Volume 2010, Article ID 728028,43pages doi:10.1155/2010/728028
Research Article
Strong Convergence for Generalized Equilibrium
Problems, Fixed Point Problems and Relaxed
Cocoercive Variational Inequalities
Chaichana Jaiboon
1, 2and Poom Kumam
11Department of Mathematics, Faculty of Science, King Mongkut’s University of Technology Thonburi,
KMUTT, Bangkok 10140, Thailand
2Department of Mathematics, Faculty of Applied Liberal Arts, Rajamangala University of Technology,
Rattanakosin, RMUTR, Bangkok 10100, Thailand
Correspondence should be addressed to Poom Kumam,[email protected]
Received 31 October 2009; Accepted 1 February 2010
Academic Editor: Jong Kim
Copyrightq2010 C. Jaiboon 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.
We introduce a new iterative scheme for finding the common element of the set of solutions of the generalized equilibrium problems, the set of fixed points of an infinite family of nonexpansive mappings, and the set of solutions of the variational inequality problems for a relaxedu, v
-cocoercive andξ-Lipschitz continuous mapping in a real Hilbert space. Then, we prove the strong convergence of a common element of the above three sets under some suitable conditions. Our result can be considered as an improvement and refinement of the previously known results.
1. Introduction
Variational inequalities introduced by Stampacchia1in the early sixties have had a great impact and influence in the development of almost all branches of pure and applied sciences. It is well known that the variational inequalities are equivalent to the fixed point problems. This alternative equivalent formulation has been used to suggest and analyze in variational inequalities. In particular, the solution of the variational inequalities can be computed using the iterative projection methods. It is well known that the convergence of a projection method requires the operator to be strongly monotone and Lipschitz continuous. Gabay 2 has shown that the convergence of a projection method can be proved for cocoercive operators. Note that cocoercivity is a weaker condition than strong monotonicity.
problems, we also have the problem of finding the fixed points of the nonexpansive mappings. It is natural to construct a unified approach for these problems. In this direction, several authors have introduced some iterative schemes for finding a common element of a set of the solutions of the equilibrium problems and a set of the fixed points of infinitely
finitelymany nonexpansive mappings; see5–7and the references therein. In this paper, we suggest and analyze a new iterative method for finding a common element of a set of the solutions of generalized equilibrium problems and a set of fixed points of an infinite family of nonexpansive mappings and the set solution of the variational inequality problems for a relaxedu, v-cocoercive mapping in a real Hilbert space.
LetHbe a real Hilbert space and letEbe a nonempty closed convex subset ofHand
PEis the metric projection ofHontoE.Recall that a mappingf :E → Eis contraction onEif there exists a constantα∈0,1such thatfx−fy ≤αx−yfor allx, y∈E.A mapping
SofEinto itself is called nonexpansive ifSx−Sy ≤ x−yfor allx, y ∈ E.We denote byFSthe set of fixed points ofS, that is,FS {x∈E: Sxx}. IfE⊂ His nonempty, bounded, closed, and convex andSis a nonexpansive mapping ofEinto itself, thenFSis nonempty; see, for example,8. We recalled some definitions as follows.
Definition 1.1. LetB:E → Hbe a mapping. Then one has the following.
1Bis calledmonotoneifBx−By, x−y ≥0,for allx, y∈E.
2Bis calledv-strongly monotoneif there exists a positive real numbervsuch that
Bx−By, x−y≥vx−y2, ∀x, y∈E. 1.1
3Bis calledξ-Lipschitz continuousif there exists a positive real numberξsuch that Bx−By≤ξx−y, ∀x, y∈E. 1.2
4Bis calledη-inverse-strongly monotone,9,10if there exists a positive real number
ηsuch that
Bx−By, x−y≥ηBx−By2, ∀x, y∈E. 1.3
If η 1,we say thatBis firmly nonexpansive. It is obvious that any η -inverse-strongly monotone mappingBis monotone and1/η-Lipschitz continuous.
5Bis calledrelaxedu, v-cocoerciveif there exists a positive real numberu, vsuch that
Bx−By, x−y ≥−uBx−By2vx−y2, ∀x, y∈E. 1.4
6A set-valued mappingT:H → 2His calledmonotoneif for allx, y∈H,f ∈Txand
g ∈Tyimplyx−y, f−g ≥0. A monotone mappingT :H → 2Hismaximalif the graph ofGTofT is not properly contained in the graph of any other monotone mapping. It is known that a monotone mapping T is maximal if and only if for
x, f∈H×H,x−y, f−g ≥0 for everyy, g∈GTimpliesf ∈Tx.
LetBbe a monotone mapping of EintoH and letNEw1 be thenormal conetoEat
w1∈E, that is,
NEw1{w∈H:ϑ−w1, w ≥0, ∀ϑ∈E}. 1.5
Define
Tw1
⎧ ⎨ ⎩
Bw1NEw1, ifw1∈E,
∅, ifw1/∈E.
1.6
ThenT is the maximal monotone and 0∈Tw1if and only ifw1∈VIE, B; see11,12
In addition, let D : E → H be a inverse-strongly monotone mapping. LetF be a bifunction ofE×E intoR, whereRis the set of real numbers. The generalized equilibrium problem forF:E×E → Ris to findx∈Esuch that
Fx, y Dx, y−x ≥0, ∀y∈E. 1.7
The set of suchx∈Eis denoted by EPF, D,that is,
EPF, D x∈E:Fx, y Dx, y−x≥0, ∀y∈E. 1.8
Special Cases
IIfD≡0:the zero mapping, then the problem1.7is reduced to the equilibrium problem:
Findx∈Esuch thatFx, y ≥0, ∀y∈E. 1.9
The set of solutions of1.9is denoted by EPF,that is,
EPF x∈E:Fx, y ≥0, ∀y∈E. 1.10
IIIfF ≡0, the problem1.7is reduced to the variational inequality problem:
Find x∈Esuch thatDx, y−x ≥0, ∀y∈E. 1.11
The set of solutions of1.11is denoted by VIE, D, that is,
The generalized equilibrium problem1.7is very general in the sense that it includes, as special case, some optimization, variational inequalities, minimax problems, the Nash equilibrium problem in noncooperative games, economics, and others see, e.g., 4, 13. Some methods have been proposed to solve the equilibrium problem and the generalized equilibrium problem; see, for instance,5,14–28. Recently, Combettes and Hirstoaga29
introduced an iterative scheme of finding the best approximation to the initial data when EPFis nonempty and proved a strong convergence theorem. Very recently, Moudafi24
introduced an itertive method for finding an element of EPF, D∩FS, whereD:E → H
is an inverse-strongly monotone mapping and then proved a weak convergence theorem. For finding a common element of the set of fixed points of a nonexpansive mapping and the set of solutions of variational inequality problem for anη-inverse-strongly monotone, Takahashi and Toyoda30introduced the following iterative scheme:
x0∈Echosen arbitrary,
xn1αnxn 1−αnSPExn−τnBxn, ∀n≥0,
1.13
whereBis anη-inverse-strongly monotone mapping,{αn}is a sequence in0, 1, and{τn} is a sequence in0,2η. They showed that ifFS∩VIE, Bis nonempty, then the sequence
{xn}generated by1.13converges weakly to somez∈FS∩VIE, B. On the other hand, Shang et al.31introduced a new iterative process for finding a common element of the set of fixed points of a nonexpansive mapping and the set of solutions of the variational inequality problem for a relaxed u, v-cocoercive mapping in a real Hilbert space. Let S : E → E
be a nonexpansive mapping. Starting with arbitrary initialx1 ∈ E,defined sequences{xn} recursively by
xn1αnfxn βnxnγnSPEI−τnBxn, ∀n≥1. 1.14
They proved that under certain appropriate conditions imposed on{αn},{βn},{γn},and{τn}, the sequence{xn}converges strongly toz∈FS∩VIE, B, wherezPFS∩VIE,Bfz.
