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Fixed Point Theory and Applications Volume 2010, Article ID 407651,26pages doi:10.1155/2010/407651

Research Article

Approximation of Common Fixed

Points of a Countable Family of Relatively

Nonexpansive Mappings

Daruni Boonchari

1

and Satit Saejung

2

1Department of Mathematics, Mahasarakham University, Maha Sarakham 44150, Thailand

2Department of Mathematics, Khon Kaen University, Khon Kaen 40002, Thailand

Correspondence should be addressed to Satit Saejung,saejung@kku.ac.th

Received 22 June 2009; Revised 20 October 2009; Accepted 21 November 2009

Academic Editor: Tomonari Suzuki

Copyrightq2010 D. Boonchari and S. Saejung. 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 introduce two general iterative schemes for finding a common fixed point of a countable family of relatively nonexpansive mappings in a Banach space. Under suitable setting, we not only obtain several convergence theorems announced by many authors but also prove them under weaker assumptions. Applications to the problem of finding a common element of the fixed point set of a relatively nonexpansive mapping and the solution set of an equilibrium problem are also discussed.

1. Introduction and Preliminaries

LetCbe a nonempty subset of a Banach spaceE, and letT be a mapping fromCinto itself. When{xn}is a sequence inE, we denote strong convergence of{xn}toxEbyxnxand

weak convergence byxn x. We also denote the weak∗convergence of a sequence{xn}to

xin the dualEbyx

n x∗ ∗. A pointpCis an asymptotic fixed point ofT if there exists

{xn}inCsuch thatxn pandxnTxn → 0. We denoteFTandFTby the set of fixed

points and of asymptotic fixed points ofT, respectively. A Banach spaceEis said to be strictly convex ifxy/2 < 1 forx, ySE {zE : z 1}andx /y. It is also said to be uniformly convex if for each ∈ 0,2, there exists δ > 0 such that xy/2 < 1−δ for

x, ySEandxy. The spaceEis said to be smooth if the limit

lim

t→0

xtxx

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exists for allx, ySE. It is also said to be uniformly smooth if the limit exists uniformly in

x, ySE.

Many problems in nonlinear analysis can be formulated as a problem of finding a fixed point of a certain mapping or a common fixed point of a family of mappings. This paper deals with a class of nonlinear mappings, so-called relatively nonexpansive mappings introduced by Matsushita and Takahashi1. This type of mappings is closely related to the resolvent of maximal monotone operatorssee2–4.

LetEbe a smooth, strictly convex and reflexive Banach space and letCbe a nonempty closed convex subset ofE. Throughout this paper, we denote byφthe function defined by

φx, yx22x, Jyy2 x, yE, 1.2

whereJis the normalized duality mapping fromEto the dual spaceE∗given by the following relation:

x, Jx x2Jx2. 1.3

We know that if Eis smooth, strictly convex, and reflexive, then the duality mappingJ is single-valued, one-to-one, and onto. The duality mappingJis said to be weakly sequentially continuous ifxn ximplies thatJxn Jx∗ see5for more details.

Following Matsushita and Takahashi6, a mappingT :CEis said to be relatively nonexpansive if the following conditions are satisfied:

R1FTis nonempty;

R2φu, Txφu, xfor alluFT,xC;

R3FT FT.

If T satisfiesR1and R2, thenT is called relatively quasi-nonexpansive 7. Obviously, relative nonexpansiveness implies relative quasi-nonexpansiveness but the converse is not true. Relatively quasi-nonexpansive mappings are sometimes called hemirelatively nonexpansive mappings. But we do prefer the former name because in a Hilbert space setting, relatively quasi-nonexpansive mappings are nothing but quasi-nonexpansive.

In2, Alber introduced the generalized projectionΠC fromEontoCas follows:

ΠCx arg min yC φ

y, xxE. 1.4

IfEis a Hilbert space, thenφy, x yx2 andΠC becomes the metric projection of E

ontoC. Alber’s generalized projection is an example of relatively nonexpansive mappings. For more example, see1,8.

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Recently, a problem of finding a common element of the set of solutions of an equilibrium problem and the set of fixed points of a relatively nonexpansive mapping is studied by Takahashi and Zembayashi in15,16. The purpose of this paper is to introduce a new iterative scheme which unifies several ones studied by many authors and to deduce the corresponding convergence theorems under the weaker assumptions. More precisely, many restrictions as were the case in other papers are dropped away.

First, we start with some preliminaries which will be used throughout the paper.

Lemma 1.1see7, Lemma 2.5. LetCbe a nonempty closed convex subset of a strictly convex and smooth Banach spaceEand letTbe a relatively quasi-nonexpansive mapping fromCinto itself. ThenFTis closed and convex.

Lemma 1.2see17, Proposition 5. LetCbe a nonempty closed convex subset of a smooth, strictly convex, and reflexive Banach spaceE. Then

φx,ΠCyφΠCy, yφx, y 1.5

for allxCandyE.

Lemma 1.3see17. LetEbe a smooth and uniformly convex Banach space and letr >0. Then there exists a strictly increasing, continuous, and convex functionh:0,2r → Rsuch thath0 0

and

hxyφx, y 1.6

for allx, yBr {zE:zr}.

Lemma 1.4see17, Proposition 2. LetEbe a smooth and uniformly convex Banach space and let

{xn}and{yn}be sequences ofEsuch that either{xn}or{yn}is bounded. Iflimn→ ∞φxn, yn 0, thenlimn→ ∞xnyn0.

