• No results found

Strong Convergence of Monotone Hybrid Algorithm for Hemi-Relatively Nonexpansive Mappings

N/A
N/A
Protected

Academic year: 2020

Share "Strong Convergence of Monotone Hybrid Algorithm for Hemi-Relatively Nonexpansive Mappings"

Copied!
8
0
0

Loading.... (view fulltext now)

Full text

(1)

Volume 2008, Article ID 284613,8pages doi:10.1155/2008/284613

Research Article

Strong Convergence of Monotone Hybrid Algorithm

for Hemi-Relatively Nonexpansive Mappings

Yongfu Su,1Dongxing Wang,1 and Meijuan Shang1, 2

1Department of Mathematics, Tianjin Polytechnic University, Tianjin 300160, China 2Department of Mathematics, Shijiazhuang University, Shijiazhuang 050035, China

Correspondence should be addressed to Yongfu Su,suyongfu@tjpu.edu.cn

Received 1 June 2007; Revised 5 September 2007; Accepted 16 October 2007

Recommended by Simeon Reich

The purpose of this article is to prove strong convergence theorems for fixed points of closed hemi-relatively nonexpansive mappings. In order to get these convergence theorems, the monotone hy-brid iteration method is presented and is used to approximate those fixed points. Note that the hybrid iteration method presented by S. Matsushita and W. Takahashi can be used for relatively nonexpansive mapping, but it cannot be used for hemi-relatively nonexpansive mapping. The re-sults of this paper modify and improve the rere-sults of S. Matsushita and W. Takahashi2005, and some others.

Copyrightq2008 Yongfu Su 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.

1. Introduction

In 2005, Shin-ya Matsushita and Wataru Takahashi1proposed the following hybrid iteration methodit is also called the CQ methodwith generalized projection for relatively nonexpan-sive mappingT in a Banach spaceE:

x0∈C chosen arbitrarily,

ynJ−1

αnJxn

1−αn

JTxn

,

Cn

zC:φz, yn

φz, xn

,

Qn

zC:xnz, Jx0−Jxn ≥0

,

xn1 Π

Cn∩Qn

x0

.

1.1

(2)

Theorem 1.1MT. LetEbe a uniformly convex and uniformly smooth real Banach space, letCbe a nonempty, closed, and convex subset ofE, letTbe a relatively nonexpansive mapping fromCinto itself, and let{αn} be a sequence of real numbers such that0 ≤ αn < 1 andlim supn→∞αn < 1. Suppose

that{xn}is given by1.1, whereJis the duality mapping onE. If the setFTof fixed points ofT is

nonempty, then{xn}converges strongly toΠFTx0, whereΠFT·is the generalized projection from

ContoFT.

The purpose of this article is to prove strong convergence theorems for fixed points of closed hemi-relatively nonexpansive mappings. In order to get these convergence theorems, the monotone hybrid iteration method is presented and is used to approximate those fixed points. Note that the hybrid iteration method presented by S.Matsushita and W. Takahashi can be used for relatively nonexpansive mapping, but it cannot be used for hemi-relatively non-expansive mapping. The results of this paper modify and improve the results of S.Matsushita and W. Takahashi1, and some others.

2. Preliminaries

LetEbe a real Banach space with dualE∗. We denote byJ the normalized duality mapping fromEto 2Edefined by

JxfE∗:x, fx2f2, 2.1

where ·,· denotes the generalized duality pairing. It is well known that ifE∗ is uniformly convex, thenJis uniformly continuous on bounded subsets ofE. In this case,Jis singe valued and also one to one.

Recall that ifCis a nonempty, closed, and convex subset of a Hilbert spaceH andPC :

HCis the metric projection ofH ontoC, thenPC is nonexpansive. This is true only when

H is a real Hilbert space. In this connection, Alber 2has recently introduced a generalized projection operator ΠC in a Banach spaceEwhich is an analogue of the metric projection in Hilbert spaces.

Next, we assume that Eis a smooth Banach space. Consider the functional defined as

2,3by

φx, y x2−2x, Jyy2 forx, yE. 2.2

Observe that, in a Hilbert spaceH,2.2reduces toφx, y xy2,x, yH.

