Adv. Fixed Point Theory, 4 (2014), No. 2, 229-244 ISSN: 1927-6303
STRONG CONVERGENCE THEOREMS FOR COMMON SOLUTIONS OF VARIATIONAL INEQUALITY AND FIXED POINT PROBLEMS
CHANGQUN WU
School of Business and Administration, Henan University, Kaifeng 475000, China
Copyright c2014 Changqun Wu. 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.
Abstract. In this paper, we introduce a viscosity approximation method for solving common solutions of vari-ational inequality and fixed point problems. Strong convergence theorems are established in the framework of Hilbert spaces.
Keywords: inverse-strongly monotone mapping; nonexpansive mapping; fixed point; Hilbert space. 2010 AMS Subject Classification:47H09, 47H10.
1. Introduction
Fixed point theory has emerged as an effective and powerful tool for studying a wide class of real world problems which arise in economics, image reconstruction, transportation, network, elasticity and optimization; see [1-5] and the references therein.
The computation of fixed points is important in the study of many real world problems, including inverse problems; for instance, it is not hard to show that the split feasibility problem and the convex feasibility problem in signal processing and image reconstruction can both be formulated as a problem of finding fixed points of certain operators, respectively; see [6-8] for more details and the references therein.
E-mail address: [email protected] Received June 3, 2013
The aim of this paper is to investigate a viscosity approximation method for solving common solutions of variational inequality and fixed point problems. Strong convergence theorems are established in the framework of Hilbert spaces. The organization of this article is as follows. In Section 2, we provide some necessary preliminaries. In Section 3, a viscosity iterative method is discussed. Strong convergence theorems of common solutions are established in Hilbert spaces
2. Preliminaries
From now on, we always assume thatH is a real Hilbert space, whose the inner product and the norm are denoted byh·,·iandk · k, respectively. LetCbe a closed convex subset ofH and letA:C→H be a mapping. We denote by PC be the projection ofH onto the closed convex subsetC. The classical variational inequality problem is to findu∈Csuch that
hAu,v−ui ≥0, ∀v∈C. (2.1)
We denoted byV I(C,A)the set of solutions of the variational inequality. One can see that the variational inequality problem (2.1) is equivalent to a fixed point problem, that is, an element u∈C is a solution of the variational inequality (2.1) if and only ifu∈C is a fixed point of the mappingPC(I−λA),whereλ >0 is a constant andIis the identity mapping.
Recall that a mapping A:C→H is said to be inverse-strongly monotone if there exists a positive real numberµ such that
hAx−Ay,x−yi ≥µkAx−Ayk2, ∀x,y∈C.
Recall that a mappingT :C→Cis said to be nonexpansive if
kT x−Tyk ≤ kx−yk, ∀x,y∈C.
In this paper, we denote byF(T)the set of fixed points ofT. Recall that a mapping f :C→C is said to be contractive if there existsα ∈(0,1)such that
Recall that a linear bounded operator B:C→C is said to be strongly positive if there exists a constant ¯γ >0 such thathBx,xi ≥γ¯kxk2,∀x∈C.Recall that monotone mappingT :H→2His
said to be maximal if the graph ofG(T)ofT is not properly contained in the graph of any other monotone mapping. Recall that set-valued mappingT :H→2H is said to be monotone if, for allx,y∈H, f ∈T xandg∈Tyimplyhx−y,f−gi ≥0.It is known that a monotone mappingT is maximal if and only if, for any(x,f)∈H×H,hx−y,f−gi ≥0 for all(y,g)∈G(T)implies
f ∈T x.
LetA be a monotone mapping ofC into H and NCvbe the normal cone toC atv∈C, i.e., NCv={w∈H:hv−u,wi ≥0, ∀u∈C}and define
T v=
Av+NCv, v∈C,
/0, v∈/C.
ThenT is maximal monotone and 0∈T vif and only ifv∈V I(C,A); see [9] and the references therein.
Iterative methods for nonexpansive mappings have recently been applied to solve convex minimization problems; see [10-15] and the references therein. A typical problem is to mini-mize a quadratic function over the set of the fixed points of a nonexpansive mapping on a real Hilbert spaceH:
min
x∈Ω
1
2hBx,xi − hx,bi, (2.2)
whereBis a linear bounded operator onH,Ωis the fixed point set of a nonexpansive mapping
Sandbis a given point inH.
