Volume 2011, Article ID 217407,27pages doi:10.1155/2011/217407
Research Article
A New Hybrid Algorithm for a System of
Mixed Equilibrium Problems, Fixed Point Problems
for Nonexpansive Semigroup, and Variational
Inclusion Problem
Thanyarat Jitpeera and Poom Kumam
Department of Mathematics, Faculty of Science, King Mongkut’s University of Technology Thonburi (KMUTT), Bangmod, Bangkok 10140, Thailand
Correspondence should be addressed to Poom Kumam,[email protected]
Received 14 December 2010; Accepted 15 January 2011 Academic Editor: Jen Chih Yao
Copyrightq2011 T. Jitpeera and P. Kumam. 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 consider a shrinking projection method for finding the common element of the set of common fixed points for nonexpansive semigroups, the set of common fixed points for an infinite family of aξ-strict pseudocontraction, the set of solutions of a system of mixed equilibrium problems, and the set of solutions of the variational inclusion problem. Strong convergence of the sequences generated by the proposed iterative scheme is obtained. The results presented in this paper extend and improve some well-known results in the literature.
1. Introduction
Throughout this paper, we assume thatHbe a real Hilbert space with inner product·,·and norm · , and letCbe a nonempty closed convex subset ofH. We denote weak convergence and strong convergence by notationsand →, respectively. LetI{Fk}k∈Γbe a countable
family of bifunctions fromC×CtoR, whereRis the set of real numbers andΓis an arbitrary index set. Letϕ :C → R ∪ {∞}be a proper extended real-valued function. Thesystem of mixed equilibrium problemsis to findx∈Csuch that
Fkx, yϕy≥ϕx, ∀k∈Γ, ∀y∈C. 1.1
The set of solutions of1.1is denoted by SMEPFk, ϕ, that is,
IfΓ is a singleton, the problem 1.1 reduces to find the followingmixed equilibrium problemsee also the work of Flores-Baz´an in1. For findingx∈Csuch that,
Fx, yϕy≥ϕx, ∀y∈C, 1.3
the set of solutions of1.3is denoted by MEPF, ϕ. Combettes and Hirstoaga2introduced the followingsystem of equilibrium problems. For findingx∈Csuch that,
Fk
x, y≥0, ∀k∈Γ, ∀y∈C, 1.4
the set of solutions of1.4is denoted by SEPI, that is,
SEPI x∈C:Fk
x, y≥0, ∀k∈Γ, ∀y∈C. 1.5
IfΓis a singleton, the problem1.4becomes the followingequilibrium problem. For finding
x∈Csuch that
Fx, y≥0, ∀y∈C. 1.6
The set of solution of1.6is denoted by EPF.
The equilibrium problem include fixed point problems, optimization problems, varia-tional inequalities problems, Nash equilibrium problems, noncooperative games, economics and themixedequilibrium problems as special casessee, e.g.,3–8. Some methods have been proposed to solve the equilibrium problem, see, for instance,9–17.
Recall that, a mappingT :C → Cis said to benonexpansiveif
Tx−Ty≤x−y, ∀x, y∈C. 1.7
We denote the set offixed pointsofT byFT, that isFT {x∈C:xTx}.
Definition 1.1. A family S {Ss : 0 ≤ s ≤ ∞} of mappings of Cinto itself is called a
nonexpansive semigrouponCif it satisfies the following conditions:
1S0xx, for allx∈C;
2Sst SsSt, for alls, t≥0;
3Ssx−Ssy ≤ x−y, for allx, y∈Cands≥0;
4for allx∈C,s→Ssxis continuous.
We denoted byFSthe set of all common fixed points ofS {Ss :s≥0}, that is,
FS s≥0FSs. It is know thatFSis closed and convex.
LetB : H → Hbe a single-valued nonlinear mapping andM : H → 2H be a
set-valued mapping. Thevariational inclusion problemis to findx∈Hsuch that
whereθis the zero vecter inH. The set of solutions of problem1.8is denoted byIB, M. A set-valued mappingM : H → 2H is calledmonotoneif for allx, y ∈ H, f ∈ Mxand g ∈Myimplyx−y, f−g ≥0. A monotone mappingMismaximalif its graphGM:
{f, x ∈ H ×H : f ∈ Mx} of Mis not properly contained in the graph of any other monotone mapping. It is known that a monotone mappingMis maximal if and only if for
x, f∈H×H,x−y, f−g ≥0 for ally, g∈GMimplyf∈Mx.
Definition 1.2. A mappingB : C → H is said to be ak-Lipschitz continous if there exists a constantk >0 such that
Bx−By≤kx−y, ∀x, y∈C. 1.9
Definition 1.3. A mappingB:C → His said to be aβ-inverse-strongly monotoneif there exists a constantβ >0 with the property
Bx−By, x−y≥βBx−By2, ∀x, y∈C. 1.10
Remark 1.4. It is obvious that any β-inverse-strongly monotone mappings B is monotone and 1/β-Lipschitz continuous. It is easy to see that for anyλconstant is in0,2β, then the mappingI−λBis nonexpansive, whereIis the identity mapping onH.
Definition 1.5. Letη : C×C → H is called Lipschitz continuous, if there exists a constant
L >0 such that
ηx, y≤Lx−y, ∀x, y∈C. 1.11
LetK:C → Rbe a differentiable functional on a convex setC, which is called:
1η-convex18if
Ky− Kx≥ Kx, ηy, x, ∀x, y∈C, 1.12
whereKxis the Fr´echet derivative ofKatx;
2η-strongly convex19if there exists a constantσ >0 such that
Ky− Kx− Kx, ηy, x≥ σ
2x−y
2
, ∀x, y∈C. 1.13
In particular, ifηx, y x−yfor allx, y∈C, thenKis said to bestrongly convex.
