Abstract and Applied Analysis

Volume 2013 (2013), Article ID 174302, 7 pages

http://dx.doi.org/10.1155/2013/174302

Research Article

Strong Convergence for a Strongly Quasi-Nonexpansive Sequence in Hilbert Spaces

Satit Saejung and Kanokwan Wongchan

Department of Mathematics, Faculty of Science, Khon Kaen University, Khon Kaen 40002, Thailand

Received 7 May 2013; Accepted 6 November 2013

Academic Editor: Simeon Reich

Copyright © 2013 Satit Saejung and Kanokwan Wongchan. 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 prove a strong convergence theorem for strongly quasi-nonexpansive sequence of mappings in Hilbert spaces. Moreover, we can improve the recent results of Tian and Jin (2011). We also give a simple proof of Marino-Xu’s result (2006).

1. Introduction

Let be a Hilbert space with inner product and induced norm . Recall that a mapping is said to be -Lipschitzian where if for all . In this paper, we are interested in nonexpansive mappings (that is, -Lipschitzian ones) and contractions (that is, -Lipschitzian ones with ). The problem of finding a fixed point of such mappings plays an important role in many nonlinear equations appearing in both pure and applied sciences. The celebrated Banach’s contraction principle is probably known as the major tool for the case of contraction mappings. However, for nonexpansive mappings, the situation is more difficult and different.

In 2000, Moudafi [1] introduced the viscosity approximation method, starting with an arbitrary initial , and defined a sequence by

where is a nonexpansive mapping, is a contraction, and is a sequence in satisfying

(M1);(M2);(M3).

It was proved that the sequence generated by (1) converges to a fixed point of and the following inequality holds:

In the literature, Moudafi’s scheme has been widely studied and extended (see [2, 3]). It should be noted that the convergence of Moudafi’s scheme is equivalent to that of its special setting with a constant contraction (see [4]). In fact, this follows from the role of the nonexpansiveness of .

In the earlier result, the following scheme was studied by Halpern [5]; starting with an arbitrary initial and a given , he defined a sequence by

where is a certain sequence in . In fact, Halpern proved in 1967 the convergence of the iterative sequence where and . Many researchers (see, e.g., [6, 7]) have improved Halpern’s result from Hilbert spaces to certain Banach spaces with the following conditions on :

(C1);(C2);(C3) or .

Halpern also showed that conditions (C1) and (C2) are necessary for the convergence of the sequence generated by (3) for any given .

On the other hand, Chidume-Chidume [8] and Suzuki [9] independently discovered that together just conditions (C1) and (C2) are sufficient for the convergence of the following iterative sequence:

where and . Recently, Saejung [10] proved that the conclusion remains true if is a strongly nonexpansive mapping. It is noted that in Hilbert spaces the mapping is strongly nonexpansive whenever . Recall that a mapping is strongly nonexpansive (see [11, 12]) if it is nonexpansive and whenever , are sequences in such that is bounded and .

In the aforementioned results, it was assumed that has a fixed point; that is, . Now we consider the following more general settings. A mapping is

(i)quasi-nonexpansive if and for all and ;(ii)strongly quasi-nonexpansive if it is quasi-nonexpansive and whenever is a bounded sequence in such that for some .

In 2010, Maingé [2] proved the convergence of the sequence defined by and

where , and is a quasi-nonexpansive mapping under the conditions (C1) and (C2). In 2011, Wongchan and Saejung [13] improved Maingé’s result by replacing with a strongly nonexpansive mapping . Hence, the restriction can be extended to .

There are also some other iterative schemes closely related to the schemes above studied by many authors. For example, inspired by the scheme studied by Yamada [14], Tian and Jin [15, 16] recently proposed the following iterative scheme, starting with an arbitrary initial and

where and are the same as Maingé’s result but is strongly monotone and Lipschitzian.

A careful reading shows that there are some connections between them. We will discuss and consolidate them into the following scheme: Started with an arbitrary initial and

where , are Lipschitzian and is a certain sequence of quasi-nonexpansive mappings.