In 2008, S. Takahashi and W. Takahashi27introduced the following iterative scheme for finding an element ofFS∩EFF, Dunder some mild conditions. LetEbe a nonempty closed convex subset of a real Hilbert spaceH. Let D be an η-inverse-strongly monotone mapping ofEintoHand letSbe a nonexpansive mapping ofEinto itself such thatFS∩
EPF, D/∅.Supposex1 σ∈Eand let{un},{yn}, and{xn}by sequences generated by
Fun, y
Dxn, y−un
1
rn
y−un, un−xn
≥0, ∀y∈C,
ynαnσ 1−αnun,
xn1βnxn
1−βn Syn,
1.15
where {αn} ⊂ 0,1,{βn} ⊂ 0,1,and {rn} ⊂ 0,2η satisfy some parameters controlling conditions. They proved that the sequence{xn} defined by 1.15converges strongly to a common element ofFS∩EFF, D.
references therein. Convex minimization problems have a great impact and influence in the development of almost all branches of pure and applied sciences.
A typical problem is to minimize a quadratic function over the set of the fixed points a nonexpansive mapping in a real Hilbert spaceH:
min x∈E
1
2Ax, x − x, b
, 1.16
whereEis the fixed point set of a nonexpansive mappingSonHandbis a given point inH. Assume thatAis astrongly positive bounded linear operatoronH; that is, there exists a constant
γ >0 such that
Ax, x ≥γx2, ∀x∈H. 1.17
In 2006, Marino and Xu36considered the following iterative method:
xn1nγfxn 1−nASxn, ∀n≥0. 1.18
They proved that if the sequence{n}of parameters satisfies appropriate conditions, then the sequence{xn} generated by1.18 converges strongly to the unique of the variational inequality
A−γf z, x−z≥0, ∀x∈FS, 1.19
which is the optimality condition for the minimization problem
min x∈FS
1
2Ax, x −hx
, 1.20
wherehis a potential function forγfi.e.,hx γfxforx∈H. In 2008, Qin et al.26proposed the following iterative algorithm:
Fun, y 1
rn
y−un, un−xn
≥0, ∀y∈H,
xn1nγfxn I−nASPEI−τnBun,
1.21
whereAis a strongly positive linear bounded operator andBis a relaxed cocoercive mapping of E into H. They prove that if the sequences {n}, {τn}, and {rn} of parameters satisfy appropriate condition, then the sequences{xn}and{un}both converge to the unique solution
zof the variational inequality
which is the optimality condition for the minimization problem
min x∈FS∩VIE,B∩EPF
1
2Ax, x −hx
, 1.23
wherehis a potential function forγfi.e.,hx γfxforx∈H.
Furthermore, for finding approximate common fixed points of an infinite family of nonexpansive mappings {Tn} under very mild conditions on the parameters, we need the following definition.
Definition 1.2see37. Let{Tn}∞n1be a sequence of nonexpansive mappings ofEinto itself
and let {μn}∞n1 be a sequence of nonnegative numbers in0,1. For each n ≥ 1, define a
mappingWnofEinto itself as follows:
Un,n1 I,
Un,nμnTnUn,n1
1−μn I,
Un,n−1 μn−1Tn−1Un,n
1−μn−1 I,
.. .
Un,kμkTkUn,k1
1−μk I,
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.
1.24
Such a mappingsWnis called theW-mapping generated byT1, T2, . . . , Tnandμ1, μ2, . . . , μn. It is obvious thatWnis nonexpansive, and ifxTnx,thenxUn,kWnx.
On the other hand, Yao et al.38introduced and considered an iterative scheme for finding a common element of the set of solutions of the equilibrium problem and the set of common fixed points of an infinite family of nonexpansive mappings onE. Starting with an arbitrary initialx1∈H, define sequences{xn}and{un}recursively by
Fun, y
1
rn
y−un, un−xn
≥0, ∀y∈H,
xn1 nγfxn βnxn
1−βn I−nA Wnun, ∀n≥1,
1.25
where{n}is a sequence in0,1. It is proved38that under certain appropriate conditions imposed on {n} and {rn}, the sequence {xn} generated by 1.25 converges strongly to
z P∞
n1FTn∩EPFI−Aγfz. Very recently, Qin et al.6introduced an iterative scheme
solutions of the variational inequality problem for a relaxed cocoercive mapping, and the set of solutions of the equilibrium problems in a real Hilbert space. Starting with an arbitrary initialx1∈H, define sequences{xn}and{un}recursively by
Fun, y
1
rn
y−un, un−xn
≥0, ∀y∈H,
xn1nγfWnxn I−nAWnPEI−τnBun, ∀n≥1,
1.26
whereBis a relaxedu, v-cocoercive mapping andAis a strongly positive linear bounded operator. They proved that under certain appropriate conditions imposed on{n},{τn},and
{rn}, the sequences{xn}and{un}generated by1.26converge strongly to some pointz ∈
∞
n1FTn∩EPF∩VIE, B, which is a unique solution of the variation inequality:
A−γf z, x−z ≥0, ∀x∈
∞
n1
FTn∩EPF∩VIE, B 1.27
and is also the optimality for some minimization problems.
