Volume 2008, Article ID 714939,15pages doi:10.1155/2008/714939
Research Article
Viscosity Approximation Methods for Generalized
Mixed Equilibrium Problems and Fixed Points of
a Sequence of Nonexpansive Mappings
Wei-You Zeng,1 Nan-Jing Huang,1and Chang-Wen Zhao2
1Department of Mathematics, Sichuan University, Chengdu, Sichuan 610064, China 2College of Business and Management, Sichuan University, Chengdu, Sichuan 610064, China
Correspondence should be addressed to Nan-Jing Huang,[email protected]
Received 17 July 2008; Accepted 11 November 2008
Recommended by Wataru Takahashi
We introduce an iterative scheme by the viscosity approximation method for finding a common element of the set of common solutions for generalized mixed equilibrium problems and the set of common fixed points of a sequence of nonexpansive mappings in Hilbert spaces. We show a strong convergence theorem under some suitable conditions.
Copyrightq2008 Wei-You Zeng et al. 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
Equilibrium problems theory provides us with a unified, natural, innovative, and general framework to study a wide class of problems arising in finance, economics, network analysis, transportation, elasticity, and optimization, which has been extended and generalized in many directions using novel and innovative techniques; see1–8. Inspired and motivated by the research and activities going in this fascinating area, we introduce and consider a new class of equilibrium problems, which is known as the generalized mixed equilibrium problems.
LetCbe a nonempty closed convex subset of a real Hilbert spaceHandT :C → 2Ha multivalued mapping. Letϕ:C×C → Rbe a real-valued function andΦ:H×C×C → R
an equilibrium-like function, that is,
Φw, u, v Φw, v, u 0, ∀w, u, v∈H×C×C. 1.1
We consider the problem of findingu∈Candw∈Tusuch that
which is called the generalized mixed equilibrium problemfor short, GMEP. IfTis a single-valued mapping, then problem1.2is equivalent to findingu∈Csuch that
ΦTu, u, vϕv, u−ϕu, u≥0, ∀v∈C. 1.3
We denote Ω for the set of solutions of GMEP 1.2. This class is a quite general and unifying one and includes several classes of equilibrium problems and variational inequalities as special cases. In recent years, several numerical techniques including projection, resolvent, and auxiliary principle have been developed and analyzed for solving variational inequalities. It is well known that projection- and resolvent-type methods cannot be extended for equilibrium problems. To overcome this drawback, one usually uses the auxiliary principle technique. Glowinski et al. 9 have used this technique to study the existence of a solution of mixed variational inequalities. The viscosity approximation method is one of the important methods for approximation fixed points of nonexpansive type mappings. It was first discussed by Moudafi10. Recently, Hirstoaga11and S. Takahashi and W. Takahashi 12 applied viscosity approximation technique for finding a common element of set of solutions of an equilibrium problem EP and set of fixed points of a nonexpansive mapping. Very recently, Yao et al. 13introduced and studied an iteration process for finding a common element of the set of solutions of the EP and the set of common fixed points of infinitely many nonexpansive mappings inH. Let{Tn}∞n1 be a sequence of
nonexpansive mappings ofCinto itself and let{λn}∞n1be a sequence of nonnegative numbers
in0,1. For anyn≥1, define a mappingSnofCinto itself as follows:
Un,n1I,
Un,nλnTnUn,n11−λn
I,
Un,n−1λn−1Tn−1Un,n
1−λn−1
I,
.. .
Un,kλkTkUn,k11−λk
I,
Un,k−1λk−1Tk−1Un,k
1−λk−1
I,
.. .
Un,2λ2T2Un,3
1−λ2I,
SnUn,1 λ1T1Un,2
1−λ1I.
1.4
Such a mappingSnis called theS-mapping generated byTn, Tn−1, . . . , T1 andλn, λn−1, . . . , λ1, see14.
