Volume 2011, Article ID 572156,14pages doi:10.1155/2011/572156
Research Article
A New Strong Convergence Theorem for
Equilibrium Problems and Fixed Point Problems in
Banach Spaces
Weerayuth Nilsrakoo
Department of Mathematics, Statistics and Computer, Faculty of Science, Ubon Ratchathani University, Ubon Ratchathani 34190, Thailand
Correspondence should be addressed to Weerayuth Nilsrakoo,[email protected]
Received 5 June 2010; Revised 28 December 2010; Accepted 20 January 2011
Academic Editor: Fabio Zanolin
Copyrightq2011 Weerayuth Nilsrakoo. 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 a new iterative sequence for finding a common element of the set of fixed points of a relatively nonexpansive mapping and the set of solutions of an equilibrium problem in a Banach space. Then, we study the strong convergence of the sequences. With an appropriate setting, we obtain the corresponding results due to Takahashi-Takahashi and Takahashi-Zembayashi. Some of our results are established with weaker assumptions.
1. Introduction
Throughout this paper, we denote by andthe sets of positive integers and real numbers,
respectively. LetEbe a Banach space,E∗the dual space ofEandCa closed convex subsets ofE. LetF :C×C → be a bifunction. Theequilibrium problemis to findx∈Csuch that
Fx, y≥0, ∀y∈C. 1.1
The set of solutions of 1.1 is denoted by EPF. The equilibrium problems include fixed point problems, optimization problems, variational inequality problems, and Nash equilibrium problems as special cases.
LetEbe a smooth Banach space andJthe normalized duality mapping fromEtoE∗. Alber1considered the following functionalϕ:E×E → 0,∞defined by
Using this functional, Matsushita and Takahashi2,3studied and investigated the following mappings in Banach spaces. A mappingS:C → Eisrelatively nonexpansiveif the following properties are satisfied:
R1FS/,
R2ϕp, Sx≤ϕp, xfor allp∈FSandx∈C,
R3FS FS,
whereFSandFSdenote the set of fixed points ofSand the set of asymptotic fixed points ofS, respectively. It is known thatSsatisfies conditionR3if and only ifI−Sis demiclosed at zero, whereI is the identity mapping; that is, whenever a sequence{xn}inCconverges weakly topand{xn−Sxn}converges strongly to 0, it follows thatp∈FS. In a Hilbert space H, the duality mappingJ is an identity mapping andϕx, y x−y2 for allx, y ∈ H. Hence, ifS : C → His nonexpansivei.e.,Sx−Sy ≤ x−yfor allx, y ∈ C, then it is relatively nonexpansive.
Recently, many authors studied the problems of finding a common element of the set of fixed points for a mapping and the set of solutions of equilibrium problem in the setting of Hilbert space and uniformly smooth and uniformly convex Banach space, respectivelysee, e.g.,4–21and the references therein. In a Hilbert spaceH, S. Takahashi and W. Takahashi
17introduced the iteration as follows: sequence{xn}generated byu, x1∈C,
Fzn, yrn1y−zn, zn−xn≥0, ∀y∈C,
xn1βnxn
1−βnSαnu 1−αnzn,
1.3
for everyn∈ , whereSis nonexpansive,{αn}and{βn}are appropriate sequences in0,1, and{rn}is an appropriate positive real sequence. They proved that{xn}converges strongly to some element in FS∩EPF. In 2009, Takahashi and Zembayashi 19proposed the iteration in a uniformly smooth and uniformly convex Banach space as follows: a sequence
{xn}generated byu1∈E,
xn∈Csuch thatFxn, y rn1 y−xn, Jxn−Jun≥0, ∀y∈C,
un1J−1αnJxn 1−αnJSxn,
1.4
for everyn∈ ,Sis relatively nonexpansive,{αn}is an appropriate sequence in0,1, and
{rn} is an appropriate positive real sequence. They proved that ifJ is weakly sequentially continuous, then{xn}convergesweaklyto some element inFS∩EPF.
