Volume 2010, Article ID 361512,33pages doi:10.1155/2010/361512
Research Article
A System of Generalized Mixed Equilibrium
Problems and Fixed Point Problems for
Pseudocontractive Mappings in Hilbert Spaces
Poom Kumam
1and Chaichana Jaiboon
21Department of Mathematics, Faculty of Science, King Mongkut’s University of Technology Thonburi,
KMUTT, Bangkok 10140, Thailand
2Department of Mathematics, Faculty of Liberal Arts, Rajamangala University of Technology Rattanakosin,
RMUTR, Bangkok 10100, Thailand
Correspondence should be addressed to Chaichana Jaiboon,[email protected]
Received 2 April 2010; Accepted 11 June 2010
Academic Editor: A. T. M. Lau
Copyrightq2010 P. Kumam and C. Jaiboon. 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 and analyze a new iterative algorithm for finding a common element of the set of fixed points of strict pseudocontractions, the set of common solutions of a system of generalized mixed equilibrium problems, and the set of common solutions of the variational inequalities with inverse-strongly monotone mappings in Hilbert spaces. Furthermore, we prove new strong convergence theorems for a new iterative algorithm under some mild conditions. Finally, we also apply our results for solving convex feasibility problems in Hilbert spaces. The results obtained in this paper improve and extend the corresponding results announced by Qin and Kang2010and the previously known results in this area.
1. Introduction
Let H be a real Hilbert space with inner product ·,· and norm · and let E be a nonempty closed convex subset ofH. We denote weak convergence and strong convergence by notationsand →, respectively. LetS:E → Ebe a mapping. In the sequel, we will use
FSto denote the set offixed pointsofS, that is,FS {x∈E:Sxx}.
Definition 1.1. LetS:E → Ebe a mapping. ThenSis called
1contractionif there exists a constantα∈0,1such that
2nonexpansive if
Sx−Sy ≤ x−y, ∀x, y∈E. 1.2
Remark 1.2. It is well known that ifE⊂His nonempty, bounded, closed, and convex andS
is a nonexpansive mapping onEthenFSis nonempty; see, for example,1.
3strongly pseudocontractivewith the coefficientτ ∈0,1if
Sx−Sy, x−y≥τx−y2, ∀x, y∈E, 1.3
4strictly pseudocontractivewith the coefficientk∈0,1if
Sx−Sy2≤x−y2kI−Sx−I−Sy2, ∀x, y∈E; 1.4
for such a case,Sis also said to be ak-strict pseudocontraction,and ifk0, thenSis a nonexpansive mapping,
5pseudocontractive if
Sx−Sy2≤
x−y2I−Sx−I−Sy2, ∀x, y∈E. 1.5
The class of strict pseudocontractions falls into the one between classes of nonex-pansive mappings and pseudocontractions. Within the past several decades, many authors have been devoting to the studies on the existence and convergence of fixed points for strict pseudocontractions.
In 1967, Browder and Petryshyn2introduced a convex combination method to study strict pseudocontractions in Hilbert spaces. On the other hand, Marino and Xu3and Zhou 4 introduced and researched some iterative scheme for finding a fixed point of a strict pseudocontraction mapping. More precisely, takek∈0,1and define a mappingSkby
Skxkx 1−kSx, ∀x∈E, 1.6
whereSis a strict pseudocontraction. Under appropriate restrictions onk, it is proved the mappingSkis nonexpansive. Therefore, the techniques of studying nonexpansive mappings can be applied to study more general strict pseudocontractions.
Thedomainof the functionϕ:E → R ∪ {∞}is the set
domϕx∈E:ϕx<∞. 1.7
Letϕ:E → R ∪ {∞}be a proper extended real-valued function and letΦbe a bifunction of
There exists thegeneralized mixed equilibrium problemfor findingx∈Esuch that
Φx, yΨx, y−xϕy−ϕx≥0, ∀y∈E. 1.8
The set of solutions of1.8is denoted by GMEPΦ, ϕ,Ψ,that is,
GMEPΦ, ϕ,Ψx∈E:Φx, yΨx, y−xϕy−ϕx≥0, ∀y∈E. 1.9
We see thatxis a solution of problem1.8implies thatx∈domϕ.
Special Examples
1IfΨ 0, problem1.8is reduced into themixed equilibrium problemfor findingx∈E
such that
Φx, yϕy−ϕx≥0, ∀y∈E. 1.10
Problem1.10was studied by Ceng and Yao5. The set of solutions of1.10is denoted by MEPΦ, ϕ.
2Ifϕ0, problem1.8is reduced into thegeneralized equilibrium problemfor finding
x∈Esuch that
Φx, yΨx, y−x≥0, ∀y∈E. 1.11
Problem1.11was studied by Takahashi and Toyoda6. The set of solutions of 1.11is denoted by GEPΦ,Ψ.
3IfΨ 0 andϕ0, problem1.8is reduced into theequilibrium problem for finding
x∈Esuch that
Φx, y≥0, ∀y∈E. 1.12
Problem1.12was studied by Blum and Oettli7. The set of solutions of1.12is denoted by EPΦ.
4IfΦ 0, problem1.8is reduced into themixed variational inequality of Browder type
for findingx∈Esuch that
Ψx, y−xϕy−ϕx≥0, ∀y∈E. 1.13
Problem1.13was studied by Browder8. The set of solutions of1.13is denoted by VIE,Ψ, ϕ.
5IfΦ 0 andϕ0, problem1.8is reduced into thevariational inequality problemfor findingx∈Esuch that
Problem1.14was studied by Hartman and Stampacchia9. The set of solutions of 1.14 is denoted by VIE,Ψ. The variational inequality has been extensively studied in the literature. See, for example,7,10,11and the references therein. 6IfΦ 0 andΨ 0, problem1.8is reduced into theminimize problemfor finding
x∈Esuch that
ϕy−ϕx≥0, ∀y∈E. 1.15
The set of solutions of1.15is denoted by Argminϕ.
The generalized mixed equilibrium problems include fixed point problems, variational inequality problems, optimization problems, Nash equilibrium problems, and the equilib-rium problem as special cases. Numerous problems in physics, optimization, and economics reduce to find a solution of 1.8. In 1997, Combettes and Hirstoaga 12 introduced an iterative scheme of finding the best approximation to initial data when EPΦis nonempty and proved a strong convergence theorem. Many authors have proposed some useful methods for solving the GMEPΦ, ϕ,Ψ, MEPΦ, ϕ,and EPΦ; see, for instance,5,12–23.