Definition 1.6. LetM:H → 2Hbe a set-valued maximal monotone mapping, then the
single-valued mappingJM,λ:H → Hdefined by
JM,λx IλM−1x, x∈H 1.14
R1The resolvent operatorJM,λis single-valued and nonexpansive for allλ >0, that is,
JM,λx−JM,λ
y≤x−y, ∀x, y∈H, ∀λ >0. 1.15
R2The resolvent operatorJM,λis 1-inverse-strongly monotone; see20, that is, JM,λx−JM,λy2≤ x−y, JM,λx−JM,λ
y, ∀x, y∈H. 1.16
R3The solution of problem 1.8is a fixed point of the operatorJM,λI−λBfor all λ >0; see also21, that is,
IB, M FJM,λI−λB, ∀λ >0. 1.17
R4If 0< λ≤2β, then the mappingJM,λI−λB:H → His nonexpansive.
R5IB, Mis closed and convex.
In 2007, Takahashi et al. 22 proved the following strong convergence theorem for a nonexpansive mapping by using the shrinking projection method in mathematical programming. ForC1 Candx1PC1x0, they define a sequence{xn}as follows:
ynαnxn 1−αnTxn,
Cn1
z∈Cn:yn−z≤ xn−z,
xn1PCn1x0, ∀n≥0,
1.18
where 0 ≤ αn < a < 1. They proved that the sequence{xn}generated by1.18converges weakly toz∈FT, wherezPFTx0.
In 2008, S. Takahashi and W. Takahashi23introduced the following iterative scheme for finding a common element of the set of solution of generalized equilibrium problem and the set of fixed points of a nonexpansive mapping in a Hilbert space. They proved the strong convergence theorems under certain appropriate conditions imposed on parameters. Next, Zhang et al.24introduced the following new iterative scheme for finding a common element of the set of solution to the problem1.8and the set of fixed points of a nonexpansive mapping in a real Hilbert space. Starting with an arbitraryx1x∈H, define a sequence{xn}
by
ynJM,λxn−λBxn,
xn1 αnx 1−αnTyn, ∀n≥1,
1.19
to the problem1.8, the set of solutions of an equilibrium problem, and the set of fixed points of a nonexpansive mapping in a Hilbert space.
In 2009, Saeidi 26 introduced a more general iterative algorithm for finding a common element of the set of solution for a system of equilibrium problems and the set of common fixed points for a finite family of nonexpansive mappings and a nonexpansive semigroup. In 2010, Katchang and Kumam27obtained a strong convergence theorem for finding a common element of the set of fixed points of a family of finitely nonexpansive mappings, the set of solutions of a mixed equilibrium problem and the set of solutions of a variational inclusion problem for an inverse-strongly monotone mapping. Let Wn be W-mapping defined by 2.8,f be a contraction mapping andA, B be inverse-strongly monotone mappings. Let JM,λ IλM−1 be the resolvent operator associated with M
and a positive numberλ. Starting with arbitrary initialx1∈H, defined a sequence{xn}by
Fun, y
ϕy−ϕun
1
rn
y−un, un−xn
≥0, ∀y∈C,
ynJM,λun−λAun,
vnJM,λ
yn−λAyn
,
xn1αnγfxn βnxn
1−βnI−αnBWnvn, ∀n≥1.
1.20
They proved that under certain appropriate conditions imposed on{αn},{βn}, and{rn}, the sequence{xn}generated by1.20 converges strongly top ∈ Ω : ∞i1FSi∩IA, M∩
MEPF, ϕ, wherepPΩI−Bγfp. Later, Kumam et al.28proved a strongly convergence theorem of the iterative sequence generated by the shrinking projection method for finding a common element of the set of solutions of generalized mixed equilibrium problems, the set of fixed points of a finite family of quasinonexpansive mappings, and the set of solutions of variational inclusion problems.
Liu et al. 29 introduced a hybrid iterative scheme for finding a common element of the set of solutions of mixed equilibrium problems, the set of common fixed points for nonexpansive semigroup and the set of solution of quasivariational inclusions with multivalued maximal monotone mappings and inverse-strongly monotone mappings. Recently, Jitpeera and Kumam 30 considered a shrinking projection method of finding the common element of the set of common fixed points for a finite family of a ξ-strict pseudocontraction, the set of solutions of a systems of equilibrium problems and the set of solutions of variational inclusions. Then, they proved strong convergence theorems of the iterative sequence generated by the shrinking projection method under some suitable conditions in a real Hilbert space. Very recently, Hao 18 introduced a general iterative method for finding a common element of solution set of quasi variational inclusion problems and of the common fixed point set of an infinite family of nonexpansive mappings.
2. Preliminaries
LetHbe a real Hilbert space andCbe a nonempty closed convex subset ofH. Recall that thenearest pointprojectionPC fromHontoCassigns to eachx ∈Hthe unique point in PCx∈Csatisfying the propertyx−PCxminy∈Cx−y.
The following characterizes the projectionPC. We recall some lemmas which will be needed in the rest of this paper.
Lemma 2.1. For a givenz∈H,u∈C,uPCz⇔ u−z, v−u ≥0, for allv∈C. It is well known thatPCis a firmly nonexpansive mapping ofHontoCand satisfies
PCx−PCy2≤ PCx−PCy, x−y
, ∀x, y∈H. 2.1
Moreover,PCxis characterized by the following properties:PCx∈Cand for allx∈H, y∈C,
x−PCx, y−PCx≤0. 2.2
Lemma 2.2see20. LetM:H → 2Hbe a maximal monotone mapping and letB :H → H be a Lipshitz continuous mapping. Then the mappingLMB:H → 2His a maximal monotone mapping.