2. Preliminaries
LetH be a real Hilbert space with inner product·,·and norm · , and letCbe a closed convex subset ofH. Then, for anyx∈H, there exists a unique nearest point inC, denoted by
PCx, such that
x−PCx≤ x−y , ∀y∈C. 2.1
PC is called metric projection of H onto C. It is well known that PC is nonexpansive. Furthermore, forx∈Handu∈C,
uPCx⇐⇒ x−u, u−y ≥0, ∀y∈C. 2.2
We denote byFTthe set of fixed points of a self-mappingTonC, that is,FT {x∈ C:Txx}. It is well known that ifC⊂His nonempty, bounded, closed, and convex andT
is nonexpansive, thenFTis nonempty; see15. Let{Tn}∞n1be a sequence of nonexpansive
mappings ofCinto itself, whereCis a nonempty closed convex subset of a real Hilbert space
H. Given a sequence{λn}∞n1in0,1, we define a sequence{Sn}∞n1of self-mappings onCby 1.4. Then we have the following lemmas which are important to prove our results.
Lemma 2.1see14. LetC be a nonempty closed convex subset of a real Hilbert spaceH. Let
{Tn}∞n1 be a sequence of nonexpansive mappings ofCinto itself such that
∞
n1FTn/∅, and let
{λn}∞n1 be a sequence in0, b for someb ∈ 0,1. Then, for every x ∈ Cand k ∈ N the limit
limn→ ∞Un,kxexists.
UsingLemma 2.1, one can define mappingSofCinto itself as follows:
Sx lim
n→ ∞Snxnlim→ ∞Un,1x, 2.3
for every x ∈ C. Such a mapping S is called the S-mapping generated byT1, T2, . . . and
λ1, λ2, . . . .Throughout this paper, we will assume that 0 < λn ≤ b < 1 for every n ≥ 1. SinceSnis nonexpansive,S:C → Cis also nonexpansive.
Lemma 2.2see14. LetC be a nonempty closed convex subset of a real Hilbert spaceH. Let
{Tn}∞n1 be a sequence of nonexpansive mappings ofCinto itself such that∞n1FTn/∅, and let
{λn}∞n1be a sequence in0, bfor someb∈0,1. Then,FS
∞
n1FTn.
LetCbe a convex subset of a real Hilbert spaceHandκ:C → Ra Fr´echet differential function. Thenκis said to beη-convex strongly convex if there exists a constantμ > 0 such that
κy−κx−κx, ηy, x≥ μ
2 x−y
2, ∀x, y∈C. 2.4
LetCbe a nonempty subset of a real Hilbert spaceH. A bifunctionϕ·,·:C×C → R
is said to be skew-symmetric if
ϕu, v ϕv, u−ϕu, u−ϕv, v≤0, ∀u, v∈C. 2.5
If the skew-symmetric bifunctionϕ·,·is linear in both arguments, then
ϕu, u≥0, ∀u∈C. 2.6
We denotefor weak convergence and →for strong convergence. A functionψ:C×C → R
is called weakly sequentially continuous atx0, y0 ∈ C×C, ifψxn, yn → ψx0, y0 as
n → ∞for each sequence{xn, yn}inC×Cconverging weakly tox0, y0. The function
ψ·,·is called weakly sequentially continuous onC×Cif it is weakly sequentially continuous at each point ofC×C.
LetCBXdenote the set of nonempty closed bounded subsets ofX. ForA, B∈CBX, define the HausdorffmetricHas follows:
HA, B max
sup a∈A
inf b∈B
da, b,sup b∈B
inf a∈A
db, a . 2.7
Lemma 2.3see16. LetA, B∈CBXanda∈A. Then for >1, there must exist a pointb∈B
such thatda, b≤HA, B.
LetCbe a nonempty closed convex subset of a real Hilbert spaceHandT :C → 2H a multivalued mapping. Forx∈C, letw∈Tx. Letϕ:C×C → Rbe a real-valued function satisfying the following:
ϕ1ϕ·,·is skew symmetric;
ϕ2for each fixedy∈C,ϕ·, yis convex and upper semicontinuous;
ϕ3ϕ·,·is weakly continuous onC×C.