In this paper, motivated by iterative schemes considered in1.15,1.25, and1.26
we will introduce a new iterative process3.4below for finding a common element of the set of fixed points of an infinite family of nonexpansive mappings, the set of solutions of the generalized equilibrium problem, and the set of solutions of variational inequality problem for a relaxedu, v-cocoercive mapping in a real Hilbert space. The results obtained in this paper improve and extend the recent ones announced by Yao et al.38, S. Takahashi and W. Takahashi27, and Qin et al.6and many others.
2. Preliminaries
LetHbe a real Hilbert space with inner product·,·and norm · . LetEbe a nonempty closed convex subset of H. We denote weak convergence and strong convergence by notations and →, respectively. Recall that the nearest point projection PE from H to
Eassigns eachx∈Hthe unique point inPEx∈Esatisfying the property
x−PExmin y∈E
x−y. 2.1
The following characterizes the projectionPE.
We need some facts tools in a real Hilbert spaceHwhich are listed as follows. Lemma 2.1. For anyx∈H,z∈E,
zPEx⇐⇒
x−z, z−y≥0, ∀y∈E. 2.2
It is well known thatPEis a firmly nonexpansive mapping ofHontoEand satisfies
PEx−PEy2≤
PEx−PEy, x−y
Moreover,PExis characterized by the following properties:PEx∈Eand for allx∈H, y∈E,
x−PEx, y−PEx ≤0. 2.4
Lemma 2.2see39. LetHbe a Hilbert space, letEbe a nonempty closed convex subset ofH,and letBbe a mapping ofEintoH. Letp∈E. Then forλ >0,
p∈VIE, B⇐⇒pPE
p−λBp , 2.5
wherePEis the metric projection ofHontoE.
It is clear from Lemma 2.2that variational inequality and fixed point problem are equivalent. This alternative equivalent formulation has played a significant role in the studies of the variational inequalities and related optimization problems.
Lemma 2.3see40. Each Hilbert space H satisfies Opials condition; that is, for any sequence
{xn} ⊂Hwithxn x, the inequality
lim inf
n→ ∞ xn−x<lim infn→ ∞ xn−y 2.6
holds for eachy∈Hwithy /x.
Lemma 2.4see36. Assume thatA is a strongly positive linear bounded operator onH with coefficientγ >0and0< ρ≤ A−1. ThenI−ρA ≤1−ργ.
For solving the equilibrium problem for a bifunctionF:E×E → R, let us assume that
Fsatisfies the following conditions:
A1Fx, x 0, for allx∈E;
A2Fis monotone, that is,Fx, y Fy, x≤0, for allx, y∈E;
A3limt↓0Ftz 1−tx, y≤Fx, y, for allx, y, z∈E;
A4for eachx∈E, y→Fx, yis convex and lower semicontinuous.
The following lemma appears implicitly in4.
Lemma 2.5see4. LetEbe a nonempty closed convex subset ofHand letF be a bifunction of
E×EintoRsatisfying (A1)–(A4). Letr >0andx∈H. Then, there existsz∈Esuch that
Fz, y 1
r
y−z, z−x≥0, ∀y∈E. 2.7
Lemma 2.6see5. Assume thatF:E×E → Rsatisfies (A1)–(A4). Forr >0andx∈H, define a mappingTr :H → Eas follows:
Trx
z∈E:Fz, y 1
r
y−z, z−x≥0, ∀y∈E
, 2.8
for allz∈H. Then, the following holds:
1Tr is single-valued;
2Tr is firmly nonexpansive, that is, for anyx, y∈H,
Trx−Try2≤
Trx−Try, x−y
; 2.9
3FTr EPF;
4EPFis closed and convex.
Remark 2.7. Replacingxwithx−rDx∈Hin2.7, then there existsz∈E, such that
Fz, y Dx, y−z1
ry−z, z−x ≥0, ∀y∈E. 2.10
Lemma 2.8see41. LetE be a nonempty closed convex subset of a real Hilbert space H. Let
T1, T2, . . . be nonexpansive mappings of E into itself such that
∞
n1FTn is nonempty, and let
μ1, μ2, . . .be real numbers such that 0 ≤ μn ≤ b < 1for everyn ≥ 1. Then, for every x ∈ Eand
k∈N, the limitlimn→ ∞Un,kxexists.
UsingLemma 2.8, one can define a mappingWofEinto itself as follows:
Wx lim
n→ ∞Wnxnlim→ ∞Un,1x, 2.11
for every x ∈ E. Such aW is called theW-mapping generated byT1, T2, . . . and μ1, μ2, . . ..
Throughout this paper, we will assume that 0≤μn≤b <1 for everyn≥1. Then, we have the following results.
Lemma 2.9see41. LetE be a nonempty closed convex subset of a real Hilbert space H. Let
T1, T2, . . .be nonexpansive mappings ofEinto itself such that∞n1FTnis nonempty, letμ1, μ2, . . .
be real numbers such that0≤μn≤b <1for everyn≥1. Then,FW
∞
n1FTn.
Lemma 2.10see7. If{xn}is a bounded sequence inE, thenlimn→ ∞Wxn−Wnxn0. Lemma 2.11see42. Let{xn}and{zn}be bounded sequences in a Banach spaceXand let{βn}be a sequence in0,1with0<lim infn→ ∞βn≤lim supn→ ∞βn<1.Supposexn1 1−βnznβnxn for all integersn≥0andlim supn→ ∞zn1−zn − xn1−xn≤0.Then,limn→ ∞zn−xn0. Lemma 2.12. LetHbe a real Hilbert space. Then the following inequality holds:
1xy2≤ x22y, xy,
2xy2≥ x22y, x
Lemma 2.13see43. Assume that{an}is a sequence of nonnegative real numbers such that
an1≤1−lnanσn, ∀n≥0, 2.12
where{ln}is a sequence in0,1and{σn}is a sequence inRsuch that
1∞n1ln∞,
2lim supn→ ∞σn/ln≤0or∞n1|σn|<∞. Thenlimn→ ∞an0.
3. Main Results
In this section, we prove a strong convergence theorem of a new iterative method3.4for an infinite family of nonexpansive mappings and relaxedu, v-cocoercive mappings in a real Hilbert space.
We first prove the following lemmas.