Lemma 2.3see31. LetCbe a closed convex subset ofH. Let{xn}be a bounded sequence inH. Assume that
1the weakω-limit setωwxn⊂C,
2for eachz∈C,limn→ ∞xn−zexists.
Then{xn}is weakly convergent to a point inC.
Lemma 2.4see32. Each Hilbert spaceH satisfies Opial’s condition, that is, for any sequence
{xn} ⊂Hwithxn x, the inequalitylim infn→ ∞xn−x<lim infn→ ∞xn−y, hold for each y∈Hwithy /x.
Lemma 2.5 see 33. Each Hilbert space H, satisfies the Kadec-Klee property, that is, for any sequence{xn}withxn xandxn → xtogether implyxn−x → 0.
For solving the system of mixed equilibrium problem, let us assume that functionFk: C×C → R,k1,2, . . . , Nsatisfies the following conditions:
H1Fkis monotone, that is,Fkx, y Fky, x≤0, for allx, y∈C;
H2for each fixedy∈C,x→Fkx, yis convex and upper semicontinuous;
H3for each fixedx∈C,y→Fkx, yis convex.
satisfying (H1)–(H3). Assume that
iη:C×C → HiskLipschitz continuous with constantk >0such that;
aηx, y ηy, x 0, for allx, y∈C,
bη·,·is affine in the first variable,
cfor each fixedx∈C,y→ηx, yis sequentially continuous from the weak topology to the weak topology,
iiK:C → Risη-strongly convex with constantσ >0and its derivativeKis sequentially continuous from the weak topology to the strong topology;
iiifor each x ∈ C, there exist a bounded subset Dx ⊂ Cand zx ∈ Csuch that for any y∈C\Dx,
Fy, zx
ϕzx−ϕ
y1 r K
y− Kx, ηz
x, y
<0. 2.3
For givenr >0, LetKF
r :C → Cbe the mapping defined by:
KFrx
y∈C:Fy, zϕz−ϕy1 r K
y− Kx, ηz, y≥0, ∀z∈C 2.4
for allx∈C. Then the following hold
1KFr is single-valued;
2KF
r is nonexpansive ifKis Lipschitz continuous with constantν >0such thatσ≥kν;
3FKF
r MEPF, ϕ;
4MEPF, ϕis closed and convex.
Lemma 2.7see35. LetV :C → Hbe aξ-strict pseudocontraction, then
1the fixed point setFVofV is closed convex so that the projectionPFVis well defined;
2define a mappingT:C → Hby
Txtx 1−tV x, ∀x∈C. 2.5
Ift∈ξ,1, thenT is a nonexpansive mapping such thatFV FT.
A family of mappings {Vi : C → H}∞i1 is called a family of uniformly ξ-strict
pseudocontractions, if there exists a constantξ∈0,1such that
Vix−Viy2≤
Let {Vi : C → C}∞i1 be a countable family of uniformly ξ-strict pseudocontractions. Let
{Ti:C → C}∞i1be the sequence of nonexpansive mappings defined by2.5, that is,
Tixtx 1−tVix, ∀x∈C, ∀i≥1, t∈ξ,1. 2.7
Let{Ti}be a sequence of nonexpansive mappings ofCinto itself defined by2.7and
let{μi}be a sequence of nonnegative numbers in0,1. For eachn≥1, define a mappingWn
ofCinto itself as follows:
Un,n1 I,
Un,nμnTnUn,n1
1−μn
I, Un,n−1 μn−1Tn−1Un,n
1−μn−1I,
.. .
Un,kμkTkUn,k1
1−μkI, Un,k−1 μk−1Tk−1Un,k
1−μk−1
I,
.. .
Un,2 μ2T2Un,3
1−μ2
I, WnUn,1 μ1T1Un,2
1−μ1
I.
2.8
Such a mappingWnis nonexpansive fromCtoCand it is called theW-mapping generated byT1, T2, . . . , Tnandμ1, μ2, . . . , μn. For eachn, k∈N, let the mappingUn,kbe defined by2.8.
Then we can have the following crucial conclusions concerningWn.
Lemma 2.8see36. LetC be a nonempty closed convex subset of a real Hilbert spaceH. Let T1, T2, . . .be nonexpansive mappings ofCinto itself such that∞i1FTiis nonempty, letμ1, μ2, . . .
be real numbers such that 0 ≤ μi ≤ b < 1 for everyi ≥ 1. Then, for every x ∈ C and k ∈ N,
limn→ ∞Un,kxexists.
Using this lemma, one can define a mappingU∞,k and W : C → Cas follows U∞,kx
limn→ ∞Un,kxand
Wx: lim
n→ ∞Wnxnlim→ ∞Un,1x, ∀x∈C. 2.9
Such a mappingW is called theW-mapping. SinceWn is nonexpansive andFW
∞ i1FTi, W:C → Cis also nonexpansive. Indeed, observe that for eachx, y∈Csuch that
Wx−Wy lim
Lemma 2.9see36. LetCbe a nonempty closed convex subset of a Hilbert spaceH,{Ti:C → C} be a countable family of nonexpansive mappings with∞i1FTi/∅,{μi}be a real sequence such that
0< μi≤b <1, for alli≥1. ThenFW ∞
i1FTi.