Letκ:C → Rbe a differentiable functional with Fr´echet derivativeκxatxsatisfying the following:
κ1κis sequentially continuous from the weak topology to the strong topology;
κ2κis Lipschitz continuous with Lipschitz constantν >0. Letη:C×C → Hbe a function satisfying the following:
η1ηx, y ηy, x 0 for allx, y∈C;
η2η·,·is affine in the first coordinate variable;
η3for each fixedy∈C,x→ηy, xis sequentially continuous from the weak topology to the weak topology.
Φ1for each fixed v ∈ C,w, u → Φw, u, vis an upper semicontinuous function fromH×CtoR, that is,wn → wandun → uimply lim supn→ ∞Φwn, un, v ≤
Φw, u, v;
Φ2for each fixedw, v∈H×C,u→Φw, u, vis a concave function;
Φ3for each fixedw, u∈H×C,v→Φw, u, vis a convex function.
Letrbe a positive parameter. For a given elementx∈Candwx ∈Tx, consider the following auxiliary problem for GMEP1.2: findu∈Csuch that
Φwx, u, v ϕv, u−ϕu, u 1
r
κu−κx, ηv, u≥0, ∀v∈C. 2.8
It is easy to see that ifux, thenuis a solution of GMEP1.2.
Lemma 2.4see6. LetCbe a nonempty closed convex bounded subset of a real Hilbert spaceH
andϕ:C×C → Ra real-valued function satisfying the conditionsϕ1–ϕ3. LetT :C → 2Hbe a
multivalued mapping andΦ:H×C×C → Rthe equilibrium-like function satisfying the conditions
Φ1–Φ3. Assume thatη:C×C → His a Lipschitz function with Lipschitz constantλ >0which satisfies the conditionsη1–η3. Letκ : C → Rbe anη-strongly convex function with constant
μ >0which satisfies the conditionsκ1andκ2. For eachx∈C, letwx∈Tx. Forr >0, define a
mappingTr :C → Cby
Trx
u∈C:Φwx, u, v ϕv, u−ϕu, u 1
r
κu−κx, ηv, u≥0, ∀v∈C . 2.9
Then one has the following:
athe auxiliary problem2.8has a unique solution;
bTr is single valued;
cifλν/μandΦw1, Trx1, Trx2 Φw2, Trx2, Trx1≤0for allx1, x2 ∈Cand all
w1∈Tx1,w2∈Tx2, it follows thatTris nonexpansive;
dFTr Ω;
e Ωis closed and convex.
We also need the following lemmas for our main results.
Lemma 2.5see17. Let{an}, {bn}, and{cn}be three sequences of nonnegative numbers such
that
an1≤bnancn, ∀n1,2, . . . . 2.10
Ifbn≥1,∞n1bn−1<∞, and∞n1cn<∞, thenlimn→ ∞anexists.
Lemma 2.6. Let{an}and{cn}be sequences of nonnegative numbers such that
an1≤Θancn, ∀n1,2, . . . . 2.11
Proof. It is easy to see that inequality2.11is equivalent to
an1 ≤Θbnancn, ∀n1,2, . . . , 2.12
whereΘ∈0,1,bn1 and
∞
n1cn<∞. It follows that
an1 ≤bnancn ∀n1,2, . . . . 2.13
Note thatLemma 2.5implies that limn→ ∞an exists. Suppose limn→ ∞an dfor somed >0. It is obvious that limn→ ∞cn 0 and so inequality2.12 implies that d ≤ Θd, which is a contradiction. Thus, limn→ ∞and0. This completes the proof.
