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Abstract and Applied Analysis
Volume 2013 (2013), Article ID 904164, 10 pages
http://dx.doi.org/10.1155/2013/904164
Research Article

Existence and Approximation of Attractive Points of the Widely More Generalized Hybrid Mappings in Hilbert Spaces

1Graduate Institute of Business and Management, College of Management, Chang-Gung University, Kwei-Shan, Taoyuan Hsien 330, Taiwan
2Department of Mathematical and Computing Sciences, Tokyo Institute of Technology, Tokyo 152-8552, Japan

Received 30 January 2013; Accepted 5 June 2013

Academic Editor: Mohamed Amine Khamsi

Copyright © 2013 Sy-Ming Guu and Wataru Takahashi. 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 the widely more generalized hybrid mappings which have been proposed to unify several well-known nonlinear mappings including the nonexpansive mappings, nonspreading mappings, hybrid mappings, and generalized hybrid mappings. Without the convexity assumption, we will establish the existence theorem and mean convergence theorem for attractive point of the widely more generalized hybrid mappings in a Hilbert space. Moreover, we prove a weak convergence theorem of Mann’s type and a strong convergence theorem of Shimizu and Takahashi’s type for such a wide class of nonlinear mappings in a Hilbert space. Our results can be viewed as a generalization of Kocourek, Takahashi and Yao, and Hojo and Takahashi where they studied the generalized hybrid mappings.

1. Introduction

Let be a real Hilbert space, and let be a nonempty subset of . For a mapping , we denote by and the sets of fixed points and attractive points of  , respectively, that is, (i); (ii).

A mapping is called nonexpansive [1] if for all . A mapping is called nonspreading [2], hybrid [3] if for all , respectively; see also [4, 5]. These three terms are independent, and they are deduced from the notion of firmly nonexpansive mapping in a Hilbert space; see [3]. A mapping is said to be firmly nonexpansive if for all ; see, for instance, Goebel and Kirk [6]. The class of nonspreading mappings was first defined in a strictly convex, smooth, and reflexive Banach space. The resolvents of a maximal monotone operator are nonspreading mappings; see [2] for more details. These three classes of nonlinear mappings are important in the study of the geometry of infinite dimensional spaces. Indeed, by using the fact that the resolvents of a maximal monotone operator are nonspreading mappings, Takahashi et al. [7] solved an open problem which is related to Ray’s theorem [8] in the geometry of Banach spaces. Motivated by these mappings, Kocourek et al. [9] introduced a broad class of nonlinear mappings in a Hilbert space which covers nonexpansive mappings, nonspreading mappings, and hybrid mappings. A mapping is said to be generalized hybrid if there exist such that for all , where is the set of real numbers. We call such a mapping an ()-generalized hybrid mapping. An ()-generalized hybrid mapping is nonexpansive for and , nonspreading for and , and hybrid for and . They proved fixed point theorems for such mappings; see also Kohsaka and Takahashi [10] and Iemoto and Takahashi [4]. Moreover, they proved the following nonlinear ergodic theorem which generalizes Baillon’s theorem [11].

Theorem 1 (see [9]). Let be a real Hilbert space, let be a nonempty closed convex subset of , let be a generalized hybrid mapping from into itself with , and let be the metric projection of onto . Then for any , converges weakly to , where .

We see that the set needs to be closed and convex in Theorem 1. As a contrast, Takahashi and Takeuchi [12] proved the following theorem which establishes the existence of attractive point and mean convergence property without the convexity assumption in a Hilbert space; see also Lin and Takahashi [13] and Takahashi et al. [14].

Theorem 2. Let be a real Hilbert space, and let be a nonempty subset of . Let be a generalized hybrid mapping from into itself. Let and be sequences defined by for all . If is bounded, then the followings hold: (1) is nonempty, closed, and convex; (2) converges weakly to , where and is the metric projection of onto .

Very recently Kawasaki and Takahashi [15] introduced a class of nonlinear mappings in a Hilbert space which covers contractive mappings [16] and generalized hybrid mappings. A mapping is called widely more generalized hybrid if there exist such that for any ; see also Kawasaki and Takahashi [17].