In [14], it is proved that the sequence{xn}defined by the iterative method below, with the
initial guessx0∈H chosen arbitrarily,
xn+1= (I−αnB)Sxn+αnb, ∀n≥0,
Recently, Marino and Xu [11] introduced a general iterative scheme by the viscosity approx-imation method:
x0∈H, xn+1= (I−αnB)Sxn+αnγf(xn), ∀n≥0,
whereSis a nonexpansive mapping onH, f is a contraction onH with the coefficient α, Bis
a bounded linear strongly positive operator onH with the coefficient ¯γ andγ is a constant such
that 0<γ <γ¯/α. They proved that the sequence{xn}generated by the above iterative scheme
converges strongly to the unique solution of the variational inequality: h(B−γf)x∗,x−x∗i ≥0,
∀x∈F(S),which is the optimality condition for the minimization problem minx∈F(S)12hBx,xi − h(x),wherehis a potential function forγf (i.e.,h0(x) =γf(x)for allx∈H.)
Recently, variational inequalities and fixed point problems have been considered by many au-thors; see [16-20] and the references therein. For finding a common element of the sets of fixed points of nonexpansive mappings and solutions of variational inequalities forµ-inverse-strongly
monotone mapping, Iiduka and Takahashi [21] proposed the following iterative scheme: x1=x∈C, xn+1=αnx+ (1−αn)SPC(xn−λnAxn), ∀n≥1,
where {αn} is a sequence in (0,1) and {λn} is a sequence in (0,2µ). They proved that the
sequence{xn}converges strongly to somez∈F(S)∩V I(C,A).
In this paper, we consider a mappingWndefined by Un,n+1=I,
Un,n=γnTnUn,n+1+ (1−γn)I,
Un,n−1=γn−1Tn−1Un,n+ (1−γn−1)I, · · ·
Un,k=γkTkUn,k+1+ (1−γk)I,
Un,k−1=γk−1Tk−1Un,k+ (1−γk−1)I, · · ·
Un,2=γ2T2Un,3+ (1−γ2)I, Wn=Un,1=γ1T1Un,2+ (1−γ1)I,
where γ1,γ2, . . . are real numbers such that 0≤γn≤1 andT1,T2,· · · be an infinite family of mappings ofCinto itself. Nonexpansivity of eachTiensures the nonexpansivity ofWn.
ConcerningWn, we have the following lemmas which are important to prove our main results. Lemma 2.1.[8]Let C be a nonempty closed convex subset of a strictly convex Banach space E. Let T1,T2,· · · be nonexpansive mappings of C into itself such that∩∞n=1F(Tn)6= /0andγ1,γ2,· · · be real numbers such that0<γn≤b<1for any n≥1. Then, for all x∈C and k∈N, the limit
limn→∞Un,kx exists.
Using Lemma 2.1, one can define the mappingW ofCinto itself as follows.
W x= lim
n→∞
Wnx= lim
n→∞
Un,1x, ∀x∈C. (2.4)
Such a mappingW is called theW-mapping generated byT1,T2,· · · andγ1,γ2,· · ·. Throughout this paper, we shall always assume that 0<γi≤b<1 for alli≥1.
Lemma 2.2.[8]Let C be a nonempty closed convex subset of a strictly convex Banach space E. Let T1,T2,· · · be nonexpansive mappings of C into itself such that∩∞
n=1F(Tn)6= /0andγ1,γ2,· · · be real numbers such that0<γn≤b<1for any n≥1. Then F(W) =∩∞n=1F(Tn).
Lemma 2.3. [6]Let C be a nonempty closed convex subset of a Hilbert space H. Let T1,T2,· · ·
be nonexpansive mappings of C into itself such that ∩∞
n=1F(Tn)6= /0 and γ1,γ2,· · · be a real sequence such that0<γn≤b<1for all n≥1. If K is any bounded subset of C, then
lim
n→∞supx∈KkW x−
Wnxk=0.
In order to prove our main results, we also need the following lemmas.
Lemma 2.4. [11]Assume that B is a strong positive linear bounded operator on a Hilbert space H with coefficientγ¯>0and0<ρ ≤ kBk−1. ThenkI−ρBk ≤1−ργ¯.
Lemma 2.5[11]Let H be a Hilbert space, B be a strongly positive linear bounded self-adjoint operator on H with the coefficient γ¯>0. Assume that 0 <γ <γ¯/α. Let T :H →H be a
{xt}converges strongly as t→0to a fixed pointx of T , which solves the variational inequality:¯ h(B−γf)x¯,x¯−zi ≤0, ∀z∈F(T).