Lemma 2.10see37. LetCbe a nonempty closed convex subset of a Hilbert spaceH,{Ti :C → C}be a countable family of nonexpansive mappings with∞i1FTi/∅,{μi}be a real sequence such that0< μi≤b <1, for alli≥1. IfDis any bounded subset ofC, then
lim
n→ ∞supx∈DWx−Wnx0. 2.11
Lemma 2.11see38. LetCbe a nonempty bounded closed convex subset of a Hilbert spaceHand letS{Ss: 0≤s <∞}be a nonexpansive semigroup onC, then for anyh≥0,
lim
t→ ∞supx∈C
1
t t
0
Tsxds−Th
1
t t
0
Tsxds
0. 2.12
Lemma 2.12see39. Let C be a nonempty bounded closed convex subset of H,{xn}be a sequence in C andS{Ss: 0≤s <∞}be a nonexpansive semigroup on C. If the following conditions are satisfied:
1xn z;
2lim sups→ ∞lim supn→ ∞Ssxn−xn0, thenz∈FS.
3. Main Results
In this section, we will introduce an iterative scheme by using a shrinking projection method for finding the common element of the set of common fixed points for nonexpansive semi-groups, the set of common fixed points for an infinite family ofξ-strict pseudocontraction, the set of solutions of a systems of mixed equilibrium problems and the set of solutions of the variational inclusions problem in a real Hilbert space.
Theorem 3.1. LetCbe a nonempty closed convex subset of a real Hilbert spaceH, let{Fk : C× C → R, k 1,2, . . . , N}be a finite family of mixed equilibrium functions satisfying conditions (H1)–(H3). Let S {Ss : 0 ≤ s < ∞} be a nonexpansive semigroup on Cand let {tn} be a positive real divergent sequence. Let{Vi : C → C}∞i1 be a countable family of uniformlyξ-strict
pseudocontractions,{Ti :C → C}∞i1 be the countable family of nonexpansive mappings defined by Tixtx 1−tVix, for allx∈C, for alli≥1,t∈ξ,1,Wnbe theW-mapping defined by2.8 andW be a mapping defined by2.9withFW/∅. LetA, B :C → H beγ, β-inverse-strongly monotone mappings andM1, M2:H → 2Hbe maximal monotone mappings such that
Θ:FS∩FW∩
N
k1
SMEPFk
Letrk >0,k 1,2, . . . , N, which are constants. Let{xn},{yn},{vn},{zn}, and{un}be sequences generated byx0∈C,C1C,x1PC1x0,un∈Cand
x0x∈Cchosen arbitrarily,
unKrFN,nNK
FN−1 rN−1,nK
FN−2 rN−2,n· · ·K
F2 r2,nK
F1 r1,nxn,
ynJM2,δnun−δnBun,
vnJM1,λn
yn−λnAyn
,
znαnvn 1−αn1 tn
tn
0
SsWnvnds,
Cn1
⎧ ⎨
⎩z∈Cn:zn−z2≤ xn−z2−αn1−αn vn−
1
tn tn
0
SsWnvnds
2⎫⎬
⎭,
xn1PCn1x0, n∈N,
3.2
whereKFk
rk :C → C,k1,2, . . . , Nis the mapping defined by2.4and{αn}be a sequence in0,1
for alln∈N. Assume the following conditions are satisfied:
C1ηk:C×C → HisLk-Lipschitz continuous with constantk1,2, . . . , Nsuch that
aηkx, y ηky, x 0, for allx, y∈C,
bx→ηkx, yis affine,
cfor each fixedy∈C,y→ηkx, yis sequentially continuous from the weak topology to the weak topology;
C2Kk:C → Risηk-strongly convex with constantσk>0and its derivativeKkis not only sequentially continuous from the weak topology to the strong topology but also Lipschitz continuous with a Lipschitz constantνk>0such thatσk> Lkνk;
C3for eachk ∈ {1,2, . . . , N}and for allx ∈ C, there exist a bounded subsetDx ⊂ Cand zx ∈Csuch that for anyy∈C\Dx,
Fk
y, zx
ϕzx−ϕ
y 1 rk K
y− Kx, ηz
x, y
<0, 3.3
C4{αn} ⊂c, d, for somec, d∈ξ,1;
C5{λn} ⊂a1, b1, for somea1, b1∈0,2γ;
C6{δn} ⊂a2, b2, for somea2, b2∈0,2β;
C7lim infn→ ∞rk,n>0, for eachk∈1,2,3, . . . , N.
Proof. Pick anyp ∈ Θ. TakingIk
n KrFk,nk K
Fk−1 rk−1,nK
Fk−2 rk−2,n· · ·K
F2 r2,nK
F1
r1,n fork ∈ {1,2,3, . . . , N}and I0
n I for alln ∈N. From the definition ofKrFk,nk is nonexpansive for eachk 1,2,3, . . . , N, thenIk
nalso andpIFrk,nk p, we note thatunI
N
nxn. If follows that
un−pINnxn−INnp≤xn−p. 3.4
Next, we will divide the proof into eight steps.
Step 1. We first show by induction thatΘ⊂Cnfor eachn≥1.
Takingp∈Θ, we get thatpJM1,λkp−λkAp JM2,δkp−δkBp. SinceJM1,λk, JM2,δk are nonexpansive. From the assumption, we see thatΘ⊂CC1. SupposeΘ⊂Ckfor some
k≥1. For anyp∈Θ Ck, we have
vk−pJM1,λk
yk−λkAyk
−JM1,λk
p−λkAp
≤yk−λkAyk−p−λkAp
≤I−λkAyk−I−λkAp
≤yk−p,
3.5
yk−pJM2,δkuk−δkBuk−JM2,δk
p−δkBp
≤uk−δkBuk−p−δkBp
≤uk−p
≤xk−p,
3.6
which yields
zk−p2
αkvk−p 1−αk
1
tk tk
0
SsWkvkds−p
2
≤αkvk−p2 1−αk 1 tk tk 0
SsWkvkds−p
2
−αk1−αk vk− 1 tk tk 0
SsWkvkds
2
≤αkvk−p2 1−αkvk−p2−αk1−αkvk− 1 tk
tk
0
SsWkvkds
2
≤vk−p2−αk1−αkvk− 1 tk
tk
0
SsWkvkds
2
.