Lemma 1.5see2. LetCbe a nonempty closed convex subset of a smooth, strictly convex, and reflexive Banach spaceE, letxE,and letzC. Then

z ΠCx⇐⇒yz, JxJz≤0,yC. 1.7

Lemma 1.6see18. LetEbe a uniformly convex Banach space and letr >0. Then there exists a strictly increasing, continuous, and convex functiong :0,2r → Rsuch thatg0 0and

tx 1ty2tx2 1ty2t

1−tgxy 1.8

for allx, yBr andt∈0,1.

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Lemma 1.7. Let LetCbe a closed convex subset of a smooth Banach spaceE. LetT be a relatively quasi-nonexpansive mapping fromEintoEand let{Si}Ni1be a family of relatively quasi-nonexpansive

mappings fromCinto itself such thatFTNi1FSi/. The mappingU:CEis defined by

UxTJ−1N

i1

ωiαiJx 1−αiJSix 1.9

for allxCand {ωi},{αi} ⊂ 0,1,i 1,2, . . . , N such thatNi1ωi 1. IfxCand z

FTN

i1FSi, then

φz, Uxφz, x. 1.10

Proof. The proof of this lemma can be extracted from that ofLemma 1.8; so it is omitted.

IfEhas a stronger assumption, we have the following lemma.

Lemma 1.8. LetC be a closed convex subset of a uniformly smooth Banach space E. Let r > 0. Then, there exists a strictly increasing, continuous, and convex functiong∗:0,6r → Rsuch that

g0 0and for each relatively quasi-nonexpansive mappingT : E Eand each finite family of

relatively quasi-nonexpansive mappings{Si}Ni1 :CCsuch thatFTN

i1FSi/,

N

i1

ωiαi1−αigJzJSizφu, zφu, Uz 1.11

for allzCBr anduFTNi1FSiBr, where

UxTJ−1N

i1

ωiαiJx 1−αiJSix 1.12

xCand{ωi},{αi} ⊂0,1,i1,2, . . . , Nsuch thatNi1ωi 1.

Proof. Let r > 0. From Lemma 1.6and E∗ is uniformly convex, then there exists a strictly increasing, continuous, and convex functiong∗:0,6r → Rsuch thatg∗0 0 and

tx1ty∗2tx2 1ty∗2t

1−tgx∗−y∗ 1.13

for allx, y∗ ∈ {z∗ ∈ E∗ : z∗ ≤ 3r}and t ∈ 0,1. LetT : EEand {Si}Ni1 : CC be relatively quasi-nonexpansive for all i 1,2, . . . , N such thatFTNi1FSi/∅. For

zCBr anduFTNi1FSiBr. It follows that

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and henceSiz ≤3r. Consequently, fori1,2, . . . , N,

αiJz 1−αiJSiz2≤αiJz2 1−αiJSiz2−αi1−αigJzJSiz. 1.15

Then

φu, Uzφ

u, J−1N

i1

ωiαiJz 1−αiJSiz

u22

u, N

i1

ωiαiJz 1−αiJSiz

N

i1

ωiαiJz 1−αiJSiz

2

N

i1

ωi

u22u, α

iJz 1−αiJSiz αiJz 1−αiJSiz2

N

i1

ωi

u22u, α

iJz 1−αiJSiz αiJz2 1−αiJSiz2

αi1−αigJzJSiz

N

i1

ωiαiφu, z 1−αiφu, Sizαi1−αigJzJSiz

φu, zN

i1

ωiαi1−αigJzJSiz.

1.16

Thus

N

i1

ωiαi1−αigJzJSizφu, zφu, Uz. 1.17

Lemma 1.9. Let C be a closed convex subset of a uniformly smooth and strictly convex Banach spaceE. LetT be a relatively quasi-nonexpansive mapping fromEintoEand let{Si}Ni1 be a family

of relatively quasi-nonexpansive mappings fromC into itself such thatFTNi1FSi/. The mappingU:CEis defined by

UxTJ−1N

i1

ωiαiJx 1−αiJSix 1.18

for allxCand{ωi},{αi} ⊂0,1,i1,2, . . . , Nsuch thatNi1ωi1. Then, the following hold:

1FU FTNi1FSi,

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Proof. 1Clearly,FTNi1FSiFU. We want to show the reverse inclusion. Letz

FUanduFTNi1FSi. Choose

r :max{u,z,S1z,S2z, . . . ,Smz}. 1.19

FromLemma 1.8, we have

N

i1

ωiαi1−αigJzJSiz 0. 1.20

Fromωiαi1−αi>0 for alli1,2, . . . , Nand by the properties ofg∗, we have

JzJSiz 1.21

for alli1,2, . . . , N. FromJis one to one, we have

zSiz 1.22

for alli1,2, . . . , N. Consider

zUzTJ−1N

i1

ωiαiJz 1−αiJSiz Tz. 1.23

ThuszFTNi1FSi.

2It follows directly from the above discussion.

2. Weak Convergence Theorem

Theorem 2.1. LetCbe a nonempty closed convex subset of a uniformly smooth and uniformly convex Banach spaceE. Let{Tn}∞n1:ECbe a family of relatively quasi-nonexpansive mappings and let

{Si}Ni1:CCbe a family of relatively quasi-nonexpansive mappings such thatF:

n1FTnN

i1FSi/. Let the sequence{xn}be generated byx1∈C,

xn1 TnJ−1 N

i1

ωn,iαn,iJxn 1−αn,iJSixn 2.1

for anyn ∈ N,{ωn,i},{αn,i} ⊂ 0,1for alln ∈N,i 1,2, . . . , N such thatNi1ωn,i 1for all

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Proof. Letu∈∞n1FTnNi1FSi. Put

Un TnJ−1 N

i1

ωn,iαn,iJ 1−αn,iJSi. 2.2

FromLemma 1.7, we have

φu, xn1 φu, Unxnφu, xn. 2.3

Therefore limn→ ∞φu, xnexists. This implies that{φu, xn},{xn}and{Sixn}are bounded

for alli1,2, . . . , N.

