Volume 2010, Article ID 836714,29pages doi:10.1155/2010/836714
Research Article
Iterative Methods for Finding Common
Solution of Generalized Equilibrium Problems
and Variational Inequality Problems
and Fixed Point Problems of a Finite Family
of Nonexpansive Mappings
Atid Kangtunyakarn
Department of Mathematics, Faculty of Science, King Mongkut’s Institute of Technology Ladkrabang, Bangkok 10520, Thailand
Correspondence should be addressed to Atid Kangtunyakarn,[email protected]
Received 7 October 2010; Accepted 2 November 2010
Academic Editor: T. D. Benavides
Copyrightq2010 Atid Kangtunyakarn. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
We introduce a new method for a system of generalized equilibrium problems, system of variational inequality problems, and fixed point problems by usingS-mapping generated by a finite family of nonexpansive mappings and real numbers. Then, we prove a strong convergence theorem of the proposed iteration under some control condition. By using our main result, we obtain strong convergence theorem for finding a common element of the set of solution of a system of generalized equilibrium problems, system of variational inequality problems, and the set of common fixed points of a finite family of strictly pseudocontractive mappings.
1. Introduction
LetHbe a real Hilbert space, and letCbe a nonempty closed convex subset ofH. LetA :
C → Hbe a nonlinear mapping, and letF:C×C → Rbe a bifunction. A mappingT ofH
into itself is callednonexpansiveifTx−Ty ≤ x−yfor allx, y∈H. We denote byFTthe set of fixed points ofT i.e.,FT {x∈H:Txx}. Goebel and Kirk1showed thatFT
is always closed convex, and also nonempty providedThas a bounded trajectory.
A bounded linear operatorAonHis calledstrongly positivewith coefficientγif there is a constantγ >0 with the property
The equilibrium problem forFis to findx∈Csuch that
Fx, y≥0, ∀y∈C. 1.2
The set of solutions of1.2is denoted by EPF. Many problems in physics, optimization, and economics are seeking some elements of EPF, see2,3. Several iterative methods have been proposed to solve the equilibrium problem, see, for instance,2–4. In 2005, Combettes and Hirstoaga3introduced an iterative scheme of finding the best approximation to the initial data when EPFis nonempty and proved a strong convergence theorem.
The variational inequality problem is to find a pointu∈Csuch that
v−u, Au ≥0 ∀v∈C. 1.3
The set of solutions of the variational inequality is denoted by VIC, A, and we consider the following generalized equilibrium problem.
Find z∈C such thatFz, yAz, y−z ≥0, ∀y∈C. 1.4
The set of suchz∈Cis denoted by EPF, A, that is,
EPF, A z∈C:Fz, yAz, y−z≥0, ∀y∈C. 1.5
In the case ofA≡0, EPF, A EPF. Numerous problems in physics, optimization, variational inequalities, minimax problems, the Nash equilibrium problem in noncooperative games reduce to find element of1.5
A mappingAofCintoHis calledinverse-strongly monotone, see5, if there exists a positive real numberαsuch that
x−y, Ax−Ay≥αAx−Ay2 1.6
for allx, y∈C.
The problem of finding a common fixed point of a family of nonexpansive mappings has been studied by many authors. The well-known convex feasibility problem reduces to finding a point in the intersection of the fixed point sets of a family of nonexpansive mapping
see6,7.
In 2008, S.Takahashi and W.Takahashi11introduced a general iterative method for finding a common element of EPF, AandFT. They defined{xn}in the following way:
u, x1∈C, arbitrarily;
Fzn, y
Axn, y−zn
1
λn
y−zn, zn−xn
≥0, ∀y∈C,
xn1βnxn
1−βn
Tanu 1−anzn, ∀n∈N,
1.7
whereAis anα-inverse strongly monotone mapping ofCintoHwith positive real numberα, and{an} ∈0,1,{βn} ⊂0,1,{λn} ⊂0,2α, and proved strong convergence of the scheme
1.7toz∈ Ni1FTi∩EPF, A, wherezP N
i1FTi∩EPF, Auin the framework of a Hilbert
space, under some suitable conditions on{an},{βn},{λn}and bifunctionF.
Very recently, in 2010, Qin, et al.12introduced a iterative scheme method for finding a common element of EPF1, A, EPF2, B and common fixed point of infinite family of
nonexpansive mappings. They defined{xn}in the following way:
x1∈C, arbitrarily;
F1un, u Axn, u−un1
ru−un, un−xn ≥0, ∀u∈C,
F2vn, v Bxn, v−vn1
sv−vn, vn−xn ≥0, ∀v∈C, ynδnun 1−δnvn,
xn1αnfxn βnxnγnWnxn, ∀n∈N,
1.8
where f : C → C is a contraction mapping andWn isW-mapping generated by infinite
family of nonexpansive mappings and infinite real number. Under suitable conditions of these parameters they proved strong convergence of the scheme1.8toz PFfz, where F ∞
i1FTi∩EPF1, A∩EPF2, B.
In this paper, motivated by11,12, we introduced a general iterative scheme{xn} defined by
Fun, u Axn, u−un 1 rnu−
un, un−xn ≥0,
Gvn, v Bxn, v−vn
1
snv−
vn, vn−xn ≥0,
ynδnPCun−λnAun 1−δnPCvn−ηnBvn, xn1 αnfxn βnxnγnSnyn, ∀n≥0,
wheref : C → CandSn isS-mapping generated byT0, . . . , Tn andαn, αn−1, . . . , α0. Under
suitable conditions, we proved strong convergence of{xn}tozPFfz, andzis solution of
Ax∗, x−x∗ ≥0,
Bx∗, x−x∗ ≥0. 1.10
2. Preliminaries
In this section, we collect and give some useful lemmas that will be used for our main result in the next section.
Let C be closed convex subset of a real Hilbert space H, and let PC be the metric
projection ofHontoC, that is, forx∈H,PCxsatisfies the property
x−PCxmin y∈C
x−y. 2.1
The following characterizes the projectionPC.