Applying3.5and3.6, we get
zk−p2≤
xk−p2−αk1−αk vk−
1
tk tk
0
SsWkvkds
2
. 3.8
Hencep∈Ck1. This implies thatΘ⊂Cnfor eachn≥1.
Step 2. Next, we show that{xn}is well defined andCnis closed and convex for anyn∈N. It is obvious thatC1 Cis closed and convex. Suppose thatCkis closed and convex
for somek≥1. Now, we show thatCk1is closed and convex for somek. For anyp∈Ck, we
obtain
zk−p2≤
xk−p2 3.9
is equivalent to
zk−xk22 zk−xk, xk−p≤0. 3.10
ThusCk1 is closed and convex. Then,Cn is closed and convex for anyn ∈ N. This implies
that{xn}is well-defined.
Step 3. Next, we show that{xn}is bounded and limn→ ∞xn−x0exists. FromxnPCnx0, we have
x0−xn, xn−y
≥0, 3.11
for eachy∈Cn. UsingΘ⊂Cn, we also have
x0−xn, xn−p
≥0, ∀p∈Θ, n∈N. 3.12
So, forp∈Θ, we observe that
0≤ x0−xn, xn−p
x0−xn, xn−x0x0−p
−x0−xn, x0−xn x0−xn, x0−p
≤ −x0−xn2x0−xnx0−p.
3.13
This implies that
Hence, we get{xn}is bounded. It follows by3.5–3.7, that{vn},{yn}, and{Wnvn}are also bounded. FromxnPCnx0, andxn1PCn1x0∈Cn1⊂Cn, we obtain
x0−xn, xn−xn1 ≥0. 3.15
It follows that, we have for eachn∈N
0≤ x0−xn, xn−xn1
x0−xn, xn−x0x0−xn1
−x0−xn, x0−xnx0−xn, x0−xn1
≤ −x0−xn2x0−xnx0−xn1.
3.16
It follows that
x0−xn ≤ x0−xn1. 3.17
Thus, since the sequence {xn − x0} is a bounded and nondecreasing sequence, so
limn→ ∞xn−x0exists, that is
m lim
n→ ∞xn−x0. 3.18
Step 4. Next, we show that limn→ ∞xn1−xn0 and limn→ ∞xn−zn0.
Applying3.15, we get
xn−xn12xn−x0x0−xn12
xn−x022xn−x0, x0−xn1x0−xn12
xn−x022xn−x0, x0−xnxn−xn1x0−xn12
xn−x02−2xn−x0, xn−x02xn−x0, xn−xn1x0−xn12
−xn−x022xn−x0, xn−xn1x0−xn12
≤ −xn−x02x0−xn12.
3.19
Thus, by3.18, we obtain
lim
On the other hand, fromxn1PCn1x0∈Cn1⊂Cn, which implies that
xn1−zn ≤ xn1−xn. 3.21
It follows by3.21, we also have
zn−xn ≤ zn−xn1xn1−xn ≤2xn−xn1. 3.22
By3.20, we obtain
lim
n→ ∞xn−zn0. 3.23
Step 5. Next, we show that
lim
n→ ∞ Ik
nxn−Ikn−1xn0 3.24
for everyk ∈ {1,2,3, . . . , N}. Indeed, forp ∈Θ, note thatKFk
rk,nis the firmly nonexpansie, so we have
Ik
nxn−Iknp
2
KFk
rk,nI
k−1
n xn−KFrk,nk p
2
≤Ik
nxn−p,Ikn−1xn−p
1
2
Ik
nxn−p
2
Ik−1 n xn−p
2
−Ik
nxn−Ikn−1xn
2
.
3.25
Thus, we get
Ik
nxn−Iknp
2
≤Ik−1 n xn−p
2
−Ik
nxn−Ikn−1xn
2
. 3.26
It follows that
un−p2≤Iknxn−Iknp
2
≤Ik−1
n xn−p
2
−Ik
nxn−Ikn−1xn
2
≤xn−p2−Iknxn−Ikn−1xn
2
.
By3.5,3.6,3.7, and3.27, we have for eachk∈ {1,2,3, . . . , N}
zn−p2≤vn−p2
≤un−p2
≤xn−p2−Inkxn−Ikn−1xn2.
3.28
Consequently, we have
Ik
nxn−Ikn−1xn
2
≤xn−p2−zn−p2
≤ xn−znxn−pzn−p.
3.29
Equation3.23implies that for everyk∈ {1,2,3, . . . , N}
lim
n→ ∞
Ik
nxn−Ikn−1xn0. 3.30
Step 6. Next, we show that limn→ ∞yn −vn 0 and limn→ ∞KnWnvn −vn 0, where
Kn 1/tn tn
0 Ssds.
For any givenp ∈Θ,λn ∈ 0,2γ,δn ∈0,2βandp JM1,λnp−λnAp JM2,δnp−
δnBp. SinceI−λnAandI−δnBare nonexpansive, we have
vn−p2 JM1,λnyn−λnAyn−JM1,λnp−λnAp
2
≤yn−λnAyn−p−λnAp2
yn−p−λnAyn−Ap2
≤yn−p2−2λnyn−p, Ayn−Apλ2nAyn−Ap2
≤xn−p2−2λnγAyn−Ap2λ2nAyn−Ap2
≤xn−p2λn
λn−2γAyn−Ap2.
