Fixed Point Theory and Applications Volume 2011, Article ID 208434,16pages doi:10.1155/2011/208434
Research Article
A Method for a Solution of Equilibrium
Problem and Fixed Point Problem of
a Nonexpansive Semigroup in Hilbert’s Spaces
Nguyen Buong
1and Nguyen Dinh Duong
21Vietnamese Academy of Science and Technology, Institute of Information Technology,
18 Hoang Quoc Viet Road, Cau Giay, Hanoi, Vietnam
2Department of Mathematics, Vietnam Maritime University, Hai Phong 35000, Vietnam Correspondence should be addressed to Nguyen Buong,[email protected]
Received 3 October 2010; Accepted 13 January 2011
Academic Editor: Ljubomir B. Ciric
Copyrightq2011 N. Buong and N. D. Duong. 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 viscosity approximation method for finding a common element of the set of solutions for an equilibrium problem involving a bifunction defined on a closed, convex subset and the set of fixed points for a nonexpansive semigroup on another one in Hilbert’s spaces.
1. Introduction
LetCbe a nonempty, closed, and convex subset of a real Hilbert spaceH. Denote the metric projection fromx∈HontoCbyPCx. LetT :C → Cbe a nonexpansive mapping onC, that
is,T :C → CandTx−Ty ≤ x−y, for allx, y∈C. We useFTto denote the set of fixed points ofT, that is,FT {x∈C:xTx}.
Let{Ts:s >0}be a nonexpansive semigroup on a closed convex subsetC, that is,
1for eachs >0,Tsis a nonexpansive mapping onC,
2T0xxfor allx∈C,
3Ts1s2 Ts1◦Ts2for alls1, s2>0,
4for eachx∈C, the mappingT·xfrom0,∞intoCis continuous.
Denote byFs>0FTs. We know1,2thatFis a closed, convex subset inHand
The equilibrium problem is for a bifunctionGu, vdefined onC×Cto findu∗ ∈ C
such that
Gu∗, v≥0, ∀v∈C. 1.1
Assume that the bifunctionGsatisfies the following set of standard properties:
A1Gu, u 0, for allu∈C,
A2Gu, v Gv, u≤0 for allu, v∈C×C,
A3for everyu ∈ C,Gu,· : C → −∞,∞is weakly lower semicontinuous and convex,
A4limt→0G1−tutz, v≤Gu, v, for allu, z, v∈C×C×C.
Denote the set of solutions of1.1by EPG. We also know3that EPGis a closed convex subset inH.
The problem studied in this paper is formulated as follows. LetC1 andC2 be closed
convex subsets in H. Let Gu, v be a bifunction satisfying conditions A1–A4with C
replaced byC1and let{Ts:s >0}be a nonexpansive semigroup onC2. Find an element
p∈EPG∩ F, 1.2
where EPG andF denote the set of solutions of an equilibrium problem involving by a bifunctionGu, vonC1×C1and the fixed point set of a nonexpansive semigroup{Ts:s >
0}on a closed convex subsetC2, respectively.
In the case thatC1≡H,Gu, v 0,C2C, andTs T, a nonexpansive mapping on
C, for alls >0,1.2is the fixed point problem of a nonexpansive mapping. In 2000, Moudafi
4proved the following strong convergence theorem.
Theorem 1.1. Let C be a nonempty, closed, convex subset of a Hilbert space H and let T be a nonexpansive mapping onCsuch thatFT/∅. Letfbe a contraction onCand let{xk}be a sequence
generated by:x1 ∈Cand
xk1 1εk
εkfxk
1
1εkTxk, k≥
1, 1.3
where{εk} ∈0,1satisfies
lim
k→ ∞εk0, ∞
k1
εk∞, lim k→ ∞
εk11 − 1
εk
0. 1.4
Such a method for approximation of fixed points is called the viscosity approximation method. It has been developed by Chen and Song5to findp∈ F, the set of fixed points for a semigroup{Ts:s >0}onC. They proposed the following algorithm:x1∈Cand
xk1μkfxk
1−μk
1
sk
sk
0
Tsxkds, k≥1, 1.5
where f : C → C, is a contraction, {μk} ⊂ 0,1and {sk} are sequences of positive real
numbers satisfying the conditions:μk → 0,∞k1μk∞, andsk → ∞ask → ∞.
