Hybrid Methods for Equilibrium Problems and Fixed Points Problems of a Countable Family of Relatively Nonexpansive Mappings in Banach Spaces

(1)

Volume 2010, Article ID 962628,17pages doi:10.1155/2010/962628

Research Article

Hybrid Methods for Equilibrium Problems

and Fixed Points Problems of a Countable Family

of Relatively Nonexpansive Mappings

in Banach Spaces

Somyot Plubtieng and Wanna Sriprad

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

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

Received 1 August 2009; Accepted 19 November 2009

Academic Editor: Tomonari Suzuki

Copyrightq2010 S. Plubtieng and W. Sriprad. 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.

The purpose of this paper is to introduce hybrid projection algorithms for finding a common element of the set of common fixed points of a countable family of relatively nonexpansive mappings and the set of solutions of an equilibrium problem in the framework of Banach spaces. Moreover, we apply our result to the problem of finding a common element of an equilibrium problem and the problem of finding a zero of a maximal monotone operator. Our result improve and extend the corresponding results announced by Takahashi and Zembayashi2008 and 2009, and many others.

1. Introduction

LetEbe a real Banach space andE∗the dual space ofE. LetCbe a nonempty closed convex subset ofEandfa bifunction fromC×CtoR, whereRdenotes the set of real numbers. The equilibrium problem is to findp∈Csuch that

fp, y≥0, ∀y∈C. 1.1

The set of solutions of1.1is denoted by EPf. Given a mappingT :C → E∗, letfx, y

Tx, y−xfor allx, y∈C. Then, p∈EPfif and only ifTp, y−p ≥0 for ally∈C, that is,

(2)

equilibrium problem; see, for instance, Blum and Oettli 1, Combettes and Hirstoaga 2,

and Moudafi3.

Recall that a mappingS:C → Cis said to be nonexpansive if

_Sx₋_Sy_≤_x₋_y_, _∀_{x, y}_∈_C. _1.2

We denote by FS the set of fixed points of S. If a Banach spaceE is uniformly convex,

C⊂Eis bounded, closed and convex, andSis a nonexpansive mapping ofCinto itself, then

FSis nonempty; see4for more details. Recently, many authors studied the problem of finding a common element of the set of fixed points of a nonexpansive mapping and the set of solutions of an equilibrium problem in the framework of Hilbert spaces and Banach spaces, respectively; see, for instance,5–13and the references therein.

A popular method is the hybrid projection method developed by Nakajo and Takahashi14, Kamimura and Takahashi 15, and Martinez-Yanes and Xu 16; see also

5, 17–20 and references therein. Recently Takahashi et al. 21 introduced an alterative projection method, which is called the shrinking projection method, and they showed several strong convergence theorems for a family of nonexpansive mappings. In 2008, Takahashi and Zembayashi12introduced two iterative sequences for finding a common element of the set of fixed points of a relatively nonexpansive mapping and the set of solution of an equilibrium problem in a Banach space. Then they prove strong and weak convergence of the sequences. Very recently, Takahashi and Zembayashi13proved a strong convergence theorem for finding a common element of the set of solutions of an equilibrium problem and the set of fixed points of a relatively nonexpansive mapping in a Banach space by using a new hybrid method.

On the other hand, motivated by Nakajo and Takahashi 14, Matsushita and

Takahashi 17 reformulated the definition of the notion and obtained weak and strong convergence theorems to approximate a fixed point of a single relatively nonexpansive mapping. Very recently, Aoyama et al. 22 introduce a Halpern type iterative sequence for finding a common fixed point of a countable family of nonexpansive mappings. Let

x1x∈Cand

xn1αnx 1−αnTnxn 1.3

for all n ∈ N,where C is a nonempty closed convex subset of a Banach space; {αn} is a sequence in0,1,and{Tn}is a sequence of nonexpansive mappings with some condition. They proved that {xn} defined by 1.3 converges strongly to a common fixed point of {Tn}.