Lemma 2.1see13. Givenx ∈ H and y ∈ C. ThenPCx y if and only if there holds the
inequality
x−y, y−z≥0 ∀z∈C. 2.2
Lemma 2.2see14. Let{sn}be a sequence of nonnegative real numbers satisfying
sn1 1−αnsnβn, ∀n≥0 2.3
where{αn},{βn}satisfy the conditions
1{αn} ⊂0,1,∞n1αn∞,
2lim supn→ ∞βn/αn≤0.
Thenlimn→ ∞sn0.
Lemma 2.3see 15. Let Cbe a closed convex subset of a strictly convex Banach spaceE. Let
{Tn :n ∈N}be a sequence of nonexpansive mappings on C. Suppose that n∞1FTnis nonempty.
Let{λn}be a sequence of positive numbers withΣ∞n1λn1. Then a mappingSonCdefined by
Sx Σ∞n1λnTnx 2.4
forx∈Cis well defined, nonexpansive, andFS n∞1FTnhold.
Lemma 2.5see17. Let{xn}and{zn}be bounded sequences in a Banach spaceX, and let{βn} be a sequence in0,1with0<lim infn→ ∞βn≤lim supn→ ∞βn<1. Suppose that
xn1βnxn
1−βn
zn 2.5
for all integern≥0and
lim supn→ ∞zn1−zn − xn1−xn≤0. 2.6
Thenlimn→ ∞xn−zn0.
For solving the equilibrium problem for a bifunctionF :C×C → R, let us assume thatFsatisfies the following conditions:
A1Fx, x 0 for allx∈C;
A2Fis monotone, that is,Fx, y Fy, x≤0, ∀x, y∈C,
A3for allx, y, z∈C,
limt→0F
tz 1−tx, y≤Fx, y, 2.7
A4for allx∈C, y→Fx, yis convex and lower semicontinuous. The following lemma appears implicitly in2.
Lemma 2.6see2. LetCbe a nonempty closed convex subset ofH, and letFbe a bifunction of
C×CintoRsatisfying (A1)–(A4). Letr >0andx∈H. Then, there existsz∈Csuch that
Fz, y 1 r
y−z, z−x 2.8
for allx∈C.
Lemma 2.7see3. Assume thatF:C×C → Rsatisfies (A1)–(A4). Forr >0andx∈H, define a mappingTr :H → Cas follows:
Trx
z∈C:Fz, y 1 r
y−z, z−x≥0, ∀y∈C
. 2.9
for allz∈H. Then, the following hold:
1Tr is single-valued;
2Tr is firmly nonexpansive, that is,
Trx−Tr
y2≤Trx−Tr
y, x−y ∀x, y∈H; 2.10
3FTr EPF;
In 2009, Kangtunyakarn and Suantai18defined a new mapping and proved their lemma as follows.
Definition 2.8. LetCbe a nonempty convex subset of real Banach space. Let{Ti}Ni1be a finite family of nonexpansive mappings ofCinto itself. For eachj1,2, . . . , N, letαj αj1, αj2, αj3∈ I×I×I, whereI∈0,1andα1jαj2α3j 1. We define the mappingS:C → Cas follows:
U0I,
U1α11T1U0α12U0α13I, U2α21T2U1α22U1α23I, U3α31T3U2α32U2α33I,
.. .
UN−1αN1−1TN−1UN−2αN2−1UN−2αN3 −1I, SUNαN1 TNUN−1αN2 UN−1αN3 I.
2.11
This mapping is calledS-mappinggenerated byT1, . . . , TNandα1, α2, . . . , αN.
Lemma 2.9. LetCbe a nonempty closed convex subset of strictly convex. Let{Ti}Ni1be a finite family of nonexpanxive mappings ofCinto itself with Ni1FTi/∅, and letαj αj1, αj2, α3j∈I×I×I, j 1,2,3, . . . , N, whereI 0,1,αj1αj2αj3 1,αj1 ∈ 0,1for allj 1,2, . . . , N−1,αN1 ∈
0,1αj2, αj3 ∈ 0,1for all j 1,2, . . . , N. Let S be the mapping generated byT1, . . . , TN and α1, α2, . . . , αN. ThenFS Ni1FTi.
Lemma 2.10. LetCbe a nonempty closed convex subset of Banach space. Let{Ti}Ni1be a finite family of nonexpansive mappings ofCinto itself andαjn αn,j1 , αn,j2 , αn,j3 , αj αj1, αj2, α3j∈I×I×I, whereI 0,1,αn,j1 αn,j2 αn,j3 1andαj1αj2αj3 1such thatαn,ji → αji ∈0,1as n →
∞fori 1,3 and j 1,2,3, . . . , N.Moreover, for everyn∈ N, letSandSnbe theS-mappings generated byT1, T2, . . . , TNandα1, α2, . . . , αNandT1, T2, . . . , TNandα1n, α2n, ..., αNn, respectively. Thenlimn→ ∞Snx−Sx0for everyx∈C.
Lemma 2.11see19. Let Cbe a nonempty closed convex subset of a Hilbert space H, and let
G:C → Cbe defined by
Gx PCx−λAx, ∀x∈C, 2.12
with∀λ >0. Thenx∗∈V IC, Aif and only ifx∗∈FG.
3. Main Result
family of nonexpansive mappings withF Ni1FTi∩EPF, A∩EPG, B∩FG1∩FG2/∅, whereG1, G2 : C → Care defined byG1x PCx−λnAx,G2x PCx−ηnBx, ∀x ∈C.