Definition 1.3. LetB:E → Hbe a nonlinear mapping. ThenBis called
1monotoneif
Bx−By, x−y≥0, ∀x, y∈E, 1.16
2β-strongly monotone if there exists a constantβ >0 such that
Bx−By, x−y≥βx−y2, ∀x, y∈E, 1.17
3η-Lipschitz continuousif there exists a positive real numberηsuch that
Bx−By≤ηx−y, ∀x, y∈E, 1.18
4β-inverse-strongly monotoneif there exists a constantβ >0 such that
Bx−By, x−y≥βBx−By2, ∀x, y∈E. 1.19
Remark 1.4. It is obvious that anyβ-inverse-strongly monotone mappingsBare monotone
and 1/β-Lipschitz continuous.
For finding a common element of the set of fixed points of a nonexpansive mapping and the set of solution of variational inequalities for aβ-inverse-strongly monotone mapping, Takahashi and Toyoda6introduced the following iterative scheme:
x0∈Echosen arbitrary,
xn1αnxn 1−αnSPExn−λnBxn, ∀n≥0,
wherePEis the metric projection ofHontoE,Bis aβ-inverse-strongly monotone mapping,
{αn}is a sequence in 0,1, and{λn} is a sequence in0,2β. They showed that if FS∩
VIE, Bis nonempty, then the sequence{xn}generated by1.20converges weakly to some q∈FS∩VIE, B.
On the other hand, Y. Yao and J.-C Yao24introduced the following iterative process defined recursively by
x1x∈Echosen arbitrary,
ynPExn−λnBxn,
xn1αnxβnxnγnSPE
yn−λnByn
, ∀n≥1,
1.21
where Bis aβ-inverse-strongly monotone mapping, {αn},{βn},and {γn}are sequences in the interval0,1, and{λn}is a sequence in0,2β. They showed that ifFS∩VIE, Bis
nonempty, then the sequence{xn}generated by1.21converges strongly to someq∈FS∩
VIE, B.
LetAbe a strongly positive linear bounded operator onHif there is a constantγ >0 with property
Ax, x ≥γx2, ∀x∈H. 1.22
A typical problem is to minimize a quadratic function over the set of the fixed points a nonexpansive mapping on a real Hilbert spaceH:
min
x∈E
1
2Ax, x − x, b, 1.23
whereAis a linear bounded operator,Eis the fixed point set of a nonexpansive mappingS
onH,andbis a given point inH.Moreover, it is shown in25that the sequence{xn}defined
by the scheme
xn1 nγfxn 1−nASxn 1.24
converges strongly to q PFSI − A γfq. Recently, Plubtieng and Punpaeng 26
proposed the following iterative algorithm:
Φun, y
1
rn
y−un, un−xn
≥0, ∀y∈H,
xn1nγfxn I−nASun.
1.25
They proved that if the sequences{n}and{rn}of parameters satisfy appropriate condition,
then the sequences{xn}and{un}both converge to the unique solutionqof the variational inequality:
which is the optimality condition for the minimization problem:
min
x∈FS∩EPφ
1
2Ax, x −hx, 1.27
wherehis a potential function forγfi.e.,hx γfxforx∈H.
Very recently, Ceng et al. 27 introduced iterative scheme for finding a common element of the set of solutions of equilibrium problems and the of fixed points of ak-strict pseudocontraction mapping defined in the setting of real Hilbert spaceH:x0∈Hand let
Φun, y
1
rn
y−un, un−xn
≥0, ∀y∈E,
xn1αnun 1−αnSun,
1.28
where{αn} ⊂a, bfor somea, b∈k,1and{rn} ⊂0,∞satisfies lim infn→ ∞rn>0. Further,
they proved that{xn}and{un}converge weakly toq∈FS∩EPΦ, whereqPFS∩EPΦx0.
On the other hand, for finding a common element of the set of fixed points of ak -strict pseudocontraction mapping and the set of solutions of an equilibrium problems in a real Hilbert space, Liu28introduced the following iterative scheme:
Φun, y
1
rn
y−un, un−xn
≥0, ∀y∈E,
ynβnun1−βnSun,
xn1nγfxn I−nAyn, ∀n≥1,
1.29
where S is a k-strict pseudocontraction mapping and {n} and {βn} are sequences in
0,1.They proved that under certain appropriate conditions over{n},{βn}, and{rn}, the
In 2008, Ceng and Yao5introduced an iterative scheme for finding a common fixed point of a finite family of nonexpansive mappings and the set of solutions of a problem1.8
in Hilbert spaces and obtained the strong convergence theorem which used the following condition.
GK:E → R is η-strongly convex with constant σ > 0 and its derivative K is sequentially continuous from the weak topology to the strong topology. We note that the conditionGfor the functionK:E → Ris a very strong condition. We also note that the conditionGdoes not cover the caseKx x2/2 andηx, y x−yfor eachx, y∈E×E. Very recently, Wangkeeree and Wangkeeree29introduced a general iterative method for finding a common element of the set of solutions of the mixed equilibrium problems, the set of fixed point of a k-strict pseudocontraction mapping, and the set of solutions of the variational inequality for an inverse-strongly monotone mapping in Hilbert spaces. They obtained a strong convergence theorem except the conditionGfor the sequences generated by these processes.
In 2009, Qin and Kang30introduced an explicit viscosity approximation method for finding a common element of the set of fixed points of strict pseudocontraction and the set of solutions of variational inequalities with inverse-strongly monotone mappings in Hilbert spaces. Let{xn}be a sequence generated by the following iterative algorithm:
x1∈E,
znPExn−μnCxn,
ynPExn−λnBxn,
xn1nfxn βnxnγn αn1Skxnαn2ynαn3zn
, ∀n≥1.
1.30
Then, they proved that under certain appropriate conditions imposed on {n}, {βn},{γn},
{αn1}, {αn2}, and {αn3}, the sequence {xn}generated by 1.30converges strongly to q ∈ FS∩VIE, B∩VIE, C, whereqPFS∩VIE,B∩VIE,Cfq.
In the present paper, motivated and inspired by Qin and Kang30, Peng and Yao21, Plubtieng and Punpaeng26, and Liu28, we introduce a new general iterative scheme for finding a common element of the set of fixed points of strict pseudocontractions, the set of common solutions of the system of generalized mixed equilibrium problems, and the set of common solutions of the variational inequalities for inverse-strongly monotone mappings in Hilbert spaces. We obtain a strong convergence theorem for the sequences generated by these processes under some parameter controlling conditions. The results in this paper extend and improve the corresponding recent results of Qin and Kang30, Peng and Yao21, Plubtieng and Punpaeng26, and Liu28and many others.
