Hybrid Iterative Methods for Convex Feasibility Problems and Fixed Point Problems of Relatively Nonexpansive Mappings in Banach Spaces

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Volume 2008, Article ID 583082,19pages doi:10.1155/2008/583082

Research Article

Hybrid Iterative Methods for Convex Feasibility

Problems and Fixed Point Problems of Relatively

Nonexpansive Mappings in Banach Spaces

Somyot Plubtieng and Kasamsuk Ungchittrakool

Department of Mathematics, Faculty of Science, Naresuan University, Phitsanulok 65000, Thailand

Correspondence should be addressed to Somyot Plubtieng,[email protected]

Received 2 July 2008; Accepted 23 December 2008

Recommended by Hichem Ben-El-Mechaiekh

The convex feasibility problemCFPof finding a point in the nonempty intersectionNi1Ciis considered, whereN1 is an integer and theCi’s are assumed to be convex closed subsets of a Banach spaceE. By using hybrid iterative methods, we prove theorems on the strong convergence to a common fixed point for a finite family of relatively nonexpansive mappings. Then, we apply our results for solving convex feasibility problems in Banach spaces.

Copyrightq2008 S. Plubtieng and K. Ungchittrakool. 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

We are concerned with theconvex feasibility problemCFP

finding an x∈

N

i1

Ci, 1.1

whereN1 is an integer, andC1, . . . , CNare intersecting closed convex subsets of a Banach

spaceE. This problem is a frequently appearing problem in diverse areas of mathematical

and physical sciences. There is a considerable investigation onCFPin the framework of

Hilbert spaces which captures applications in various disciplines such as image restoration

1–4, computer tomography5, and radiation theraphy treatment planning6. In computer

tomography with limited data, in which an unknown image has to be reconstructed from a priori knowledge and from measured results, each piece of information gives a constraint

which in turn, gives rise to a convex set Ci to which the unknown image should belong

see7. The advantage of a Hilbert spaceHis that thenearest pointprojectionPK onto

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So projection methods have dominated in the iterative approaches toCFPin Hilbert spaces;

see 6, 8–11 and the references therein. In 1993, Kitahara and Takahashi 12 deal with

the convex feasibility problem by convex combinations of sunny nonexpansive retractions

in uniformly convex Banach spaces see also Takahashi and Tamura 13, O’Hara et al.

14, and Chang et al.15 for the previous results on this subject. It is known that if C

is a nonempty closed convex subset of a smooth, reflexive, and strictly convex Banach

space E, then the generalized projection ΠC see, Alber 16 or Kamimura and Takahashi

17 from E ontoC is relatively nonexpansive, whereas the metric projection PC from E

onto C is not generally nonexpansive. Our purpose in the present paper is to obtain an

analogous result for a finite family of relatively nonexpansive mappings in Banach spaces.

This notion was originally introduced by Butnariu et al. 18. Recently, Matsushita and

Takahashi 19 reformulated the definition of the notion and obtained weak and strong

convergence theorems to approximate a fixed point of a single relatively nonexpansive

mapping. Motivated by Nakajo and Takahashi20, Matsushita and Takahashi21studied

the strong convergence of the sequence{xn}generated by

x0 x∈C,

yn J−1αnJxn1−αnJTxn,

Hn z∈C:φz, ynφz, xn,

Wn z∈C:xn−z, Jx−Jxn 0,

xn1 ΠHn∩Wnx, n0,1,2, . . . ,

1.2

whereJis the duality mapping onE,{αn} ⊂0,1,T is a relatively nonexpansive mapping

fromCinto itself, andΠFT·is the generalized projection fromContoFT.

Very recently, Plubtieng and Ungchittrakool22studied the strong convergence to a

common fixed point of two relatively nonexpansive mappings of the sequence{xn}generated

by

x0x∈C,

ynJ−1αnJxn1−αnJzn,

znJ−1βn1Jxnβn2JTxnβn3JSxn,

Hnz∈C:φz, ynφz, xn,

Wnz∈C:xn−z, Jx−Jxn 0,

xn1PHn∩Wnx, n0,1,2, . . . ,

1.3

whereJ is the duality mapping on E, andPF·is the generalized projection from Conto

F:FS∩FT.