Letf :C → Cbe a contraction with the coefficientθ ∈0,1. LetSnbe the S-mappings generated
byT1, T2, . . . , TNandα1n, α2n, . . . , αNn, whereαjn αn,j1 , αn,j2 , αn,j3 ∈I×I×I, I 0,1,αn,j1 αn,j2 αn,j3 1and0 < η1 ≤ αn,j1 ≤θ1 < 1∀n∈N,∀j 1,2, . . . , N−1, 0< ηN ≤αn,N1 ≤1and
0≤αn,j2 , αn,j3 ≤θ3 <1 ∀n∈N, ∀j1,2, . . . , N. Let{xn},{un},{vn},{yn}be sequences generated byx1, u, v∈C
Fun, u Axn, u−un 1
rnu−un, un−xn ≥0, Gvn, v Bxn, v−vn
1
snv−
vn, vn−xn ≥0,
ynδnPCun−λnAun 1−δnPC
vn−ηnBvn
, xn1αnfxn βnxnγnSnyn, ∀n≥1,
3.1
where {αn},{βn},{γn} ∈ 0,1such thatαnβnγn 1,rn ∈ a, b ⊂ 0,2α,sn ∈ c, d ⊂
0,2β,λn∈e, f⊂0,2α,ηn∈g, h⊂0,2β. Assume that
ilimn→ ∞n0andΣ∞n0αn∞,
ii0<lim infn→ ∞βn≤lim supn→ ∞βn<1,
iiilimn→ ∞δn δ∈0,1,
iv Σ∞n0|sn1−sn|,Σ∞n0|rn1−rn|,Σ∞n0|λn1−λn|,Σ∞n0|ηn1−ηn|,Σ∞n0|αn1−αn|,Σ∞n0|βn1− βn|<∞,
v|αn11,j−α1n,j| → 0, and|α3n1,j−αn,j3 | → 0as n → ∞, for allj∈ {1,2,3, . . . , N}. Then the sequence{xn},{yn},{un},{vn}converge strongly tozPFfz, andzis solution of
Ax∗, x−x∗ ≥0,
Bx∗, x−x∗ ≥0. 3.2
Proof. First, we show thatI−λnA,I−ηnBI−rnAandI−snBare nonexpansive. Let
x, y∈C. SinceAisα-strongly monotone andλn<2αfor alln∈N, we have I−λnAx−I−λnAy2x−y−λn
Ax−Ay2
x−y2−2λnx−y, Ax−Ayλ2nAx−Ay2 ≤x−y2−2αλnAx−Ay2λ2nAx−Ay
2
x−y2λnλn−2αAx−Ay2 ≤x−y2.
ThusI−λnAis nonexpansive. By using the same proof, we obtain thatI−ηnB I−rnA
andI−snBare nonexpansive.
We will divide our proof into 6 steps.
Step 1. We will show that the sequence{xn}is bounded. Since
Fun, u Axn, u−un 1
rnu−un, un−xn ≥0, ∀u∈C, 3.4
then we have
Fun, u 1 rnu−
un, un−I−rnAxn ≥0. 3.5
ByLemma 2.7, we haveunTrnI−rnAxn. By the same argument as above, we obtaine that
vnTsnI−snBxn
Letz∈F. ThenFz, y y−z, Az ≥0 andGz, y y−z, Bz ≥0. Hence
Fz, y 1 rn
y−z, z−zrnAz
≥0,
Gz, y 1 sn
y−z, z−zsnBz
≥0.
3.6
Again byLemma 2.7, we havez Trnz−rnAz Tsnz−snBz. Sincez ∈ F, we have
zPCI−λnAzPCI−ηnBz. By nonexpansiveness ofTrn,Tsn,I−rnA,I−snB, we have
xn1−z ≤αnfxn−zβnxn−zγnSnyn−z
≤αnfxn−fzαnfz−zβnxn−zγnyn−z ≤αnθxn−zαnfz−zβnxn−z
γnδnPCun−λnAun−z 1−δn
PC
vn−ηnBvn
−z
≤αnθxn−zαnfz−zβnxn−zγnδnun−z 1−δnvn−z
αnθxn−zαnfz−zβnxn−z
γnδnTrnI−rnAxn−TrnI−rnAz
1−δnTsnI−snBxn−TsnI−snBz
≤αnθxn−zαnfz−zβnxn−zγnxn−z
αnθxn−zαnfz−z 1−αnxn−z
1−αn1−θxn−zαnfz−z
≤max
xn−z,fz−z
1−θ
.
By induction we can prove that{xn}is bounded and so are{un},{vn},{yn},{Snyn}. Without of generality, assume that there exists a bounded setK⊂Csuch that
{un},{vn},yn
,Snyn
∈K. 3.8
Step 2. We will show that limn→ ∞xn1−xn0.
Puttingkn xn1−βnxn/1−βn, we have
xn1
1−βnknβnxn, ∀n≥0. 3.9
From definition ofkn, we have
kn1−kn
xn2−βn1xn1
1−βn1 −
xn1−βnxn
1−βn
αn1fxn1 γn1Sn1yn1
1−βn1 −
αnfxn γnSnyn
1−βn
αn1fxn1
1−βn1−αn1
Sn1yn1
1−βn1 −
αnfxn 1−βn−αnSnyn
1−βn
αn1
1−βn1
fxn1−Sn1yn1
− αn
1−βn
fxn−SnynSn1yn1−Snyn
≤ αn1
1−βn1
fxn1−Sn1yn1 αn
1−βn
fxn−SnynSn1yn1−Snyn
≤ αn1
1−βn1
fxn1−Sn1yn1 αn
1−βnfxn−Snyn
Sn1yn1−Snynxn1−xn.