Equivalently,x¯=PF(γf+I−B)x¯.
Lemma 2.6. [22] Let H be a Hilbert space, C a closed convex subset of H, f :C →C a
contraction with the coefficient α ∈(0,1) and B a strongly positive linear bounded operator
with the coefficientγ¯>0. Then, for any0<γ <αγ¯,
hx−y,(B−γf)x−(B−γf)yi ≥(γ¯−γ α)kx−yk2, ∀x,y∈C.
That is, B−γf is strongly monotone with the coefficientγ¯−α γ.
Lemma 2.7. [23]Assume that{αn}is a sequence of nonnegative real numbers such that
αn+1≤(1−γn)αn+δn,
where{γn}is a sequence in(0,1)and{δn}is a sequence such that
(i) ∑∞n=1γn=∞;
(ii) lim supn→∞δn
γn ≤0or∑
∞
n=1|δn|<∞.
Thenlimn→∞αn=0.
3. Main Results
Theorem 3.1. Let H be a real Hilbert space and let C be be a nonempty closed convex subset of
H such that C±C⊂C. Let A:C→H be aµ-inverse-strongly monotone mapping. Let f :C→C
be a contraction with the coefficientα and let T1,T2,· · · be a sequence of nonexpansive
self-mappings on C. Let B be a strongly positive linear bounded self-adjoint operator of C into itself
with the coefficientγ¯>0. Assume that0<γ <αγ¯. Let the sequence{xn}be generated by
x1∈C, xn+1=αnγf(Wnxn) + (I−αnB)WnPC(I−λnA)xn, ∀n≥1,
where the mapping Wnis defined by (2.3),{αn}is a sequence in(0,1)and{λn}is a sequence
in(0,2µ). If F=∩∞i=1F(Ti)∩V I(C,A)6= /0and{αn}and{λn}are chosen such that
(b) ∑∞n=1αn=∞;
(c) ∑∞n=1|λn+1−λn|<∞and∑∞n=1|αn+1−αn|<∞;
(d) {λn} ⊂[u,v] f or some u,v with0<u<v<2µ,
then the sequence{xn}converges strongly to some x∗∈F, which uniquely solves the following
variation inequality:
hBx∗−γf(x∗),x∗−pi ≤0, ∀p∈F.
Equivalently, we have x∗=PF(γf+I−B)x∗.
Proof.First, we show that{xn}is bounded. Note that the mappingI−λnAis nonexpansive for
eachn≥1. Indeed, from the condition (d), for∀x,y∈C, we have k(I−λnA)x−(I−λnA)yk2
≤ kx−yk2−2λnhAx−Ay,x−yi+λn2kAx−Ayk2
≤ kx−yk2−2λnµkAx−Ayk2+λn2kAx−Ayk2 =kx−yk2+λn(λn−2µ)kAx−Ayk2
≤ kx−yk2.
This shows that I−λnA is a nonexpansive mapping. Noticing that condition (a), we may
as-sume, with no loss of generality, that αn≤ kBk−1 for all n≥1. Using Lemma 2.4, we know
that, if 0<αn≤ kBk−1for alln≥1, thenkI−αnBk ≤1−αnγ¯. Fixing p∈F, we have
kxn+1−pk
=kαn(γf(Wnxn)−Bp) + (I−αnB)(WnPC(I−λnA)xn−p)k
≤αnkγf(Wnxn)−Bpk+ (1−αnγ¯)kWnPC(I−λnA)xn−pk
≤αnγkf(Wnxn)−f(p)k+αnkγf(p)−Bpk+ (1−αnγ¯)kxn−pk
≤[1−αn(γ¯−γ α)]kxn−pk+αnkγf(p)−Bpk.
By simple inductions, we obtain
kxn−pk ≤max{kx1−pk,
kBp−γf(p)k
¯
which yields that the sequence{xn}is bounded. Puttingρn=PC(I−λnA)xn, we have
kρn+1−ρnk
=kPC(I−λn+1A)xn+1−PC(I−λnA)xnk
≤ k(I−λn+1A)xn+1−(I−λnA)xnk
=k(I−λn+1A)xn+1−(I−λn+1A)xn+ (I−λn+1A)xn−(I−λnA)xnk
=kxn+1−xnk+|λn+1−λn|kAxnk.