Lemma 3.1. LetHbe a real Hilbert space, letEbe a nonempty closed convex subset ofH, and let
D:E → Hbeη-inverse-strongly monotone. It0≤rn≤2η, thenI−rnDis a nonexpansive mapping inH.
Proof. For allx, y∈Eand 0≤rn≤2η, we have
I−rnDx−I−rnDy2 x−y−rnDx−Dy2
x−y2−2rnx−y, Dx−Dyrn2Dx−Dy
2
≤x−y2−2rnηDx−Dyrn2Dx−Dy
2
x−y2rn
rn−2η Dx−Dy2
≤x−y2.
3.1
So,I−rnDis a nonexpansive mapping ofEintoH.
Lemma 3.2. LetH be a real Hilbert space, letEbe a nonempty closed convex subset ofH,and let
B :E → Hbe a relaxedu, v-cocoercive andξ-Lipschitz continuous. If0 ≤ τn ≤ 2v−uξ2/ξ2,
v > uξ2, thenI−τ
nBis a nonexpansive mapping inH.
Proof. For anyx, y∈Eandτn ≤2v−uξ2/ξ2,v > uξ2. Puttingr12τnuξ2−2τnvτn2ξ2, we obtain
τn≤
2v−uξ2
ξ2 ⇐⇒τnξ
22uξ2−2v≤0
⇐⇒τn2ξ22τnuξ2−2τnv≤0
⇐⇒1τn2ξ22τnuξ2−2τnv≤1,
that is,r≤1. It follows that
I−τnBx−I−τnBy2x−y−τnBx−By2
x−y2−2τnx−y, Bx−Byτn2Bx−By2
≤x−y2−2τn
−uBx−By2vx−y2τn2Bx−By2
≤x−y22τnuξ2x−y2−2τnvx−y2τn2ξ2x−y
2
12τnuξ2−2τnvτn2ξ2
x−y2
rx−y2
≤x−y2,
3.3
for allx, y∈E. ThusI−τnBx−I−τnBy ≤ x−y. So,I−τnBis a nonexpansive mapping ofEintoH. Now, we prove the following main theorem.
Theorem 3.3. LetEbe a nonempty closed convex subset of a real Hilbert spaceH, and letF:E×E → Rbe a bifunction satisfying (A1)–(A4). Let
1{Tn}be an infinite family of nonexpansive mappings ofEintoE;
2Dbe anη-inverse strongly monotone mappings ofEintoH;
3Bbe relaxedu, v-cocoercive andξ-Lipschitz continuous mappings ofEintoH.
Assume thatΘ:∞n1FTn∩EPF, D∩VIE, B/∅. Letf :E → Ebe a contraction mapping
with0< α <1and letAbe a strongly positive linear bounded operator onHwith coefficientγ >0
and0< γ < γ/α. Let{xn},{yn},{kn},and{un}be sequences generated by
x1∈Echosen arbitrary,
Fun, y Dxn, y−un
1
rn
y−un, un−xn ≥0, ∀y∈E,
ynϕnun
1−ϕn WnPEun−δnBun,
knαnxn 1−αnWnPE
yn−λnByn ,
xn1nγfWnxn βnxn
1−βn I−nA WnPEkn−τnBkn, ∀n≥1,
3.4
where {Wn}is the sequence generated by 1.24and{n},{αn},{ϕn},and {βn}are sequences in
0,1satisfy the following conditions:
C1limn→ ∞n0,
∞
n1n∞,
C30<lim infn→ ∞βn≤lim supn→ ∞βn<1,
C4lim infn→ ∞rn>0andlimn→ ∞|rn1−rn|0,
C5limn→ ∞|λn1−λn|limn→ ∞|δn1−δn|limn→ ∞|τn1−τn|0,
C6{τn},{λn},{δn} ⊂a, bfor somea, bwith0≤a≤b≤2v−uξ2/ξ2,v > uξ2,
C7{rn} ⊂c, dfor somec, dwith0< c < d <2η.
Then,{xn}and{un}converge strongly to a pointz∈Θ, wherezPΘI−Aγfz, which solves the variational inequality
A−γf z, x−z ≥0, ∀x∈Θ, 3.5
which is the optimality condition fot the minimization problem
min x∈Θ
1
2Ax, x −hx
, 3.6
wherehis a potential function forγf(i.e.,hx γfxforx∈H).
Proof. Since limn→ ∞n 0 by the condition C1and lim supn→ ∞βn < 1, we may assume, without loss of generality, thatn ≤ 1−βnA−1. SinceAis a strongly positive bounded linear operator onH, then
Asup{|Ax, x|:x∈H,x1}. 3.7
Observe that
1−βn I−nA x, x1−βn−nAx, x
≥1−βn−nA
≥0,
3.8
that is to say1−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.
3.9
We will divide the proof ofTheorem 3.3into six steps. Step 1. We prove that there existsz∈Esuch thatzP∞
LetI P∞
n1FTn∩EPF,D∩VIE,B. Note that f is a contraction mapping ofEinto itself
with coefficientα∈0,1. Then, we have I
I−Aγf x−II−Aγf y ≤I−Aγf x−I−Aγf y
≤ I−Ax−yγfx−fy
≤1−γ x−yγαx−y
1−γ−αγ x−y, ∀x, y∈H.
3.10
Therefore,II−Aγfis a contraction mapping ofEinto itself. Therefore by the Banach Contraction Mapping Principle guarantee thatII−Aγf has a unique fixed point, say
z∈E. That is,zII−Aγfz P∞
n1FTn∩EPF,D∩VIE,BI−Aγfz.
Step 2. We prove that{xn}is bounded. Since
Fun, y Dxn, y−un
1
rn
y−un, un−xn ≥0, ∀y∈E, 3.11
we obtain
Fun, y
1
rn
y−un, un−I−rnDxn ≥0, ∀y∈E. 3.12
FromLemma 2.6, we haveunTrnxn−rnDxn,for alln∈N.
For anyp∈Θ:n∞1FTn∩EPF, D∩VIE, B, it follows fromp∈EPF, Dthat
Fp, y y−p, Dp≥0, ∀y∈E. 3.13
So, we have
Fp, y 1
rn
y−p, p−p−rnDp ≥0, ∀y∈E. 3.14
ByLemma 2.6again, we havepTrnp−rnDp,for alln∈N. If follows that
un−pTr
nxn−rnDxn−Trn
p−rnDp
≤xn−rnDxn−
p−rnDp
I−rnDxn−I−rnDp≤xn−p.