2. Preliminaries
We collect together some definitions and preliminaries which are needed in this paper. We say that a Banach spaceEisstrictly convexif the following implication holds forx, y∈E:
xy1, x /y implyxy 2
<1. 2.1
It is also said to beuniformly convexif for anyε >0, there existsδ >0 such that
xy1, x−y≥εimplyxy 2
≤1−δ. 2.2
It is known that ifE is a uniformly convex Banach space, then E is reflexive and strictly convex. We say thatE isuniformly smoothif the dual spaceE∗ ofEis uniformly convex. A Banach spaceEissmoothif the limit limt→0xty−x/texists for all norm one elements
xandyinE. It is not hard to show that ifEis reflexive, thenEis smooth if and only ifE∗is strictly convex.
LetEbe a smooth Banach space. The functionϕ:E×E → see1is defined by
ϕx, yx2−2x, Jyy2 x, y∈E, 2.3
where theduality mappingJ:E → E∗is given by
x, Jxx2Jx2 x∈E. 2.4
It is obvious from the definition of the functionϕthat
x −y2≤ϕx, y≤xy2, 2.5
ϕx, J−1λJy 1−λJz ≤λϕx, y 1−λϕx, z, 2.6
for allλ ∈0,1andx, y, z∈E. The following lemma is an analogue of Xu’s inequality22, Theorem 2with respect toϕ.
Lemma 2.1. LetEbe a uniformly smooth Banach space andr > 0. Then, there exists a continuous, strictly increasing, and convex functiong :0,2r → 0,∞such thatg0 0and
ϕx, J−1λJy 1−λJz ≤λϕx, y 1−λϕx, z−λ1−λgJy−Jz, 2.7
for allλ∈0,1,x∈E, andy, z∈Br.
Lemma 2.2see23, Proposition 2. LetEbe a uniformly convex and smooth Banach space, and let{xn}and{yn}be two sequences ofEsuch that{xn}or{yn}is bounded. Ifϕxn, yn → 0, then xn−yn → 0.
Remark 2.3. For any bounded sequences{xn}and{yn}in a uniformly convex and uniformly smooth Banach spaceE, we have
ϕxn, yn−→0 ⇐⇒ xn−yn−→0 ⇐⇒ Jxn−Jyn−→0. 2.8
LetCbe a nonempty closed convex subset of a reflexive, strictly convex, and smooth Banach spaceE. It is known that1,23for anyx∈E, there exists a unique pointx∈Csuch that
ϕx, x min y∈C ϕ
y, x. 2.9
Following Alber1, we denote such an elementxbyΠCx. The mappingΠC is called the
generalized projectionfromEontoC. It is easy to see that in a Hilbert space, the mappingΠC coincides with the metric projectionPC. Concerning the generalized projection, the following are well known.
Lemma 2.4 see 23, Propositions 4 and 5. Let C be a nonempty closed convex subset of a reflexive, strictly convex and smooth Banach spaceE,x∈E, andx∈C. Then,
ax ΠCxif and only ify−x, Jx −Jx ≤0for ally∈C, bϕy,ΠCx ϕΠCx, x≤ϕy, xfor ally∈C.
Remark 2.5. The generalized projection mapping ΠC above is relatively nonexpansive and
FΠC C.
LetEbe a reflexive, strictly convex and smooth Banach space. The duality mapping J∗fromE∗ontoE∗∗ Ecoincides with the inverse of the duality mappingJfromEontoE∗, that is,J∗J−1. We make use of the following mappingV :E×E∗ →
studied in Alber1
Vx, x∗ x2−2x, x∗x∗2, 2.10
for allx ∈Eandx∗ ∈ E∗. Obviously,Vx, x∗ ϕx, J−1x∗for allx ∈Eandx∗ ∈E∗. We
know the following lemmasee1and24, Lemma 3.2.
Lemma 2.6. LetEbe a reflexive, strictly convex and smooth Banach space, and letV be as in2.10. Then,
Vx, x∗ 2J−1x∗−x, y∗≤Vx, x∗y∗, 2.11
Lemma 2.7see25,Lemma 2.1. Let{an}be a sequence of nonnegative real numbers. Suppose that
an1≤
1−γnanγnδn, 2.12
for alln ∈ , where the sequences{γn}in 0,1and{δn}in satisfy conditions:limn→ ∞γn 0,
∞
n1γn∞, andlim supn→ ∞δn≤0. Then,limn→ ∞an0.