(3.1)
It follow that
kxn+2−xn+1k
=k(I−αn+1B)(Wn+1ρn+1−Wnρn)−(αn+1−αn)BWnρn
+γ[αn+1(f(Wn+1xn+1)−f(Wnxn)) +f(xn)(αn+1−αn)]k
≤(1−αn+1γ¯)(kρn+1−ρnk+kWn+1ρn−Wnρnk) +|αn+1−αn|kBWnρnk +γ[αn+1αkxn+1−xnk+kWn+1xn−Wnxnk+kf(xn)k|αn+1−αn|].
(3.2)
SinceTiandUn,iare nonexpansive, we find that
kWn+1ρn−Wnρnk=kγ1T1Un+1,2ρn−γ1T1Un,2ρnk
≤γ1kUn+1,2ρn−Un,2ρnk
=γ1kγ2T2Uu+1,3ρn−γ2T2Un,3ρnk
≤γ1γ2kUu+1,3ρn−Un,3ρnk
≤ · · ·
≤γ1γ2· · ·γnkUn+1,n+1ρn−Un,n+1ρnk
≤M1
n
∏
i=1
γi,
whereM1≥0 is an appropriate constant such thatkUn+1,n+1ρn−Un,n+1ρnk ≤M1for alln≥1. Using (3.1) and (3.2), we arrive at
kxn+2−xn+1k ≤[1−αn+1(γ¯−α γ)]kxn+1−xnk +M2(2
n
∏
i=1
γi+2|αn+1−αn|+|λn+1−λn|),
whereM2is an appropriate constant such that M2=max{M1,sup
n≥1
{kAxnk},γsup
n≥1
{kf(xn)k},sup n≥1
{kBWnρnk}}.
Using the restrictions (a), (b) and (c), we find that lim
n→∞kxn+1−
xnk=0. (3.4)
Note that
kρn−pk2≤ k(xn−p)−λn(Axn−Ap)k2
=kxn−pk2−2λnhxn−p,Axn−Api+λn2kAxn−Apk2
≤ kxn−pk2−2λnµkAxn−Apk2+λn2kAxn−Apk2 =kxn−pk2+λn(λn−2µ)kAxn−Apk2.
(3.5)
On the other hand, we have kxn+1−pk2
=kαnγf(xn) + (I−αnB)Wnρn−pk2
≤(αnkγf(Wnxn)−Bpk+ (1−αnγ¯)kWnρn−pk)2
≤(αnkγf(Wnxn)−Bpk+ (1−αnγ¯)kρn−pk)2
≤αnkγf(Wnxn)−Bpk2+kρn−pk2+2αnkγf(Wnxn)−Bpkkρn−pk.
(3.6)
Substituting (3.5) into (3.6), we obtain
kxn+1−pk2≤αnkγf(Wnxn)−Bpk2+kxn−pk2+λn(λn−2µ)kAxn−Apk2 +2αnkγf(xn)−Bpkkρn−pk.
It follows that
u(2µ−v)kAxn−Apk2
≤αnkγf(xn)−Bpk2+kxn−pk2− kxn+1−pk2
+2αnkγf(xn)−Bpkkρn−pk
≤αnkγf(xn)−Bpk2+ (kxn−pk+kxn+1−pk)kxn−xn+1k
Using (2.4), we see that
lim
n→∞
kAxn−Apk=0. (3.7)
SincePC is firmly nonexpansive, we find that
kρn−pk2≤h(I−λnA)xn−(I−λnA)p,ρn−pi
=1
2{k(I−λnA)xn−(I−λnA)pk 2+k
ρn−pk2
− k(I−λnA)xn−(I−λnA)p−(ρn−p)k2}
≤1
2{kxn−pk 2+k
ρn−pk2− k(xn−ρn)−λn(Axn−Ap)k2}
=1
2{kxn−pk 2+k
ρn−pk2− kxn−ρnk2−λn2kAxn−Apk2
+2λnhxn−ρn,Axn−Api},
which yields that
kρn−pk2≤kxn−pk2+2λnkxn−ρnkkAxn−Apk − kxn−ρnk2. (3.8)
This in turn implies that
kxn+1−pk2≤αnkγf(Wnxn)−Bpk2+kxn−pk2+2λnkxn−ρnkkAxn−Apk +2αnkγf(Wnxn)−Bpkkρn−pk − kxn−ρnk2.