Recently, Yao and Noor6proposed a new viscosity approximation method
xk1 μkfxk βkxkγkTskxk, k≥0, x0∈C, 1.6
where{μk},{βk}, and {γk} are in0,1,sk → ∞, for findingp ∈ F, when{Ts : s > 0}
satisfies the uniformly asymptotically regularity condition
lim
s→ ∞sup
x∈C
TtTsx−Tsx0, 1.7
uniformly int, andCis any bounded subset ofC. Further, Plubtieng and Pupaeng in7
studied the following algorithm:
xk1μkfxk βkxk
1−βk−μk sk
0
Tsxkds, k≥0, x0∈C, 1.8
where{μk}and{βk}are in0,1satisfying the following conditions:μkβk<1, limk→ ∞μk
limk→ ∞βk0,
k≥1μk∞, and{sk}is a positive divergent real sequence.
There were some methods proposed to solve equilibrium problem 1.1; see for instance 8–12. In particular, Combettes and Histoaga 3 proposed several methods for solving the equilibrium problem.
In 2007, S. Takahashi and W. Takahashi13combinated the Moudafi’s method with the Combettes and Histoaga’s result in3to find an elementp∈EPG∩FT. They proved the following strong convergence theorem.
Theorem 1.2. Let C be a nonempty, closed, convex subset of a Hilbert space H, let T be a nonexpansive mapping onC and letG be a bifunction from C×Cto−∞,∞ satisfying (A1)– (A4) such thatEPG∩FT/∅. Letf be a contraction onCand let{xk}and{uk}be sequences
generated by:x1 ∈Hand
Guk, y
1
rk
uk−xk, y−uk
≥0, ∀y∈C,
xk1μkfxk
1−μk
Tuk, k≥1,
where{μk} ∈0,1and{rk} ⊂0,∞satisfy
lim
k→ ∞μk0, ∞
k1
μk∞, lim inf
k→ ∞rk>0, ∞
k1
μk1−μk<∞, ∞
k1
|rk1−rk|<∞.
1.10
Then,{xk}and{uk}converge strongly top∈EPG∩FT, wherepPEPG∩FTfp.
Very recently, Ceng and Wong in14combined algorithm1.6with the result in3
to propose the following procudure:
Guk, y
1
rk
uk−xk, y−uk
≥0, ∀y∈C,
xk1μkfxk βkxkγkTskuk, k≥1,
1.11
for finding an element p ∈ EPG∩ F in the case thatC1 C2 Cunder the uniformly
asymptotic regularity condition on the nonexpansive semigroup{Ts:s >0}onC.
In this paper, motivated by the above results, to solve1.2, we introduce the following algorithm:
x1∈H, any element,
uk∈C1:G
uk, y
1
rk
uk−xk, y−uk
≥0, ∀y∈C1,
xk1μkfuk βkxkγkTkPC2uk, k≥1,
1.12
wheref is a contraction onH, that is,f : H → H andfx−fy ≤ ax−y, for all
x, y∈H, 0≤a <1,
Tkx
1
sk
sk
0
Tsxds, 1.13
for allx∈C2,{μk},{βk}, and{γk}be the sequences in0,1, and{rk},{sk}are the sequences
in0,∞satisfy the following conditions:
iμkβkγk1,
iilimk→ ∞μk0,
k≥1μk∞,
iii0<lim infk→ ∞βk≤lim supk→ ∞βk<1,
ivlimk→ ∞sk∞with bounded supk≥1|sk−sk1|,
vlim infk→ ∞rk>0 and limk→ ∞|rk−rk1|0.
2. Main Results
We formulate the following facts needed in the proof of our results.
Lemma 2.1. LetHbe a real Hilbert spaceH. There holds the following identity:
xy2
≤ x2 2y, xy, ∀x, y∈H. 2.1
Lemma 2.2see15. LetCbe a nonempty, closed, convex subset of a real Hilbert spaceH. For any
x∈H, there exists a uniquez∈Csuch thatz−x ≤ y−x, for ally ∈C, andz∈PCxif and
only ifz−x, y−z ≥0for ally∈C.
