be weakly convergent if there is an such that for every bounded linear functional on . The element is called the weak limit of , and we say that converges weakly to . In this paper, we use and to denote the strong convergence and weak convergence, respectively.

A Banach space is said to have the Kadec-Klee property if, for every sequence in , and together imply ; for more details on Kadec-Klee property, the reader is referred to [29, 30] and the references therein.

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

Lemma 1.9 (see [31]). Let , be two fixed numbers. Then, a Banach space is uniformly convex if and only if there exists a continuous, strictly increasing, and convex function , such that for all , in , , where

Lemma 1.10 (see [24]). Let be a uniformly convex Banach space and a nonempty closed convex subset of , and let be an asymptotically nonexpansive mapping with a sequence and as . Then, is demiclosed at zero; that is, if weakly and strongly, then .

Lemma 1.11 (see [26]). Let be a uniformly convex Banach space and a nonempty closed convex nonexpansive retract of with as a nonexpansive retraction. Let be two asymptotically nonexpansive nonself-mappings of with sequences such that , , respectively, and . Suppose that and are real sequences in . From an arbitrary , define the sequence by (1.14). If , then exists.

Lemma 1.12 (see [26]). Let be a uniformly convex Banach space and a nonempty closed convex nonexpansive retract of with as a nonexpansive retraction. Let be two asymptotically nonexpansive nonself-mappings of with sequences such that , , respectively, and . Suppose that and are real sequences in for some . From an arbitrary , define the sequence by (1.14). Then, .

Lemma 1.13 (see [16]). Let be a real reflexive Banach space such that its dual has the Kadec-Klee property. Let be a bounded sequence in and , where denotes the set of all weak subsequential limits of . Suppose that exists for all . Then, .

We denote by the set of strictly increasing, continuous convex functions with . Let be a convex subset of the Banach space . A mapping is said to be type [32] if and , for all , in . Obviously, every type   mapping is nonexpansive. For more information about mappings of type , see [33–35].

Lemma 1.14 (see [36, 37]). Let be a uniformly convex Banach space and a convex subset of . Then, there exists such that for each mapping with Lipschitz constant , for all and .

2. Main Results

In this section, we prove weak convergence theorems of the projection type Ishikawa iteration scheme (1.14) for two asymptotically nonexpansive nonself-mappings in uniformly convex Banach spaces.

Firstly, we deal with the weak convergence of the sequence defined by (1.14) in a real uniformly convex Banach space whose dual has the Kadec-Klee property. In order to prove our main results, the following lemma is needed.

Lemma 2.1. Let be a real uniformly convex Banach space and a nonempty closed convex nonexpansive retract of with as a nonexpansive retraction. Let be two asymptotically nonexpansive nonself-mappings of with sequences such that , , respectively, and . Suppose that and are real sequences in for some . Let and be the sequences defined by (1.14). Then, for all , the limit exists for all .

Proof. It follows from Lemma 1.11 that the sequence is bounded. Then, there exists such that . Let where . Then, and, by Lemma 1.11, exists. Without loss of the generality, we may assume that for some positive number . Let and .  For each , define by where Setting and . For , we have Set , and , where . Also, for all and , for all .Applying Lemma 1.14 with , , and using the facts that , , and exist for all , we obtain . Observe that Consequently, This implies that exists for all . This completes the proof.

Theorem 2.2. Let be a real uniformly convex Banach space which has a Fréchet differentiable norm and a nonempty closed convex nonexpansive retract of with as a nonexpansive retraction. Let be two asymptotically nonexpansive nonself-mappings of with sequences such that , respectively, and . Suppose that and are real sequences in for some . Let and be the sequences defined by (1.14). Then, converges weakly to a fixed point of and .

Proof. Set and in (1.18). By using Lemmas 1.11, and 2.1 and the same proof of Lemma  4 of Osilike and Udomene [7], we can show that, for every , for all . Since is reflexive and is bounded, we from Lemma 1.13 conclude that for each . Let . It follows that ; that is, Therefore, . This completes the proof.