Lemma 2.7see6. Let{xn}be a sequence in a normed spaceX, · such that
xn1−xn2≤Θxn−xn1bncn, ∀n1,2, . . . , 2.14
whereΘ∈0,1, and{bn}and{cn}are sequences satisfy the following conditions:
ibn≥1for alln1,2, . . .and∞n1bn−1<∞;
iicn≥0for alln1,2, . . .and∞n1cn<∞.
Then{xn}is a Cauchy sequence.
Lemma 2.8see18. Let{an}be a sequence of nonnegative real numbers such that
an1≤1−λn
anλnσnξn, ∀n1,2, . . . , 2.15
where{λn},{σn}and{ξn}are sequences of real numbers satisfying the following conditions:
i{λn} ⊂0,1,limn→ ∞λn0and
∞
n1λn∞;
iilim supn→ ∞σn≤0;
iiiξn ≥0for alln1,2, . . .and
∞
n1ξn<∞.
Then,limn→ ∞an0.
3. Iterative algorithm and convergence theorem
Let Cbe a nonempty closed convex subset of a real Hilbert spaceH,T : C → CBHa multivalued mapping, f : C → C a contraction mapping with constant α ∈ 0,1, and
Sn : C → Can S-mapping generated by T1, T2, . . . and λ1, λ2, . . ., where sequence {Tn} is nonexpansive. Let{αn}be a sequence in0,1and{rn}a sequence in0,∞. We can develop
Algorithm 3.1. For givenx1 ∈ Candw1 ∈ Tx1, there exist sequences{xn},{un}inCand
{wn:wn∈Txn}inHsuch that for alln1,2, . . .,
wn−wn1≤1 1
n
HTxn
, Txn1;
Φwn, un, v
ϕv, un
−ϕun, un
1
rn
κun
−κxn
, ηv, un
≥0, ∀v∈C;
xn1αnf
xn
1−αn
Sn
un
.
3.1
We now prove the strong convergence of iterative sequence {xn}, {un}, and {wn} generated byAlgorithm 3.1.
Theorem 3.2. Let Cbe a nonempty closed convex bounded subset of a real Hilbert space H, T :
C → CBHa multivaluedH-Lipschitz continuous mapping with constantL >0,f :C → Ca contraction mapping with constantα∈0,1. Letϕ:C×C → Rbe a real-valued function satisfying the conditionsϕ1–ϕ3and letΦ : H×C×C → R be an equilibrium-like function satisfying conditionsΦ1–Φ3andΦ4:
Φ4 Φw, Trx, Tsy Φw, T sy, Trx ≤ −γ Trx−Tsy 2 for all x, y ∈ Cand
r, s∈0,∞, whereγ >0,w∈Txandw ∈Ty.
Assume thatη :C×C → His a Lipschitz function with Lipschitz constantλ >0which satisfies the conditionsη1∼η3. Letκ : C → R be anη-strongly convex function with constantμ > 0
which satisfies conditionsκ1andκ2withλν/μ≤1. LetSn :C → Cbe anS-mapping generated
byT1, T2, . . .andλ1, λ2, . . .and∞n1FTn∩Ω/∅, where sequence{Tn}is nonexpansive. Let{xn},
{un}and{wn}be sequences generated byAlgorithm 3.1, where{αn}is a sequence in0,1and{rn}
in0,∞satisfying the following conditions:
C1limn→ ∞αn0,∞n1αn∞and∞n1|αn−αn1|<∞; C2lim infn→ ∞rn>0andn∞1|rn−rn1|<∞;
C3∞n11−αnεn<∞whereεnsupx∈C Snx−Sn1x .
Then the sequences{xn}and{un}converge strongly tox∗ ∈
∞
n1FTn∩Ω, and{wn}converges
strongly tow∗∈Tx∗, wherex∗P∩∞
n1FTn∩Ωfx ∗.