Lemma 2.3see16. Let{ak}be a sequence of nonnegative real numbers satisfying the following
condition:
ak1 ≤1−bkakbkck, 2.2
where {bk} and {ck} are sequences of real numbers such that bk ∈ 0,1, ∞k1bk ∞, and
lim supk→ ∞ck≤0. Then,limk→ ∞ak0.
Lemma 2.4see9. LetCbe a nonempty, closed, convex subset ofHandGbe a bifunction ofC×C
into−∞,∞satisfying the conditions (A1)–(A4). Letr >0andx∈H. Then, there existsz∈C
such that
Gz, v 1
rz−x, v−z ≥0, ∀v∈C. 2.3
Lemma 2.5see9. Assume thatG:C×C → −∞,∞satisfies the conditions (A1)–(A4). For
r >0andx∈H, define a mappingTr :H → Cas follows:
Trx
z∈C:Gz, v 1
rz−x, v−z ≥0,∀v∈C
. 2.4
Then, the following statements hold:
iTr is single-valued,
iiTr is firmly nonexpansive, that is, for anyx, y∈H,
Trx−Try2≤
Trx−Tr
y, x−y, 2.5
iiiFTr EPG,
Lemma 2.6see17. LetCbe a nonempty bounded closed convex subset in a real Hilbert spaceH
and let{Ts:s >0}be a nonexpansive semigroup onC. Then, for anyh >0,
lim sup
t→ ∞supy∈C
Th
1
t
t
0
Tsyds
−1
t
t
0
Tsyds0. 2.6
Lemma 2.7 Demiclosedness Principle 18. If C is a closed convex subset of H, T is a nonexpansive mapping onC,{xk}is a sequence inCsuch thatxk x∈Candxk−Txk → 0, then
x−Tx0.
Lemma 2.8see19. Let{xk}and{zk}be bounded sequences in a Banach spaceEand{βk}be a
sequence in0,1with0<lim infk→ ∞βk≤lim supk→ ∞βk<1. Supposexk1βkxk 1−βkzk
for allk≥1andlim supk→ ∞zk1−zk − xk1−xk ≤0. Then,limk→ ∞zk−xk0.
Now, we are in a position to prove the following result.
Theorem 2.9. LetC1andC2be two nonempty, closed, convex subsets in a real Hilbert spaceH. LetG
be a bifunction fromC1×C1to−∞,∞satisfying conditions (A1)–(A4) withCreplaced byC1, let
{Ts:s >0}be a nonexpansive semigroup onC2such thatEPG∩ F/∅and letfbe a contraction
ofHinto itself. Then,{xk}and{uk}generated by1.12-1.13converge strongly top∈EPG∩F,
wherepPEPG∩Ffp.
Proof. LetQPEPG∩F. Then,Qfis a contraction ofHinto itself. In fact, fromfx−fy ≤
ax−yfor allx, y∈Hand the nonexpansive property ofPCfor a closed convex subsetCin
H, it implies that
Qfx− Qfy ≤ fx−fy ≤ax−y. 2.7
Hence,Qfis a contraction ofHinto itself. SinceHis complete, there exists a unique element
p∈Hsuch thatpQfp. Such apis an element ofC1∩C2, because EPG∩ F/∅.
ByLemma 2.4,{uk} and{xk}are well defined. For eachu ∈ EPG∩ F, by putting
ukTrkxkand usingiiandiiiinLemma 2.5, we have that
uk−uTrkxk−Trku ≤ xk−u. 2.8
PutMumax{x1−u,1/1−afu−u}. Clearly,x1−u ≤Mu. Suppose thatxk−u ≤
Mu. Then, we have, from the nonexpansive property ofTkPC2, conditioniand2.8, that
xk1−uμk
fuk−u
βkxk−u γkTkPC2uk−u
≤μkfuk−uβkxk−uγkTkPC2uk−TkPC2u
≤μk
fuk−fufu−u
βkxk−uγkuk−u
≤μk
auk−ufu−u
1−μk
≤1−μk1−a
xk−uμk1−a
1
1−afu−u
≤1−μk1−a
Muμk1−aMuMu.
2.9
So,xk−u ≤ Mufor allk ≥1 and hence{xk}is bounded. Therefore,{uk},{TkPC2uk}, and
{fuk}are also bounded.