(3)

2. Preliminaries

LetEbe a real Banach space with norm · and let E∗be the dual ofE. For allx ∈ Eand

x∗_∈_E∗_{, we denote the value of}_x∗_at_x_by_{x, x}∗_{. The normalized duality mapping}_J_from_E

toE∗is defined by

Jxx∗_∈_E∗_:_{x, x}∗_x2_x∗2_, _2.1

forx∈ E. By Hahn-Banach theorem,Jxis nonempty; see4for more details. We denote the strong convergence and the weak convergence of a sequence{xn}toxinEbyxn → x andxn x, respectively. We also denote the weak∗convergence of a sequence{xn∗}tox∗in

E∗_by_x∗

n∗x∗.A Banach spaceEis said to be strictly convex ifxy/2<1 for allx, y∈E withxy1 andx /y. It is also said to be uniformly convex if for eachε∈0,2, there existsδ > 0 such thatxy/2 ≤1−δforx, y ∈Ewithx y 1 andx−y ≥ε. A uniformly convex Banach space has the Kadec-Klee property, that is,xn xandxn → x implyxn → x. LetSE {x∈E:x1}be the unit sphere ofE. Then the Banach spaceE is said to be smooth provided that

lim t→0

_x_ty₋_x

t 2.2

exists for eachx, y ∈ SE. It is also said to be uniformly smooth if the limit is attained uniformly forx, y ∈SE. It is well know that ifEis smooth, strictly convex and reflexive, then the duality mappingJis single valued, one-to-one and onto.

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

φy, xy2₋

2y, Jxx2 ∀x, y∈C. 2.3

It is obvious from the definition of the functionφthatx − y2φy, xy2x2, for allx, y∈E. Following Alber23, the generalized projectionΠCfromEontoCis defined by

ΠCx arg min y∈C φ

y, x, ∀x∈E. _2.4

IfEis a Hilbert space, thenφy, x y−x2andΠCis the metric projection ofHontoC. We know the following lemmas for generalized projections.

Lemma 2.1Alber 23, Kamimura and Takahashi15. LetC be a nonempty closed convex subset of a smooth, strictly convex and reflexive Banach spaceE. Then

(4)

Lemma 2.2Alber 23, Kamimura and Takahashi15. LetC be a nonempty closed convex subset of a smooth, strictly convex and reflexive Banach spaceE;letx∈E,and letz∈C.Then

z ΠCx⇐⇒y−z, Jx−Jz≤0, ∀y∈C. 2.6

LetCbe a nonempty closed convex subset of a smooth, strictly convex and reflexive Banach spaceE; letTbe a mapping fromCinto itself. We denote byFTthe set of fixed point ofT. A pointp∈Cis said to be an asymptotic fixed point ofT if there exists{xn}inCwhich converges weakly top and limn→ ∞xn−Txn 0. The set of asymptotic fixed points ofT will be denoted byFT. Following Matsushita and Takahashi17, a mappingT fromCinto itself is said to be relatively nonexpansive ifFTis nonempty,φu, Tx≤φu, x, for allu∈

FT, x∈C,andFT FT.

The following lemma is according to Matsushita and Takahashi17.

Lemma 2.3 Matsushita and Takahashi 17. Let Cbe a nonempty closed convex subset of a smooth, strictly convex and reflexive Banach spaceE, and letT be a relatively nonexpansive mapping fromCinto itself. ThenFTis closed and convex.

We also know the following three lemmas.

Lemma 2.4 Kamimura and Takahashi15. LetEbe a uniformly convex and smooth Banach space, and let {xn}, {yn} be sequences in E such that either {xn} or {yn} is bounded. If limn→ ∞φxn, yn 0, thenlimn→ ∞xn−yn0.

Lemma 2.5Xu24, Z˘alinescu25,26. LetEbe a uniformly convex Banach space, and letr >0. Then there exists a strictly increasing, continuous, and convex functiong :0,2r → Rsuch that

g0 0and

tx 1−ty2≤tx2 1−ty2−t1−tgx−y, 2.7

for allx, y∈Br andt∈0,1, whereBr {z∈E:z ≤r}.

Lemma 2.6 Kamimura and Takahashi15. LetEbe a smooth and uniformly convex Banach space and letr > 0.Then there exists a strictly increasing, continuous, and convex function g : 0,2r → Rsuch thatg0 0and

gx−y≤φx, y 2.8

for allx, y∈Br.

For solving the equilibrium problem, let us assume that a bifunctionf satisfies the following conditions:

A1fx, x 0 for allx∈C;

(5)

A3for eachx, y, z∈C,

lim t→0f

tz 1−tx, y≤fx, y; _2.9

A4for eachx∈C, y→fx, yis convex and lower semicontinuous.

The following result is in Blum and Oettlli1.

Lemma 2.7Blum and Oettlli1. LetCbe a nonempty closed convex subset of a smooth, strictly convex and reflexive Banach spaceE, and letfbe a bifunction fromC×C → Rsatisfying (A1)–(A4), and letr >0andx∈E. Then, there existsz∈Csuch that

fz, y1_ry−z, Jz−Jx≥0, ∀y∈C. 2.10

We also know the following lemmas.

