Strong Convergence Theorems by Shrinking Projection Methods for Class Mappings

Abstract

We prove a strong convergence theorem by a shrinking projection method for the class of mappings. Using this theorem, we get a new result. We also describe a shrinking projection method for a nonexpansive mapping on Hilbert spaces, which is the same as that of Takahashi et al. (2008).

1. Introduction

Let be a real Hilbert space with inner product and norm , and let be a nonempty closed convex subset of . Recall that a mapping is said to be nonexpansive if for all . The set of fixed points of is Fix.

is said to be quasi-nonexpansive if Fix is nonempty and for all and Fix.

Given , let

(11)

be the half-space generated by . A mapping is said to be the class (or a cutter) if .

Remark 1.1.

The class is fundamental because it contains several types of operators commonly found in various areas of applied mathematics and in particular in approximation and optimization theory (see [1] for details).

Combettes [2], Bauschke, and Combettes [1] studied properties of the class mappings and presented several algorithms. They introduced an abstract Haugazeau method in [1] as follows: starting ,

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Using Lemma 1.2 given below and the fact that a nonexpansive mapping is quasi-nonexpansive, one can easily obtain hybrid methods introduced by Nakajo and Takahashi [3] for a nonexpansive mapping.

Recently, Takahashi et al. [4] proposed a shrinking projection method for nonexpansive mappings . Let , , , and

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where , denotes the metric projection from onto a closed convex subset of .

Inspired by Bauschke and Combettes [1] and Takahashi et al. [4], we present a shrinking projection method for the class of mappings. Furthermore, we obtain a shrinking projection method for a nonexpansive mapping on Hilbert spaces, which is the same as presented by Takahashi et al. [4].

We will use the following notations:

(1) for weak convergence and for strong convergence;

(2) denotes the weak -limit of .

We need some facts and tools in a real Hilbert space which are listed below.

Lemma 1.2 (see [1]).

Let be a Hilbert space. Let be the identity operator of .

(i)If dom, then is quasi-nonexpansive if and only if .

(ii)If , then , for all .

Definition 1.3.

Let for each . The sequence is called to be coherent if, for every bounded sequence in , there holds

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Definition 1.4.

is called demiclosed at if whenever , and .

Next lemma shows that nonexpansive mappings are demeiclosed at 0.

Lemma 1.5 (Goebel and Kirk [5]).

Let be a closed convex subset of a real Hilbert space , and let be a nonexpansive mapping such that . If a sequence in is such that and , then .

Lemma 1.6 (see [6]).

Let be a closed convex subset of . Let be a sequence in and . Let . If is such that and satisfies the condition

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then .

Lemma 1.7 (Goebel and Kirk [5]).

Let be a closed convex subset of real Hilbert space , and let be the (metric or nearest point) projection from onto (i.e., for , is the only point in such that ). Given and , then if and only if there holds the relation

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2. Main Results

In this section, we will introduce a shrinking projection method for the class of mappings and prove strong convergence theorem.

Theorem 2.1.

Let for each such that . Suppose that the sequence is coherent. Let . For and , define a sequence as follows:

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Then, converges strongly to .

Proof.

We first show by induction that for all . is obvious. Suppose for some . Note that, by the definition of , we always have , that is,

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From the definition of and , we obtain . This implies that

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It is obvious that is closed and convex. So, from the definition, is closed and convex for all . So we get that is well defined.

Since is the projection of onto which contains , we have

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Taking , we get

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The last inequality ensures that is bounded. From and , using Lemma 1.7, we get

(26)

It follows that

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Thus is increasing. Since is bounded, exists. From (2.7), it follows that

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and . On the other hand, by , we have

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Hence,

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We therefore get . Since the sequence is coherent, we have . From Lemma 1.6 and (2.5), the result holds.

Remark 2.2.

We take so that is satisfied.

Theorem 2.3.

Let for each such that . Suppose that the sequence is coherent. Let . For and , define a sequence as follows:

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where . Then, converges strongly to .

Proof.

Set . By Lemma 1.2 (ii), we have that . From , it follows that which implies that the sequence is coherent. It is obvious that Fix() = Fix(), for all . Hence . Using Theorem 2.1, we get the desired result.

3. Deduced Results

In this section, using Theorem 2.3, we obtain some new strong convergence results for the class of mappings, a quasi-nonexpansive mapping and a nonexpansive mapping in a Hilbert space.

Theorem 3.1.

Let such that and satisfying that is demiclosed at 0. Let . For and , define a sequence as follows:

(31)

where . Then, converges strongly to .

Proof.

Let in (2.11) for all . Following the proof of Theorem 2.1, we can easily get (2.5) and . By (2.5), we obtain that is bounded and is nonempty. For any , there exists a subsequence of the sequence such that . From , it follows that . Since is demiclosed at 0, we get Fix. Thus Fix which together with Lemma 1.6 and (2.5) implies that .

Theorem 3.2.

Let be a Hilbert space. Let be a quasi-nonexpansive mapping on such that and satisfying that is demiclosed at 0. Let . For and , define a sequence as follows:

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where . Then, converges strongly to .

Proof.

By Lemma 1.2 (i), . Substitute in (3.1) by . Then . Set , then . So, we have

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Since is demiclosed at 0, is demiclosed at 0. So we can obtain the result by using Theorem 3.1.

Since a nonexpansive mapping is quasi-nonexpansive, using Lemma 1.5 and Theorem 3.2, we have following corollary.

Corollary 3.3.

Let be a Hilbert space. Let be a nonexpansive mapping such that . Let . For and , define a sequence as follows:

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where . Then, converges strongly to .

Remark 3.4.

Corollary 3.3 is a special case of Theorem  4.1 in [4] when .