2. Preliminaries
LetHbe a real Hilbert space and letEbe a nonempty closed convex subset ofH. In a real Hilbert spaceH, it is well known that
λx 1−λy2
For anyx∈H, there exists aunique nearest pointinE, denoted byPEx, such that
x−PEx ≤x−y, ∀y∈E. 2.2
The mappingPEis called themetric projectionofHontoE.
It is well known thatPEis a firmly nonexpansive mapping ofHontoE, that is,
x−y, PEx−PEy≥PEx−PEy2, ∀x, y∈H. 2.3
Moreover,PExis characterized by the following properties:PEx∈Eand
x−PEx, y−PEx≤0,
x−y2≥ x−PEx2y−PEx2
2.4
for allx∈H, y∈E.
Lemma 2.1. LetEbe a nonempty closed convex subset of a real Hilbert spaceH. Givenx∈Hand
z∈E,then,
zPEx⇐⇒
x−z, y−z≤0, ∀y∈E. 2.5
Lemma 2.2. LetHbe a Hilbert space, letEbe a nonempty closed convex subset ofH,and letBbe a mapping ofEintoH.Letu∈E. Then forλ >0,
u∈VIE, B⇐⇒uPEu−λBu, 2.6
wherePEis the metric projection ofHontoE.
A set-valued mappingT :H → 2His called amonotone if for allx, y∈H,f∈Txand g ∈Tyimplyx−y, f −g ≥ 0. A monotone mappingT :H → 2H is calledmaximalif the
graphGTofT is not properly contained in the graph of any other monotone mapping. It is known that a monotone mappingTis maximal if and only if forx, f∈H×H,x−y, f−g ≥
0 for everyy, g∈GTimpliesf ∈Tx. LetBbe a monotone map ofEintoH,η-Lipschitz continuous mappings and letNEvbe thenormal conetoEwhenv∈E, that is,
NEv{w∈H:v−u, w ≥0, ∀u∈E}, 2.7
and define a mappingTonEby
Tv
⎧ ⎨ ⎩
BvNEv, v∈E,
∅, v /∈E.
2.8
Lemma 2.3. LetH be a Hilbert space, letEbe a nonempty closed convex subset ofH,and letΨ :
E → Hbeρ-inverse-strongly monotone. It0 < r ≤ 2ρ, thenI−ρΨis a nonexpansive mapping in
H.
Proof. For allx, y∈Eand 0< r≤2ρ, we have
I−rΨx−I−rΨy2 x−y−rΨx−Ψy2
x−y2−2rx−y,Ψx−Ψyr2Ψx−Ψy2
≤x−y2−2rρΨx−Ψyr2Ψx−Ψy2
x−y2rr−2ρΨx−Ψy2
≤x−y2.
2.9
So,I−ρΨis a nonexpansive mapping ofEintoH.
Lemma 2.4see32. LetE,·,·be an inner product space. Then, for allx, y, z∈Eandα, β, γ ∈
0,1withαβγ1,one has
αxβyγz2
αx2βy2γz2−αβx−y2−αγx−z2−βγy−z2. 2.10
Lemma 2.5see25. LetE be a nonempty closed convex subset ofH,let f be a contraction of
H into itself withα ∈ 0,1, and letAbe a strongly positive linear bounded operator on Hwith coefficientγ >0. Then, for0< γ < γ/α,
x−y,A−γfx−A−γfy≥γ−γαx−y2, x, y∈H. 2.11
That is,A−γfis strongly monotone with coefficientγ−γα.
Lemma 2.6see25. Assume thatA is a strongly positive linear bounded operator onH with coefficientγ >0and0< ϑ≤ A−1. ThenI−ϑA ≤1−ϑγ.
Lemma 2.7see4. LetEbe a nonempty closed convex subset of a real Hilbert spaceH and let
S : E → Ebe a k-strict pseudocontraction mapping with a fixed point. ThenFSis closed and
convex. DefineSk:E → EbySkkx 1−kSxfor eachx∈E. ThenSkis nonexpansive such
thatFSk FS.
Lemma 2.8see33. LetEbe a closed convex subset of a Hilbert spaceHand letS:E → Ebe a nonexpansive mapping. ThenI−Sis demiclosed at zero, that is,
xn x, xn−Sxn−→0 implies xSx. 2.12
Lemma 2.9see34. LetEbe a nonempty closed convex subset of a strictly convex Banach space
nonempty. Letδnbe a sequence of positive number with∞n1δn1. Then a mappingSonEdefined by
Sx
∞
n1
δnTnx 2.13
forx∈Eis well defined and nonexpansive andFS ∞n1FTnholds.
For solving the mixed equilibrium problem, let us give the following assumptions for the bifunctionΦ, the functionϕ,and the setE:
A1 Φx, x 0 for allx∈E;
A2 Φis monotone, that is,Φx, y Φy, x≤0 for allx, y∈E; A3for eachx, y, z∈E, limt→0Φtz 1−tx, y≤Φx, y;
A4for eachx∈E, y→Φx, yis convex and lower semicontinuous; A5for eachy∈E, x→Φx, yis weakly upper semicontinuous;
B1for eachx ∈Handr >0, there exists a bounded subsetDx ⊆Eandyx ∈Esuch
that for anyz∈E\Dx,
Φz, yx
ϕyx
−ϕz 1 r
yx−z, z−x
<0; 2.14
B2Eis a bounded set.
By similar argument as in the proof of Lemma 2.10 in 35, we have the following lemma appearing.
Lemma 2.10. LetEbe a nonempty closed convex subset ofH. LetΦ:E×E → Rbe a bifunction
satisfies (A1)–(A5) and letϕ:E → R∪{∞}be a proper lower semicontinuous and convex function.
Assume that either (B1) or (B2) holds. Forr >0andx∈H, define a mappingTrΦ:H → Eas follows:
TrΦx
z∈E:Φz, yϕy−ϕz 1 r
y−z, z−x≥0, ∀y∈E
2.15
for allz∈H. Then, the following holds:
ifor eachx∈H, TrΦx/∅;
iiTrΦis single-valued;
iiiTrΦis firmly nonexpansive, that is, for anyx, y∈H,
TrΦx−TrΦy2≤TrΦx−TrΦy, x−y; 2.16
ivFTrΦ MEPΦ, ϕ;
Remark 2.11. We remark thatLemma 2.10is not a consequence of Lemma 3.1 in5, because the condition of the sequential continuity from the weak topology to the strong topology for the derivativeKof the functionK:E → Rdoes not cover the caseKx x2/2.