Proof. It is easy to see fromΦ4that
Φw, Trx, Tsy
Φw, T sy, Trx
≤0 3.2
for allx, y ∈Candr, s ∈0,∞, whereγ >0,w ∈Tx, andw ∈Ty. All the conclusions
a–eofLemma 2.4hold. LetQP∩∞
n1FTn∩Ω. ThenQfis a contraction ofCinto itself. In fact,
Qfx−Qfy≤fx−fy< α x−y , ∀x, y∈C. 3.3
Hence there exists a unique elementq∈ Csuch thatq Qfq. Noting thatfq ∈ Cand
Now, we prove that xn−xn1 → 0 and un−un1 → 0 asn → ∞. Observe that
xn1−xnαnf xn
1−αn
Sn
un
−αn−1fxn−1
−1−αn−1
Sn−1
un−1
αnf
xn
−αnf
xn−1
αnf
xn−1
−αn−1fxn−1
1−αn
Sn
un
−1−αn
Sn−1
un−1
1−αn
Sn−1
un−1
−1−αn−1
Sn−1
un−1
≤αnf
xn
−fxn−1αn−αn−1f
xn−1Sn−1
un−1
1−αnSn
un
−Sn−1
un−1
≤ααnxn−xn−12αn−αn−1diamC 1−αnSn
un
−Sn−1
unSn−1
un
−Sn−1
un−1
≤ααnxn−xn−12αn−αn−1diamC
1−αnun−un−1εn−1
. 3.4
Noting thatunTrnxnandun1 Trn1xn1, it follows from3.1that
Φwn, un, v
ϕv, un
−ϕun, un
1
rn
κun
−κxn
, ηv, un
≥0, 3.5
Φwn1, un1, v
ϕv, un1−ϕun1, un1 1 rn1
κun1−κxn1, ηv, un1≥0 ∀v∈C. 3.6
Puttingvun1in3.5andvunin3.6, respectively, we have
Φwn, un, un1ϕun1, un
−ϕun, un
1
rn
κun
−κxn
, ηun1, un
≥0,
Φwn1, un1, un
ϕun, un1−ϕun1, un1 1 rn1
κun1−κxn1, ηun, un1≥0. 3.7
Adding up those inequalities, we obtain from2.5,η1, andΦ4that
−1 rn
κun
−κxn
, ηun, un1 1 rn1
It follows that
γrnun−un12
≤
κun
−κxn
− rn
rn1
κun1−κxn1, ηun1, un
≤κun
−κun1, ηun1, un
κun1−κxn1κxn1−κxn
− rn
rn1
κun1−κxn1, ηun1, un
≤ −μun−un12
κxn1
−κxn
1− rn
rn1
κun1
−κxn1
ηun1, un
≤ −μun−un12λνxn1−xn
rn1−rn rn1
un1−xn1un1−un,
3.9
sinceηandκare Lipschitz continuous wiht Lipschitz constantsλandν, respectively. Noting that lim infn→ ∞rn >0, without loss of generality, we assume that there exists a real number
r >0 such thatrn≥r >0 for alln1,2, . . . .Thus,
γrun1−un≤ −μun1−unλν
xn1−xn
rn1−rn
r un1−xn1
, 3.10
which implies that
1γr
μ
un1−un≤xn1−xn
rn1−rn
r diamC, 3.11
and hence
un1−un≤δxn1−xnrn1−rn
r δdiamC, 3.12
whereδ1/1γr/μ∈0,1. SetΘ:max{α, δ} ∈0,1. Combining3.4and3.12yields
xn1−xn≤ ααn
1−αn
δxn−xn−1
1−αn
εn−1
2αn−αn−1
1−αn
δrn−rn−1 r
diamC
≤Θxn−xn−1
1−αn
εn−1
2αn−αn−1
rn−rn−1 r
diamC.
From conditionsC1andC3,
∞
n1
1−αn1εn
∞
n1
1−αn
εn
αn−αn1εn
≤∞
n1
1−αn
εnαn−αn1sup
n∈N
εn
<∞.