If we appliedLemma 3.2, we getI−λnBandI−δnBare nonexpansive. Sincep ∈ VIE, B andWnis a nonexpansive, we havepWnPEp−λnBp WnPEp−δnBp, and we have
yn−p≤ϕnun−p
1−ϕn WnPEun−δnBun−WnPE
p−δnBp
≤ϕnun−p
1−ϕn un−δnBun−
p−δnBp
ϕnun−p
1−ϕn I−δnBun−I−δnBp
≤ϕnun−p
1−ϕn un−p
≤un−p≤xn−p.
3.16
It follows that
kn−p≤αnxn−p 1−αnWnPE
yn−λnByn −WnPE
p−λnBp
≤αnxn−p 1−αnyn−λnByn −
p−λnBp
αnxn−p 1−αnI−λnByn−I−λnBp
≤αnxn−p 1−αnyn−p
≤αnxn−p 1−αnxn−pxn−p,
3.17
which yields that xn1−pn
γfxn−Ap βn
xn−p
1−βn I−nA WnPEkn−τnBkn−p
≤1−βn−nγ PEI−τnBkn−pβnxn−pnγfxn−Ap
≤1−βn−nγ kn−pβnxn−pnγfxn−Ap
≤1−βn−nγ xn−pβnxn−pnγfxn−Ap
≤1−nγ xn−pnγfxn−f
p nγf
p −Ap
≤1−nγ xn−pnγαxn−pnγf
p −Ap
1−γ−αγ n xn−p
γ−αγ n
γfp −Ap
γ−αγ .
3.18
This in turn implies that
xn−p≤maxx1−p,γfp −Ap
γ−αγ
Therefore, {xn} is bounded. We also obtain that {un}, {kn},{yn}, {Bun}, {Bkn},{Byn},
{Wnun},{Wnkn},{Wnyn},and{fWnxn}are all bounded.
Step 3. We claim that limn→ ∞xn1−xn0 and limn→ ∞Wnθn−xn0.
FromLemma 2.6, we haveunTrnxn−rnDxnandun1Trn1xn1−rn1Dxn1. Let
n xn−rnDxn, we getunTrnn,un1Trn1n1, and so
Fun, y 1
rn
y−un, un−n
≥0, ∀y∈E, 3.20
Fun1, y 1
rn1
y−un1, un1−n1
≥0, ∀y∈E. 3.21
Puttingyun1in3.20andyunin3.21, we have
Fun, un1 1
rnun1−un, un−n ≥0,
Fun1, un 1
rn1
un−un1, un1−n1 ≥0.
3.22
So, from the monotonicity ofF, we get
un1−un,un−n
rn −
un1−n1
rn1
≥0, 3.23
and hence
un1−un, un−un1un1−n− rn
rn1un1−n1
≥0. 3.24
Without loss of generality, let us assume that there exists a real numbercsuch thatrn> c >0 for alln∈N.Then, we have
un1−un2≤
un1−un, n1−n
1− rn
rn1
un1−n1
≤ un1−un
n1−n
1− rn
rn1
un1−n1
,
and hence
un1−un ≤ n1−n 1
c|rn1−rn|un1−n1
xn1−rn1Dxn1−xn−rnDxn 1
c|rn1−rn|un1−n1
≤ xn1−rn1Dxn1−xn−rn1Dxn|rn1−rn|Dxn
1
c|rn1−rn|un1−n1
≤ xn1−xn|rn1−rn|Dxn 1
c|rn1−rn|un1−n1
≤ xn1−xnM1|rn1−rn|,
3.26
whereM1sup{Dxn 1/cun1−n1:n∈N}.
PutθnPEkn−τnBkn,φn PEyn−λnByn,andψnPEun−δnBun. SinceI−τnB, I−λnB,andI−δnBare nonexpansive, then we have the following some estimates:
ψn1−ψn≤ PEun1−δn1Bun1−PEun−δnBun
≤ un1−δn1Bun1−un−δnBun
un1−δn1Bun1−un−δn1Bun δn−δn1Bun
≤ un1−δn1Bun1−un−δn1Bun|δn−δn1|Bun
I−δn1Bun1−I−δn1Bun|δn−δn1|Bun
≤ un1−un|δn−δn1|Bun.
3.27
Similarly, we can prove that
φn1−φn≤yn1−yn|λn−λn1|Byn, 3.28
θn1−θn ≤ kn1−kn|τn−τn1|Bkn. 3.29
SinceTiandUn,iare nonexpansive, we deduce that, for eachn≤1,
Wn1ψn−Wnψnμ1T1Un1,2ψn−μ1T1Un,2ψn
≤μ1Un1,2ψn−Un,2ψn
μ1μ2T2Un1,3ψn−μ2T2Un,3ψn
.. .
≤n
i1
μiUn1,n1ψn−Un,n1ψn
≤M2
n
i1
μi,
3.30
whereM2≥0 is a constant such thatUn1,n1ψn−Un,n1ψn ≤M2for alln≥0.
Similarly, we can obtain that there exist nonnegative numbersM3,M4such that
Un1,n1ϕn−Un,n1ϕn≤M3, Un1,n1θn−Un,n1θn ≤M4, 3.31
and so are
Wn1φn−Wnφn≤M3n i1
μi, Wn1θn−Wnθn ≤M4
n
i1
μi. 3.32
Observing that
ynϕnun
1−ϕn Wnψn,
yn1ϕn1un1
1−ϕn1 Wn1ψn1,
3.33
we obtain
yn−yn1ϕnun−un1
1−ϕn Wnψn−Wn1ψn1
Wn1ψn1−un1 ϕn1−ϕn ,
3.34
which yields that
yn−yn1≤ϕnun−un1
1−ϕn Wn1ψn1−Wnψnϕn1−ϕnWn1ψn1−un1
≤ϕnun−un1
1−ϕn Wn1ψn1−Wn1ψnWn1ψn−Wnψn
ϕn1−ϕnWn1ψn1−un1
≤ϕnun−un1
1−ϕn ψn1−ψnWn1ψn−Wnψn
ϕn1−ϕnWn1ψn1−un1
≤ϕnun−un1
1−ϕn ψn1−ψnWn1ψn−Wnψn
ϕn1−ϕnWn1ψn1−un1.