The generalized projectionΠC :ECis a map that assigns to an arbitrary pointxE the minimum point of the functionalφy, x, that is,ΠCx x,wherexis the solution to the minimization problem

φx, x min

y∈Cφy, x, 2.3

existence and uniqueness of the operator ΠC follow from the properties of the functional

φy, xand strict monotonicity of the mappingJsee, e.g.,2–4. In Hilbert space,ΠCPC.It is obvious from the definition of the functionφthat

(3)

Remark 2.1. IfEis a reflexive strict convex and smooth Banach space, then forx, yE,φx, y

0 if and only ifxy. It is sufficient to show that ifφx, y 0, thenxy. From2.4, we have

xy. This impliesx, Jyx2Jy2.From the definition ofJ,we haveJxJy, that is,xy; see5for more details.

We refer the interested reader to the 6, where additional information on the duality mapping may be found.

Let C be a closed convex subset of E, and Let T be a mapping from C into itself. We denote by FT the set of fixed points of T. T is called hemi-relatively nonexpansive if

φp, Txφp, xfor allxCandpFT.

A pointpinCis said to be an asymptotic fixed point ofT7ifCcontains a sequence{xn} which converges weakly topsuch that the strong limn→∞Txnxn 0.The set of asymptotic fixed points ofT will be denoted byFT. A hemi-relatively nonexpansive mappingT fromC

into itself is called relatively nonexpansive1,7,8ifFT FT.

We need the following lemmas for the proof of our main results.

Lemma 2.2Kamimura and Takahashi4,1, Proposition 2.1. LetEbe a uniformly convex and

smooth real Banach space and let{xn},{yn}be two sequences ofE. Ifφxn, yn → 0and either{xn}

or{yn}is bounded, thenxnyn → 0.

Lemma 2.3Alber2,1, Proposition 2.2. LetCbe a nonempty closed convex subset of a smooth

real Banach spaceEandxE. Then,x0 ΠCxif and only if

x0−y, JxJx0

≥0 ∀yC. 2.5

Lemma 2.4 Alber 2, 1, Proposition 2.3. Let E be a reflexive, strict convex, and smooth real

Banach space, letCbe a nonempty closed convex subset ofEand letxE.Then

φ y,Π

cx

φ Π

cx, x

φy, xyC. 2.6

By using the similar method as1, Proposition 2.4, the following lemma is not hard to prove.

Lemma 2.5. LetEbe a strictly convex and smooth real Banach space, letCbe a closed convex subset

ofE, and letT be a hemi-relatively nonexpansive mapping fromCinto itself. ThenFTis closed and convex.

Recall that an operatorT in a Banach space is called closed, ifxnx, Txny, then

Txy.

3. Strong convergence for hemi-relatively nonexpansive mappings

Theorem 3.1. Theorem 3.1LetEbe a uniformly convex and uniformly smooth real Banach space, let

Cbe a nonempty closed convex subset ofE, letT : CCbe a closed hemi-relatively nonexpansive mapping such thatFT/. Assume that{αn}is a sequence in0,1such thatlim supn→∞αn < 1.

(4)

x0∈C chosen arbitrarily,

ynJ−1

αnJxn

1−αn

JTxn , Cn

zCn−1∩Qn−1:φ

z, yn

φz, xn

, C0

zC:φz, y0

φz, x0

,

Qn

zCn−1∩Qn−1:

xnz, Jx0−Jxn

≥0, Q0C,

xn1 Π

Cn∩Qn

x0

,

3.1

where J is the duality mapping on E. Then{xn} converges strongly to ΠFTx0, whereΠFT is the generalized projection fromContoFT.

Proof. We first show thatCnandQnare closed and convex for eachn≥0. From the definition of Cn andQn, it is obvious thatCnis closed andQnis closed and convex for eachn ≥ 0. We show thatCnis convex for anyn≥0. Since

φz, yn

φz, xn

3.2

is equivalent to

2z, JxnJyn

xn2−yn2, 3.3

it follows thatCnis convex.

Next, we show thatFTCnfor alln≥0. Indeed, we have for allpFTthat

φp, yn

φp, j−1αnjxn

1−αn

jtxn

p2−2p, αnjxn

1−αn

jtxn

αnxn2

1−αntxn2

αnφ

p, xn

1−αn

φp, txn

αnφ

p, xn

1−αn

φp, xn

φp, xn

.

3.4

That is,pCnfor alln≥0.