Letyn≡ΠFxn. From2.3andm∈N, we have

φyn, xnmφyn, xn. 2.4

Consequently,

φyn, ynmφynm, xnmφyn, xnmφyn, xn. 2.5

In particular,

φyn1, xn1

φyn, xn. 2.6

This implies that limn→ ∞φyn, xnexists. This together with the boundedness of{xn}gives

r : supnNyn < ∞. UsingLemma 1.3, there exists a strictly increasing, continuous, and

convex functionh:0,2r → Rsuch thath0 0 and

hynynmφyn, ynmφyn, xnφynm, xnm. 2.7

Since{φyn, xn}is a convergent sequence, it follows from the properties ofgthat{yn}is a

Cauchy sequence. SinceFis closed, there existszFsuch thatynz.

We first establish weak convergence theorem for finding a common fixed point of a countable family of relatively quasi-nonexpansive mappings. Recall that, for a family of mappings{Tn}∞n1 :CEwith

n1FTn/∅, we say that{Tn}satisfies the NST-condition

19if for each bounded sequence{zn}inC,

lim

n→ ∞znTnzn0 impliesωw{zn} ⊂ ∞

n1

FTn, 2.8

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Theorem 2.2. Let C be a nonempty closed convex subset of a uniformly smooth and uniformly convex Banach spaceE. Let{Tn}∞n1 :ECbe a family of relatively quasi-nonexpansive mappings

satisfying NST-condition and let{Si}Ni1 : CCbe a family of relatively nonexpansive mappings

such thatF:∞n1FTnNi1FSi/and suppose that

φu, Tnx φTnx, xφu, x 2.9

for allu∈∞n1FTn,n∈NandxE. Let the sequence{xn}be generated byx1∈C,

xn1 TnJ−1 N

i1

ωn,iαn,iJxn 1−αn,iJSixn 2.10

for anyn ∈ N,{ωn,i},{αn,i} ⊂ 0,1for alln ∈N,i 1,2, . . . , N such thatNi1ωn,i 1for all

n∈N,lim infn→ ∞ωn,iαn,i1−αn,i>0for alli1,2, . . . , N. IfJis weakly sequentially continuous, then{xn}converges weakly tozF, wherezlimn→ ∞ΠFxn.

Proof. LetuF. FromTheorem 2.1, limn→ ∞φu, xnexists and hence{xn}and {Sixn}are

bounded for alli1,2, . . . , N. Let

rsup

n∈N{xn,S1xn,S2xn, . . . ,SNxn}. 2.11

By Lemma 1.8, there exists a strictly increasing, continuous, and convex function g∗ :

0,2r → Rsuch thatg∗0 0 and

N

i1

ωn,iαn,i1−αn,igJxnJSixnφu, xnφu, xn1. 2.12

In particular, for alli1,2, . . . , N,

ωn,iαn,i1−αn,igJxnJSixnφu, xnφu, xn1. 2.13

Hence,

n1

ωn,iαn,i1−αn,igJxnJSixn<∞ 2.14

for all i 1,2, . . . , N. Since lim infn→ ∞ωn,iαn,i1−αn,i > 0 for alli 1,2. . . , N and the

properties ofg, we have

lim

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for alli 1,2. . . , N. SinceJ−1 is uniformly norm-to-norm continuous on bounded sets, we have

lim

n→ ∞xnSixn0 2.16

for alli1,2. . . , N. Since{xn}is bounded, there exists a subsequence{xnk}of{xn}such that

xnk zC. SinceSiis relatively nonexpansive,zFSi FSifor alli1,2. . . , N.

We show thatz∈∞n1FTn. Let

yn J−1 N

i1

ωn,iαn,iJxn 1−αn,iJSixn. 2.17

We note from2.15that

N

i1

ωn,iαn,iJxn 1−αn,iJSixnJxn

N

i1

ωn,i1−αn,iJSixnJxn −→0. 2.18

SinceJ−1is uniformly norm-to-norm continuous on bounded sets, it follows that

lim

n→ ∞ynxnnlim→ ∞ J−1

N

i1

ωn,iαn,iJxn 1−αn,iJSixn

J−1Jx

n

0. 2.19

Moreover, by2.9and the existence of limn→ ∞φu, xn, we have

φTnyn, ynφu, ynφu, Tnyn

φ

u, J−1N

i1

ωn,iαn,iJxn 1−αn,iJSixn

φu, xn1

φu, xnφu, xn1−→0.

2.20

It follows fromLemma 1.4that limn→ ∞Tnynyn 0. From2.19andxnk z, we have

ynk z. Since{Tn}satisfies NST-condition, we havez

n1FTn. HencezF. Letzn ΠFxn. FromLemma 1.5andzF, we have

znkz, JxnkJznk ≥0. 2.21

FromTheorem 2.1, we know thatznzF. SinceJis weakly sequentially continuous, we

have

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Moreover, sinceJis monotone,

zz, JzJz0. 2.23

Then

zz, JzJz0. 2.24

SinceE is strictly convex, z z. This implies that ωw{xn} {z} and hence xn z

limn→ ∞ΠFxn.