Substitution of3.27and3.30into3.35yields that
yn−yn1≤ϕnun−un1
1−ϕn {un1−un|δn−δn1|Bun}
M2
n
i1
μiϕn1−ϕnWn1ψn1−un1
un−un1
1−ϕn |δn−δn1|Bun
M2
n
i1
μiWn1ψn1−un1ϕn1−ϕn
≤ un−un1M5
|δn−δn1|ϕn1−ϕn M2
n
i1
μi,
3.36
whereM5is an appropriate constant such thatM5max{supn≥1Bun,supn≥1Wnψn−un}. Observing that
knαnxn 1−αnWnφn,
kn1αn1xn1 1−αn1Wnφn1,
3.37
we obtain
kn−kn1αnxn−xn1 1−αn
Wnφn−Wn1φn1
Wn1φn1−xn1 αn1−αn,
3.38
which yields that
kn−kn1 ≤αnxn−xn1 1−αnWnφn−Wn1φn1|αn1−αn|Wn1φn1−xn1
≤αnxn−xn1 1−αnWn1φn1−Wn1φnWn1φn−Wnφn
|αn1−αn|Wn1φn1−xn1
≤αnxn−xn1 1−αnφn1−φnWn1φn−Wnφn
|αn1−αn|Wn1φn1−xn1.
Substitution of3.28and3.32into3.39yields that
kn−kn1 ≤αnxn−xn1 1−αnyn1−yn|λn−λn1|Byn
M3
n
i1
μi|αn1−αn|Wn1φn1−xn1
αnxn−xn1 1−αnyn1−yn 1−αn|λn−λn1|Byn
M3
n
i1
μi|αn1−αn|Wn1φn1−xn1
≤αnxn−xn1 1−αnyn1−ynM3
n
i1
μi
M6|λn−λn1||αn1−αn|,
3.40
whereM6is an appropriate constant such thatM6max{supn≥1Byn,supn≥1Wnφn−xn}. Substituting3.26and3.36into3.40, we obtain
kn−kn1 ≤αnxn−xn11−αn
un−un1M5
|δn−δn1|ϕn1−ϕn M2
n
i1
μi
M3
n
i1
μiM6|λn−λn1||αn1−αn|
αnxn−xn1 1−αnun−un1 1−αnM5
|δn−δn1|ϕn1−ϕn
1−αnM2
n
i1
μiM3
n
i1
μiM6|λn−λn1||αn1−αn|
≤αnxn−xn1 1−αn{xn1−xnM1|rn1−rn|}
1−αnM5
|δn−δn1|ϕn1−ϕn 1−αnM2
n
i1
μi
M3
n
i1
μiM6|λn−λn1||αn1−αn|
αnxn−xn1 1−αnxn1−xn 1−αnM1|rn1−rn|
1−αnM5
|δn−δn1|ϕn1−ϕn 1−αnM2
n
i1
μi
M3
n
i1
μiM6|λn−λn1||αn1−αn|
≤ xn−xn1M1|rn1−rn|M2
n
i1
μiM3
n
i1
μi
M5
|δn−δn1|ϕn1−ϕn M6|λn−λn1||αn1−αn|
≤ xn−xn1M2
n
i1
μiM3
n
i1
μi
K1
|rn1−rn||δn−δn1|ϕn1−ϕn|λn−λn1||αn1−αn| ,
Substituting3.41into3.29, we obtain
θn1−θn ≤ kn1−kn|τn−τn1|Bkn
≤ xn−xn1M2
n
i1
μiM3
n
i1
μi
K1
|rn1−rn||δn−δn1|ϕn1−ϕn|λn−λn1||αn1−αn|
|τn−τn1|Bkn
≤ xn−xn1M2
n
i1
μiM3
n
i1
μi
K2
|rn1−rn||δn−δn1|ϕn1−ϕn|λn−λn1||αn1−αn||τn−τn1| ,
3.42
whereK2is an appropriate constant such thatK2max{supn≥1Bkn, K1}.
Define
xn1
1−βn znβnxn, n≥1. 3.43
Observe that from the definitionzn, we obtain
zn1−zn
n1γfWn1xn1
1−βn1 I−n1A Wn1θn1
1−βn1
− nγfWnxn
1−βn I−nA Wnθn 1−βn
n1
1−βn1
γfWn1xn1−
n
1−βn
γfWnxn Wn1θn1−Wnθn
n 1−βn
AWnθn− n1
1−βn1
AWn1θn1
n1
1−βn1
γfWn1xn1−AWn1θn1
n
1−βn
AWnθn−γfWnxn
Wn1θn1−Wn1θnWn1θn−Wnθn.
It follows from3.32,3.42, and3.44that
zn1−zn − xn1−xn
≤ n1
1−βn1
γfWn1xn1AWn1θn1 n 1−βn
AWnθnγfWnxn
Wn1θn1−Wn1θnWn1θn−Wnθn − xn1−xn
≤ n1
1−βn1
γfWn1xn1AWn1θn1 n 1−βn
AWnθnγfWnxn
θn1−θnWn1θn−Wnθn − xn1−xn
≤ n1
1−βn1
γfWn1xn1AWn1θn1
n
1−βn
AWnθnγfWnxn
M2
n
i1
μiM3
n
i1
μiM4
n
i1
μi
K2
|rn1−rn||δn−δn1|ϕn1−ϕn|λn−λn1||αn1−αn||τn−τn1|
≤ n1
1−βn1
γfWn1xn1AWn1θn1 n 1−βn
AWnθnγfWnxn
3K
n
i1
μi
K2
|rn1−rn||δn−δn1|ϕn1−ϕn|λn−λn1||αn1−αn||τn−τn1| ,
3.45
whereKis an appropriate constant such thatKmax{M2, M3, M4}.
It follows from conditionsC1,C2,C3,C4,C5, and 0< μi≤b <1,for alli≥1
lim sup
n→ ∞ zn1−zn − xn1−xn≤0. 3.46
Hence, byLemma 2.11, we obtain
lim
n→ ∞zn−xn0. 3.47
It follows that
lim
n→ ∞xn1−xnnlim→ ∞
Applying3.48and conditions inTheorem 3.3to3.26,3.41, and3.42, we obtain that
lim
n→ ∞un1−unnlim→ ∞kn1−knnlim→ ∞θn1−θn0. 3.49
From3.49,C2,C5, and 0< μi≤b <1,for alli≥1, we also have
lim
n→ ∞yn1−yn0. 3.50
Sincexn1nγfWnxn βnxn 1−βnI−nAWnθn, we have
xn−Wnθn ≤ xn−xn1xn1−Wnθn
xn−xn1nγfWnxn βnxn
1−βn I−nA Wnθn−Wnθn
xn−xn1n
γfWnxn−AWnθn βnxn−Wnθn
≤ xn−xn1nγfWnxnAWnθn βnxn−Wnθn,
3.51
that is,
xn−Wnθn ≤ 1 1−βn
xn−xn1
n
1−βn
γfWnxnAWnθn . 3.52
ByC1,C3, and3.48it follows that
lim
n→ ∞Wnθn−xn0. 3.53
Step 4. We claim that the following statements hold:
ilimn→ ∞un−θn0;
iilimn→ ∞xn−un0;
iiilimn→ ∞Wnθn−θn0.