Lemma 2.12see36. Let{xn}and{ln}be bounded sequences in a Banach spaceXand let{βn}be a sequence in0,1with0<lim infn→ ∞βn≤lim supn→ ∞βn<1.Supposexn1 1−βnlnβnxn
for all integersn≥1andlim supn→ ∞ln1−ln − xn1−xn≤0.Then,limn→ ∞ln−xn0.
Lemma 2.13see37. Assume that{an}is a sequence of nonnegative real numbers such that
an1≤
1−n
anσn, n≥1, 2.17
where{n}is a sequence in0,1and{σn}is a sequence inRsuch that
1∞n1n∞,
2lim supn→ ∞σn/n≤0or
∞
n1|σn|<∞.
Thenlimn→ ∞an 0.
Lemma 2.14. LetHbe a real Hilbert space. Then for allx, y∈H,
xy2≤ x22y, xy. 2.18
3. Main Results
In this section, we will use the new approximation iterative method to prove a strong convergence theorem for finding a common element of the set of fixed points of strict pseudocontractions, the set of common solutions of the system of generalized mixed equilibrium problems, and the set of a common solutions of the variational inequalities for inverse-strongly monotone mappings in a real Hilbert space.
Theorem 3.1. LetEbe a nonempty closed convex subset of a real Hilbert spaceH. LetΦ1andΦ2be
two bifunctions fromE×EtoRsatisfyingA1–A5and letϕ:E → R ∪ {∞}be a proper lower semicontinuous and convex function with either (B1) or (B2). LetC:E → Hbe aξ-inverse-strongly
monotone mapping, letΨ1:E → Hbe a ρ-inverse-strongly monotone mapping, letΨ2:E → H
be anω-inverse-strongly monotone mapping, and letB:E → Hbe aβ-inverse-strongly monotone
mapping. Letf:E → Ebe anα-contraction with coefficientα0 ≤α < 1and letAbe a strongly positive linear bounded operator onHwith coefficientγ >0and0 < γ < γ/α. LetS:E → Ebe a
k-strict pseudocontraction with a fixed point. Define a mappingSk:E → EbySkxkx1−kSx,
for allx∈E. Suppose that
Θ:FS∩VIE, B∩VIE, C∩GMEPΦ1, ϕ,Ψ1
∩GMEPΦ2, ϕ,Ψ2
/
Let{xn}be a sequence generated by the following iterative algorithm:
x1∈E, un∈E, vn∈E,
Φ1un, u ϕx−ϕun Ψ1xn, u−un
1
ru−un, un−xn ≥0, ∀u∈E,
Φ2vn, v ϕx−ϕvn Ψ2xn, v−vn1
sv−vn, vn−xn ≥0, ∀v∈E, knαn1Skxnαn2PExn−λnBxn αn3PE
xn−μnCxn
αn4unαn5vn,
xn1nγfxn βnxn
1−βnI−nAkn, ∀n≥1,
3.2
where {n},{βn},{γn}, and{αni}are sequences in0,1, wherei 1,2,3,4,5,r ∈ 0,2ρ,s ∈
0,2ω, and{λn} and {μn} are positive sequences. Assume that the control sequences satisfy the following restrictions:
C15i1αni1,
C2limn→ ∞n0and∞n1n∞,
C30<lim infn→ ∞βn≤lim supn→ ∞βn<1,
C4limn→ ∞|λn1−λn|limn→ ∞|μn1−μn|0,
C5 d≤λn≤2βande≤μn≤2ξ, whered, eare two positive constants,
C6limn→ ∞αniαi∈0,1, wherei1,2,3,4,5.
Then, {xn} converges strongly to a point q ∈ Θwhich is the unique solution of the variational
inequality:
A−γfq, x−q≥0, ∀x∈Θ. 3.3
Equivalently, one hasqPΘI−Aγfq.
Proof. Sincen → 0, asn → ∞, we may assume, without loss of generality, thatn ≤1−
βnA−1for alln∈N. ByLemma 2.6, we know that if 0≤ϑ≤ A−1, thenI−ϑA ≤1−ϑγ.
We will assume thatI−A ≤ 1−γ. SinceAis a strongly positive bounded linear operator onH,we have
Asup{|Ax, x|:x∈H,x1}. 3.4
Observe that
1−βnI−nAx, x1−βn−nAx, x
≥1−βn−nA
≥0,
and so this shows that1−βnI−nAis positive. It follows that
1−βn
I−nAsup1−βn
I−nA
x, x:x∈H,x1
sup1−βn−nAx, x:x∈H,x1
≤1−βn−nγ.
3.6
Sincefis a contraction ofHinto itself withα∈0,1, then, we have
PΘI−Aγfx−PΘI−Aγfy≤I−Aγfx−I−Aγfy
≤ I−Ax−yγfx−fy
≤1−γx−yγαx−y
1−γ−γαx−y, ∀x, y∈H.
3.7
Since 0<1−γ−γα<1, it follows thatPΘI−Aγfis a contraction ofHinto itself. Therefore
the Banach Contraction Mapping Principle implies that there exists a unique elementq∈H
such thatqPΘI−Aγfq.
Next, we will divide the proof into five steps.
Step 1. We claim that{xn}is bounded.
Indeed, letp∈Θand byLemma 2.10, we obtain
pPEp−λnBpPEp−μnCpTΦ1
r I−rΨ1pTsΦ2I−sΨ2p. 3.8
Note thatunTΦ1
r I−rΨ1xn∈domϕandvnTsΦ2I−sΨ2xn∈domϕ; we have
un−pTΦ1
r I−rΨ1xn−TrΦ1I−rΨ1p≤xn−p,
vn−pTsΦ2I−sΨ2xn−TsΦ2I−sΨ2p≤xn−p.
PutznPExn−μnCxnandynPExn−λnBxn. For eachλn≤2βandμn≤2ξbyLemma 2.3, we get thatI−λnBandI−μnBare nonexpansive. Thus, we have
zn−pPE
xn−μnCxn−PEp−μnCp
≤xn−μnCxn−p−μnCp
I−μnCxn−I−μnCp
≤xn−p,
yn−pPExn−λnBxn−PE
p−λnBp
≤xn−λnBxn−
p−λnBp
I−λnBvn−I−λnBp
≤vn−p≤xn−p.