3.31
Similarly, we can show that
yn−p2≤
Observe that
zn−p2
αnvn−p 1−αn
1
tn tn
0
SsWnvnds−p
2
≤αnvn−p2 1−αn
1
tn tn
0
SsWnvnds−p
2
−αn1−αn vn−
1
tn tn
0
SsWnvnds
2
≤αnvn−p2 1−αn1 tn
tn
0
SsWnvnds−p
2
≤αnxn−p2 1−αnvn−p2.
3.33
Substituting3.31into3.33and using conditionsC4andC5, we have
zn−p2≤αnxn−p2 1−αn
xn−p2λn
λn−2γAyn−Ap2
xn−p2 1−αnλn
λn−2γAyn−Ap2.
3.34
It follows that
1−da1
2γ−b1Ayn−Ap2≤1−αnλn
2γ−λnAyn−Ap2
≤xn−p2−zn−p2
≤ xn−znxn−pzn−p.
3.35
By3.23, we obtain
lim
Since the resolvent operatorJM1,λn is 1-inverse-strongly monotone, we obtain
vn−p2JM1,λn
yn−λnAyn
−JM1,λn
p−λnAp2
JM1,λnI−λnAyn−JM1,λnI−λnAp
2
≤ I−λnAyn−I−λnAp, vn−p
1
2
I−λnAyn−I−λnAp2vn−p2
−I−λnAyn−I−λnAp−vn−p2
≤ 1
2
yn−p2vn−p2−yn−vn
−λn
Ayn−Ap2
≤ 1
2
xn−p2vn−p2−yn−vn2
−λ2nAyn−Ap22λnyn−vn, Ayn−Ap,
3.37
which yields
vn−p2 ≤
xn−p2−yn−vn22λnyn−vnAyn−Ap. 3.38
Similarly, we can obtain
yn−p2 ≤
xn−p2−un−yn22δnun−ynBun−Bp. 3.39
Substituting3.38into3.33, and using conditionC4andC5, we have
zn−p2≤αnxn−p2 1−αnvn−p2
≤αnxn−p2 1−αnxn−p2−yn−vn22λnyn−vnAyn−Ap
xn−p2−1−αnyn−vn221−αnλnyn−vnAyn−Ap.
3.40
It follows that
1−αnyn−vn2 ≤xn−p2−zn−p221−αnλnyn−vnAyn−Ap
≤ xn−znxn−pzn−p21−αnλnyn−vnAyn−Ap.
3.41
By3.23and3.36, we get
lim
From3.8andC4, we also have
αn1−αn vn−
1
tn tn
0
SsWnvnds
2
≤xn−p2−zn−p2
≤ xn−znxn−pzn−p.
3.43
SinceKn 1/tn tn
0 Ssds, we obtain3.23, we have
lim
n→ ∞KnWnvn−vn0. 3.44
Since{Wnvn}is a bounded sequence inC, fromLemma 2.11for allh≥0, we have
lim
n→ ∞KnWnvn−ShKnWnvn
lim
n→ ∞
1
tn tn
0
SsWnvnds−Sh
1
tn tn
0
SsWnvnds
0.
3.45
From3.44and3.45, we get
vn−Ssvn ≤ vn− KnWnvnKnWnvn−SsKnWnvnSsKnWnvn−Ssvn
≤2vn− KnWnvnKnWnvn−SsKnWnvn.
3.46
So, we have
lim
n→ ∞vn−Ssvn0. 3.47
Step 7. Next, we show thatq∈Θ:FS∩FW∩Nk1SMEPFk∩IA, M1∩IB, M2/∅.
Since{vni} is bounded, there exists a subsequence {vnij} of {vni} which converges weakly toq∈C. Without loss of generality, we can assume thatvni q.
1 First, we prove thatq ∈ FS. Indeed, from Lemma 2.12and 3.47, we getq ∈ FS, that is,qSsq, for alls≥0.
2We show thatq ∈ FW n∞1FWn, whereFWn ∞
i1FTi, for alln ≥ 1
andFWn1 ⊂ FWn. Assume thatq /∈ FW, then there exists a positive integermsuch
thatq /∈ FTmand soq /∈ mi1FTi. Hence for anyn≥ m,q /∈ ni1FTi FWn, that is,
therefore we haveq /KnWnq, for alln≥m. It follows from the Opial’s condition and3.44
that
lim inf
i→ ∞ vni−q<lim inf
i→ ∞ vni− KniWniq
≤lim inf
i→ ∞
vni− KniWnivniKniWnivni− KniWniq
≤lim inf
i→ ∞ vni−q,
3.48
which is a contradiction. Thus, we getq∈FW.
3We prove thatq∈kN1SMEPFk, ϕ. SinceIkn KFrkk,k1,2, . . . , Nandu
k
nIknxn,
we have
Fk
Ik nxn, x
ϕx−ϕIknxn
1
rk
KIk nxn
− KIk−1
n xn
, ηx,Iknxn
≥0, ∀x∈C.
3.49
It follows that
1
rk
KIk nixni
− KIk−1
ni xni
, ηx,Iknixni
≥ −FkIknixni, x
−ϕx ϕIknixni
3.50
for allx∈C. From3.30and by conditionsC1candC2, we get
lim
ni→ ∞ 1
rk
KIk nixni
− KIk−1
ni xni
, ηx,Iknixni
0. 3.51
By the assumption and by conditionH1, we know that the function ϕand the mapping
x → −Fkx, yboth are convex and lower semicontinuous, hence they are weakly lower
semicontinuous.
These together withKIk
nixni− KIkn−1i xni/rk → 0 andIknixni q, we have
0lim inf
ni→ ∞
KIk nixni
− KIk−1
ni xni
rk , η
x,Ik
nixni
≥lim inf
ni→ ∞
−Fk
Ik nixni, x
−ϕx ϕIknixni
.
