Strong Convergence Theorems by Shrinking Projection Methods for Class Mappings

(1)

Volume 2011, Article ID 681214,7pages doi:10.1155/2011/681214

Research Article

Strong Convergence Theorems by Shrinking

Projection Methods for Class

T

Mappings

Qiao-Li Dong,

1, 2

_{Songnian He,}

1, 2

_{and Fang Su}

3

1_{College of Science, Civil Aviation University of China, Tianjin 300300, China}

2_{Tianjin Key Laboratory for Advanced Signal Processing, Civil Aviation University of China,}

Tianjin 300300, China

3_{Department of Mathematics and Systems Science, National University of Defense Technology,}

Changsha 410073, China

Correspondence should be addressed to Qiao-Li Dong,[email protected]

Received 9 December 2010; Accepted 2 February 2011

Academic Editor: S. Al-Homidan

Copyrightq2011 Qiao-Li Dong et al. 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.

We prove a strong convergence theorem by a shrinking projection method for the class ofT 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

LetHbe a real Hilbert space with inner product·,·and norm · , and letCbe a nonempty closed convex subset ofH. Recall that a mappingT :H → His said to be nonexpansive if

Tx−Ty ≤ x−yfor allx, y∈H. The set of fixed points ofTis FixT:{x∈H:Txx}.

T : H → His said to be quasi-nonexpansive if FixTis nonempty andTx−p ≤

x−pfor allx∈Handp∈FixT. Givenx, y∈H, let

Hx, y:z∈H:z−y, x−y≤0 1.1

be the half-space generated byx, y. A mappingT :H → His said to be the classTor a cutterifT ∈T{T :H → H|domT H and FixT⊂Hx, Tx, for allx∈H}.

(2)

Combettes 2, Bauschke, and Combettes 1 studied properties of the class T mappings and presented several algorithms. They introduced an abstract Haugazeau method in1as follows: startingx0∈H,

xn1PHx0,xn∩Hxn,Tnxnx0. 1.2

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

3for a nonexpansive mapping.

Recently, Takahashi et al.4proposed a shrinking projection method for nonexpan-sive mappingsTn:C → C. Letx0 ∈H,C1C,x1PC1x0, and

ynαn 1−αnTnxn,

Cn1

z∈Cn:yn−z ≤ xn−z,

xn1PCn1x0, n1,2, . . . ,

1.3

where 0≤αn ≤a <1,PKdenotes the metric projection fromHonto a closed convex subset

KofH.

Inspired by Bauschke and Combettes 1 and Takahashi et al. 4, we present a shrinking projection method for the class ofTmappings. 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:

1for weak convergence and → for strong convergence;

2ωwxn {x:∃xnj x}denotes the weakω-limit of{xn}.

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

Lemma 1.2see1. LetHbe a Hilbert space. LetIbe the identity operator ofH.

iIfdomT H, then2T−Iis quasi-nonexpansive if and only ifT ∈T.

iiIfT ∈T, thenλI 1−λT ∈T, for allλ∈0,1.

Definition 1.3. LetTn ∈Tfor eachn. The sequence{Tn}is called to be coherent if, for every

bounded sequence{v_n}inH, there holds

∞

n0

vn1−vn2<∞,

∞

n0

vn−Tnvn2<∞,

⇒ωwvn⊂

∞

n0

FixTn. 1.4

Definition 1.4. T is called demiclosed aty ∈ Hif Tx ywhenever{x_n} ⊂ H,x_n xand

Txn → y.

(3)

Lemma 1.5Goebel and Kirk5. LetCbe a closed convex subset of a real Hilbert space H, and letT :C → Cbe a nonexpansive mapping such thatFixT_/∅. If a sequence{x_n}inCis such that

xn zandxn−Txn → 0, thenzTz.

Lemma 1.6see6. LetKbe a closed convex subset ofH. Let{xn}be a sequence inHandu∈H.

LetqP_Ku. Ifx_nis such thatω_wx_n⊂Kand satisfies the condition

xn−u ≤ u−q, ∀n, 1.5

thenxn → q.