We note that the block iterative method is a method which often used by many authors

to solve the convex feasibility problem CFP see, 23, 24, etc.. In 2008, Plubtieng and

Ungchittrakool25established strong convergence theorems of block iterative methods for

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process by using the shrinking method proposed, whose studied by Takahashi et al.26,

which is diﬀerent from the method in25. Let C be a closed convex subset ofE and for

each i 1,2, . . . , N, let Ti : C → C be a relatively nonexpansive mapping such that

F:N_i₁FTi/∅. Define{xn}in the two following ways:

x0∈E, C1C, x1 ΠC1x0,

znJ−1

β1

n Jxn N

i1

βi1

n JTixn

,

Cn1

z∈Cn:φz, ynφz, xn,

xn1 ΠCn1x0, n0,1,2, . . . ,

1.4

and

x0∈C,

znJ−1

β1

n Jxn N

i1

βi1

n JTixn

,

Wnz∈C:xn−z, Jx0−Jxn 0,

xn1 ΠHn∩Wnx0, n0,1,2, . . . ,

1.5

where{αn},{βni} ⊂0,1, Ni11β i

n 1 satisfy some appropriate conditions.

We will prove that both iterations1.4and1.5converge strongly to a common fixed

point ofN_i₁FTi. Using this results, we also discuss the convex feasibility problem in Banach

spaces. Moreover, we apply our results to the problem of finding a common zero of a finite family of maximal monotone operators and equilibrium problems.

Throughout the paper, we will use the notations:

i → for strong convergence andfor weak convergence;

iiωwxn {x:∃xnr x}denotes the weakω-limit set of{xn}.

2. Preliminaries

LetEbe a real Banach space with norm·and letE∗be the dual ofE. Denote by ·,·the

duality product. The normalized duality mappingJfromEtoE∗is defined by

Jxx∗_∈_E∗_:_{x, x}∗ _x2_x∗2 2.1

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A Banach spaceEis said to be strictly convex ifxy/2 <1 for allx, y ∈Ewith

xy1 andx /y. It is also said to be uniformly convex if limn→ ∞xn−yn0 for any

two sequences{xn}, {yn}inEsuch thatxnyn 1 and limn→ ∞xnyn/21. Let

U{x∈E:x1}be the unit sphere ofE. Then the Banach spaceEis said to be smooth

provided that

lim

t→0

xty − x

t 2.2

exists for eachx, y∈U. It is also said to be uniformly smooth if the limit is attained uniformly

for x, y ∈ U. It is well known that p and Lp 1 < p < ∞ are uniformly convex and

uniformly smooth; see Cioranescu 27 or Diestel28. We know that ifE is smooth, then

the duality mappingJis single valued. It is also known that ifEis uniformly smooth, then

J is uniformly norm-to-norm continuous on each bounded subset ofE. Some properties of

the duality mapping have been given in27,29,30. A Banach spaceEis said to have the

Kadec-Klee property if a sequence{xn}of Esatisfying thatxn x ∈ Eandxn → x,

thenxn → x. It is known that ifEis uniformly convex, thenEhas the Kadec-Klee property;

see27,30for more details. LetEbe a smooth Banach space. The functionφ:E×E → Ris

defined by

φy, x y2₋₂ _{y, Jx}_x2 _2.3

for allx, y∈E. It is obvious from the definition of the functionφthat

1 y − x2φy, xyx2,

2φx, y φx, z φz, y 2 x−z, Jz−Jy,

3φx, y x, Jx−Jy y−x, JyxJx−Jyy−xy,

for allx, y, z∈E. LetEbe a strictly convex, smooth, and reflexive Banach space, and letJbe

the duality mapping fromEintoE∗. ThenJ−1is also single-valued, one-to-one, and surjective,

and it is the duality mapping fromE∗ into E. We make use of the following mapping V

studied in Alber16:

Vx, x∗_x2₋₂_{x, x}∗ _x∗2 _2.4

for allx∈Eandx∗ ∈E∗. In other words,Vx, x∗ φx, J−1x∗for allx∈Eandx∗ ∈E∗.