Lemma 2.8see26, Lemma 3.1. Let{an}be a sequence of real numbers such that there exists
a subsequence{ni} of{n}such that ani < ani1 for alli ∈ . Then, there exists a nondecreasing sequence{mk} ⊂ such thatmk → ∞,
amk ≤amk1, ak ≤amk1, 2.13
for allk∈ . In fact,mkmax{j ≤k:aj< aj1}.
For solving the equilibrium problem, we usually assume that a bifunctionF:C×C →
satisfies the following conditions:
A1Fx, x 0 for allx∈C,
A2Fis monotone, that is,Fx, y Fy, x≤0,for allx, y∈C,
A3for allx, y, z∈C, lim supt→0Ftz 1−tx, y≤Fx, y,
A4for allx∈C,Fx,·is convex and lower semicontinuous.
The following lemma gives a characterization of a solution of an equilibrium problem.
Lemma 2.9see19,Lemma 2.8. LetCbe a nonempty closed convex subset of a reflexive, strictly convex, and uniformly smooth Banach spaceE. Let F : C×C → be a bifunction satisfying
conditionsA1–A4. Forr > 0, define a mappingTr : E → C so-called the resolvent ofF as
follows:
Trx
z∈C:Fz, y 1 r
y−z, Jz−Jx≥0 ∀y∈C
, 2.14
for allx∈E. Then, the following hold: iTr is single-valued,
iiTr is a firmly nonexpansive-type mapping27, that is, for allx, y∈E
Trx−Try, JTrx−JTry≤Trx−Try, Jx−Jy, 2.15
iiiFTr EPF,
ivEPFis closed and convex,
Lemma 2.10see4, Lemma 2.3. LetCbe a nonempty closed convex subset of a Banach spaceE,
Fa bifunction fromC×C → satisfying conditionsA1–A4andz∈C. Then,z∈EPFif and
Remark 2.11see27. LetCbe a nonempty subset of a smooth Banach spaceE. IfS:C → E is a firmly nonexpansive-type mapping, then
ϕz, Sx≤ϕz, Sx ϕSx, x≤ϕz, x, 2.16
for allx∈Candz∈FS. In particular,Ssatisfies conditionR2.
Lemma 2.12see3, Proposition 2.4. LetC be a nonempty closed convex subset of a strictly convex and smooth Banach spaceEandS:C → Ea relatively nonexpansive mapping. Then,FS is closed and convex.
3. Main Results
In this section, we prove a strong convergence theorem for finding a common element of the fixed points set of a relatively nonexpansive mapping and the set of solutions of an equilibrium problem in a uniformly convex and uniformly smooth Banach space.
Theorem 3.1. LetCbe a nonempty closed convex subset of a uniformly convex and uniformly smooth Banach spaceEandF :C×C → a bifunction satisfying conditionsA1–A4andS:C → E
a relatively nonexpansive mapping such that FS∩EPF/. Let {un} and {xn} be sequences
generated byu∈C,u1∈Eand
Fxn, yr1 n
y−xn, Jxn−Jun≥0, ∀y∈C,
yn ΠCJ−1αnJu 1−αnJxn,
un1J−1
βnJxn1−βnJSyn,
3.1
for alln∈ , where{αn} ⊂0,1satisfyinglimn→ ∞αn0and∞n1αn∞,{βn} ⊂a, b⊂0,1, and{rn} ⊂c,∞⊂0,∞. Then,{un}and{xn}converge strongly toΠFS∩EPFu.
Proof. Note thatxn can be rewritten asxn Trnun. SinceFS∩EPFis nonempty, closed,
and convex, we putu ΠFS∩EPFu. SinceΠC,Trn, andSsatisfy conditionR2, by2.6, we
get
ϕu, y n≤ϕ
u, J−1α
nJu 1−αnJxn
≤αnϕu, u 1−αnϕu, x n
≤αnϕu, u 1−αnϕu, un ,
and so
ϕu, un 1≤βnϕu, xn
1−βnϕu, Syn
≤βnϕu, u n 1−βnϕu, y n
≤αn1−βnϕu, u 1−αn1−βnϕu, un
≤maxϕu, u , ϕu, u n.