By definition ofSn, fork∈ {2,3, . . . , N}, we have
Un1,kyn−Un,kynαn11,kTkUn1,k−1ynαn21,kUn1,k−1ynαn31,kyn
−αn,k
1 TkUn,k−1yn−αn,k2 Un,k−1yn−αn,k3 yn
αn11,kTkUn1,k−1yn−TkUn,k−1yn
αn11,k−αn,k1 TkUn,k−1yn
αn31,k−α3n,kynαn21,k
Un1,k−1yn−Un,k−1yn
αn21,k−αn,k2 Un,k−1yn
≤αn11,kUn1,k−1yn−Un,k−1ynαn11,k−α1n,kTkUn,k−1yn
αn31,k−αn,k3 ynαn21,kUn1,k−1yn−Un,k−1yn
αn21,k−αn,k2 Un,k−1yn
αn11,kα2n1,kUn1,k−1yn−Un,k−1yn
αn11,k−αn,k1 TkUn,k−1ynα3n1,k−αn,k3 yn
αn21,k−αn,k2 Un,k−1yn
≤Un1,k−1yn−Un,k−1ynαn11,k−αn,k1 TkUn,k−1yn
αn31,k−αn,k3 yn1−α1n1,kαn31,k
−1−αn,k1 αn,k3 Un,k−1yn
Un1,k−1yn−Un,k−1ynαn11,k−α n,k
1 TkUn,k−1yn
αn31,k−αn,k3 ynαn,k1 −αn11,kαn,k3 −αn31,kUn,k−1yn
≤Un1,k−1yn−Un,k−1ynαn11,k−αn,k1 TkUn,k−1yn
αn31,k−αn,k3 ynαn,k1 −α1n1,kUn,k−1yn
αn,k3 −α3n1,kUn,k−1yn
Un1,k−1yn−Un,k−1ynαn11,k−α n,k
1 TkUn,k−1ynUn,k−1yn
αn31,k−αn,k3 ynUn,k−1yn.
By3.11, we obtain that for eachn∈N, Sn1yn−SnynUn1,Nyn−Un,Nyn
≤Un1,N−1yn−Un,N−1ynαn11,N−αn,N1 TNUn,N−1yn
Un,N−1ynαn31,N−αn,N3 ynUn,N−1yn
≤Un1,N−2yn−Un,N−2ynαn11,N−1−αn,N1 −1 ×TN−1Un,N−2yn Un,N−2yn
αn31,N−1−α3n,N−1ynUn,N−2yn
αn11,N−αn,N1 TNUn,N−1ynUn,N−1yn
αn31,N−α3n,NynUn,N−1yn
Un1,N−2yn−Un,N−2yn N
jN−1
αn11,j−αn,j1 TjUn,j−1yn
Un,j−1yn N
jN−1
αn31,j−αn,j3 ynUn,j−1yn
≤
.. .
≤Un1,1yn−Un,1yn N
j2
αn11,j−αn,j1 TjUn,j−1ynUn,j−1yn
N
j2
αn31,j−αn,j3 ynUn,j−1yn
1−αn11,1ynαn11,1T1yn−
1−αn,11yn−αn,11T1yn
N
j2
αn11,j−αn,j1 TjUn,j−1ynUn,j−1yn
N
j2
αn31,j−αn,j3 ynUn,j−1yn
αn11,1−α1n,1T1yn−yn
N
j2
αn11,j−αn,j1 TjUn,j−1ynUn,j−1yn
N
j2
αn31,j−αn,j3 ynUn,j−1yn.
This together with the conditioniv, we obtain
lim
n→ ∞Sn1yn−Snyn0. 3.13
By3.10,3.13and conditionsi,ii,iii,iv, it implies that
lim sup
n→ ∞ kn1−kn − xn1−xn≤0. 3.14
FromLemma 2.5,3.9,3.14and conditionii, we have
lim
n→ ∞xn−kn0. 3.15
From3.9, we can rewrite
xn1−xn
1−βn
kn−xn. 3.16
By3.15, we have
lim
n→ ∞xn1−xn0. 3.17
On the other hand, we have
xn−Snyn≤ xn−xn1xn1−Snyn
xn−xn1αnfxn βnxnγnSnyn−Snyn
xn−xn1αn
fxn−Snynβnxn−Snyn
xn−xn1αnfxn−Snynβnxn−Snyn.
3.18
This implies that
1−βnxn−Snyn≤ xn−xn1αnfxn−Snyn. 3.19
By3.17and conditionii, we have
lim
n→ ∞xn−Snyn0. 3.20
Step 3. Letz∈F; we show that
lim
From definition ofyn, we have
yn−z2
δnPCun−λnAun−PCI−λnAz 1−δnPCvn−ηnBvn
−PCI−ηnBz2
≤δnPCun−λnAun−PCI−λnAz2
1−δnPC
vn−ηnBvn
−PC
I−ηnB
z2
≤δnun−λnAun−zλnAz2 1−δnvn−ηnBvn−zηnBz2
δnun−z−λnAun−Az2 1−δnvn−z−ηnBvn−Bz2
δn
un−z2λ2nAun−Az2−2λnun−z, Aun−Az
1−δnvn−z2ηn2Bvn−Bz2−2ηnvn−z, Bvn−Bz
≤δn
un−z2λ2nAun−Az2−2λnαAun−Az2
1−δnvn−z2ηn2Bvn−Bz2−2ηnβBvn−Bz2
δn
un−z2−λn2α−λnAun−Az2
1−δnvn−z2−ηn2β−ηnBvn−Bz2
3.22
δnTrnI−rnAxn−Trnz−rnAz
2−λn2α−λnAun−Az2
1−δn
TsnI−snBxn−Tsnz−snBz
2−η n
2β−ηn
Bvn−Bz2
≤δnxn−z2−λn2α−λnAun−Az2
1−δn
xn−z2−ηn
2β−ηn
Bvn−Bz2
xn−z2−λnδn2α−λnAun−Az2
−ηn1−δn2β−ηnBvn−Bz2.
By3.23, we have
xn1−z2αn
fxn−z
βnxn−z γn
Snyn−z2
≤αnfxn−z2βnxn−z2γnSnyn−z2
≤αnfxn−z2βnxn−z2γnyn−z2
≤αnfxn−z2βnxn−z2γn
xn−z2−λnδn2α−λnAun−Az2
−ηn1−δn2β−ηnBvn−Bz2
αnfxn−z2βnxn−z2γnxn−z2−λnγnδn2α−λnAun−Az2
−ηnγn1−δn2β−ηnBvn−Bz2
≤αnfxn−z2xn−z2−λnγnδn2α−λnAun−Az2
−ηnγn1−δn
2β−ηn
Bvn−Bz2.