Next, we show thatxk1−xk → 0 ask → ∞. For this purpose, we define a sequence
{xk}by
xk1βkxk
1−βk
zk. 2.10
Then, we observe that
zk1−zk
μk1fuk1 γk1Tk1PC2uk1 1−βk1
−μkfuk γkTkPC2uk 1−βk
μk1
1−βk1fuk1−
μk
1−βkfuk
γk1
1−βk1Tk1PC2uk1−Tk1PC2uk
γk1
1−βk1Tk1PC2uk−
γk
1−βkTkPC2uk
μk1
1−βk1fuk1−
μk
1−βkfuk
γk1
1−βk1Tk1PC2uk1−Tk1PC2uk Tk1PC2uk
− μk1
1−βk1Tk1PC2uk−TkPC2uk
μk
1−βkTkPC2uk,
2.11
and, hence,
zk1−zk − xk1−xk ≤
μk1
1−βk1
fuk1Tk1PC2uk
μk
1−βk
fukTkPC2uk
γk1
1−βk1uk1−uk
Tk1PC2uk−TkPC2uk − xk1−xk.
Now, we estimate the valueuk1−uk by usinguk Trkxk anduk1 Trk1xk1. We have from2.4that
Guk, y
1
rk
uk−xk, y−uk
≥0, ∀y∈C1, 2.13
Guk1, y
1
rk1
uk1−xk1, y−uk1
≥0, ∀y∈C1. 2.14
Puttingy uk1 in2.13andy uk in2.14, adding the one to the other obtained result
and usingA2, we obtain that
uk−xk
rk −
uk1−xk1
rk1 , uk1−uk
≥0 2.15
and, hence,
uk−uk1uk1−xk− rk
rk1uk1−xk1, uk1−uk
≥0. 2.16
Without loss of generality, let us assume that there exists a real numberbsuch thatrk> b >0
for allk≥1. Then, we have
uk1−uk2≤
xk1−xk
1− rk
rk1
uk1−xk1, uk1−uk
≤
xk1−xk
1− rk
rk1
uk1−xk1
uk1−uk
2.17
and, hence,
uk1−uk ≤ xk1−xk
1
rk1|rk1−rk|uk1−xk1
≤ xk1−xk2Mu
b |rk1−rk|.
2.18
On the other hand,
TkPC2uk−Tk1PC2uk
1
sk
sk
0
TsPC2ukds− 1
sk1 sk1
0
TsPC2ukds
1 sk sk 0
TsPC2uk−TsPC2uds− 1
sk1 sk1
0
TsPC2uk−TsPC2uds
1
sk −
1
sk1
sk1
0
TsPC2uk−TsPC2uds 1
sk
sk
sk1
TsPC2uk−TsPC2uds
≤1
sk −
1
sk1
sk1Mu|sk−sk1|
sk Mu
≤ supk≥1|sk1−sk|
sk
2Mu.
2.19
So, we get from2.10,2.12,2.18,2.19, and the nonexpansive property ofTk1PC2that
zk1−zk − xk1−xk ≤ μk1
1−βk1
fuk1Tk1PC2uk
μk
1−βk
fukTkPC2uk
γk12Mu
1−βk1
b|rk1−rk|
supk≥1|sk1−sk|
sk 2Mu.
2.20
So,
lim sup
k→ ∞
zk1−zk − xk1−xk ≤0, 2.21
and byLemma 2.8, we have
lim
k→ ∞zk−xk0. 2.22
Consequently, it follows from2.10and conditioniiithat
lim
k→ ∞xk1−xkklim→ ∞
1−βk
zk−xk0. 2.23
By2.18,2.23, and
lim
k→ ∞|rk−rk1|0, 2.24
we also obtain
lim
We have, for everyu∈EPG∩ F, fromiiiinLemma 2.5, that
uk−u2Trkxk−Trku 2
≤ Trkxk−Trku, xk−u
uk−u, xk−u
1
2
uk−u2xk−u2− uk−xk2
2.26
and, hence,
uk−u2≤ xk−u2− uk−xk2. 2.27
Therefore, from the convexity of · 2and conditioni, we have
xk1−u2≤μkfuk−u2βkxk−u2γkTkPC2uk−u
2
≤μkfuk−u2βkxk−u2γkuk−u2
≤μkfuk−u2βkxk−u2γk
xk−u2− uk−xk2
≤μkfuk−u2
1−μk
xk−u2−γkuk−xk2
≤μkfuk−uxk−u2−γkuk−xk2
2.28
and, hence,
γkuk−xk2≤μkfuk−uxk−u2− xk1−u2
≤μkfuk−u2Muxk−xk1.