A mapping is called quasi-nonexpansive if and for all and . It is well known that if is closed and convex and is quasi-nonexpansive, then is closed and convex; see Itoh and Takahashi [18]. For a simpler proof of such a result in a Hilbert space, see, for example, [19]. A generalized hybrid mapping with a fixed point is quasi-nonexpansive. However, a widely more generalized hybrid mapping is not quasi-nonexpansive generally even if it has a fixed point. In [15], they proved fixed point theorems and nonlinear ergodic theorems of Baillon’s type for such new mappings in a Hilbert space.

In this paper, motivated by these results, we establish the attractive point theorem and mean convergence theorem without the commonly required convexity for the widely more generalized hybrid mappings in a Hilbert space. Moreover, we prove a weak convergence theorem of Mann’s type [20] and a strong convergence theorem of Shimizu and Takahashi’s type [21] for such a class of nonlinear mappings in a Hilbert space which generalize Kocourek et al. [9] and Hojo and Takahashi [22] for generalized hybrid mappings, respectively.

2. Preliminaries

Throughout this paper, we denote by the set of positive integers. Let be a real Hilbert space with inner product and norm . We denote the strong convergence and the weak convergence of to by and , respectively. Let be a nonempty subset of . We denote by the closure of the convex hull of . In a Hilbert space, it is known [1] that for any and ,

Furthermore, we have that for any .

Let be a nonempty closed convex subset of and . Then we know that there exists a unique nearest point such that . We denote such a correspondence by . The mapping is called the metric projection of onto . It is known that is nonexpansive and for any and ; see [1] for more details. For proving a nonlinear ergodic theorem in this paper, we also need the following lemma proved by Takahashi and Toyoda [23].

Lemma 3. Let be a nonempty closed convex subset of . Let be the metric projection from onto . Let be a sequence in . If for any and , then converges strongly to some .

To prove a strong convergence theorem in this paper, we need the following lemma.

Lemma 4 (Aoyama et al. [24]). Let be a sequence of nonnegative real numbers, let be a sequence of with , let be a sequence of nonnegative real numbers with , and let be a sequence of real numbers with . Suppose that for all . Then .

Let be the Banach space of bounded sequences with supremum norm. Let be an element of (the dual space of ). Then we denote by the value of at . Sometimes, we denote by the value . A linear functional on is called a mean if , where . A mean is called a Banach limit on if . We know that there exists a Banach limit on . If is a Banach limit on , then for , In particular, if and , then we have . See [25] for the proof of existence of a Banach limit and its other elementary properties. Using means and the Riesz theorem, we can obtain the following result; see [25, 26].

Lemma 5. Let be a real Hilbert space, let be a bounded sequence in , and let be a mean on . Then there exists a unique point such that for any .

The following result obtained by Takahashi and Takeuchi [12] is important in this paper.

Lemma 6. Let be a Hilbert space, let be a nonempty subset of , and let be a mapping from into . Then is a closed and convex subset of .

We also know the following result from [14].

Lemma 7. Let be a Hilbert space, let be a nonempty subset of , and let be a quasi-nonexpansive mapping from into . Then .

3. Attractive Point Theorems

Let be a real Hilbert space, and let be a nonempty subset of . Recall that a mapping from into is said to be widely more generalized hybrid [15] if there exist such that for any . Such a mapping is called ()-widely more generalized hybrid. An ()-widely more generalized hybrid mapping is generalized hybrid in the sense of Kocourek et al. [9] if and . We first prove an attractive point theorem for widely more generalized hybrid mappings in a Hilbert space.

Theorem 8. Let be a real Hilbert space, let be a nonempty subset of , and let be an ()-widely more generalized hybrid mapping from into itself which satisfies either of the following conditions: (1), , and ; (2), , and .
Then has an attractive point if and only if there exists such that is bounded.