3.24
By3.24, we have
λnγnδn2α−λnAun−Az2≤αnfxn−z2xn−z2− xn1−z2
−ηnγn1−δn
2β−ηn
Bvn−Bz2
≤αnfxn−z2 xn−zxn1−zxn1−xn.
3.25
From conditionsi–iiiand3.17, we have
lim
n→ ∞Aun−Az
20.
3.26
By using the same method as3.26, we have
lim
n→ ∞Bvn−Bz 20.
By nonexpansiveness ofTrn, Tsn, I−λnA, I−ηnBand3.23, we have
yn−z2≤δn
PCun−λnAun−PCI−λnAz2
1−δnPC
vn−ηnBvn
−PC
I−ηnB
z2 ≤δnI−λnAun−I−λnAz2 1−δnI−ηnB
vn−
I−ηnB
z2 ≤δnun−z2 1−δnvn−z2
δnTrnI−rnAxn−TrnI−rnAz
2 1−δ
nTsnI−snBxn
−TsnI−snBz
2
≤δnI−rnAxn−I−rnAz2 1−δnI−snBxn−I−snBz2
δnxn−rnAxn−zrnAz2 1−δnxn−snBxn−zsnBz2
δnxn−z−rnAxn−Az2 1−δnxn−z−snBxn−Bz2
δn
xn−z2rn2Axn−Az2−2rnxn−z, Axn−Az
1−δn
xn−z2s2nBxn−Bz2−2snxn−zBxn−Bz
δnxn−z2r2
nδnAxn−Az2−2δnrnxn−z, Axn−Az
1−δnxn−z2s2n1−δnBxn−Bz2−2sn1−δnxn−zBxn−Bz ≤ xn−z2r2
nδnAxn−Az2−2δnrnαAxn−Az2
s2n1−δnBxn−Bz2−2sn1−δnβBxn−Bz2
xn−z2−δnrn2α−rnAxn−Az2−sn1−δn2β−snBxn−Bz2.
3.28
By3.28, we have
xn1−z2αn
fxn−zβnxn−z γnSnyn−z2 ≤αnfxn−z2βnxn−z2γnSnyn−z2 ≤αnfxn−z2βnxn−z2γnyn−z2 ≤αnfxn−z2βnxn−z2
γn
xn−z2−δnrn2α−rnAxn−Az2
−sn1−δn2β−snBxn−Bz2
−snγn1−δn2β−snBxn−Bz2
≤αnfxn−z2xn−z2−δnγnrn2α−rnAxn−Az2 −snγn1−δn
2β−sn
Bxn−Bz2.
3.29
By3.29, we have
δnγnrn2α−rnAxn−Az2≤αnfxn−z2xn−z2− xn1−z2 −snγn1−δn2β−snBxn−Bz2
≤αnfxn−z2 xn−zxn1−zxn1−xn.
3.30
From3.17and conditionsi–iii, we have
lim
n→ ∞Axn−Az0. 3.31
By using the same method as3.31, we have
lim
n→ ∞Bxn−Bz0. 3.32
Step 4. We will show that
lim
n→ ∞yn−xn0. 3.33
PuttingMnPCun−λnAunandNnPCvn−ηnBvn, we will show that
lim
n→ ∞un−xnnlim→ ∞vn−xnnlim→ ∞Mn−unnlim→ ∞Nn−vn0. 3.34
Letz∈F; by3.28, we have
yn−z2≤δnMn−z2 1−δnNn−z2
By nonexpansiveness ofI−rnA, we have
un−z2 Tr
nxn−rnAxn−Trnz−rnAz
2
≤ xn−rnAxn−z−rnAz, un−z
1
2
xn−rnAxn−z−rnAz2un−z2
−xn−rnAxn−z−rnAz−un−z2
≤ 1
2
xn−z2un−z2− xn−un−rnAxn−Az2
1
2
xn−z2un−z2− xn−un2
2rnxn−un, Axn−Az −rn2Axn−Az2.
3.36
This implies
un−z2≤ xn−z2− xn−un22rnxn−un, Axn−Az −rn2Axn−Az2. 3.37
By using the same method as3.37, we have
vn−z2≤ xn−z2− xn−vn22snxn−vn, Bxn−Bz −sn2Bxn−Bz2. 3.38
Substituting3.37and3.38into3.35, we have
yn−z2≤δnun−z2 1−δnvn−z2
≤δnxn−z2− xn−un22rnxn−un, Axn−Az −rn2Axn−Az2
1−δnxn−z2− xn−vn22snxn−vn, Bxn−Bz −s2nBxn−Bz2 ≤δnxn−z2−δnxn−un22δnrnxn−unAxn−Az 1−δnxn−z2
−1−δnxn−vn22sn1−δnxn−vnBxn−Bz
xn−z2−δnxn−un22δnrnxn−unAxn−Az −1−δnxn−vn2
2sn1−δnxn−vnBxn−Bz.
By3.39, we have
xn1−z2≤αnfxn−z2βnxn−z2γnyn−z2
≤αnfxn−z2βnxn−z2
γnxn−z2−δnxn−un2
2δnrnxn−unAxn−Az −1−δnxn−vn2
2sn1−δnxn−vnBxn−Bz
αnfxn−z2βnxn−z2γnxn−z2−γnδnxn−un2
2γnδnrnxn−unAxn−Az −1−δnγnxn−vn2
2snγn1−δnxn−vnBxn−Bz
≤αnfxn−z2xn−z2−γnδnxn−un2
2γnδnrnxn−unAxn−Az −1−δnγnxn−vn2
2snγn1−δnxn−vnBxn−Bz.
3.40
It follows that
γnδnxn−un2≤αnfxn−z2xn−z2− xn
1−z2
2γnδnrnxn−unAxn−Az −1−δnγnxn−vn2
2snγn1−δnxn−vnBxn−Bz
≤αnfxn−z2 xn−zxn1−zxn1−xn
2γnδnrnxn−unAxn−Az2snγn1−δnxn−vnBxn−Bz.