2.29
Without loss of generality, we assume that 0 < β∗ ≤ βk ≤ β < 1 for all k ≥ 1. Then, for
sufficiently largek,
0≤
1−β−μk
uk−xk2≤μkfuk−u2Muxk−xk1. 2.30
So, we have
lim
Further, sincexk1μkfuk βkxkγkTkPC2uk, by conditioni,2.19and
xk1−Tk1PC2uk1μkfuk βkxkγkTkPC2uk
−μkβkγk
TkPC2ukTkPC2uk−Tk1PC2uk1
μk
fuk−TkPC2uk
βkxk−TkPC2uk
TkPC2uk−Tk1PC2uk1,
2.32
we obtain that
xk1−Tk1PC2uk1 ≤μkfuk−TkPC2ukβkxk−TkPC2uk
uk1−uk
supk≥1|sk1−sk|
sk
2Mu.
2.33
Then, from2.25,2.33and the conditions on{μk}and{sk}, it implies that
1−β
lim sup
k→ ∞xk−TkPC2uk ≤0, 2.34
and so
lim sup
k→ ∞xk−TkPC2uk ≤
0. 2.35
Since
TkPC2uk−uk ≤ TkPC2uk−xkxk−uk, 2.36
we obtain from2.31that
lim
k→ ∞TkPC2uk−uk0. 2.37
Next, we show that
lim sup
k→ ∞
fp−p, xk−p
≤0. 2.38
We choose a subsequence{uki}of the sequence{uk}such that
lim sup
k→ ∞
fp−p, xk−p
lim
i→ ∞
fp−p, xki−p
. 2.39
First, we prove thatz∈EPG. From2.4it follows that
Guk, y
1
rk
uk−xk, y−uk
≥0, ∀y∈C1, 2.40
and, hence, by using conditionA2, we get
1
rk
uk−xk, y−uk
≥Gy, uk
, ∀y∈C1. 2.41
Therefore,
ukj−xkj
rkj
, y−ukj
≥Gy, ukj
, ∀y∈C1. 2.42
This together with conditionA3and2.31imply that
0≥Gy, z, ∀y∈C1. 2.43
So,Gz, y≥0 for ally∈C1. It means thatz∈EPG.
Next we show thatz∈ F. SinceTkPC2uk∈C2, we have
TkPC2uk−PC2ukPC2TkPC2uk−PC2uk
≤ TkPC2uk−uk,
2.44
and, hence, from2.31it follows that
lim
k→ ∞TkPC2uk−PC2uk0. 2.45
Thus,2.37together with2.45imply
lim
k→ ∞uk−PC2uk0. 2.46
On the other hand, for eachh >0, we have that
ThPC2uk−PC2uk ≤
ThPC2uk−Th
1
sk
sk
0
TsPC2ukds
Th
1
sk
sk
0
TsPC2ukds
− 1
sk
sk
0
TsPC2ukds
1 sk sk 0
TsPC2ukds−PC2uk
≤21
sk
sk
0
TsPC2ukds−PC2uk
Th
1
sk
sk
0
TsPC2ukds
− 1
sk
sk
0
TsPC2ukds
.
2.47
LetC02{x∈C2:x−p ≤Mp}. SincepPF∩EQGfp∈C2, we have from2.33that
PC2uk−pPC2uk−PC2p ≤ uk−p ≤ xk−p ≤Mp. 2.48
So,C20is a nonempty bounded closed convex subset. It is easy to verify that{Ts:s >0}is a nonexpansive semigroup onC02. ByLemma 2.6, we get
lim
k→ ∞ Th
1
sk
sk
0
TsPC2ukds
− 1
sk
sk
0
TsPC2ukds
0, 2.49
for every fixedh >0, and hence, by2.45–2.47, we obtain
lim
n→ ∞ThPC2uk−uk0 2.50
for eachh > 0. ByLemma 2.7,z ∈ FThPC2 FThfor allh > 0, becauseFTPC
FT for any mappingT : C → C. It means thatz ∈ F. Therefore,z ∈ F ∩EPG. Since
pPEPG∩Ffp, we have fromLemma 2.2that
lim sup
k→ ∞
fp−p, xk−p
lim
i→ ∞
fp−p, xki−p
fp−p, z−p≤0.