3.14
Setan: xn−xn−1 and
cn:
1−αn
εn−1
2αn−αn−1
rn−rn−1 r
diamC. 3.15
Then Lemmas2.6and2.7imply that limn→ ∞ xn1−xn 0 and{xn}is a Cauchy sequence inC. Hence from3.12, we get
lim
n→ ∞un1−un0. 3.16
We know fromC3that limn→ ∞εn0. It follows that
xn1−Sn1
un1≤xn1−Sn
unSn
un
−Sn
un1Sn
un1−Sn1un1 ≤αnf
xn
−Sn
unun1−unεn
≤αndiamC un1−unεn.
3.17
Thus, limn→ ∞ xn−Snun 0.
Next, we prove that there existsx∗∈C, such thatxn → x∗,un → x∗, andwn → w as
n → ∞, wherew ∈Tx∗.
Letp∈∞n1FTn∩Ω. Then
un−p2 Trn
xn
−Trnp
2
≤Trn
xn
−Trnp, xn−p
≤un−p, xn−p
≤ 1
2un−p
2
xn−p2−un−xn
2 ,
3.18
and so
un−p2≤
By the convexity of · , we have
xn1−p2≤ αnf
xn
−p21−αnSn
un
−p2
≤αnf
xn
−p21−αnun−p2
≤αndiamC2xn−p2−un−xn2.
3.20
It follows that
un−xn2≤
αndiamC2xn−p2−xn1−p2
≤αndiamC2xn−pxn1−pxn−xn1 ≤αndiamC22xn−xn1diamC.
3.21
This implies that
lim
n→ ∞un−xn0. 3.22
Since{xn}is a Cauchy sequence inC, there exists an elementx∗∈Csuch that limn→ ∞xnx∗. Now limn→ ∞ un−xn 0 implies that limn→ ∞unx∗. From3.1, we have
wn−wn1≤1 1
n
HTxn
, Txn1
≤2HTxn
, Txn1 ≤2Lxn−xn1
3.23
and form > n≥1,
wm−wn≤m−1
in
wi−wi1≤2L
m−1
in
xi−xi1, 3.24
m−1
in
xi−xi1m−1
in
ai1≤
m−1
in
Θaici
Θm−1
in
ai m−1
in
ci
Θm−1
in
ai1 Θan−am
m−1
in
ci
≤Θm−1
in
ai1 Θan m−1
in
ci.
Thus,
m−1
in
xi−xi1≤ Θ
1−Θxn−xn−1
m−1
in ci
1−Θ . 3.26
By3.24and3.26, we have
lim
m,n→ ∞wm−wn0. 3.27
It follows that{wn}is a Cauchy sequence inHand so there exists an elementw inHsuch that limn→ ∞wnw:
dw, T x∗ inf b∈Tx∗d
w, b
≤ w−wnd
wn, T
x∗
≤ w−wnH
Txn
, Tx∗
≤ w−wnLxn−x∗−→0 asn−→ ∞,
3.28
that is,dw, Tx ∗ 0. We conclude thatw ∈Tx∗asTx∗∈CBH. It follows that
x∗−S
n
x∗≤x∗−unun−xnxn−Sn
unSn
un
−Sn
x∗
≤2x∗−unun−xnxn−Sn
un−→0 asn−→ ∞
3.29
and so x∗ limn→ ∞Snx∗ Sx∗, that is, x ∈ FS
∞
n1FTn. Since xn → x∗ and
un → x∗, we know thatκun−κxn → 0. From3.1andΦ1, we have
Φw, x ∗, vϕv, x∗−ϕx∗, x∗≥0, 3.30
that is,x∗∈Ω. Thus,x∗∈∞n1FTn∩Ω.