Lemma 2.8Takahashi and Zembayashi12. LetCbe a nonempty closed convex subset of a uniformly smooth, strictly convex and reflexive Banach spaceE, and letfbe a bifunction fromC×C →

RsatisfyingA1–A4. Forr >0andx∈E, define a mappingTr :E → Cas follows:

Trx

z∈C:fz, y1

r

y−z, Jz−Jx≥0,∀y∈C

2.11

for allx∈E, Then, the following holds:

1Tr is single-valued;

2Tr is a firmly nonexpansive-type mapping, that is, for allx, y∈E,

Trx−Try, JTrx−JTry ≤ Trx−Try, Jx−Jy; 2.12

3FTr FTr EPf;

4EPfis closed and convex.

Lemma 2.9Takahashi and Zembayashi12. LetCbe a nonempty closed convex subset of a smooth, strictly convex and reflexive Banach space E, and letf be a bifunction fromC×C → R satisfying (A1)–(A4). Then forr >0,x∈E,andq∈FTr,

(6)

3. Main Results

Let C be a nonempty closed convex subset of a Banach space E; let {Tn} be a family of mappings ofCinto itself withF : ∞_n₁FTn/∅ and ωwzndenotes the set of all weak subsequential limits of a bounded sequence {zn} in C. {Tn} is said to satisfy the NST∗ -condition27if for every bounded sequence{zn}inC,

lim

n→ ∞zn1−znnlim→ ∞zn−Tnzn0 impliesωwzn⊂F. 3.1

In this section, by using the NST∗-condition, we prove two strong convergence theorems for finding a common element of the set of solutions of an equilibrium problem and the set of fixed points of a countable family of relatively nonexpansive mappings in a Banach space.

Theorem 3.1. Let E be a uniformly convex and uniformly smooth Banach space, and let Cbe a nonempty closed convex subset ofE. Letfbe a bifunction fromC×CtoRsatisfying (A1)–(A4), and let {Sn}be a family of relatively nonexpansive mappings fromCinto itself such thatΩ: ∞n1FSn∩ EPf/∅. Let{xn}be a sequence generated byx0x∈C, C0 C,and

yn J−1αnJxn 1−αnJSnxn,

un∈Csuch thatfun, y_r1 n

y−un, Jun−Jyn≥0, ∀y∈C,

Cn1

z∈Cn :φz, unφz, xn,

xn1 ΠCn1x,

3.2

for everyn∈N∪ {0}, whereJis the duality mapping onE,{αn} ⊂0,1satisfieslim infn→ ∞αn1−

αn>0and{rn} ⊂a,∞for somea >0.Suppose that{Sn}satisfy the NST∗-condition. Then{xn} converges strongly toΠ_Ωx, whereΠ_Ωis the generalized projection ofEontoΩ : ∞_n₁FSn∩ EPf.

Proof. Putting un Trnyn for all n ∈ N, we have from Lemma 2.9that Trn are relatively

nonexpansive.

We first show thatCnis closed and convex. It is obvious thatCnis closed. Since

φz, un≤φz, xn⇐⇒ un2− xn2−2z, Jun−Jxn ≥0, 3.3

Cnis convex. So,Cnis a closed convex subset ofEfor alln∈N∪ {0}.

(7)

SinceTrk andSkare relatively nonexpansive, it follows that

φu, uk φu, Trkyk

≤φu, yk

φu, J−1_α

kJxk 1−αkJSkxk

u2₋₂_{u, α}

kJxk 1−αkJSkxkαkJxk 1−αkJSkxk2 ≤ u2₋₂_α

ku, Jxk −21−αku, JSkxkαkxk2 1−αkSkxk2 αkφu, xk 1−αkφu, Skxk≤φu, xk.

3.4

Henceu∈Ck1. This implies thatΩ⊂Cnfor alln∈N∪ {0}. So,{xn}is well defined. Next, we show that{xn}is bounded. From the definition ofxn, we have

φxn, x φΠCnx, x≤φu, x−φu,ΠCnx≤φu, x, 3.5

for all u ∈ Ω ⊂ Cn. Thus {φxn, x} is bounded and therefore {xn} and {Snxn} are also bounded.