So,2.38is proved. Further, sincexk1−pμkfuk−p βkxk−p γkTkPC2uk−p, by usingLemma 2.1, we have that
xk1−p2 ≤
βkxk−p γkTkPC2uk−p
2
2μk
fuk−p, xk1−p
≤βkxk−pγkuk−p
2
2μk
fuk−f
p, xk1−p
2μk
fp−p, xk1−p
≤1−μk
2
xk−p22μkauk−pxk1−p
2μk
fp−p, xk1−p
≤1−μk
2
xk−p2μka
uk−p2xk1−p2
2μk
fp−p, xk1−p
.
2.52
This with2.8implies that
xk1−p2≤
1−μk
2
μka
1−μka
xk−p2 2μk
1−μka
fp−p, xk1−p
1−2μkμka
1−μka
xk−p2 μ2k
1−μka
xk−p2
2μk
1−μka
fp−p, xk1−p
1−21−aμk 1−μka
xk−p2
21−aμk
1−μka
×
μkM2p
21−a
1 1−a
fp−p, xk1−p
1−bkxk−p2bkck,
2.53
where
bk
21−aμk
1−μka , ck
μkMp2
21−a
1 1−a
fp−p, xk1−p
. 2.54
UsingLemma 2.3, we get
lim
k→ ∞xk−p0. 2.55
Remarks. aNote that the following parametersμk1/3k,βkμk1/4,γk−2μk3/4,
rkμka0for any fixed numbera0>0, andsk b0kc0withb0, c0 >0 for allk≥1 satisfy
all conditions inTheorem 2.9.
bIfTs Tfor alls >0 andC1C2C, then we have the following corollary.
Corollary 2.10. LetCbe a nonempty, closed, convex subsets in a real Hilbert spaceH. Let G be a bifunction fromC×Cto−∞,∞satisfying conditions (A1)–(A4), letTbe a nonexpansive mapping onCsuch thatEPG∩FT/∅and letf be a contraction ofHinto itself. Let {xk}and{uk}be
sequences generated byx1∈Hand
uk∈C, G
uk, y
1
rk
uk−xk, y−uk
≥0, ∀y∈C,
xk1μkfuk βkxkγkTuk, k≥1,
2.56
where{μk},{βk},{γk}, and{rk}satisfy conditions (i)–(v). Then,{xk}and{uk}converge strongly
top∈EPG∩FT, wherepPEPG∩FTfp.
Proof. From the proof of the theorem, TkPC2uk−1−Tk−1PC2uk−1 Tuk−1−Tuk−1 0 in
2.12.
cIn the case thatC1 C2 C, a closed convex subset in H,Gu, v 0 for all
u, v∈C×C, we have the following result.
Corollary 2.11. LetCbe a nonempty, closed, convex subsets in a real Hilbert spaceH. Let{Ts :
s >0}be a nonexpansive semigroup onCsuch thatF/∅and letfbe a contraction ofHinto itself. Let{xk}and{uk}be sequences generated byx1∈Hand
ukPCxk,
xk1μkfuk βkxkγkTkuk, k≥1,
2.57
whereTkxis defined by1.13for allx∈Cand{μk},{βk},{γk}, and{sk}satisfy conditions (i)–(v).
Then, the sequences{xk}and{uk}converge strongly top∈ F, wherepPFfp.
Proof. ByLemma 2.2,ukPCxkif and only if
uk−xk, y−uk
≥0, ∀y∈C. 2.58
Clearly, in addition, iffis a contraction ofCinto itself andx1∈C, then we obtain the
algoritm
xk1 μkfxk βkxkγkTkxk, k≥1, 2.59
whereTk is defined by1.13and{μk},{βk},{γk}, and{sk}satisfy conditionsi–v. This
algorithm is different from Yao and Noor’s algorithm1.6, in whichTkx Tskxfor all
Acknowledgment
This work was supported by the Vietnamese National Foundation of Science and Technology Development.
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