We next apply our result for finding a common element of a fixed point set of a relatively nonexpansive mapping and the solution set of an equilibrium problem. This problem is extensively studied in11, 14–16. LetC be a subset of a Banach space Eand letf:C×C → Rbe a bifunction. The equilibrium problem for a bifunctionfis to findxC such thatfx, y≥0 for allyC. The set of solutions above is denoted by EPf, that is

x∈EPf ifffx, y≥0 ∀yC. 2.25

To solve the equilibrium problem, we usually assume that a bifunction f satisfies the following conditionsCis closed and convex:

A1fx, x 0 for allxC;

A2fis monotone, that is,fx, y fy, x≤0, for allx, yC;

A3for allx, y, zC, lim supt0ftz 1−tx, yfx, y;

A4for allxC,fx,·is convex and lower semicontinuous.

The following lemma gives a characterization of a solution of an equilibrium problem.

Lemma 2.3. LetCbe a nonempty closed convex subset of a Banach spaceE. Letfbe a bifunction from

C×C → Rsatisfying (A1)–(A4). Suppose thatpC. ThenpEPfif and only iffy, p≤0for allyC.

Proof. Letp∈EPf, thenfp, y≥0 for allyC. FromA2, we get thatfy, p≤ −fp, y≤ 0 for allyC.

Conversely, assume thatfy, p≤0 for allyC. For anyyC, let

xtty 1−tp, fort∈0,1. 2.26

Thenfxt, p≤0 and hence

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Sofxt, y≥0 for allt∈0,1. FromA3, we have

0≤lim sup

t↓0

fty 1−tp, yfp, yyC. 2.28

Hencep∈EPf.

Takahashi and Zembayashi proved the following important result.

Lemma 2.4see15, Lemma 2.8. LetCbe a nonempty closed convex subset of a uniformly smooth, strictly convex and reflexive Banach spaceE. Letfbe a bifunction fromC×C → Rsatisfying (A1)– (A4). Forr >0andxE, define a mappingTr :ECas follows:

Trx

zC:fz, y1

r

yz, JzJx≥0∀yC

2.29

for allxE. Then, the following hold:

1Tr is single-valued;

2Tr is a firmly nonexpansive-type mapping [20], that is, for allx, yE

TrxTry, JTrxJTryTrxTry, JxJy; 2.30

3FTr EPf;

4EPfis closed and convex.

We now deduce Takahashi and Zembayashi’s recent result fromTheorem 2.2.

Corollary 2.5see15, Theorem 4.1. LetCbe a nonempty closed convex subset of a uniformly smooth and uniformly convex Banach spaceE. Letfbe a bifunction fromC×CtoRsatisfying (A1)– (A4) and letSbe a relatively nonexpansive mapping fromCinto itself such thatFSEPf/. Let the sequence{xn}be generated byu1 ∈E,

xnCsuch thatfxn, yr1 n

yxn, JxnJun≥0 ∀yC,

un1 J−1αnJxn 1−αnJSxn

2.31

for everyn ∈ N,{αn} ⊂ 0,1satisfyinglim infn→ ∞αn1−αn > 0and {rn} ⊂ a,for some

a >0. IfJ is weakly sequentially continuous, then{xn}converges weakly toz ∈ΠFSEPf, where

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Proof. Put TnTrn where Trn is defined by Lemma 2.4. Then

n1FTn EPf. By reindexing the sequences {xn} and {un} of this iteration, we can apply Theorem 2.2 by

showing that the family {Tn} satisfies the condition 2.9and NST-condition. It is proved

in15, Lemma 2.9that

φu, Tnx φTnx, xφu, xxE, u

n1

FTn. 2.32

To see that {Tn} satisfies NST-condition, let {zn} be a bounded sequence in C such that

limn→ ∞znTnzn 0 andpωw{zn}. Suppose that there exists a subsequence{znk} of {zn}such thatznk p. ThenTnkznk pC. SinceJ is uniformly continuous on bounded

sets andrnka, we have

lim

k→ ∞ 1

rnk

JznkJTnkznk0. 2.33

From the definition ofTrnk, we have

fTnkznk, y

r1

nk

yTnkznk, JTnkznkJznk

≥0 ∀yC. 2.34

Since

fy, Tnkznk

≤ −fTnkznk, y

≤ 1

rnk

yTnkznk, JTnkznkJznk

r1

nk

yTnkznkJTnkznkJznk

2.35

andfis lower semicontinuous and convex in the second variable, we have

fy, p≤lim inf

k→ ∞ f

y, Tnkznk

≤0. 2.36

Thusfy, p≤0 for allyC. FromLemma 2.3, we havep∈EPf. Then{Tn}satisfies the

NST-condition. FromTheorem 2.2whereN1,{xn}converges weakly tozFTnFS

EPfFS, wherezlimn→ ∞ΠEPfFSxn.

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Corollary 2.6see11, Theorem 3.5. LetCbe a nonempty and closed convex subset of a uniformly convex and uniformly smooth Banach spaceE. Letf be a bifunction fromC×CtoRsatisfies (A1)– (A4) and letT, S:CCbe two relatively nonexpansive mappings such thatF :FTFS

EPf/. Let the sequence{xn}be generated by the following manner:

xnCsuch thatfxn, yr1 n

yxn, JxnJun≥0 ∀yC,

un1J−1

αnJxnβnJTxnγnJSxnn≥1.

2.37

Assume that{αn},{βn},and{γn}are three sequences in0,1satisfying the following restrictions:

aαnβnγn1;

blim infn→ ∞αnβn>0,lim infn→ ∞αnγn>0;

c{rn} ⊂a,for somea >0.

IfJis weakly sequentially continuous, then{xn}converges weakly tozF, wherezlimn→ ∞ΠFxn.

The following result also follows fromTheorem 2.2.