Fromxn1∈Cn1⊂Cnandxn ΠCnx, we have

φxn, x≤φxn1, x, ∀n∈N∪ {0}. 3.6

This implies that{φxn, x}is nondecreasing and so limn→ ∞φxn, xexists. Since

φxn1, xn φxn1,ΠCnx≤φxn1, x−φΠCnx, x≤φxn1, x−φxn, x, 3.7

for alln∈N∪ {0}, it follows that limn→ ∞φxn1, xn 0.Fromxn1 ΠCn1x∈Cn1, we have

φxn1, un≤φxn1, xn, ∀n∈N∪ {0}. 3.8

Therefore, we also have

lim

n→ ∞φxn1, un 0. 3.9

Since limn→ ∞φxn1, xn limn→ ∞φxn1, un 0 andEis uniformly convex and smooth, it follows fromLemma 2.4that

lim

n→ ∞xn1−xnnlim→ ∞xn1−un0. 3.10

Thus, we have

lim

(8)

SinceJis uniformly norm-to-norm continuous on bounded sets, it follows by3.11that

lim

n→ ∞Jxn−Jun0. 3.12

Letrsup_n_∈_N{xn,Snxn}. SinceEis a uniformly smooth Banach space, we note thatE∗is a uniformly convex Banach space. Therefore, byLemma 2.5, there exists a continuous, strictly increasing, and convex functiongwithg0 0 such that

_αx∗ ₁₋_α_y∗2 _≤_α_x_∗2 ₁₋_α_y∗2₋_α₁₋_α_g_x_∗₋_y_∗ _3.13

forx∗, y∗∈B_r∗{z∈E∗:z ≤r}andα∈0,1.So, we have

φu, un φu, Trnyn

≤φu, ynφ

u, J−1_α

nJxn 1−αnJSnxn

u2₋₂_{u, α}

nJxn 1−αnJSnxnαnJxn 1−αnJSnxn2 ≤ u2₋₂_α

nu, Jxn −21−αnu, JSnxnαnxn2 1−αnSnxn2 −αn1−αngJxn−JSnxn

αnφu, xn 1−αnφu, Snxn−αn1−αngJxn−JSnxn ≤φu, xn−αn1−αngJxn−JSnxn

3.14

for allu∈Ω.Therefore, we have

αn1−αngJxn−JSnxn≤φu, xn−φu, un, ∀n∈N∪ {0}. 3.15

Since

φu, xn−φu, un xn2− un2−2u, Jxn−Jun

≤xn2− un22|u, Jxn−Jun|

≤ |xn − unxnun|2uJxn−Jun ≤ |xn−unxnun|2uJxn−Jun,

3.16

it follows that

lim n→ ∞

φu, xn−φu, un0. 3.17

From lim infn→ ∞αn1−αn>0,we have

lim

(9)

Therefore, we note from the property ofgthat

lim

n→ ∞Jxn−JSnxn0. 3.19

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

lim

n→ ∞xn−Snxnnlim→ ∞

J−1_Jx

n−J−1JSnxn0. 3.20

Since{Sn}satisfy the NST∗-condition, we haveωwxn⊂F:∞n1FSn. So, we assume that a subsequence{xnk}of{xn}converges weakly tox∈F.We shall show thatx∈EPf. From unTrnynandLemma 2.9, we have

φun, ynφTrnyn, yn

≤φu, yn−φu, Trnyn

≤φu, xn−φu, Trnyn

φu, xn−φu, un.

3.21

So, we note from3.17that

lim n→ ∞φ

un, yn0. 3.22

SinceEis uniformly convex and smooth and{un}is bounded, it follows fromLemma 2.4that

lim

n→ ∞un−yn0. 3.23

Fromxnk x,un−yn → 0 andxn−un → 0, we haveynk x.SinceJis uniformly

norm-to-norm continuous on bounded sets and3.23, it follows that

lim

n→ ∞Jun−Jyn0. 3.24

Fromrn≥a,we have

lim n→ ∞

Jun−Jyn

rn 0. 3.25

By the definition ofun Trnyn, we have

fun, y_r1 n

y−un, Jun−Jyn≥0, ∀y∈C. 3.26

Replacingnbynk, we have fromA2that

1

rnk

y−unk, Junk−Jynk

≥fy, unk

(10)

Sincefx,·is convex and lower semicontinuous, it is also weakly lower semicontinuous. Lettingk → ∞, we note from3.25andA4that

fy,x≤lim inf k→ ∞ f

y, unk

≤lim inf k→ ∞

y−unk,

Junk−Jynk rnk

0, ∀y∈C. 3.28

Fortwith 0< t≤1 andy∈C, letytty 1−tx.Sincey ∈Candx∈C, we haveyt∈C and hencefyt,x≤0.So, fromA1, we have

0fyt, yt≤tfyt, y 1−tfyt,x≤tfyt, y. 3.29

This implies that

fyt, y≥0, ∀y∈C. 3.30

Lettingt → 0fromA3, we have

fx, y≥0, ∀y∈C. 3.31

Therefore, we obtainx∈EPf.Finally, we will show thatxn → ΠΩx. Letw ΠΩx. From

xn ΠCnxandw∈Ω⊂Cn,we note that

φxn, x≤φw, x. 3.32

Since the norm is weakly lower semicontinuous, it follows that

φx, x x2₋₂_{x, Jx}_x2

≤lim inf k→ ∞

xnk2−2xnk, Jxx2

lim inf

k→ ∞ φxnk, x

≤lim sup k→ ∞

φxnk, x≤φw, x.