3.52
Then, we obtain
Fk
q, xϕx−ϕq≥0, ∀x∈C, ∀k,1,2, . . . , N. 3.53
Thereforeq∈Nk1SMEPFk, ϕ.
We observe that A is an 1/γ-Lipschitz monotone mapping and DA H. From
Lemma 2.2, we know thatM1 A is maximal monotone. Letv, g ∈ GM1 A that is,
g−Av∈M1v. Sincevni JM1,λniyni−λniAyni, we have
yni−λniAyni ∈IλniM1vni, 3.54
that is,
1
λni
yni−vni−λniAyni
∈M1vni. 3.55
By virtue of the maximal monotonicity ofM1A, we have
v−vni, g−Av− 1
λni
yni−vni−λniAyni
!
≥0, 3.56
and so
v−vni, g ≥ v−vni, Av 1
λni
yni−vni −λniAyni
!
v−vni, Av−AvniAvni−Ayni 1
λni
yni−vni
!
≥0 v−vni, Avni−Ayni
v−vni, 1
λni
yni−vni
! .
3.57
By3.42,vni qandAis inverse-strongly monotone, we obtain that limn→ ∞Ayn−Avn0 and it follows that
lim
ni→ ∞
v−vni, g
v−q, g≥0. 3.58
It follows from the maximal monotonicity ofM1Athatθ∈M1Aq, that is,q∈IA, M1.
Since{yni}is bounded, there exists a subsequence{ynij}of{yni}which converges weakly to
q∈C. Without loss of generality, we can assume thatyni q. In similar way, we can obtain
q∈IB, M2, henceq∈IA, M1∩IB, M2.
Step 8. Finally, we show thatxn → zandun → z, wherezPΘx0.
SinceΘis nonempty closed convex subset ofH, there exists a uniquez∈Θsuch that
zPΘx0. Sincez∈Θ⊂CnandxnPCnx0, we have
x0−xn ≤ x0−PCnx0 ≤x0−z 3.59
By the weakly lower semicontinuous of the norm, we have
x0−z ≤lim inf
ni→ ∞
x0−xni ≤x0−z. 3.60
However, sincez∈ωwxn⊂Θ, we have
x0−z≤ x0−PCnx0 ≤ x0−z. 3.61
Using3.59and3.60, we obtainzz. Thusωwxn {z}andxn z. So, we have
x0−z≤ x
0−z ≤lim inf
n→ ∞ x0−xn ≤lim supn→ ∞ x0−xn ≤x0−z .
3.62
Thus, we obtain that
x0−z lim
n→ ∞x0−xnx0−z
. 3.63
Fromxn z, we obtainx0−xnx0−z. Using the Kadec-Klee property, we obtain that
xn−zxn−x0−z−x0 −→0 asn−→ ∞ 3.64
and hencexn → zin norm. Finally, noticingun−z INnxn−INnz ≤ xn−z. We also
conclude thatun → zin norm. This completes the proof.
Theorem 3.2. LetCbe a nonempty closed convex subset of a real Hilbert spaceH, let{Fk : C× C → R, k 1,2, . . . , N} be a finite family of mixed equilibrium functions satisfying conditions (H1)–(H3). Let S {Ss : 0 ≤ s < ∞} be a nonexpansive semigroup on Cand let {tn} be a positive real divergent sequence. Let{Vi : C → C}∞i1 be a countable family of uniformlyξ-strict
pseudocontractions,{Ti :C → C}∞i1 be the countable family of nonexpansive mappings defined by
Tixtx 1−tVix, for allx∈C, for alli≥1, t ∈ξ,1,Wn be theW-mapping defined by2.8
andW be a mapping defined by2.9withFW/∅. LetA, B :C → H beγ, β-inverse-strongly monotone mapping. Such that
Θ:FS∩FW∩ N
k1
SMEPFk
Letrk >0,k 1,2, . . . , N, which are constants. Let{xn},{yn},{vn},{zn}, and{un}be sequences generated byx0∈C,C1C,x1PC1x0,un∈Cand
x0x∈Cchosen arbitrarily,
unKFrN,nN K
FN−1 rN−1,nK
FN−2 rN−2,n· · ·K
F2 r2,nK
F1 r1,nxn,
ynPCun−δnBun,
vnPC
yn−λnAyn
,
znαnvn 1−αn1 tn
tn
0
SsWnvnds,
Cn1
⎧ ⎨
⎩z∈Cn:zn−z2≤ xn−z2−αn1−αn vn−
1
tn tn
0
SsWnvnds
2⎫⎬
⎭,
xn1PCn1x0, n∈N,
3.66
whereKFk
rk : C → C, k 1,2, . . . , N is the mapping defined by2.4and{αn}be a sequence in
0,1for alln∈N. Assume the following conditions are satisfied:
C1ηk:C×C → HisLk-Lipschitz continuous with constantk1,2, . . . , Nsuch that
aηkx, y ηky, x 0, for allx, y∈C,
bx→ηkx, yis affine,
cfor each fixedy∈C,y→ηkx, yis sequentially continuous from the weak topology to the weak topology;
C2Kk:C → Risηk-strongly convex with constantσk>0and its derivativeKkis not only sequentially continuous from the weak topology to the strong topology but also Lipschitz continuous with a Lipschitz constantνk>0such thatσk> Lkνk;
C3for eachk ∈ {1,2, . . . , N}and for allx ∈ C, there exist a bounded subsetDx ⊂ Cand zx ∈Csuch that for anyy∈C\Dx,
Fk
y, zx
ϕzx−ϕ
y 1 rk K
y− Kx, ηz
x, y
<0, 3.67
C4{αn} ⊂c, d, for somec, d∈ξ,1;
C5{λn} ⊂a1, b1, for somea1, b1∈0,2γ;
C6{δn} ⊂a2, b2, for somea2, b2∈0,2β;
C7lim infn→ ∞rk,n>0, for eachk∈1,2,3, . . . , N.