Corollary 2.7see9, Theorem 5.3. LetEbe a uniformly smooth and uniformly convex Banach space and letCbe a nonempty closed convex subset ofE. Let{Si}Ni1 be a finite family of relatively

nonexpansive mappings fromCinto itself such thatF Ni1FSiis a nonempty and let {αn,i :

n, i ∈ N, 1 ≤ iN} ⊂ 0,1and{ωn,i : n, i ∈ N, 1 ≤ iN} ⊂ 0,1be sequences such that

lim infn→ ∞αn,i1−αn,i>0andlim infn→ ∞ωn,i>0for alli∈ {1,2, . . . , N}andNi1ωn,i1for

alln∈N. LetUnbe a sequence of mappings defined by

Unx ΠCJ−1 N

i1

ωn,iαn,iJx 1−αn,iJSix 2.38

for allxCand let the sequence{xn}be generated byx1xCand

xn1Unxn n1,2, . . .. 2.39

Then the following hold:

1the sequence {xn} is bounded and each weak subsequential limit of {xn} belongs to N

i1FSi;

2if the duality mapping J from E into Eis weakly sequentially continuous, then {xn} converges weakly to the strong limit ofFxn}.

Proof. Since ΠC is relatively nonexpansive, the family {ΠC} satisfies the NST-condition.

Moreover,FΠC Cand

φx,ΠCyφΠCy, yφx, yyE, xC. 2.40

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3. Strong Convergence Theorem

In this section, we prove strong convergence of an iterative sequence generated by the hybrid method in mathematical programming. We start with the following useful common tools.

Lemma 3.1. LetCbe a nonempty closed convex subset of a uniformly smooth and uniformly convex Banach space E. Let {Tn}∞n1 : EE and {Si}Ni1 : CC be families of relatively

quasi-nonexpansive mappings such thatF:∞n1FTnNi1FSi/, and

φu, Tnx φTnx, xφu, x 3.1

for allu∈∞n1FTn,n∈NandxE. Let{xn} ⊂Cbe such that{xn}and{Sixn}are bounded for alli1,2, . . . , N, and

ynJ−1 N

i1

ωn,iαn,iJxn 1−αn,iJSixn,

unTnyn,

3.2

where{ωn,i},{αn,i} ⊂ 0,1for alln ∈Nandi 1,2, . . . , N satisfyNi1ωn,i 1for alln ∈ N, lim infn→ ∞ωn,i1−αn,i>0for alli1,2, . . . , Nandlimn→ ∞xnun0. Then the following statements hold:

1limn→ ∞φu, xnφu, un 0for alluC,

2limn→ ∞unyn0,

3ωw{xn}ωw{yn},

4iflimn→ ∞xn1−xn0, thenlimn→ ∞xnSixn0for alli1,2, . . . , N,

5ifxnz, thenunzandynz.

Proof. 1 Since limn→ ∞xnun 0 and J is uniformly norm-to-norm continuous on

bounded sets,

lim

n→ ∞JxnJun0. 3.3

We note here that{un}is also bounded. For anyuC, we have

φu, xnφu, unxn2− un2−2u, JunJxn

xn2− u22|u, JunJxn |

xnunxnun 2uJunJxn −→0.

3.4

2LetuF. Using3.1and the relative quasi-nonexpansiveness of eachTn, we have

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ByLemma 1.4and the boundedness of{un}, we have

lim

n→ ∞unyn0. 3.6

3Since

xnynxnununynxnunTnynyn−→0, 3.7

we haveωw{xn}ωw{yn}.

4 Assume that limn→ ∞xn1 −xn 0. From limn→ ∞xnyn 0, we get that

limn→ ∞xn1−yn0. SinceJis uniformly norm-to-norm continuous on bounded sets, we

have

lim

n→ ∞Jxn1−Jxnnlim→ ∞Jxn1−Jyn0. 3.8

So,

Jxn1−Jyn Jxn1−

N

i1

ωn,iαn,iJxn 1−αn,iJSixn

N

i1

ωn,i1−αn,iJxn1−JSixnωn,iαn,iJxn1−Jxn.

3.9

From3.8, we have

N

i1

ωn,i1−αn,iJxn1−JSixnJxn1−Jyn N

i1

ωn,iαn,iJxn1−Jxn −→0. 3.10

It follows from lim infn→ ∞ωn,i1−αn,i>0 for alli1,2, . . . , Nthat

lim

n→ ∞Jxn1−JSixn0 3.11

for alli1,2, . . . , N. SinceJ−1 is uniformly norm-to-norm continuous on bounded sets and limn→ ∞xn1−xn0, we have

lim

n→ ∞xnSixn0 3.12

for alli1,2, . . . , N, as desired.

5Assume thatxnz. From the assumption and2, we have

lim

n→ ∞xnunnlim→ ∞unyn0. 3.13

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Lemma 3.2see21, Lemma 2.4. LetFbe a closed convex subset of a strictly convex, smooth and reflexive Banach spaceEsatisfying Kadec-Klee property. LetxEand{xn}be a sequence inEsuch thatωw{xn} ⊂Fandφxn, xφΠFx, xfor alln∈N. Thenxnz ΠFx.

Recall that a Banach space E satisfies Kadec–Klee property if whenever {un} is a

sequence inEwithxn xandxnx, it follows thatxnx.