3.33

From the definition ofΠΩ, we havexw. Hence limk→ ∞φxnk, x φw, x.Therefore, we

obtain

0 lim k→ ∞

φxnk, x−φw, x

lim k→ ∞

xnk

2₋_w2₋₂_x

nk−w, Jx

lim k→ ∞

xnk2− w2

.

(11)

SinceEhas the Kadec-Klee property, it follows thatxnk → w ΠΩ.Therefore,{xn}converges

strongly toΠ_Ωx.

As direct consequences ofTheorem 3.1, we can obtain the following corollaries.

Corollary 3.2 Takahashi and Zembayashi 13. Let E be a uniformly convex and uniformly smooth Banach space, and letCbe a nonempty closed convex subset ofE. Letf be a bifunction from

C×CtoRsatisfying (A1)–(A4), and letSbe a relatively nonexpansive mapping fromCinto itself such thatΩ:FS∩EPf/∅. Let{xn}be a sequence generated byx0 x∈C, C0C,and

ynJ−1αnJxn 1−αnJSxn,

Cn1

xn1 ΠCn1x,

3.35

αn> 0,and{rn} ⊂a,∞for somea >0.Then{xn}converges strongly toΠΩ, whereΠΩ is the

generalized projection ofEontoΩ:FS∩EPf.

Proof. PutSn≡S. Let{zn}be a bounded sequence inCwith limn→ ∞zn1−znlimn→ ∞zn−

Szn0,and letz∈ωwzn. Then there exists subsequence{znk}of{zn}such thatznk z.

It follows directly from the definition ofS thatz ∈ FS FS. HenceS satisfies NST∗ -condition, byTheorem 3.1;{xn}converges strongly toΠΩ.

Corollary 3.3 Takahashi and Zembayashi 12. Let E be a uniformly convex and uniformly smooth Banach space; letCbe a nonempty closed convex subset ofE. Letfbe a bifunction fromC×C toRsatisfying (A1)–(A4). Let{xn}be a sequence generated byx0x∈C, C0C,and

y−un, Jun−Jxn≥0, ∀y∈C,

Cn1

xn1 ΠCn1x,

3.36

for everyn∈N∪ {0}, whereJis the duality mapping onEand{rn} ⊂a,∞for somea >0.Then, {xn}converges strongly toΠEPfx.

Proof. PuttingSn≡IinTheorem 3.1, we obtainCorollary 3.3.

(12)

Cinto itself such thatF:∞_n₁FSn/∅. Let{xn}be a sequence generated byx0x∈C, C0 C, and

yn ΠCJ−1αnJxn 1−αnJSnxn,

Cn1

z∈Cn:φz, ynφz, xn,

xn1 ΠCn1x,

3.37

αn>0. Suppose that{Sn}satisfy the NST∗-condition. Then{xn}converges strongly toΠF, where ΠF is the generalized projection ofEontoF:n∞1FSn.

Proof. Puttingfx, y 0 for allx, y∈Candrn 1 inTheorem 3.1, we obtainCorollary 3.4.

Similarly as in the proof ofTheorem 3.1, we can prove the following theorem.

Theorem 3.5. Let E be a uniformly convex and uniformly smooth Banach space, and let Cbe a nonempty closed convex subset ofE. Letfbe a bifunction fromC×CtoRsatisfying (A1)–(A4) and let {Sn}be a family of relatively nonexpansive mappings fromCinto itself such thatΩ: ∞n1FSn∩ EPf/∅. Let{xn}be a sequence generated byx0x∈Cand

yn J−1αnJxn 1−αnJSnxn,

Hnz∈C:φz, unφz, xn,

Wn{z∈C:xn−z, Jx−Jxn ≥0},

xn1 ΠHn∩Wnx,

3.38

αn>0and{rn} ⊂a,∞for somea >0.Suppose that{Sn}satisfy the NST∗-condition. Then{xn} converges strongly toΠΩx, whereΠΩis the generalized projection ofEontoΩ : ∞n1FSn∩ EPf.