2. Preliminaries

In this section, we collect together some known lemmas which are our main tool in proving our results. Let be a closed and convex subset of . Recall that the metric projection is defined as follows: for , is the only one point in satisfying

Lemma 1 (see [17]). Let be a nonempty closed convex subset of a Hilbert space . Then for and , if and only if for all .

Lemma 2. Let be a Hilbert space. Then

for all .

We also need the following lemma.

Lemma 3 (see [18, Lemma 2.5]). Let , , and , be such that

(i) is a bounded sequence;(ii) for all ;(iii)whenever is a subsequence of satisfying , it follows that ;(iv) and .

Then .

Lemma 4 (see [19, Lemma 2.3]). Let be a sequence of nonnegative real numbers, a sequence of real numbers in with , a sequence of nonnegative real numbers with , and a sequence of real numbers with . Suppose that

Then .

3. Main Results

Recall that is a strongly quasi-nonexpansive sequence if it satisfies the following conditions:

(1);(2) for all and and for all ;(3) whenever is a bounded sequence in such that for some .

We also say that satisfies the NST-condition if whenever is a bounded sequence in such that it follows that every weak cluster point of belongs to .

Remark 5.

(1)Being strongly nonexpansive the sequence and NST-condition are apparently inherited by subsequences.(2)Suppose that for all .

(i)If is a strongly nonexpansive mapping, then is a strongly nonexpansive sequence.(ii)If is demiclosed at zero, then satisfies NST-condition.

Recall that is demiclosed at zero if is a sequence in such that and ; then .

We now state our main theorem.

Theorem 6. Let be a strongly quasi-nonexpansive sequence satisfying the NST-condition. Let be - and -Lipschitzian, respectively. Suppose that is given by and

where is a sequence in satisfying the conditions (C1) and (C2). Suppose that . Then converges strongly to .

Before we give the proof, we note that is closed and convex. It follows from that is an -contraction. Then the mapping is a contraction. By Banach’s contraction principle, there exists a unique element such that . It follows then from Lemma 1 that for all .

Let us consider the following three lemmas first.

Lemma 7. The sequence is bounded. Hence, so are the sequences , , and .

Proof. We consider the following inequality:

Since each is quasi-nonexpansive and , we have

It follows from the Lipschitzian conditions of and , respectively that,

Then, we have

By induction, for all , we have

In particular, the sequence is bounded.

Lemma 8. The following inequality holds for all :

Proof. It follows from Lemma 2 that

Since each is quasi-nonexpansive and ,

Next, we consider

Hence, the result follows.

Lemma 9. If there is a subsequence of such that , then

Proof. We note that . We consider the following inequality:

Then . Since is strongly quasi-nonexpansive, so is . This implies that . Moreover,

Then . Since is bounded, there exists a subsequence of such that and

As , we have     . Since satisfies NST-condition, we have and hence . Therefore,

as desired.

Proof of Theorem 6. We are ready to apply Lemma 3. Set

It follows that

(i) is a bounded sequence (by Lemma 7);(ii) for all (by Lemma 8);(iii)whenever is a subsequence of satisfying , it follows that (by Lemma 9).

Hence, . This completes the proof.

4. Deduced Results

4.1. Wongchan and Saejung’s Result

Setting and for all in the proof of Theorem 6, we immediately have the following result of Wongchan and Saejung ([13, Theorem 6]).

Corollary 10. Let be a closed convex subset of a Hilbert space and a strongly quasi-nonexpansive mapping such that is demiclosed at zero. Suppose that is a contraction and a sequence is generated by and

where is a sequence in satisfying the conditions (C1) and (C2). Then converges strongly to .

4.2. Tian and Jin’s Result I

Recall that a mapping is -strongly monotone if for all .

Lemma 11. Let be an -strongly monotone and -Lipschitzian mapping. Then where for all . In particular, if , then is a contraction.

Proof. Let . Then

Theorem 12. Let be a strongly quasi-nonexpansive mapping such that is demiclosed at zero. Let be an -strongly monotone and -Lipschitzian mapping. Let be an -Lipschitzian mapping and let a sequence be generated by and

where the sequence satisfies the conditions (C1) and (C2). Suppose that and , where . Then converges to .