For eachx∈E, the mappingVx,·:E∗ → Ris a continuous and convex function fromE∗

intoR.

LetCbe a nonempty closed convex subset of a smooth, strictly convex, and reflexive

Banach spaceE, for anyx∈E, there exists a pointx0∈Csuch thatφx0, x miny∈Cφy, x.

The mappingΠC:E → Cdefined byΠCxx0is called thegeneralized projection16,17,31.

The following are well-known results. For example, see16,17,31.

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Lemma 2.1see27,30,32. IfEis a strictly convex and smooth Banach space, then forx, y∈E,

φy, x 0if and only ifxy.

Proof. It is suﬃcient to show that ifφy, x 0 thenx y. From1, we havex y.

This implies y, Jx y2 _Jx2_{. From the definition of}_J_{, we have}_Jx _Jy_{. Since}_J _is

one-to-one, we havexy.

Lemma 2.2 Kamimura and Takahashi17. LetEbe a uniformly convex and smooth Banach space and let{yn},{zn}be two sequences ofE. Ifφyn, zn → 0and either{yn}or{zn}is bounded,

thenyn−zn → 0.

LetCbe a nonempty closed convex subset of a smooth, strictly convex, and reflexive

Banach spaceE, letTbe a mapping fromCinto itself, and letFTbe the set of all fixed points

ofT. Then a pointp ∈ Cis said to be anasymptotic fixed pointofT see Reich33if there

exists a sequence{xn}inCconverging weakly topand limn→ ∞xn−Txn 0. We denote

the set of all asymptotic fixed points ofTbyFTand we say thatTis arelatively nonexpansive

mappingif the following conditions are satisfied:

R1FTis nonempty;

R2φu, Txφu, xfor allu∈FTandx∈C;

R3FT FT.

Lemma 2.3 Alber 16, Alber and Reich 31, Kamimura and Takahashi 17. Let C be a nonempty closed convex subset of a smooth Banach spaceE, letx∈E, and letx0∈C. Then,x0 ΠCx

if and only if x0−y, Jx−Jx00for ally∈C.

Lemma 2.4 Alber 16, Alber and Reich31, Kamimura and Takahashi 17. Let E be a reflexive, strictly convex and smooth Banach space, letCbe a nonempty closed convex subset ofEand letx∈E. Thenφy,ΠCx φΠCx, xφy, xfor ally∈C.

Lemma 2.5. LetEbe a uniformly convex Banach space and letBr0 {x∈E:xr}be a closed

ball ofE. Then there exists a continuous strictly increasing convex functiong :0,∞ → 0,∞with

g0 0such that

N

i1

ωi_x_i

2

N

i1

ωi_x_i2₋_ωj_ωk_g_x_j₋_x_k_, _{for any}_{j, k}_{∈ {}₁_,₂_{, . . . , N}_}_, _2.5

where{xi}Ni1⊂Br0and{ωi} N

i1 ⊂0,1with

N

i1ωi1.

Proof. It suﬃcient to show that

N

i1

ωi_x_i

2

N

i1

ωi_x_i2₋_ω1_ω2_g_x

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It is obvious that2.6holds forN 1,2 see34for more details. Next, we assume that

2.6is true forN−1. It remains to show that2.6holds forN. We observe that

N

i1

ωi_x_i

2

ωN_x_N₁₋_ωN

_N₋₁

i1

ωi

1−ωNxi

2

ωN_x

N21−ωN

N−1

i1

ωi

1−ωNxi 2

ωN_x_N2

1−ωN

_N₋₁

i1

ωi

1−ωNxi

2₋ ω1ω2

1−ωN2g

x1−x2

N

i1

ωi_x

i2− ω

1_ω2

1−ωNgx1−x2

N

i1

ωi_x_i2₋_ω1_ω2_g_x

1−x2.