Proof. We first show thatHn∩Wnis closed and convex. It is obvious thatHnis closed andWn is closed and convex. Sinceφz, un≤φz, xn⇔ un2−xn2−2z, Jun−Jxn ≥0,it follows thatHn is convex. HenceHn∩Wn is a closed and convex subset ofEfor alln ∈ N∪ {0}. Similarly as in proof ofTheorem 3.1, we note thatΩ⊂Hnfor alln∈N∪ {0}.Next, we show by induction thatΩ⊂Hn∩Wnfor alln∈N∪ {0}. FromW0C, we note thatΩ⊂H0∩W0. Suppose thatΩ⊂Hk∩Wkfor somek∈N∪ {0}.Then there existsxk1 ∈Hk∩Wksuch that

xk1 ΠHk∩Wkx.From the definition ofxk1, we have

(13)

for allz∈Hk∩Wk. SinceΩ⊂Hk∩Wk, we have

xk1−z, Jx−Jxk1 ≥0, ∀z∈Ω 3.40

and hencez ∈ Wk1. So, we haveΩ ⊂ Wk1 and therefore,Ω ⊂ Hk1∩Wk1.Hence Ω ⊂

Hn∩Wnfor alln∈N∪ {0}. This implies that{xn}is well defined. By the same argument as in proof ofTheorem 3.1, we can prove that the sequence{xn}converges strongly toΠΩx.

SettingSn≡SinTheorem 3.5, we have the following result.

Corollary 3.6Takahashi and Zembayashi12, Theorem 3.1. LetEbe a uniformly convex and uniformly smooth Banach space; letCbe a nonempty closed convex subset ofE. Letfbe a bifunction fromC×CtoRsatisfying (A1)–(A4), and letSbe a relatively nonexpansive mapping fromCinto itself such thatFS∩EPf/∅. Let{xn}be a sequence generated by

x0x∈C,

ynJ−1αnJxn 1−αnJSxn,

xn1 ΠHn∩Wnx,

3.41

for everyn∈N∪ {0}, whereJis the duality mapping onE,{αn} ∈0,1satisfieslim infn→ ∞αn1−

αn > 0and{rn} ⊂ a,∞for somea >0.Then, {xn}converges strongly toΠFS∩EPfx,where ΠFS∩EPfis the generalized projection ofEontoFS∩EPf.

4. Applications

In this section, we apply our result to the problem of finding a common element of an equilibrium problem and the problem of finding a zero of a maximal monotone operator in a Banach space by using the shrinking projection method.

LetEbe a real Banach space. An operatorT ⊂E×E∗is said to bemonotoneifx−y, x∗−

y∗_≥_{0 whenever}_{x, x}∗_,_{y, y}∗_∈_T_{. We denote the set}_{_x_∈_E_{: 0}_∈_Tx_}_by_T−1₀_._{A monotone}

T is said to bemaximalif its graphGT {x, y:y ∈Tx}is not properly contained in the graph of any other monotone operator. IfT is maximal monotone, then the solution setT−1₀ is closed and convex.

LetEbe a smooth, strictly convex and reflexive Banach space, and letT ⊂ E×E∗be a maximal monotone operator. Then for eachλ > 0 andx ∈E, there corresponds a unique elementxλ∈DTsatisfying

(14)

see Barbu28or Takahashi4. We define theresolventofT byJλx xλ. In other words,

Jλ JλT−1Jfor allλ >0. We know thatJλis relatively nonexpansive andT−10FJλfor allλ >0see4,17, whereFJλdenotes the set of all fixed points ofJλ. We can also define, for eachλ >0, theYosida approximationofTbyAλλ−1J−JJλ.We know thatJλx, Aλx∈T for allλ >0.

We now consider the strong convergence theorem for finding a common element of the solution set of an equilibrium problem and the problem of finding a zero of a maximal monotone operator.

Theorem 4.1. LetEbe a uniformly convex and uniformly smooth Banach space. Let T ⊂ E×E∗ be a maximal monotone operator, and let Jλ J λT−1J for all λ > 0. LetC be a nonempty closed convex subset ofEsuch thatDT ⊂ C ⊂ J−1

λ>0RJ λT.Letf be a bifunction from

C×CtoRsatisfying (A1)–(A4) withΩ : EPf∩T−1₀_/_∅_{. Let}_{_x

n}be a sequence generated by

x0x∈C, C0C,and

ynJ−1αnJxn 1−αnJJλnxn,

Cn1

xn1 ΠCn1x,

4.2

for everyn ∈ N∪ {0}, whereJ is the duality mapping onE,{αn} ⊂ 0,1,{λn} ∈ 0,∞satisfy lim infn→ ∞αn1−αn > 0,lim infn→ ∞λn > 0, and{rn} ⊂ a,∞ for somea > 0.Then {xn} converges strongly toΠΩx, whereΠΩis the generalized projection ofEontoΩ:EPf∩T−10.