Next, we show thatFTQnfor alln≥ 0, we prove this by induction. Forn 0,we haveFTCQ0.Assume thatFTQn.Sincexn1is the projection ofx0ontoCnQn, by Lemma 2.3, we have

xn1−z, Jx0−Jxn1

≥0,zCnQn. 3.5

AsFTCnQnby the induction assumptions, the last inequality holds, in particular, for all

zFT. This together with the definition ofQn1implies thatFTQn1. Sincexn1 ΠCn∩Qnx0andCnQnCn−1∩Qn−1for alln≥1, we have

φxn, x0

φxn1, x0

3.6

for alln≥0. Therefore,{φxn, x0}is nondecreasing. In addition, it follows from the definition ofQnandLemma 2.3thatxn ΠQnx0. Therefore, byLemma 2.4, we have

φxn, x0

φ Π

Qn x0, x0

φp, x0

φp, xn

φp, x0

(5)

for eachpFTQn for alln ≥ 0.Therefore,φxn, x0is bounded, this together with 3.6 implies that the limit of{φxn, x0}exists. Put

lim n→∞φ

xn, x0

d. 3.8

FromLemma 2.4, we have, for any positive integerm, that

φxnm, xn

φ xnm,Π Cnx0

φxnm, x0

φ Π

Cn x0, x0

φxnm, x0

φxn, x0

, 3.9

for alln≥0.Therefore,

lim n→∞φ

xnm, xn

0. 3.10

We claim that{xn}is a Cauchy sequence. If not, there exists a positive real numberε0>0 and subsequence{nk},{mk} ⊂ {n}such that

xnkmkxnkε0, 3.11

for allk≥1.

On the other hand, from3.8and3.9we have

φxnkmk, xnk

φxnkmk, x0

φxnk, x0

φxnkmk, x0

d|dφxnk, x0

| −→0, k −→ ∞. 3.12

Because from3.8we know thatφxn, x0is bounded, this and2.4imply that{xn} is also bounded, so byLemma 2.2we obtain

lim

k→∞xnkmkxnk0. 3.13

This is a contradiction, so that{xn}is a Cauchy sequence, therefore there exists a pointpC such that{xn}converges strongly top.

Sincexn1 ΠCn∩Qnx0∈Cn, from the definition ofCn, we have

φxn1, yn

φxn1, xn

. 3.14

It follows from3.10,3.14that

φxn1, yn

−→0. 3.15

By usingLemma 2.2, we have

lim

(6)

SinceJis uniformly norm-to-norm continuous on bounded sets, we have

lim

n→∞Jxn1−Jynn→∞limJxn1−Jxn0. 3.17

Noticing that

Jxn1JynJxn1

αnJxn

1−αn

JTxn

αn

Jxn1−Jxn

1−αn

Jxn1−JTxn

1−αn

Jxn1−Jtxn

αn

JxnJxn1

≥1−αnJxn1−JtxnαnJxnJxn1,

3.18

which implies that

Jxn1JTxn 1 1−αn

Jxn1JynαnJxnJxn1. 3.19

This together with3.17and lim supn→∞αn<1 implies that

lim

n→∞Jxn1−JTxn0. 3.20

SinceJ−1is also uniformly norm-to-norm continuous on any bounded sets, we have

lim

n→∞xn1−Txn0. 3.21

Observe that

xnTxnxnxn1xn1Txn. 3.22

It follows from3.16and3.21that

lim

n→∞xnTxn0. 3.23

SinceT is a closed operator andxnp, thenpis a fixed point ofT. Finally, we prove thatp ΠFTx0. FromLemma 2.4, we have

φ p, Π

FTx0

φ Π

FTx0, x0

φp, x0

. 3.24

On the other hand, sincexn1 ΠCn∩Qn andCnQnFT, for all n, we get fromLemma 2.4

that

φ Π

FTx0, xn1

φxn1, x0

φ Π

FTx0, x0

. 3.25

By the definition of φx, y, it follows that both φp, x0 ≤ φΠFTx0, x0 and φp, x0 ≥

φΠFTx0, x0, whenceφp, x0 φΠFTx0, x0. Therefore, it follows from the uniqueness of

(7)

Theorem 3.2. Let E be a uniformly convex and uniformly smooth real Banach space, let C be a nonempty, closed, and convex subset of E, and let T : CC be a closed relative nonexpansive mapping such thatFT/. Assume that{αn}is a sequences in0,1such thatlim supn→∞αn<1.