3.41
By conditionsi–iii,3.41,3.31,3.32, and3.17, we have
lim
n→ ∞xn−un0. 3.42
By using the same method as3.42, we have
lim
By nonexpansiveness ofTrnI−rnA, we have
Mn−z2PCun−λnAun−PCz−λnAz2 ≤ un−αnAun−z−αnAz, Mn−z
1
2
un−αnAun−z−αnAz2Mn−z2− un−αnAun
−z−αnAz−Mn−z2
≤ 1
2
un−z2Mn−z2− un−Mn−αnAun−Az2
1
2
TrnI−rnAxn−TrnI−rnAz
2M
n−z2− un−Mn2
2αnun−Mn, Aun−Az −α2nAun−Az2
≤ 1
2
xn−z2Mn−z2− un−Mn22αnun−Mn, Aun−Az
−α2
nAun−Az2
.
3.44
Hence, we have
Mn−z2≤ xn−z2− un−Mn22αnun−Mn, Aun−Az −α2nAun−Az2.
3.45
By using the same method as3.45, we have
Nn−z2≤ xn−z2− vn−Nn22ηnvn−Nn, Bvn−Bz −η2
nBvn−Bz2. 3.46
Substituting3.45and3.46into3.35, we have yn−z2≤δnMn−z2 1−δnNn−z2
≤δnxn−z2− un−Mn22αnun−Mn, Aun−Az −α2nAun−Az2
1−δnxn−z2− vn−Nn22ηnvn−Nn, Bvn−Bz −η2nBvn−Bz2 ≤δnxn−z2−δnun−Mn22δnαnun−MnAun−Az
1−δnxn−z2−1−δnvn−Nn221−δnηnvn−NnBvn−Bz
xn−z2−δnun−Mn22δnαnun−MnAun−Az −1−δnvn−Nn2
21−δnηnvn−NnBvn−Bz.
By3.47, we have
xn1−z2≤αnfxn−z2βnxn−z2γnyn−z2
≤αnfxn−z2βnxn−z2
γnxn−z2−δnun−Mn2
2δnαnun−MnAun−Az −1−δnvn−Nn2
21−δnηnvn−NnBvn−Bz
αnfxn−z2βnxn−z2γnxn−z2−δnγnun−Mn2
2δnγnαnun−MnAun−Az −1−δnγnvn−Nn2
21−δnγnηnvn−NnBvn−Bz
≤αnfxn−z2xn−z2−δnγnun−Mn2
2δnγnαnun−MnAun−Az −1−δnγnvn−Nn2
21−δnγnηnvn−NnBvn−Bz.
3.48
It follows that
δnγnun−Mn2≤αnfxn−z2xn−z2− xn
1−z2
2δnγnαnun−MnAun−Az −1−δnγnvn−Nn2
21−δnγnηnvn−NnBvn−Bz
≤αnfxn−z2 xn−zxn1−zxn1−xn
2δnγnαnun−MnAun−Az21−δnγnηnvn−NnBvn−Bz.
3.49
From3.17,3.26,3.27, and conditionsi–iii, we have
lim
n→ ∞un−Mn0. 3.50
By using the same method as3.50, we have
lim
By3.42and3.50, we have
lim
n→ ∞Mn−xn0. 3.52
By3.43and3.51, we have
lim
n→ ∞Nn−xn0. 3.53
SinceMnPCun−λnAunandNnPCvn−ηnBvn, we have
yn−xnδnMn−xn 1−δnNn−xn. 3.54
By3.52and3.53, we obtain
lim
n→ ∞yn−xn0. 3.55
Note that
xn−Snxn ≤xn−SnynSnyn−Snxn ≤xn−Snynyn−xn.
3.56
From3.20and3.55, we have
lim
n→ ∞xn−Snxn0. 3.57
Step 5. We will show that
lim sup
n→ ∞
fz−z, xn−z≤0, 3.58
wherezPFfz. To show this inequality, take subsequence{xni}of{xn}such that
lim sup
n→ ∞
fz−z, xn−zlim sup
i→ ∞
fz−z, xni−z
. 3.59
Since{xni}is bounded, there exists a subsequence{xnij}of {xni} which converges weakly
toq. Without loss of generality, we can assume thatxni q. SinceCis closed convex,Cis
weakly closed. So, we haveq∈C. Let us show thatq∈F Ni1FTi∩EPF, A∩EPG, B∩ FG1∩FG2. We first show thatq∈EPF, A∩EPG, B∩FG1∩FG2. From3.42, we
haveuni q. SinceunTrnI−rnAxn, for anyy∈C, we have
Fun, yAxn, y−un 1 rn
FromA2, we have
Axn, y−un
1
rn
y−un, un−xn
≥Fy, un
. 3.61
This implies that
Axni, y−uni
1
rni
y−uni, uni−xni
≥Fy, uni
. 3.62
Putzt ty 1−tqfor allt∈ 0,1andy ∈C. Then, we havezt ∈C. So, from3.62, we have
zt−uni, Azt ≥ zt−uni, Azt − zt−uni, Axni −
zt−uni,
uni −xni
rni
Fzt, uni
zt−uni, Azt−Aunizt−uni, Auni −Axni
−
zt−uni,
uni −xni
rni
Fzt, uni.
3.63
Sinceuni−xni → 0, we haveAuni−Axni → 0. Further, from monotonicity ofA, we have
zt−uni, Azt−Auni ≥0. So, fromA4, we have
zt−q, Azt≥Fzt, q asi−→ ∞. 3.64
FromA1,A4, and3.64, we also have
0Fzt, zt≤tF
zt, y
1−tFzt, q
≤tFzt, y
1−tzt−q, Azt
tFzt, y 1−tty−q, Azt.