Proof. InTheorem 3.1, takeMi iC : H → 2H, whereiC : 0 → 0,∞is the indicator
function ofC, that is,
iCx ⎧ ⎨ ⎩
0, x∈C,
∞, x /∈C,
3.68
fori 1,2. Then1.8is equivalent to variational inequality problem, that is, to findx ∈ C
such that
Ax, y −x≥0, ∀y∈C. 3.69
Again, sinceMiiC, fori1,2, then
JM1,λn PC JM2,δn. 3.70
So, we have
vnPC
yn−λnAyn
JM1,λn
yn−λnAyn
,
ynPCun−δnBun JM2,δnun−δnBun.
3.71
Hence, we can obtain the desired conclusion fromTheorem 3.1immediately.
Next, we consider another class of important mappings.
Definition 3.3. A mapping S : C → C is called strictly pseudocontraction if there exists a constant 0≤κ <1 such that
Sx−Sy2≤x−y2κI−Sx−I−Sy2, ∀x, y∈C. 3.72
If κ 0, then S is nonexpansive. In this case, we say that S : C → C is a κ-strictly pseudocontraction. PuttingBI−S. Then, we have
I−Bx−I−By2 ≤
x−y2κBx−By2, ∀x, y∈C. 3.73
Observe that
I−Bx−I−By2
x−y2Bx−By2−2 x−y, Bx−By, ∀x, y∈C. 3.74
Hence, we obtain
x−y, Bx−By≥ 1−κ
2 Bx−By
2
, ∀x, y∈C. 3.75
Now, we obtain the following result.
Theorem 3.4. LetCbe a nonempty closed convex subset of a real Hilbert spaceH, let{Fk : C× C → R, k 1,2, . . . , N}be a finite family of mixed equilibrium functions satisfying conditions (H1)–(H3). Let S {Ss : 0 ≤ s < ∞} be a nonexpansive semigroup on Cand let {tn} be a positive real divergent sequence. Let{Vi : C → C}∞i1 be a countable family of uniformlyξ-strict
pseudocontractions,{Ti :C → C}∞i1 be the countable family of nonexpansive mappings defined by
Tixtx 1−tVix, for allx∈C, for alli≥1,t∈ξ,1,Wnbe theW-mapping defined by2.8
andW be a mapping defined by2.9withFW/∅. LetA, B :C → H beγ, β-inverse-strongly monotone mapping andSA, SB beκγ, κβ-strictly pseudocontraction mapping of Cinto Cfor some
0≤κγ<1,0≤κβ<1such that
Θ:FS∩FW∩ N
k1
SMEPFk
∩FSA∩FSB/∅. 3.76
Letrk >0,k 1,2, . . . , N, which are constants. Let{xn},{yn},{vn},{zn}, and{un}be sequences generated byx0∈C,C1C,x1PC1x0,un∈Cand
x0x∈Cchosen arbitrarily,
un KFN
rN,nK
FN−1 rN−1,nK
FN−2 rN−2,n· · ·K
F2 r2,nK
F1 r1,nxn,
yn 1−δnunδnSBun,
vn 1−λnynλnSAyn,
znαnvn 1−αn
1
tn tn
0
SsWnvnds,
Cn1
⎧ ⎨
⎩z∈Cn:zn−z2≤ xn−z2−αn1−αn vn−
1
tn tn
0
SsWnvnds
2⎫⎬
⎭,
xn1PCn1x0, n∈N,
3.77
whereKFk
rk :C → C,k1,2, . . . , Nis the mapping defined by2.4and{αn}be a sequence in0,1
for alln∈N. Assume the following conditions are satisfied:
C1ηk:C×C → HisLk-Lipschitz continuous with constantk1,2, . . . , Nsuch that
aηkx, y ηky, x 0, for allx, y∈C,
bx→ηkx, yis affine,
cfor each fixedy∈C,y→ηkx, yis sequentially continuous from the weak topology to the weak topology;
C3for eachk ∈ {1,2, . . . , N}and for allx ∈ C, there exist a bounded subsetDx ⊂ Cand zx ∈Csuch that for anyy∈C\Dx,
Fk
y, zx
ϕzx−ϕ
y 1 rk K
y− Kx, ηz
x, y
<0; 3.78
C4{αn} ⊂c, d, for somec, d∈ξ,1;
C5{λn} ⊂a1, b1, for somea1, b1∈0,2γ;
C6{δn} ⊂a2, b2, for somea2, b2∈0,2β;
C7lim infn→ ∞rk,n>0, for eachk∈1,2,3, . . . , N
Then,{xn}and{un}converge strongly tozPΘx0.
Proof. TakingA≡I−SAandB≡I−SB, then we see thatA, Bis1−κγ/2,1−κβ/ 2-inverse-strongly monotone mapping, respectively. We haveFSA VIC, AandFSB VIC, B.
So, we have
yn PCun−δnBun PC1−δnunδnSBun 1−δnunδnSBun∈C,
vnPC
yn−λnAyn
PC
1−λnynλnSAyn
1−λnynλnSAyn∈C.
3.79
By usingTheorem 3.2, it is easy to obtain the desired conclusion.
Acknowledgments
The authors would like to thank the Faculty of Science, King Monkut’s University of Tech-nology Thonburi for its financial support. Moreover, P. Kumam was supported by the Com-mission on Higher Education and the Thailand Research Fund under Grant MRG5380044.
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