Define a sequence{xn}inCby the following algorithm:

x0∈C chosen arbitrarily,

ynJ−1

αnJxn

1−αn

JTxn

, Cn

zCn−1∩Qn−1:φ

z, yn

φz, xn

, C0

zC:φz, y0

φz, x0

,

Qn

zCn−1∩Qn−1:

xnz, Jx0−Jxn

≥0, Q0C,

xn1 Π

Cn∩Qn

x0

,

3.26

where J is the duality mapping on E. Then{xn} converges strongly to ΠFTx0, whereΠFT is the

generalized projection fromContoFT.

Proof. Since every relatively nonexpansive mapping is a hemi-relatively one,Theorem 3.2 is implied byTheorem 3.1.

Remark 3.3. In recent years, the hybrid iteration methods for approximating fixed points of nonlinear mappings have been introduced and studied by various authors1,8–11. In fact, all hybrid iteration methods can be replacedor modifiedby monotone hybrid iteration methods, respectively. In addition, by using the monotone hybrid method we can easily show that the iteration sequence {xn} is a Cauchy sequence, without the use of the Kadec-Klee property, demiclosedness principle, and Opial’s condition or other methods which make use of the weak topology.

Acknowledgments

The authors would like to thank the referee for valuable suggestions which helped to improve this manuscript. This project is supported by the National Natural Science Foundation of China under Grant no. 10771050.

References

1S.-Y. Matsushita and W. Takahashi, “A strong convergence theorem for relatively nonexpansive map-pings in a banach space,”Journal of Approximation Theory, vol. 134, no. 2, pp. 257–266, 2005.

2Ya. I. Alber, “Metric and generalized projection operators in Banach spaces: properties and applica-tions,” inTheory and Applications of Nonlinear Operators of Accretive and Monotone Type, A. G. Kartsatos, Ed., vol. 178 ofLecture Notes in Pure and Appl. Math., pp. 15–50, Marcel Dekker, New York, NY, USA, 1996.

3Ya. I. Alber and S. Reich, “An iterative method for solving a class of nonlinear operator equations in Banach spaces,”PanAmerican Mathematical Journal, vol. 4, no. 2, pp. 39–54, 1994.

4S. Kamimura and W. Takahashi, “Strong convergence of a proximal-type algorithm in a Banach space,” SIAM Journal on Optimization, vol. 13, no. 3, pp. 938–945, 2002.

(8)

6S. Reich, “Review of geometry of Banach spaces, duality mappings and nonlinear problems by Ioana Cioranescu,”Bulletin of the American Mathematical Society, vol. 26, pp. 367–370, 1992.

7D. Butnariu, S. Reich, and A. J. Zaslavski, “Asymptotic behavior of relatively nonexpansive operators in Banach spaces,”Journal of Applied Analysis, vol. 7, no. 2, pp. 151–174, 2001.

8C. Martinez-Yanes and H.-K. Xu, “Strong convergence of the CQ method for fixed point iteration processes,”Nonlinear Analysis: Theory, Methods & Applications, vol. 64, no. 11, pp. 2400–2411, 2006.

9T.-H. Kim and H.-K. Xu, “Strong convergence of modified Mann iterations for asymptotically nonex-pansive mappings and semigroups,”Nonlinear Analysis: Theory, Methods & Applications, vol. 64, no. 5, pp. 1140–1152, 2006.

10K. Nakajo and W. Takahashi, “Strong convergence theorems for nonexpansive mappings and non-expansive semigroups,”Journal of Mathematical Analysis and Applications, vol. 279, no. 2, pp. 372–379, 2003.

References

Related documents

We first identify the causal impacts of coresidence and nearby residence with elderly parents on female labor force participation and hours of work, taking into account

While many studies have demonstrated that viruses re- program cell metabolism and rely on metabolic changes for optimal virus replication in vitro, significant work re- mains

Thus, this paper presents the result of an experimental investigation on predicting surface roughness (Ra) of tool steel using Taguchi’s parameter design and response

In this paper, a joint lot sizing and marketing problem was investigated by considering production of various types of non-perfect products. The products are classified in four

We investigated the relationship between different assem- blies of Aβ — freshly prepared, oligomeric, and fibrillar — and tau seeding using several cell-culture models,

Thus, the data obtained through comprehensive surveillance indicate that JEV is hyperen- demic in Bali, that it causes substantial human illness, and that it circulates year-round..

The approaches to critical visual theory adopted by the authors of this issue may be subsumed under the following headings (3.1) critical visu- al discourse and visual memes