Wnθn−p2≤PEkn−τnBkn−PEp−τnBp2
≤kn−τnBkn−p−τnBp2
kn−p−τnBkn−Bp2
≤kn−p2−2τnkn−p, Bkn−Bpτn2Bkn−Bp2
≤xn−p2−2τnkn−p, Bkn−Bpτn2Bkn−Bp2
≤xn−p2−2τn
−uBkn−Bp2vkn−p2
τ2
nBkn−Bp2
≤xn−p22τnuBkn−Bp2−2τnvkn−p2τn2Bkn−Bp2
≤xn−p22τnuBkn−Bp2− 2τnv
ξ2 Bkn−Bp
2
τn2Bkn−Bp2
xn−p2
2τnuτn2− 2τnv
ξ2
Bkn−Bp2.
3.54
Similarly, we have
Wnφn−p2≤xn−p2
2λnuλ2n− 2λnv
ξ2
Byn−Bp2,
Wnψn−p2≤xn−p2
2δnuδn2− 2δnv
ξ2
Bun−Bp2.
3.55
Observe that
xn1−p21−βnI−nAWnθn−p βnxn−p nγfWnxn−Ap2
1−βnI−nAWnθn−p βnxn−p22nγfWnxn−Ap2
2βnn
xn−p, γfWnxn−Ap
2n
1−βn I−nA Wnθn−p , γfWnxn−Ap
≤1−βn−nγ Wnθn−pβnxn−p 22nγfWnxn−Ap2
2βnn
xn−p, γfWnxn−Ap
2n
1−βn I−nA Wnθn−p , γfWnxn−Ap
≤1−βn−nγ Wnθn−pβnxn−p 2cn
≤1−βn−nγ 2Wnθn−p2β2nxn−p2
21−βn−nγ βnWnθn−pxn−pcn
≤1−βn−nγ 2Wnθn−p2β2nxn−p2
1−βn−nγ βn
Wnθn−p2xn−p2
1−nγ 2−2
1−nγ βnβ2n
Wnθn−p2β2nxn−p2
1−nγ βn−β2n
Wnθn−p2xn−p2
cn
1−nγ 2−
1−nγ βn
Wnθn−p2
1−nγ βnxn−p2cn
1−nγ 1−βn−nγ Wnθn−p2
1−nγ βnxn−p2cn,
3.56
where
cnn2γfxn−Ap22βnnxn−p, γfxn−Ap
2n
1−βn I−nA Wnθn−p , γfxn−Ap.
3.57
It follows from conditionC1that
lim
n→ ∞cn0. 3.58
Substituting3.54into3.56, and using conditionC6, we have xn1−p2≤
1−nγ 1−βn−nγ xn−p2
2τnuτn2− 2τnv
ξ2
Bkn−Bp2
1−nγ βnxn−p2cn
1−nγ 2xn−p2
1−nγ 1−βn−nγ
×
2τnuτn2− 2τnv
ξ2
Bkn−Bp2cn
≤xn−p2
2τnuτn2− 2τnv
ξ2
Bkn−Bp2cn.
3.59
It follows that
2av
ξ2 −2bu−b
2Bk
n−Bp2≤
2τnv
ξ2 −2τnu−τ
2
n
Bkn−Bp2
≤xn−p2−xn1−p2cn
xn−p−xn1−p xn−pxn1−p cn
≤ xn−xn1xn−pxn1−p cn.
Sincecn → 0 asn → ∞and3.48, we obtain
lim
n→ ∞Bkn−Bp0. 3.61
Note that kn−p2≤
αnxn−p 1−αnWnφn−p2
≤αnxn−p2 1−αn
xn−p2
2λnuλ2n− 2λnv
ξ2
Byn−Bp2
≤xn−p2 1−αn
2λnuλ2n− 2λnv
ξ2
Byn−Bp2,
3.62
yn−p2≤ϕnun−p
1−ϕn Wnψn−p2
≤ϕnxn−p2
1−ϕn xn−p2
2δnuδn2− 2δnv
ξ2
Bun−Bp2
≤xn−p2
1−ϕn 2δnuδ2n− 2δnv
ξ2
Bun−Bp2.
3.63
Using3.56again, we have
xn1−p2≤
1−nγ 1−βn−nγ Wnθn−p2
1−nγ βnxn−p2cn
≤1−nγ 1−βn−nγ θn−p2
1−nγ βnxn−p2cn
≤1−nγ 1−βn−nγ kn−p2
1−nγ βnxn−p2cn.
3.64
Substituting3.62into3.64and using conditionC2andC6, we have
xn1−p2≤
1−nγ 1−βn−nγ xn−p2 1−αn
2λnuλ2n− 2λnv
ξ2
Byn−Bp2
1−nγ βnxn−p2cn
1−nγ 1−βn−nγ 1−αn
2λnuλ2n− 2λnv
ξ2
Byn−Bp2
1−nγ 2xn−p2cn
≤xn−p2 1−αn
2λnuλ2n− 2λnv
ξ2
Byn−Bp2cn.
It follows that
1−αn
2av
ξ2 −2bu−b
2By
n−Bp2≤1−αn
2τnv
ξ2 −2τnu−τ
2
n
Byn−Bp2
≤xn−p2−xn1−p2cn
≤ xn−xn1xn−pxn1−p cn.