3.10
FromLemma 2.7, we have thatSkis nonexpansive withFSk FS. It follows that
kn−pα1
n Skxnαn2ynαn3znαn4unαn5vn
≤αn1Skxn−pαn2yn−pαn3zn−pαn3un−pαn3vn−p
≤αn1xn−pαn2xn−pαn3xn−pαn3xn−pαn3xn−p
xn−p,
3.11
which yields that
xn1−pn
γfxn−Ap
βn
xn−p
1−βn
I−nA
kn−p
≤1−βn−nγkn−pβnxn−pnγfxn−Ap
≤1−βn−nγxn−pβnxn−pnγfxn−Ap
≤1−nγxn−pnγfxn−f
pnγf
≤1−nγxn−pnγαxn−pnγfp−Ap
1−γ−αγnxn−pγ−αγnγf
p−Ap γ−αγ
≤max
xn−p,γfp−Ap γ−αγ
≤ ...
≤max
x1−p,
γfp−Ap γ−αγ
, ∀n∈N.
3.12
Hence,{xn}is bounded, and so are{un},{vn},{zn},{yn},{kn},{fxn},{Cxn},and{Bxn}.
Step 2. We claim that limn→ ∞xn1−xn0 and limn→ ∞kn−xn0.
Observing thatunTΦ1
r I−rΨ1xn∈domϕandun1TrΦ1I−rΨ1xn1∈domϕ, by
the nonexpansiveness ofTΦ1
r , we get
un1−unTrΦ1I−rΨ1xn1−TrΦ1I−rΨ1xn≤ xn1−xn. 3.13
Similarly, letvnTsΦ2I−sΨ2xn ∈domϕandvn1TsΦ2I−sΨ2xn1∈domϕ; we have
vn1−vnTsΦ2I−sΨ2xn1−TsΦ2I−sΨ2xn≤ xn1−xn. 3.14
FromznPExn−μnCxnandynPExn−λnBxn; thus, we compute
zn1−znPE
xn1−μn1Cxn1
−PE
xn−μnCxn
≤xn1−μn1Cxn1
−xn−μnCxn
xn1−μn1Cxn1
−xn−μn1Cxn
μn−μn1
Cxn
≤xn1−μn1Cxn1
−xn−μn1Cxnμn−μn1Cxn
I−μn1C
xn1−
I−μn1C
xnμn−μn1Cxn ≤ xn1−xnμn−μn1Cxn.
3.15
Similarly, we have
yn1−ynPExn1−λn1Bxn1−PExn−λnBxn
≤ xn1−xn|λn−λn1|Bxn.
Also noticing that
knαn1Skxnαn2ynαn3znαn4unαn5vn
kn1αn11Skxn1αn21yn1αn31zn1αn41un1αn51vn1,
3.17
we compute
kn1−kn ≤αn11Skxn1−Skxnαn11−αn1Skxnαn21yn1−yn
αn21−αn2ynαn31zn1−znαn31−αn3zn
αn41un1−unαn41−αn4unαn51vn1−vnαn51−αn5vn
≤αn11xn1−xnαn11−αn1Skxnαn21yn1−yn
αn21−αn2ynαn31zn1−znαn31−αn3zn
αn41un1−unαn41−αn4unαn51vn1−vnαn51−αn5vn.
3.18
Substitution of3.13,3.14,3.15, and3.16into3.18yields that
kn1−kn ≤αn11xn1−xnαn11−αn1Skxn
αn21{xn1−xn|λn−λn1|Bxn}αn21−αn2yn
αn31xn1−xnμn−μn1Cxn
αn31−αn3zn
αn41xn1−xnαn41−αn4unαn51xn1−xnαn51−αn5vn
≤ xn1−xnM1αn11−αn1αn21−αn2αn31−αn3αn41−αn4
αn51−αn5|λn−λn1|μn−μn1,
3.19
whereM1is an appropriate constant such that
M1≥max
sup
n≥1
{Skxn},sup
n≥1
yn,sup
n≥1
{zn}, sup
n≥1
{un},
sup
n≥1
{vn},sup
n≥1
{Bxn},sup
n≥1
{Cxn}
.
Puttingxn1 1−βnlnβnxn,for alln≥1, we have
ln
xn1−βnxn
1−βn
nγfxn 1−βnI−nAkn
1−βn
. 3.21
Then, we compute
ln1−ln
n1γfxn1
1−βn1
I−n1A
kn1
1−βn1 −
nγfxn 1−βnI−nAkn
1−βn
n1
1−βn1
γfxn1−
n
1−βnγfxn kn1−kn n
1−βnAkn− n1
1−βn1
Akn1
n1
1−βn1
γfxn1−Akn1
n
1−βn
Akn−γfxnkn1−kn.
3.22
It follows from3.19and3.22that
ln1−ln − xn1−xn ≤ n1
1−βn1
γfxn1Akn1
n
1−βn
Aknγfxn
kn1−kn − xn1−xn
≤ n1
1−βn1
γfxn1Akn1
n 1−βn
Aknγfxn
M1αn11−αn1αn21−αn2αn31−αn3αn41−αn4
αn51−αn5|λn−λn1|μn−μn1.
3.23
This together withC2,C3,C4, andC6implies that
lim sup
n→ ∞ ln1−ln − xn1−xn≤0. 3.24
Hence, byLemma 2.12, we obtainln−xn → 0 asn → ∞. It follows that
lim
n→ ∞xn1−xnnlim→ ∞
Moreover, we also get
lim
n→ ∞un1−unnlim→ ∞vn1−vnnlim→ ∞zn1−znnlim→ ∞yn1−yn
lim
n→ ∞kn1−kn0.
3.26
Observe that
xn1−xnn
γfxn−Axn1−βn−nγkn−xn. 3.27
By conditionsC2,C3, and3.25, we have
limn→ ∞kn−xn0. 3.28
Step 3. We claim that the following statements hold:
s1limn→ ∞xn−un0;
s2limn→ ∞xn−yn0;
s3limn→ ∞xn−zn0;
s4limn→ ∞xn−vn0.
Forp∈Θ, we compute
zn−p2PE
xn−μnCxn
−PE
p−μnCp2
≤xn−μnCxn−p−μnCp2
xn−p
−μn
Cxn−Cp2
≤xn−p2−2μn
xn−p, Cxn−Cp
μ2
nCxn−Cp2
≤xn−p2μnμn−2ξCxn−Cp2
xn−p2−μn
2ξ−μnCxn−Cp2.