3.3
By induction, we have
ϕz, un1≤max
ϕu, u , ϕu, u 1
, 3.4
for alln∈ . This implies that{un}is bounded and so are{xn},{yn}, and{Syn}. Put
zn ≡J−1αnJu 1−αnJxn. 3.5
Then,yn≡ΠCzn. UsingLemma 2.6gives
ϕu, yn ≤ϕu,zn Vu,Jzn
≤Vu, Jz n−αnJu−Ju−2zn−u, −αnJu−Ju
ϕu, J −1αnJu 1−αnJxn 2αnzn−u, Ju −Ju
≤αnϕu, u 1−αnϕu, xn 2αnzn−u, Ju −Ju
≤1−αnϕu, u n 2αnzn−u, Ju −Ju.
3.6
Let g : 0,2r → 0,∞ be a function satisfying the properties ofLemma 2.1, wherer sup{xn,Syn:n∈ }. Then, byRemark 2.11and3.6, we get
ϕu, u n1≤βnϕu, x n 1−βnϕu, Sy n−βn1−βngJxn−JSyn
≤βnϕu, un −ϕxn, un1−βnϕu, yn
−βn1−βngJxn−JSyn
≤βnϕu, un 1−βn1−αnϕu, un 2αnzn−u, Ju −Ju
−βnϕxn, un−βn1−βngJxn−JSyn
1−γnϕu, u n 2γnzn−u, Ju −Ju
3.7
−βnϕxn, un−βn1−βngJxn−JSyn
≤1−γnϕu, un 2γnzn−u, Ju −Ju, 3.8
whereγn αn1−βnfor alln ∈ . Notice that{γn} ⊂ 0,1satisfying limn→ ∞γn 0 and
∞
The rest of the proof will be divided into two parts.
Case 1. Suppose that there existsn0 ∈ such that {ϕu, u n}∞nn0 is nonincreasing. In this
situation,{ϕu, un }is then convergent. Then,
ϕu, un −ϕu, un 1−→0. 3.9
It follows from3.7andγn → 0 that
βnϕxn, un βn1−βngJxn−JSyn−→0. 3.10
Since{βn} ⊂a, b⊂0,1,
ϕxn, un−→0, gJxn−JSyn−→0. 3.11
Consequently, byRemark 2.3,
xn−un−→0, Jxn−JSyn−→0, xn−Syn −→0. 3.12
From2.6andαn → 0, we obtain
ϕxn, yn≤ϕxn, zn≤αnϕxn, u 1−αnϕxn, xn αnϕxn, u−→0. 3.13
This implies that
xn−yn−→0, zn−yn−→0. 3.14
Therefore,
yn−Syn−→0. 3.15
Since{yn}is bounded andEis reflexive, we choose a subsequence{yni}of{yn}such that
yni zand
lim sup n→ ∞
yn−u, Ju −Juilim → ∞
yni−u, Ju −Ju
. 3.16
Then,xni z. Sincexn−un → 0 andrn≥c >0, byRemark 2.3,
lim n→ ∞
1
rnJxn−Jun0. 3.17
Notice that
Replacingnbyni, we have fromA2that
1 rni
y−xni, Jxni−Juni
≥ −Fxni, y
≥Fy, xni
, ∀y∈C. 3.19
Lettingi → ∞, we have from3.17andA4that
Fy, z≤0, ∀y∈C. 3.20
FromLemma 2.10, we havez ∈EPF. SinceSsatisfies conditionR3and3.15,z∈FS. It follows thatz∈FS∩EPF. ByLemma 2.4a, we immediately obtain that
lim sup
n→ ∞ yn−u, Ju −Juz−u, Ju −Ju ≤0. 3.21
Sincezn−yn → 0,
lim sup
n→ ∞ zn−u, Ju −Ju ≤0. 3.22
It follows fromLemma 2.7and3.8thatϕu, un → 0. Then,un → uand soxn → u.
Case 2. Suppose that there exists a subsequence{ni}of{n}such that
ϕu, un i< ϕu, un i1, 3.23
for alli∈ . Then, byLemma 2.8, there exists a nondecreasing sequence{mk} ⊂ such that mk → ∞,
ϕu, u mk≤ϕu, u mk1, ϕu, u k≤ϕu, u mk1 3.24
for allk∈ . From3.7andγn → 0, we have
βmkϕxmk, umk βmk
1−βmk
gJxmk −JSymk
≤ϕu, um k−ϕu, um k1
−γmkϕu, um k 2γmkzmk−u, Ju −Ju
≤ −γmkϕu, u mk 2γmkzmk−u, Ju −Ju −→0.