Proof. Suppose that has an attractive point . Then for all and . Therefore is bounded.
Conversely suppose that there exists such that is bounded. Since is an ()-widely more generalized hybrid mapping from into itself, we obtain that for any and . By (9) we obtain that Thus we have that From we have that and hence By , we have that From this inequality and we obtain that Applying a Banach limit to both sides of this inequality, we obtain that and hence Since there exists by Lemma 5 such that for any , we obtain from (24) that We obtain from (9) that and hence Since , we obtain that Since , we obtain that This implies that is an attractive point.
In the case of , , , and , we can obtain the result by replacing the variables and . This completes the proof.

Using Theorem 8, we can show the following attractive point theorem for generalized hybrid mappings in a Hilbert space.

Theorem 9 (Takahashi and Takeuchi [12]). Let be a Hilbert space, let be a nonempty subset of , and let be a generalized hybrid mapping from into ; that is, there exist real numbers and such that for all . Then has an attractive point if and only if there exists such that is bounded.

Proof. An -generalized hybrid mapping is an ()-widely more generalized hybrid mapping. Furthermore, the mapping satisfies the condition (2) in Theorem 8, that is, Then we have the desired result from Theorem 8.

4. Nonlinear Ergodic Theorems

In this section, using the technique developed by Takahashi [26], we prove a mean convergence theorem without convexity for widely more generalized hybrid mappings in a Hilbert space. Before proving the result, we need the following two lemmas.

Lemma 10. Let be a nonempty subset of a real Hilbert space . Let be an ()-widely more generalized hybrid mapping from into itself such that . Suppose that it satisfies either of the following conditions: (1), and ; (2), and . For any , define . Then, . In particular, if is bounded, then

Proof. Let . Since is nonempty, we obtain that for any and . Then we have that is bounded. Since for any and , is also bounded. Using and , as in the proof of Theorem 8 we have that for any and . Summing up these inequalities with respect to and dividing by , we obtain that Replacing by , we obtain that Since , , and are bounded, we have that Since , we have that . In particular, if is bounded, then and hence .
Similarly, we can obtain the desired result for the case of , , and . This completes the proof.

Lemma 11. Let be a Hilbert space, and let be a nonempty subset of . Let be an ()-widely more generalized hybrid mapping. Suppose that it satisfies either of the following conditions: (1), and ; (2), and . If and , then .

Proof. Let be an ()-widely more generalized hybrid mapping, and suppose that and . Replacing by in (14), we have that From this inequality, we have that We apply a Banach limit to both sides of this inequality. We have that and hence Thus we have From , we also have From we obtain that and hence Since , we have that Using (9), we have that and hence Since and , we have that for all . This implies that .
Similarly, we can obtain the desired result for the case of , , and . This completes the proof.

Now we have the following nonlinear ergodic theorem for widely more generalized hybrid mappings in a Hilbert space.

Theorem 12. Let be a real Hilbert space, let be a nonempty subset of , let be an ()-widely more generalized hybrid mapping from into itself such that , and let be the metric projection of onto . Suppose that satisfies either of the conditions: (1), , and ; (2), , and . Then for any , is weakly convergent to an attractive point of  , where .

Proof. Let . Since is nonempty, we obtain that for any and . Thus is bounded. Since for any and , is also bounded. Therefore there exists a strictly increasing sequence and such that converges weakly to . Using , , , and , we have from Lemma 10 that We have from Lemma 11 that . Since is closed and convex from Lemma 6, the metric projection from onto is well defined. By Lemma 3, there exists such that converges strongly to . To complete the proof, we show that . Note that the metric projection satisfies for any and for any ; see [25]. Therefore for any and . Since is the metric projection from onto and , we obtain that That is, is nonincreasing. Therefore we obtain Summing up these inequalities with respect to and dividing by , we obtain Since converges weakly to and converges strongly to , we obtain that Putting , we obtain and hence . This completes the proof.
Similarly, we can obtain the desired result for the case of , , , and .

As the proof of Theorem 9, we can prove Takahashi and Takeuchi’s mean convergence theorem for generalized hybrid mappings in a Hilbert space.

Theorem 13. Let be a Hilbert space, let be a nonempty subset of , and let be a generalized hybrid mapping from into itself; that is, there exist such that for all . Suppose that , and let be the metric projection from onto . Then for any , converges weakly to , where .