3.29
By the same way, we can get
yn−p2 ≤
We note that
un−p2
TΦ1
r I−rΨ1xn−TrΦ1I−rΨ1p 2
≤I−rΨ1xn−I−rΨ1p2
xn−p
−rΨ1xn−Ψ1p2
xn−p2−2rxn−p,Ψ1xn−Ψ1p
r2Ψ1xn−Ψ1p2
≤ xn−p2−2rρΨ1xn−Ψ1pr2Ψ1xn−Ψ1p2
xn−p2r
r−2ρΨ1xn−Ψ1p2
xn−p2−r2ρ−rΨ1xn−Ψ1p2.
3.31
Similarly, we have
vn−p2≤xn−p2−s2ω−sΨ2xn−Ψ2p2. 3.32
Observe that
kn−p2≤αn1Skxn−p2αn2yn−p2αn3zn−p2αn4un−p2αn5vn−p2
≤αn1xn−p2αn2yn−p2αn3zn−p2αn4un−p2αn5vn−p2.
3.33
Substituting3.29,3.30,3.31, and3.32into3.33, we obtain
kn−p2 ≤αn1xn−p2αn2
xn−p2−λn2β−λnBxn−Bp2
αn3
xn−p2−μn
2ξ−μnCxn−Cp2
αn4
xn−p2−r
2ρ−rΨ1xn−Ψ1p2
αn5
xn−p2−s2ω−sΨ2xn−Ψ2p2
xn−p2−αn2λn
2β−λnBxn−Bp2−αn3μn
2ξ−μnCxn−Cp2
− αn4r
2ρ−rΨ1xn−Ψ1p2−αn5s2ω−sΨ2xn−Ψ2p2.
It follows from3.2and3.34that
xn1−p2
nγfxn βnxn1−βnI−nAkn−p2
≤nγfxn−Ap2βnxn−p2
1−βn−nγkn−p2
≤nγfxn−Ap2βnxn−p2
1−βn−nγ
×xn−p2−αn2λn
2β−λnBxn−Bp2−αn3μn
2ξ−μnCxn−Cp2
−αn4r
2ρ−rΨ1xn−Ψ1p2−αn5s2ω−sΨ2xn−Ψ2p2
nγfxn−Ap2
1−nγxn−p2−
1−βn−nγ
αn2λn
2β−λnBxn−Bp2
−1−βn−nγαn3μn
2ξ−μnCxn−Cp2
−1−βn−nγ
αn4r
2ρ−rΨ1xn−Ψ1p2
−1−βn−nγ
αn5s2ω−sΨ2xn−Ψ2p2
≤nγfxn−Ap2xn−p2−1−βn−nγαn2λn
2β−λnBxn−Bp2
−1−βn−nγ
αn3μn
2ξ−μnCxn−Cp2
−1−βn−nγαn4r
2ρ−rΨ1xn−Ψ1p2
−1−βn−nγ
αn5s2ω−sΨ2xn−Ψ2p2
≤nγfxn−Ap2xn−p2−1−βn−nγαn3μn
2ξ−μnCxn−Cp2.
3.35
It follows fromC5that
1−βn−nγαn3μn
2ξ−μnCxn−Cp2
≤nγfxn−Ap2xn−p2−xn1−p2
nγfxn−Ap2xn−p−xn1−pxn−pxn1−p
≤nγfxn−Ap2xn1−xnxn−pxn1−p.
3.36
FromC2,C6, and3.25, we have
lim
Sinces∈0,2ω, we also have
1−βn−nγαn5s2ω−sΨ2xn−Ψ2p2
≤nγfxn−Ap2xn−p2−xn1−p2
≤nγfxn−Ap2xn1−xnxn−pxn1−p.
3.38
FromC2,C6, and3.25, we obtain
lim
n→ ∞Ψ2xn−Ψ2p0. 3.39
Similarly, from3.37and3.39, we can prove that
lim
n→ ∞Bxn−Bpnlim→ ∞Ψ1xn−Ψ1p0. 3.40
On the other hand, letp ∈Θfor eachn≥ 1; we getp TΦ1
r I−rΨ1p. ByLemma 2.10iii,
that is,TΦ1
r is firmly nonexpansive, we obtain
un−p2
TΦ1
r I−rΨ1xn−TrΦ1I−rΨ1p 2
≤I−rΨ1xn−I−rΨ1p, un−p
1 2
I−rΨ1xn−I−rΨ1p2un−p2−I−rΨ1xn−I−rΨ1p−
un−p2
≤ 1
2
xn−p2un−p2−xn−un−rΨ1xn−Ψ1p2
≤ 1
2
xn−p2un−p2− xn−un22rxn−unΨ1xn−Ψ1p−r2Ψ1xn−Ψ1p2
.
3.41
So, we obtain
un−p2≤
Observe that
yn−p2
PExn−λnBxn−PEp−λnBp2
≤I−λnBxn−I−λnBp, yn−p
1 2
I−λnBxn−I−λnBp2yn−p2
−I−λnBxn−I−λnBp−
yn−p2
≤ 1
2
xn−p2yn−p2−xn−yn
−λn
Bxn−Bp2
≤ 1
2
xn−p2yn−p2−xn−yn2−λ2nBxn−Bp2
2λn
xn−yn, Bxn−Bp
,
3.43
and hence
yn−p2 ≤xn−p2−xn−yn22λnxn−ynBxn−Bp. 3.44
By using the same argument in3.42and3.44, we can prove that
vn−p2 ≤xn−p2− xn−vn22sxn−vnΨ2xn−Ψ2p,
zn−p2 ≤
xn−p2− xn−zn22μnxn−znCxn−Cp.
3.45
Substituting3.42,3.44, and3.45into3.33, we obtain
kn−p2≤αn1xn−p2αn2yn−p2αn3zn−p2αn4un−p2αn5vn−p2
≤αn1xn−p2αn2
xn−p2−xn−yn22λnxn−ynBxn−Bp
αn3
xn−p2− xn−zn22μnxn−znCxn−Cp
αn4
xn−p2− xn−un22rxn−unΨ1xn−Ψ1p
αn5
xn−p2− xn−vn22sxn−vnΨ2xn−Ψ2p
xn−p2−αn2xn−yn22λnαn2xn−ynBxn−Bp
−αn3xn−zn22μnαn3xn−znCxn−Cp
−αn4xn−un22rαn4xn−unΨ1xn−Ψ1p
−αn5xn−vn22sαn5xn−vnΨ2xn−Ψ2p.