Lemma 1.7Goebel and Kirk5. LetKbe a closed convex subset of real Hilbert spaceH, and let

PKbe the (metric or nearest point) projection fromHontoK(i.e., forx∈H,PKxis the only point

inKsuch thatx−PKxinf{x−z:z∈K}). Givenx∈Handz∈K, thenzPKxif and

only if there holds the relation

x−z, y−z≤0, ∀y∈K. 1.6

2. Main Results

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

Theorem 2.1. LetT_n∈Tfor eachnsuch thatF:_n∞₁FixT_n_/∅. Suppose that the sequence{T_n} is coherent. Letx0∈H. ForC1Handx1x0, define a sequence{xn}as follows:

xn1PCn1x0, n1,2, . . . , Cn1{z∈Cn:z−Tnxn, xn−Tnxn ≤0}.

2.1

Then,{x_n}converges strongly toP_Fx0.

Proof. We first show by induction thatF ⊂ Cn for all n ∈ N.F ⊂ C1 is obvious. Suppose

F ⊂Ckfor somek∈N. Note that, by the definition ofTk∈T, we always haveF ⊂FixTk⊂

Hxk, Tkxk, that is,

z−Tkxk, xk−Tkxk ≤0, ∀z∈ F. 2.2

From the definition ofCk1andF ⊂Ck, we obtainF ⊂Ck1. This implies that

F ⊂Cn, ∀n∈N. 2.3

It is obvious thatC1His closed and convex. So, from the definition,Cnis closed and convex for alln∈N. So we get that{xn}is well defined.

Sincex_nis the projection ofx0ontoCnwhich containsF, we have

(4)

TakingyPFx0 ∈ F, we get

x0−xn ≤ x0−PFx0. 2.5

The last inequality ensures that{x0−xn}is bounded. FromxnPCnx0andxn1PCn1x0∈

Cn1⊂Cn, usingLemma 1.7, we get

xn1−xn, x0−xn ≤0. 2.6

It follows that

x0−xn12x0−xn−xn1−xn2

x0−xn2−2x0−xn, xn1−xnxn1−xn2

≥ x0−xn2xn1−xn2

≥ x0−xn2.

2.7

Thus{xn−x0}is increasing. Since{xn−x0}is bounded, limn→ ∞xn−x0exists. From

2.7, it follows that

xn1−xn2≤ x0−xn12− x0−xn2, 2.8

and∞_n₁xn1−xn2<∞. On the other hand, byxn1PCn1x0∈Cn1, we have

xn1−Tnxn, xn−Tnxn ≤0. 2.9

Hence,

xn1−xn2xn1−Tnxn−xn−Tnxn2

xn1−Tnxn2−2xn1−Tnxn, xn−Tnxnxn−Tnxn2

≥ xn1−Tnxn2xn−Tnxn2.

2.10

We therefore get ∞_n₁x_n−T_nx_n2 < ∞. Since the sequence {T_n} is coherent, we have

ωwxn⊂ F. FromLemma 1.6and2.5, the result holds.

(5)

Theorem 2.3. LetTn∈Tfor eachnsuch thatF:n∞1FixTn/∅. Suppose that the sequence{Tn} is coherent. Letx0∈H. ForC1Handx1x0, define a sequence{xn}as follows:

ynαnxn 1−αnTnxn,

Cn1

z∈Cn:z−yn, xn−yn≤0,

xn1PCn1x0, n1,2, . . . ,

2.11

where0≤αn≤a <1. Then,{xn}converges strongly toPFx0.

Proof. SetS_n α_nI 1−α_nT_n. ByLemma 1.2ii, we have thatS_n ∈T. Fromx_n−S_nx_n

1−αnxn −Tnxn, it follows that1−axn−Tnxn ≤ xn−Snxn ≤ xn −Tnxnwhich

implies that the sequence{Sn}is coherent. It is obvious that FixSn FixTn, for alln∈N.

HenceF∞_n₁FixS_n ∞_n₁FixT_n. UsingTheorem 2.1, we get the desired result.