2.7

This completes the proof.

Lemma 2.6. LetCbe a closed convex subset of a smooth Banach spaceEand letx, y ∈E. Then the setK:{v∈C:φv, yφv, x}is closed and convex.

Proof. As a matter of fact, the defining inequality inKis equivalent to the inequality

v,2Jx−Jy x2− y2. 2.8

This inequality is aﬃne invand hence the setKis closed and convex.

3. Main result

In this section, we prove strong convergence theorems for finding a common fixed point of a finite family of relatively nonexpansive mappings in Banach spaces by using the hybrid method in mathematical programming.

Theorem 3.1. Let E be a uniformly convex and uniformly smooth Banach space, and let Cbe a nonempty closed convex subset ofE. Let{Ti}Ni1be a finite family of relatively nonexpansive mappings

fromCinto itself such thatF:_iN₁FTiis nonempty and letx0∈E. ForC1Candx1 ΠC1x0, define a sequence{xn}ofCas follows:

znJ−1

β1

n Jxn N

i1

βi1

n JTixn

,

Cn1

xn1 ΠCn1x0, n0,1,2, . . . ,

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where{αn},{βni} ⊂0,1satisfy the following conditions:

i0αn<1for alln∈N∪ {0}andlim supn→ ∞αn<1,

ii0βni1for alli1,2, . . . , N1,Ni11β i

n 1for alln∈N∪ {0}. If either

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

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

Then the sequence{xn}converges strongly toΠFx0, whereΠF is the generalized projection fromE

ontoF.

Proof. We first show by induction thatF ⊂Cnfor alln∈N.F ⊂ C1is obvious. Suppose that

F⊂Ckfor somek∈N. Then, we have, foru∈F ⊂Ck,

φu, ykφu, J−1αkJxk1−αkJzkVu, αkJxk1−αkJzk

αkVu, Jxk1−αkVu, Jzkαkφu, xk1−αkφu, zk,

φu, zkV

u, β1

k Jxk N

i1

βi1

k JTixk

β1

k V

u, Jxk N

i1

βi1

k V

u, JTixk

φu, xk.

3.2

It follow that

φu, ykφu, xk 3.3

and henceu∈Ck1. This implies thatF ⊂ Cnfor alln ∈N. Next, we show thatCnis closed

and convex for alln∈N. Obvious thatC1Cis closed and convex. Suppose thatCkis closed

and convex for somek∈N. Forz∈Ck, we note byLemma 2.6thatCk1is closed and convex.

Then for anyn ∈ N,Cn is closed and convex. This implies that{xn}is well-defined. From

xn ΠCnx0, we have

φxn, x0

φu, x0

−φu, xnφu, x0

∀u∈Cn. 3.4

In particular, letu∈F, we have

φxn, x0

φu, x0

∀n∈N. 3.5

Thereforeφxn, x0 is bounded and hence{xn} is bounded by 1. Fromxn ΠCnx0 and

xn1∈Cn1 ⊂Cn, we have

φxn, x0

min

y∈Cn

φy, x0

φxn1, x0

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Therefore{φxn, x0}is nondecreasing. So there exists the limit ofφxn, x0. ByLemma 2.4, we have

φxn1, xnφxn1,ΠCnx0

φxn1, x0

−φΠCnx0, x0

φxn1, x0

−φxn, x0

.

3.7

for eachn∈N. This implies that limn→ ∞φxn1, xn 0. Sincexn1∈Cn1it follows from the

definition ofCn1that

φxn1, ynφxn1, xn ∀n∈N. 3.8

Lettingn → ∞, we have limn→ ∞φxn1, yn 0. ByLemma 2.2, we obtain

lim

n→ ∞xn1−ynnlim→ ∞xn1−xn0. 3.9

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

lim

n→ ∞Jxn1−Jynnlim→ ∞Jxn1−Jxn0. 3.10

SinceJxn1−JynJxn1−αnJxn−1−αnJzn1−αnJxn1−Jzn −αnJxn−Jxn1

for eachn∈N∪ {0}, we get that

Jxn1−Jzn

1 1−αn

_Jx_n₁₋_Jy_nαnJxn−Jxn1

1

1−αn

Jxn1−JynJxn−Jxn1.