3.65
Thus
0≤Fzt, y 1−ty−q, Azt. 3.66
Lettingt → 0, we have, for eachy∈C,
0≤Fq, yy−q, Aq. 3.67
This implies that
From3.43, we havevni q. SincevnTsnI−snBxn, for anyy∈C, we have
Gvn, y
Bxn, y−vn
1
sn
y−vn, vn−xn
≥0. 3.69
FromA2, we have
Bxn, y−vn 1 sn
y−vn, vn−xn≥Gy, vn. 3.70
This implies that
Bxni, y−vni
1
sni
y−vni, vni−xni
≥Gy, vni
. 3.71
Putztty 1−tqfor allt∈0,1andy∈C. Then, we havezt∈C. So, from3.71we have
zt−vni, Bzt ≥ zt−vni, Bzt − zt−vni, Bxni −
zt−vni,
vni−xni
sni
Gzt, vni
zt−vni, Bzt−Bvnizt−vni, Bvni−Bxni −
zt−vni,
vni−xni
sni
Gzt, vni.
3.72
Sincevni−xni → 0, we haveBvni−Bxni → 0. Further, from monotonicity ofB, we have
zt−vni, Bzt−Bvni ≥0. So, fromA4, we have
zt−q, Bzt≥Gzt, q. 3.73
FromA1,A4, and3.64, we also have
0Gzt, zt≤tGzt, y 1−tGzt, q ≤tGzt, y 1−tzt−q, Bzt
tGzt, y
1−tty−q, Bzt
,
3.74
hence
0≤Gzt, y
1−ty−q, Bzt
. 3.75
Lettingt → 0, we have, for eachy∈C,
This implies that
q∈EPG, B. 3.77
Define a mappingQ:C → Cby
QxδPCI−λnAx 1−δPC
I−ηnB
x, ∀x∈C, 3.78
where limn→ ∞δnδ∈0,1. FromLemma 2.3, we have thatQis nonexpansive with
FQ FPCI−λnAFPCI−ηnB. 3.79
Next, we show that
lim
n→nxn−Qxn0. 3.80
By nonexpansiveness ofI−ηnBandI−λnA, we have
xn−Qxn ≤xn−ynyn−Qxn
xn−ynδnPCun−λnAun 1−δnPC
vn−ηnBvn
−δPCI−λnAxn −1−δPCI−ηnBxn
xn−ynδnPCI−λnAun−δnPCI−λnAxnδnPCI−λnAxn
1−δnPC
I−ηnB
vn−1−δnPC
I−ηnB
xn
1−δnPCI−ηnBxn−δPCI−λnAxn−1−δPCI−ηnBxn
xn−ynδnPCI−λnAun−PCI−λnAxn δn−δPCI−λnAxn
1−δnPCI−ηnBvn−PCI−ηnBxn
δ−δnPCI−ηnBxn
≤xn−ynδnPCI−λnAun−PCI−λnAxn|δn−δ|PCI−λnAxn
1−δnPCI−ηnBvn−PCI−ηnBxn|δn−δ|PCI−ηnBxn ≤xn−ynδnun−xn|δn−δ|PCI−λnAxn 1−δnvn−xn
|δn−δ|PC
I−ηnB
xn
≤xn−ynδnun−xn2|δn−δ|M1 1−δnvn−xn,
whereM1 supn≥0{PCI−λnAxnPCI−ηnBxn}. From3.17,3.42,3.43,3.55,
and conditioniii, we have limn→nxn−Qxn0. Sincexni q, it follows from3.80that,
limi→ ∞xni−Qxni0. ByLemma 2.4, we obtain that
q∈FQ FPCI−λnA∩FPCI−ηnBFG1∩FG2. 3.82
Assume thatq /Sq. Using Opials,property,3.57andLemma 2.10we have
lim inf
i→ ∞ xni −q<lim inf
i→ ∞ xni−Sq
≤lim inf
i→ ∞
xni−SnixniSnixni−SniqSniq−Sq
≤lim inf
i→ ∞ xni−q.
3.83
This is a contradiction, so we have
q∈ N
i1
FTi FS. 3.84
From 3.68, 3.77 3.82, and 3.84, we have q ∈ F. Since PFf is contraction with the
coefficient θ ∈ 0,1, PF has a unique fixed point. Letz be a fixed point of PFf, that is
zPFfz. Sincexni qandq∈F, we have
lim sup
n→ ∞
fz−z, xn−zlim sup
i→ ∞
fz−z, xni−z
fz−z, q−z≤0.
3.85
Step 6. Finally, we will show thatxn → zasn → ∞. By nonexpansiveness ofTrn, Tsn, I −
λnA, I−ηnB, I−rnA, I−snB, we can show thatyn−z ≤ xn−z. Then
xn1−z2
αn
fxn−z
βnxn−z γn
Snyn−z
, xn1−z
αn
fxn−z, xn1−z
βnxn−z, xn1−zγn
Snyn−z, xn1−z
≤αn
fxn−fz, xn1−z
αn
fz−z, xn1−z
βnxn−zxn1−z
γnSnyn−zxn1−z
≤αnfxn−fzxn1−zαn
fz−z, xn1−z
βnxn−zxn1−z
≤αnθxn−zxn1−zαn
fz−z, xn1−z
βnxn−zxn1−z
γnxn−zxn1−z
1−αn1−θxn−zxn1−zαn
fz−z, xn1−z
≤1−αn1−θ
xn−z2xn1−z2
2
αnfz−z, xn1−z
≤ 1−αn1−θ
2 xn−z
2xn1−z2
2 αn
fz−z, xn1−z
;
3.86
we have
xn1−z2≤1−αn1−θxn−z22αnfz−z, xn1−z. 3.87
ByStep 5,3.87, andLemma 2.2, we have limn→ ∞xn z, wherez PFfz. It easy to see
that sequences{yn},{un}, and{vn}converge strongly tozPFfz.
4. Application
Using our main theorem Theorem 3.1, we obtain the following strong convergence theorems involving finite family ofκ-strict pseudocontractions.
To prove strong convergence theorem in this section, we need definition and lemma as follows.