3. Deduced Results

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

Theorem 3.1. LetT ∈Tsuch thatFixT_/∅and satisfying thatI−Tis demiclosed at 0. Letx0∈H. ForC1Handx1x0, define a sequence{xn}as follows:

ynαnxn 1−αnTxn,

Cn1

z∈Cn:z−yn, xn−yn ≤0,

xn1PCn1x0, n1,2, . . . ,

3.1

where0≤α_n≤a <1. Then,{x_n}converges strongly toPFixTx0.

Proof. LetTn T in2.11for alln ∈ N. Following the proof ofTheorem 2.1, we can easily

get2.5and∞_n₁xn−Txn2 <∞. By2.5, we obtain that{xn}is bounded andωwxnis

nonempty. For anyx ∈ω_wx_n, there exists a subsequence{x_n_j}of the sequence{x_n}such thatxnj x. From

_∞

n1xn−Txn2 < ∞, it follows thatxn−Txn → 0. Since I−T is demiclosed at 0, we getx ∈FixT. Thusω_wx_n ⊂FixTwhich together withLemma 1.6

and2.5implies thatxn → PFixTx0.

Theorem 3.2. LetH be a Hilbert space. Let Sbe a quasi-nonexpansive mapping on H such that FixS_/∅and satisfying thatI−Sis demiclosed at 0. Letx0∈H. ForC1Handx1x0, define a sequence{xn}as follows:

unαnxn 1−αnSxn,

Cn1{z∈Cn:z−un ≤ xn−z},

xn1PCn1x0, n1,2, . . . ,

3.2

(6)

Proof. ByLemma 1.2i,SI/2 ∈ T. Substitute T in 3.1bySI/2. Then yn 1

αn/2xn 1−αn/2Sxn. Setun2yn−xnαnxn 1−αnSxn, thenyn unxn/2. So,

we have

Cn1

z∈Cn:z−yn, xn−yn≤0

{z∈Cn:2z−xnun, xn−un ≤0}

{z∈Cn:z−un ≤ xn−z}.

3.3

SinceI−Sis demiclosed at 0,I−SI/2 I−S/2 is demiclosed at 0. So we can obtain the result by usingTheorem 3.1.

Since a nonexpansive mapping is quasi-nonexpansive, using Lemma 1.5 and

Theorem 3.2, we have following corollary.

Corollary 3.3. LetHbe a Hilbert space. LetSbe a nonexpansive mappingHsuch thatFixS/∅. Letx0∈H. ForC1Handx1x0, define a sequence{xn}as follows:

unαnxn 1−αnSxn,

Cn1{z∈Cn: z−un ≤ xn−z},

xn1PCn1x0, n1,2, . . . ,

3.4

where0≤αn≤a <1. Then,{xn}converges strongly toPFixSx0.

Remark 3.4. Corollary 3.3is a special case of Theorem 4.1 in4whenC1H.

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 Grant08QD10Xand Fundamental Research Funds for the Central Universities GrantZXH2009D021.

References

1 H. H. Bauschke and P. L. Combettes, “A weak-to-strong convergence principle for Fej´er-monotone methods in Hilbert spaces,”Mathematics of Operations Research, vol. 26, no. 2, pp. 248–264, 2001. 2 P. L. Combettes, “Quasi-Fej´erian analysis of some optimization algorithms,” in Inherently Parallel

Algorithms in Feasibility and Optimization and Their Applications, vol. 8, pp. 115–152, North-Holland, Amsterdam, The Netherlands, 2001.

3 K. Nakajo and W. Takahashi, “Strong convergence theorems for nonexpansive mappings and nonexpansive semigroups,”Journal of Mathematical Analysis and Applications, vol. 279, no. 2, pp. 372– 379, 2003.

(7)

5 K. Goebel and W. A. Kirk,Topics in Metric Fixed Point Theory, vol. 28 ofCambridge Studies in Advanced Mathematics, Cambridge University Press, Cambridge, UK, 1990.