3.11

From 3.10 and lim sup_n_{→ ∞}αn < 1, we have limn→ ∞Jxn1−Jzn 0.Since J−1 is also

uniformly norm-to-norm continuous on bounded sets, it follows that

lim

n→ ∞xn1−znnlim→ ∞J −1_Jx

n1

−J−1_Jz

n0. 3.12

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Next, we show thatxn−Tixn → 0 for alli1,2, . . . , N. Since{xn}is bounded and

φp, Tixnφp, xnfor alli1,2, . . . , N, wherep∈F. We also obtain that{Jxn}and{JTixn}

are bounded for alli1,2, . . . , N. Then there existsr >0 such that{Jxn},{JTixn} ⊂Br0for

alli1,2, . . . , N. ThereforeLemma 2.5is applicable. Assume thataholds, we observe that

φp, znp2−2

p, β1 n Jxn

N

i1

βi1 n JTixn

β1 n Jxn

N

i1

βi1 n JTixn

2

p2₋₂_β1

n p, Jxn N

i1

βi1

n p, JTixn βn1xn2 N

i1

βi1

n Tixn2

−β1

n βni1gJxn−JTixn

β1

n p2−2p, Jxn xn2 N

i1

βi1

n p22p, JTixn Tixn2

−β1

n βni1gJxn−JTixn

β1

n φp, xn N

i1

βi1

n φp, Tixn−βn1βni1gJxn−JTixn

φp, xn−βn1βni1gJxn−JTixn

3.13

and hence

β1

n βni1gJxn−JTixnφp, xn−φp, zn

2p, zn−xn xnznxn−zn

2pzn−xnxnznxn−zn

−→0,

3.14

whereg :0,∞ → 0,∞is a continuous strictly increasing convex function withg0 0 in

Lemma 2.5. Bya, we have limn→ ∞gJxn−JTixn 0 and then limn→ ∞Jxn−JTixn0

for alli1,2, . . . , N. SinceJ−1is also uniformly norm-to-norm continuous on bounded sets,

we obtain

lim

n→ ∞xn−Tixnnlim→ ∞J

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for alli1,2, . . . , N. Ifbholds, we get

φp, znp2−2

p, β1

n Jxn N

i1

βi1

n JTixn

β1

n Jxn N

i1

βi1

n JTixn

2

p2₋₂_β1

n p, Jxn N

i1

βi1

n p, JTixn βn1xn2 N

i1

βi1

n Tixn2

−βk1

n βnl1gJTkxn−JTlxn

β1

n p2−2p, Jxn xn2 N

i1

βi1

n p22p, JTixn Tixn2

−βk1

n βnl1gJTkxn−JTlxn

β1

n φp, xn N

i1

βi1

n φp, Tixn−βnk1βnl1gJTkxn−JTlxn

φp, xn−βnk1βnl1gJTkxn−JTlxn

3.16

and hence

βk1

n βnl1gJTkxn−JTlxnφp, xn−φp, zn

2p, zn−xn xnznxn−zn

2pzn−xnxnznxn−zn

−→0.

3.17

Then by the same argument above, we have limn→ ∞Tkxn−Tlxn0 for allk, l1,2, . . . , N.

Next, we observe that

φTkxn, zn V

Tkxn, βn1Jxn N

i1

βi1

n JTixn

β1

n VTkxn, Jxn N

i1

βi1

n VTkxn, JTixn

β1

n φTkxn, xn N

i1

βi1

n φTkxn, Tixn

−→0.