Hence, we have
kxn−ρnk2≤αnkγf(Wnxn)−Bpk2+kxn−pk2− kxn+1−pk2
+2λnkxn−ρnkkAxn−Apk+2αnkγf(Wnxn)−Bpkkρn−pk
≤αnkγf(Wnxn)−Bpk2+ (kxn−pk+kxn+1−pk)kxn+1−xnk
+2λnkxn−ρnkkAxn−Apk+2αnkγf(Wnxn)−Bpkkρn−pk.
This yields that
lim
n→∞kxn−ρnk=0. (3.9)
Notice that
kρn−Wnρnk ≤ kxn+1−Wnρnk+kxn−xn+1k+kxn−ρnk
Using the restriction (a), we find from (3.4) and (3.9) that lim
n→∞
kρn−Wnρnk=0. (3.10)
Since the sequence{xn}is bounded, we see that{ρn}is also a bounded sequence inC. Without
loss of generality, we can assume that there exists a bounded setK⊂Csuch thatρn∈Kfor all
n≥1.On the other hand, we have
kWρn−ρnk ≤ kWρn−Wnρnk+kWnρn−ρnk
≤sup ρ∈K
kWρ−Wnρk+kWnρn−ρnk.
Using Lemma 1.3, we obtain from (3.10) that lim
n→∞
kWρn−ρnk=0. (3.11)
Now, we are in a position to show thatxn→x∗asn→∞. First, we prove that the uniqueness
of the solution of the variational inequality (2.1), which is indeed a consequence of the strong monotonicity ofB−γf. Suppose that x∗∈F andx∗∗ ∈F both are solutions to (2.1). Then we
have
h(B−γf)x∗,x∗−x∗∗i ≤0 and
h(B−γf)x∗∗,x∗∗−x∗i ≤0.
Adding up the two inequalities, we see that
h(B−γf)x∗−(B−γf)x∗∗,x∗−x∗∗i ≤0.
The strong monotonicity ofB−γf implies thatx∗=x∗∗ and the uniqueness is proved. Let x∗
be the unique solution of (2.1). That is,x∗=PF(γf+ (I−B))x∗.
Next, we show that
lim sup
n→∞
hBx∗−γf(x∗),x∗−xni ≤0. (3.12)
To show it, we choose a subsequence{xni}of{xn}such that
lim sup
n→∞
hBx∗−γf(x∗),x∗−xni=lim i→∞
Since {xni} is bounded, it follows that that there is a subsequence {xn
i j} of {xni} converges
weakly to p. We may assume that, without loss of generality, thatxni *p.Therefore, we have p∈F. Indeed, let us first show thatp∈V I(C,A).Put
Tw=
Av+NCv, v∈C
/0, v∈/C.
Then T is maximal monotone. Let (v,w)∈G(T). Since w−Av∈NCv and ρn∈C, we have
hv−ρn,w−Avi ≥0.
On the other hand, from ρn =PC(I−λnA)xn, we have hv−ρn,ρn−(I−λnA)xni ≥0 and
hencehv−ρn,ρnλ−xn n+Axni ≥0.It follows that
hv−ρni,wi
≥ hv−ρni,Avi ≥ hv−ρni,Avi − hv−ρni,
ρni−xni
λni
+Axnii
≥ hv−ρni,Av−
ρni−xni
λni
−Axnii
=hv−ρni,Av−Aρnii+hv−ρni,Aρni−Axnii − hv−ρni,
ρni−xni
λni
i
≥ hv−ρni,Aρni−Axnii − hv−ρni,
ρni−xni
λni
i,
which implies thathv−p,wi ≥0. We have p∈A−10 and hencep∈V I(C,A).