3.1. The CQ-Method

Theorem 3.3. LetCbe a nonempty closed convex subset of a uniformly smooth and uniformly convex Banach spaceE. Let{Tn}∞i1:EEbe a family of relatively quasi-nonexpansive mappings satisfying

NST-condition and let{Si}Ni1 :CCbe a family of relatively nonexpansive mappings such that

F:∞n1FTnNi1FSi/, and

φu, Tnx φTnx, xφu, x 3.14

for allu∈∞n1FTn,n∈NandxE. Let the sequence{xn}be generated by

x1xC,

unTnJ−1 N

i1

ωn,iαn,iJxn 1−αn,iJSixn,

Cn zC:φz, unφz, xn,

Qn{zC:xnz, JxJxn ≥0},

xn1 ΠCnQnx

3.15

for everyn∈N,{ωn,i},{αn,i} ⊂0,1for alln∈Nandi1,2, . . . , NsatisfyingNi1ωn,i 1for

alln∈N,lim infn→ ∞ωn,i1−αn,i>0for alli1,2, . . . , N. Then{xn}converges strongly toΠFx.

Proof. The proof is broken into 4 steps.

Step 1{xn}is well defined. First, we show thatCnQnis closed and convex. Clearly,Qnis

closed and convex. Since

φz, unφz, xn⇐⇒ un2− xn2−2z, JunJxn ≤0, 3.16

thenCnis closed and convex. ThusCnQnis closed and convex.

We next show thatFCnQn. LetuF. Then, fromLemma 1.7,

φu, unφu, xn. 3.17

(17)

Next, we show by induction thatFCnQnfor alln∈N. SinceQ1C, we have

FC1∩Q1. 3.18

Suppose thatFCkQkfor somek∈N. Fromxk1 ΠCkQkxCkQkand the definition

of the generalized projection, we have

xk1−z, JxJxk1 ≥0 3.19

for allzCkQk. FromFCkQk,

xk1−p, JxJxk1

≥0 3.20

for allpF. HenceFQk1, and we also haveFCk1∩Qk1. So, we have∅/FCnQn

for alln∈Nand hence the sequence{xn}is well defined.

Step 2 ωw{xn} ⊂ Ni1FSi. From the definition of Qn, we have xn ΠQnx. Using

Lemma 1.2, we get

φxn, x φΠQnx, x

φu, xφu,ΠQnx

φu, x 3.21

for alluQn. In particular, sincexn1∈QnandΠFxFQn,

φxn, xφxn1, x, 3.22

φxn, xφΠFx, x 3.23

for alln ∈N. This implies that limn→ ∞φxn, xexists and{xn}is bounded. Moreover, from

3.21andxn1∈Qn,

φxn, xφxn1, xφxn1, xn. 3.24

Hence

φxn1, xnφxn1, xφxn, x−→0. 3.25

It follows fromxn1 ΠCnQnxCnthat

φxn1, unφxn1, xn−→0. 3.26

From3.25,3.26, andLemma 1.4, we have

lim

(18)

So limn→ ∞xnun0. UsingLemma 3.14, we get that

lim

n→ ∞xnSixn0 3.28

for alli1,2, . . . , N. Since eachSiis relatively nonexpansive,

ωw{xn} ⊂ N

i1

FSi N

i1

FSi. 3.29

Step 3 ωw{xn} ⊂ ∞n1FTn. Let yn J−1 N

i1ωn,iαn,iJxn 1 − αn,iJSixn. From Lemma 3.12, we have

lim

n→ ∞Tnynyn0, 3.30

andωw{xn}ωw{yn}. It follows from NST-condition thatωw{xn}ωw{yn} ⊂∞n1FTn.

Step 4xn → ΠFx. From Steps2and3, we haveωw{xn} ⊂ F. The conclusion follows by

Lemma 3.2and3.23.

We applyTheorem 3.3and the proof ofCorollary 2.5 and then obtain the following result.

Corollary 3.4. LetC,E,f,Sbe as inCorollary 2.5. Let the sequence{xn}be generated by

x1xC,

ynJ−1αnJxn 1−αnJSxn,

unCsuch thatfun, yr1 n

yun, JunJyn≥0 ∀yC,

Cn zC:φz, unφz, xn,

Qn{zC:xnz, JxJxn ≥0},

xn1 ΠCnQnx

3.31

for everyn ∈ N,{αn} ⊂ 0,1satisfyinglim supn→ ∞αn < 1and{rn} ⊂a,for somea > 0. Then,{xn}converges strongly toΠFSEPfx, whereΠFSEPfis the generalized projection ofE

ontoFSEPf.

Remark 3.5. Corollary 3.4improves the restriction on{αn}of15, Theorem 3.1. In fact, it is

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3.2. The Monotone CQ-Method

LetCbe a closed subset of a Banach spaceE. Recall that a mappingT :CCis closed if for each{xn}inC, ifxnxandTxny, thenTx y. A family of mappings{Tn}:CE

with∞n1FTn/∅is said to satisfy the∗-condition if for each bounded sequence{zn}inC,

lim

n→ ∞znTnzn0, zn−→zimplyz∈ ∞

n1

FTn. 3.32

Remark 3.6. 1If{Tn}satisfies NST-condition, then{Tn}satisfies∗-condition.

2IfTnT andTis closed, then{Tn}satisfies∗-condition.

Theorem 3.7. Let C be a nonempty closed convex subset of a uniformly smooth and uniformly convex Banach spaceE. Let{Tn}∞n1 :EEbe a family of relatively quasi-nonexpansive mappings

satisfying-condition and let{Si}Ni1 :CCbe a family of closed relatively quasi-nonexpansive

mappings such thatF:∞n1FTnNi1FSi/, and

φu, Tnx φTnx, xφu, x 3.33

for allu∈∞n1FTn,n∈N,andxE. Let the sequence{xn}be generated by

x0xC, Q0C,

unTnJ−1 N

i1

ωn,iαn,iJxn 1−αn,iJSixn,

C0

zC:φz, u0≤φz, x0

,

CnzCn−1∩Qn−1:φz, unφz, xn,

Qn{zCn−1∩Qn−1:xnz, JxJxn ≥0},

xn1 ΠCnQnx

3.34

for everyn∈N,{ωn,i},{αn,i} ⊂0,1satisfyingiN1ωn,i1andlim infn→ ∞ωn,i1−αn,i>0for

alli1,2, . . . , N. Then{xn}converges strongly toΠFx.