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asβn1 → 0. By Lemma 2.2, we have limn→ ∞Tkxn −zn 0 for allk 1,2, . . . , N, and

hence

Tixn−xnTixn−znzn−xn−→0 asn−→ ∞, 3.19

for alli1,2, . . . , N. Thenωwxn⊂Ni1FTi

_N

i1FTi F.

Finally, we show thatxn → ΠFx0. Let{xnk}be a subsequence of{xn}such thatxnk

v∈ωwxn⊂F. Putw: ΠFx0∈F⊂Cnk, we observe that

φxnk, x0

φΠC_nkx0, x0

min

y∈C_nkφ

y, x0

φw, x0

min

z∈F φ

z, x0

φv, x0

. 3.20

Sinceφ·, x0is weakly lower semicontinuous, we obtain

φv, x0

lim inf

k→ ∞ φ

xnk, x0

lim sup

k→ ∞

φxnk, x0

φw, x0

φv, x0

. 3.21

This implies that v w and limk→ ∞xnk w and then the Kadec-Klee property ofE

yieldsxnk → w. Since{xnk}is an arbitrary,xn → w. This completes the proof.

Corollary 3.2. LetE be a uniformly convex and uniformly smooth Banach space, and let C be a nonempty closed convex subset ofE. Let{Ωi}Ni1be a finite family of nonempty closed convex subset of

Csuch thatΩ:N_i₁Ωiis nonempty and letx0∈E. ForC1Candx1 ΠC1x0, define a sequence {xn}ofCas follows:

znJ−1

β1

n Jxn N

i1

βi1

n JΠΩixn

,

Cn1

xn1 ΠCn1x0, n0,1,2, . . . ,

3.22

i0αn<1for alln∈N∪ {0}andlim sup_n_{→ ∞}αn<1,

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Theorem 3.3. Let E be a uniformly convex and uniformly smooth Banach space, and let Cbe a nonempty closed convex subset ofE. Let{Ti}Ni1be a finite family of relatively nonexpansive mappings

fromCinto itself such thatF:N_i₁FTiis nonempty. Let a sequence{xn}defined by

x0∈C,

znJ−1

β1

n Jxn N

i1

βi1

n JTixn

,

Wnz∈C:xn−z, Jx0−Jxn 0,

xn1 ΠHn∩Wnx0, n0,1,2, . . . ,

3.23

Then the sequence{xn}converges strongly toΠFx0, whereΠF is the generalized projection fromE

ontoF.

Proof. From the definition ofHnandWn, it is obviousHnandWnare closed and convex for

eachn ∈N∪ {0}. Next, we show thatF ⊂Hn∩Wnfor eachn∈ N∪ {0}. Letu ∈F and let

n∈N∪ {0}. Then, as in the proof ofTheorem 3.1, we have

φu, znφu, xn 3.24

for alln∈N∪ {0}, and thenφu, ynφu, xn. Thus, we haveu∈Hn. Therefore we obtain

F ⊂ Hn for eachn ∈ N∪ {0}. We note by21, Proposion 2.4that eachFTiis closed and

convex and so is F. Using the same argument presented in the proof of21, Theorem 3.1;

page 261-262, we haveF ⊂Hn∩Wnfor eachn∈N∪ {0},{xn}is well defined and bounded,

and

lim

n→ ∞xn1−ynnlim→ ∞xn1−xn0. 3.25

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

lim

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As in the proof ofTheorem 3.1, we also have that

Jxn1−Jzn

1 1−αn

_Jx_n₁₋_Jy_nαnJxn−Jxn1

1

1−αn

Jxn1−JynJxn−Jxn1.

3.27

From 3.26 and lim sup_n_{→ ∞}αn < 1, we have limn→ ∞Jxn1−Jzn 0.Since J−1 is also

uniformly norm-to-norm continuous on bounded sets, we obtain

lim

n→ ∞xn1−znnlim→ ∞J −1_Jx_n

1

−J−1_Jz_n₀_. _3.28

Fromxn−znxn−xn1xn1−znwe have limn→ ∞xn−zn0. By the same argument

as in the proof ofTheorem 3.1, we have{xn}converges strongly toΠFx0.