3.25
Using the same proof of Case1, we also obtain
lim sup k→ ∞
zmk−u, Ju −Ju ≤0. 3.26
From3.8, we have
ϕu, u mk1≤
1−γmk
Sinceϕu, um k≤ϕu, um k1, we have
γmkϕu, um k≤ϕu, um k−ϕu, um k1 2γmkzmk−u, Ju −Ju
≤2γmk
ymk−u, Ju −Ju
. 3.28
In particular, sinceγmk >0, we get
ϕu, um k≤2zmk−u, Ju −Ju. 3.29
It follows from3.26thatϕu, um k → 0. This together with3.27gives
ϕu, um k1−→0. 3.30
Butϕu, uk ≤ϕu, um k1for allk∈ , we conclude thatuk → u, andxk → u.
From two cases, we can conclude that{un}and{xn}converge strongly touand the proof is finished.
ApplyingTheorem 3.1and28, Theorem 3.2, we have the following result.
Theorem 3.2. LetCbe a nonempty closed convex subset of a uniformly convex and uniformly smooth Banach spaceE,F :C×C → a bifunction satisfying conditions (A1)–(A4), and{Ti:C → E}
∞ i1 a sequence of relatively nonexpansive mappings such that∞i1FTi∩EPF/. Let{un}and{xn}
be sequences generated by3.1, whereS:C → Eis defined by
SxJ−1
∞
i1
αiJTix
for eachx∈C. 3.31
Then,{un}and{xn}converge strongly toΠ∞
i1FTi∩EPFu.
SettingF≡0 andrn≡1 inTheorem 3.1, we have the following result.
Corollary 3.3. LetCbe a nonempty closed convex subset of a uniformly convex and uniformly smooth Banach spaceEandS:C → Ea relatively nonexpansive mapping. Let{un}and{xn}be sequences
generated byu∈C,u1∈Eand
xn ΠCun,
yn ΠCJ−1αnJu 1−αnJxn,
un1J−1
βnJxn1−βnJSyn,
3.32
LettingS:C → CinCorollary 3.3, we have the following result.
Corollary 3.4. LetCbe a nonempty closed convex subset of a uniformly convex and uniformly smooth Banach spaceEandS : C → Ca relatively nonexpansive mapping. Let{xn}be a sequence inC defined byu∈C, x1 ∈Cand
yn ΠCJ−1αnJu 1−αnJxn,
xn1J−1
βnJxn1−βnJSyn,
3.33
for alln∈ , where{αn} ⊂0,1satisfyinglimn→ ∞αn0and∞n1αn∞,{βn} ⊂a, b⊂0,1. Then{xn}converges strongly toΠFSu.
LetSbe the identity mapping inTheorem 3.1, we also have the following result.
Corollary 3.5. LetCbe a nonempty closed convex subset of a uniformly convex and uniformly smooth Banach spaceEandF:C×C → a bifunction satisfying conditions (A1)–(A4) such thatEPF/.
Let{un}and{xn}be sequences generated byu∈C, u1 ∈Eand
Fxn, yr1 n
y−xn, Jxn−Jun≥0, ∀y∈C,
yn ΠCJ−1αnJu 1−αnJxn,
un1J−1
βnJxn1−βnJyn,
3.34
for alln∈ , where{αn} ⊂0,1satisfyinglimn→ ∞αn0and∞n1αn∞,{βn} ⊂a, b⊂0,1, and{rn} ⊂c,∞⊂0,∞. Then,{un}and{xn}converge strongly toΠEPFu.
4. Deduced Theorems in Hilbert Spaces
In Hilbert spaces, every nonexpansive mappings are relatively nonexpansive, and J is the identity operator. We obtain the following result.
Theorem 4.1. LetCbe a nonempty closed convex subset of a Hilbert space H,F : C×C →
a bifunction satisfying conditions (A1)–(A4), andS : C → H a nonexpansive mapping such that
FS∩EPF/. Let{xn}be a sequence inCdefined byu∈C, x1∈Hand
xn1βnTrnxn
1−βnSαnu 1−αnTrnxn, 4.1
for all n ∈ , where Trn is the resolvent of F, {αn} ⊂ 0,1 satisfying limn→ ∞αn 0 and
∞
n1αn ∞,{βn} ⊂a, b⊂0,1, and{rn} ⊂c,∞⊂0,∞. Then,{xn}converges strongly to
PFS∩EPFu.
