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Abstract and Applied Analysis
Volume 2012 (2012), Article ID 315835, 13 pages
http://dx.doi.org/10.1155/2012/315835
Research Article

Computing the Fixed Points of Strictly Pseudocontractive Mappings by the Implicit and Explicit Iterations

Department of Information Management, Cheng Shiu University, Kaohsiung 833, Taiwan

Received 16 February 2012; Accepted 1 March 2012

Academic Editor: Khalida Inayat Noor

Copyright © 2012 Yeong-Cheng Liou. 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

It is known that strictly pseudocontractive mappings have more powerful applications than nonexpansive mappings in solving inverse problems. In this paper, we devote to study computing the fixed points of strictly pseudocontractive mappings by the iterations. Two iterative methods (one implicit and another explicit) for finding the fixed point of strictly pseudocontractive mappings have been constructed in Hilbert spaces. As special cases, we can use these two methods to find the minimum norm fixed point of strictly pseudocontractive mappings.

1. Introduction

In this paper, we devote to study computing the fixed points of strictly pseudocontractive mappings by the iterations. Our motivations are mainly in two respects.

Motivation 1
Iterative methods for finding fixed points of nonexpansive mappings have received vast investigations due to its extensive applications in a variety of applied areas of inverse problem, partial differential equations, image recovery, and signal processing; see [135] and the references therein. It is known [36] that strictly pseudocontractive mappings have more powerful applications than nonexpansive mappings in solving inverse problems. Therefore it is interesting to develop the algorithms for strictly pseudocontractive mappings.

Motivation 2
In many problems, it is needed to find a solution with minimum norm. In an abstract way, we may formulate such problems as finding a point with the property where is a nonempty closed convex subset of a real Hilbert space . A typical example is the least-squares solution to the constrained linear inverse problem [37]. Some related works for finding the minimum-norm solution (or fixed point of nonexpansive mappings) have been considered by some authors. The reader can refer to [3841].
In the present paper, we present two iterative methods (one implicit and another explicit) for finding the fixed point of strictly pseudocontractive mappings in Hilbert spaces. As special cases, we can use these two methods to find the minimum norm fixed point of strictly pseudocontractive mappings.

2. Preliminaries

Let be a real Hilbert space with inner product and norm , respectively. Let be a nonempty closed convex subset of .

2.1. Some Concepts

Recall that a mapping is called nonexpansive, if for all . And a mapping is said to be strictly pseudocontractive if there exists a constant such that for all . For such a case, we also say that is a -strictly pseudocontractive mapping. It is clear that, in a real Hilbert space , (2.2) is equivalent to for all . It is clear that the class of strictly pseudocontractive mappings strictly includes the class of nonexpansive mappings.

Recall that the nearest point (or metric) projection from onto is defined as follows: for each point , is the unique point in with the property: Note that is characterized by the inequality: Consequently, is nonexpansive.

2.2. Several Useful Lemmas

Lemma 2.1 (see [42]). Let be a real Hilbert space. There holds the following identity: for all .

Lemma 2.2 (see [43]). Let be a nonempty closed convex subset of a real Hilbert space . Let be a -strict pseudocontraction. Then, (i) is closed convex so that the projection is well defined; (ii) for , is nonexpansive.

Lemma 2.3 2.3 (see [42]). Let be a nonempty closed convex of a real Hilbert space . Let be a -strictly pseudocontractive mapping. Then is demiclosed at that is if and , then .

Lemma 2.4 (see [44]). Let and be bounded sequences in a Banach space and let be a sequence in with . Suppose that for all and Then .

Lemma 2.5 (see [45]). Let be a sequence of nonnegative real numbers satisfying where and satisfy (i),(ii)either or . Then converges to .

We use the following notation: (i) stands for the set of fixed points of ; (ii) stands for the weak convergence of to ; (iii) stands for the strong convergence of to .

3. Iterations and Convergence Analysis

Theorem 3.1. Let be a nonempty closed convex subset of a real Hilbert space . Let a -strictly pseudocontractive mapping with . Let be a constant. For and any , let be the sequence defined by the following implicit manner: Then the sequence converges strongly to .