Definition 4.1. A mappingT :C → Cis said to be aκ-strongly pseudo contraction mapping, if there existκ∈0,1such that
Tx−Ty2 ≤
x−y2κI−Tx−I−Ty2, ∀x, y∈C. 4.1
Lemma 4.2see20. LetC be a nonempty closed convex subset of a real Hilbert spaceH and
T :C → Caκ-strict pseudo contraction. DefineS:C → CbySxαx 1−αTxfor eachx∈C. Then, asα∈κ,1Sis nonexpansive such thatFS FT.
Theorem 4.3. LetCbe a nonempty closed convex subset of a Hilbert spaceH. LetFandGbe two bifunctions fromC×CintoRsatisfying (A1)–(A4), respectively. LetA : C → His aα-inverse strongly monotone mapping andB:C → Hbe aβ-inverse strongly monotone mapping. Let{Ti}Ni1 be a finite family ofκi-psuedo contractions withF Ni1FTi∩EPF, A∩EPG, B∩FG1∩ FG2/∅, where G1, G2 : C → C are defined by G1x PCx−λnAx, G2x PCx− ηnBx, for allx∈C. Define a mappingTκibyTκi κix1−κiTix, for all x∈C, i∈ {1,2, . . . , N}.
Letf :C → Cbe a contraction with the coefficientθ ∈0,1. LetSnbe the S-mappings generated
by Tκ1, Tκ2, . . . , TκN andα
n
1 , α
n
2 , . . . , α
n
N , whereα
n
j α
n,j 1 , α
n,j 2 , α
n,j
αn,j1 αn,j2 αn,j3 1and0< η1 ≤αn,j1 ≤θ1<1for alln∈N,for allj 1,2, . . . , N−1, 0< ηN ≤ αn,N1 ≤1and0≤α2n,j, αn,j3 ≤θ3<1for alln∈N,for allj1,2, . . . , N. Let{xn},{un},{vn},{yn} be sequences generated byx1, u, v∈C
Fun, u Axn, u−un
1
rnu−
un, un−xn ≥0,
Gvn, v Bxn, v−vn
1
snv−
vn, vn−xn ≥0,
ynδnPCun−λnAun 1−δnPCvn−ηnBvn,
xn1αnfxn βnxnγnSnyn, ∀n≥1,
4.2
where{αn},{βn},{γn} ∈0,1such thatαnβnγn 1, rn ∈a, b⊂ 0,2α, sn ∈ c, d ⊂
0,2β,λn∈e, f⊂0,2α,ηn∈g, h⊂0,2β. Assume that
ilimn→ ∞αn0 andΣ∞n0αn∞,
iilim infn→ ∞βn≤lim supn→ ∞βn<1,
iiilimn→ ∞δn δ∈0,1,
iv Σ∞n0|sn1 − sn|,Σ∞n0|rn1 − rn|, Σ∞n0|λn1 − λn|, Σ∞n0|ηn1 − ηn|, Σ∞n0|αn1 − αn|, Σ∞n0|βn1−βn|<∞,
v|αn11,j−α1n,j| → 0 and |α3n1,j−αn,j3 | → 0 asn → ∞, for allj ∈ {1,2,3, . . . , N}.
Then the sequence{xn},{yn},{un},{vn}converges strongly tozPFfz, andzis solution of
Ax∗, x−x∗ ≥0,
Bx∗, x−x∗ ≥0.
4.3
Proof. For every i ∈ {1,2, . . . , N}, by Lemma 4.2, we haveTκi is nonexpansive mappings.
FromTheorem 3.1, we can concluded the desired conclusion.
Theorem 4.4. LetCbe a nonempty closed convex subset of a Hilbert spaceH. LetFandGbe two bifunctions fromC×CintoRsatisfying (A1)–(A4), respectively. LetA : C → H be aα-inverse strongly monotone mapping. Let{Ti}Ni1 be a finite family ofκi-strict pseudo contractions withF
N
i1FTi∩EPF, A∩FG1/∅, whereG1:C → Cdefined byG1x PCx−λnAx, for allx∈ C. Define a mapping Tκi by Tκi κix 1 −κiTix, for allx ∈ C, i ∈ N. Let f : C → C a
contraction with the coefficient θ ∈ 0,1. LetSn be the S-mappings generated byTκ1, Tκ2, . . . , TκN
andα1n, α2n, . . . , αNn, whereαjn αn,j1 , αn,j2 , αn,j3 ∈ I×I×I, I 0,1,αn,j1 αn,j2 αn,j3 1
0 ≤ αn,j2 , αn,j3 ≤ θ3 < 1 for alln ∈ N,for all j 1,2, . . . , N. Let{xn},{un},{yn} be sequences generated byx1, u,∈C
Fun, u Axn, u−un
1
rnu−
un, un−xn ≥0,
ynPCun−λnAun,
xn1αnfxn βnxnγnSnyn, ∀n≥1,
4.4
where{αn},{βn},{γn} ∈ 0,1such thatαnβnγn 1,rn ∈ a, b ⊂ 0,2α,λn ∈ e, f ⊂
0,2α. Assume that
ilimn→ ∞αn0 andΣ∞n0αn∞,
ii0< lim infn→ ∞βn ≤lim supn→ ∞βn<1,
iii Σ∞n0|rn1−rn|, Σ∞n0|λn1−λn|, Σ∞n0|αn1−αn|, Σn∞0|βn1−βn|<∞,
iv|αn11,j−α1n,j| → 0 and|α3n1,j−αn,j3 | → 0 asn → ∞, for allj ∈ {1,2,3, . . . , N}. Then the sequence{xn},{yn},{un}converges strongly tozPFfz, andzis solution of
Ax∗, x−x∗ ≥0. 4.5
Proof. For everyi∈ {1,2, . . . , N}, byLemma 4.2, we have thatTκi is nonexpansive mappings,
puttingF ≡G,A≡B,sn rn,λnηn, andun vn. FromTheorem 3.1, we can conclude the desired conclusion.
Acknowledgment
The authors would like to thank Professor Dr. Suthep Suantai for his suggestion in doing and improving this paper.
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