Remark 4.2. InTheorem 4.1, we have the same conclusion if the mappingS:C → His only quasinonexpansivei.e.,FS/andp−Sx ≤ p−xfor allx∈Candp∈FSsuch that
LettingF≡0 inTheorem 4.1, we have the following result.
Corollary 4.3. LetCbe a nonempty closed convex subset of a Hilbert spaceHandS :C → Ha nonexpansive mapping such thatFS/. Let{xn}be a sequence inCdefined byu∈ C, x1 ∈H
and
xn1 βnPCxn
1−βnSαnu 1−αnPCxn, 4.2
for alln ∈ , where{αn} ⊂ 0,1satisfyinglimn→ ∞αn 0,∞n1αn ∞, and{βn} ⊂ a, b ⊂ 0,1. Then,{xn}converges strongly toPFSu.
LetSbe the identity mapping inTheorem 4.1, we have the following result.
Corollary 4.4. LetCbe a nonempty closed convex subset of a Hilbert spaceHandF:C×C → a
bifunction satisfying conditions (A1)–(A4). Let{xn}be a sequence inHdefined byu, x1∈Hand
xn1γnu1−γnTrnxn, 4.3
for alln∈ , whereTrn is the resolvent ofF,{γn} ⊂ 0,1satisfyinglimn→ ∞γn 0,
∞
n1γn ∞, and{rn} ⊂c,∞⊂0,∞. Then{xn}converges strongly toΠEPFu.
Proof. We may assume without loss of generality thatγn<1/2 for alln∈ . Settingαn 2γn andβn1/2 for alln∈ , we get
xn1
1 2Trnxn
1
2Iαnu 1−αnTrnxn, 4.4
limn→ ∞αn0, and∞n1αn∞. ApplyingTheorem 4.1,{xn}converges strongly toPEPFu.
Remark 4.5. Corollary 4.4 improves and extends 29, Corollary 5.3. More precisely, the conditions limn→ ∞γn1/γn 1 and∞n1|rn1−rn|<∞are removed.
ApplyingCorollary 4.4and30, Theorem 8, we have the following result.
Corollary 4.6. LetCbe a nonempty closed convex subset of a Hilbert spaceH,F : C×C → a
bifunction satisfying conditions (A1)–(A4), andf :C → Ca contraction ofHinto itself. Let{xn} be a sequence inHdefined byu, x1∈Hand
xn1γnfxn
1−γnTrnxn, 4.5
for alln∈ , whereTrnis the resolvent ofF,{γn} ⊂0,1satisfyinglimn→ ∞γn0and
∞
n1γn∞ and{rn} ⊂c,∞⊂0,∞. Then,{xn}converges strongly tozPEPFfz.
Acknowledgment
The author would like to thank the referees for their comments and helpful suggestions.
References
1 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.
2 S.-Y. Matsushita and W. Takahashi, “Weak and strong convergence theorems for relatively nonexpansive mappings in Banach spaces,”Fixed Point Theory and Applications, no. 1, pp. 37–47, 2004.
3 S.-Y. 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.
4 D. Boonchari and S. Saejung, “Approximation of common fixed points of a countable family of relatively nonexpansive mappings,”Fixed Point Theory and Applications, vol. 2010, Article ID 407651, 26 pages, 2010.
5 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.
6 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.
7 A. Kangtunyakarn and S. Suantai, “A new mapping for finding common solutions of equilibrium problems and fixed point problems of finite family of nonexpansive mappings,”Nonlinear Analysis. Theory, Methods & Applications, vol. 71, no. 10, pp. 4448–4460, 2009.
8 W. Nilsrakoo and S. Saejung, “Weak and strong convergence theorems for countable Lipschitzian mappings and its applications,”Nonlinear Analysis. Theory, Methods & Applications, vol. 69, no. 8, pp. 2695–2708, 2008.
9 W. Nilsrakoo and S. Saejung, “Weak convergence theorems for a countable family of Lipschitzian mappings,”Journal of Computational and Applied Mathematics, vol. 230, no. 2, pp. 451–462, 2009.