5. Weak Convergence Theorems of Mann’s Type

In this section, we prove a weak convergence theorem of Mann’s type [20] for widely more generalized hybrid mappings in a Hilbert space by using Lemma 11 and the technique developed by Ibaraki and Takahashi [27, 28].

Theorem 14. Let be a Hilbert space, and let be a convex subset of . Let be a widely more generalized hybrid mapping with such that it satisfies either of the conditions: (1), and ; (2), and .
Let be the metric projection of onto . Let be a sequence of real numbers such that and . Suppose that is the sequence generated by and Then converges weakly to , where .

Proof. Let . We have that for all . Hence exists. Then is bounded. We also have from (8) that
Thus we have Since exists and , we have that Since is bounded, there exists a subsequence of such that . By Lemma 11 and (70), we obtain that . Let and be two subsequences of such that and . To complete the proof, we show . We know that , and hence and exist. Put Note that for , From and , we have Combining (73) and (74), we obtain . Thus we obtain . This implies that converges weakly to an element . Since for all and , we obtain from Lemma 3 that converges strongly to an element . On the other hand, we have from the property of that for all and . Since and , we obtain for all . Putting , we obtain . This means . This completes the proof.

Using Theorem 14, we can show the following weak convergence theorem of Mann’s type for generalized hybrid mappings in a Hilbert space.

Theorem 15 (Kocourek et al. [9]). Let be a Hilbert space, and let be a closed convex subset of . Let be a generalized hybrid mapping with . Let be a sequence of real numbers such that and . Suppose is the sequence generated by and Then the sequence converges weakly to an element .

Proof. As in the proof of Theorem 9, a generalized hybrid mapping is a widely more generalized hybrid mapping. Since and is closed and convex, we have from Theorem 14 that . A generalized hybrid mapping with is quasi-nonexpansive, we have from Lemma 7 that . Thus converges weakly to an element .

6. Strong Convergence Theorem

In this section, using an idea of mean convergence by Shimizu and Takahashi [21, 29], we prove the following strong convergence theorem for widely more generalized hybrid mappings in a Hilbert space.

Theorem 16. Let be a nonempty convex subset of a real Hilbert space . Let be a widely more generalized hybrid mapping of into itself with such that it satisfies either of the following conditions: (1), , and ; (2), , and .
Let , and define sequences and in as follows: and for all , where , and . If is nonempty, then and converge strongly to , where is the metric projection of onto .

Proof. Since is a widely more generalized hybrid mapping, there exist such that for any . Since , we have that for all and , Then we have Hence, by induction, we obtain for all . This implies that and are bounded. Since , we have also that is bounded.
Let . Using and , as in the proof of Theorem 8 we have that for any and . Summing up these inequalities with respect to and dividing by , we obtain that Since is bounded, there exists a subsequence of such that . Replacing by , we have that Since and are bounded, we have that as . Using (9), we have that and hence Since and , we have that for all . This implies that .
On the other hand, since , is bounded, and , we have . Let us show We may assume without loss of generality that there exists a subsequence of such that and . From , we have . From the above argument, we have . Since is the metric projection of onto , we have This implies Since , from (7) and (80) we have Putting , , and in Lemma 4, we have from and (93) that By , we also obtain as .
Similarly, we can obtain the desired result for the case of , , , and .

Using Theorem 16, we can show the following result obtained by Kurokawa and Takahashi [30].

Theorem 17 (Hojo and Takahashi [22]). Let be a nonempty closed convex subset of a real Hilbert space . Let be a generalized hybrid mapping of into itself. Let and define two sequences and in as follows: and for all , where , and . If is nonempty, then and converge strongly to , where is the metric projection of onto .

Proof. As in the proof of Theorem 9, a generalized hybrid mapping is a widely more generalized hybrid mapping. Since and is closed and convex, we have from Theorem 16 that . A generalized hybrid mapping with is quasi-nonexpansive, we have from Lemma 7 that . Thus and converge strongly to an element .

Acknowledgments

Research for Sy-Ming Guu is partially supported by NSC 100-2221-E-182-072-MY2. Wataru Takahashi is partially supported by Grant-in-Aid for Scientific Research no. 23540188 from Japan Society for the Promotion of Science.

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