Corollary 3.4. LetE be a uniformly convex and uniformly smooth Banach space, and let C be a nonempty closed convex subset ofE. Let{Ωi}Ni1be a finite family of nonempty closed convex subset

ofCsuch thatΩ:N_i₁Ωiis nonempty. Let a sequence{xn}defined by

x0∈C,

znJ−1

β1 n Jxn

N

i1

βi1 n JΠΩixn

,

Wnz∈C:xn−z, Jx0−Jxn 0

, xn1 ΠHn∩Wnx0, n0,1,2, . . . ,

3.29

Then the sequence{xn}converges strongly toΠΩx0, whereΠΩ is the generalized projection fromE ontoΩ.

IfN2,T1T andT2S, thenTheorem 3.3reduces to the following corollary.

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be two relatively nonexpansive mappings fromCinto itself withF :FS∩FTis nonempty. Let a sequence{xn}be defined by

x0x∈C,

znJ−1βn1Jxnβn2JTxnβn3JSxn,

Wnz∈C:xn−z, Jx−Jxn 0,

xn1PHn∩Wnx, n0,1,2, . . . ,

3.30

with the following restrictions:

ii0 βn1, βn2, βn3 1,βn1βn2βn3 1for all n∈ N∪ {0},limn→ ∞βn1 0and

lim infn→ ∞βn2βn3>0.

Then the sequence{xn}converges strongly toPFx, wherePFis the generalized projection fromConto

F.

4. Applications

4.1. Maximal monotone operators

LetAbe a multivalued operator fromEtoE∗ with domainDA {z ∈ E : Az /∅}and

rangeRA ∪{Az:z∈DA}. An operatorAis said to be monotone if x1−x2, y1−y20

for eachxi ∈ DAandyi ∈ Axi,i 1,2. A monotone operatorAis said to be maximal if

its graphGA {x, y : y ∈ Ax} is not properly contained in the graph of any other

monotone operator. We know that ifAis a maximal monotone operator, thenA−10is closed

and convex. LetE be a reflexive, strictly convex and smooth Banach space, and letAbe a

monotone operator fromEtoE∗, we known from Rockafellar35thatAis maximal if and

only ifRJrA E∗for allr >0. LetJr :E → DAdefined byJr JrA−1Jand such

aJr is called the resolvent ofA. We know thatJr is a relatively nonexpansive; see21and

A−1₀ _F_J_r_{for all}_{r >}_{0; see}₃₀_,₃₂_{for more details.}

Theorem 4.1. LetEbe a uniformly convex and uniformly smooth Banach space. LetAi ⊂E×E∗be

a maximal monotone operator for eachi1,2, . . . , Nsuch thatΛ:N_i₁A−1_i 0is nonempty and let

x0∈E. ForC1 E, define a sequence{xn}as follows:

znJ−1

β1

n Jxn N

i1

βi1

n JJrAiixn

,

Cn1

xn1 ΠCn1x0, n0,1,2, . . . ,

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whereJAi

ri is the resolvent ofAi withri >0for eachi1,2, . . . , N, and{αn},{β i

n } ⊂0,1satisfy

the following conditions:

Then the sequence{xn}converges strongly toΠΛx0, whereΠΛis the generalized projection fromE

ontoΛ.

Theorem 4.2. LetEbe a uniformly convex and uniformly smooth Banach space. LetAi ⊂E×E∗be

a maximal monotone operator for eachi1,2, . . . , Nsuch thatΛ:N_i₁A−1_i 0is nonempty. Let a sequence{xn}defined by

x0∈E,

znJ−1

β1 n Jxn

N

i1

βi1 n JJrAiixn

,

Hnz∈E:φz, ynφz, xn,

Wnz∈E:xn−z, Jx0−Jxn 0

, xn1 ΠHn∩Wnx0, n0,1,2, . . . ,

4.2

whereJAi

ri is the resolvent ofAi withri >0for eachi1,2, . . . , N, and{αn},{β i

n } ⊂0,1satisfy

the following conditions:

Then the sequence{xn}converges strongly toΠΛx0, whereΠΛis the generalized projection fromE ontoΛ.