SinceqQfq, we havefq−q, p−q ≤0 for allp∈∞n1FTn∩Ω. Fromxn → x∗, we have
lim n→ ∞
fq−q, xn−q
and so
xn1−q2
1−αn
Sn
un
−qαn
fxn
−q2
≤1−αn2Sn
un
−q22αn
fxn
−q, xn1−q
≤1−αn
2
un−q22αn
fxn
−fq fq−q, xn1−q
≤1−αn
2
un−q22ααnxn−qxn1−q2αn
fq−q, xn1−q
≤1−αn
2
un−q2ααnxn−q2xn1−q2
2αn
fq−q, xn1−q
. 3.32
It follows from3.19that
xn1−q2≤
1−αn
2 ααn 1−ααn
xn−q2 2αn 1−ααn
fq−q, xn1−q
≤
1−2αn1−α 1−ααn
xn−q2
2αn1−α 1−ααn
×
αn 1−αsupn∈N
xn−q2 1
1−α
fq−q, xn1−q.
3.33
Set
λn:
2αn1−α
1−ααn , ξn:0,
σn: 1α−n
αsupn∈Nxn−q 2 1
1−α
fq−q, xn1−q.
3.34
Then, limn→ ∞λn 0,∞n1λn ∞, and lim supn→ ∞σn ≤ 0. It follows fromLemma 2.8that limn→ ∞xnqand sox∗q. This completes the proof.
Remark 3.3. Theorem 3.2 improves and extends the main results of S. Takahashi and W. Takahashi12.
We now give some applications of Theorem 3.2. If the set-valued mapping T in
Theorem 3.2is single-valued, then we have the following corollary.
Corollary 3.4. LetCbe a nonempty closed convex bounded subset of a real Hilbert spaceH,T :C → Ha Lipschitz continuous mapping with constantL >0,f : C → Ca contraction mapping with constantα∈0,1. Letϕ:C×C → Rbe a real-valued function satisfying the conditionsϕ1–ϕ3
and letΦ:H×C×C → Rbe an equilibrium-like function satisfying the conditionsΦ1–Φ3and
Φ4:
Φ4ΦTx, Trx, Tsy ΦTy, Tsy, Trx≤ −γ Trx−Tsy 2for allx, y∈Cand
Assume thatη:C×C → His a Lipschitz function with Lipschitz constantλ >0which satisfies the conditionsη1∼η3. Letκ:C → Rbe anη-strongly convex function with constantμ >0which satisfies the conditionsκ1andκ2withλν/μ≤1. LetSn :C → Cbe anS-mapping generated
byT1, T2, . . .andλ1, λ2, . . .and∞n1FTn∩Ω/∅, where sequence{Tn}is nonexpansive. Let{xn},
{un}, and{wn}be sequences generated by
ΦTxn
, un, v
ϕv, un
−ϕun, un
1
rn
κun
−κxn
, ηv, un
≥0, ∀v∈C,
xn1αnf
xn
1−αn
Sn
un
, n1,2, . . . ,
3.35
where {αn} is a sequence in0,1 and {rn} in 0,∞ satisfying conditions C1–C3. Then the
sequences{xn}and{un}converge strongly tox∗∈∞n1FTn∩Ω, wherex∗P∩∞
n1FTn∩Ωfx∗. Corollary 3.5. Let Cbe a nonempty closed convex bounded subset of a real Hilbert spaceH,T :
C → CBHa multivaluedH-Lipschitz continuous mapping with constantL >0,f :C → Ca contraction mapping with constantα∈0,1. Letϕ:C×C → Rbe a real-valued function satisfying the conditionsϕ1–ϕ3and letΦ:H×C×C → Rbe an equilibrium-like function satisfying the conditionsΦ1–Φ4andΩ/∅. Assume thatη:C×C → His a Lipschitz function with Lipschitz constantλ >0 which satisfies the conditionsη1∼η3. Letκ : C → R be anη-strongly convex function with constantμ >0which satisfies the conditionsκ1andκ2withλν/μ≤1. Let{xn},
{un}, and{wn}be sequences generated by
wn∈T
xn
, wn−wn1 ≤
1 1
n
HTxn
, Txn1,
Φwn, un, v
ϕv, un
−ϕun, un
1
rn
κun
−κxn
, ηv, un
≥0, ∀v∈C,
xn1αnf
xn
1−αn
un, n1,2, . . . ,
3.36
where{αn}is a sequence in0,1and{rn}in0,∞satisfying conditionsC1andC2. Then the
sequences{xn}and{un}converge strongly tox∗∈Ω, and{wn}converges strongly tow∗∈Tx∗,
wherex∗PΩfx∗.