FromLemma 2.4,3.2, and3.46, we obtain
xn1−p2 n
γfxn−Ap
βn
xn−p
1−βn
I−nA
kn−p2
≤nγfxn−Ap2βnxn−p21−βn−nγkn−p2
≤nγfxn−Ap2βnxn−p2
1−βn−nγxn−p2−αn2xn−yn22λnαn2xn−ynBxn−Bp
−αn3xn−zn22μnαn3xn−znCxn−Cp
−αn4xn−un22rαn4xn−unΨ1xn−Ψ1p
−αn5xn−vn22sαn5xn−vnΨ2xn−Ψ2p
nγfxn−Ap21−nγxn−p2
−1−βn−nγ
αn2xn−yn22
1−βn−nγ
λnαn2xn−ynBxn−Bp
−1−βn−nγαn3xn−zn22
1−βn−nγμnαn3xn−znCxn−Cp
−1−βn−nγ
αn4xn−un22r
1−βn−nγ
αn4xn−unΨ1xn−Ψ1p
−1−βn−nγ
αn5xn−vn22s
1−βn−nγ
αn5xn−vnΨ2xn−Ψ2p
≤nγfxn−Ap2xn−p2−1−βn−nγαn2xn−yn2
21−βn−nγ
λnαn2xn−ynBxn−Bp−
1−βn−nγ
αn3xn−zn2
21−βn−nγμnαn3xn−znCxn−Cp−
1−βn−nγαn4xn−un2
2r1−βn−nγ
αn4xn−unΨ1xn−Ψ1p−
1−βn−nγ
αn5xn−vn2
2s1−βn−nγαn5xn−vnΨ2xn−Ψ2p.
3.47
It follows that
1−βn−nγ
αn4xn−un2 ≤nγfxn−Ap2xn1−xnxn−pxn1−p
21−βn−nγ
λnαn2xn−ynBxn−Bp
21−βn−nγμnαn3xn−znCxn−Cp
2r1−βn−nγ
αn4xn−unΨ1xn−Ψ1p
2s1−βn−nγαn5xn−vnΨ2xn−Ψ2p.
FromC2,C6,3.37,3.39,3.40, andxn1−xn → 0 asn → ∞, we also have
lim
n→ ∞xn−un0. 3.49
From3.47and by using the same argument above, we can prove that
lim
n→ ∞xn−ynnlim→ ∞xn−znnlim→ ∞xn−vn0. 3.50
Applying3.28,3.49, and3.50, we obtain
lim
n→ ∞kn−unnlim→ ∞kn−ynnlim→ ∞kn−znnlim→ ∞kn−vn0. 3.51
Step 4. We claim that lim supn→ ∞A−γfq, q−xn ≤0,whereqPΘI−Aγfqis the
unique solution of the variational inequalityA−γfq, x−q ≥0, for allx∈Θ.
To show the above inequality, we choose a subsequence{xni}of{xn}such that
lim sup
n→ ∞
A−γfq, q−xn
lim
i→ ∞
A−γfq, q−xni. 3.52
Since{xni}is bounded, there exists a subsequence{xnij}of{xni}which converges weakly to
z∈E. Without loss of generality, we can assume thatxni z.We claim thatz∈Θ. That is, we will prove that
z∈FS∩VIE, C∩VIE, B∩GMEPΦ1, ϕ,Ψ1
∩GMEPΦ2, ϕ,Ψ2
. 3.53
Assume also thatλn → λ∈d,2βandμn → μ∈e,2ξ. Define a mappingQ:E → Eby
Qxα1Skxα2PE1−λBxα3PE1−μCxα4TΦ1
r I−rΨ1x
α5TΦ2
s I−rΨ2x, ∀x∈E,
3.54
where limn→ ∞αni αi ∈0,1, wherei1,2,3,4,5. Since
5
i1α
i
n 1 and byLemma 2.9,
we have thatQis nonexpansive and
FQ FSk∩FPE1−λB∩F
PE
1−μC∩FTΦ1
r I−rΨ1
∩FTΦ2
s I−rΨ2
FS∩VIE, C∩VIE, B∩GMEPΦ1, ϕ,Ψ1
∩GMEPΦ2, ϕ,Ψ2
.
Notice that
Qxn−xn
≤ Qxn−knkn−xn
α1Skxnα2PE1−λBxnα3PE1−μCxnα4TΦ1
r I−rΨ1xn
α5TΦ2
s I−rΨ2xn
− αn1Skxnαn2PE1−λnBxnαn3PE
1−μnC
xn
αn4TrΦ1I−rΨ1xnαn5TsΦ2I−rΨ2xn kn−xn
≤α1−αn1Skxnα2PEI−λBxn−PEI−λnBxnα2−αn2PEI−λnBxn
α3PE
I−μCxn−PE
I−μnC
xnα3−αn3PE
I−μnC
xn
α4−αn4TrΦ1I−rΨ1xnα5−αn5TsΦ2I−rΨ2xnkn−xn
≤α1−αn1Skxnα2|λn−λ|Bxnα2−αn2PEI−λnBxn
α3μn−μCxnα3−αn3PE
I−μnC
xn
α4−αn4TrΦ1I−rΨ1xnα5−αn5TsΦ2I−rΨ2xnkn−xn
≤K1
5
i1
αi−αni|λn−λ|μn−μ
kn−xn,
3.56
whereK1is an appropriate constant such that
K1≥max
sup
n≥1
TΦ1
r I−rΨ1xn
,sup
n≥1
TΦ2
s I−rΨ2xn
,sup
n≥1
{PEI−λnBxn},
sup
n≥1
PE
I−μnC
xn,sup n≥1
{Bxn},sup n≥1
{Cxn},sup n≥1
{Skxn}
.
3.57
FromC4,C6, and3.28, we obtain
lim
n→ ∞xn− Qxn0. 3.58
SincePΘI−Aγfqis a contraction with the coefficientα∈0,1, we have that there exists a
unique fixed point. We useqto denote the unique fixed point to the mappingPΘI−Aγfq.
which converges weakly toz. Without loss of generality, we may assume that{xni} z. It follows from3.58, that
lim
n→ ∞xni− Qxni0. 3.59
It follows fromLemma 2.8thatz∈FQ. By3.55, we havez∈Θ. Hence from3.52and2.4, we arrive at
lim sup
n→ ∞
A−γfq, q−xnlim sup
n→ ∞
A−γfq, q−xni
A−γfq, q−z≤0.
3.60
On the other hand, we have
A−γfq, q−xn1
A−γfq, xn−xn1
A−γfq, q−xn
≤A−γfqxn−xn1
A−γfq, q−xn
. 3.61
From3.25and3.60, we obtain that
lim sup
n→ ∞
A−γfq, q−xn1
≤0. 3.62
Step 5. We claim that limn→ ∞xn−q0.