Proof. Step 1. The sequence is well defined.
Set . It is easily to check that . Then, we can rewrite (3.1) as which is equivalent to the following: Note that is nonexpansive (see Lemma 2.2). For fix , we define a mapping by For , we have which implies that is a self-contraction of for every . Hence has a unique fixed point which is the unique solution of the fixed point equation (3.3).
Step 2. The sequence is bounded.
Pick up any . From (3.3), we have It follows that
Hence, is bounded and so is .
Step 3. .
From (3.3), we have
It follows that
Step  4.  .
Since is bounded, there exists a subsequence of , which converges weakly to a point . Noticing (3.9) we can use Lemma 2.3 to get .
By using the convexity of the norm and Lemma 2.1, for any , we have It turns out that where is some constant such that Therefore we can substitute for in (3.11) to get However, . This together with (3.13) guarantees that . It is clear that . As a matter of fact, in (3.11), if we let , then we get This is equivalent to Hence, . Therefore, . This completes the proof.

Corollary 3.2. Let be a nonempty closed convex subset of a real Hilbert space . Let a nonexpansive mapping with . Let be a constant. For and any , let be the sequence defined by the following implicit manner: Then the sequence converges strongly to .

Corollary 3.3. Let be a nonempty closed convex subset of a real Hilbert space . Let a -strictly pseudocontractive mapping with . Let be a constant. For any , let be the sequence defined by the following implicit manner: Then the sequence converges strongly to which is the minimum norm fixed point of .

Corollary 3.4. Let be a nonempty closed convex subset of a real Hilbert space . Let a nonexpansive mapping with . Let be a constant. For any , let be the sequence defined by the following implicit manner: Then the sequence converges strongly to which is the minimum norm fixed point of .

Next, we introduce an explicit algorithm for finding the fixed point of .

Theorem 3.5. Let be a nonempty closed convex subset of a real Hilbert space . Let a -strictly pseudocontractive mapping with . Let and be two constants in satisfying . For and any , let be the sequence defined by the following explicit manner: where satisfies the following conditions: ), ().
Then the sequence converges strongly to .

Proof. Step 1. The sequence is bounded.
First, we can rewrite (3.19) as Take . From (3.20), we have By induction, Hence, the sequence is bounded and is also bounded.
Step 2. .
We can rewrite (3.20) as where It follows that Thus, This together with Lemma 2.4 implies that Note that It follows that Thus,
Step 3.  , where .
To see this, we can take a subsequence of satisfying the properties By the demiclosed principle (see Lemma 2.3) and (3.30), we have that . So, Step 4. .
From (3.20), we get where and It is easy to see that and . We can therefore apply Lemma 2.5 to (3.34) and conclude that as . This completes the proof.

Corollary 3.6. Let be a nonempty closed convex subset of a real Hilbert space . Let a nonexpansive mapping with . Let and be two constants in satisfying . For and any , let be the sequence defined by the following explicit manner: where satisfies the following conditions: (), (). Then the sequence converges strongly to .

Corollary 3.7. Let be a nonempty closed convex subset of a real Hilbert space . Let a -strictly pseudocontractive mapping with . Let and be two constants in satisfying . For any , let be the sequence defined by the following explicit manner: where satisfies the following conditions: (), (). Then the sequence converges strongly to which is the minimum norm fixed point of .

Corollary 3.8. Let be a nonempty closed convex subset of a real Hilbert space . Let a nonexpansive mapping with . Let and be two constants in satisfying . For any , let be the sequence defined by the following explicit manner: where satisfies the following conditions: (); (). Then the sequence converges strongly to which is the minimum norm fixed point of  .

4. Conclusion

Finding fixed points of nonlinear mappings (especially, nonexpansive mappings) has received vast investigations due to its extensive applications in a variety of applied areas of inverse problem, partial differential equations, image recovery and signal processing. It is wellknown that strictly pseudocontractive mappings have more powerful applications than nonexpansive mappings in solving inverse problems. In this paper, we devote to construct the methods for computing the fixed points of strictly pseudocontractive mappings. Two iterative methods have been presented. Especially, we can use these two methods to find the minimum norm fixed point of strictly pseudocontractive mappings. The ideas contained in the present paper can help us to solve the minimum norm problems in the applied science.

Acknowledgment

The author was supported in part by NSC 100-2221-E-230-012.

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