10 W. Nilsrakoo and S. Saejung, “Strong convergence theorems for a countable family of quasi-Lipschitzian mappings and its applications,”Journal of Mathematical Analysis and Applications, vol. 356, no. 1, pp. 154–167, 2009.
11 W. Nilsrakoo and S. Saejung, “Equilibrium problems and Moudafi’s viscosity approximation methods in Hilbert spaces,” Dynamics of Continuous, Discrete & Impulsive Systems. Series A. Mathematical Analysis, vol. 17, no. 2, pp. 195–213, 2010.
12 S. Plubtieng and W. Sriprad, “Hybrid methods for equilibrium problems and fixed points problems of a countable family of relatively nonexpansive mappings in Banach spaces,”Fixed Point Theory and Applications, vol. 2010, Article ID 962628, 17 pages, 2010.
13 S. Plubtieng and W. Sriprad, “A viscosity approximation method for finding common solutions of variational inclusions, equilibrium problems, and fixed point problems in Hilbert spaces,”Fixed Point Theory and Applications, vol. 2009, Article ID 567147, 20 pages, 2009.
14 X. Qin, Y. J. Cho, and S. M. Kang, “Convergence theorems of common elements for equilibrium problems and fixed point problems in Banach spaces,” Journal of Computational and Applied Mathematics, vol. 225, no. 1, pp. 20–30, 2009.
15 A. Tada and W. Takahashi, “Weak and strong convergence theorems for a nonexpansive mapping and an equilibrium problem,”Journal of Optimization Theory and Applications, vol. 133, no. 3, pp. 359–370, 2007.
16 S. Takahashi and W. Takahashi, “Viscosity approximation methods for equilibrium problems and fixed point problems in Hilbert spaces,”Journal of Mathematical Analysis and Applications, vol. 331, no. 1, pp. 506–515, 2007.
17 S. Takahashi and W. Takahashi, “Strong convergence theorem for a generalized equilibrium problem and a nonexpansive mapping in a Hilbert space,”Nonlinear Analysis. Theory, Methods & Applications, vol. 69, no. 3, pp. 1025–1033, 2008.
19 W. Takahashi and Kei 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.
20 K. Wattanawitoon and P. Kumam, “Strong convergence theorems by a new hybrid projection algorithm for fixed point problems and equilibrium problems of two relatively quasi-nonexpansive mappings,”Nonlinear Analysis. Hybrid Systems, vol. 3, no. 1, pp. 11–20, 2009.
21 Y. Yao, Y.-C. Liou, and J.-C. Yao, “Convergence theorem for equilibrium problems and fixed point problems of infinite family of nonexpansive mappings,”Fixed Point Theory and Applications, vol. 2007, Article ID 64363, 12 pages, 2007.
22 H. K. Xu, “Inequalities in Banach spaces with applications,”Nonlinear Analysis. Theory, Methods & Applications, vol. 16, no. 12, pp. 1127–1138, 1991.
23 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.
24 F. Kohsaka and W. Takahashi, “Strong convergence of an iterative sequence for maximal monotone operators in a Banach space,”Abstract and Applied Analysis, no. 3, pp. 239–249, 2004.
25 H.-K. Xu, “Another control condition in an iterative method for nonexpansive mappings,”Bulletin of the Australian Mathematical Society, vol. 65, no. 1, pp. 109–113, 2002.
26 P.-E. Maing´e, “Strong convergence of projected subgradient methods for nonsmooth and nonstrictly convex minimization,”Set-Valued Analysis, vol. 16, no. 7-8, pp. 899–912, 2008.
27 F. Kohsaka and W. Takahashi, “Existence and approximation of fixed points of firmly nonexpansive-type mappings in Banach spaces,”SIAM Journal on Optimization, vol. 19, no. 2, pp. 824–835, 2008.
28 W. Nilsrakoo and S. Saejung, “On the fixed-point set of a family of relatively nonexpansive and generalized nonexpansive mappings,”Fixed Point Theory and Applications, vol. 2010, Article ID 414232, 14 pages, 2010.
29 Y. Song and Y. Zheng, “Strong convergence of iteration algorithms for a countable family of nonexpansive mappings,”Nonlinear Analysis. Theory, Methods & Applications, vol. 71, no. 7-8, pp. 3072– 3082, 2009.