4.2. Equilibrium problems

For solving the equilibrium problem, let us assume that a bifunctionfsatisfies the following

conditions:

A1fx, x 0 for allx∈C;

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A3fis upper-hemicontinuous, that is, for eachx, y, z∈C,

lim sup

t↓0 f

tz 1−tx, yfx, y; 4.3

A4fx,·is convex and lower semicontinuous for eachx∈C.

The following result is in Blum and Oettli36.

Lemma 4.3Blum and Oettli36. LetCbe a nonempty closed convex subset of a smooth, strictly convex and reflexive Banach spaceE, letf be a bifunction ofC×CintoRsatisfying (A1)–(A4). Let

r >0andx∈E. Then, there existsz∈Csuch that

fz, y 1

r y−z, Jz−Jx0 ∀y∈C. 4.4

The following result is in Takahashi and Zembayashi37.

Lemma 4.4Takahashi and Zembayashi37. LetCbe a closed convex subset of a uniformly smooth, strictly convex, and reflexive Banach spaceEand letf:C×C → Rsatisfies (A1)–(A4). For

r >0andx∈E, define a mappingTr :E → Cas follows:

Trx

z∈C:fz, y 1

r y−z, Jz−Jx0, ∀y∈C

4.5

for allx∈E. Then, the following hold;

1Tr is single-valued;

2Tr is firmly nonexpansive-type mapping [38], that is, for anyx, y∈E,

Trx−Try, JTrx−JTry Trx−Try, Jx−Jy ; 4.6

3FTr EPf;

4EPfis closed and convex.

Lemma 4.5Takahashi and Zembayashi37. LetCbe a closed convex subset of a smooth, strictly convex, and reflexive Banach spaceE. letfbe a bifunction fromC×CtoRsatisfying (A1)–(A4), and letr >0. Then forx∈Eandq∈FTr,

φq, TrxφTrx, xφq, x. 4.7

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for eachi1,2, . . . , N, andΘ : N_i₁EPfi/∅, and letx0 ∈ E. ForC1 Candx1 ΠC1x0, define a sequence{xn}ofCas follows:

uni∈C such thatf_iuni, y

1

ri

y−uni, Juni−Jxn 0 ∀y∈C, for eachi1,2, . . . , N,

znJ−1

β1 n Jxn

N

i1

βi1 n Juni

,

Cn1

xn1 ΠCn1x0, n0,1,2, . . . ,

4.8

Then the sequence{xn}converges strongly toΠΘx0, whereΠΘ is the generalized projection fromE ontoΘ.

Theorem 4.7. Let E be a uniformly convex and uniformly smooth Banach space, and let Cbe a nonempty closed convex subset ofE. Letfibe a bifunction fromC×CintoRsatisfying (A1)–(A4) for eachi1,2, . . . , N, andΘ:N_i₁EPfi/∅. Let a sequence{xn}defined by

x0 ∈E,

uni ∈C such that fi

uni, y

1

ri

y−uni, Juni−Jxn 0 ∀y∈C, for each i1,2, . . . , N,

yn J−1αnJxn1−αnJzn,

zn J−1

β1

n Jxn N

i1

βi1

n Juni

,

Hn z∈E:φz, ynφz, xn,

Wn z∈E:xn−z, Jx0−Jxn 0,

xn1 ΠHn∩Wnx0, n0,1,2, . . . ,

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where{αn},{βni} ⊂0,1andri>0for alli1,2, . . . , N, satisfy the following conditions:

Then the sequence{xn}converges strongly toΠΘx0, whereΠΘ is the generalized projection fromE ontoΘ.

Acknowledgment

The authors would like to thank The Thailand Research Fund for financial support.

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