Indeed, by3.2and using Lemmas2.6and2.14, we observe that
xn1−q2nγfxn βnxn
1−βn
I−nA
kn−q2
1−βn
I−nA
kn−q
βn
xn−q
n
γfxn−Aq2
≤1−βn
1−βn
I−nA
1−βn
kn−qβnxn−q
2
2nγfxn−Aq, xn1−q
≤1−βn
1−βn
I−nA
1−βn
kn−q
2
2nγfxn−fq, xn1−q
2nγfq−Aq, xn1−q
≤1−βn
1−βnI−nA
1−βn
kn−q
2
βnxn−q2
2nγαxn−qxn1−q2n
γfq−Aq, xn1−q
≤ 1−βn
I−nA2
1−βn
kn−q2βnxn−q2
nγα
xn−q2xn1−q2
2n
γfq−Aq, xn1−q
≤ 1−βn
I−nA2
1−βn
xn−q2
βnxn−q2
nγα
xn−q2xn1−q2
2n
γfq−Aq, xn1−q
≤
1−βn
−γn
2
1−βn βnnγα
xn−q2
nγαxn1−q22n
γfq−Aq, xn1−q
≤
1−2γ−αγn γ22
n
1−βn
xn−q2
nγαxn1−q22n
γfq−Aq, xn1−q
, 3.63
which implies that
xn1−q2≤
1− 2
γ−αγn
1−αγn
xn−q2
n 1−αγn
γ2n
1−βn
xn−q22
γfq−Aq, xn1−q
. 3.64 Taking σn n
1−αγn
γ2n
1−βn
xn−q22
γfq−Aq, xn1−q
n 2
γ−αγn
1−αγn ,
3.65
then, we can rewrite3.64as
xn1−q2≤
and we can see that ∞n1n ∞ and lim supn→ ∞σn/n ≤ 0. Applying Lemma 2.13 to 3.66, we conclude that{xn}converges strongly toqin norm. This completes the proof.
If the mappingSis nonexpansive, thenSkS0S. We can obtain the following result
fromTheorem 3.1immediately.
Corollary 3.2. LetEbe a nonempty closed convex subset of a real Hilbert spaceH. LetΦ1andΦ2be
two bifunctions fromE×EtoRsatisfying (A1)–(A5) and letϕ:E → R ∪ {∞}be a proper lower
semicontinuous and convex function with either (B1) or (B2). LetC:E → Hbe aξ-inverse-strongly
monotone mapping, letΨ1:E → Hbe a ρ-inverse-strongly monotone mapping, letΨ2:E → H
be anω-inverse-strongly monotone mapping and letB:E → Hbe aβ-inverse-strongly monotone
mapping. Letf :E → Ebe anα-contraction with coefficientα0≤ α <1and letAbe a strongly positive linear bounded operator onHwith coefficientγ >0and0 < γ < γ/α. LetS :E → Ebe a nonexpansive mapping with a fixed point. Suppose that
Θ:FS∩VIE, B∩VIE, C∩GMEPΦ1, ϕ,Ψ1
∩GMEPΦ2, ϕ,Ψ2
/
∅. 3.67
Let{xn}be a sequence generated by the following iterative algorithm3.2, where{n},{βn},{γn},
and{αni}are sequences in0,1, wherei1,2,3,4,5,r∈0,2ρ,s∈0,2ω, and{λn}and{μn}
are positive sequences. Assume that the control sequences satisfy (C1)–(C6) in Theorem 3.1. Then,
{xn}converges strongly to a pointq∈Θwhich is the unique solution of the variational inequality:
A−γfq, x−q ≥0, ∀x∈Θ. 3.68
Equivalently, one hasqPΘI−Aγfq.
Ifϕ 0,Ψ1 Ψ2 0, A I, γ ≡ 1,andγn 1−n−βn inTheorem 3.1, then we can
obtain the following result immediately.
Corollary 3.3. LetEbe a nonempty closed convex subset of a real Hilbert spaceH. LetΦ1andΦ2
be two bifunctions fromE×EtoRsatisfying (A1)–(A4). LetC :E → H be aξ-inverse-strongly
monotone mapping and letB:E → Hbe aβ-inverse-strongly monotone mapping. Letf :E → E
be anα-contraction with coefficientα0 ≤ α < 1and let Abe a strongly positive linear bounded operator onHwith coefficientγ >0and0< γ < γ/α. LetS:E → Ebe ak-strict pseudocontraction
with a fixed point. Define a mappingSk :E → EbySkxkx 1−kSx,for allx∈E. Suppose
that
Let{xn}be a sequence generated by the following iterative algorithm:
x1∈E, un∈E, vn∈E,
Φ1un, u
1
ru−un, un−xn ≥0, ∀u∈E,
Φ2vn, v 1
sv−vn, vn−xn ≥0, ∀v∈E, znPExn−μnCxn,
ynPExn−λnBxn,
knαn1Skxnαn2ynαn3znαn4unαn5vn,
xn1nfxn βnxnγnkn, ∀n≥1,
3.70
where {n}, {βn}, {γn}, and {αni} are sequences in 0,1, where i 1,2,3,4,5, r ∈ 0,∞, s ∈ 0,∞, and{λn}and {μn}are positive sequences. Assume that the control sequences satisfy
the condition (C1)–(C6) inTheorem 3.1andnβnγn1. Then,{xn}converges strongly to a point q∈Θ, whereqPΘfq.
IfB0, C0,andΦ1un, u Φ1vn, v 0 inCorollary 3.3, thenPEIand we get unynxnandvnznxn; hence we can obtain the following result immediately.
Corollary 3.4. LetEbe a nonempty closed convex subset of a real Hilbert spaceH. LetS:E → E
be ak-strict pseudocontraction with a fixed point. Define a mappingSk:E → EbySkxkx 1−
kSx,for allx∈E. Suppose thatFS/∅.Let{xn}be a sequence generated by the following iterative
algorithm:
x1∈E,
kn αnSkxn 1−αnxn,
xn1nfxn βnxnγnkn, ∀n≥1,
3.71
where{n},{βn},{γn}, and{αn}are sequences in0,1. Assume that the control sequences satisfy
the conditions (C2) and (C3),limn→ ∞αn α ∈0,1inTheorem 3.1, andnβnγn 1. Then,
{xn}converges strongly to a pointq∈FS, whereqPFSfq.
Finally, we consider the followingConvex Feasibility ProblemCFP:
finding an x∈ M
i1
Ci, 3.72
where M ≥ 1 is an integer and each Ci is assumed to be the of solutions of equilibrium