Proof. Let Tn I in Theorem 3.2 for n 1,2, . . ., where I is an identity mapping. Then
Sn I forn1,2, . . . .Thus, the conditionC3is satisfied. NowCorollary 3.5follows from
Theorem 3.2. This completes the proof.
Acknowledgments
The authors would like to thank the referees very much for their valuable comments and suggestions. This work was supported by the National Natural Science Foundation of China
10671135and Specialized Research Fund for the Doctoral Program of Higher Education
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 N.-J. Huang, H.-Y. Lan, and K. L. Teo, “On the existence and convergence of approximate solutions for equilibrium problems in Banach spaces,”Journal of Inequalities and Applications, vol. 2007, Article ID 17294, 14 pages, 2007.
3 F. Giannessi, A. Maugeri, and P. M. Pardalos, Eds.,Equilibrium Problems: Nonsmooth Optimization and Variational Inequality Models, vol. 58 ofNonconvex Optimization and Its Applications, Kluwer Academic Publishers, Dordrecht, The Netherlands, 2001.
4 F. Flores-Baz´an, “Existence theorems for generalized noncoercive equilibrium problems: the quasi-convex case,”SIAM Journal on Optimization, vol. 11, no. 3, pp. 675–690, 2000.
5 U. Mosco, “Implicit variational problems and quasi variational inequalities,” inNonlinear Operators and the Calculus of Variations (Summer School, Univ. Libre Bruxelles, Brussels, 1975), vol. 543 ofLecture Notes in Mathematics, pp. 83–156, Springer, Berlin, Germany, 1976.
6 D. R. Sahu, N.-C. Wong, and J.-C. Yao, “On convergence analysis of an iterative algorithm for finding common solution of generalized mixed equilibrium problems and fixed point problemes,” to appear inMathematical Inequalities & Applications.
7 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.
8 J.-W. Peng and J.-C. Yao, “Some new iterative algorithms for generalized mixed equilibrium problems with strict pseudo-contractions and monotone mappings,” to appear inTaiwanese Journal of Mathematics.
9 R. Glowinski, J.-L. Lions, and R. Tremolieres,Numerical Analysis of Variational Inequalities, vol. 8 of Studies in Mathematics and Its Applications, North-Holland, Amsterdam, The Netherlands, 1981.
10 A. Moudafi, “Viscosity approximation methods for fixed-points problems,”Journal of Mathematical Analysis and Applications, vol. 241, no. 1, pp. 46–55, 2000.
11 S. A. Hirstoaga, “Iterative selection methods for common fixed point problems,” Journal of Mathematical Analysis and Applications, vol. 324, no. 2, pp. 1020–1035, 2006.
12 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.
13 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.
14 K. Shimoji and W. Takahashi, “Strong convergence to common fixed points of infinite nonexpansive mappings and applications,”Taiwanese Journal of Mathematics, vol. 5, no. 2, pp. 387–404, 2001.
15 W. Takahashi, Nonlinear Functional Analysis. Fixed Point Theory and Its Application, Yokohama, Yokohama, Japan, 2000.
16 S. B. Nadler Jr., “Multi-valued contraction mappings,”Pacific Journal of Mathematics, vol. 30, pp. 475– 488, 1969.
17 M. O. Osilike and S. C. Aniagbosor, “Weak and strong convergence theorems for fixed points of asymptotically nonexpansive mappings,”Mathematical and Computer Modelling, vol. 32, no. 10, pp. 1181–1191, 2000.