3.66
Sincecn → 0 asn → ∞and3.48, we obtain
lim
n→ ∞Byn−Bp0. 3.67
In a similar way, we can prove
lim
n→ ∞Bun−Bp0. 3.68
By2.3, we also have
θn−p2
PEkn−τnBkn−PEp−τnBp2
PEI−τnBkn−PEI−τnBp2
≤ I−τnBkn−I−τnBp, θn−p
1
2
I−τnBkn−I−τnBp2θn−p2
−I−τnBkn−I−τnBp−
θn−p 2
≤ 1
2kn−p
2
θn−p2−kn−θn−τnBkn−Bp2
≤ 1
2
xn−p2θn−p2− kn−θn2−τn2Bkn−Bp22τn
kn−θn, Bkn−Bp
,
3.69
which yields that
θn−p2≤
Substituting3.70into3.56, we have xn1−p2≤
1−nγ 1−βn−nγ Wnθn−p2
1−nγ βnxn−p2cn
≤1−nγ 1−βn−nγ θn−p2
1−nγ βnxn−p2cn
≤1−nγ 1−βn−nγ xn−p2− kn−θn22τnkn−θnBkn−Bp
1−nγ βnxn−p2cn
1−nγ 2xn−p2−
1−nγ 1−βn−nγ kn−θn2
21−nγ 1−βn−nγ τnkn−θnBkn−Bpcn
≤xn−p2−
1−nγ 1−βn−nγ kn−θn2
21−nγ 1−βn−nγ τnkn−θnBkn−Bpcn.
3.71
It follows that
1−nγ 1−βn−nγ kn−θn2≤xn−p2−xn1−p2
21−nγ 1−βn−nγ τnkn−θnBkn−Bpcn
≤ xn−xn1xn−pxn1−p
21−nγ 1−βn−nγ τnkn−θnBkn−Bpcn.
3.72
Applyingxn1−xn → 0,Bkn−Bp → 0 andcn → 0 asn → ∞to the last inequality, we have
lim
n→ ∞kn−θn0. 3.73
On the other hand, we have
Wnθn−p2≤PEkn−τnBkn−PEp−τnBp2
PEI−τnBkn−PEI−τnBp2
≤ I−τnBkn−I−τnBp, Wnθn−p
1
2
I−τnBkn−I−τnBp2Wnθn−p2
−I−τnBkn−I−τnBp−
Wnθn−p 2
≤ 1
2kn−p
2
Wnθn−p2−kn−Wnθn−τnBkn−Bp2
≤ 1
2
xn−p2Wnθn−p2− kn−Wnθn2
−τn2Bkn−Bp22τn
kn−Wnθn, Bkn−Bp
,
3.74
which yields that
Wnθn−p2≤xn−p2− kn−Wnθn22τnkn−WnθnBkn−Bp. 3.75
Similarly, we can prove Wnφn−p2≤
xn−p2−yn−Wnφn22λnyn−WnφnByn−Bp, 3.76
Wnψn−p2 ≤xn−p2−un−Wnψn22δnun−WnψnBun−Bp. 3.77
Substituting3.75into3.56, we have xn1−p2≤
1−nγ 1−βn−nγ Wnθn−p2
1−nγ βnxn−p2cn
≤1−nγ 1−βn−nγ
×xn−p2− kn−Wnθn22τnkn−WnθnBkn−Bp
1−nγ βnxn−p2cn
1−nγ 2xn−p2−
1−nγ 1−βn−nγ kn−Wnθn2
21−nγ 1−βn−nγ τnkn−WnθnBkn−Bpcn
≤xn−p2−
1−nγ 1−βn−nγ kn−Wnθn2
21−nγ 1−βn−nγ τnkn−WnθnBkn−Bpcn,
3.78
which yields that
1−nγ 1−βn−nγ kn−Wnθn2
≤xn−p2−xn1−p22
1−nγ 1−βn−nγ τnkn−WnθnBkn−Bpcn
≤ xn−xn1xn−pxn1−p
21−nγ 1−βn−nγ τnkn−WnθnBkn−Bpcn.
Applying3.48and3.61to the last inequality, we have
lim
n→ ∞kn−Wnθn0. 3.80
Using3.64again, we have xn1−p2≤
1−nγ 1−βn−nγ kn−p2
1−nγ βnxn−p2cn
1−nγ 1−βn−nγ αnxn−p 1−αnWnφn−p2
1−nγ βnxn−p2cn
≤1−nγ 1−βn−nγ αnxn−p2 1−αnWnφn−p2
1−nγ βnxn−p2cn
1−nγ 1−βn−nγ αnxn−p2
1−nγ 1−βn−nγ 1−αnWnφn−p2
1−nγ βnxn−p2cn
≤1−nγ 1−βn−nγ αnxn−p2
1−nγ 1−βn−nγ 1−αn
×xn−p2−yn−Wnφn22λnyn−WnφnByn−Bp
1−nγ βnxn−p2cn
1−nγ 1−βn−nγ αnxn−p2
1−nγ 1−βn−nγ 1−αnxn−p2
−1−nγ 1−βn−nγ 1−αnyn−Wnφn2
1−nγ 1−βn−nγ 1−αn2λnyn−WnφnByn−Bp
1−nγ βnxn−p2cn
1−nγ 1−βn−nγ xn−p2
−1−nγ 1−βn−nγ 1−αnyn−Wnφn2
1−nγ 1−βn−nγ 1−αn2λnyn−WnφnByn−Bp
1−nγ βnxn−p2cn
1−nγ 2xn−p2−
1−nγ 1−βn−nγ 1−αnyn−Wnφn2
1−nγ 1−βn−nγ 1−αn2λnyn−WnφnByn−Bpcn
≤xn−p2−
1−nγ 1−βn−nγ 1−αnyn−Wnφn2
1−nγ 1−βn−nγ 1−αn2λnyn−WnφnByn−Bpcn,
which implies that
1−nγ 1−βn−nγ 1−αnyn−Wnφn2
≤xn−p2−xn1−p2
21−nγ 1−βn−nγ 1−αnλnyn−WnφnByn−Bpcn
≤ xn−xn1xn−pxn1−p
21−nγ 1−βn−nγ 1−αnλnyn−WnφnByn−Bpcn.
3.82
From3.48and3.67, we obtain
lim
n→ ∞yn−Wnφn0. 3.83
By using the same argument, we can prove that
lim
n→ ∞un−Wnψn0. 3.84
Note that
kn−Wnφnαn
xn−Wnφn ,
yn−Wnψnϕn
un−Wnψn .
3.85
Sinceαn → 0 andϕn → 0 asn → ∞, respectively, we also have
lim
n→ ∞kn−Wnφnnlim→ ∞yn−Wnψn0. 3.86
On the other hand, we observe
un−θn ≤un−WnψnWnψn−ynyn−Wnφn Wnφn−knkn−θn.
3.87
Applying3.73,3.83,3.84, and3.86, we have
lim