Theorem 2.3. Let be a real uniformly convex Banach space such that its dual has the Kadec-Klee property and a nonempty closed convex nonexpansive retract of with as a nonexpansive retraction. Let be two asymptotically nonexpansive nonself-mappings of with sequences such that , , respectively, and . Suppose that and are real sequences in for some . Let and be the sequences defined by (1.14). Then, converges weakly to a fixed point of and .

Proof. It follows from Lemma 1.11 that the sequence is bounded. Then, there exists a subsequence of converging weakly to a point . By Lemma 1.12, we have Now, using Lemma 1.10, we have ; that is, . Thus, . It remains to show that converges weakly to . Suppose that is another subsequence of converging weakly to some . Then, and so . By Lemma 2.1, exists for all . It follows from Lemma 1.13 that . As a result, is a singleton, and so converges weakly to a fixed point of .

In the remainder of this section, we deal with the weak convergence of the sequences generated by the projection type Ishikawa iteration scheme (1.14) for two asymptotically nonexpansive nonself-mappings in a uniformly convex Banach space without any of the Opial's condition, Kadec-Klee property, or Fréchet differentiable norm.

Let and be two asymptotically nonexpansive nonself-mappings of with , , and , , respectively. In the sequel, we take , where .

We start with proving the following lemma for later use.

Lemma 2.4. Let be a uniformly convex Banach space and a nonempty bounded closed convex nonexpansive retract of with as a nonexpansive retraction. Let be two asymptotically nonexpansive nonself-mappings of with sequences such that , as , respectively, and . Suppose that and are real sequences in for some . Then, for the sequence given in (1.14), we have that

Proof. By setting , then if . Let . Since is bounded, there exists such that for some . Applying Lemma 1.9 for scheme (1.14), we have and so, From (2.14), we obtain the following two important inequalities: Now, we prove that Assume that . Then, there exists a subsequence (use the same notation for subsequence as for the sequence) of and such that By definition of , we have From (2.15), we have In addition, and ; there exists such that for all . From (2.20), we obtain for all .Let . It follows from (2.21) that By letting in (2.22), we obtain which contradicts the reality. This proves that . Thus, . Consequently, we have Similarly, using (2.16), we may show that Using (2.24), we have From (2.25), (2.26), and the uniform equicontinuous of (see Remark 1.1), we have Since it follows from (2.26), (2.27), and the uniform equi-continuity of (see Remark 1.1) that Since and again from the fact that is uniformly equicontinuous mapping, by Using (2.29), we have In addition, where . It follows from (2.29) and (2.30) that We denote to be the identity maps from onto itself. Thus, by the inequality (2.30) and (2.32), we have which implies that . Similarly, we may show that . The proof is completed.

Our weak convergence theorem is as follows. We do not use the rate of convergence conditions, namely, and in its proof.

Theorem 2.5. Let be a uniformly convex Banach space and a nonempty bounded closed convex nonexpansive retract of with as a nonexpansive retraction. Let be two asymptotically nonexpansive nonself-mappings of with sequences such that , as , respectively, and . Suppose that and are real sequences in for some . Then, the sequence given in (1.14) converges weakly to a common fixed point of and .

Proof. Since is a nonempty bounded closed convex subset of a uniformly convex Banach space , there exists a subsequence of such that converges weakly to , where denotes the set of all weak subsequential limits of . This show that and, by Lemma 2.4, . Since and are demiclosed at zero, using Lemma 1.10, we have . Therefore, .For any , there exists a subsequence of such that It follows from (2.24) and (2.34) that Now, from (1.14), (2.34), and (2.35), Also, from (2.25) and (2.36), we have It follows from (2.36) and (2.37) that Continuing in this way, by induction, we can prove that, for any , By induction, one can prove that converges weakly to as ; in fact, gives that as . This completes the prove.

Acknowledgments

The author would like to thank the Thailand Research Fund, The Commission on Higher Education(MRG5380226), and University of Phayao, Phayao, Thailand, for financial support during the preparation of this paper. Thanks are also extended to the anonymous referees for their helpful comments which improved the presentation of the original version of this paper.

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