Proof.

Step 1 {xn} is well defined. This step is almost the same as Step 1 of the proof of

Theorem 3.3, so it is omitted.

Step 2{xn}is a Cauchy sequence inC. We can follow the proof ofTheorem 3.3and conclude

that

lim

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exists. Moreover, asxnmQnfor alln, mandxn ΠQnx,

φxnm, xn φxnm,ΠQnx

φxnm, xφΠQnx, x

φxnm, xφxn, x.

3.36

Since {xn} is bounded, it follows from Lemma 1.3 that there exists a strictly increasing,

continuous, and convex functionhsuch thath0 0 and

hxnmxnφxnm, xφxn, x. 3.37

Since limn→ ∞φxn, xexists, we have that{xn}is a Cauchy sequence. Therefore,xnzfor

somezC.

Step 3zNi1FSi. Sincexn1 ΠCnQnxCn, we have

φxn1, unφxn1, xn−→φz, z 0. 3.38

ByLemma 1.4and the boundedness of{xn}, we have

lim

n→ ∞xn1−un0. 3.39

So, we have limn→ ∞xnun0. UsingLemma 3.14, we get that

lim

n→ ∞xnSixn0 3.40

for alli1,2, . . . , N. Since eachSiis closed,zNi1FSi.

Step 4z∈∞n1FTn. Letyn J−1Ni1ωn,iαn,iJxn 1−αn,iJSixn. FromLemma 3.12, we have limn→ ∞ynTnyn0 andynz. It follows from∗-condition thatzn∞1FTn.

Step 5xn → ΠFx. From Steps3and4, we haveωw{xn} ⊂ F. The conclusion follows by

Lemma 3.2and3.23.

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Corollary 3.8see12, Theorem 3.1. LetCbe a nonempty closed convex subset of a uniformly convex and uniformly smooth real Banach spaceE. Let T : CCbe a closed relatively quasi-nonexpansive mapping such that FT/. Assume that {αn} is a sequence in 0,1 such that

lim supn→ ∞αn <1. Define a sequence{xn}inCby the following algorithm:

x0∈Cchosen arbitrarily,

ynJ−1αnJxn 1−αnJTxn,

CnzCn−1∩Qn−1:φ

z, ynφz, xn,

C0

zC:φz, y0

φz, x0

,

Qn{zCn−1∩Qn−1:xnz, Jx0−Jxn ≥0},

Q0C,

xn1 ΠCnQnx0.

3.41

Then{xn}converges strongly toΠFTx0.

LettingTnidentity andN2 yield the following result.

Corollary 3.9see13, Theorem 3.1. LetCbe a nonempty closed convex subset of a uniformly convex and uniformly smooth real Banach spaceE. LetT, Sbe two closed relatively quasi-nonexpansive mappings from Cinto itself such thatF : FTFS/. Define a sequence{xn}in Cbe the following algorithm:

x0∈Cchosen arbitrarily,

zn J−1

β1

n Jxnβn2JTxnβn3JSxn

,

ynJ−1αnJxn 1−αnJzn,

C0

zC:φz, y0

φz, x0

,

CnzCn−1∩Qn−1:φ

z, ynφz, xn,

Qn{zCn−1∩Qn−1:xnz, Jx0−Jxn ≥0},

Q0C,

xn1 ΠCnQnx0

3.42

with the conditions:βn1, βn2, βn3∈0,1withβn1βn2βn31and

1lim infn→ ∞βn1βn2>0;

2lim infn→ ∞βn1βn3>0;

30≤αnα <1for someα∈0,1.

(22)

Remark 3.10. UsingTheorem 3.7, we can show that the conclusion ofCorollary 3.9remains true under the more general restrictions on{αn},{βn1},{βn2},and{βn3}:

1αn, βn1∈0,1are arbitrary;

2lim infn→ ∞βn2 >0 and lim infn→ ∞βn3>0.

3.3. The Shrinking Projection Method

Theorem 3.11. LetC, E, {Tn}∞n1, {Si}Ni1be as inTheorem 3.7. Let the sequence{xn}be generated

by

x0∈Echosen arbitrarily,

C1C,

x1 ΠC1x0,

unTnJ−1 N

i1

ωn,iαn,iJxn 1−αn,iJSixn,

Cn1

zCn:φz, unφz, xn,

xn1 ΠCn1x0

3.43

for everyn∈N,{ωn,i},{αn,i} ⊂0,1for alln∈Nandi1,2, . . . , NsatisfiesNi1ωn,i1for all

n∈N,lim infn→ ∞ωn,i1−αn,i>0for alli1,2, . . . , N. Then{xn}converges strongly toΠFx.

Proof. The proof is almost the same as the proofs of Theorems3.3and3.7; so it is omitted.

In particular, applyingTheorem 3.11gives the following result.

Corollary 3.12. LetC, E, f, Sbe as inCorollary 2.5. Let the sequence{xn}be generated byx0

xC,C0Cand

ynJ−1αnJxn 1−αnJSxn,

unCsuch thatfun, yr1 n

yun, JunJyn≥0 ∀yC,

Cn1

zCn:φz, unφz, xn,

xn1 ΠCn1x0

3.44

for everyn ∈ N∪ {0}, where J is the duality mapping on E. Assume that {αn} ⊂ 0,1satisfies

lim supn→ ∞αn <1and{rn} ⊂a,for somea >0. Then{xn}converges strongly toΠFSEPfx,

whereΠFSEPfis the generalized projection ofEontoFSEPf.