Proof. Let {zn}be a bounded sequence in C such that limn→ ∞zn1 −zn limn→ ∞zn −

Jλnzn 0,and let z ∈ ωwzn. Then, there exists a subsequence{znk}of {zn}such that znk z. By the uniform smoothness ofE, we have

lim

n→ ∞Jzn−JJλnzn0. 4.3

Since lim infn→ ∞λn>0, we have

lim

n→ ∞Aλnzn_nlim_{→ ∞}

1

λnJzn−JJλnzn0. 4.4

Letu, u∗∈T. Then it holds from the monotonicity ofTthat

u−Jλ_nkznk, u∗−Aλ_nkznk

≥0, 4.5

for allk ∈ N. Lettingk → ∞, we getu−z, u∗ ≥ 0. Then, the maximality of T implies

z∈T−1_{0 :}∞

n1FJλn. Hence byTheorem 3.1,{xn}converges strongly toΠΩx.

(15)

Corollary 4.2. LetEbe a uniformly convex and uniformly smooth Banach space. LetT ⊂E×E∗be a maximal monotone operator, and letJλ JλT−1J for allλ > 0, withT−10/∅. Let{xn}be a sequence generated byx0x∈E, C0E,and

Cn1

z∈Cn:φz, ynφz, xn,

xn1 ΠCn1x,

4.6

for everyn ∈ N∪ {0}, whereJ is the duality mapping onE,{αn} ⊂ 0,1,{λn} ∈ 0,∞satisfy lim infn→ ∞αn1−αn>0,lim infn→ ∞λn>0. Then{xn}converges strongly toΠΩx, whereΠΩis

the generalized projection ofEontoΩ:T−1₀_.

Similarly as in the proof ofTheorem 4.1, we can prove the following theorem.

Theorem 4.3. LetEbe a uniformly convex and uniformly smooth Banach space. LetT ⊂E×E∗be a maximal monotone operator and letJλ JλT−1J for allλ > 0. LetCbe a nonempty closed convex subset ofEsuch thatDT⊂C⊂J−1

λ>0RJλT. Letfbe a bifunction fromC×Cto

Rsatisfying (A1)–(A4) withΩ:EPf∩T−1₀_/_∅_{. Let}_{_x

n}be a sequence generated byx0 x∈C and

xn1 ΠHn∩Wnx,

4.7

for everyn ∈ N∪ {0}, whereJ is the duality mapping onE,{αn} ⊂ 0,1,{λn} ∈ 0,∞satisfy lim infn→ ∞αn1−αn > 0,lim infn→ ∞λn > 0, and {rn} ⊂ a,∞for some a > 0.Then {xn} converges strongly toΠΩx, whereΠΩis the generalized projection ofEontoΩ:EPf∩T−10.

Corollary 4.4. LetEbe a uniformly convex and uniformly smooth Banach space. LetT ⊂E×E∗be a maximal monotone operator and letJλ JλT−1J for allλ > 0, withT−10/∅. Let{xn}be a sequence generated byx0x∈Eand

ynJ−1αnJxn 1−αnJJλnxn, Hnz∈C:φz, ynφz, xn,

xn1 ΠHn∩Wnx,

(16)

for everyn ∈ N∪ {0}, whereJ is the duality mapping onE,{αn} ⊂ 0,1,{λn} ∈ 0,∞satisfy lim infn→ ∞αn1−αn>0,lim infn→ ∞λn>0. Then{xn}converges strongly toΠΩx, whereΠΩis

the generalized projection ofEontoΩ:T−1₀_.

Acknowledgments

The first author thanks the National Research Council of Thailand to Naresuan University, 2009 for the financial support. Moreover, the second author would like to thank the National Centre of Excellence in Mathematics, PERDO, under the Commission on Higher Education, Ministry of Education, Thailand. This work is dedicated to Professor Wataru Takahashi on his retirement.

References

1 E. Blum and W. Oettli, “From optimization and variational inequalities to equilibrium problems,”The Mathematics Student, vol. 63, no. 1–4, pp. 123–145, 1994.

2 P. L. Combettes and S. A. Hirstoaga, “Equilibrium programming in Hilbert spaces,” Journal of Nonlinear and Convex Analysis, vol. 6, no. 1, pp. 117–136, 2005.

3 A. Moudafi, “Second-order diﬀerential proximal methods for equilibrium problems,” Journal of Inequalities in Pure and Applied Mathematics, vol. 4, no. 1, article 18, p. 7, 2003.

