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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 ,

(12)

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

(13)

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

(14)

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

(15)

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

(16)

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:

(21)

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,

(22)

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

(23)

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

(24)

Taking , we get

(25)

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

(26)

It follows that

(27)

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

(28)

and . On the other hand, by , we have

(29)

Hence,

(210)

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:

(211)

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:

(32)

where . Then, converges strongly to .

Proof.

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

(33)

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:

(34)

where . Then, converges strongly to .

Remark 3.4.

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

References

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    Bauschke HH, Combettes PL: A weak-to-strong convergence principle for Fejér-monotone methods in Hilbert spaces. Mathematics of Operations Research 2001,26(2):248–264. 10.1287/moor.26.2.248.10558

  2. 2.

    Combettes PL: Quasi-Fejérian analysis of some optimization algorithms. In Inherently Parallel Algorithms in Feasibility and Optimization and Their Applications. Volume 8. North-Holland, Amsterdam, The Netherlands; 2001:115–152.

  3. 3.

    Nakajo K, Takahashi W: Strong convergence theorems for nonexpansive mappings and nonexpansive semigroups. Journal of Mathematical Analysis and Applications 2003,279(2):372–379. 10.1016/S0022-247X(02)00458-4

  4. 4.

    Takahashi W, Takeuchi Y, Kubota R: Strong convergence theorems by hybrid methods for families of nonexpansive mappings in Hilbert spaces. Journal of Mathematical Analysis and Applications 2008,341(1):276–286. 10.1016/j.jmaa.2007.09.062

  5. 5.

    Goebel K, Kirk WA: Topics in Metric Fixed Point Theory, Cambridge Studies in Advanced Mathematics. Volume 28. Cambridge University Press, Cambridge, UK; 1990:viii+244.

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    Martinez-Yanes C, Xu H-K: Strong convergence of the CQ method for fixed point iteration processes. Nonlinear Analysis. Theory, Methods & Applications 2006,64(11):2400–2411. 10.1016/j.na.2005.08.018

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Acknowledgments

The authors would like to express their thanks to the referee for the valuable comments and suggestions for improving this paper. This paper is supported by Research Funds of Civil Aviation University of China Grant (08QD10X) and Fundamental Research Funds for the Central Universities Grant (ZXH2009D021).

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Correspondence to Qiao-Li Dong.

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Dong, Q., He, S. & Su, F. Strong Convergence Theorems by Shrinking Projection Methods for Class Mappings. Fixed Point Theory Appl 2011, 681214 (2011). https://doi.org/10.1155/2011/681214

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Keywords

  • Hilbert Space
  • Applied Mathematic
  • Differential Geometry
  • Convex Subset
  • Optimization Theory