Next, let us show p∈ ∩∞
i=1F(Ti). Using the Opial’s condition, we find that
lim inf
i→∞
kρni−pk<lim inf i→∞
kρni−W pk
=lim inf
i→∞
kρni−Wρni+Wρni−W pk
≤lim inf
i→∞
kWρni−W pk
≤lim inf
i→∞
kρni−pk,
which is a contradiction. Thus we have p∈F(W) =∩∞
i=1F(Ti).On the other hand, we have
lim sup
n→∞
hBx∗−γf(x∗),x∗−xni= lim i→∞
hBx∗−γf(x∗),x∗−xnii
Note that
kxn+1−x∗k2
≤(1−αnγ¯)2kWnρn−x∗k2+2αnhγf(Wnxn)−Bx∗,xn+1−x∗i
≤(1−αnγ¯)2kxn−x∗k2+α γ αn(kxn−x∗k2+kxn+1−x∗k2)
+2αnhγf(x∗)−Bx∗,xn+1−x∗i. Therefore, we have
kxn+1−x∗k2
≤ (1−αnγ¯) 2+α
nγ α
1−αnγ α kxn−
x∗k2+ 2αn
1−αnγ αhγ
f(x∗)−Bx∗,xn+1−x∗i
= (1−2αnγ¯+αnα γ)
1−αnγ α
kxn−x∗k2+
αn2γ¯2
1−αnγ α
kxn−x∗k2
+ 2αn
1−αnγ α
hγf(x∗)−Bx∗,xn+1−x∗i
≤[1−2αn(γ¯−α γ) 1−αnγ α ]kxn−
x∗k2
+2αn(γ¯−α γ)
1−αnγ α
[ 1
¯
γ−α γhγf(x
∗)−Bx∗,x
n+1−x∗i+
αnγ¯2
2(γ¯−α γ)M3],
whereM3is an appropriate constant such thatM3≥supn≥1kxn−x∗k2. Put kn= 2αn(γ¯−α γ)
1−αnα γ ,
dn= 1
¯
γ−α γhγf(x
∗)−Bx∗,x
n+1−qi+
αnγ¯2
2(γ¯−α γ)M3.
Hence, we have
kxn+1−x∗k2≤(1−kn)kxn−x∗k+bndn.
It follows that
lim
n→∞
kn=0,
∞
∑
n=1
kn=∞, lim sup n→∞
dn≤0.
Using (2.7), we conclude the desired conclusion immediately. This completes the proof. Puttingγ =1 andB=Iin Theorem 3.1, we have the following results.
Corollary 3.2. Let H be a real Hilbert space and let C be be a nonempty closed convex subset
with the coefficientα and let T1,T2,· · · be a sequence of nonexpansive self-mappings on C. Let
the sequence{xn}be generated by
x1∈C, xn+1=αnf(Wnxn) + (1−αn)WnPC(I−λnA)xn, ∀n≥1,
where the mapping Wnis defined by (2.3),{αn}is a sequence in(0,1)and{λn}is a sequence
in(0,2µ). If F=∩∞i=1F(Ti)∩V I(C,A)6= /0and{αn}and{λn}are chosen such that
(a) limn→∞αn=0;
(b) ∑∞n=1αn=∞;
(c) ∑∞n=1|λn+1−λn|<∞and∑∞n=1|αn+1−αn|<∞;
(d) {λn} ⊂[u,v] f or some u,v with0<u<v<2µ,
then the sequence{xn}converges strongly to some x∗∈F, which uniquely solves the following
variation inequality:
hx∗−γf(x∗),x∗−pi ≤0, ∀p∈F.
Equivalently, we have x∗=PFf(x∗).
For a single mapping, we have the following.
Corollary 3.3. Let H be a real Hilbert space and let C be be a nonempty closed convex subset
of H such that C±C⊂C. Let A:C→H be a µ-inverse-strongly monotone mapping. Let
f :C→C be a contraction with the coefficient α and let T be a nonexpansive self-mappings
on C. Let B be a strongly positive linear bounded self-adjoint operator of C into itself with the
coefficientγ¯>0. Assume that0<γ < γ¯
α. Let the sequence{xn}be generated by
x1∈C, xn+1=αnγf(T xn) + (I−αnB)T PC(I−λnA)xn, ∀n≥1,
{αn}is a sequence in(0,1)and{λn}is a sequence in(0,2µ). If F=F(T)∩V I(C,A)6= /0and
{αn}and{λn}are chosen such that
(a) limn→∞αn=0;
(b) ∑∞n=1αn=∞;
(c) ∑∞n=1|λn+1−λn|<∞and∑∞n=1|αn+1−αn|<∞;
then the sequence{xn}converges strongly to some x∗∈F, which uniquely solves the following
variation inequality:
hBx∗−γf(x∗),x∗−pi ≤0, ∀p∈F.
Equivalently, we have x∗=PF(γf+I−B)x∗.
Conflict of Interests
The authors declare that there is no conflict of interests. Acknowledgements
The author was grateful to the reviewer for the useful suggestions which improved the contents of the article.
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