Proof. First we rewrite the iteration (29) as follows:

where and . Note that is a -Lipschitzian and is a -Lipschitzian. Using and putting for all in Theorem 6 imply that converges to , where

Lemma 13 (see [12]). If is a quasi-nonexpansive mapping, then the mapping is strongly quasi-nonexpansive wherever .

Using Theorem 12 and Lemma 13, we immediately have the following result which is an improvement of Tian and Jin’s result ([15, Theorem 3.1]).

Theorem 14. Let be a quasi-nonexpansive mapping such that is demiclosed at zero. Let be an -strongly monotone and -Lipschitzian mapping. Let be an -Lipschitzian mapping and let the sequence be generated by and

where , and the sequence satisfies the conditions (C1) and (C2). Suppose that and where . Then converges to .

Remark 15. Theorem 14 improves the result of Tian and Jin ([15, Theorem 3.1]) in the following ways.

(i)We assume that while [15, Theorem 3.1] is proved under the assumptions . We note that .(ii)Our result allows us to choose in the wider interval while [15, Theorem 3.1] is proved under the assumptions .

4.3. Tian and Jin’s Result II

Recall that a mapping is strongly positive with the coefficient if

for all .

Lemma 16 (see [20]). Let be a strongly positive self-adjoint linear bounded operator with coefficient on and . Then .

Theorem 17. Let be a strongly quasi-nonexpansive mapping such that is demiclosed at zero. Let be a bounded linear self-adjoint operator and strongly positive with the coefficient . Let be an -contraction mapping and let a sequence be generated by and

where the sequence satisfies the conditions (C1) and (C2). Suppose that . Then converges to .

Proof. By Lemma 16, we can choose such that . Rewrite the iteration (34) as follows:

where , and for all . Note that is -Lipschitzian and is ()-Lipschitzian. It follows from that

Setting for all in Theorem 6 implies that converges to such that ; that is, for all . This implies that for all ; that is, . This completes the proof.

Using Lemma 13 and Theorem 17, we immediately have the following result which is an improvement of Tian and Jin's result ([16, Theorem 3.1]).

Theorem 18. Let be a quasi-nonexpansive mapping such that is demiclosed at zero. Let be a bounded linear self-adjoint operator and strongly positive with the coefficient . Let be an -contraction mapping, and let the sequence be generated by and

where , and the sequence satisfies the conditions (C1) and (C2). Suppose that . Then converges to .

Remark 19. Theorem 18 improves the result of Tian and Jin ([16, Theorem 3.1]). In fact, their result was proved under the assumption while our result allows us to choose in the wider interval .

5. A Discussion on Marino-Xu’s Result

The following theorem is studied by many authors; for example, see [3].

Theorem 20. Let be a closed convex subset of a Hilbert space . Suppose that

(i) is a nonexpansive mapping and ;(ii) is a sequence satisfying the conditions (C1), (C2), and (C3).

Define the following iterative sequence:

Then converges to .

Using the technique in [4], we can give a simple proof of the following result proved by Marino and Xu [20].

Theorem 21. Suppose that

(i) is a bounded linear self-adjoint operator and it is strongly positive with the coefficient ;(ii) is a nonexpansive mapping and ;(iii) is an -contraction;(iv) is a positive number such that ;(v) is a sequence satisfying the conditions (C1), (C2), and (C3).

Define the following iterative sequence:

Then converges to and for all .

Proof. Choose such that . First we show that is a contraction. To see this, let . Then

It follows from that is a contraction. Note that is nonexpansive and hence is a contraction from into itself. It follows from the closedness of and the Banach’s contraction principle that there exists a unique element such that

Therefore,

Now we define the following iterative sequence:

It follows from Theorem 20 that the sequence converges to . Observe that

We next consider the following expression:

It follows from Lemma 4 that . Therefore, we conclude that converges to and for all . This completes the proof.

Acknowledgments

The authors thank Professor Simeon Reich and the referee for their comments and suggestions. The first author is supported by the TRF Research Career Development Grant RSA 5680002 and the second author is supported by grant fund under the program Strategic Scholarships for Frontier Research Network for the Ph.D. Program Thai Doctoral degree from the Office of the Higher Education Commission, Thailand.

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