Remark 3.13. Corollary 3.12improves the restriction on{αn}of16, Theorem 3.1. In fact, it is

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Corollary 3.14 see 11, Theorem 3.1. Let C be a nonempty and closed convex subset of a uniformly convex and uniformly smooth Banach spaceE. Let f be a bifunction from C×C toR satisfying (A1)–(A4) and letT, S : CCbe two closed relatively quasi-nonexpansive mappings such thatF:FTFSEPf/. Let the sequence{xn}be generated by the following manner:

x0∈Echosen arbitrarily,

C1C,

x1 ΠC1x0,

ynJ−1αnJxnβnJTxnγnJSxn,

unCsuch thatfun, yr1 n

yun, JunJyn≥0 ∀yC,

Cn1

zCn:φz, unφz, xn,

xn1 ΠCn1x0.

3.45

Assume that{αn},{βn},and{γn}are three sequences in0,1satisfying the restrictions:

aαnβnγn1;

blim infn→ ∞αnβn>0,lim infn→ ∞αnγn>0;

c{rn} ⊂a,for somea >0. Then{xn}converges strongly toΠFx0.

Remark 3.15. The conclusion of Corollary 3.14 remains true under the more general assumption; that is, we can replacebby the following one:

blim infn→ ∞βn>0 and lim infn→ ∞γn>0.

We also deduce the following result.

Corollary 3.16 see 14, Theorem 3.1. Let C, E, f, T, S be as in Corollary 3.14. Let the sequences{xn},{yn},{zn},and{un}be generated by the following:

x0∈Echosen arbitrarily,

C1C,

x1 ΠC1x0,

ynJ−1δnJxn 1−δnJzn,

znJ−1αnJxnβnJTxnγnJSxn,

unCsuch thatfun, yr1 n

yun, JunJzn≥0 ∀yC,

Cn1

zCn:φz, unφz, xn,

xn1 ΠCn1x0.

(24)

Assume that{αn},{βn},and{γn}are three sequences in0,1satisfying the following restrictions:

aαnβnγn1;

b0≤αn<1for alln∈N∪ {0}andlim supn→ ∞αn<1;

clim infn→ ∞αnβn>0,lim infn→ ∞αnγn>0;

d{rn} ⊂a,for somea >0.

Then{xn}and{un}converge strongly toΠFx0.

Remark 3.17. The conclusion of Corollary 3.16 remains true under the more general restrictions; that is, we replacebandcby the following one:

blim infn→ ∞βn>0 and lim infn→ ∞γn>0.

Corollary 3.18see10, Theorem 3.1. LetCbe a nonempty closed convex subset of a uniformly convex and uniformly smooth Banach space E. Let {Ti}Ni1 : CC be a family of relatively

nonexpansive mappings such thatF :Ni1FTi/and letx0 ∈E. ForC1 Candx1 ΠC1x0,

define a sequence{xn}ofCas follows:

ynJ−1αnJxn 1−αnJzn,

znJ−1

β1

n Jxn N

i1

βi1

n JTixn

,

Cn1

zCn :φz, ynφz, xn,

xn1 ΠCn1x0,

3.47

where{αn},{βni} ⊂0,1satisfies the following restrictions:

i0≤αn<1for alln∈N∪ {0}andlim supn→ ∞αn<1;

ii0≤βni≤1for alli1,2, . . . , N1,Ni11β

i

n 1for alln∈N∪ {0}. If

aeitherlim infn→ ∞βn1βni1>0for alli1,2, . . . , Nor

blimn→ ∞βn10andlim infn→ ∞βnk1βnl1>0for alli /j, k, l1,2, . . . , N.

then the sequence{xn}converges strongly toΠFx0.

Remark 3.19. The conclusion of Corollary 3.18 remains true under the more general restrictions on{αn},{βni}:

1αn, βn1∈0,1are arbitrary.

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Acknowledgments

The authors would like to thank the referee for their comments on the manuscript. The first author is supported by grant fund under the program Strategic Scholarships for Frontier Research Network for the Ph.D. Program Thai Doctoral degree from the Office of the Higher Education Commission, Thailand, and the second author is supported by the Thailand Research Fund under Grant MRG5180146.

References

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2 Y. I. Alber, “Metric and generalized projection operators in Banach spaces: properties and applications,” inTheory and Applications of Nonlinear Operators of Accretive and Monotone Type, vol. 178 ofLecture Notes in Pure and Applied Mathematics, pp. 15–50, Marcel Dekker, New York, NY, USA, 1996.

3 W. Takahashi,Convex Analysis and Approximation Fixed points, vol. 2 ofMathematical Analysis Series, Yokohama Publishers, Yokohama, Japan, 2000.

4 W. Takahashi, Nonlinear Functional Analysis, Fixed Point Theory and Its Application, Yokohama Publishers, Yokohama, Japan, 2000.

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Mathematics and Its Applications, Kluwer Academic Publishers, Dordrecht, The Netherlands, 1990.

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9 F. Kohsaka and W. Takahashi, “Block iterative methods for a finite family of relatively nonexpansive mappings in Banach spaces,”Fixed Point Theory and Applications, vol. 2007, Article ID 21972, 18 pages, 2007.

10 S. Plubtieng and K. Ungchittrakool, “Hybrid iterative methods for convex feasibility problems and fixed point problems of relatively nonexpansive mappings in Banach spaces,”Fixed Point Theory and Applications, vol. 2008, Article ID 583082, 19 pages, 2008.

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