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

5 L.-C. Ceng and J.-C. Yao, “Hybrid viscosity approximation schemes for equilibrium problems and fixed point problems of infinitely many nonexpansive mappings,” Applied Mathematics and Computation, vol. 198, no. 2, pp. 729–741, 2008.

6 L.-C. Ceng and J.-C. Yao, “A hybrid iterative scheme for mixed equilibrium problems and fixed point problems,”Journal of Computational and Applied Mathematics, vol. 214, no. 1, pp. 186–201, 2008.

7 L.-C. Ceng, S. Schaible, and J.-C. Yao, “Implicit iteration scheme with perturbed mapping for equilibrium problems and fixed point problems of finitely many nonexpansive mappings,”Journal of Optimization Theory and Applications, vol. 139, no. 2, pp. 403–418, 2008.

8 L.-C. Ceng, A. Petrus¸el, and J.-C. Yao, “Iterative approaches to solving equilibrium problems and fixed point problems of infinitely many nonexpansive mappings,”Journal of Optimization Theory and Applications, vol. 143, no. 1, pp. 37–58, 2009.

9 J.-W. Peng and J.-C. Yao, “Strong convergence theorems of iterative scheme based on the extragradient method for mixed equilibrium problems and fixed point problems,” Mathematical and Computer Modelling, vol. 49, no. 9-10, pp. 1816–1828, 2009.

10 S. Plubtieng and R. Punpaeng, “A general iterative method for equilibrium problems and fixed point problems in Hilbert spaces,”Journal of Mathematical Analysis and Applications, vol. 336, no. 1, pp. 455– 469, 2007.

11 S. Plubtieng and R. Punpaeng, “A new iterative method for equilibrium problems and fixed point problems of nonexpansive mappings and monotone mappings,” Applied Mathematics and Computation, vol. 197, no. 2, pp. 548–558, 2008.

12 W. Takahashi and K. Zembayashi, “Strong and weak convergence theorems for equilibrium problems and relatively nonexpansive mappings in Banach spaces,” Nonlinear Analysis: Theory, Methods & Applications, vol. 70, no. 1, pp. 45–57, 2009.

13 W. Takahashi and K. Zembayashi, “Strong convergence theorem by a new hybrid method for equilibrium problems and relatively nonexpansive mappings,”Fixed Point Theory and Applications, vol. 2008, Article ID 52846, 11 pages, 2008.

14 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.

15 S. 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.

(17)

17 S. Matsushita and W. Takahashi, “A strong convergence theorem for relatively nonexpansive mappings in a Banach space,”Journal of Approximation Theory, vol. 134, no. 2, pp. 257–266, 2005.

18 J.-W. Peng and J.-C. Yao, “A modified CQ method for equilibrium problems, fixed points and variational inequality,”Fixed Point Theory, vol. 9, no. 2, pp. 515–531, 2008.

19 J.-W. Peng and J.-C. Yao, “A new hybrid-extragradient method for generalized mixed equilibrium problems, fixed point problems and variational inequality problems,”Taiwanese Journal of Mathemat-ics, vol. 12, no. 6, pp. 1401–1432, 2008.

20 S. Plubtieng and K. Ungchittrakool, “Strong convergence theorems for a common fixed point of two relatively nonexpansive mappings in a Banach space,”Journal of Approximation Theory, vol. 149, no. 2, pp. 103–115, 2007.

21 W. Takahashi, Y. Takeuchi, and R. Kubota, “Strong convergence theorems by hybrid methods for families of nonexpansive mappings in Hilbert spaces,” Journal of Mathematical Analysis and Applications, vol. 341, no. 1, pp. 276–286, 2008.

22 K. Aoyama, Y. Kimura, W. Takahashi, and M. Toyoda, “Approximation of common fixed points of a countable family of nonexpansive mappings in a Banach space,”Nonlinear Analysis: Theory, Methods & Applications, vol. 67, no. 8, pp. 2350–2360, 2007.

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

24 H. K. Xu, “Inequalities in Banach spaces with applications,”Nonlinear Analysis: Theory, Methods & Applications, vol. 16, no. 12, pp. 1127–1138, 1991.

25 C. Z˘alinescu, “On uniformly convex functions,”Journal of Mathematical Analysis and Applications, vol. 95, no. 2, pp. 344–374, 1983.

26 C. Z˘alinescu,Convex Analysis in General Vector Spaces, World Scientific, River Edge, NJ, USA, 2002.

27 K. Nakajo, K. Shimoji, and W. Takahashi, “On strong convergence by the hybrid method for families of mappings in Hilbert spaces,”Nonlinear Analysis: Theory, Methods & Applications, vol. 71, no. 1-2, pp. 112–119, 2009.